the analogue gravity challenge...way. after introducing the nuts and bolts we motivate the utility...

INTERNATIONAL SCHOOL FOR ADVANCED STUDIESAstroparticle Physics Sector

Academic Year 2011/2012

The Analogue Gravity ChallengeThe bridge from theory

to experiment

thesis submitted for the degree of

Doctor of Philosophy

Advisor: Candidate:Prof. Stefano Liberati Angus Prain

September 19, 2012

ABSTRACT

Analogue models of gravity have for some time been an excellent arena within whichto improve our theoretical understanding of several crucial phenomena at the boundaryof gravity and quantum field theory. Recently, however, the field has reached a levelof maturity which opens up the possibility of directly probing the purely theoreticalresults experimentally.

In this thesis I describe the gravitational analogy in three different experimentalscenarios and their aims to detect three different phenomena normally associated withwave propagation in strong gravitational fields.

To this end we shall review the basic features of the gravitational analogy, the basictools and material useful for modeling and go into some detail to the theoretical basisfor the analogy in the three systems discussed.

At the end of this thesis we shall discuss the current perspectives of the field andsurvey the most promising lines of research which can be pursued in the future.

4

Acknowledgements

I warmly thank my supervisor, Stefano Liberati, for his patience and guidance duringthe course of my PhD as well as all his good ideas for projects that resulted in sucha varied thesis. I also would like to thank all the collaborators I have worked withwhile completing the work in this thesis. These people include Serena Fagnocchi, MattVisser, Silke Weinfurtner, Mauricio Richartz, Jason Penner and Joe Niemela. I wouldespecially like to thank Silke without whom the last two chapters of this thesis wouldhave been missing!

Contents

1 Analogue Gravity Introduced 11

1.1 Analogue gravity, explained . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Geometric acoustics in fluids . . . . . . . . . . . . . . . . . . . . 12

1.1.2 Physical acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.3 The nitty gritty details of physical acoustics . . . . . . . . . . . 14

1.2 The motivation - what’s the problem with real gravity? . . . . . . . . 16

1.3 The analogy developed . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.1 Modeling spacetime with fluids . . . . . . . . . . . . . . . . . . 20

1.3.2 Fluid black holes . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3.3 The expanding universe . . . . . . . . . . . . . . . . . . . . . . 25

1.3.4 Practical considerations . . . . . . . . . . . . . . . . . . . . . . 25

1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 A Toolbox for Analogue Gravity 29

2.1 Quantum fields and gravity together . . . . . . . . . . . . . . . . . . . 29

2.1.1 Dynamical Casimir Effect - Cosmological particle creation . . . 30

2.1.2 Bogoliubov coefficients - more details . . . . . . . . . . . . . . . 36

2.1.3 Hawking effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.4 The many faces of superradiance . . . . . . . . . . . . . . . . . 43

2.2 Bose Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.1 Non-interacting Bosons . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.2 Interactions change things slightly . . . . . . . . . . . . . . . . . 51

2.2.3 A BEC as an analogue spacetime . . . . . . . . . . . . . . . . . 53

6 Contents

2.3 Theory of surface waves on fluids . . . . . . . . . . . . . . . . . . . . . 56

2.3.1 Background flow . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3.2 Wave propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4 Non-liner optics, superluminal pulses . . . . . . . . . . . . . . . . . . . 60

2.5 Wave dispersion, tools and basics . . . . . . . . . . . . . . . . . . . . . 61

2.5.1 Motivation for considering dispersive theories . . . . . . . . . . 61

2.5.2 Dispersion diagrams . . . . . . . . . . . . . . . . . . . . . . . . 62

2.5.3 Moving media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.5.4 Hawking ‘scattering’ with modified dispersion . . . . . . . . . . 65

3 Particle Creation and Correlations in Expanding Bose Einstein Con-densates 69

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 The ingredients for cosmology in the lab . . . . . . . . . . . . . . . . . 72

3.2.1 Scaling solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.2 Dictionary of spacetimes and trapping frequencies . . . . . . . 75

3.2.3 Cigar shaped expanding and suddenly released BEC . . . . . . . 76

3.2.4 Validity of the hydrodynamic approximation during expansion . 78

3.3 Quantum excitations and correlations during expansion . . . . . . . . . 78

3.3.1 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.2 Measuring quantum correlations - Wightman functions . . . . . 82

3.3.3 Exactly Soluble Expanding Spacetime . . . . . . . . . . . . . . 83

3.4 Axial correlations in an expanding elongated condensate . . . . . . . . 92

3.4.1 Conformal symmetry approximation . . . . . . . . . . . . . . . 94

3.4.2 Spatially Homogeneous Ω(Z) approximation . . . . . . . . . . . 96

3.4.3 Actual finite expansion of an elongated BEC . . . . . . . . . . 103

3.5 Loose ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5.1 Relation to Hawking radiation . . . . . . . . . . . . . . . . . . . 105

3.5.2 Modified dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.6 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 107

Contents 7

4 Quantum Vacuum Radiation in Optical Systems 109

4.1 Optical black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.1.1 The Experiment Distilled . . . . . . . . . . . . . . . . . . . . . 111

4.1.2 Some physics questions from the experiment . . . . . . . . . . . 113

4.2 An alternative explanation – DCE . . . . . . . . . . . . . . . . . . . . . 116

4.2.1 Evaluation of βk directly in perturbation theory . . . . . . . . . 119

4.2.2 Spectrum and flux . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.2.3 Putting numbers to the model . . . . . . . . . . . . . . . . . . . 122

4.2.4 The effect of a finite pulse width . . . . . . . . . . . . . . . . . . 129

4.3 Alternative models for the observed emission . . . . . . . . . . . . . . 133

4.3.1 A possibly distinctive test: correlations in the radiation . . . . . 135

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5 Superradiance in Dispersive Theories 139

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Warmup problem - zero flow scattering . . . . . . . . . . . . . . . . . . 140

5.2.1 Non-dispersive preliminary . . . . . . . . . . . . . . . . . . . . . 141

5.2.2 Introducing dispersion . . . . . . . . . . . . . . . . . . . . . . . 145

5.2.3 Subluminal scattering . . . . . . . . . . . . . . . . . . . . . . . 150

5.2.4 Superluminal scattering . . . . . . . . . . . . . . . . . . . . . . 156

5.3 Non-zero flow (or, curved geometry) . . . . . . . . . . . . . . . . . . . . 158

5.3.1 Equation of motion, dispersion . . . . . . . . . . . . . . . . . . . 160

5.3.2 The Wronskian and inner product reconsidered . . . . . . . . . 161

5.3.3 Subluminal scattering . . . . . . . . . . . . . . . . . . . . . . . 162

5.3.4 Superluminal scattering . . . . . . . . . . . . . . . . . . . . . . 168

5.3.5 Some exact solutions for non-zero flow in the large Λ approxima-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.3.6 Loose ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.4 Conclusions summary and outlook . . . . . . . . . . . . . . . . . . . . . 180

5.4.1 What can we do next? . . . . . . . . . . . . . . . . . . . . . . . 181

8 Contents

6 Experimental Black Holes and a Bathtub Vortex 183

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.2 The bathtub vortex theory . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.2.1 Vortex geometry with flat free surface . . . . . . . . . . . . . . . 185

6.2.2 Axisymmetrically draining fluid: curved free surface . . . . . . . 186

6.2.3 Equation of motion for perturbations . . . . . . . . . . . . . . . 190

6.2.4 Non-linear fluid dynamics of a draining vortex . . . . . . . . . 191

6.3 Method to experimentally determine superradiance . . . . . . . . . . . 193

6.4 The laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.4.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . 195

6.5 The results so far . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.6 Conclusion and future directions . . . . . . . . . . . . . . . . . . . . . . 199

Conclusion 206

A Miscellaneous useful results 207

A.1 General method for getting rid of the friction term . . . . . . . . . . . 207

A.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

A.3 Gamma functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

B Superradiance Appendices 209

B.1 A pole in the scattering solution and its resolution . . . . . . . . . . . 209

B.1.1 The exact solution for modified and linear dispersion . . . . . . 209

B.1.2 Smooth model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

B.2 Conserved inner product with flow velocities . . . . . . . . . . . . . . . 212

B.3 Perturbation theory for the roots of the modified dispersion relations . 213

B.3.1 Critical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

B.4 Cylindrically symmetric dimensional reduction . . . . . . . . . . . . . . 216

C Optics Appendices 219

C.1 Exact homogeneous model for DCE – tanh(·) profile. . . . . . . . . . . 219

C.2 Homogeneous perturbation expansion for βk . . . . . . . . . . . . . . . 221

C.3 Perturbation theory with finite size effects . . . . . . . . . . . . . . . . 222

Contents 9

D BEC Appendices 228

D.1 Calculating the exact form of a Wightman function by contour inteation 228

D.2 Adiabatic limit of the entanglement contribution to the Wightman function233

D.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

D.3 Glossary of basic concepts and approximations in the theory of BEC . . 235

Chapter 1

Analogue Gravity Introduced

Gravity is hard. Specifically, experimenting with gravity is hard. In this chapter I willintroduce the intriguing possibility that is very recently becoming a realization, thatgravitational kinematics and the behaviour of matter and waves in strong gravitationalfields be simulated in laboratory sized systems without experimenting directly withgravity itself.

This chapter serves to introduce the topic of analogue gravity from its historicalfirst beginnings as a description of sound wave propagation in fluids in terms of ad’Alembert equation for a curved spacetime. This is done mainly in the first sectionof this chapter where we also explore the extent and completeness of the analogy,how to incorporate the vocabulary and how much of gravity can be simulated in thisway. After introducing the nuts and bolts we motivate the utility of analogy, whichgoes beyond a simulation of Einstein gravity, to whet the appetite for the subsequenttechnical chapters in this thesis.

1.1 Analogue gravity, explained

In real spacetime we have the concept of a metric which describes the causal relationship between events where physical processes are carried out. I will now describe a situ-ation in which a metrical relation which describes the causal and physical relationshipspresent in an acoustic system which involves only sound waves.

The essential first example of an analogue geometry is the the geometry ‘experi-enced’ by propagating linearized sound waves in an inhomogeneous1 fluid.

The dynamical equation of motion for linearized perturbations (sound waves) ψ ina homogeneous fluid is given by the standard no-frills wave equation(

∂2t − c2∇

)ψ = 0, (1.1)

1Invicid, irrotational barotropic (see below for details).

12 Chapter 1. Analogue Gravity Introduced

Figure 1.1: Spatial contours of constant loudness in a flowing fluid. These are the‘sound cones’, tilted beyond 90 degs by a super-sonic flow on the right.

where c is the sound velocity. As will be demonstrated in detail below, the dynamicalequation on motion for such sound waves in an inhomogeneous fluid is given by

gψ =1√

g∂a(gab√

g∂bψ)

= 0, (1.2)

where g is the d’Alembertian differential operator associated with the metric

g =ρ

cs

[−(c2

s − v2)dt2 − 2vidt dxi + δijdx

i dxj],

=ρ

cs

[−c2

sdt2 + (dx− vdt)2

], g = |Det g|, (1.3)

where v(x) is the flow velocity field of the fluid and cs is the sound speed. That is, thegeneralization for sound waves to an inhomogeneous background fluid is mathematicallyequivalent to a generalization of the background geometry on which the sound wavestravel. The ‘game’ of acoustic geometry at this level is to construct fluid configurationswhich give rise to interesting acoustic metrics and to study small perturbations onthese configurations which behave as if propagating in the acoustic metric geometry.

Below we derive equation (1.2) from the equations of motion for fluids.

1.1.1 Geometric acoustics in fluids

The geometric acoustics limit is entirely analogous to the geometric optics limit in opticswhereby a sound wave is modeled as a sound ray. Of course in all wave phenomena,as in quantum mechanics2, there is an intrinsic ‘particle wave duality’ manifested byan uncertainty relation between position and velocity. The geometric acoustics limitis essentially the approximation provided by the particle picture alone. That is, wave-like perturbations are assumed to move along trajectories traced out by their definiteposition as a function of time.

2Although in quantum mechanics this duality has far reaching consequences not implied by thesame duality in ordinary wave dynamics.

1.1. Analogue gravity, explained 13

Figure 1.2: Non-trivial sound geodesics curve around a region of high flow rate. Theflow of the river is from left to right in this image.

Concretely the geometric acoustics limit is imposed by considering only the highfrequency sound waves in the fluid. Such high frequency modes follow the null geodesicsof the acoustic geometry in a manner entirely analogous to the null trajectories of highfrequency light rays in spacetime. As null trajectories, their form is independent of theconformal factor ρ/c in the metric 1.3.

Consider that a sound particle as it propagates at the sound speed cs in a flowingfluid with a position dependent flow speed v(x). Then as far as the a stationary observeris concerned the sound particle’s motion in the unit n direction is given by

dx

dt= v(x) + csn. (1.4)

Hence the possible trajectories are defined by

||dx− vdt|| = csdt, (1.5)

which are precisely the null geodesics of the metric 1.3.

As an illustration of curved nature of effective geometry experienced by sound, con-sider a source of sound waves down-stream of a submerged boulder in a river schemati-cally shown in Figure 1.2. One can imagine that the surface flow directly on top of theboulder is faster than the flow on either side. Intuitively, then, an observer upstreamand co-linear with the source and boulder will receive first those sound waves thatactually propagate around the boulder and not over the top of it. This observation isstrongly reminiscent of the curvature of the ‘shortest path’ between points in spacetimeby the presence of a non-trivial gravitational field. Therefore we can appreciate that, at


the very least, we should not be surprised to find utility in the methods of differentialgeometry for the description of sound waves in inhomogeneous flowing fluids.

Of course, at extremely high frequency / short wavelengths the continuum ap-proximation for the fluid breaks down and the picture of dispersion-less sound rayspropagating in the fluid should fail. In fact, it is in this limit that we may learn ourmost important lessons from the analogy; in the space of all possible modifications towave propagation there should exist a subspace of ‘most plausible’ modifications whichwe might glimpse through the analogy.

1.1.2 Physical acoustics

Physical acoustics corresponds to the full dynamical description of the ‘field’ of soundfluctuations in the fluid including all the wave-like features in the propagation. Theanalogy for physical acoustics is therefore more comprehensive than for the null geodesicsof geometric acoustics. In the physical acoustics picture we linearize perturbationsaround a solution to the full fluid mechanical equations of motion to obtain a scalarwave equation identical to that of a scalar field in a curved spacetime. This is done inthe next section in full detail where we reproduce the full equation of motion (1.2).

1.1.3 The nitty gritty details of physical acoustics

Historically the first and also the simplest example an analogue geometry is the physicalacoustics picture for sound propagation in fluids [1]. We will derive the result (1.2) in3+1 dimensions, the result being dimension dependent in general.

It should be made clear as an aside at this point that the analogy is not restrictedonly to acoustics in flowing fluids. We will include at the end of this chapter a partiallist of systems for which the analogy has been established and investigated.

Consider an irrotational barotropic perfect fluid. Barotropic means that the pres-sure is a function of only density and irrotational means dv = ∇ × v = 0 where v isthe velocity vector field. The Poincare lemma for the plane ensures then that v is thegradient of a scalar: v = df for some function f . This is the simplifying assumption ofirrotationality; the fluid dynamics reduces to that of a scalar field. The assumption ofbarotropicity is a statement that the pressure should depend only on the scalar densityprofile and prevents the generation of vorticity due to non-scalar pressure effects. Abarotropic fluid has surfaces of constant pressure that coincide with surfaces of con-stant density. More generally in non-barotropic fluids the surfaces of constant pressureintersect the surfaces of constant density and the character of these intersections pro-vides information about the fluid. The equations of motion for a invicid perfect fluid

1.1. Analogue gravity, explained 15

are given by the coupled partial differential equations

∂ρ

∂t+∇ · (ρv) = 0 (Continuity equation), (1.6)

ρ

(∂v

∂t+ (v · ∇)v

)= −∇p (Euler equation). (1.7)

Which under the assumptions listed above reduce to

∂ρ

∂t+ (∇ρ) · (∇f) + ρ∇2f = 0 (1.8)

ρ

(∂

∂t∇f +

1

2∇(|∇f |2

))= −∇p (1.9)

These equations are supplemented by the barotropic equation of state p = p(ρ). Ofcourse the assumption of irrotationality can only be applied to the initial velocity config-uration; it is up to the equations of motion to forbid the generation of a curl componentunder time evolution from an initially irrotational flow. We now demonstrate this fact.We have by (B.1)

∂

∂t∇× v = ∇× ∂v

∂t(1.10)

= −∇× ∇pρ− 1

2∇×∇

(|∇f |2

). (1.11)

Now, the second term vanishes by the well known fact that the curl of a gradientvanishes3. For the first term we need to do some work. As mentioned above, for abarotropic fluid the surfaces of constant pressure coincide with the surfaces of constantdensity and hence that the vectors ∇p and ∇ρ are parallel. We have

∇× ∇pρ

= ∇ρ−1 ×∇p = ρ−2∇ρ×∇p = 0, (1.12)

so that we have demonstrated the assertion.

So, we are ready to proceed. Consider an exact barotropic solution to (B.1) and(2.121) (f0, ρ0, p0) and small perturbations thereof

f = f0 + εf1, ρ = ρ0 + ερ1. (1.13)

We have

p(ρ0 + ερ1) = p(ρ0) + ερ1p′(ρ0) +O(ε2) =: p0 + εp1, (1.14)

3This is an expression of the general fact that d2 = 0


so that p1 = p′(ρ0)ρ1 = c2ρ1 where we define the sound speed c2 = p′(ρ0). Insertingthis ansatz into (B.1) and (2.121) and collecting terms one obtains

∂ρ1

∂t+∇ · (ρ1∇f0 + ρ0∇f1) = 0 (1.15)

∂f1

∂t+∇f0 · ∇f1 = −p1

ρ0

. (1.16)

The last equation is derived by using the expansion

∇(p0 + εp1)

ρ0 + ερ1

=∇p0

ρ0

+ ε∇(p1

ρ0

), (1.17)

and also we have neglected any zero mode constant fluctuations in the fluid by settingan integration constant to zero.

Eliminating both p1 and ρ1 we arrive at

0 = ∂tρ0

c2∂tf1 + ∂t

ρ0

c2δijvi∂jf1 + ∂iδij

ρ0

c2vj∂tf1∂iδijρ0∂jf1 + ∂i

ρ0

c2vivj∂jf1, (1.18)

which we rewrite, after dropping the subscript on f , in the surprisingly familiar form

1√

g∂a(√

ggab∂b)f = 0, (1.19)

where we define the matrix of coefficients

g−1 :=1

cρ0

−1... −v

· · · · · · · · · · · · · · · ·−v

... c2113 − v ⊗ v

, g =ρ0

c

−(c2 − v2)... −v

· · · · · · · · · ·−v

... 113

, (1.20)

and the determinant g := det g4.

This result provides a demonstration of the claims laid in the introduction of thissection, namely that traveling waves in a sufficiently idealized inhomogeneous fluid areexactly described by a Klein Gordon equation on a non-flat Lorentzian spacetime.

1.2 The motivation - what’s the problem with real

gravity?

At first the fact that the causal structure of sound waves is encoded in a non-flat curvedgeometry might seem to be a curiosity with limited physics implications.

4In 1+D spacetime dimensions the metric differs in the power of the conformal factor appearingin the definition of g. It can be shown that the conformal factor in 1+D dimensions is given by(ρ0/c)

2/(1−D) being undefined in 1+1 dimensions.

1.2. The motivation - what’s the problem with real gravity? 17

By now the notions of analogue gravity or analogue geometry are mainstream inthe gravity community. Many approaches to the problems faced by our current theoryof gravity attempt to go beyond the continuum Lorentz invariant picture and replacespacetime with some discrete (and inherently quantum) structures. Indeed, it is widelyexpected that the continuum is manifested merely as a coarse grained representationof some more fundamental degrees of freedom in the deep ultraviolet. The subject ofanalogue gravity probes the realm between the discrete picture of quantum spacetimeand the continuum ‘real world’ approximation. In particular analogue gravity aims toshed light on the semi-classical5 regime where matter fields are described by quantumfield theories but gravity is assumed to be a classical background field. This regimeis expected to cover a vast terrain and moreover is the fertile seed-bed for an array ofexotic quant-gravitational phenomena, such a black hole radiance, cosmological particlecreation and superradiant scattering, which serve as way-stones on the road to realquantum gravity.

Observability

One might ask why it is not enough to describe these phenomena directly, why wewould want to simulate them in some complicated terrestrial experiments. The answerlies in the glaring fact that none of the results of semi-classical gravity, where gravityand quanta are simultaneously involved, have been probed experimentally. This lackof observability is due to many reasons:

• Many of the process are quantum spontaneous emissions from the vacuum6 oc-curring in far off places in the universe,

• They usually occur in strong gravitational fields (not Earth-strength ones) orhighly accelerated frames,

• The fluxes are small,

• The spectra are usually quasi-thermal and the temperatures typically low

Analogues alleviate most if not all these difficulties:

• Powerful modern experimental techniques are capable of directly observing thequantum nature of matter7

5As we will see, in fact we only probe semi-classical gravity in the test-field approximation wherethe back-reaction of quantum fields on the classical metric, through the renormalised stress energytensor is neglected.

6Even non-gravitational spontaneous quantum emission from the vacuum has proved to be ex-tremely difficult to probe experimentally – see [2] for a recent breakthrough in this area.

7In particular we have in mind here Bose–Einstein condensates and quantum optical systems.


• Analogue gravitational fields have nothing to do with large masses or high flowrates8.

• Since analogue horizons are not causal horizons for the experimenter, he canuse other means to directly observe ‘beyond the horizon’ allowing for correlationmeasurements between emitted partners9

• Cold atom experiments are capable of measuring very low temperature thermalensembles.

Separation of kinematic from dynamic gravity

Instead of solving complicated non-linear dynamical equations for the gravitationalfield (for example the metric tensor and possibly also an independent connection ortorsion), in analogue gravity one finds an effective geometry (read: gravitational field)directly in the equations of motion of some analogue system upon which some dynam-ical processes are taking place, usually wave propagation. Hence the analogy isolatesonly the kinematic aspects of gravity. We cannot expect, therefore, to find some equiv-alent of the Einstein equations describing the interaction between the matter fields andthe analogue geometric tensors.

The central questions of the analogue gravity program are most succinctly summa-rized by Visser [4]

• How much of general relativity can you carry over into these condensed matteranalogues?

• How easy is it to actually build these things in the laboratory and do a fewexperiments?

The natural secondary but highly significant question to ask once we agree that atleast some relativity is contained in the analogy is

• Can we model quantum fields in curved spacetime on these analogue geometries?

It turns out that, indeed, many of the important results of semi-classical gravityare not directly dependent on the dynamical form of the Einstein equations and areinstead, really kinematic results putting us in a position to model them with analogues.Importantly, the Hawking emission falls into this category (see 2.1.3 for the details ofthis argument).

8Very high surface gravities are evidently achievable – see [3]9This is partly the theme of Chapter 3

1.2. The motivation - what’s the problem with real gravity? 19

The problem with black holes

There is also a significant quantity of theoretical work on these more general analoguespacetimes, and their implications regarding the Hawking effect [5, 6, 7, 8, 9, 10].

Probably the most significant semi-classical gravity result is the establishment ofHawking radiation first described by Hawking in 1974 [11]. Arguably, Hawking’s re-sults on black hole radiation and their embedding into the more general framework ofblack hole thermodynamics serves as a benchmark for any quantum theory of gravity.However, the derivation of the existence of Hawking radiation is not strictly semi-classical: Hawking photons observed at late times at null infinity necessarily possessedan arbitrarily high frequency and hence energy in their past making the neglect ofback-reaction questionable. These Hawking photons are virtual photons and hence oneshould carry out a more complete analysis of the renormalised stress energy tensor inorder to understand back-reaction effects beyond the heuristic picture of the redshiftargument.

In this sense Hawking radiation serves as a microscope which is sensitive, witharbitrary precision, to the microstructure of spacetime and quantum field theory. Itis probably too much to believe in the validity of a classical smooth spacetime onarbitrarily short length scales: one expects both ordinary quantum field theory andour current geometric gravitational theories to break down or at least be significantlymodified in the deep ultraviolet.

In fact there are good reasons to have doubts that physics is invariant under arbi-trarily large boosts. Firstly there are the UV divergences of ordinary QFT – renormal-isation is the idea that one can ignore very high energy processes during low energyexperiments and it is well known that renormalisability of gravitational theories is prob-lematic, possibly requiring a breakdown of boost invariance in the high UV [12, 13].Secondly, and more in the specific context of black holes, there is an infinity in the en-tanglement entropy of a quantum field [14, 15] which again, would be tamed by someform of Lorentz violation in the UV [16, 17].

Therefore, the possibility of producing in the laboratory the geometric (kinematic)features pertaining to the onset of a Hawking flux is of great interest. Crucially, in thefluid and condensed matter analogues the continuum picture of sound quanta prop-agating on an effective geometry will break down for high frequency phonons in anentirely predictable way. That is, we know for sure that the fluid mechanical equationsof motion should be replaced by molecular dynamical ones for disturbances approach-ing the inter-molecular spacing. Thus in analogue gravity we are able to isolate thosefeatures of black hole physics that depend on the unknown ultraviolet regime.


Quantum gravity phenomenology

The analogy between gravity and condensed matter systems is useful for two reasons:on the one hand it allows us to experimentally test some predictions of Lorentz invariant(LI) physics by analogy; on the other hand it provides for us the possibility of testingthe physical and mathematical consequences of relaxing exact LI. As we will see, ananalogue LI description is always approximate, applying only in a window of scalesin the analogue system, with unavoidable violations near the edge of the window. Itwould be very convenient if we could model LI physics entirely inside the analogueLI window but, as we shall see, this is not always possible. This can either be seenas a stumbling block for the simulation of LI physics or as an opportunity to test,observe and experiment with Lorentz violating (LV) versions of some semi-classicalgravity phenomena. The general wisdom would be that if we can see these phenomenain fluids, then their spacetime counterparts do not depend in a essential way on thedetails of physics in the UV.

The idea that LI might not be an exact symmetry of nature has a long and con-voluted history, enjoying a strong revival recently [18]. However, while spacetime LVis not a topic which we shall be considering explicitly in this thesis, it suffices to saythat analogue models are an arena where exact LI is broken and in an understand-able, predictable and testable way. Indeed LV, black-hole physics, quantum gravityphenomenology all find a natural expression in analogue models and should be kept inthe back of every analogue gravity-est’s mind.

1.3 The analogy developed

The next step in developing the analogy is to try to find background fluid configurationswhose associated analogue geometry corresponds to a known exact solution to theEinstein gravitational field equations. This is the content of the rest of the followingfew short subsections.

1.3.1 Modeling spacetime with fluids

It is one thing to rewrite fluid mechanics in a fancy new way but is there any physicsat stake here? In fact, as we will show below, it is possible to construct analogue(equatorial) Kerr and Schwarzschild geometries in fluids allowing us, in the very least, toimport the techniques and known results from spacetime physics on these backgroundsto the study of fluid perturbations in the lab. Our real interest in this analogy, ofcourse, is in the information flow in the opposite direction. The question is ‘what canwe learn about spacetime physics by studying fluid perturbations?’. For example, itmight be useful to find the analogue of the concepts of event horizon, apparent horizon

1.3. The analogy developed 21

or ergo-region in fluids, independently from their realisation in the exact Kerr andSchwarzschild cases, in order to elucidate the nature of the logical interconnectednessof the network of ideas that paint the picture of black hole physics - local Lorentzinvariance, continuum approximation, causal boundaries, etc - in the ‘real world’.

Consider the generic acoustic line element given above in equation (1.3) The firstpiece of vocabulary that can be imported to the fluid dynamics is that of the ergo region.We see that the metric coefficient gtt becomes singular for x such that c2(x) = v2(x).It can be shown that the analogue spacetime curvature evaluated at these points isnon-singular and hence, as in the case of the Schwarzschild horizon, these points arenot true singularities of the geometry. We define the ergo region as the set of points

E := x | ||v(x)|| > c(x), (1.21)

the boundary of which is called an ergo surface. In the ergo region the co-movingfrequency of a wave is negative when the laboratory frequency is positive and wewill see in the next chapter that ergo-regions are associated with the phenomenon ofsuperradiance.

Another important concept in mathematical relativity is that of a trapped surface.In the fluid analogues here we define a trapped surface as a two dimensional surfacewhere the normal component of the fluid velocity is everywhere greater in magnitudethan the local speed of sound. Compare this to the rather sophisticated definition oftrapped surface in relativity [19]. The reason we are allowed such a simple definitionhere is that we have access to a preferred background ‘lab’ frame which provides anobjective definition of velocity.

A horizon is defined in a similarly straightforward way as the boundary of the regionfrom which null geodesics of the metric (1.3) can emerge. They are a kind of causalsingularity beyond which sound waves cannot emerge.

Assuming that the flow v is time independent, the metric (1.2) can be brought intothe manifestly static form

ds2 =ρ

c

[−(c2 − v2)dτ 2 +

(δij +

vivjc2 − v2

)dxidxj

], (1.22)

by way of the coordinate transformation

dτ = dt+vidx

i

c2 − v2. (1.23)

It is well known how to define the surface gravity of such a static geometry at ahomogeneous space-like hyper-surface. The idea is to take the obvious time-like Killingvector possessed by this metric

k = ∂τ , (1.24)


normalize it

K =k√

ds(k,k)=

k√(ρ/c)(c2 − v2)

, (1.25)

compute its acceleration,

a = (K · ∇)K =1

2

(∇(c2 − v2)

c2 − v2+∇(ρ/c)

ρ/c

), (1.26)

and finally finish the arithmetic by taking the limit as one approaches the surface ofinterest

surface gravity = limv2→c2√ds(a, a) ds(k,k) (1.27)

=1

2v · ∇(c2 − v2) (1.28)

We can re-write this in terms of the normal derivative to the acoustic horizon

surface gravity = c∂(c− v)

∂n. (1.29)

Now that we have surface gravity (essentially, as mentioned above, a measure of theacceleration of the fluid at the acoustic horizon) and horizon we have all the ingredientsto follow the Hawking derivation of Hawking radiation including an ignorance of themicrophysics at extremely short distances.

1.3.2 Fluid black holes

Can we make a fluid flow such that the acoustics are those of a scalar field in a blackhole geometry? Here we describe how the answer is ‘yes, morally’ for non-rotatingblack holes but ‘no’ for rotating ones changing to ‘yes’ when we relax the requirementto modeling only the equatorial plane of the Kerr solution.

Schwarzschild geometry

Assume a spherically symmetric velocity potential with a constant (that is positionindependent) speed of sound c and a background velocity field

v =

√2GM

r∂r. (1.30)

The continuity equation reduces to the divergence

∇ · (ρv) = 0, (1.31)


which in polar coordinates10 and purely radial flow becomes

2

rρ

√2GM

r+

∂

∂r

√2GM

r= 0, (1.32)

which is easily solved to yieldρ ∝ r−3/2. (1.33)

Integrating the constancy of the speed of sound equation c2 = dp/dρ we deduce theequation of state

p = p∞ − c2ρ ⇒ p− p∞ ∝ c2r−3/2, (1.34)

providing us with a concrete expression for the metric coefficients as functions of r

ds2 ∝ r−3/2

−dt2 +

(dr ±

√2GM

rdt

)2

+ dΩ2

, (1.35)

where the ingoing and outgoing perturbations see the + and − forms of the met-ric respectively. Without specifying the exact solution to the continuity equation weare stuck with the proportionality equation. Now, as gravitational physicists we aresupposed to recognize in this metric the Schwarzschild metric written in the slightlynon-standard Pailleve-Gullstrand coordinate system multiplied by the conformal factorr−3/2.

We can see that we have an acoustic horizon at the usual place r = 2GM in theSchwarzschild geometry. There is only one caveat: As mentioned above, we have notreally reproduced the Schwarzschild geometry but actually one that is conformallyrelated to it.

It can be shown that the Hawking effect is conformally invariant, the temperaturebeing related to the surface gravity of the horizon which is insensitive to the conformalfactor. The only place where the conformal factor enters the physics is in the backscattering from the spacetime curvature, the so-called ‘grey-body’ factor. Hence, prac-tically, the conformal factor should not be such a big problem if all we want to do isre-produce the Hawking effect in the lab.

Kerr geometry

The generalization of acoustic Schwarzschild to acoustic Kerr is highly non-trivial. Infact it has been shown [20] that the simulation of Kerr geometry in fluids, even up to aconformal factor, is impossible. The basic idea is that the acoustic metric must neces-sarily possess conformally flat spatial sections as can be seen in its form (1.20) whereas

10In a general coordinate system the divergence of a vector field is written as g−1/2(√

g1/2ξa)a.


the Kerr metric does not possess this algebraic feature. However, it can be shown thenthe (2+1 dimensional) equatorial plane of the Kerr geometry is implementable as ananalogue geometry as in (1.3) by some judicious coordinate changes.

In Boyer-Lindquist coordinates the equatorial sections are described by the metric

ds2 = dt2+2m

r(dt− adφ)2 − dr2

1− 2mr

+ a2

r2

−(r2 + a2

)dφ2, (1.36)

which can be manipulated into the form (1.3) by the change of coordinates from r tor defined by [21]

1

r

dr

dr=

1√(1− 2m

r+ a2

r2)(r2 + a2 + 2ma2

r). (1.37)

One finds the form (1.3) in 2+1 dimensions with

vr = 0

vφ =2am

H2(r)r(r)

1

1 + a2

r2+ 2ma2

r3

, (1.38)

where the function H(r) describes the inverse transformation to (1.37) between r andr11

r = rH(r). (1.39)

We also have a position dependent sound speed cs

c2s =

1− 2mr

+ a2

r2

H2(1 + a2

r2+ 2ma2

r3

) , (1.40)

where r is to be thought of as a function of r here.

The point of writing these complicated formulae down is to convince the reader that,indeed it is possible to find some fluid configuration which simulates the equatorial Kerrgeometry but that its form is rather complicated and very likely not going to occur fornaturally flowing fluids without external driving forces.

Crucially, however, the concepts of ergo-region and horizon do not depend on repro-ducing exactly the Kerr solution of GR: we expect to be able to reproduce ergo-regionsand horizons in a ‘natural’ fluid configuration which does not match the Kerr solutionand which can support an analogue Hawking or superradiance process. We will discusssuch a possibility later in 6 in the case of a ‘naturally’ occurring bathtub vortex flow.

11To get a handle on H we note that when m → 0 we have r(r) =√r2 − a2 in which case H(r)

approaches the function√r2 − a2/r.


1.3.3 The expanding universe

We will make use of the analogy later on to model an expanding universe described bya homogeneous FRW line element

ds2 = −dt2 + a2(t)dx2. (1.41)

One way to achieve this geometry experimentally is to find a homogeneous fluid inwhich the sound speed is controllable in time. We could then work with zero flow andmake a coordinate transformation

ds2 = Ω2(t)[−c2

s(t)dt2 + dx2

](1.42)

= −dη2 + Ω2(η)dx2, where, η(t) =

∫ t

Ω2(t′)c2s(t′)dt′. (1.43)

Indeed such fluids exist with a controllable sound speed, we will discuss them inChapter 3. In that chapter we will also show how an inhomogeneous cosmologicalmetric can arise from the naturally occurring flow of a particular kind of fluid associatedwith Bose Einstein condensation and its controlled release from a confining trappingpotential.

1.3.4 Practical considerations

Naturalness – It should be stressed that, although we have presented the analogyand how it can be used to model real spacetime configurations of ‘practical’ interestto a gravity theorist, the problem of initializing and sustaining the fluid or condensedmatter configuration which induces a particular acoustic geometry has been completelyunaddressed to this point. We should not expect to be able find fluids in nature thatbehave in these non-standard ways under the influence of, say, gravity alone. In practicewe should expect to have to force the fluid to behave as we require. In the presence ofan external driving force the Euler equation (1.7) is modified to include the extra termon the right hand side

ρ

(∂v

∂t+ (v · ∇)v)

)= −∇p+ Fdriving, (1.44)

which can be used to drive the fluid into a particular flow state such as those requiredabove for the simulation of the Kerr equatorial plane.

Another potential difficulty in practice for these fluids is to initiate and sustainthese high velocity flows without introducing turbulence. Throughout we have beenusing fluid equations suitable for laminar flow - flow without turbulence. It has beensuggested that such flow conditions should be easiest to construct at extremely lowtemperatures.


Completeness – Immediately we can see from (1.3) that the analogue metric doesnot have enough degrees of freedom to simulate an arbitrary gravitational field. Ingeneral in n spacetime dimensions the gravitational field is described by n(n − 1)/2independent degrees of freedom which in 3+1 dimensions are 6. The analogue metricpossesses only 4 degrees of freedom, the 3 components of the velocity v and the densityρ.

Quanta – We have blithely been talking periodically up to this point about classicalfluids and quantum spontaneous emissions in the same breath. We shall see belowthat there exists a wider class of models for analogue geometry than those for a simpleflowing fluid presented up to this point, including models which utilize the quantumnature of the constituent matter in an essential way. Further on in this thesis we willbe discussing the restricted problem of understanding some of the modified physicsmechanisms that come part-and-parcel with broken (analogue) Lorentz invariance whenit comes to the Hawking process or superradiance. This is part of a more generalprogram of research which seeks to probe the principles, conditions and mechanismsnecessary for a spontaneous emission, even if, in practice, there will be no spontaneousemission process in the system. In this direction one experiments directly with classicalscattering in the inhomogeneous fluids (numerically or physically), inferring the physicsof the quantum spontaneous emission by general arguments.

Another angle to the question of quanta is given in the recent article [22] whereUnruh has demonstrated the equivalence between quantum and classical amplifiers inthe context of analogue models. According to Unruh, a black hole is nothing butan amplifier of the quantum field vacuum fluctuations, the black hole (or white hole)radiance can be understood as a continual scattering of vacuum modes by the geometry.When dispersion is included, an additional mechanism for producing outgoing particlesis available (‘mode conversion’) which intrinsically involves the dispersive nature ofthe propagation, but the analogy follows through identically: dispersive spontaneousemission and dispersive stimulated classical emission are two examples of the sameamplification process. The question is, what can we hope to learn about the quantumspontaneous emission by looking at classical stimulated emission (scattering)? Unruhhas answered this question by saying that the Bogoliubov coefficients in the classicalscattering experiment match those for the quantum spontaneous emission since both arebased on the same underlying amplifier model. Indeed, in the article [23], the authorsstudy the scattering (and mode conversion) of an incident, classical water wave ontoa white hole flow profile and claim to be confirming the ‘Hawking mechanism’. Thisis possible since the Bogoliubov coefficients of the scattering process βk measuring theamount of mode conversion and scattering by the flow exactly match the Bogoliubovcoefficients measuring the spontaneously emitted ‘quanta’, even though we never hopeto observe such quanta, nor do we really believe in the quantization of water waves; thequantum amplification is determined exactly by the classical behaviour of the amplifier.

Finally, there can be cases where a perfectly well defined analogue metric and geom-

1.4. Summary 27

etry emerges for the perturbations of a system but where the commutation relations forthe fields associated with such perturbations are not the canonical ones (for example,see the article on the ‘slow-light’ proposal [24] where the highly non-linear dispersionrelation can interfere with the emergence of the correct commutation relations for thequantum theory to pass through the analogue). In general, one requires both the met-ric and commutation relations to emerge correctly in an analogue in order to carry outthe analogue derivation of a QFT process on the curved geometry.

1.4 Summary

In this chapter we have introduced the analogy in simple and intuitive terms derivingthe basic first example of the analogy for sound in a moving fluid.

The motivation and larger picture into which the analogue program should beembedded has been pointed out and we have entered into the discussion of the practicalaspect of analogue gravity which, after all, is the crucial element and perhaps onlyreason to be thinking about analogues at all.

Cast in a different light, this possibility of describing terrestrial experiments usinga curved spacetime analogue provides one with a new set of tools to tackle difficultproblems in condensed matter physics and is able to suggest new and interesting non-equilibrium quantum effects. Moreover, a successful application of these alternativetechniques and consequent observation of the effects predicted would serve as furthertheoretical support for the validity of the semiclassical theory.

Chapter 2

A Toolbox for Analogue Gravity

In this chapter we collect and dissect a toolbox for the aspiring analogue gravity-est.Often we point the reader to the relevant literature where a complete discussion isunnecessary. Due to the interdisciplinary nature of analogue gravity we necessarilyrequire a very general toolbox touching on a wide range of topics from fluid mechanics,Bose-Einstein Condensation (BEC), nonlinear optics, semi-classical gravity phenomenaand non-linear wave phenomena. In no way is this toolbox intended to be complete,however, and we include a partial list of excluded topics at the end of the chapter. Thechoice of topics reflects the necessary background for what is presented in the followingchapters.

In the first section we introduce some phenomena from quantum field theory incurved spacetime. These are the main actors in analogue gravity since these are thephenomena which we are trying to simulate1 in condensed matter systems throughthe analogy. We discuss the Dynamical Casimir effect (DCE) / cosmological particlecreation (isolating a conflict of terminology that exists in the literature), Hawkingradiation and superradiant scattering. In the second section we dissect Bogoliubovcoefficients a little, expanding on some alternative ways to compute them. In thefollowing three sections we present basic material on three different analogue systemsand how there can exists a notion of an effective metric for excitations. In the finalsection we discuss modified dispersion in general, the useful tools and some of thedynamical consequences.

2.1 Quantum fields and gravity together

In this section we discuss three separate phenomena of quantum fields in curved space-time. General references for quantum fields in curved spacetime are [25, 26, 27, 28].

1Equivalently, generalise.

30 Chapter 2. A Toolbox for Analogue Gravity

For black hole radiation the contemporary treatments of Jacobson is most illuminat-ing [27]; for superradiance the reviews [29] and the works of [30] are recommended;for dynamical Casimir effect / cosmological particle production the text of Birrell andDavies is still the best reference [25].

2.1.1 Dynamical Casimir Effect - Cosmological particle cre-ation

Within the framework of quantum field theory in curved spacetime where fields prop-agate on a fixed background, some interesting – and sometimes surprising – quantuminstabilities are predicted. Among them a crucial role has been played by the so calledcosmological particle creation: a quantum emission of particles as a consequence of anexpansion of spacetime.

Also known as Dynamical Casimir effect (DCE), the perhaps confusingly namedquantum emission is a very general phenomenon, being associated with wave propaga-tion over a dynamical space and not (when back-reaction is neglected) to a propertyof the Einstein equations, it is not confined to gravitational physics. Originally dis-cussed in the gravitational context by Parker [31, 25] who studied particle productionin FLRW spacetimes, the Dynamical Casimir Effect (DCE) has found applications inan extremely diverse range of subjects (see for example the excellent text [32] and thereview article [33] which is mainly focussed on the moving boundary type DCE whichwe do not discuss in this thesis).

Very simply put, as far as we are concerned here DCE is the observation that awave equation with some time dependent parameters do not admit global plane wavesas solutions. This is a very trivial and unsurprising fact but the physical implicationsare rather far-reaching when it is embedded into the language and formalism of QFT.

To illustrate the physics at stake here let us take the very simple example of theordinary wave equation in flat spacetime

∂2t f = c2∇2f. (2.1)

The dynamics described by 2.1 is equivalent to that of a collection of harmonic oscil-lators. Expanding f in eigenfunctions of the spatial Laplacian ∇2

f(t, x) =∑k

Akλk(t, x), λk(x, t) = fk(t)eik·x, (2.2)

each coefficient fk satisfies the equation of motion for a simple harmonic oscillator(∂2t + ω2

k

)fk = 0, (2.3)

2.1. Quantum fields and gravity together 31

where ω2k = c2k · k depends only on the magnitude k =

√k · k of the wavenumber k.

Crucially, when c is space and time independent 2.3 admits global plane wave solutions

fk = e±iωkt, (2.4)

providing the global solution to (2.1) as a linear combination of traveling plane waves

f(t, x) =∑k, ±

A±k e±iωkt+ix·k, (2.5)

the constant coefficients being determined by the initial condition. Note that for each kthere are two frequencies ±ωk (where we always assume ω > 0 and indicate the negativepartner with an explicit minus sign) — this doubling of the degrees of freedom will becrucial for the physics of particle production. That an arbitrary initial condition can bereconstructed is a non-trivial functional analytic fact made possible by the completenessof the plane wave ‘basis’ eix·k which we can express as a ‘decomposition of unity’relation ∑

k

eik·(x−x′) = δ(x− x′). (2.6)

Frequency or norm? — Associated with the equation of motion (2.1) is an innerproduct

(f, g) = i

∫dx [f ∗∂tg − g∂tf ∗] , (2.7)

which is independent of time when both f and g satisfy the equation of motion(2.1). This inner product is not really an inner product, however, since it is notpositive definite: for fixed constant frequency k the solutions with time dependenceexp(+iωt) have negative norm whereas for exp(−iωt) the norm is positive. Hencewe usually abuse language a bit and say that such negative norm modes are neg-ative frequency. In some interesting situations, for example in the ergo-region ofa rotating black hole, the frequency can be positive while the norm be negative.The splitting of the Hilbert space is always done with respect to the positive andnegative norm states. Hence we will use negative norm to describe the componentssuch as exp(iωkt) here and in what follows in this thesis but occasionally write‘negative frequency’ when no confusion is possible.

Now, when c is time dependent (which could happen, for example, when the back-ground spacetime is expanding), the steps leading up to (2.3) remain valid but theequation of motion now includes the time dependent frequency(

∂2t + ω2

k(t))fk = 0, (2.8)

where ω2k(t) = c2(t)k · k. Usually we are interested in situations where the time de-

pendence is confined to a finite time interval so that (2.8) admits solutions which in


the asymptotic past and future look locally (in time) like plane waves with frequenciesωink and ωout

k . Global solutions to 2.8 in that case are not plane waves but are typicallyexpressible in terms of Hypergeometric functions. We can, however, choose bound-ary conditions either in the past or future of the time dependent region such that theglobal solution (for each k) and its first derivative converge point-wise, as t → ±∞appropriately, to a single complex exponential plane wave exp(iω

in/outk t). The fact that

this plane wave is a not global solution to the equation of motion implies that in thefuture region it cannot be simply exp(iωout

k t) but must contain also a component withexp(−iωout

k t) – the only other function which is a solution in the future region, theplane wave of opposite frequency.

Now comes the interesting part: at times when c(t) is constant particles are definedin QFT with respect to a plane wave basis of the wave equation at that time, the fieldoperator φ at some time t being expressed as

φ =∑

eik·xfkak + e−ik·xf ∗ka†k, (2.9)

where fk are the positive norm ωin/outk > 0 solutions to (2.3) and the n-particle states

of momentum k being defined as the states a†kn|0〉. What is important here is that

the particle content is tied to a decomposition of the solution space into positive andnegative norm solutions to (2.3), here indicated by the complex conjugate for thenegative norm. When this decomposition is modified, the definition of the n particlestates is also modified.

This can happen, for example, when an initially positive frequency fk propagatesthrough a time dependent region emerging on the other side with a negative frequencycomponent

e−iωink t ←− fk(t) −→ αe−iωout

k t + βeiωoutk t, (2.10)

where α and β are some constants. Note that the solution must be of this form onthe right hand side since the only solutions to (2.8) when ωk is constant are these twoexponentials. This structure (2.10) is the basic crucial ingredient of particle productioneffects, its significance for DCE cannot be overstated.

Since we work in the Heisenberg picture of quantum evolution we need to choose thevacuum state once, which we do at −∞ in terms of the solutions fk which are purelypositive norm at −∞, that is, with the time dependence exp(−iωin

k t). An observerwho decomposes his field in to positive frequencies ωout

k at +∞ should use a differentdecomposition in terms of solutions gk to (2.8) which converge to positive frequencyplane waves at +∞

gk(t) −→ e−iωoutk t, as t→ +∞. (2.11)

Of course the is only one unique quantum field and its unique quantum evolution, itis only its interpretation in terms of particle content which changes from −∞ to +∞.


Hence we can write φ in one of two ways

φ =∑k

eik·xfkak + e−k·xf ∗ka†k (2.12)

=∑k

eik·xgkbk + e−k·xg∗kb†k, (2.13)

the first representing a decomposition in terms of −∞ particles the second in terms of+∞ particles.

In terms of the gk (2.10) becomes

fk = αkgk + βkg∗k, for t→∞, (2.14)

one arrives at

φ =∑k

gk

(αkakeik·x + β∗ka

†ke−ik·x

)+ g∗k

(βkakeik·x + α∗ka

†ke−ik·x

)(2.15)

so that we may identifybk = αkak − β∗ka

†−k. (2.16)

and its adjoint relation. Here one uses the fact that the mode functions are parametrizedonly by the magnitude of the wavenumber k so that the coefficients αk and βk alsoonly depend on the magnitude and that, in particular, β−k = βk but the exponentialfactor changes sign along with an over all sign change from the integration measure.Importantly the creation operators depend on the angle of k as well as its magnitude.This kind of relationship between different sets of annihilation and creation operatorsis known as a Bogoliubov transformation, here presented in this simplified context.

The number of b†k quanta contained in the vacuum state at late times is easilycomputed to be

Numberk = 〈0|b†kbk|0〉 = |βk|2 = |βk|2, (2.17)

since in this case βk only depends on the magnitude k of the vector wavenumber k.The spectrum is usually quoted in terms of the number density Numberk at fixed normk, given by including the phase space integral over the angles in k space, for examplein 3+1 dimensions we have

Numberk = k2|βk|2. (2.18)

The spectrum Numberk typically has the bell shape shown in Figure 2.1 which canbe understood intuitively by considering two separate limits: the adiabatic and sud-den limits. The adiabatic limit applies to frequencies much higher than the typicaltimescale of the variation of c(t) in (2.8) — these frequencies should not notice theslow time dependence of the parameter and hence should not be excited in the spec-trum. This is the exponentially decaying tail to the spectrum in 2.1. The other extreme


Figure 2.1: The characteristic ‘bell shape’ for DCE emission spectrum k2|βk|2 in 3+1dimensions with three regions clearly visible: low k where the sudden approximation isaccurate and the spectrum is approximately quadratic in k; the turnover region wherethe particle production is most abundant; the exponentially decays adiabatic region athigh k.

applies to frequencies for which the variation of c(t) is very fast with respect to theperiod of oscillation for the mode. These modes, ‘see’ the time variation as essentiallyinstantaneous. In that case the equation of motion can be solved exactly by assumingcontinuity at the jump giving

e−iωink t ←− fk −→

1

2

ωoutk + ωin

k

ωoutk

e−iωoutk t − 1

2

ωink − ωout

k

ωoutk

e+iωoutk t, (2.19)

which, using the dispersion relation ω2k = c2k2 associated with (2.8), leads to the

momentum independent Bogoliubov coefficient

|βk|2 =1

4

(cin + cout

cin

)2

. (2.20)

and therefore the quadratic Numberk ∝ k2. This is the quadratically increasing partof the low k (that part for which the frequencies are low and hence the sudden limitapplies) end of the spectrum shown in 2.1.

The peak wavenumber of emission is the wavenumber most efficiently producesby the time dependence. Intuitively this peak wavenumber should be such that the


corresponding frequency is approximately equal to the relative time change in the timedependent parameter

ωpeak 'c

c(2.21)

This ‘rule of thumb’ generally plays out but there can exist deviations from this rulein certain situations [34].

It can be shown a complete description of |0〉 in terms of the gk basis and the stateannihilated by all the bk, that is, bk|Ω〉 = 0 is given by

|0〉 =∏k

Nk exp

[−(β∗k2αk

)b†kb†k

]|Ω〉

=∏k

Nk∑n

√2n!

n!

(β∗k2αk

)n|2k〉, (2.22)

where the product is also supposed to indicate a tensor product of decoupled k states|k〉 = b†k|Ω〉. Here the Nk are normalisation factors. We note that the initial vacuumstate is expressed as a product state of excited states in the future region and thateach k eigenmode is doubly excited. This structure of the quantum state defines isas a so-called ‘squeezed state’, typically found in many quantum emission processes.We will see later that this double excitation corresponds to a back-to-back emission ofoppositely traveling quanta. This kind of phenomenon is known as an in-equivalenceof vacua and is the source many mathematical difficulties in the problem of trying todefine what QFT is in a general curved spacetime.

We point out here that the only initial assumption required to derive all of whatwe have above was the relation (2.10). Many and diverse physical systems are capableof giving rise to such a frequency mixing, including the cosmological particle creationalready mentioned, a variation of the refractive index for electromagnetic waves (equiv-alent to a time varying wave speed) or time varying boundary conditions for a waveproblem. It is this latter example which is often assumed to be the extent of theterminology ‘Dynamical Casimir effect’ due to its close relationship with the originalCasimir effect and Casimir force between two conducting plates for the electromagneticfield. In the case of cosmological particle creation commonly referred to, we note herefor completeness that the scalar wave equation on an expanding background spacetimecan be cast in the form (

∂2t + a2(t)∇2

)f = 0, (2.23)

where a is the scale factor of cosmological expansion entering the metric tensor

ds2 = −dτ 2 + a2(τ)dx2, (2.24)

(τ being a modified time coordinate) so that the time dependent wave speed in (2.8)is directly related to the expansion (or contraction) of the background space itself.


We will see (see 3) that particle production in this way can arise also in Bose–Einstein condensates and we will have time to discuss DCE in more detail there as wellas in a latter chapter (4) on optics.

2.1.2 Bogoliubov coefficients - more details

In this short section we provide a more technical motivation for the relation (2.34)based on the inner product, show that it is equivalent to the intuitive definition anddiscuss some alternative method for their characterisation based on some simple localmeasurements which do not require a global solution to the wave equation.

Define the asymptotic wave speeds as ci = c(t → −∞) and cf = c(t → +∞) andlet the equation of motion be

d2φ

dt2− c2(t)∇2φ = 0. (2.25)

The equation (2.25) is solved by linear combinations of eigenfunctions to the spatialLaplacian

φk = fk(t)eik·x. (2.26)

The Bogoliubov coefficients are defined as inner products between different collec-tions of exact solutions to the wave equation (2.25). For example between the twocollections (one for each k) of Hypergeometric global solutions to(2.25), one set whichconverges to a single plane positive frequency plane waves in the past φin

k and oneset which converges similarly in the future φout

k .

The Bogoliubov coefficient is a function of two momenta and is defined in the timedependent case as

βij = (φout∗i , φin

j ), (2.27)

where the inner product ( , ) between two functions φ1 and φ2 is defined slightlydifferently than the time independent case as

(φ1, φ2) = −i∫

Σ

c2(t) (φ∗1∂tφ2 − φ2∂tφ∗1) , (2.28)

and can be non-zero even if the momenta i and j differ in magnitude. The correctlynormalised plane waves, for example in the ‘in’ region (t→ −∞) should satisfy

(ψink , ψ

ink′) = δ(k − k′) (2.29)

so that they are given by

ψink (t, x) =

eikx−iωit

ci√

2ωi, (2.30)


where ω2i = c2

i k2. We have, for example

φink (t)→ ψin

k (t), as t→ −∞. (2.31)

Then at a time tf when c(t) has relaxed to the constant value cf the inner productbetween an out solution with momenta kout and an in solution with momentum kin isgiven by

βkin,kout = −i e−iωkout t√2ωkout

cf (∂tfkin + iωkoutfkin)

∫Σ

dx ei(kin+kout)x

= −i e−iωkout t√2ωkout

cf (∂tfkin + iωkoutfkin) δ(kin + kout)

=: βk δ(kin + kout), (2.32)

where we defined the isotropic Bogoliubov coefficient βk with k = |kin| = |kout|. Thatis

βk := −i e−iωoutt

√2ωout

cf (∂tfkin + iωoutfkin) , (2.33)

where ω2out = n2

fk2out. We see that for this simple wave equation, kin must be equal to

−kout as we anticipated earlier when we discussed the squeezed nature of the vacuumstate and the back-to-back particle production.

This form of the Bogoliubov coefficient is particularly useful for numerical computa-tions for example: Choose an appropriately normalised initial condition vkin propagateit through the non-trivial time dependent region and compute the derivative ∂tfkin .

Consider the global solution which behaves as in (2.10) converging to the a singleparticle (positive frequency) state in the asymptotic past but which is a mix of positiveand negative frequency properly normalised plane waves in the future (necessarily ofthe same magnitude as the initial frequency as constrained by the δ functional in thegeneral form for βkinkout)

e−iωint

ci√

2ωin

←−−−−t→−∞

f(t) −−−→t→∞

Ae−iωoutt

cf√

2ωout

+Beiωoutt

cf√

2ωout

. (2.34)

Then

βk = −i e−iωoutt

√2ωout

nf (∂tfkin + ikcffkin)

= −i e−iωoutt

√2ωout

cfcf√

2ωout

(iωout(−Ae−iωoutt +Beiωoutt) + iωk(Ae−iωoutt +Be−ωoutt)

)= −i e−iωoutt

2ωout

(2iωoutBeiωoutt

)= B, (2.35)


which is the intuitive definition of the Bogoliubov coefficient (as the component of thenegative frequency part of the asymptotic waveform) that we gave at the beginning ofthis section.

We see that when the initial condition is a global solution to the wave equation(there is no non-trivial time dependence in the wave speed, cf = ci and the initialcondition remains of plane wave form) the term in round brackets on the second line of2.32 vanishes and β = 0 identically. When there is a time dependence the initial condi-tion will evolve out of the monochromatic plane wave form, the term in round bracketswill be non-vanishing and βk will be non-vanishing, indicating particle production.

It is important to note that any initial condition will propagate through the timedependent region emerging in the future static region as a function which necessarilyoscillates with frequency ωout since the frequencies are constrained by the future dis-persion relation ω2 = c2

fk2. That is, the initial positive norm solution (2.34) is required

to oscillate at the ‘out’ frequency in the future region but in a way in which the realand imaginary parts possibly do not recombine to constitute a pure exponential planewave. An ‘in’ solution will be deformed only in its relationship between the real andimaginary parts of the waveform, a seemingly trivial deformation which is responsiblefor all the particle production effects.

If α = a+ ia and β = b+ ib, a, a, b b all real, then

αe−iωt + βeiωt =

√(a+ b)2 +

(a− b

)2

cos (ωt+ φ1)

+ i

[√(a+ b

)2

+ (−a+ b)2 cos (ωt+ φ2)

], (2.36)

where

φ1 := arctan

(a− ba+ b

), (2.37)

φ2 := arctan

(−a+ b

a+ b

). (2.38)

Therefore by reading the phase shift and amplitude difference with respect to a purecosine of the real and imaginary parts of an ‘in’ state in the ‘out’ region we can readdirectly off the Bogoliubov coefficients without doing any complicated integrals like(2.28)!

We can invert these equations to express the Bogoliubov coefficient componentsas functions of the amplitudes and phase shifts as follows: For a real amplitude A1,


imaginary amplitude A2, and phases φ1 and φ2 we have that

4ab =A2

1

1 + tan2 φ1

− A22

1 + tan−2 φ2

, (2.39)

4ab =A2

2

1 + tan2 φ2

− A21

1 + tan−2 φ1

. (2.40)

Writing this as a = Γ1/b and a = Γ2/b we have

Γ2

b− b = tan φ1

(Γ1

b+ b

), (2.41)

− Γ1

b+ b = tan φ2

(Γ2

b+ b

), (2.42)

and hence (tan φ2 −

1

tanφ1

)b2 − 2bb+ Γ2

(tanφ2 +

1

tanφ1

)= 0, (2.43)

so that,

b =b tanφ1

tanφ1 tanφ2 − 1± 1

2

tanφ1

tanφ1 tanφ2 − 1

√4b2 − 4Γ2 (tan2 φ2 − tan−2 φ1), (2.44)

leaving us with the last equation to solve for b

− Γ1 + b2 − b tanφ2

(Γ2

b(b)+ b(b)

)= 0, (2.45)

which can be done numerically. After obtaining b we can back substitute to get band hence |β|2. In this way one could write a very simple numerical recipe to get theBogoliubov coefficients from just the data of the amplitudes and phases of the ‘in’ modefunctions when propagating in the ‘out’ region. The hard part of this algorithm wouldbe obtaining the phase and amplitude shifts for each k separately in order to plot |βk|2as a function of k.

From the general Bogoliubov βkin,kout we can compute the number density, totalnumber and total energy

dN

d3kout

=

∫|βkin,kout |2d3kin, (2.46)

N =

∫dN

dωout

dωout, (2.47)

E = ~∫ωout

dN

dωout

dωout. (2.48)


Spatial variation of propagation speed

In the case of a spatially varying wave speed the analysis is somewhat more involved.Firstly we wish to find the equivalent of the plane wave basis in the asymptotic regions.In the case when the parameters are both time and space independent in the asymptoticregions then the analysis is very similar to above. The only real difference is that theassumption of separability is modified to the assumption that

φin(t, x) = vkin(t)ψkin(x), (2.49)

where the functions ψk are eigenfunctions of the differential operator

D =1

c2(x)

d2

dx2. (2.50)

Under mild assumptions on the function c2(x) these spatial mode functions satisfy theusual completeness relations∫

dx ψn(x)ψn′(x) = δn,n′ , φ(x) =∑n

(φ, ψn)ψn(x), (2.51)

where the simplified summation notation is understood to represent an integral inthe case of non-compact regions and it can be shown that the eigenvalues are strictlynegative whence we label them as −n2. For example, exact solutions are known formany different choices of c(x) including tanh, sech2, square wells and delta functions.It is the functions ψn which play the role of the plane waves exp(ikx) in the Fourierexpansion, allowing for an exact solution to be found.

Under decomposition of the general type in (2.51) the equation of motion, in theasymptotic past when the time dependence not yet acting, becomes

∂2t (vψ)− ∂2

x(vψ) =(∂2t v + n2v

)ψ = 0⇒ ∂2

t v + n2v = 0, (2.52)

and we are back to the same mathematical problem as before. Everything followsthrough identically except for the fact that ‘single particle states’ are now representedby wave functions which are not necessarily plane monochromatic waves but insteadare eigenfunctions to the operator D.

2.1.3 Hawking effect

The term ‘black hole’ was coined by John Wheeler in 1967 to describe the exact spheri-cally symmetric vacuum solutions to Einstein’s equations which possess trapped regionssuch as the Schwarzschild, Kerr and the electromagnetically charged generalisations aswell as more general black holes. Since we do not explicitly compute any Hawking


radiation in this thesis, this description is intentionally brief but included for com-pleteness due to the significant role played by the Hawking process in the developmentof analogue gravity.

Classically nothing can come out of a black hole: it is black because the eventhorizon is a one-way membrane allowing only ingoing geodesics. When one includesthe quantum effects, however, the black hole is no longer black but, for example inthe case of Schwarzschild hole, glows with a color corresponding to its temperature THwhich is related to the surface gravity κ by (in unite where c = ~ = G = 1)

TH =κ

2π. (2.53)

However, the Hawking effect can also be understood on the level of classical func-tions and ordinary differential equations and it is the purpose of this small subsectionto provide such a presentation in order to isolate the non-trivial details from stan-dard methods which differentiate the hawking process from other effects which persistalso in Minkowski space. We consider here the more physical description in terms ofgravitational collapse.

Consider a wave packet which travels through the surface of a collapsing star, justmanaging to escape the other side as the surface falls to within the Schwarzschildradius. Such a packet will be ‘almost’ trapped by the hole as it forms but managesto escape after spending some time close to the horizon2. It can be shown that therelationship between the outgoing null coordinates u before the star collapse (at I −)and the outgoing null coordinate U after the collapse, far from the star (at I +) isgiven by the particular exponential form

U ' UH + Ae−κu, (2.54)

where UH labels the outgoing null curve which sits exactly on the horizon (the generatorof the horizon), A is some constant and κ is the surface gravity. We see the large red-shifting effect of the horizon for those modes which eventually escape to I + where alarge change in u gives rise to a small change in U .

In Figure. 2.2 schematically how in-falling null geodesics which escape to I + tearingaway from the horizon formed from gravitational collapse. The relationship betweenthe affine coordinates at I − and those on I + is approximately exponential at latetimes (large (u) giving rise to the Hawking spectrum as encoded in the Bogoliubovcoefficients.

In the article in reference [35] it is shown that this exponential relation is all that isrequired in order to see a thermal flux at I +, generalising the derivation of Hawking

2This is the apparent horizon: the real event horizon might in fact be further if in the future, alarge chuck of matter falls into the hole, increasing its mass and hence its Schwarzschild radius.


Singularity

Figure 2.2: Schematic diagram of the geodesics which propagate from I − to I +

by passing close to the formed horizon around a collapsed star. These geodesics areclassically the ones followed by any wavepacket which are observed at late times fromthe black hole.

radiation to the case of a slowly varying κ or a finite lifetime horizon for which theexponential relation applies only for a finite time. In this way the idea of Hawkingradiation can be seen to not strictly depend on the existence of an absolute horizonbut can also apply, for example, to apparent horizons.

Another way to understand Hawking radiation is through the difference betweenthe frequencies as measured by freely falling observers and the Killing frequency asmeasured far from the hole. This viewpoint brings the discussion closer to that ofDCE, discussed above. Very simply, the idea is that, to a freely falling observer anoutgoing mode close to the horizon has a time dependence which drops to zero as theobserver passes the event horizon. Such a function, to the observer, cannot containonly positive frequencies (positive norm modes, we are not in a case where these twoconcepts diverge) since a function which vanishes in space on an interval must containall frequencies (see [27] for a detailed account). This highlights that the quantumvacuum as defined by freely falling observers and their notion of positive norm modes


differs from the vacuum state as defined with respect to the Killing frequencies. Hence,as in 2.1.1 there is a Bogoliubov transformation between the creation ad annihilationoperators which define the two vacua and Fock spaces. It can be shown that the relation(2.54) implies that

|βk|2 ∝1

1− e2πω/κ, (2.55)

which is a Planckian spectrum of temperature T = κ/2π.

2.1.4 The many faces of superradiance

As a third and final QFT process amenable to experimental probes through the Analogyand which we will discuss at some length in a subsequent chapter in the context on theanalogy, we introduce here the phenomenon of superradiant scattering, again initiallyfound to pertain to wave scattering by a rotating Kerr black hole [36, 37] but laterunderstood to be a very general physical process [29]. It is related to various instabilitiesdiscovered earlier known as Klein instabilities or Schiff-Snyder-Weinberg effects [26].We will have time to discuss how the phenomenon is robust against modificationsto the linear dispersion relation for wave propagation, paving the way for a futureexperimental observation in real world dispersive flowing fluids.

Put simply, the phenomenon of superradiance is the occurrence of a wave scatteringprocess in which the total reflected amplitude of wave is greater than the total incidentamplitude. At the quantum level it represents an instability in a quantum field if thescattering equation which gives ride to superradiance is related to the field’s equationof motion.

The traditional scattering problem is defined as follows: one has an equation ofmotion with some potential

d2f

dx2+(ω2 + V (x)

)f = 0, V (x) =

Vi, x→ −∞Vf , x→ +∞

(2.56)

to which is associated the functional

W [f ] = f ′f ∗ − ff ′∗, (2.57)

which is conserved in space when acting on solutions to (2.56). The equation of motionoften arises after the separation of the time dependence exp(−iωt) of a stationary 1+1dimensional wave equation where ω is interpreted as frequencies. Then the ‘scatteringsolution’ fs representing an incoming wave from the right being both transmitted andreflected by the potential decomposes asymptotically as

T eiktx ←− fs(x) −→ eikinx +Reikrx, (2.58)


where the momenta ki kr and kf satisfy the dispersion relation in each region obtainedby inserting the plane wave ansatz into the equation of motion (2.56)

− k2 + ω2 + Vi,f = 0, =⇒ k = ±√ω2 + Vi,f , (2.59)

in the two asymptotic regions. The signs of the momenta are determined by therequirement that the reflected and transmitted modes have positive and negative groupvelocities respectively. For example

0 <dω

dkr=krω, =⇒ kr =

√ω2 + Vi > 0 (2.60)

Similarly we have kt, kin < 0 determined by the negativity of their group velocities.

The conservation of W [fs] implies the relation between the reflection and transmis-sion coefficients R and T respectively

1− |R|2 = −ktkr|T |2. (2.61)

From relation (2.61) and the fact that kt < 0 andkr > 0 we can conclude that in theordinary scattering process we must have |R| < 1.

The important ingredients are the scattering solution and the dispersion relation:we choose which modes to include consistently in the scattering solution by their groupvelocity dω/dk in the asymptotic regions, the mode with the coefficient R must be rightmoving at +∞ while the the mode with the coefficient T must be left moving at −∞.

Superradiant scattering must therefore have an unusual kind of potential V in orderto avoid the arguments here leading to |R| < 1.

Spacetime superradiance – Equatorial Kerr metric

As we saw in the introduction, the geometry of the equatorial sections of a rotating Kerrblack hole are described in Boyer-Lindquist coordinates (in units where G = c = 1) bythe metric

ds2 = dt2+2M

r(dt− adφ)2 − dr2

1− 2Mr

+ a2

r2

−(r2 + a2

)dφ2, (2.62)

where a is the rotation parameter and m is the mass which completely characterise thesolution.

The wave equation for a massless scalar field f in the full 3+1 dimensional Boyer-Lindquist coordinates including the polar angular coordinate θ is complicated and we donot write it down here. However a coordinate change [38] of the Kerr metric results ina separated equation of motion for radial modes from the polar angular part justifying


the dimensional reduction to the equatorial θ = π/2 sections. That is, expanding thefield f in eigenmodes fω,m = R(r)Θ(θ)e−iωteimφ of constant angular momentum mand frequency ω, the resulting equation of motion for the function R separates fromthe equation of motion of Θ. Furthermore, under the redefinition of R and change ofvariables to the radial coordinate r∗ as

u(r) =√r2 + a2R(r), (2.63)

dr∗

dr=

r2 + a2

r2 − 2Mr + a2, (2.64)

the equation of motion for u(r∗) is written in the simple form

d2u

dr∗2+ Vω,m(r∗)u = 0, (2.65)

where Vω,m is a function whose precise form is rather complex

Vω,m = −4Mramω − a2m2 + ∆ (λm,l + ω2) a2

(r2 + a2)2(2.66)

− ∆ (3r2 − rMr + a2)

(r2 + a2)3+

3∆2r2

(r2 + a2)4+ ω2. (2.67)

Here ∆ = r2 +2Mr+a2 and the λm,l are eigenvalues for the Θ angular part of the waveequation. The role of the variable r∗ is similar to a tortoise coordinate, covering onlythe region r > r+

h , where r+h = M +

√M2 − a2 is the outer horizon radius (see [39] for

a concise summary of the Kerr solution) where the component grr of the metric (2.62)becomes singular. This region r ∈ (r+

h ,∞) is mapped in terms of r∗ onto the real liner∗ ∈ (−∞,+∞).

We note, however, that as r →∞ the function Vω,m tends to a constant Vω,m → ω2.The coordinate r∗ has a nice interpretation as a generalised ‘tortoise’ coordinate suchthat r∗ → −∞ as r approaches the event horizon. The other asymptotic is given by

Vω,m →(ω − am

2M(M +√M2 − a2)

)2

(2.68)

as r → r+h or equivalently r∗ → −∞. The quantity subtracting ω inside the bracket

can be expressed in terms of the angular velocity of the Kerr black hole ΩH = ΩH =−(gtφ/gφφ)|H as simply mΩH giving the simple form of the potential

Vω,m =

ω2, r∗ → +∞(ω −mΩH)2 , r∗ → −∞

(2.69)

The allowed wave-numbers and those chosen for the scattering solution can beconcisely understood using a dispersion diagram as shown in Fig.2.3. We note that


the scattering solution chooses only a subset of the set of all possible wave-numbersallowed by the dispersion relation

k2 =

ω2 r∗ → +∞(ω −mΩH)2 r∗ → −∞

. (2.70)

in such a way that the solution can be interpreted as a scattering process: A localisedwave packet with momentum centered around kin traveling from +∞ converting intoa reflected wavepacket with momentum centered around kr which travels back to +∞and a transmitted wavepacket with momentum centered around kt which travels on to−∞. The direction of motion is given by the group velocity dω/dk of the packet andthe momenta are determined by the dispersion relation. In the r∗ → −∞ region wehave

dω

dk=

k

ω −mΩH

, (2.71)

and the sign does not coincide with the sign of k when ω < mΩH : crucially the groupand phase velocities are of opposite sign in the r∗ → −∞ region when ω < mΩH .Explicitly solving the dispersion relation we have

kin = −ωkr = ω

kt = −(ω −mΩH). (2.72)

In this case, the Wronskian (2.57) is still conserved when acting on solutions to theequation of motion and acting on the scattering solution (2.58) with the above definedmomenta gives

1− |R|2 =(ω −mΩH)

ω|T |2, (2.73)

and hence

|R|2 = 1− (ω −mΩH)

ω|T |2. (2.74)

Since we choose positive frequencies ω > 0 there is a window of frequencies ω < ΩH

for which |R| > 1 is possible.

This is one of the historical routes to the discovery of superradiance [36, 37] butthe basic key ingredients can be distilled by the equation of motion (2.65) with thepotential like (2.69) and the requirement of a scattering solution of the type definedabove.

Electromagnetic potential formulation

An equation of the form (2.65) can also arise as the wave equation in flat spacetime fora charged scalar field in the presence of a electromagnetic vector potential Vµ which


Figure 2.3: The dispersion relation (2.70) shown in the two asymptotic regions, thegreen dash-dotted line at +∞ and the blue dashed line at −∞. We have shown thecase where ω < mΩH , specifically ω = 0.4 and mΩH = 0.7. Note that the intersectionsof the blue dashed line (the effective frequency at −∞) lie below the k axis, allowingfor a negative norm transmitted mode and consequently superradiant scattering backto +∞.

is non-zero only in the time direction Vµ = δ0µeV [30]. The potential modifies the

derivative operators ∂µ to Dµ = ∂µ − iVµ and consequently the wave equation to

0 = DµDµf

= ∂2xf − (∂t − ieV )2 f, (2.75)

leading tod2fωdx2

+ (ω − eV (x))2 fω = 0, (2.76)

when acting on fixed frequency modes.

If one choses the potential to have the correct form

eV (x) =

0 x→ +∞eV0 x→ −∞

, (2.77)


we are led directly, from the analysis above to the conclusion that

|R|2 = 1− ω − eV0

ω|T |2 > 1, (2.78)

and hence superradiance for ω < eV0.

Acoustic superradiance

Alternatively we can understand the the equation of motion

d2fωdx2

+ (ω − eV (x))2 fω = 0, (2.79)

as the equation of motion for a dimensionally reduced analogue model. For the 2+1dimensional fluid metric given in the previous chapter (1.20) with a flow vector fieldV = (0, V ) which is zero in the x direction but which is non-zero and a function ofonly x in the, say, z direction, the analogue metric is given by

g ∝

1− V 2(x) 0 V (x)0 −1 0

V (x) 0 −1

(2.80)

As long as V < 1 we can always make a change of frame to the frame in which V = 0and the metric is revealed as that of flat space. The determinant is 1 and the waveequation is [

(∂t + V ∂z)2 + ∂2

z + ∂2x

]f = 0, (2.81)

reducing for fixed frequency ω and fixed momenta p in the z direction to

d2fω,pdx2

+[(ω − pV )2 − p2

]fω,p = 0 (2.82)

If the flow is such that

V (x) =

0 x→ +∞V0 x→ −∞

, (2.83)

then we are in a situation very similar to that of the Kerr black hole once again butthis time with the additional ‘mass’ term p2.

According to the analysis of [30] superradiance still exists for such a wave equationbut only in the regime known as the Klein region, when,

|p| ≤ ω ≤ pV0 − |p|, (2.84)

if such a region exists. We see that the frequencies that are superradiated must not betoo small for the mass term to effect their propagation but must not be too large forthe y flow to be negligible.

Note that we cannot choose p = 0 since p also comes as a pre-factor on the velocityV inside the squared bracket which is ultimately responsible for the superradiance.


Generalised non-dispersive superradiant scattering

In the article in Reference [40] superradiant scattering is generalised to include also thecase when the potential contains an imaginary part

d2fωdx2

+ [V (x) + iΓ(x)] fω = 0 (2.85)

where V (x)→ ω2 as x→ +∞.

In this case, the ordinary Wronskian (2.57) is not conserved and its derivative withrespect to x can be related to the imaginary part of the potential

id

dxW [f ] = 2Γ|f |2 (2.86)

Assuming only the for of the scattering solution at large x

fs → eikinx +Reikrx, (2.87)

we can integrate the condition (2.86) up to an arbitrary point x0 (which is supposedto be analogous to an horizon) finding

|R|2 = 1− i

2ωW [f ]

∣∣∣∣x=x0

− 1

ω

∫ ∞x0

Γ(x)|f(x)|2 dx (2.88)

The authors conclude that superradiance is possible in such cases when the righthand side is greater than 1, implying certain conditions on the form of the imaginarypart of the potential and the evaluation of the Wronskian at the ‘horizon’ x0.

The quantum spontaneous emission

Superradiance can be understood alone as a classical scattering phenomena. However,if one considers a quantum field whose fixed frequency modes satisfy an equation ofmotion such as

d2fωdx2

+ (ω − eV (x))2 fω = 0, (2.89)

with the appropriate asymptotic behaviour of the potential, then one might ask whatsuperradiance implies for the stability of such systems. For example the zero-pointvacuum fluctuations could possibly be continually amplified in this manner leading toan instability of the vacuum and particle production.

This line of reasoning indeed plays out as expected [41, 42, 36], leading to the ideathat quantum fields are capable of extracting angular momentum from the spinningblack hole through amplification of the vacuum fluctuations.


2.2 Bose Einstein Condensation

The physics of Bose Einstein Condensation (BEC) is complex and was first describedby Bose and Einstein in 1924. Nevertheless it may be discussed using very basicphysics as we will show below. As with all good theories, by making use of carefullychosen approximations, one can get to the underlying physics of BEC relatively easily.The limits of these approximations mark the boundary of the validity of the analoguedescription, of which we will make use later, and it is for this reason that we shallspend some time here discussing them.

The main aim of this short discussion is to remove the veil of mystery from thephysics of BEC sufficiently to render transparent the description of elementary BECexcitations in terms of an analogue metric. All too often the important message froman analogue gravity computation is shrouded in the details of the background physics.We make no attempt here to use hyper-precise terminology and stick to intuitive, basicdescriptions as far as possible.

2.2.1 Non-interacting Bosons

At its root a BEC is nothing but a collection of Bosons in their lowest energy state fora given temperature. Bosons, of course, distribute themselves throughout the energylevels εi according to the Bose distribution

f(εn) =1

exp β(εn − µ)− 1, (2.90)

where µ, the chemical potential, is the increase in energy of a system of particles whenan additional particle is added and β = 1/T , is the inverse temperature. The energiesεi are kinematically determined by the properties of the Hamiltonian operator. Forexample, in the time independent hard potential well the εn are the eigenvalues withthe well known discrete spectrum(

− ~2

2m∇2 + V

)ψn = εnψn (2.91)

=⇒ εn =~2n2π2

2mL2. (2.92)

From the distribution (2.90) one can calculate the mean number of particles andmean energy

N =∑n

f(εn), E =∑n

εnf(εn). (2.93)

2.2. Bose Einstein Condensation 51

The generalisation to include an arbitrary spectrum we introduce the density ofstates

g(ε) =∑j

δ(ε− εj). (2.94)

For example, for bosons in a harmonic potential well in 3D the energy levels are

εijk = ~∑i

niωi, (2.95)

so that for small ε the density of states g(ε) is quite discrete and becomes more con-tinuous at higher ε and is approximated by the integral

g(ε) ≈ 1

~3ω30

∫ ∞0

dudvdw δ(ε− u− v − w) =ε2

2~3ω30

. (2.96)

For a general potential V the energy eigenvalues are difficult if not impossible toobtain explicitly so one usually resorts to asymptotic, scaling, or inequality-based datausually resulting in a thermodynamic (high occupation number) description. However,we will discuss a particular class of time dependent potentials later on (see Section3.2.1 in the following chapter) for which we can compute the eigenvalues exactly andwhich constitutes an analogue model for an expanding spacetime.

The key physics result that underpins the entire phenomenon of BEC is that thereis a phase transition in the occupation of the ground state associated with the lowestenergy eigenvalue ε0 at some point as one lowers the temperature of the bosons. Atthis phase transition the ground state attains a macroscopic occupation number andthe BEC is well approximated by a perfect fluid.

More precisely, it is known that the chemical potential µ is a complicated functionof temperature T = β−1 and particle number N although it is known that at high tem-perature µ is large and negative, reducing the Bose distribution (2.90) to the classicalBoltzman factor, while below a certain critical temperature Tc the chemical potentialis nothing but the ground state energy of a Boson ε0. From (2.90) we see that thisimplies a singular occupation of the ground state ψ0 when T < Tc. This singularityis resolved when interactions between the bosons are included which are neglected in(2.90).

2.2.2 Interactions change things slightly

Fundamentally we have a Hamiltonian operator H describing the N-body dynamics ofthe collections of bosons written as a function of creation and annihilation operators.This Hamiltonian does not diagonalise into N free particles but instead there is aninteraction term with two annihilation and two creation operators and hence there is


a scattering of the atoms from one another. These interactions will introduce severalnon-trivial new features which will turn out to have important interpretations in termsof curved spacetime kinematics.

First approximation – Dilute Gas approximation. This approximation allows us toneglect the full dynamics of scattering, keeping only the two body s-wave scatteringinteraction. This interaction is modeled by the ‘pseudo-potential’

V (r) = gδ(r) (2.97)

It is called a ‘pseudo’-potential since there really is no potential function for the par-ticles, they are free particles that interact ‘at a point’.

Then the Hamiltonian can finally be written down simply as

H =∑n

εna†nan +

g

2V

∑n

a†na†nananδn1+n2,n3+n4 (2.98)

Second approximation – Macroscopic occupation of ground state. This assumptionis essentially the assumption that the occupation number of the ground state is so highthat the state is an approximate eigenstate of the creation a†0 and annihilation operatora0. That is, at high occupation adding one particle or removing one makes ‘practically’no difference to the state. Hence weakly (that is, under expectation values), we replacethese operators by the eigenvalues

√N0. After this substitution the Hamiltonian is a

sum over excited state quantum numbers and takes the general form

H = A+∑n

Bna†a+N0A

∑n

(a†na†−n + ana−n) (2.99)

which contains particle non-conserving terms in the last sum, where A is a numbervalued constant. However, perhaps miraculously, it is possible to diagonalize thisHamiltonian with the non-perturbative ‘Bogoliubov’ transformation, mixing creationand annihilation operators in new linear combinations of them

an = unαn + v∗−nb†−n, a†n = u∗nb

†n + v−nα−n (2.100)

Then,

H = const +∑n6=0

Enb†nbn, (2.101)

with the famous Bogoliubov dispersion relation energy eigenvalues

En =√ε2n + 2εngN0 =

√gN0

mp2 +

p4

(2m)2. (2.102)


Here we have included the atomic Bosonic dispersion ε(p) = p2/(2m) in the last line.The coefficients of the Bogoliubov linear combinations (2.100) are not arbitrary, ofcourse, but are constrained to satisfy

v2n = u2

n − 1 =1

2

(εn + gN0

En− 1

), (2.103)

which uniquely determine them as

un, v−n = ±

√N2

0/2m+ gN0

2εn± 1

2. (2.104)

This is similar to what we described in 2.1.1 where the choice of vacuum state uniquelydetermines the mode functions. This constraint is exactly the condition on the modefunctions which cancel the off diagonal terms in the Hamiltonian. It turns out thatthese mode functions are in fact real and even functions of n so that, for example,un = u−n = u∗n as can be seen from (2.104).

Now it is interesting to note that the dispersion relation (2.102) exhibits relativisticbehaviour for low momenta

E(p) ≈ cp for small p, (2.105)

where the wave propagation speed c is given by c2 = gN0/m, but exhibits non-relativistic free particle like behaviour for high p

E(p) ≈ p2

2mfor high p. (2.106)

In this way, the quantum theory described by (2.101) describes elementary quantumexcitations on top of the ‘classical’ background of the macroscopically occupied groundstate. These elementary excitations, somewhat surprisingly, are not single particleexcited states, but instead are collective degrees of freedom which behave relativisticallyat low energies. It is these exitations whose dynamics are subject to a description interms of an analogue spacetime and metric as we will in the next sections.

2.2.3 A BEC as an analogue spacetime

In a later chapter 3 we will consider a weakly interacting BEC trapped in an attractiveharmonic potential and the computation of a correlation signal for the produced quanta.

Here in this sub-section we derive the analogue metric for low momentum BECperturbations which we will make use of later on and describe the approximationsnecessary for such a geometric description. We introduce here the fluid dynamicaldescription of a BEC and show how the basic perturbations of the canonical variables,


fluid flow and density, are described by a wave equation equivalent to that whichdescribes the propagation of a scalar field on a curved spacetime.

We will not discuss any of the manifold phenomena associated with dispersion;these are ‘advanced topics’ which we will have time to look at in separate sectionsand chapters. General references for this by now standard material can be found inReferences [43, 44, 45].

In the dilute gas approximation, one can describe a Bose gas through a secondquantised field Ψ satisfying

i~∂

∂tΨ =

(− ~2

2m∇2 + Vext(x) + g(a) Ψ†Ψ

)Ψ. (2.107)

Here m is the mass of the constituents, g introduced in (2.97), parametrizes the strengthof the interactions between pairs of bosons in the gas and a is the scattering length ofthe atoms in an s-wave approximation. The former can be re-expressed in terms of thescattering length as

g(a) =4πa~2

m. (2.108)

Being a non-relativistic field, Ψ satisfies the commutation[Ψ(x), Ψ†(x′)

]= δ(x,x′),

[Ψ(x), Ψ(x′)

]= 0. (2.109)

In a mean field approximation, the quantum field can be separated into a macro-scopic (classical) condensate (called the wave function of the condensate) and a fluc-

tuation: Ψ = ψ + ϕ, with 〈Ψ〉 = ψ. Collecting the leading order terms in ψ, equation(2.107) leads to the Gross–Pitaevskii (GP) equation of motion for the wave function ofthe condensate

i~∂ψ

∂t= − ~2

2m∇2ψ + Vextψ + g(a)|ψ|2ψ. (2.110)

Neglecting the terms proportional to the fluctuation ϕ amounts to neglecting back-reaction effects of the quantum fluctuations on the condensate.

Representing the wave function of the condensate as [46]

ψ(t,x) =√ρ(t,x) exp[−iθ(t,x)/~], (2.111)

where ρ = ψ∗ψ is the condensate density, and defining an irrotational ‘velocity field’ byv := ∇θ/m, the Gross–Pitaevskii equation can be rewritten as a continuity equationplus an Euler equation reminiscent of the fluid dynamical equations of motion:

∂

∂tρ+∇ · (ρv) = 0, (2.112)

m∂

∂tv +∇

(mv2

2+ Vext(t,x) + gρ− ~2

2m

∇2√ρ√ρ

)= 0. (2.113)


These equations are completely equivalent to those of an irrotational and inviscid fluidapart from the presence of the so-called quantum potential

Vquantum = −~2∇2√ρ/(2m√ρ), (2.114)

which has the dimensions of an energy. When relatively small gradients of the back-ground density ρ are involved, this term can be safely neglected and a hydrodynamicalapproximation (also known as the Thomas Fermi (TF) approximation) holds.

This constitutes the derivation of the perfect fluid description of the backgroundBEC dynamics on top of which are played out the dynamics of the quantum pertur-bations. Near the edges of the BEC, where the density is low, the quantum pressurebecomes non-negligible and the fluid description breaks down.

Inserting the ansatz Ψ = ψ+ ϕ into Eq. (2.107) and again neglecting back-reaction,one alternatively obtains the equation for the operator valued perturbation

i~∂

∂tϕ(t,x) =

(− ~2

2m∇2 + Vext(x) + 2gρ

)ϕ(t,x) + g ρ ϕ†(t,x). (2.115)

The elementary excitations of this dynamical system are described by variables ob-tained by a Bogoliubov transformation from the fundamental field. This can be alsoshortcut by adopting the so called quantum acoustic representation [45]

ϕ(t,x) = e−iθ/~(

12√ncρ1 − i

√ρ

~ θ1

), (2.116)

where ρ1, θ1 are real quantum fields which describe the excitations in density and phaserespectively. Inserting the acoustic representation ansatz above into Eq. (2.115) oneagain obtains a pair of hydrodynamic-like equations for the quantum excitations

∂tρ1 +1

m∇ ·(ρ1 ∇θ + ρ ∇θ1

)= 0, (2.117)

∂tθ1 +∇θ · ∇θ1

m+ g(a) ρ1 −

~2

2mD2ρ1 = 0, (2.118)

where D2 represents a second-order differential operator derived from the linearizationof the quantum potential [45]. As mentioned earlier, we will be concerned with theacoustic excitations which are wave-like. These excitations are of sufficiently low fre-quency as to render the quantum potential negligible, allowing for a bona-fide perfectfluid description and equations (2.117) and (2.118) are identical to the equations (1.16)with θ1 and ρ1 playing the roles of ρ1 and f1 in (1.16). Hence the derivation of theequivalence between this set of equations and a Klein Gordon equation for θ1 in acurved background follows immediately as in section 1.1.2. The metric in this case isgiven by

ds2 =

√ρ

gm

[−(c2

s − v2)dt2 − 2vidxi dt+ dx2

]. (2.119)


Here c2s(t,x) = gρ(t,x)/m is the squared local sound speed of the perturbations and

v(t,x) is the background flow velocity of the BEC introduced above [6, 43]. It isthen clear that in this long wavelength regime the quasi-particles satisfy a relativistic,phononic dispersion relation ω2 = c2

sk2 (with ω and k equal respectively to the energy

and wavenumber of the quasi-particle).

Alternatively one could adopt an eikonal (short wavelength) approximation in theequations (2.117) and (2.118), isolating also the high frequency part and show that thecomplete dispersion relation is indeed the well known Bogoliubov dispersion given byω2Bog = c2

sk2+~k4/(2m) as is done explicitly in ??. It is evident at this point that in this

context the gravitational analogy is valid only in the long wavelength approximation.More specifically, for wavelengths of the order of the healing length ξ = ~/√mgρ, whenthe quartic term becomes comparable to the quadratic one in ω2

Bog(k), a geometricaldescription is no longer available.

2.3 Theory of surface waves on fluids

In fluid dynamics, gravity waves are waves created at the interface between two fluidsas a result of a gravitational restoring force. The system considered in this section andthat which will be the subject os Chapter 6 is the interface between an incompressiblefluid with density ρ and another incompressible fluid with negligible density ρ ( ρ),specifically the air-water interface in an open channel flow.

The subject of open channel flows is classical, dating back to Airy in the 19thCentury [47]. The plethora of gravity wave phenomena is not entirely captured byany one single wave equation where various different approximations can be adoptedresulting in different partial differential equations for some restricted dynamics. Toname a few, Boussinesq approximation results in the Boussinesq-type equations fromwhich the Kortewegde Vries equation results in the approximation to unidirectionalpropagation. The Airy wave theory we will use here is a linear theory, essentiallymaking use of the first order perturbation theory of surface disturbances. Even in thesimplified context of Airy wave theory, the diversity of phenomena is extremely richand includes dispersive effects.

Linearised gravity waves have received considerable attention recently from theanalogue community being the subject of at least two analogue gravity experiments[48, 49, 23] and a series of theoretical works [50, 51, 52].

In general, the open channel flow problem consists of two steps (i) Solving for thebackground flow velocity profile and free surface and (ii) solving the equation of motionfor perturbations on top of this background.

Below we will demonstrate the meaning of ‘long wavelengths’ and show that theysatisfy a wave equation in an analogue curved spacetime while also discussing thenon-linear dispersion for shorter wavelengths.

2.3. Theory of surface waves on fluids 57

2.3.1 Background flow

We start our analysis again here with the basic equations of fluid dynamics as we wrotethem in (1.1.3), but now with the additional assumption of incompressibility

∇ · v = 0, (2.120)

and the Euler equation,

dv

dt=∂v

∂t+ (v · ∇)v = −∇p

ρ− gz, (2.121)

where g, v and P are, respectively, the gravitational acceleration, the velocity and thepressure of the fluid. Note that we are assuming that the gravitational force is the onlyexternal body force acting on the fluid.

In Refs. [50, 52] the authors relax the assumption of flat bottom and height as wellas that of incompressibility. In Ref. [53] the authors study gravity waves when theassumption of irrotationality of the background flow is relaxed. For simplicity here weassume incompressibility, irrotationality and flat bottom.

Let x‖ = (x, y) and z (or, equivalently, x‖ = (r, φ) and z) be the spatial coordinatesand assume the fluid tank to be flat such that the z-direction is perpendicular to thebottom of the tank (z = 0). Also, let h

(t,x‖

)be the height of the fluid at x||. Besides

the basic equations above, to determine the flow of water in an open channel, we needthree boundary conditions, namely:

1. The normal speed must vanish at the bottom of the tank, i.e. vz|z=0 = 0;

2. The rate of change in the height of the fluid must be equal to the vertical velocityof the fluid at the surface,

vz|z=h =dh

dt

∣∣∣∣z=h

=∂h

∂t+(v‖∣∣z=h· ∇‖

)h; (2.122)

3. The pressure must be continuous at the interface. Since ρ, the pressure of theair component, is negligible, the pressure at the surface of the fluid must vanish(observe that we are neglecting surface tension effects here),

P (z = h) = 0⇒ P = P1(z − h) +O(z − h)2. (2.123)

As usual (recall the first example in 1.1.3) the flow is assumed to be irrotational,so that the velocity of the fluid v can be written as the gradient of a scalar field ψ,

v = ∇ψ. (2.124)


The continuity equation, therefore, reduces to Laplace’s equation for the field,

∇ · v = ∇2ψ = ∇2‖ψ + ∂2

zψ = 0. (2.125)

Euler’s equation under the irrotational assumption can be written as Bernoulli’sequation,

∂ψ

∂t+

1

2∇ψ · ∇ψ = −P

ρ− gz. (2.126)

Expanding the equation above in powers of z − h, we obtain from the 0th-order term,

0 =∂ψ

∂t

∣∣∣∣z=h

+ v2‖∣∣z=h

+ v2z

∣∣z=h

+ gh, (2.127)

where we have used the boundary condition (3).

2.3.2 Wave propagation

When the free surface of an open channel flow is perturbed, gravity acts as a restoringforce resulting in oscillations around the background flow. The mathematical descrip-tion of these oscillations, called gravity waves, is given in terms of perturbations of thebackground quantities, ψ and h, in the following way.

Taylor expanding ψ around z = 0,

ψ(t,x‖, z

)=∞∑n=0

zn

n!ψn(t,x‖

), (2.128)

and substituting into Laplace’s equation (2.125) above yields the recursion relation forthe ψn

ψn+2 = −∇2‖ψn. (2.129)

The boundary condition (1) implies that ψ1 = 0 and, therefore, from the recursionrelation, we write the field ψ only in terms of ψ0 [54],

ψ(t,x‖, z

)=∞∑n=0

(−1)nz2n

(2n)!∇2n‖ ψ0

(t,x‖

). (2.130)

Substituting equation. (2.130) in the boundary condition for stationary surface heightequation. (2.122), we obtain, after some algebraic manipulation, an equation involvingonly h and ψ0,

0 =∂h

∂t+∇‖ ·

∫ h

0

v‖dz

=∂h

∂t+∇‖ ·

(∞∑n=0

(−1)nh2n+1

(2n+ 1)!∇2n+1‖ ψ0

((t,x‖

))(2.131)

2.3. Theory of surface waves on fluids 59

In terms of the variables h and ψ0 by using eq. (2.130) the Bernoulli equation (2.126)becomes,

∞∑n=0

(−1)nh2n

(2n)!

∂

∂t∇2n‖ ψ0 +

1

2

(∞∑n=0

(−1)nh2n

(2n)!∇2n+1‖ ψ0

)2

+1

2

(∂h

∂t+∇‖h ·

∞∑n=0

(−1)nh2n

(2n)!∇2n+1‖ ψ0

)2

= −gh. (2.132)

Hence we may describe surface perturbations by the functions h and ψ0 alone. Let

ψ0 → ψ0 + δψ0, h→ h+ δh, (2.133)

where δψ0 ψ0 and δh h. By substituting this ansatz into the equations whichdetermine the background flow, i.e. equations (2.131) and (2.132) assuming that ψ0 andh are exact solutions to the background equations, we obtain equations that describethe perturbations themselves,

∂δh

∂t+∇‖ ·

(v‖∣∣z=h

δh)

+∇‖ ·

(∞∑n=0

(−1)nh2n+1

(2n+ 1)!∇2n+1‖ δψ0

)= 0, (2.134)

and (∂

∂t+ v‖

∣∣z=h· ∇‖

)( ∞∑n=0

(−1)nh2n

(2n)!∇2n‖ δψ0

)= −gδh, (2.135)

where g is given by,

g = g +

(∂

∂t+ v‖

∣∣z=h· ∇‖

)2

h. (2.136)

Assuming that the height of the fluid is much smaller than the horizontal scales onwhich the flow profile changes significantly, we can keep only the lowest order term inthe sums appearing in the perturbation equations above. In other words, if the heighth is much smaller than the wavelength λ of the waves, we can neglect higher orderterms since ∇2

‖ = O (1/λ2) and h/λ 1, see ref. [50]. Therefore, eqs. (2.134) and

(2.135) reduce, respectively, to

∂δh

∂t+∇‖ ·

(v‖∣∣z=h

δh)

+∇‖ ·(h∇‖δψ0

)= 0, (2.137)

and (∂

∂t+ v‖

∣∣z=h· ∇‖

)δψ0 = −gδh. (2.138)

Eliminating δh we arrive at an equation of motion for δψ0 which compactly written asa Klein Gordon equation

1√−g

∂µ(√−ggµν∂νδψ) = 0, (2.139)


with an effective metric, dropping the |z=h subscripts, given by,

gµν =h

g

−(c2 − v2||)

... −v

· · · · · · · · · ·−v

... 113

(2.140)

Shallow water gravity waves, therefore, propagate in the analogue spacetime abovewith a velocity given by c2 = gh. Note that when the surface is exactly flat and timeindependent then g = g and we recover the standard result that the surface wavepropagation speed is

√gh.

2.4 Non-liner optics, superluminal pulses

This section deals with some elementary optics theory which will be necessary later onin Chapter 4. Most of the material from this section comes from the thesis [55].

When an electromagnetic wave travels in a dielectric medium, it excites the atomsin the medium, altering the way it propagates. Importantly, the reaction of the atomsto the wave is not instantaneous. This implies that the action of the atoms back on theelectromagnetic wave depends on how the wave was interacting with the atoms sometime prior. This is a non-local interaction and is responsible for some very interestingnon-linear effects in optics.

The Polarisation of a dielectric material by an time varying electric field can bemodeled as the linear but non-instantaneous in time form

P = ε0

∫ t

−∞χ(t− t′)E(t′)dt′ (2.141)

where ε0 is the dielectric constant and χ is a response function modeling the relaxationtime of the bound charges.

Using the Maxwell equations this implies equation of motion for the electric field

∇2Eω +ω2

c2(1 + χω)Eω = ∇2Eω +

ω2

c2nωEω = 0 (2.142)

the ω subscript refers to the time Fourier transform which acts simply on the convolu-tion (2.141) yielding a product. Here we have defined the refractive index nω = 1 +χω.

The assumption (2.141), however, is only accurate for low power electric fields andshould be replaces by a higher order effect depending on multiple powers of the electricfield

P = P (1) + P (2) + . . . (2.143)

2.5. Wave dispersion, tools and basics 61

where

P (n) = ε0

∫ t

dt1 . . .

∫ t

dtnχ(n)a1,...an

Ea1(t1) . . . Ean(tn) (2.144)

It can be shown [55] that the next non-zero contribution comes at order P (3) whichinduces a modification to the equation of motion (2.142).

These higher order effects play a role when the electric power of a signal is strong.We will be interested later on in this thesis in the case where an independent field modepropagates in a medium in which a second, high power signal is also propagating. Thenon-linear effect of the high power signal, through relation (2.143) is to locally modifythe refractive index experienced by the first mode. This effect, known as the non-linearKerr effect is the focus of a concentrated experimental effort, which will will discusslater in Chapter 4 seeking to directly observe an analogue of the Hawking effect.

2.5 Wave dispersion, tools and basics

In this section we discuss the role of modified dispersion relations in analogue gravity.The motivation is twofold. On the one hand real analogue systems are linearly disper-sive only in a limited frequency band, being subject to an analogue description only inthis restricted sense [56]. On the other hand, there is a growing voice in the gravitycommunity that local Lorentz invariance (LLI) might be broken in the UV for variousreasons [12, 13] . In fact one of the main motivations for considering modified disper-sion comes from black hole physics where many of the ‘UV problems’ arise (recall ourdiscussion in Sec. 1.2) such as the Transplanckian problem and infinite entanglemententropy.

2.5.1 Motivation for considering dispersive theories

When waves travel on a medium which possesses some microstructure at small scalestheir propagation senses this microstructure and this is reflected in a modified disper-sion relation. The prototypical example of this phenomenon is wave propagation on aregular 1D lattice of spacing a [57]; solutions to the finite difference wave equation areplane waves exp(−iωt+ ikx) with

ω2 =2

a2(1− cos(ka)) ' k2 − a2

12k4 +O

(k6). (2.145)

We see that when the wavelength 2π/k becomes comparable with the spacing a thewaves sense the lattice and slow down i.e: the group velocity dω/dk becomes lessthan 1 but when |ka|

√6 (the turning point of the quartic dispersion relation) the

waves propagate as in the continuum with linear dispersion. We also saw above that


BEC perturbations satisfy according to the non-linear Bogoliubov dispersion, indicatingthat the very short wavelength perturbations sense the atomic nature of the superfluidbackground. The situation in optics is slightly different in that the electromagneticfield does not depend on the presence of the background to be well defined as is thecase in BEC and the lattice example given above: at long wavelengths and ‘short’frequencies there is a chance for resonances in the material and the modes see a non-trivial background structure while at short wavelengths modes do not see the dielectricand propagate as in flat space. This is reflected in the Sellmeier dispersion relationgiven above in 2.4 which converges to linear dispersion for very high frequencies.

Of course it would be desirable to have at hand a micro-theory of spacetime andquantum fields but we are not so lucky yet to have a consistent theory including bothgravity and quantum theory at short length scales. One approach taken in the blackhole literature, motivated by the analogue gravity picture, is to remain ignorant tothe true microphysics while modeling it with a modified dispersion relation in the UV.One then can go about testing the predictions for black hole radiance of the modifiedtheory, comparing them to the linear dispersive Hawking result.

One major result from the analogue gravity community has been the demonstra-tion that the inclusion of a modified dispersion relation in the UV changes dramaticallythe derivation of Hawking’s result but leaves the conclusion unscathed [58, 59, 60, 61,62, 9, 63, 64]. This has been essential groundwork necessary before real experimentalattempts are made to directly observe an analogue of Hawking radiation in dispersivemedia which naturally possess such a cut-off scale where the linear dispersion breaksdown (the length scale where the picture of a fluid dissolves into that of propagat-ing atoms). On the other hand this conclusion about the robustness of the Hawkingspectrum settles an old question about whether black hole evaporation depends on theUV completion of GR, necessarily involving some quantum or discrete properties ofspacetime, showing that one can have black hole radiance in these modified spacetimescenarios. We will have more to say on this later on in the chapter on DCE in opticalsystems as well as chapter 5 where we explicit work out a dispersive model.

2.5.2 Dispersion diagrams

Consider the polynomial dispersion relations

ω2 = k2 ± k4

Λ2, (2.146)

where the ± refers to super- and subluminal dispersion and Λ is some momentum scalecontrolling the onset of the non-linearity. Physical propagating degrees of freedompossess a frequency and wavenumber pair (ω, k) which satisfies the dispersion relation.

One can visualise the allowed wave-numbers and frequencies by drawing the dis-persion diagram, shown in Figure 2.4. We note that the subluminal dispersion relation


(a) (b)

Figure 2.4: A superluminal 2.4(a) and subluminal 2.4(b) dispersion relation with thephysical degrees of freedom marked with black dots. In both plots the blue dashed linerepresents a line of constant frequency ω.

admits 4 physical degrees of freedom as long as the frequency ω is small enough com-pared with Λ.

When one is dealing with a non-linear dispersion relation there arises a distinctionbetween group and phase velocity defined respectively as

group velocity =dω

dk, (2.147)

phase velocity =ω

k. (2.148)

The group velocity can be read from the dispersion plot as the gradient of the redcurves at the intersections with the blue dashed line. We see that the group velocitycan point in the opposite direction to the phase velocity. It is the group velocity whichis the carrier of information in a wave form and the quantity which satisfies normalcausality requirements.

2.5.3 Moving media

The existence of a discrete substrate implies that the frame of reference which is at restwith respect to the substrate play a special role; it is the one in which the modified


(a) (b)

Figure 2.5: A superluminal 2.5(a) and subluminal 2.5(b) dispersion relation when themedium is moving with respect to an observer. In both plots the blue dashed linerepresents a line of constant observer frequency ω.

dispersion is defined. Transforming to a moving frame, moving at speed V with respectto the substrate introduces a doppler factor in the dispersion relation as

(ω − kV )2 = Ω2 = k2 ± k4

Λ2, (2.149)

which modifies the dispersion diagram as shown in the two frames: 2.5 shows thedispersion relation as seen in the frame of the moving observer while in 2.6 we showthe same plots from the frame of the substrate.

In the figures 2.5 it is possible to obtain many data associated with the propagatingdegrees of freedom. The intersections, as discussed, correspond to physical degrees offreedom. The gradient of the dispersion curve at this point provides us with the groupvelocity in the frame of the medium. The difference in the gradients of the dispersioncurve and the straight line given by dΩ/dk − (−V ) give us the group velocity in themoving frame since

dΩ = dω − V dk ⇒ dΩ

dk=dω

dk− V. (2.150)

The group velocities in the frame of the medium are given by the sums


(a) (b)

Figure 2.6: A superluminal 2.6(a) and subluminal 2.6(b) dispersion relation when themedium is moving with non-zero speed V with respect to an observer . In both plotsthe blue dashed line represents a line of constant observer frequency ω.

2.5.4 Hawking ‘scattering’ with modified dispersion

Classically, nothing can come out of a black hole — they are truly black. As wesaw, with the introduction of quantum effects black holes radiate at a temperatureproportional to their surface gravity. Quantum modes are observed at I + whosehistory places them arbitrarily close to the horizon at early times but not inside3.

However, when one tries to generalise the derivation of Hawking radiation to thecase of non-linear dispersion, one discovers the fact that, due to a momentum dependentgroup and phase velocity for dispersive waves the ‘absolute blocking’ property of thenon-dispersive horizon no longer exists.

Let us assume that the dispersion is modified in the frames freely falling towardsthe black hole. These frames move with velocity V (x) at different distances x from theblack hole, defining a position dependent doppler shift as in (2.149). Then, assuming aWKB approximation where the velocity changes are mild over lengths scales large withrespect to the wavelength of the mode, we can define the position dependent dispersion

3It should be mentioned that there exists a consistent derivation of the Hawking effect from thequantum tunneling of modes from behind the horizon. See [65] and references therein for a descriptionof that interpretation of Hawking radiation.


Figure 2.7: The propagation, backwards in time of a mode escaping a region of non-zeroand negative flow. The black dot indicating the wave number and effective frequency inthe frame of the medium slides up the curve approaching the point where the blue lineis tangent to the dispersion curve and the group velocity with respect to the mediumis zero. This is where the mode conversion takes place

relation, for example for subluminal dispersion:

ω − kV (x) = k2 − k4

Λ2. (2.151)

A mode which is is propagating away from the black hole at large distances, withpositive group velocity can be shown to have originated, if one traces the history backin time, as an incoming mode at I − before approaching the hole and being ‘converted’into an outgoing mode by the spacetime curvature! This is illustrated in Figure 2.7where as we track back in time, the group velocity in the frame of the hole goes tozero since the straight line becomes tangent to the red curve at some point. Thispoint is unstable [60, 61, 62] and in fact, modes turn around at this point, propagatingback away from the hole (backwards in time). Hence we see that the outgoing modeoriginated far from the black hole, propagated as an in-coming mode towards the holebefore being converted to an outgoing mode at late times.

Therefore for dispersive fields, the possibility of an entirely classical analogue of the


Hawking process is provided, corresponding to the scattering of a classical incomingwave through the mode conversion process into an out going mode.

For superluminal dispersion, the mode can come from behind the horizon and ar-bitrarily close to the singularity prompting the study of analogue models with a flowrate which tapers off to a constant value inside the horizon, extending the interior highflow region to arbitrarily large negative positions and avoiding the singularity presentin spacetime black holes.

Chapter 3

Particle Creation and Correlationsin Expanding Bose EinsteinCondensates

In this chapter we will be looking at one of the most studied analogue systems, Bose-Einstein Condensates (BEC) and an analogue of cosmological particle creation in them.Although BEC have been extensively studied by the analogues community, the noveltyof the material in this chapter is in the use of quantum correlations as an observable,and in particular the quantum correlations as are created by analogue cosmologicalparticle production.

We consider both a 3+1 isotropically expanding BEC as well as the experimentallyrelevant case of an elongated, effectively 1+1 dimensional, expanding condensate. Inthis case we include the effects of inhomogeneities in the condensate, a feature rarelyincluded in the analogue gravity literature.

In both cases we link the BEC expansion to a simple model for an expandingspacetime and then study the correlation structure numerically and analytically (insuitable approximations). We also discuss the expected strength of such correlationpatterns and experimentally feasible BEC systems in which these effects might bedetected in the near future.

3.1 Introduction

As described in sec 1.1, the propagation of linearized acoustic perturbations on an invis-cid and irrotational fluid can be shown to be described by the same equation of motionas that which describes the propagation of a scalar field on a curved spacetime. Ofcourse such ideal systems are not readily available in nature but it is possible to produce

70Chapter 3. Particle Creation and Correlations in Expanding Bose

Einstein Condensates

systems which, to a very good degree of approximation, do actually behave as perfectfluids. In recent years a special role in this sense has been played by Bose-Einsteincondensates, the basic observation being in this case that phase perturbations of thewave function describing a weakly interacting BEC are, in a certain limit, describedby a massless quantum scalar field propagating on a non-trivially curved backgroundLorentzian space-time [43, 44, 45]. This observation stems from the demonstration,given below, that the macroscopically occupied ground state of a BEC can be thoughtof as a perfect fluid under some appropriate approximations.

A BEC can therefore be used to simulate special gravitational backgrounds and theassociated particle production such as Hawking radiation of black holes [66, 67, 68, 69],or cosmological particle production. For example, releasing the trapping potential ofa BEC appropriately [70, 71, 72, 73, 74, 75, 76] or modulating external parametersof the experimental setting [77, 78, 79, 80, 81], it is possible to induce the necessarydynamics which give rise to interesting background geometries. Due to the non-trivialtime evolution, the system undergoes a non-equilibrium phase to which are associatedquantum emissions whose spectrum strictly depends on the parameters modulatingthe evolution. This opens up the possibility of experimentally accessing new kindsof phenomena directly involving the quantum nature of cold atom systems as wellas providing additional insights on the general mechanisms of such emissions in thegravitational case.

We highlight at this point the dual nature of the analogy: on one hand, the analogyprovides us with useful mathematical tools with which to describe interesting condensedmatter phenomena while on the other, the analogy provides us with models upon whichthe dynamics of some semiclassical gravity phenomena are played out.

As the title suggests, the central focus of this chapter will be on dynamical particleproduction in an expanding BEC. This emission is of the dynamical Casimir type,arising from a time dependence in some parameters in the equation of motion. It couldbe said that such an emission is a ‘regular’ QFT emission: the predicted spectra andphysics mechanisms involved, generally speaking, occur at the same energy scale. Inthis way a modification of the high energy part of the physics of a low energy processwill not effect the obtained results1. In contrast, the Hawking emission in its regularguise could be said to be an ‘exceptional’ emission, seemingly depending on arbitrarilyhigh frequencies in the prediction of a low frequency emission: although the spectrumof Hawking quanta is peaked, under ordinary circumstances, at sensible frequencieswhile decaying to zero emission in the UV, the black hole horizon2 and the exponential

1An notable counterexample to this would be if the time dependence arose from an expansion of thephysical background, either spacetime or condensate itself. In those cases a produced particle mighthave a wavelength which, going back in time through the contraction, have been ‘Trans-Planckian’at early times. However, the idea that the particle was ‘produced’ at a certain moment in the pastwhen its wavelength should have been short is problematic, with particles being well defined only inthe static regions after the expansion has ceased.

2Note that strictly speaking, a trapping horizon is not a necessary ingredient for a Hawking emis-

3.1. Introduction 71

blue-shifting to the past it entails ensures that each Hawking quanta observed at I +

spent a part of its lifetime as an exponentially large frequency. The lifetime of the blackhole (or apparent horizon) determines the magnitude of exponential blue-shifting of theHawking quanta. In this way a black hole in the ordinary sense (with linear dispersion)acts as an arbitrarily strong microscope, sensitive to all scales in the background3. Now,the cosmological emission being, morally speaking, insensitive to the microphysics ofthe background geometry (be it spacetime or an analogue thereof), we will not beconcerned in this chapter with the modifications induced by a non-linear dispersionrelation, preferring to restrict our attention to the linear dispersive low frequency partof the Bogoliubov spectrum, keeping track of the UV scale of the modifications andchecking that the results lie within the linear regime (see (2.105)). As mentionedabove, at the end of this chapter we relax this simplification and briefly show how thespectrum really is unaffected by a high frequency modification to the dispersion for‘slow’ processes.

The issue of how to practically observe these quantum emissions and verify theirorigin is a crucial point in the condensed matter literature. From the perspective ofanalogue models a direct observation of the created particles seems quite difficult inthe light of the fact that often they possess a quasi-thermal and spectrum, somehowobservable on top of a background thermal noise. There has been recently a large inter-est in the calculation and measurement, instead, of correlation patterns in expandingcondensates to access these quantum effects (for example see [82, 83, 84]). In the article[85] the author proposes studying the density correlations in a toroidally shaped BECwith time varying interaction strength as a method to uncover some micro-physics, inparticular the character of the quantum squeezing of phononic modes as the interactionstrength is varied. In [67] it has been proposed that evidence for analogue Hawkingemission in a sonic black hole manifested in a BEC might be solicited by measuringthe two point correlation structure associated with the quantized density or phase per-turbation fields. The idea is that the correlation pattern should display a peculiarsignature of the type of emission expected from black holes. In [68] a numerical studyof the full dispersive dynamics confirmed the robustness of the results of [67] outsideof the linear dispersive regime while in [66, 69] a more general method was presentedfor studying the correlations due to analogue Hawking radiation in BEC. Followingthe same line of reasoning, we apply this idea in this chapter to analogue cosmologicalemission and attempt to isolate a cosmological signal in the correlation pattern in anexpanding BEC.

In particular this chapter is mainly concerned with the experimentally accessiblecorrelations between the phase and density perturbations in a BEC which are bornduring the expansion of the underlying gas both in the widely studied effectively 1+1dimensional “cigar” shaped geometry and in a homogeneous expanding BEC. Such

sion, see [10, 35]3See chapter 4 for further discussion on this point.



BEC systems are of theoretical interest in their own right and correlation measurementsare experimentally accessible with the current technology [82, 83, 84]. Therefore it ispossible that these systems will provide in the near future an arena within which toexperimentally address interesting and new non-equilibrium quantum effects while atthe same time shed light on the physics of quantum fields in expanding spacetime.

In the next section we collect and contextualize some results relevant for the study ofcorrelations in an expanding BEC, reviewing how the analogy is applied to the dynamicsof the system. In the third section we compute the correlation structure for a masslessscalar field in a particular expanding background spacetime in 3+1D and show how tointerpret such a result in terms of both an isotropically expanding 3+1D condensateand a BEC with a time dependent scattering length. We shall develop general formulasfor probing the effect of particle production on correlations as embodied in the equal-time Wightman functions. We shall derive results which are completely generic toparticle production in any spatially translation invariant system, which can be appliedto standard cosmologies as much as to analogue spacetimes.

In the fourth section which represents the main result of this chapter using bothanalytic and numerical methods we discuss the structure of correlations in an anisotrop-ically expanding, effectively 1 + 1-dimensional, BEC. In particular we shall derive thegeneral, finite size, 1 + 1-dimensional, equal time Wightman function relevant for the“cigar” shaped expanding BEC presently realized in experiments. We conclude withsome proposals for improving the observability in practice of the correlation signal andsome comments on how such emissions are related to Hawking radiation.

3.2 The ingredients for cosmology in the lab

In this section we give a more detailed look at the analogue description of a BEC, andin particular that which pertains to an expanding condensate relevant for modeling,through the analogy, phenomena in an expanding universe.

We refer the reader to the introductory section 2.2 for the basic theory as well asthe appendix D.3 for additional details and a glossary of terms and concepts relevantto BEC physics.

3.2.1 Scaling solution

Condensates undergoing expansion or contraction in D spatial dimensions as a resultof a time varying confining harmonic potential Vext admit a dynamic generalization tothe static Thomas Fermi approximation for the Gross Pitaevskii equation (see equation(2.110) in chapter 2) known as the scaling solution [86, 87]. In this case the external

3.2. The ingredients for cosmology in the lab 73

potential is of harmonic type with time varying trapping frequencies

Vext(t,x) =1

2

D∑i

mω2i (t)x

2i . (3.1)

It can be shown that with the density undergoes a simple scaling transformation ineach spatial dimension xi parametrized by the so-called scaling functions λi(t) as

ρ(t,x) =ρs(xi/λi(t))∏D

i λi(t), (3.2)

where ρs is the static TF ground state solution4

ρs =

ρ0 (1−

∑i ωix

2i ) , for

∑i ωix

2i < 1

0, otherwise, (3.3)

where ωi = mω2i /2µ (µ being the chemical potential) and ρ0 is the maximum density

at the centre of the BEC. The scaling functions are determined by the dynamics ofthe varying harmonic frequencies ωi(t), controlled by the experimenter, as we will seebelow.

In the static TF approximation the density and chemical potential can be expressedalgebraically in terms of the physical parameters as

ρ0 =µ

g(a)=

1

8π

(15m3N

∏i ωi

a3/2~3

)2/5

, (3.4)

and

µ =1

2~$(

15Na

√m$

~

)2/5

. (3.5)

respectively, where N is the number of condensed bosons. Here we have introduced thenotation $ = (

∏i ωi)

1/d in d spatial dimensions, the geometric mean of the trappingfrequencies. For the phase, the scaling transformation takes the form

θ(t,x) =1

2m∑i

x2i

λiλi, (3.6)

where the dot ˙ represents the lab time derivative ∂t.

This scaling induces a background flow velocity profile in the condensate given by

v(t,x) =λ(t)

λ(t)x. (3.7)

4Note that the TF approximation is not a good approximation near the boundary of the condensatewhere the density drops to zero. The condensate boundary, here given by the parabola sharplydropping to zero, in the full solution will be smooth.



With this velocity and the evolution of the density given in (3.2) one can start studyingthe analogue geometry associated with such a scaling background solution using themetric (2.119), reproduced here for reference

ds2 =

√ρ

gm

[−(c2

s − v2)dt2 − 2vidxi dt+ dx2

]. (3.8)

Written in terms of co-moving coordinates xi = xi/λi(t) the metric (3.8) for thescaling solution BEC in D spatial dimensions reads

ds2 =

(ρ(t,x)

gm

) 1D−1 (

−c2s(t,x) dt2 + λ2

i (t) dxi2), (3.9)

from which a number of standard representations for a FRW background are easilyobtained. The cosmological assumption of homogeneity is satisfied for a BEC in thecentral region where the density is almost constant. Working in such a central region,ρ and cs become functions only of t and, assuming an isotropic expansion λi(t) = λ(t)for all i, the metric can be written in standard FRW form as

ds2 = −dτ 2 + a2(τ)dx2, τ(t) =

∫ t

dt′ρ1/2(D−1)(t′)c(t′)

(gm)1/2(D−1), (3.10)

with the scale factor defined by

a2(τ) =

(ρ(τ)

gm

) 1D−1

λ2(τ) ∝ λ2−D/(D−1) =

λ(τ)0 in 2+1 dimensions

λ(τ)1/2 in 3+1 dimensions,(3.11)

where we have used the scaling solution ρ(η) = ρ0/λD(η) for the density.

What one can see by (3.11) is that in 2+1 dimensions, the effective geometry is ascaled Minkowski space and all particle creation effects are hidden in the definition ofthe time coordinate τ . Of course in 1+1 dimensions, the effective geometry does notexist (as shown in Sec. 1.1.3), so that 3+1 dimensions is the first dimension when weobtain an interesting geometry.

Alternatively, still within the assumptions of homogeneity and isotropy, we definea conformal time coordinate

η(t) =

∫ t

dt′c(t′)

λ(t′), (3.12)

with respect to which the metric is written

ds2 =

(ρ(η)

gm

) 1D−1

λ2(dη2 − dx2

). (3.13)

Note that we require a homogeneous sound speed c for this definition.


On the other hand, if the expansion is explicitly anisotropic we may define thefunction

τ(t) =

∫ t

dt′(ρ(t′)

gm

) 12(D−1)

c(t′), (3.14)

also requiring a homogeneous sound speed but with respect to which the metric takeson the anisotropic Bianchi I or Kasner form

ds2 = dτ 2 −(ρ

gm

) 1D−1

λ2 · dx2 =: dτ 2 −∑i

a2i (τ)(dxi)2, (3.15)

where we have defined the independent FRW scale factors ai.

Note that here, with the analogue picture available, we have two notions of propercoordinates in the game. On the one hand, there are the lab coordinates, the true propercoordinates given by xl = λ(t)xc, and the analogue geometric ‘proper’ coordinatesxip := ai(τ)xi.

3.2.2 Dictionary of spacetimes and trapping frequencies

As stated above, for time dependent harmonic traps the scaling functions λi(t) are notfree parameters but are constrained to be solutions to the auxiliary equations of motion

λi(t) + ω2i (t)λi(t) =

ω2i (0)

λi(t)∏

j λj(t), (3.16)

with boundary conditions λi(0) = 1 and λi(0) = 0, determined by a self consistencycondition on the dynamics of the expanding condensate [86].

Equation (3.16) represents a kind of dictionary mapping scaling functions to trap-ping potentials; in practice, as an analogue-gravity-ist, one has in mind some particu-larly interesting spacetime one wishes to simulate by an expanding BEC. The scalingparameters λi(t) determine the time dependence of the analogue spacetime metric com-pletely, albeit in a variety of ways depending on things such as the dimension of the BECand analogue spacetime, so that in order to get a BEC to behave like your favouritespacetime one needs to control λi which are in turn are completely determined byωi(t) through (3.16). In this way, one can custom-make some time dependent analoguegeometry though experimental control of the trapping frequencies ωi(t).

This dictionary is one-to-one and not always intuitive, the inverse transformationbeing

ωi(t) =

√ω2i (0)

λ2i (t)

∏j λj(t)

− λi(t)

λi(t). (3.17)



An example of a counterintuitive translation is as follows. Intuitively, the most naturaldevelopment for the scaling functions is that of complete and sudden release of thetrapping potential. The characteristic expansion rate in that case λ is given by theinitial frequency ω(0). If we ask ‘What is the trapping potential which gives a smooth(e.g.. tanh(·)) expansion for the λi?’ the answer depends on the steepness of the tanh:to force a condensate to expand faster than the free expansion rate requires a repulsivepotential while slower expansions are possible with a nice slowly decaying potential.Indeed if the term inside the square root in (3.17) becomes negative, we understandthe associated λ(t) to be unphysical as it would require a complex trapping frequency.

The example of a forced expansion will be pertinent later on for analytical consid-erations but at this stage we focus on the sudden release scenario since this is closestto what is being performed in experiments.

3.2.3 Cigar shaped expanding and suddenly released BEC

Of particular experimental interest is the ‘cigar’ shaped condensate possessing oneelongated dimension (‘||’) and two tightly trapped orthogonal dimensions (‘⊥’). Inwhat follows, unless otherwise stated, we work with the cigar background geometry.Since our main focus later on will be on such elongated condensates and in particularon experimentally realistic condensates, we list here the following realistic values forthe physical parameters describing the cigar BEC

ω|| = 102Hz, ω⊥ = 103Hz (3.18)

m = 1.57× 10−25 kg (Rubidium 87) (3.19)

a = 42 Bohr = 2.221× 10−9 m (3.20)

N = 105. (3.21)

These are respectively the trapping frequencies, atomic mass, scattering length andnumber of condensed atoms. Assuming these numbers, the following derived quantitiesare obtained

L ≈ 5.45× 10−5m (3.22)

µ ≈ 5.8× 10−31 kgm2s−2 ≈ 5500~ s−1 (3.23)

ρ0 ≈ 2.9× 1020 m−3 (3.24)

cs,0 ≈ 2 mms−1 (3.25)

corresponding to the transverse condensate size (where, in the TF approximation thedensity drops to zero), the (static) chemical potential, the maximum condensate densityand the maximum initial sound speed respectively.

Starting from such a cigar shaped initial geometry, let us consider the case whenthe ‘||’ potential is suddenly switched off while keeping the ‘⊥’ potentials constant. A


Figure 3.1: The solution of the scaling problem for an anisotropic expansion of aninitially cigar shaped configuration for both the radial/orthogonal (blue-dashed line)and axial/parallel (red-solid line) directions to the axis of release. Note that the scalingfunction of the direction in which the trapping frequency remains fixed (blue, dashed)in fact decreases during the expansion. The parameters used in this solution are ω|| =100Hz, ω⊥ = 103Hz and a sudden release of the || direction only.

similar problem has been studied in [83] and observed in [82] where the authors insteadconsider a BEC freely expanding in all the three spatial dimensions after completelyreleasing the trapping potential of an initially cigar shaped condensate. In these anal-yses, due to the expansion in all three dimensions, the density drops very quickly andthe gas becomes collision-less signaling the failure of the hydrodynamical descriptionshortly after the release of the trap. In the case we consider here instead, the perpen-dicular trapping is kept tight and the density drops more slowly and the condensateremains longer in a collisional regime where the spacetime analogy pertains.

In Fig. 3.1 we show the numerical solution to (3.16) for cigar-like initial geometry



after the potential in the ‘||’ direction is suddenly switched off while keeping the ‘⊥’potentials constant. Notice that the tightly constrained ‘⊥’ dimension in fact pullseven tighter during the expansion in the ‘||’ dimension as indicated by the decay ofthe scaling functions λ⊥. In this way one can understand how the volume of thecigar geometry, scaling as (λ||λ

2⊥)−1, decays much more slowly than in an isotropically

expanding BEC which scales as λ−3|| .

3.2.4 Validity of the hydrodynamic approximation during ex-pansion

As already mentioned above, it is also important to keep track of the hydrodynamicalapproximation throughout the expansion. Since the quantum pressure scales differentlywith time than the density, it is of interest to understand whether the hydrodynamicapproximation improves or degrades as the expansion proceeds and, if it loses accuracy,when the approximation breaks down. In Fig. 3.2 we display the quantum pressureVquantum and interaction Vinteraction (= gρ) contributions to GP before the release ofthe axial trapping potential. In Fig. 3.3 we display the ratio Vquantum/Vinteraction as afunction of co-moving axial position z = zλ|| for four different times after the expansion.

Note that although there is always a region near the edge of the condensate wherethe ratio is greater then 1, as time progresses the proportion of the condensate whichremains within the hydrodynamical regime gets smaller until approximately t = 0.25swhen the quantum pressure becomes important globally in the condensate. The failureof the hydrodynamic approximation near the boundary of the condensate is expectedin general since the TF approximation is known to break down on scales on the orderof a healing length from the boundary where the density becomes exponentially small.Indeed the true exact solution, being a smooth exponential decay to zero, differs fromthe TF approximate solution, given in (3.3), near the edge which sharply goes to zerothere. Our analysis in the following will be concentrated on the more central part ofthe condensate where the TF approximation holds.

3.3 Quantum excitations and correlations during

expansion

We now intend to consider particle creation associated with the expansion of the con-densate from a sudden release in the cigar configuration and its signature in the cor-relation structure. Before doing so however, one might wonder if such expansion issufficiently rapid to lead to any relevant particle production at all. Also, given that weshall work in the analogue gravity framework, one would also like to be sure that theostensible excitations can be meaningfully described as phonons rather than having

3.3. Quantum excitations and correlations during expansion 79

Figure 3.2: The quantum pressure and interaction energy expressed in Joules as afunction of axial distance in meters. Note that the quantum pressure is certainly non-negligible near the boundary of the TF approximate ground state density wave-function(at z ' 2.7× 10−5m).

to deal with the complicated issues associated with non-linear dispersion. We shalldiscuss these issues first and then introduce the correlation function that will be thesubject of our studies in subsequent sections.

3.3.1 Timescales

Let us now analyze and compare the relevant inverse time scales involved in the prob-lem of an expanding condensate to make an order of magnitude estimate of whetherexcitation effects can be expected to be observable.

Firstly there is an intrinsic infrared cutoff given by the size of the condensate. Thislength scale translates into a frequency ωs using the (central maximum) sound speedωs = cs/(Lλ(t)).

Secondly there exists an intrinsic ultraviolet cutoff associated with the healinglength, the healing frequency ωh = 2πcs/ξ where ξ = ~/(mcs) is the healing length. The



Figure 3.3: The ratio R = Vquantum/Vinteraction as a function of co-moving axial positionz = zλ|| at various times t after release of the trapping potential. Note that the axialhalf-length of the condensate at time t = 0 (which of course is the co-moving lengthfor all times) is Lz ' 2.7× 10−5m.

healing length, as already mentioned, is the length scale at which the quantum poten-tial, neglected in the hydrodynamical approximation, becomes intrinsically comparablewith the interaction energy (which is proportional to the density of the condensate andhence becomes less relevant at lower densities). This is also the scale at which the Bo-goliubov dispersion relation enters in the quadratic regime and hence the descriptionin terms of non-interacting phonons breaks down. The healing frequency constraintis a constraint on the speed of expansion before the quantum potential begins to besignificant for the dynamical process.

Thirdly there is the characteristic frequency associated with the expansion itself,ωe ≈ λ/λ, which fixes the frequency of the most abundantly created excitations in anexpanding condensate. This is generic to any particle creation effect by a time-varyingexternal field (see e.g. [77] and Section 2.1.1).

It is of interest to compare these three frequencies in order to test the validity ofour approximations and to estimate if the expected excitations are created within thenatural ultraviolet and infrared cutoffs. In Fig. 3.4 we compare these three inverse


Figure 3.4: A comparison of the relative magnitudes of ωe and ωs (red-solid and blue-dashed respectively) and with the healing frequency (inset plot) in s−1 as a functionsof time (in s). Note that, for the majority of the duration of the expansion, ωs ismuch smaller than the expansion frequency ωe which is much smaller than the healingfrequency. Hence the bulk of the particle production spectrum lies within the lowfrequency cutoff provided by the total size of the condensate and the high frequencycutoff provided by the healing length.

time scales using the accurate parameters listed above for the cigar shaped condensatesuddenly released.

We see that indeed there exists a regime in which the typical excitation frequencylies entirely within the natural ultraviolet and infrared cutoff frequencies. Quantita-tively this regime begins after about 0.005s. Let us also note the vastly higher healingfrequency representing the UV cutoff for a phononic treatment. This is shown in thesmaller subfigure in Fig. 3.4 for comparison.

Therefore we conclude that there is a viable regime within which the excitationshave a phononic (linear) dispersion relation and wavelengths much shorter than thesize of the BEC.



3.3.2 Measuring quantum correlations - Wightman functions

Now that we have attained that the expanding BEC system is of the right size andcharacter to dynamically produce particles which propagate on the BEC and withlinear dispersion, we would like to know exactly how to detect such quantum emissions.As mentioned in the introduction, a direct measurement of produced particles seemsdifficult to achieve given the quasi-thermal nature of the spectrum and the low fluxesinvolved. A key observation is that quantum particle creation tends to produce particlesin pairs which propagate in opposite directions. This was shown in Section 2.1.1 byequation (2.22). These pairs of particles are naturally entangled and will carry aparticular correlation signature.

The Wightman function G(x, y) := 〈φ(x)φ(y)〉 for a quantum field φ is a measureof the correlation between two separate points x and y in spacetime. If the field φwere a truly random variable the correlation would be zero everywhere. In general theequal-time Wightman function for a massless scalar field on a flat 3+1 dimensionalFRW spacetime written in the form

ds2 = −a6(η)dη2 + a2(η)dx2, (3.26)

with scale factor a (which includes Minkowski space as the a = 1 special case) only de-pends on the magnitude of the co-moving spatial difference x = ||x−y|| and (possibly)on time. Since any function on such a spacetime decouples into independent co-movingmomenta k modes we have

G(η,x,y) =

∫d3kd3k′

(2π)3〈0|φkφk′|0〉ei(k·x+k′·y)

=

∫d3k

(2π)3eik·(x−y)uku

∗k

=

∫ ∞0

dk

2π2

k

xsin (kx) uk(η)u∗k(η)

= G(η, x), (3.27)

where k =√

k · k, uk are the mode-function solutions to u = 0 and

φk = (bkuk + b†−ku∗k)/√

2, (3.28)

is the quantum degree of freedom associated with k. In Minkowski spacetime the in-tegral 3.27 is computed exactly as G(η, x) = ~/(4π2x2) for non-zero x while on thelight-cone the Wightman function also possesses a singular imaginary part [26]. In1+1 dimensions the equal time Wightman function becomes logarithmic and time in-dependent G(x, y) = −~/4π ln (x − y)2 and, owing to the conformal invariance in1+1 dimensions for the massless scalar field and the conformal flatness of FRW space-times, also takes this form in an arbitrary 1+1 FRW background in terms of co-movingcoordinates and the conformal time variable [25].


3.3.3 Exactly Soluble Expanding Spacetime

The main purpose of this section is to gain an intuitive understanding of correlationsdue to particle production and provide a useful toolbox for the more challenging inves-tigation of the anisotropic scaling solution. In order to do so we shall study the idealcase of a homogeneous 3+1D isotropically expanding BEC. In this case, the analoguemetric is that of a simple FRW spacetime geometry. We shall choose a scale factorthat increases from an initial constant value ai to a final constant value af smoothlyas in this case the structure of correlations can be solved for analytically. Our interestin these exactly soluble models derives from the fact that in these systems one can seethe appearance in the Wightman function of the characteristic features expected fromthe creation of particles in a transparent way. We shall also discuss how to simplyinterpret such models in terms of a BEC with a varying scattering length.

We will work with the form for the metric given by (3.26) which can be obtainedform the more traditional form (3.10) by the coordinate transformation

dη =

(gm

ρ0

)3/21

λ3/2(τ)dτ. (3.29)

The functional relationship between the scale factor a the scaling function λ is un-changed (see (3.11)) in this representation as a2(η) ∝

√λ(η).

Furthermore we will consider the specific form for the scale factor

a4(η) =a4i + a4

f

2+a4f − a4

i

2tanh

(η

η0

), (3.30)

where η0, ai and af are constant numerical parameters. Such a metric describes anisotropic homogeneous spacetime with flat spatial sections which expands by a finiteamount over a finite time period.

This spacetime has been studied previously in the literature for example in [77]where it was shown, to arise as the analogue geometry experienced by phase pertur-bations in a BEC with time varying interaction strength g. In that article the particlecontent of the initial Minkowski vacuum state after expansion is studied in detail andthe observability of the created particles discussed. This spacetime is also studied in[75] where it is shown to arise as the analogue geometry associated with a ring oftrapped ions. A very similar geometry is studied in [81] where instead of tanh, the au-thors employ an error function5 temporal profile for the variable scattering length andthe correlation structure is studied semi-analytically in the so-called “sudden limit”(which we will discuss later), and numerically in the general case, with the full Bo-goliubov spectrum. In [88] the author studies the controlled release of a condensate

5Recall that the error function is defined as Erf(t) ∝∫ t

exp(−x2) dx and smoothly interpolatesbetween constant values.



such that the trapping frequency varies exponentially in time and the associated par-ticle production in terms of the number of particles produced. The separate case ofan exponential variation of a time dependent coupling constant g(t) is also studied aswell as the mixed analysis where both the trapping frequency and coupling constantare varied again in terms of the number of particles produced. In all those analyses,however, the correlation structure is not computed instead they focus solely on thespectrum of produces particles alone (given by the magnitude of βk only).

The field equation for a massless minimally coupled scalar field φ propagating onthe background geometry (3.26) decouples into independent modes and is solved bythe ansatz

φ(η,x) =

∫d3k

(2π)3/2eik·x

(bkuk(η) + b†−ku

∗k(η)

), (3.31)

where uk(η) satisfy the k-isotropic mode equation

∂2ηuk + a4k2uk = 0, (3.32)

and k2 = k · k is the squared magnitude of k.

For a(η) as (3.30) the field equation (3.32) is solved by a rather complicated prod-uct of elementary and hypergeometric functions [77]. Conveniently for us, it is notnecessary to work explicitly with these cumbersome functions: For each co-movingmomentum k the physics of cosmological particle production is contained only in innerproducts (which are labelled traditionally as αk and βk) between two particular solu-tions φin

k and φoutk to (3.32), solutions which converge respectively in the past (η → −∞)

and future (η → +∞) to plane waves6 as was shown in Section 2.1.1.

Physically, these two particular solutions represent a description of two differentvacuum states, |in〉 and |out〉, in the two asymptotic regimes where the function a(η)becomes constant and the definition of particle state is unambiguous. For example, thequantum state |in〉 described by the choice φin

k (the vacuum state for η → −∞) is nolonger a vacuum (zero particle) state as η →∞ since that state, |out〉, is described bythe mode functions φout

k and in general one has

φink = αkφ

outk + βkφ

outk∗, (3.33)

with βk 6= 0. The spectrum of particle content of the state |in〉 at late times interms of the late time particle states is given by the modulus |βk|2 which has thecharacteristic bell shape shown in Fig. 3.5 here shown as a function of the isotropicmodulus k =

√k · k appropriate for the isotropic mode equations (3.32) and hence

isotropic coefficients αk and βk.

6Specifically αk is the innerproduct between positive norm (in this case equivalent to positivefrequency, see the DCE section in Chapter 2) plane waves whereas βk is the inner product betweenone positive and one negative norm (frequency) plane wave.


Figure 3.5: The characteristic shape of the spectrum of produced particles (scaled tothe maximum flux at the peak) in an expanding spacetime described by (3.26), as afunction of the modulus k =

√k · k. Recall that the number spectrum is proportional

to the volume of space V/2π as Nk = V k2β2k/2π so that what we plot here is in fact

particle number density. The peak wave number is fixed by the typical timescale of theexpansion kpeak ∝ 1/η0.

Of course, at η → +∞ the solutions φoutk are nothing but plane waves at the

appropriate frequency (modulated by the scale factor a, and normalised by that samefrequency) so we have

φink (η,x) =

1√2ka2

f

(αke

−ika2fη + βkeika2fη

)for η →∞. (3.34)

The inner products αk and βk are given exactly in this model by the simple expres-sions

αk =2√AB

A+B

Γ(−iA)Γ(−iB)

Γ2 (−i(A+B)/2),

(3.35)

βk =−2√AB

B − AΓ(−iA)Γ(iB)

Γ2 (i(B − A)/2),

where Γ is the Euler function and the dimensionless variables A and B are

A = kη0a2i , B = kη0a

2f . (3.36)



They are constrained to satisfy |αk|2− |βk|2 = 1 by the constancy of the Klein Gordonnorm of the (unique) solution φ written in the two different mode function bases.

Since the mode-functions are simple linear combinations of plane waves long afterthe expansion has taken place, the equal time Wightman function (3.27) there can bewritten down in a relatively simple way in terms of αk and βk alone as

G(η, x) =4π

2a2f

∫ ∞0

dk

[sin kx

x

(1 + 2|βk|2

)+ I + I∗

], (3.37)

where

I =sin kx

xαkβ

∗ke−2ika2fη. (3.38)

Of course, such an expression cannot be considered exact for a realistic finite-sized BECbut will break down nearby the boundaries due to finite size effects. Furthermore, in thecase of a finite volume “cigar”-shaped BEC and anisotropic expansion the expression(3.37) will have to be replaced by a more complicated formula as we shall see in sect. 3.4.

We note a few things about relation 3.37. Firstly, when βk = 0 the expressionreduces to the familiar flat-space Wightman function. Secondly, due to the presence ofthe sum η + η′ of the times in the non-equal time Wightman function the equal timetwo point function is not time independent whereas, as is clear from the expression,the equal position two point function is spatially homogeneous. In general one expectsthe phases as well as the magnitudes of αk and βk to be relevant for the correlations.The Bogoliubov coefficients have time reversal symmetry in the magnitudes but not inthe phases.

The expression (3.37) is the sum of three physically distinct terms. The 1 termrepresents the background zero point fluctuations, the |βk|2 term an enhancement ofthe background correlations due to particle production 7 and the terms labelled Iand I∗ represent additional non-trivial structure propagating on top of the (enhanced)Minkowski-like correlations due to the entanglement between the created pairs of parti-cles. The I+I∗ term can be integrated exactly as an unwieldy infinite sum of functionswhich we calculate in appendix ??. In Fig. 3.6 we plot the result of numerically inte-grating the contribution from I + I∗

Ent =4π

2a2f

∫ ∞0

dk (I + I∗) , (3.39)

in G for four successive times as a function of separation x = ||x − x′||. We see thatgoing backwards in time a bump of correlations merges into the correlation singularity8

7This enhancement of background fluctuations from a non-vacuum excited state is discussed in[89]. There the author, instead of using the dynamically produced non-vacuum state, shows such theenhancement from the choice of an initial thermal state.

8Since we always work under the assumption that the field theory is an effective field theory validonly above a small length scale, for example the healing length in the case of a BEC, the coincidencesingularity does not exist in practice. In the BEC example, higher order contributions to the dispersionrelation become non-negligible and in fact dominate at very short length scales resolving the singularity.


Ent

x

Figure 3.6: The exact numerical integration of (minus) the entanglement contributionto G as a function of point separation x for four successive times η = 0 (red-solid),η = 1 (green-dashed), η = 2 (blue-dotted) and η = 4 (black-dot-dashed) for the choiceof parameters a2

i = 1, a2f = π and η0 = 1.

at the origin. The (coordinate) speed of the bump is determined by the final value ofthe scale factor a2

f which in the displayed plot was taken to be π. We note that thebump moves with a velocity of close to 7.5 units per second which is approximatelythe speed at which two particles moving at speed π units per second would be recedingfrom one another. Note that the coordinate speed is not constrained to be the speedof sound cs, the coordinate speed in lab variables.

The contribution to the Wightman function G (3.37) from the 1 and |βk|2 is non-propagating and contribute a divergence at the coincidence limit x = 0 which decaysin x.

In the next section we show that the essential structure of these correlations arepresent already in the so-called adiabatic approximation to the integrand I + I∗.



Adiabatic and sudden approximations

The adiabatic approximation for a mode k is accurate when kη0 1. It is a goodapproximation for high wave number, or short wavelength modes. These are the modeswhich feel the expansion as a ‘slow’ process. In the adiabatic limit one can show, from(3.35) and (3.38), that

I + I∗ ' 2

xsin (kx) sin

[k(Φ− 2a2

fη)]

e−πa2i kη0 , (3.40)

where the phase is given, when af > ai, by (see Appendix D.2)

Φ = ln

(a2i + a2

f

a2f − a2

i

)a2i η0 + ln

(a4f − a4

i

4a4f

)a2fη0. (3.41)

On the other hand, the sudden approximation is accurate for a given k wheneverkη0 1. Hence it should be a good approximation to the exact result for low wave-numbers. In this sudden limit one can similarly show (again from (3.35) and (3.38))that the Bogoliubov coefficients (3.35) become real and momentum independent

αk = α =a2f + a2

i

2aiaf,

βk = β =a2f − a2

i

2afai. (3.42)

In this limit the contribution I + I∗ becomes

I + I∗ ' −1

2

a4f − a4

i

a2fa

2i

sin kx

xcos 2ka2

fη. (3.43)

which represents an unbounded contribution to G if assumed accurate over the entirepositive k axis (as would be the case in the formal limit η0 → 0). In Fig. 3.7 wecompare the exact, adiabatic approximate and sudden approximate forms of I + I∗ fora representative choice of x and η.

The Wightman function in the sudden limit may be calculated assuming the suddenapproximation for all k. By comparison with the flat space computation, we have forx− y non-null and η 6= −η′,

〈φ(x)φ(y)〉 = − 1 + 2β2

4π[−(η − η′)2 + (x− x′)2]− αβ

4π2[−(η + η′) + (x− x′)]. (3.44)

We see the effect of particle creation β2 and a time dependent correlation αβ even inthe equal time Wightman function, predicting a propagating singularity. Alternatively


Figure 3.7: A comparison of the exact expression (in red-solid) with both the adiabaticapproximation (in blue-dots) and the sudden approximation (in green-dashes) near theturn over region (kη0 ≈ 0.3) where neither approximation is very accurate. We use therepresentative values of t = 2 and x = 3. Note that the sudden approximation is mostaccurate for small k and the adiabatic for large k. Also note that the adiabatic approx-imation remains relatively accurate (at least in mimicking the qualitative behavior ofthe exact function) all the way down to k = 0 where the approximation in principleshould be poor.

we note that the adiabatic approximation is accurate almost uniformly over all k, evenclose to the origin, where it is superseded by the sudden approximation while stillcapturing the qualitative behavior of the I + I∗ term.

Integrating the adiabatic approximation I + I∗ over k we observe a ‘bump’ of cor-relations traveling out from the origin and decaying as displayed in Fig. 3.8. This isin close agreement with the exact result as shown in Fig. 3.6. We note that the onlyfeature of the exact result not captured by the adiabatic approximation is the negative“tail” to the bump as it propagates outwards from the origin.

The role of the parameter η0 is visible in Fig. 3.9 where we again show the resultof integrating the adiabatic approximation9 to I + I∗ for the same values a2

i = 1 anda2f = π and times η = 0, 2 , 4. One sees that as the rate of expansion increases (asη0 becomes small) the shape of the bump becomes sharply peaked and that the peakoccurs slightly closer to the origin. This is consistent with our intuitive understandingof the bump as arising from propagating particles created during the expansion and is

9For computational reasons we restrict our attention to the adiabatic approximation in this section.



Figure 3.8: The integrated I + I∗ contribution to the Wightman function in the adia-batic approximation at three successive times η = 0 (red-solid), η = 2 (blue-dots) andη = 4 (green-dashes). Note that since the expression (3.37) is only valid after the ex-pansion has taken place (the midpoint of which occurs around η = 0) the contributionfor t < 0 are not shown. To make these plots and for comparison with the exact resultagain we have used the values a2

i = 1, a2f = π, and η0 = 1. Also plotted (in black-

dash-dot) is the standard flat spacetime correlations (4π2x2)−1.

also consistent with the singular sudden result (3.44).

For small η0 the expansion and hence the particle production takes place over a shorttime interval. Assume for the purposes of illustration that all the particle productionoccurs at a one point in space, say at X. (Particle production occurs at every pointbut this simplification serves to capture the relevant physical mechanism) Then, twoobservers at a fixed spatial separation symmetric to X will simultaneously observe aburst of particles passing by. For large η0 the particle production is spread out over alonger time interval since the scale factor a is changing significantly over a longer periodand hence the observers will see a more diffuse burst of particles. In this way it is notsurprising that attempting to integrate the sudden approximate solution over all k (thatis, in the formal η0 → 0 limit) one finds a propagating singularity at approximatelyx = 2a2

fη as can be inferred from the structure of (3.44).


Figure 3.9: Comparison of the bump structure of correlations for three different valuesof the parameter η0 = 2, 1, 1/2 and 1/4 (from top left to bottom right) at three differenttimes η = 0 (red-solid), η = 2 (blue-dots) and η = 4 (green-dashes). Also plotted (inblack- dash-dot) is the standard flat spacetime correlations for comparison.

.

Varying scattering length interpretation

As mentioned at the beginning of this section, the metric (3.26) can be used to describephase perturbations in a BEC with a time dependent scattering length a(t) (and hencecoupling parameter g(t) and sound speed cs) as discussed in the article [77]. In suchvarying scattering length models one works in a homogeneous approximation wherethe background condensate density ρ is assumed constant and the flow velocity v



everywhere zero. The FRW scale factor (which we temporarily write as as for clarity)is related to the scattering length a by

as(η) =( ρ

4π~2

)1/4 1

a1/4(η). (3.45)

Hence a decrease in the scattering length corresponds to an expansion of the FRWspacetime and vice versa. In [77] the authors use this form for the metric and also thetanh scale factor profile (3.30) described above.

We interpret our result in this context as a propagating bump of correlations whenthe scattering length is varied according to

a(η) =ρ

4π~2

(a4i + a4

f

2+a4f − a4

i

2tanh

(η

τ0

))−1

. (3.46)

Note that this time dependence is not determined by any internal dynamics of the BECbut instead is a free experimental choice. This is in contrast with the expanding BECcase where the time dependent parameter, the scaling function λ, is determined by theinternal dynamics of the BEC.

3.4 Axial correlations in an expanding elongated

condensate

We shall now consider the experimentally relevant case of an anisotropic harmonicallytrapped BEC, where the trap is released only along the z dimension. This sectionconstitutes the main new results in this chapter.

Included in the analysis are the inhomogeneities of the condensate, so that althoughwe only use the results of the previous section on the homogeneous condensate as aguide for the intuition we expect the results to qualitatively match.

We write λ(t) := λz(t) and λ⊥(t) := λx(t) = λy(t). The simple form and cylindricalsymmetry of the scaling solution in this case allows us to separate the variables in thebackground velocity as cs(t, z, r) = c(z, r)/

√λ(t)λ2

⊥(t) where c = cs√

1− ωz z2 − ωrr2

and ωi was defined in section 3.2.1.

Let us now introduce new coordinates through the exact differential expressions

dT =dt

λ3/2λ⊥, dZ =

dz

c(z, r)+ f dr, (3.47)

with f(r, z) = −∫ z∂rc/c

2dz′. Then the metric (3.9) is written as

ds2 =

√ρsgm

λ2c2

[−dT 2 + dZ2 +

((λ⊥λc

)2

+ f 2

)dr2 +

(rλ⊥λc

)2

dθ2 − 2f dZdr

].

(3.48)

3.4. Axial correlations in an expanding elongated condensate 93

Note that f vanishes in the limit where we can neglect the radial derivatives of c. Inthat limit note that the metric takes a particularly simple diagonal form.

Relabeling φ := θ1 the action for phase perturbations written in these coordinatesis

S = −1

2

∫d4x√

g gµν∂µφ ∂νφ

= −1

2

∫dTdZ

[−Ω(T, Z)(∂Tφ)2 +H(T, Z)(∂Zφ)2

]+O(∂rφ) +O(∂θφ), (3.49)

where

Ω =

∫drdθ

√ρsgm

rλ2⊥ , (3.50)

and

H =

∫drdθ

√ρsgm

rλ2⊥

[1 +

(fλc

λ⊥

)2]≡ Ω + δΩ. (3.51)

The action (3.49) has been written up to terms involving the radial and angular deriva-tives of the field φ which we will discard at this stage. The justification for this ap-proximation is transparent: we consider condensates sufficiently tightly trapped in theradial direction that the infrared cutoff provided by the finite radial size and the ultra-violet cutoff provided by the minimal healing length in fact preclude the appearance ofany radial or angular modes at all. In other words, in order for a mode of the field φ tosatisfy the ultraviolet band-limitation it necessarily must not contain any (non-zero)radial or angular frequencies at all10. Or even more simply, due to the small radial sizeof the condensate radial and angular modes necessarily would oscillate on length scalesshorter than the healing length and are hence absent. In fact, in this kind of regimethe scaling solution in the radial direction is not available since it relies on the staticTF approximation. The form we use above for the sound speed also is derived from thestatic TF approximation and is also unavailable in this tightly trapped regime. Henceabove we are making an error in the numerical factors in Ω and H since we are usingan approximate solution outside its regime of applicability and not the exact solutionfor the background density profile. To be consistent with the TF approximation condi-tions, one should take c to be a function of only time and axial z [46] effectively settingthe radial sound speed to zero and reducing the problem to a 1+1 dimensional one.We shall do this in what follows as well as set the scaling function λ⊥ to unity, freezingthe radial dynamics. In this approximation δΩ becomes zero (since it involves a radialderivative of the sound speed) so that H = Ω and the function Ω becomes explicitly

Ω(t, z) = Ω01√λ

(1− ω||

z2

λ2

)1/2

, (3.52)

10Recall that evanescent waves are capable of beating a band limitation when quantum tunneling(and hence an exponentially decaying mode with an imaginary wavenumber) is present. It would beinteresting to investigate possible evanescent wave contributions to the correlation structure in thiscontext.



where

Ω0 =2πA⊥g

( µm

)3/2

. (3.53)

Here it is understood that Ω is zero if the right hand side becomes negative.

Under these assumptions the 1 + 1 dimensional action (3.49) gives rise to the fieldequation

− ∂T (Ω∂Tφ) + ∂Z (Ω∂Zφ) = 0 . (3.54)

which we shall work with in what follows.

3.4.1 Conformal symmetry approximation

As a zeroth order attempt at uncovering a signature of ‘cosmological’ particle creationin the Wightman function one might be tempted to discard derivatives of the functionΩ. Under such an approximation the field equation reduces to the flat 1+1 dimen-sional wave equation for the (massless) scaled field φ = Ωφ whose solutions are simpleexponentials. The Wightman function for the phase field φ is then simply related tothe standard Green function as

〈φ(T, Z)φ(T ′, Z ′)〉 =〈φ(T, Z)φ(T ′, Z ′)〉Ω(T, Z)Ω(T ′, Z ′)

= − ~4π

ln [∆x+∆x−]

Ω(T, Z)Ω(T ′, Z ′)(3.55)

where x± are the characteristic null coordinates. However this is immediately seento be too strong an approximation to capture any particle production effects. In factthe above correlator can be easily plotted as in Fig. 3.10, which shows no sign of thetypical transient propagating over-correlation feature normally associated with particleproduction (see section 3.3.3).

This observation seems to be at odds with several results found in some of theliterature concerning the correlation structure in the case of Hawking radiation. Forexample, in [67, 68] an analogous calculation is performed to the one here, dimensionallyreducing a 3+1D BEC dynamical problem down to one in 1+1D in the case of anacoustic black hole background geometry for phase perturbations. The important pointfor the current discussion is that the authors of [67, 68] in fact do observe numerically asignature of created Hawking quanta in the same approximation which neglects all thederivatives of the function Ω. However, there the non-trivial ‘cross-horizon’ correlationsare contained entirely in the relationship between the lab coordinates x and t and thenull co-ordinates x±. Indeed also in [69] these correlations across the horizon aredemonstrated for an explicitly 1+1D black hole geometry11 which, being conformally

11Note the difference between the analyses of [69] and [67, 68]: whereas [69] works exclusively withthe 1+1D conformally flat geometry the minimally coupled field the 1+1D action of [67, 68] arises


Figure 3.10: The field (phase) correlations Gφ as a function of time t for three differentfixed laboratory spatial separations 0.2Lz, 0.5Lz/2 and 0.8Lz/2.

flat, possesses the standard logarithmic functional form for the correlator. Again theHawking signal is contained exclusively in the relationship between the null and labcoordinate functions.

The reason for this discrepancy is rooted in the rather different causal structureof the black hole and cosmological analogue spacetimes and in the inherently differentnatures of the associated particle creation processes. In the black hole spacetime thepresence of the acoustic horizon is associated to an ergo-region which allows for negativeenergy states (w.r.t. an asymptotic observer) and these states in turn allow for the sta-tionary Hawking flux on the black hole static spacetime. In the case of the cosmologicalanalogue geometry there is no such ergo-region and particle production is permittedonly as a result of the time dependence of the geometry. This time dependence is fullyencoded in the conformal factor describing the difference from Minkowski space of theanalogue geometry. It is then clear that when one introduces an approximation whichinduces a conformal symmetry in the field equation, no particle production can take

from a dimensional reduction of a 3+1D model. This dimensional reduction is encoded in the 1+1Ddynamics by the factor Ω which renders the field non-minimally coupled to the geometry.



place.

3.4.2 Spatially Homogeneous Ω(Z) approximation

As discussed above, the approximation in which we discard both temporal and spatialderivatives of Ω is too strong to capture the physics of particle creation in the 1+1dimensional massless case. However, we shall now argue that it is sufficient to keepthe T derivatives while neglecting the Z derivatives of Ω to observe a particle creationsignal.

Firstly, since the relationship between Z and the co-moving lab coordinate z isgiven by integrating (3.47) (in the approximation which neglects radial and angularderivatives, i.e. f = 0) to

Z(z) =

√2

ωzarcsin

(√mω2

z

2µz

), (3.56)

we see that ∂ZΩ is naturally suppressed. One has

∂ZΩ =∂z

∂Z∂zΩ, (3.57)

where the first factor ∂z/∂Z is naturally small near the boundary (but still withinthe region in which the TF approximation remains valid) and the second factor ∂zΩ isnaturally small in the central region where the condensate density is almost constant.Therefore, the factor ∂ZΩ is naturally small everywhere on the condensate.

No such natural cancellation is available for ∂TΩ since ∂t/∂T = λ3/2λ⊥, which isnot naturally a small factor. Hence in what follows we will keep only T derivatives ofΩ and discard the Z derivatives.

Note that this approximation is not the same as assuming Ω to be constant inlab coordinates, the difference being in the extra factor of dz/dZ which suppresses ∂Zat the boundary of the condensate. An important feature of our analysis here is theinclusion of the effects of non-constant density and finite condensate size.

Phase-Phase Correlations

Consider, then, (3.54) and assume Ω is a function only of the variable T . Then (3.54)is written

− ∂2Tφ−

Ω,T

Ω∂Tφ+ ∂2

Zφ. (3.58)

Let η be defined by the differential expression dT = Ω dη. Then (3.58) separates to

∂2ηφk + Ω2(η)k2φk = 0, (3.59)


where φk = (2π)−1∫LdZ φ e−ikZ is the kth Fourier component of the function φ and L

is the Z-size of the condensate.

Since we work in the compact region L, inverse Fourier integrals will be replacedby discrete Fourier series in this section. Hence k in (3.59) is constrained to satisfyk = 2nπ/L after choosing the boundary conditions φ(−L/2) = φ(L/2) = 0.

Such an equation of motion can be solved exactly, as we did in the previous sectionwhenever Ω2 is contained in the three parameter family of functions

S :=

a2i + a2

f

2+a2f − a2

i

2tanh

(η

η0

) ∣∣∣∣ ai, af , η0 ∈ R

(3.60)

Our strategy here will be to make use of the available exact solutions and try to finda suitable element in S labelled by (ai, af , η0) which most accurately reproduces theexact function Ω associated with free expansion of the BEC. We will see that such aclass of functions is indeed appropriate for an approximation to the free expansion casewith the key benefit of a description in terms of asymptotic particle states.

An alternative way to think about this approximation methodology is that one isstudying the correlations in a BEC which really has this particular form for the functionΩ. As will be discussed later, there is a rather direct link between Ω and the scalingfunctions λ. In fact from (3.52) we see that morally Ω2 ∝ λ−1/2 so that Ω2 ∈ S aremodels for “finite expansion” condensates (or “finite contraction” depending on therelative magnitudes of ai and af ). Such finite expansion (contraction) condensates areachievable in the lab with a suitable ramp-down or ramp-up trapping potential (see sec3.4.3). Note that the finite expansion case qualitatively differs from the free expansioncase only in the future asymptotically static region.

Recall from Fig. 3.4 that the typical frequency of produced particles during expan-sion is λ(t)/λ(t) which converges to zero at late times for a linear expansion λ(t) ∝ t.For this reason we expect all the particle production in the infinite expansion case tooccur only during the “accelerating” phase and that all of the interesting physics tohave ceased by the time the scaling function becomes a linear function of time whichoccurs at late time. In this way we expect the finite expansion approximation to freeexpansion to capture the relevant physics (with the possible doubling of the particlecreation due to a second decelerating phase relative to the free expansion) neglectingonly the irrelevant linear phase of expansion at late times.

The strategy to understand particle production effects in the correlator is as follows.We choose the quantum state to be |in〉 (and since we work in the Heisenberg picturethe system remains in this state for all time) and express our results in the η → ∞limit in terms of the number eigenstates of the asymptotic Hamiltonian which are thephysical particle states in that region.

Let us now move back to the θ notation for the phase perturbation field. The field



operator is written in 1+1D as the sum

θ =∑k

bkfk + b†kf∗k , (3.61)

where the functions fk satisfy the equation of motion (3.59) and are proportional toplane waves in the past and linear combinations of plane waves in the future as

fk(η →∞, Z) ∝ sin kZ(αke

−iωkη + βkeiωkη). (3.62)

The normalization for the functions fk is not determined by (3.59) but instead by theconsistency between the commutation relations for bk with the commutation relationsbetween the field operator θ and ρ, the density perturbation field. To fix this normal-ization we will use the commutator [θ(Z), ∂tθ(Z

′)]. In 3+1D in the TF approximationwith zero background velocity flow in the BEC one has ρ = −∂tθ/g as well as the funda-mental commutator [ θ(x), ρ(x′) ] = −i~δ3(x, x′) implying [ θ(x), ∂tθ(x

′) ] = i~gδ3(x, x′).Recall that we assume the perturbation field to be constant over the cross sectionalarea of the condensate (which was related to the existence of a UV cutoff at the healinglength). Hence integrating this commutator we get for the 1+1D phase field an extrafactor of the area from ρ(3+1)A⊥ = ρ(1+1) and hence

[ θ(x), ∂tθ(x′) ] =

i~gA⊥

δ(x, x′). (3.63)

Further, expressing this commutator in Z coordinates we recall that the conjugatemomentum is a density of weight one whereas the field θ is a scalar so that one picksup a factor of the Jacobian under a coordinate transformation

[ θ(Z), ∂tθ(Z′) ] = [ θ(x), ∂tθ(x

′) ]× dZ

dx(3.64)

Therefore

[ θ(Z), ∂tθ(Z′) ] =

i~gλfcs,0A⊥

δ(Z,Z ′), (3.65)

where cs,0 is the initial sound velocity (which appears as a result of the use of csinstead of cs in the definition of the coordinate Z) and λf is the final value of the scalefactor λ (which comes from the relationship between co-moving x and lab x). Thedispersion relation is easily computed to be ω2

k = Ω2k2 → a2fk

2 using the metric in(η, Z) coordinates. Hence we arrive at the expression

ig~λfcs,0A⊥

δ(Z,Z ′) = [ θ(Z), ∂tθ(Z′) ]

=∑k,k′

[ bk, b†k ] (fk∂tf

∗k − f ∗k∂tfk) , (3.66)


which relates the commutators and the Wronskian for the mode functions allowing usto fix the normalization factor.

For a Fock space (particle) representation we require [ bk, b†k′ ] = δkk′ . Hence we

arrive at the Wronskian constraint on the mode functions∑k

(fk∂tf∗k′ − f ∗k∂tfk′) =

ig~λfcs,0A⊥

δ(Z,Z ′). (3.67)

In order to take the laboratory time derivatives above we require the relationshipbetween t and η. We have

dη =dT

Ω=

1

Ω

dt

λ3/2=

Ω2

Ω30

dt, (3.68)

where we have firstly transformed to the variable T using dT = Ωdη and consequentlyto t using (3.47), expressing the result in terms of Ω alone. Ω and λ are related by(3.52) as Ω = Ω0/

√λ where Ω0 is the spatial part of Ω, numerically Ω0 ≈ 1.5 × 1037.

Then

dt =Ω3

0

Ω2dη =

Ω30 dη

a2i + a2

f

2 +a2f − a2

i

2 tanh(ηη0

) . (3.69)

Now, we wish to approximate the free expansion scale factor shown in Fig. 3.1 forapproximately the first t = 0.15s. This choice of the approximation region is motivatedby two constraints: firstly, after about t = 0.15s, a large proportion of the condensateno longer is described accurately by a TF approximation as can be seen in Fig. 3.3 ;Secondly the natural shape of the exact free expansion scaling function is less accuratelyfit by the tanh functions we use here over longer time periods since the scaling functionenters a linear regime at late times.

Over this t = 0.15s time interval the scaling function λ increases from λi = 1 toapproximately λf = 25. Since Ω = Ω0/

√λ we see that the variable Ω must decrease

from Ω0 to Ω0/5. Therefore we must have af = ai/5. Hence by (3.69) we have theasymptotic relations between the time variables

t(η) =

λiΩ0 η, for t→ −∞λf Ω0 η, for t→∞

, (3.70)

where the function t(η), obtained by integrating exactly the differential expression(3.69), varies smoothly between these two constant asymptotes in the intermediateregion. Taking the average gradient in the intermediate region for t(η) we see that a tinterval of 0.15s corresponds to an η interval of approximately 12×Ω0 ≈ 8.2×10−40 (inunits of [Ω−1]s or [Energy]·[Time]2). Therefore, we require the function Ω(η) to vary



over this time scale between the initial and final values ai and af . That is we shouldchoose

η0 ≈ 8.2× 10−40 [Energy] · [Time]2 (3.71)

With these choices one finds that ωk η = kt/λ3/2f .

For the un-normalized fk we have∑k

(fk∂tf∗k − f ∗k∂tfk) = 2

∑k

ik

λ3/2f

sinkZ sinkZ ′. (3.72)

Scaling the functions fk as

fk =

√λ

3/2f ~g

VolA⊥ciskfk, (3.73)

where Vol is the axial Z-length of the condensate (which has the dimension of a time),the Wronskian provides the correct numerical and functional form for the commutator.

Hence the correctly normalized mode functions are given by

fk =

√λ

3/2f ~g√

cisA⊥Vol

sin(kZ)

ke−ikt/λ

3/2f . (3.74)

Here we have transformed back to lab t coordinates in which we take the time deriva-tives.

In 1+1D at late times the equal time Wightman function is written in terms of αkand βk in a very similar way to the 3+1D example above in terms of the discrete (andinhomogeneous) Fourier series

Gθ(η, Z, Z ′) =N∞∑n=1

1

nsin (knZ) sin (knZ

′)

×(1 + 2|βkn|2 + 2Reαknβ

∗kne−2iknafη

)(3.75)

where the normalization factor N is given by

N =λ

3/2f ~g

A⊥cs,0π. (3.76)

Taking symmetrically spaced points from the center of the condensate, the transienttime dependent contribution is given by

I = N

∞∑n=1

1

nsin

[nπ

(1 +

2Z

L

)]sin

[nπ

(1− 2Z

L

)]× 2Reαknβ

∗kne−2iknafη, (3.77)


Figure 3.11: The bump of correlations propagating outwards from the center of theBEC cloud. The curves are for t = 0s (red, solid), t = 0.3s (blue, dotted), t = 0.8s(green dashed) and t = 1.2s (black, dash dots) Here we plot the truncated sum up tothe first 90 terms. The horizontal axis of this plot is the fraction of the half length Lz/2of the condensate after expansion. The vertical axis is the integrated contribution Entin units of the normalization factor N .

is easily computed as a truncated sum.

With the choices made above for the parameters ai, af and η0 we find the entan-glement structure shown in Fig. 3.11. The time scales of the propagating bump areunderstood in terms of the crossing time for the massless modes: after the expansionthe sound speed has decreased to one fifth its initial value cs/5 due to the dilution ofthe BEC gas (λ(t → ∞) = 25). The size of the condensate has increased by a factorof 25 so the crossing time increases by a factor of 125. Initially with a axial length ofLz = 5× 10−5m and cs = 2× 10−3ms−1 the crossing time is tcrossing = 5× 10−2s whichincreases to the order of seconds after expansion. (Note again as in the 3+1D case,that the entanglement propagation speed is twice the sound speed since it representspairs of phonons traveling in opposite directions). This is a slight overestimation sincethe sound speed actually decreases from a maximum at the center of the condensate tozero at the edge. However we have neglected this slowing in the analysis. This order



]

(a) (b)

Figure 3.12: The contributions to the phase Wightman function from the ‘1’ and ‘|βk|2’terms in (3.75) truncated to the first 200 term of the series. Such a truncation serves asan ultraviolet regulator on the correlation singularity at the origin of the ‘1’ backgroundcontribution.

of magnitude estimate is confirmed in the plot of Fig. 3.11.

In Fig. 3.12 we show also the integrated contributions from the ‘1’ and ‘|βk|2’ termsin the integrand, in the same units as Fig. 3.11. We note that the magnitude of thebump structure is in general much larger than the background (‘1’) and enhancement(‘|βk|2’) contributions. This is expected since the produced particles propagate fromsmall distances where there is a high intrinsic correlation to long distances where thenatural background fluctuations are no longer correlated.

Density-Density Correlations

Experimentally, although there are proposals to directly measure the phase correlations,the density correlations are those of major observational interest. Again, in the TFapproximation the density field is written in terms of the phase as

ρ(3+1) = −1

g∂tθ −

1

gvz∂zθ. (3.78)

The 1+1D density is related to the 3+1D density by ρ(3+1) = ρ(1+1)A⊥ where A⊥ is thecross sectional area of the condensate (which we assume constant). Then the density


correlator is derived according to

Gρ(z, z′) = limt→t′DGθ(t, t′, z, z′), (3.79)

where we define the differential operator D as

D =A2⊥g2

(∂t + vz∂z) (∂t′ + vz′∂z′) . (3.80)

In the 1+1D finite expansion case, D reduces to simply the time derivative since vz = 0after the expansion has finished. In Fig. 3.13 we plot the normalized density correlatorGρ(t, z, z′) := Gρ(t, z, z′)/(ρ(1+1))2 for symmetrically placed points z, z′ about the mid-dle of the condensate in units of N ×A2

⊥/g2. We note again the propagating structure

but of a slightly more complicated shape to the phase correlation bump.

3.4.3 Actual finite expansion of an elongated BEC

The availability of an asymptotically static regime in the future is a necessary ingredientfor the application of the Bogoliubov formalism. As discussed above there are two waysto use the tools contained in that formalism to understand a system which does notpossess such a future static regime such as the case of indefinite expansion of a releasedBEC. Either one considers the finite expansion as a theoretical approximation to the‘true’ unconstrained expansion case, as shown in Fig. 3.14 or alternatively one canimagine actually performing a finite expansion experiment in the lab with a condensatewhich is at first released and consequently ‘caught’ after a finite expansion. Above wehave essentially been discussing the first interpretation. Here we briefly discuss thesecond.

In order for λ(t) to follow the correct step up profile with a future non-dynamicalregion, one might expect a simple two level potential would suffice. However, theequations of motion (3.16) for the scaling function admit oscillating solutions so thatin order to arrive at a static condensate in the asymptotic future a rather finely tunedpotential is necessary.

Again assuming the constancy of the radial scaling functions during the axial ex-pansion we have for the trapping frequency

ω2||(t) =

ω2||(0)

λ3(t)− λ(t)

λ(t). (3.81)

The necessary trapping frequency is obtained by inserting the desired expansion profileinto this expression. One finds that in fact, the squared trapping frequency becomesnegative for a short time near the end of the transition from expansion to staticityindicating a temporarily repulsive force is necessary to stop the BEC from contractingfurther.



(a)

(b)

(c)

Figure 3.13: The density correlation function after expansion has taken place, forthe finite expansion case, in units of N × A2

⊥/g2.. The curves from top to bottom

correspond to t = 0.3s, t = 0.8s and t = 1.2s after the onset of expansion (which wetake, as described above, to last for 0.15s).

In this way it would be possible to actually create in the lab a BEC system withparticular time dependent trapping frequencies such that the scaling function λ behavesin the way appropriate (as discussed above) to induce the kind of finite expansion orcontraction analogue geometry for perturbations. It is worth pointing out again atthis point that we expect the general structure of a propagating bump of correlationsresulting from a finite scaling to not depend in any crucial way on the fine details ofthe scaling. We expect the result to be insensitive for example to the fact that a precisereproduction of the tanh shape profile is practically very difficult and most probablywould only be approximated in any real experiment.

3.5. Loose ends 105

Figure 3.14: The exact scaling function λ (red, solid) and its approximation by thetanh function Ω2

0/Ω (blue, dashed) over the interval t = 0.15s as described in the text.

3.5 Loose ends

In this short section we collect some loose ends from the discussion above on Hawkingradiation and modified dispersion.

3.5.1 Relation to Hawking radiation

It is interesting to compare the above analysis with a similar one for a black hole geome-try such as in [67]. In [67] an analogous calculation is performed dimensionally reducinga 3+1 dimensional BEC dynamical problem down to one in 1+1 dimensions. The cru-cial difference between cosmological particle creation from a finite expansion and blackhole Hawking radiation is that whereas the cosmological radiation is a transient timedependent effect the Hawking flux (after discarding the transient contribution) is astationary radiation. Hawking radiation is a geometric effect resulting from expressinga pure global state on a reduced physical domain. The fact that such a partial traceresults in a mixed thermal state is a specific case of a very general result in quantum



theory [90]. On the other hand, cosmological radiation resulting from parametric ex-citation of the decoupled quantum modes is more fickle and does not survive when anapproximation is made which results in an emergent conformal symmetry in the action.As is discussed above, one cannot neglect the time dependence on the conformal factorΩ if one it to capture the physics of particle creation. The fact that, as is done in[67], one can capture Hawking radiation within an approximation in which the actionis conformally invariant is related to the existence of a conformal anomaly whereby theclassical conformal symmetry is broken in the quantum theory.

3.5.2 Modified dispersion

As anticipated in the introduction, we here demonstrate that a cosmological emissionor DCE is not affected by ‘mildly’ modifying the dispersion relation outside the typicalphysics energy scale of the problem. For simplicity we assume spatial homogeneity andhence work directly in momentum space and assume the modification takes the form

∂2ηφk + a4F 2(k)φk = 0, (3.82)

which should be compared with the non-dispersive mode equation (3.32). Here weassume

F 2(k)→ k2 as k → 0+ (3.83)

Solutions to (3.82) are identical to those of (3.32) with k replaced by F (k), since kplays the role of an external parameter due to the decoupling of k modes. Thereforethe Bogoliubov coefficient of the modified equation at momentum F (k) is equal to thatof the unmodified equation at momentum k and hence the two spectra coincide asfunctions of these two separate variables respectively. Recall that the typical spectrumfor DCE takes the Planckian-type form 3.5, dropping to zero emission at high momenta.We conclude that if the variables k and F (k) start to become different only after thespectrum of produced particles has dropped to almost zero the two spectra will beindistinguishable as functions of k. That is, if the dispersion is modified only aftersome high momentum value kcutoff and that the dynamical process giving rise to theproduced particles predicts, using the linear dispersion relation a spectrum peaked ata momentum scale kpeak kcutoff then the effects of dispersion are negligible for theprocess.

It could be possible, however, by including ‘un-mild’ modified dispersion such ascomplex, non-monotonic and multiple-valued F (k) to drastically modify the spectrum.These features of the dispersion relation would correspond respectively to absorption,additional production channels, and resonances in a physical analogue model. We notehere that all these features are present in non-linear optical systems which we willdiscuss in Chapter 4.

3.6. Conclusion and discussion 107

3.6 Conclusion and discussion

In this chapter we have considered the correlation structure of expanding BECs bothin the idealized case of isotropic and homogeneous expansion as well as in the experi-mentally relevant case of a finite size, cigar-shaped, BEC anisotropically expanding.

In the first case we have developed a general formalism for probing the effects ofparticle production on correlations as embodied in the equal-time Wightman function.Specifically, for any spatially translation invariant system we have shown that it makessense to define the correlator as in (3.27) which in the case of particle production (usingBogoliubov coefficients) takes the form (3.37). This result is completely generic toparticle production in any spatially translation invariant system, applying to standardcosmologies as much as analogue spacetimes. Specifically, the general cosmologicalresult was shown to be directly implementable in a 3+1D BEC system with a timedependent scattering length as well as a 3+1D isotropically expanding BEC, openingtwo alternative windows on a possible observation of this correlation signal.

In the second case (anisotropic BEC) we have shown that the condensate dynamicsreduces to an effectively 1 + 1-dimensional system and that the Wightman functioncan be generically cast in the form (3.75) for these systems. Using this result we havestudied the correlation structure and shown how it carries the signature of analoguecosmological particle production possibly observable in future experiments. In particu-lar, we have studied the density correlations in such BEC systems, which are of currentexperimental interest.

The magnitude of the normalized correlations is of pertinent experimental concern.The overall scale of the propagating bump is given in terms of the normalization factorfor phase perturbation along with the additional factors which relate the 1+1D densityto the phase field 〈ρρ〉

ρ2' N

g21

(ρ(1+1))2≈ 10−7. Although this is a very small number,

it should be noted that this magnitude is in fact much larger than the magnitudeof the background correlation structure (present in flat spacetime also). The relativemagnitude of the background to entanglement correlations is independent of the nor-malization of the mode function solutions but instead depends on the structure of theBogoliubov coefficients which derive from the wave equation alone.

The main purpose of this chapter was to investigate the characteristic signaturerelated to the cosmological particle emission in an expanding (or more generally time-dependent) BEC, and analyze it without specific attention to its actual measurability.However, one can of course imagine altering the parameters of the background BEC insuch a way to improve the signal. For example, one parameter which can be alteredby an order of magnitude is the number of atoms N which affects the value of ρ0

entering into the denominator of the normalized correlations. Another possibility ofimproving the signal would be to alter the scattering length. We have discussed abovethe use of the analogue FRW geometry when one induces a time dependence in the



scattering length. Here we have something different in mind. Since the backgrounddensity depends on the scattering length as a3/5 (see equation (3.4)), by decreasingand holding the scattering length constant before expansion, one might be able toimprove the signal. Another possibility is to choose a longer time scale on which toapproximate the eternal expansion with a finite expansion: by stopping the expansionlater, the correlations will be propagating on a background density which is lower whichwould improve the ratio in the normalized correlations. All these ways of altering ρ0

will also affect the values of η0, ai and af possibly giving a more pronounced signal.

A separate analysis should regard the role of a non-vanishing initial temperature ofthe BEC (or more generally the role of a non-vacuum initial state). In fact, as alreadypointed out in [68, 81, 82], the net effect of such an initial temperature (i.e. a non“empty” initial state, but thermally populated) is to significantly increase the quantityof created excitations, therefore enhancing the signal without affecting the mechanismresponsible for the particle production.

Chapter 4

Quantum Vacuum Radiation inOptical Systems

This chapter represents a partial summary of a very exciting and recent interactionbetween experiment and theory in the analogue gravity community. The material iscentered around the articles in Refs. [91, 92] (hereafter referred to as the ‘BF’ experi-ment) by Belgiorno et. al which appeared in 2010 describing an experiment performedwith a non-linear optical system in which radiation was observed and was claimed to berelated to an analogue of Hawking radiation (see section 2.4 and 2.1.3). The articlesin Refs. [92, 91] and the more detailed follow up in Ref. [93] were controversial inthe community, stimulating much activity in the literature with a series of commentsand replies [94, 95, 96] as well as taking up major parts of two summer schools in thepast year1. Although at this stage the community has not reached a consensus on theclaims of BF, the complex nature of the experimental system has proven to be a richsource of interesting effects [97, 98, 99, ?] which are in some way or another closelyrelated to an an analogue of the Hawking effect.

The main technical results in this chapter are based on those presented in [97] on analternative explanation of the BF experimental findings in terms of a dynamical quan-tum instability (DCE) (see section 2.1.1). However, we will have time and space hereto discuss the broader picture of optical analogue models in the presence of dispersionand the nature of DCE in general, in particular the issue of finite size effects.

In the next section we will describe the experimental situation, relying on materialfound in the recent literature immediately surrounding the BF articles, referring thereader to some excellent reviews (see for example [55]) already present in the literatureand Sec. 2.4 for a detailed account of the optics theory. We will focus on those aspectswhich are most relevant for the model we will present for the emission.

1The first school in was the 2011 SIGRAV school in Como http://www.centrovolta.it/sigrav2011/while the second “Analogue horizons in fluids and superfluids” was in ICTP, Trieste.

110 Chapter 4. Quantum Vacuum Radiation in Optical Systems

In the second section we present a model for the DCE in optical glass in an attemptto explain the BF observations and extending them to more general situations thanthose directly relevant for the BF experiment. In some related works authors haveproposed an alternative computations for the BF observations and we comment onthose proposals in the third section.

We will end with some comments on the future outlook for such experiments andsome speculations for analogue gravity in optical systems.

4.1 Optical black holes

It has been theoretically demonstrated that electromagnetic field modes propagating ina dielectric medium with a spacetime dependent refractive index can closely resemblefield modes in a black hole background geometry [100, 101, 102, 103, 104, 105, 106, 107].The idea was further developed (and analyzed in more realistic contexts) in the sequenceof papers [108, 109, 91, 110, 55, 94] as we introduced in 2.4.

Specifically, in the BF experiment an ultrashort laser pulse was used to createa traveling RIP in a transparent dielectric medium (fused silica glass) through theso-called non-linear Kerr effect (see e.g. [107]). The authors of Refs. [92, 93] reportexperimental evidence of unpolarised photon emission, for which they suggest the mostcompelling explanation is a Hawking-like radiation due to the presence of the abovementioned blocking region which simulates a black hole-white hole system.

As we discussed in Sec. 2.4, the dispersion relation in the visible electromagneticspectrum is approximately linear, opening the door for an analogue description in thevisible band. However, as discussed in Sec. 2.1.3 the Hawking process is an ‘excep-tional’ emission, apparently depending on energies well outside the frequency bandof ‘predicted’ (assuming linear dispersion) Hawking quanta. One novel feature of theoptical dispersion (shown in Fig. 4.1) is that it is neither purely sub nor purely super-luminal in character, and in fact is not even single valued over momenta nor real valued(see below for a more precise description of this dispersion relation). Thus far in theliterature, the Hawking mechanism has been investigated only in the presence thesesimplified monotonic, real and single-valued dispersion relations, and the robustnessof the spectrum predicted from the linear theory shown. Only very recently [98, 111]has a systematic study of the Hawking process in the presence of the detailed opticaldispersion begun. Therefore it is of interest to consider alternative physics mechanismscapable of spontaneously producing particles to the Hawking mechanism in order tomore fully understand the BF results.

It is not the purpose of this chapter to go into the details of the interesting questionof whether or not an analogue of the black hole geometry pertains to these opticalsystems, whether any notion of a horizon is present, or indeed whether or not the con-cept of a horizon or Hawking radiation even makes sense when the dispersion relation

4.1. Optical black holes 111

Figure 4.1: The optical dispersion in the lab frame n2(ω)ω2 = c2k2. Note the horizontalasymptotes at the absorption resonances above which the dispersion is complex as wellas the multi-valuedness and the complicated sub- and super-luminal structure.

is the optical one (however see Section 4.3). Below we describe the BF experimentin terms which are independent of the vocabulary of black hole physics, distilling themain features which any model should seek to explain.

4.1.1 The Experiment Distilled

Later on in this chapter we will present a DCE model in an attempt to explain the BFobservations. To this end we here collect the most salient features of the observationswhich should be explained by any model of the observed emission. As a preview,remarkably, our model is capable of matching almost all of the features presentedbelow, but not all.

The central physics idea of the works cited and described above is that a sufficientlyintense, localized and moving refractive index perturbation (RIP) can give rise to a re-gion in the interior of the perturbation where electromagnetic modes cannot propagatein the direction of motion of the RIP. That is, a mode slows down while climbing the


trailing edge of the RIP, and if the RIP is sufficiently intense and fast, it is possible forthe mode’s group velocity to vanish in the frame of the RIP.

The condition for the existence of such a “blocking region”, associated with a RIPof peak refractive index nmax = n0 +η traveling at velocity vRIP on top of a backgroundrefractive index n0, would be

c

n0 + η< vRIP <

c

n0

(4.1)

as represented in Fig [not done].

This can equivalently be rephrased as a rather tight constraint on the refractiveindex

n0 ∈[

c

vRIP

− η, c

vRIP

]. (4.2)

The most salient features of the experimental observation are the following: Thefused silica was illuminated with a laser pulse of 1 ps timescale, and the input energy wasvaried in the 100–1200 µJ range. Radiation from the filament was then collected at 90degrees with respect to the laser pulse propagation axis. This arrangement was chosenin order to strongly suppress, or eliminate, known spurious effects (e.g. Cherenkovradiation, fluorescence, etc.) [92]. The observed emission spectra were reasonably wellfitted by Gaussian profiles. These Gaussian fits showed an increasing peak wavelengthof emission, and increasing overall flux, with increasing pulse intensity. Characteristicnumbers for the observations were the following: the emission is centered around apeak wavelength of about (850 ± 25) nm (see Fig. 3 of [92]) and has a bandwidththat appears to depend on the pulse intensity. While the determination of the actualsize of such a bandwidth is fitting dependent, it is evident that the most of the photoncount is concentrated in the range from 700 to 1000 nm.

The interpretation of the observed spectrum as an optical analogue of spontaneousquantum Hawking radiation is based mainly on the above mentioned window condition.In fact, while the observed radiation is clearly non-thermal, one can nevertheless claimit to be related to the blocking regions via a modification of Eq. (4.1). However, thisblocking condition, which at first glance seems rather straightforward, in fact containsa number of subtleties once optical dispersion is introduced. Indeed in situations ofdispersive propagation with n0 = n0(λ), where λ is now the wavelength of the specificmode under consideration, the simple refractive index blocking condition of Eq. (4.1)becomes a wavelength-dependent ‘window condition’ defining those modes for which ablocking region exists:

n0(λ) ∈[

c

vRIP

− η, c

vRIP

]. (4.3)

The window condition (4.3) will be satisfied by a certain set of wavelengths, a setwhich depends on which form for the dispersion relation one chooses as well as on


the experimental parameters such as the character of the RIP itself. However, not alldispersion relations will necessarily lead to windows — the ‘window region’ might (andoften does) define an empty set.

The relevant wavelength dependence of the refractive index for fused silica is ade-quately approximated by the Sellmeier relation2

n(λ) =

√√√√1 +m∑i=1

Bi λ2

λ2 − λ2i

. (4.4)

(Except in the immediate vicinity of resonances where the unphysical poles are regu-lated in real physical systems by other effects such as absorption as indicated by thefact that n(λ) becomes complex near each pole.) In Eq. (4.4) the Bi and λi are a setof coefficients determined by matching experimental data for the dispersion. Whentruncating the sum to three poles, the data is best fit with the values

λ1 = 64.25nm, λ2 = 114.00nm, λ3 = 9938.24nm. (4.5)

which fix the location of the poles in the optical dispersion.

Typical values of the induced perturbation in the refractive index η are of the orderof 10−3, and higher, increasing with the intensity of the RIP. Calculating the vRIP

relevant for the experiment requires not only the knowledge of the group refractiveindex at the typical RIP frequency, but also the characteristic Bessel beam factor,which was measured to be θB = 6.8 [93, 112]. The factor enters the formula for thevelocity as a function of refractive index as vsignal = c/(q nsignal) where q = cos(θB).

4.1.2 Some physics questions from the experiment

In Refs. [92, 93] the window condition of Eq. (4.3) was applied with the understandingthat vRIP was the group velocity of the pulse, whereas the the n0(λ) was taken to be therefractive index for phase velocities of modes in the fused silica. With this interpretationthe window was found to be approximately (870, 940) nm. We stress that this resultis very strongly sensitive to the precise numbers used for the relevant quantities inthe window condition of Eq. (4.3) due to the relative flatness of the function n0(λ)

2Note there that we quote the dispersion relation, as is standard, in terms of the wavelength invacuum (outside the glass). In practice since the modified dispersion arises from a non-instantaneous-in-time response of the polarisation to an electric field, the susceptibility and hence the refractiveindex are intrinsically frequency dependent objects (see section 2.4) and hence the resonance polesare in fact frequency resonances and not wavelength resonances. Our results in this chapter are alsogiven in terms of the wavelengths as they would be outside the glass, lying in between the describedresonances and hence in a region where the non-linear dispersion is approximately linear. Hence weconsistently work with the wavelength representation. Any dynamical model intrinsically involvingthe non-linear dispersion should be carried out in the frequency representation (see Section 4.3)


over the range of wavelengths of interest here. (The Cauchy approximation providesinsufficient accuracy for determining the window condition and it is essential to usethe Sellmeier approximation for this particular purpose.) Such a window appears tobe in good agreement with the observed spectra, hence supporting the authors’ claimof the Hawking-like nature of the observed radiation. Furthermore, the peak shift andspectral broadening with intensity are also claimed to be expected in this interpretativeframework. In particular, the emission bandwidth is predicted to depend on η whichin turn is a linear function of the pulse intensity.

While these aspects of the evidence are surely supportive of a Hawking-like inter-pretation for the origin of the observed radiation, it was immediately noticed by severalresearchers in the analogue gravity community that certain other features of the exper-iment are problematic from this point of view (see e.g. the Comment [95], and Reply[96]).

We do not wish here to further enter into that on-going debate, but we shall insteadlimit ourselves to stressing two points of main concern from our point of view:

• First of all we find odd the observation that photons are emitted at 90 degreeswith respect to the ostensible horizons set by the window condition (4.1). In fact,the window condition only holds for modes aligned with the RIP direction ofpropagation, so one would not expect at all in this model any photon productionat 90 degrees to the beam axis.

• Furthermore, we think that some concern has to be raised with the applicationof the window condition itself, as it rather oddly mixes the group velocity of theRIP with the phase velocities of the vacuum modes in a dispersive medium. Onthe contrary, we feel that there are good physics arguments for expecting thewindow condition to be determined by only considering group velocities.

Specifically, let us consider an observer in the rest frame of the pulse as it movesdown through the glass at velocity vRIP < c. Then a probe signal chasing the RIP frombehind will be seen to be approaching with a velocity given by the relativistic formulafor the combination of velocities

vcomovingsignal =

vRIP ± vlabsignal

1± vRIPvlabsignal/c

2. (4.6)

Including the effect of a Bessel beam factor, we have vRIP = c/(qnRIP). Then

vcomovingsignal

c=

nsignal − qnRIP

qnRIPnsignal − 1. (4.7)

From this we see that, in the optical band, zero approach velocity (in the RIP frame)is achieved at the location x∗ such that

nsignal(λ, x∗) = qnRIP = qn(λRIP). (4.8)


This defines a wavelength-dependent blocking region since the probe sees the presenceof the RIP as a local increase in the ambient refractive index. These velocities aregroup velocities Since nsignal(λ, x) varies between the background value n0(λ) and thevalue at the peak of the RIP n0(λ) + η, the above condition Eq. (4.8) is equivalent toEq. (4.3).

However, from a theoretical perspective, use of the relativistic combination of ve-locities formula seems to us to be well-motivated only if vRIP and vlab

signal are eitherboth group velocities or both phase velocities. Certainly vRIP is a group velocity sincethe RIP is really a propagating wave packet. This rather strongly suggests that indetermining the window region vlab

signal should also be interpreted as a group velocity.

If we now adopt a group velocity interpretation of the window condition, we areobliged to consider the group velocity refractive index which is defined as

ng(λ) :=d[ωn(ω)]

dω= n+ ω

dn

dω= n− λdn

dλ, (4.9)

where ω is a frequency. Using this formula the window condition becomes

ng(λ) ∈[c q

vRIP

− η, c qvRIP

]. (4.10)

The problem is that using this (more physically justified) formula, and the previouslygiven values for the characteristic quantities, there is no range of wavelengths for whichthe window conditions is satisfied. As a function of θB the window is non-empty onlyfor θB . 3 when η = 10−3 and λRIP = 1050 nm.

In summary, we feel that these two facts alone are quite sufficient for motivating se-rious consideration of alternative physical explanations for the interesting observationsreported in Ref. [92]. Specifically, we shall argue here that the DCE is a competitiveexplanation that certainly deserves further theoretical and experimental investigation.

In particular, we conjecture that the rapidly varying refractive index in the fusedsilica excites vacuum modes leading to substantial photon creation. This would benaturally isotropic in the subspace orthogonal to the direction of propagation of theRIP, and would not be directly dependent on the above mentioned window condi-tion, being independent of the presence or not of a blocking region. Furthermore the‘non-exceptional’ nature of DCE places one in a much safer position when neglectingdispersion effects. That is, if one can show that the RIP in the lab frame can be seenas a dynamical variation of the refractive index in such a way to predict an emissionlying in a band of frequencies where the dispersion is approximately constant, then onecan safely assume that dispersion is negligible in the lab frame for the computation.However, this does not imply that dispersion is irrelevant for the physics in the frameof the pulse, from which one would see the optical glass as a ‘river’ of atoms flowingrapidly past. Microscopically, the dispersion relation reflects the physics of the absorp-tion and re-emission of photons by the atoms of the glass itself, and hence it is natural


to be able to neglect the dispersion in the lab frame while in the pulse frame have thedispersion play a crucial role in the dynamics. We will discuss this issue of differentframes describing the same observations using vastly different physics after the nextsection in which we present computations done in the lab frame neglecting dispersion.

In the following discussion we shall present an analytically solvable 1+1 modelwhich, in spite of its simplicity, is able to reproduce many of the salient features of theexperimentally observed photon emission.

4.2 An alternative explanation – DCE

In this section we present a model for the DCE in the lab frame in an attempt to explainthe BF experimental observations. The DCE was introduced in general in 2.1.1 whilehere we apply the techniques to the particular case of electromagnetic modes travelingin a time varying dielectric. As mentioned at the end of the previous section, we hereneglect the effects of dispersion in the wave equation for reasons also discussed there.Essentially, since the BF experiments observe photons at around the 700 nm to 1000nm range and the optical dispersion

c2k2 = n2(ω)ω2 (4.11)

where n is given in terms of wavelength3 in (4.4) is approximately linear in this band(see Figure 4.1), we can safely assume that dispersion plays no role in the productionof such photons in a DCE mechanism.

As is standard, we simplify the problem to the scalar magnitude of the electric fieldcomponent of the full electromagnetic field, with the understanding that the resultsderived will apply to both photon polarisations independently, and will contribute halfof the total flux expected from a full calculation.

Below we will bootstrap our initially 1+1 dimensional results to 3+1 dimensionsusing the interpretation of the equation of motion as applying to the magnitude k ofthe wave vector k. This is common procedure in dynamical Casimir effect calculationsdue to the isotropic nature of the emitted radiation. We have in mind the idea that thedetector which observes the emitted photons actually watches only a small element ofthe silica glass which, as the pulse passes through, appears to the detector to undergo arapid time variation in its refractive index. Within our simple model this bootstrappingcan be effectively achieved by integrating for the flux our Bogoliubov coefficient |βk|2over a three dimensional k space, as we will do below. The variable x in the reduced 1+1dimensional problem represents the orthogonal coordinate to the propagation axis ofthe RIP – it is the one dimensional subspace seen by the detector. In this approximation

3Again, this is wavelength in vacuum and hence we can convert to the (conserved) frequency byω2 = 2πc2/λ2. Note that from glass to air, the wavelength is not conserved.

4.2. An alternative explanation – DCE 117

we also assume a homogeneous in x perturbation in the refractive index and do notconsider finite size effects beyond their inclusion in the volume pre-factor for the flux.We have checked that the inclusion of finite size effects using an inhomogeneous (inx) refractive index perturbation does not qualitatively nor quantitatively change ourresults and conclusions.

One might (mistakenly) argue based on a misinterpretation of boost invariance thatsuch an ‘artificial’ introduction of a time dependence in the field equation is unjustifiedsince one might naively expect a moving dielectric, (for example a pair of spectacles),to not emit radiation. However, it is important to keep in mind that optical systemsof the kind considered here are highly dispersive and do not, in general, enjoy exactboost invariance. Indeed, the physics as described in the frame of reference of theRIP itself is highly non-trivial and one expects particle production even in these staticconfigurations, entirely through a dispersive mechanism alone [16]. In the laboratoryframe of reference (detector frame), in the window of frequencies considered here, therefractive index is approximately constant and we can neglect the effects of dispersionin this frame. By moving to the pulse frame, one can render negligible the particleproduction due to time dependence,4 but necessarily one moves out of the regime inwhich dispersive effects are negligible. (See Sec. 4.3 for more comments on this point.)

The equation of motion for the 1+1 dimensional electric field φ with a time variablerefractive index n is

1

c2

(n2(t)φ,t

),t

= φ,xx. (4.12)

In momentum space this reads

φk,ττ + c2k2n2(τ)φk = 0, (4.13)

where we have introduced the “conformal time” variable

τ(t) =

∫ t dt′

n2(t′). (4.14)

For the refractive index we will initially choose both a sech2(·) time-dependentperturbation on top of momentum-dependent background value

n2(t) = n0(k)2 + 2η n0(k) sech2(t/t0) +O(η2), (4.15)

where η is a small parameter and n0(k) would be a model for the refractive index such asthe Sellmeier approximation. In an appendix we consider a tanh2(·) profile, noting thatthe two profiles essentially give the same results apart from one or two caveats which we

4Even this is an oversimplification: The authors have also made some preliminary investigationsof a residual time dependence which remains in the pulse frame, and which might also be a viableparticle production channel.


will point out. These choices are motivated not only by exact solubility of the model,but also for their close approximation to the experimental Gaussian profile reported in[92] (in the tanh case, we are motivated by the phenomena of ‘pulse steepening’ whichis known to occur in optical fibres which we discuss in more depth below) .

Then the equation of motion becomes

φk,ττ + c2k2[n2

0(k) + 2η n0(k) sech2(t(τ)/t0)]φk = 0, (4.16)

which can immediately be recognized as describing a time-independent Schrodingerscattering problem

ψ,xx +2m

~2[E − V (x)]ψ = 0, (4.17)

with

V (x) = V0 sech2(x/x0), m =1

2c2k2~2, (4.18)

whileE = n0(k)2, and V0 = −2η n0(k). (4.19)

This analogy allows us to directly write down the solutions to our equation of motionby appealing to well-known textbook results.

The Bogoliubov coefficient βk for our mode equation (4.16) is related to the trans-mission coefficient T for the scattering problem (4.17) by |βk|2 = 1/T − 1. From theliterature (see for example Ref. [113] and many references therein) we find the trans-mission probability

T =

sinh2

(π

√2mEx20

~2

)sinh2

(π

√2mEx20

~2

)+ cos2

(π2

√1− 8mV0x20

~2

) , (4.20)

so that

|βk|2 =cos2

(π2

√1 + 8ηΓ2

k/n0(k))

sinh2 (πΓk), (4.21)

with Γk = ckτ0 n0(k). The function |βk| rises from zero at k = 0 to a maximum βmax

at kmax before again decaying to zero for large k. Physically this structure reflects thetransition from low k modes which experience the RIP as a sudden, approximatelyinstantaneous influence, to high k modes for which the RIP represents an adiabaticprocess. There are also features at the poles (resonances) of the refractive index, buthere we focus on the region between the resonances, which is the region of physicalinterest due to it being the only region where observations in the RIP experiments arepreformed.

Furthermore, as we will show below, all of the features of the function |βk| derivingfrom the time dependence of n(t) occur for values of k for which n0(k) is essentially


constant, whence the effects of dispersion can be ignored for the purposes of this cal-culation. Accordingly, we henceforth set n0(k) = n0 ' 1.458 for what follows.

Due to the smallness of η we can expand the cosine about π/2 giving

|βk|2 ' η2 4π2c4k4t40n20

sinh2(πckt0n0)− η3 16π2c6k6t60n

30

sinh2(πckt0n0)+O(η4) , (4.22)

where we see explicitly the quadratic dependence of the spectrum on η at lowest order.

Bogoliubov tricks: Transmission and reflection coefficients are related to Bo-goliubov coefficients. Bogoliubov coefficients are associated with an ‘in’ frequencybeing converted into two out frequencies, one positive and one negative while thetransmission and reflection coefficients are associated with spatial scattering prob-lems where a positive momentum mode is converted to a positive (transmitted) andnegative (reflected) momentum modes. It is known that

T =1

|α|2(4.23)

which implies in particular since |α|2 − |β|2 = 1 and R + T = 1 that

R +1

|α|2= 1, ⇒ |α|2R + 1 = |α|2 = |β|2 + 1, ⇒ R =

|β|2

|α|2(4.24)

Alternatively we have the inverse relations such as

|β|2 =R

T(4.25)

4.2.1 Evaluation of βk directly in perturbation theory

The equation of motion 4.13 above

∂2t φk + c2k2n2(t)φk = 0, (4.26)

(here we have re-labeled τ by t) can be tackled using a perturbative method for the caseof the small RIP which we are considering. This approach can be considered as a warmup problem for a later section where a perturbative solution is obtained in the morecomplicated case of a spatially and time dependent refractive index. Here, instead ofusing a sinh function to approximate the gaussian RIP shape we use a gaussian directlyin order for an integral to be exactly done later on.


Define ωk := n0ck. Thenn2(t) = n2

0 + ηf(t), (4.27)

where η is a small parameter as previously and we choose

f(t) := 2c2k2n0 e−(t/t0)2 , (4.28)

and expand the solution to (4.26) in a power series in η as

φk(t) = φ(0)k (t) + ηφ

(1)k +O(η2). (4.29)

Collecting zeroth order terms (independent of η) we find the equation of motion

∂2t φ

(0)k (t) + ω2

k φ(0)k (t) = 0. (4.30)

Then the correctly normalised homogeneous solution to this equation (which is theasymptotic ‘in’ state of a traditional Bogoliubov analysis) is given by

φ(0)k =

e−iωkt

n0

√2ωk

. (4.31)

At first order in η we find the driven harmonic oscillator problem

∂2t φ

(1)k (t) + ω2

kφ(1)k (t) = −f(t)φ

(0)k (t). (4.32)

where the driving force depends on the zeroth order solution φ(0)k . The initial condition

for φ(1) is φ(1)(t→ −∞) = 0 since the perturbation is absent in the infinite past. Then,using the retarded Greens function the solution can be written as

φ(1)k (t) =

1

ωk

∫ t

−∞dt′ f(t)φ

(0)k (t) sin ωk(t− t′)

=

√2c2k2

ω3/2k

∫ t

−∞dt′ sin ωk(t− t′) e−iωkt−(t/t0)2 . (4.33)

Note that the dimension of φ(1)k coincides with that of φ

(0)k as it should by relation 4.29.

The integral in (4.33) can be done exactly and is given by∫ t

−∞dt′ sin ω(t− t′)e−iωt−(t/t0)2 =

i√πt04

[e−iωt (erf (t/t0) + 1)

− eiωte−ω2t20 (erf (iωt0 + t/t0) + 1)

]. (4.34)

The error function terms (erf) add to a real quantity due to the property erf(z) = erf(z).

For t→ −∞ we have φ(1)k → 0 whereas using the above integral and 4.33 we find

φ(1)k (t)→ i

√π

2ωk

c2k2t0ωk

(e−iωkt − e−ω

2kt

20eiωkt

)as t→∞, (4.35)


which is the time when we wish to compute the Bogoliubov coefficients, that is, afterthe perturbation has subsided.

From this result we see that, to first order in η, the solution up to O(η2) containsonly the pure positive and pure negative norm (frequency) pieces

φk =

(1

n0

√2ωk

+ iη

√π

2ωk

c2k2t0ωk

)e−iωkt −

(iη

√π

2ωk

c2k2t0ωk

e−ω2kt

20

)eiωkt +O(η2),

(4.36)as expected. We conclude that the Bogoliubov coefficient βk is given by the coefficientof the negative norm component of the correctly normalised mode function. Given thatthe initial wave functions come with a denominator of n0

√2ωk we have

βk = −iηt0√πωkn0

e−ω2kt

20 +O(η2), (4.37)

and hence

|βk|2 = η2πt20c2k2e−2c2k2n2

0t20 +O

(η3). (4.38)

We see the characteristic exponential tail of the adiabatic regime for high frequencies aswell as the quasi linear sudden regime for low frequencies (see Section 2.1.1 in Chapter 2for the general structure expected of DCE spectra). It is remarkable that these featuresare present at this level of approximation.

For completeness we note that

αk = 1 + iηt0√πω2k

n20

, (4.39)

so that

|αk|2 − |βk|2 = 1 + η2t20πω2k

n20

(1− e2ω2

kt20

)= 1 +O(η3). (4.40)

and the perturbative coefficients satisfy the Wronskian condition up to the appropriateorder in η.

4.2.2 Spectrum and flux

In what follows we concentrate exclusively on the result from the exact calculationgiven in equation (4.21) and the expansion (4.22). In the appendix we compare theresults to those obtained with the perturbative method as in the previous section.

The 3 + 1 dimensional photon flux density is given by

4πk2|βk|2 =dN

dVol dk. (4.41)


Special care should be taken when converting to wavelengths (which is relevant for thecomparison with the literature). The peak wavelength of emission λpeak is not simplygiven by λ(kpeak) since λpeak is the maxima of the function which enters the integralover λ, including the Jacobian factor

dN

dVol dλ=

∣∣∣∣dkdλ∣∣∣∣ dN

dVol dk=

2π

λ2

dN

dVol dk

∣∣∣∣k=2π/λ.

(4.42)

From the lowest order term in Eq. (4.22) we get

dN

dVol=

∫ ∞0

F (λ)dλ , (4.43)

where

F (λ) =8

15625

1

λ8

π9c4t40n20

sinh2(2π2ct0n0/λ). (4.44)

Noting that the function 1/(u8sinh2(1/u)) has its maximum at a point extremely closeto u = 1/4 we obtain

λpeak '1

2π2ct0n0 . (4.45)

Calculating the total flux we have

dN

dVol= 4π

∫dk k2|βk|2 =

8π2

21

η2

c3t30n50

. (4.46)

The emitting region in this 3+1 dimensional interpretation would approximately bethe cylindrical region defined by the width of the laser beam and length of the regionover which the refractive index is rapidly varying. Observationally the radius of thelaser beam is W ' 10−5 m [112], and we can estimate the length of the active region asct0n0, so the physical volume of the active region is V = πx2

0(ct0n0). The total numberof emitted photons is estimated to be

N =8π3

21

η2W 2

c2t20n40

. (4.47)

4.2.3 Putting numbers to the model

The parameters given explicitly in Ref. [109] are

n0 = 1.458, t0 = 2.5× 10−14 s, η = 10−3, (4.48)

where t0 is calculated from the reported initial spatial size of the pulse at its productionx0 ' 10−5 m and the RIP velocity. Note that, up to a small perturbation, τ(t) = t/n2

0


which we use inside the sech2(·) function to maintain the integrability of the equationof motion. This effectively changes the above initial value of t0 to t0/n

20 = 1.2×10−14 s.

In principle these factors are all we need to derive our estimate of the peak fre-quency and flux. However, it is well known that the RIP shape is not invariant alongits propagation (see e.g. Ref. [107]). Indeed, there are two additional effects whichare certainly necessary to take into account when fitting the data. These are “pulsesteepening” and “pulse cresting” which we shall discuss below.

Pulse steepening

Pulse steepening is a non-linear optical effect, akin to wave shoaling in water waves,which can be responsible for altering the size of the spatial region over which the pulsegoes from its background value to its peak value and back by a factor of ten or more.This effect works oppositely on the trailing and leading edges of the RIP; while thetrailing edge becomes steeper the leading edge typically becomes shallower.

An important question is what, if any, physical mechanism is responsible for settingan ultimate limit to the pulse steepening. This point is not completely clear in theextant literature, although in many experimental situations one adopts the rule ofthumb that pulse steepening saturates at “about twice the carrier frequency” [107, 112].

We believe, however, that it is reasonable to expect that the limiting mechanismshould be more closely related to physical properties of the silica glass. Furthermore wewish here to explore which effects can provide absolute limits to the pulse steepeningand hence to the peak frequency predicted by our model. In this respect there aretwo obvious candidates. One is the plasma frequency ωp and the other is optical ab-sorption, known to be particularly effective near poles in the dispersion relation (herecharacterized in terms of the Sellmeier relation).

The plasma frequency definitely provides an ultimate limit to pulse steepening, asit is fundamentally the electrons in the optical glass which communicate the propa-gating electromagnetic wave, and which themselves interact on a time scale given bythe plasma frequency. From the Sellmeier relation, and the definition of the plasmafrequency as the coefficient of the leading order deviation as the refractive index goesto 1,

n2 → 1−ω2p

ω2= 1− λ2

λ2p

, (4.49)

one obtains

ωp =∑i

√Bi ω2

i ; λp =1√∑iBi/λ2

i

. (4.50)

For the suprasil glass used in the experiment one gets

ωp = 2.6× 1016 Hz, λp = 72.7 nm . (4.51)


With the above values one gets a typical timescale for the steepening of about 2π/ωp '2.4×10−16 s. This would be the ultimate limit to pulse steepening and we would expectthe maximum steepness to closely approach, if not exactly saturate, this bound.

The function sech2(t/t0) varies between its minimum and its maximum over a timescale of 2.4× 10−16 s when the parameter t0 is 2× 10−16 s.

The above estimate represents an absolute upper bound to the steepening. How-ever, one can argue that before the plasma frequency can start to play any role, pulsesteepening will be limited by dispersive effects and in particular by the pole of the Sell-meier relation at approximately 114 nm (see Eq. (4.4)). Such poles represent resonancesinside the silica glass and their role in limiting the time variation or propagating signalsis essentially identical to that of the plasma frequency. Nonetheless, if one takes, as weshall do, twice the time scale set by the plasma frequency t0 ' 4×10−16 s, the resultantsteepness is just below that set by the Sellmeier pole scale, exactly in the right ballparkto model a limiting scale set by the pole’s presence. Specifically the propagating RIPvaries between its maximum and minimum over a length scale of approximately 140nm when t0 ' 4× 10−16 s while the Sellmeier pole sits at 114 nm.

Let us here anticipate that to match the data our model requires that the pulsesteepening saturates near the above scale t0 rather than, say, at twice the carrierfrequency given set by the above-mentioned rule of thumb. This is a relevant point asit might be crucial in dismissing or supporting the present proposal against others.

Crest amplification

Another important effect which we expect to occur in a propagating and steepeningpulse is crest amplification, whereby the maximum amplitude of the pulse increases asthe RIP steepens. The cause of this effect can be understood by a conservation of energyargument. First, the integral of the intensity is constant (and given by the energy ofthe RIP). Second, due to the non-linear Kerr effect the intensity is proportional to theperturbation in the refractive index δn(x). Therefore∫

δn(x) dx = constant, (4.52)

so that

η

∫[shape of pulse](x) dx = constant. (4.53)

That is, if the pulse changes shape in such a way as to make the area under its normal-ized curve lower, we can expect the maximum amplitude, controlled by the parameter ηto increase. Note that the total flux of emitted quanta is (approximately) quadraticallydependent on η.


While in the current context we lack any detailed understanding of the size of thiseffect, we argue that one can easily expect η to increase up to one order of magnitude.We shall hence use η ' 10−2 in what follows.

In the closely related but distinct context of fibre-optic amplifiers, power amplifica-tions of 20 dB, 30 dB, and 40 dB are not uncommon [114, 115, 116]. These are poweramplifications of 102, 103, and 104 respectively, corresponding to amplitude increasesof 10, 32, and 100 respectively. So a cresting factor of up to 100 (corresponding toη ∼ 10−1) is not inconceivable.

Returning to the current context, the pulse steepening we have argued for, fromt0 ∼ 10−14 s to t0 ∼ 10−16 s, will (provided most of the pulse is concentrated in a narrowspike with a long shallow leading edge) squeeze the pulse temporally by up to a factorof 100, thereby potentially increasing the amplitude by a factor of up to 100. Again,this implies that a cresting factor of up to 100 (corresponding to η ∼ 10−1) is notinconceivable. Therefore, we argue that our choice of η ' 10−2 is rather conservative.It is important to note that crest amplification mainly affects the total flux, at O(η2),and that effects on the peak wavelength and width of the spectrum are relativelysubdominant, at O(η3).

Peak frequency and flux matching

We are now in shape for computing the salient features predicted by our model. Fromthe previous calculation of the peak frequency we get (using the least amount of pulsesteepening, that is the largest value of t0 in the above given interval)

λpeak =1

2π2ct0n0 ' 858 nm . (4.54)

This is in striking agreement with the observational data.

Note again, however, that to get this nice matching for the peak frequency it isessential that the pulse steepening occur on a timescale and distance scale close tothose set by the pole in the Sellmeier relation discussed above. In fact we have beengenerous and chosen a time scale slightly longer than the Sellmeier resonance timescale. This has the side effect of avoiding the resonance at 114 nm — which is the firstresonance one encounters in the Sellmeier approximation, (moving up in frequency fromthe carrier frequency as the pulse steepens). Furthermore if one adopts the numericalmodel discussed in [93], then Fig. 10 of that reference implies a timescale of ≈ 1 fs at thetrailing edge of the RIP; this is within a factor 21

2of our preferred value. In contrast,

if one simply uses some small multiple of the carrier frequency as an estimate of thepulse steepening, the fit is quite bad. It is therefore important to develop a clearerunderstanding of the amount of pulse steepening to be expected in this particularexperimental setup.


l meters0 5#10 - 7 1#10 - 6 1.5#10 - 6 2#10 - 6

d Nd l dVol

0

5#1020

1#1021

1.5#1021

2#1021

2.5#1021

Casimir spectrum as a function of wavelength

Figure 4.2: Emission spectrum for the dynamical Casimir effect, as given by Eq. (4.44)and using the parameter values discussed in the text: n0 = 1.45, t0 = 4× 10−16 s andη = 10−2. Recall that the total flux is given by the area under this curve multiplied bythe volume of the emitting region.

On the other hand the total flux becomes

N =8π3

21

η2W 2

c2t20n40

' 0.6 photons. (4.55)

This estimate should be supplemented by a factor two to take into account the twophoton polarizations and with an additional solid angle scaling Ω/(4π) to account forthe finite size detector. In the experiment the detector was a 5 cm diameter lens placed10 cm from the filament [112]. In terms of the half-angle subtended by the detectorone has Ω = 2π(1 − cos θ) whence Ω/4π ≈ 0.016. Unfortunately we do not have anaccurate account of the observed flux.

We consider the above numbers to be somewhat encouraging and supportive of thedynamical Casimir effect explanation. In Fig. 4.2 we show the emission spectrum forthe dynamical Casimir effect as a function of wavelength as given by the function F (·)in Eq. (4.44).

Peak shift and broadening

Of course, having shown that the relatively unsophisticated and crude model hereproposed can can get anywhere near reproducing the observed peak frequency, and


h0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

lpeak nm

860

861

862

863

864

865

866

867

868

Peak wavelength of emission as a function of RIP energy -- sech^2 profile

Figure 4.3: Shift of the location of the maxima of the emission spectrum as a functionof wavelength (in nm) for the sech2(·) profile as the input energy is increased, heremodeled by increasing the parameter η from 2.5× 10−3 to 10−1.

provide anywhere near a reasonable flux, while encouraging, is certainly not enough forclaiming to have explained the experiment under investigation. In particular, we havenot yet explained the above-mentioned shift of the peak frequency and broadening ofthe spectrum. We shall now argue that our model can account for the former, and weshall critically discuss the relevance of the latter.

Experimentally, working from Fig. 3 in Ref. [109], we can summarize the above twopoints with the following two empirical formulas:

λpeak ' 850 nm + 25 nm/mJ× E. (4.56)

σλ ' 100 nm/mJ× E. (4.57)

where σ is the full-width-at-half-maximum (FWHM) of the Gaussian fits and E is theinput beam energy.

Peak shift – From Eq. (4.22) we see that the peak wavenumber (and hence also thepeak wavelength of emission) depends on η when we move to higher orders in the ηexpansion of the exact result. In Fig. 4.3 we plot the maxima of the spectral functionF (λ) for a range of values of η between 0.25× 10−2 and 10−1. Note that this shiftingis consistent with the shift observed in the Belgiorno et al article [109]; increasing theinput energy shifts the peak emission wavelength to a higher value. We also note thatthe shifting is not exactly linear once we also take into account the η4 term in theexpansion of |βk|2.


From Fig. 3 of Ref. [109] one might argue that the peak wavelengths of emissionare approximately

λpeak ' 860 nm at I = 270 µJ, (4.58)

λpeak ' 875 nm at I = 1280 µJ. (4.59)

That is, the experiments observe a 15 nm shift in peak wavelength from a factor of 5increase of the input energy. From our result shown in Fig. 4.3 we see that our modelpredicts a spectral shift of about 10 nm from a factor of 10 increase of the input energy(given by η).

We note that with an increase of the input energy by a factor of 10, our modelpredicts an increase in flux by a factor of (approximately) 100. In the case of a factorof 5 increase, as in the experimental data, we would expect from our model a fluxincrease of a factor of (approximately) 25. This predicted behaviour is qualitativelyconsistent with the observed spectra shown in Fig. 3 of Ref. [109] where one sees anincrease in peak flux from approximately 2 photoelectron counts at the lowest inputenergy (270 µJ) to approximately 50 at the highest input energy (1280 µJ).

Peak broadening – Peak broadening, as we have mentioned above, is a featureobserved in the data by matching the measured spectrum to Gaussian curves by eye.It is interesting to note that although the magnitude of the effect does not seem tobe in agreement with the observational data, our oversimplified model also predicts aspectral broadening as the input energy (η) is increased. We use the FWHM definitionfor bandwidth, and find that σλ behaves as

σλ ' 767 nm for η = 10−3,

σλ ' 771 nm for η = 10−2,

σλ ' 787 nm for η = 10−1. (4.60)

As an increasing function of η, we see that the predicted broadening is in qualitative(though not quantitative) agreement with the data. The broadening predicted by oursimple model is shown in Fig. 4.4, where we plot the scaled spectrum:

Fη(λ)

[Fη0(λ)]peak

×(η0

η

)2

. (4.61)

That is, we plot the spectrum scaled to the maximum of the spectrum at η0 = 10−3,multiplied by an extra factor of (η0/η)2. This is done in order for the spectral curves tobe visually comparable, and to observe the non-quadratic dependence of the spectrumon η. Note that in this figure we have used an exaggerated range of values for theparameter η to highlight the effect. We also note that this figure shows the shiftingpeak wavelength of emission, where we see the peak moving to higher wavelengths as ηis increased. We again emphasize that the reported broadening in Fig. 3 of Ref. [109]


h =1

1000h =

1100

h =1

10

l meters2#10 - 7 4#10 - 7 6#10 - 7 8#10 - 7 1#10 - 6 1.2#10 - 6 1.4#10 - 6 1.6#10 - 6 1.8#10 - 6

Scaled Flux

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Scaled flux spectra as a function of wavelength

Figure 4.4: Normalized spectra Fη(λ) × (η0/η)2/[Fη0(λ)]peak for η = 10−3 (red solidcurve), η = 10−2 (green dashed curve) and η = 10−1 (bue dash dotted curve). Here F (·)is obtained from Eq. (4.44), but modified to include the higher order terms indicatedin Eq. (4.22). We normalize to the peak of the spectrum for η0 = 10−3, and display theresult as a function of wavelength in meters. By multiplying by (η0/η)2, we can see howthe shape and maximum of the spectrum as a function of η is not exactly quadratic inη, but in fact depends on the higher order terms in indicated in (4.22). A numericalroutine to find the FWHM bandwidths of these curves (which are independent of theamplitude) give the results shown in Eq. (4.60).

arises from matching noisy data to purely phenomenologically chosen Gaussian curves(there is no physics reason underlying the choice of Gaussian curves), and that thereported magnitudes might, in a slightly more conservative analysis, be lower thanthose reported.

4.2.4 The effect of a finite pulse width

In the model presented above we have been considering the ‘homogeneous’ approx-imation for the physical time dependent region. That is, although we work in the(stationary with respect to the lab) orthogonal 1+1 dimensional subspace to the prop-agating pulse, we assume the finite time dependent perturbation to be of infinite extentin the spatial direction. This kind of approximation does not effect the particle numberdensity in the active region, which is the number of particles divided by the ‘volume’of that region and this is the reason why we can still get accurate total flux results in


the homogeneous approximation by simply multiplying the homogeneous result for theflux density by the (finite) size of the active region (see sec 4.2.2 for this discussion). Amore accurate way of carrying out the calculation would be to assume from the start aspatially finite (and of course temporally finite) pulse, calculating the Bogoliubov coef-ficients directly in the inhomogeneous theory. In what follows in this short subsectionwe report on the results of carrying out this kind of finite size computation using aperturbative method.

The equation of motion we wish to work with here is

φ,tt − c2n2(t, x)φ,xx = 0, (4.62)

where cn is a time and spatially dependent wave speed. This kind of problem isinteresting mathematically since the ordinary methods such a Fourier transform andthe inner product need to modified in order to find, for example, the normal modeseven in the time independent asymptotic regions.

One simplifying feature of the RIP system that allows us to make quick progresswith minimal mathematical complications, however, is the fact that the time and spacedependence of the wave speed cn is a small perturbation on top of a homogeneousbackground

n2(t, x) = n20 + ηf 2(t, x); η 1. (4.63)

Hence we can calculate the first terms of the power series expansion of the Bogoliubovcoefficients βk in η using perturbation theory.

Assuming the perturbative ansatz that φ(t, x) = φ(0)(t, x) + ηφ(1)(t, x) + O(η2),inserting it into (4.62) and collecting terms in powers of η we find at zeroth order theequation of motion

φ(0),tt − c2n2

0φ(0),xx = 0. (4.64)

The boundary condition corresponding to the assumption that the quantum state of thefield is the vacuum state in the asymptotic past (only positive norm modes) demandsthe form of the solution5

φ(0)(t, x) =

∫ ∞−∞

dk√2π

eikx−iωkt

n0

√2ωk

, (4.65)

where ω2k = c2n2

0k2.

Collecting the first order terms we have

∂2t

(φ(0) + ηφ(1)

)− c2

(n0(k)2 + ηf 2

)∂2x

(φ(0) + ηφ(1)

)= 0, (4.66)

so that the equation of motion for the perturbation is

∂2t φ

(1) − c2n20∂

2xφ

(1) = c2f 2∂2xφ

(0) =: F (t, x), (4.67)

5This integral can be done explicitly and comes out to be proportional to φ(0) = (1 −i)/√

8n30c(x+ n0ct) + (1 + i)/√

8n30c(x− n0ct).


where we have defined the function F . Fourier transforming in the variable x we have

∂2t φ

(1)k + ω2

kφ(1)k = F (t, k). (4.68)

Since the boundary condition for the first order term is φ(1)(t) → 0 for t → −∞ (noperturbation before the RIP acts) it is appropriate to use the retarded Green function,giving the solution

φ(1)k (t) =

1

ωk

∫ t

−∞dt′F (t′, k) sin ωk(t− t′). (4.69)

Modeling the RIP by the Gaussian bump function

f 2(t, x) = e−x2/x20−t2/t20 , (4.70)

the function F is given by

F (t, x) = c2f 2∂2xφ

(0) = −e−x2/x20−t2/t20

∫dk√2πc2k2 eikx−iωkt

n0

√2ωk

, (4.71)

leading to the Fourier transformed object

F (t, k) =

∫dx√2π

e−ikx

[e−x

2/x20−t2/t20

∫dk√2πc2k2 eikx−iωkt

n0

√2ωk

]

= e−t2/t20

∫dk√2π

e−iωkt

n0

√2ωk

c2k2

[∫dx√2π

ei(k−k)xe−x2/x20

]= e−t

2/t20

∫dk√2π

e−iωkt

n0

√2ωk

c2k2[x0e−(k−k)2x20/4

], (4.72)

upon changing the order of integration.

Equation (4.68) with F as above in (4.72) is exactly soluble in terms of so-called‘parabolic cylinder functions’. In the appendix ?? we compute this solution and showthat the Bogoliubov coefficient βk up to O(η2) terms is

βk = βk(ξ) (4.73)

given by a function of k and the dimensionless ratio

ξ =x0

t0cn0

, (4.74)

which controls the finite size effects. When ξ is large then the finite size of the RIP doesnot effect the spectrum whereas for small ξ the size of the RIP starts to be comparablewith the characteristic wavelength of emitted quanta which is determined by t0. This


Figure 4.5: Here we show the deformation of the finite size spectrum, clockwise fromtop left, as the parameter ξ is decreased from a regime in which the emission spectrumdoes not feel the finite size effects (large ξ) into one in which the spectrum is stronglydistorted (small ξ). The respective values for ξ are 5, 2, 1, 1/4. Note that even at therelatively small value ξ = 5 the spectrum is already very similar to the homogeneousresult.

is shown in Figure 4.5 where we plot the exact spectrum derived from the homogeneouscomputation given in equation (4.21) against the first order perturbative result fromthe inhomogeneous computation here (4.73) for various values of the parameter ξ.

Using the physical parameters given above in Section 4.2.3 we find that for theexperimental results one has

ξ ' 47.89, (4.75)

putting us in the ‘large ξ’ regime where the spectrum is unaffected by the finite size.In Fig. 4.6 we compare the first order finite size result 4.73 using the accurate valueξ ' 47.89 with the first order homogeneous result 4.38 observing the almost identityof the spectra.

We conclude therefore that the finite size effects play little or no role in the inter-pretation of the experimental RIP results, that we are in a regime in which the timedependent system is naturally emitting particles whose wavelengths are much smallerthan the spatial size of the time dependent region. This is the regime in which the

4.3. Alternative models for the observed emission 133

Figure 4.6: The emission spectra results for the finite size calculation as given bythe function k2β2

k using the βk in Eq. C.43 and the homogeneous result in 4.38. Wenote that very close agreement of the two spectra at this level of perturbation theoryindicated by the almost perfect overlap between the two curves.

homogeneous approximation is a good one. This means that we can interpret all the re-sults we have obtained using the homogeneous approximation as accurate descriptionsof the true finite size physical scenario.

4.3 Alternative models for the observed emission

Recently in the literature several other works, apart from the one [97] which was thefocus of this chapter, have appeared which seek to explain the BF experimental results.We here report three important contributions to the ongoing debate briefly to highlightthe activity and interest of the community on the issues involved in the propagatingRIP type scenarios.

• Cosmological particle production by time dependent mass — The article in Refer-ence [117] chooses to adopt a DCE interpretation of the emitted radiation fromthe moving RIP, noting the difficulties of applying the Hawking mechanism tothe highly dispersive optical systems. The authors present two calculations, one


very similar to ours here but without making a detailed analysis including pulsesteepening or crest amplification while in the second model a time dependentmass is used to model the particle production. Both models conclude that aparticle emission is possible and that a Hawking mechanism is not necessary forthe interpretation.

• Anomalous Doppler effect — The article of Reference [99] discusses the sponta-neous emission from the vacuum of correlated pairs of photons due to a super-luminal RIP in a very similar scenario to the BF experiment in a way similarto the anomalous doppler effect where spontaneous emission occurs due to themovement of a rapidly moving source alone (see the article [99] itself and refer-ence therein). The authors find emission in the forward and backward directionswith respect to the pulse propagation. Although no direct attempt to interpretthe BF experimental results is made, the particular emission spectrum predictedby such a mechanism is discussed in order to potentially distinguish the signalfrom others due to different mechanisms.

• Dispersive effects and horizonless emission from stationary systems — In thetwo works in References [98, 111] the authors carry out firstly a kinematic, thensubsequently a dynamical study of the spontaneous emission spectrum due to thenon-linear optical dispersion alone in the moving frame of the RIP. The analysisis similar in spirit to the study of Hawking radiation in general in the presenceof non-linear dispersion (see for example [118]) and moving media. Here a higherorder equation of motion is used to model the non-linear dispersion and generalscattering processes investigated. Although the authors use a very simple stepfunction model for the shape of the RIP they conclude that an emission is possibleeven in situations in which no notion of horizon can be defined naturally and henceargue against a Hawking mechanism interpretation for the experimental resultsof BF.

This model can be understood as somehow complimentary to the one we havediscussed in the bulk of this chapter: in the pulse frame of [98, 111], the timedependence of the RIP system is negligible6 while the non-linear dispersion iscertainly not, being an effect related to the absorption and re-emission of thephotons by the atoms which are flowing past the RIP at near luminal speeds. In-stead, in our model, which is carried out in the lab frame, the effects of dispersionare negligible (as we discussed, due to the separation of scales between the onsetof non-linear dispersion and the peak frequency of emission due to the particulartime dependence of the passing RIP) while the time dependence is certainly not,being responsible for all the particle production effects. In this way we can seethat the physics result can be invariant under a change of the reference frame

6Note that even in the pulse frame, it is known that the shape of the RIP is not static but undergoesdeformation on a timescale similar to that which occurs in the lab frame [119]

4.4. Conclusions 135

(particle emission from the vacuum, potentially with the same spectrum) whilethe mechanism and particularities are quite different in the different frames. Itis possible that both pictures could be seen as two faces of the same underly-ing quantum vacuum emission (see the conclusion chapter 6.6 for more on thispossibility).

4.3.1 A possibly distinctive test: correlations in the radiation

Based on the results on quantum correlations of Chapter 3 and various works on thissubject in the extant literature on Hawking process [85, 67, 68, 66, 69], one couldargue that the spectrum and total particle flux are not clean enough observables todistinguish the various production mechanisms and the one should instead be lookingat quantum correlations in the observed photons.

It is known that the quantum state of the spontaneously emitted quanta of theDCE type emission is a squeezed state and one should expect a correlation signaturesimilar to that predicted in Chapter 3 for the BEC emission. However for a horizonof finite lifetime it is known that the spectrum is not exactly thermal [35, 120] andit is possible that the quantum state may possess different correlation functions andhigher order moments, which could lead to a divergence in the predictions for thesetwo mechanisms for the observed spectra.

Using correlation measurements should also differentiate the observed emission fromother, non-spontaneous effects (see [121] for more on this discussion) with an otherwisesimilar spectrum (e.g. thermal noise).

4.4 Conclusions

The model provided in this chapter is very simple, in many ways overly simple, butnevertheless provides a tolerable fit to many of the known aspects of the BF experi-mental situation. The dynamical Casimir effect envisaged in this model has a long andconvoluted history in its own right (see for instance [122, 123, 124, 34, 125, 121, 126]).Particularly attractive features of the dynamical Casimir model are that it does notneed any spectral window, that there is no preferred direction, and that in performingan experiment to observe it one could look from any angle.

In the present context, the most pressing aspect of the model is to more fully un-derstand peak broadening, and to undertake a detailed confrontation of the model withempirical reality. (Specifically, it would be important to develop precise quantitativeestimates of the amount of pulse steepening and crest amplification to be expected inthe relevant experimental context, perhaps involving exact solutions to the Maxwell


equations.) Another potential line of development would be to go to a full 3+1 di-mensional calculation. We have also undertaken a computation in perturbation theorywhich includes the finite size of the pulse in the 1+1 dimensional subspace of the de-tector (the width of the beam) but, again the results were essentially identical to thehomogeneous results presented here – the calculated spectrum was peaked at wave-lengths which were much smaller than the size of the emitting region by an order ofmagnitude.

From a theoretical perspective, as we have already mentioned above, one mightargue that the natural frame of reference for understanding this experimental result isthe RIP frame. Of course, in this frame the simple picture of a time dependent refractiveindex seems to be lacking, but one instead has to view the photons as being created bythe optical glass rapidly flowing through the RIP; with the quantum ground state ofthe electromagnetic field being defined by the rest frame of the optical glass, quantumexcitations are now generated by motion through an inhomogeneous medium. Modesendowed with modified dispersion relation will see a sequence of subluminal regionsof different intensity – the asymptotic past to the left, the intermediate region wherethe refractive index is increased (the RIP) and the asymptotic future region to theright. This possibility, that modes be excited from the vacuum in a time independentsystem solely through dispersion and a background, subsonic flow has been investigatedpartly in [118, 63, 127, 128, 129] even in absence of any horizon/blocking region forthe “dual” situation of modes subject to a supersonic dispersion relation in a “flow”with two supersonic regions of different speeds. Indeed there seems to be mathematicalcorrespondence that leads to the same behavior when passing from super- to sub-sonicin the dispersion relation and in the character of the flow regions [118]. This wasalso the topic of the two recent articles [98, 111]. If confirmed this analysis wouldcorroborate our claim that a particle production from the quantum vacuum could stilltake place in our system even in the absence of analogue horizons.

Finally, we emphasize that the dynamical Casimir effect is still quantum vacuumradiation, just not Hawking radiation. Both the dynamical Casimir effect and Hawkingradiation produce squeezed state outputs. Direct tests of the difference between thesetwo effects should focus on the different correlation structure in the emitted photons.Specifically, it would be interesting to consider 2-photon correlations, and back-to-backphotons (see e.g. [121, 67]). These effects are likely to be different in detail. One mightalso need to do considerably more than just looking at the squeezed state properties.At a very practical level, the output radiation might not be “clean”, due for instanceto interaction with silica defects.

Overall, we consider this RIP-based experimental scenario to be a very promisingavenue in the ongoing search for a direct observation of some analogue of quantumvacuum radiation in condensed matter systems, and we hope that the analysis herewill shed some light in this direction. In this context quantum vacuum radiation refersto any spontaneous emission from the vacuum and not specifically to the dynamical

4.4. Conclusions 137

Casimir effect nor the analogue Hawking effect. It is our opinion that in highly disper-sive systems such as that of a propagating RIP in optical glass many of the phenomenawhich are distinct in the relativistic context can be seen as limiting idealizations ofmore general quantum vacuum emissions.

Chapter 5

Superradiance in DispersiveTheories

In this chapter we will generalise the idea of superradiance to dispersive theories bothsub- and super-luminal. We will develop general formulas describing scattering pro-cesses which admit our dispersive notion of superradiance based on the conservation ofa generalised Wronskian for a class of higher order wave equations suitable to describewave scattering on flowing fluids. We will also describe some specific models and theirexact solutions therein as a way to get a tighter grip on the general case. We workwith the motivation that such dispersive superradiant scattering might be observablein doable analogue gravity experiments in the near future. Indeed in the followingchapter we will describe one such experimental effort currently in progress.

In the first section we make a general introduction and motivation for the chapter.The second section consists of a detailed analysis of the simplest case of zero backgroundflow (this terminology will be explained below) but is to be regarded as a warmupexercise for the following section which generalises to non-zero flows. The analysisof the non-zero flow scattering represents the bulk of the interesting results of thischapter. We finish in the fourth section with some concluding remarks and possiblefuture research directions.

5.1 Introduction

As described in Section 2.1.4, superradiance is a general phenomenon in physics ex-pressed in gravitational systems as superradiant scattering from a Kerr black hole aswell as the related particle phenomenon of the Penrose process. The Penrose processand superradiance can be understood as the two faces, particle-like and wave-like, of aquantum spontaneous superradiant emission.

140 Chapter 5. Superradiance in Dispersive Theories

Through the analogy with condensed matter systems, superradiant scattering hasthe possibility of being observed in the lab directly. Assuming one could produce inthe lab a flow profile in a fluid system such that the acoustic wave equation supportssuperradiant scattering then superradiance moves from a hypothetical to a feasiblyobservable phenomenon.

As we have seen many times in this thesis, typically condensed matter systemsexhibit a non-linear dispersion relation for the excitation spectrum. We have seen howsome systems with non-linear dispersion can be manipulated into a regime in whichthe excitations satisfy a linear dispersion and can be described by an acoustic metricand wave equation. Of course this is only an approximation to the true equation ofmotion in the real system and there will always be corrections to the linear dispersiveresult due to this approximation. Therefore it is of interest, as far as analogue gravityis concerned, to attain whether some phenomenon one is studying through the analogy,for example superradiance, is robust against modifications to the dispersion relation.This might be seen as the practical side of such a theoretical investigation.

Alternatively, one might be intrinsically interested in the theoretical question ofwhether a spacetime phenomenon, such as superradiance, is robust against modifica-tions of the dispersion, with the idea that deep in the UV spacetime might not be assmooth and Lorentz invariant as low energy physics would lead us to believe.

It is in this spirit that we move in the next section to the problem of superradiancewith modified dispersion.

5.2 Warmup problem - zero flow scattering

In this section we will make the first generalisation of superradiant scattering to non-linear dispersion. This involves two steps, the first of which is to introduce the languagein a non-dispersive context and secondly to generalise that language to non-lineardispersion. In a subsequent section we will generalise further to the case of non-zeroflow velocity (equivalently non-flat background geometry). This is a slight abuse ofterminology, however, since the zero flow case presented below can be understood (aswe will describe) as arising from higher dimensional acoustic geometry with a non-zeroflow in the extra dimension only as we shall show below. The meaning of zero flow inthis context is that the flow is zero in the scattering direction. For example, this wouldattain to radial scattering in a cylindrically symmetric system with zero radial flow.

This section will be very detailed since the same steps essentially apply to thenon-zero flows computations and we will skip them there in the interests of brevity.

5.2. Warmup problem - zero flow scattering 141

5.2.1 Non-dispersive preliminary

As we saw in 2.1.4 The equation of motion

d2f

dx2+ (ω − eV (x))2f = 0. (5.1)

is capable of modeling a superradiant scattering process when the function V has aparticular asymptotic structure

eV (x) =

0, x→ −∞,eV0, x→ +∞,

(5.2)

The asymptotically flat (constant potential) regions at ±∞ are necessary in order tohave well defined plane waves (equivalently ‘particles’) there and hence a well definedscattering problem.

By way of motivating the form of equation (5.1) we note that it can be thought ofas arising from the d’Alembert equation for the 2+1 dimensional geometry

ds2 = dt2 − (dx− vdt)2 (5.3)

and the assumption that only the second component V of v = (U, V ) be nonzero andthat this non-zero component V only depend on the first spatial variable x. This iswritten

∂2f

∂x2−(∂

∂t+ ieV (x)

)2

f = 0 (5.4)

after neglecting the effective mass term induced from the momentum in the y direction.

At first these assumptions on the flow might seem unnatural for describing super-radiance when we imagine that superradiant scattering should be occurring in the nearhorizon region around a rotating black hole. One might argue that ‘surely we needradial flow, and, surely there is no asymptotically flat region at the horizon’. However,we saw in Section 2.1.4 that, by a suitable change of coordinates and variables, thehorizon region around a black hole can be regarded as asymptotically flat in the abovesense, being pushed out to −∞ and allowing for a plane wave decomposition there.The ‘radial flow’ is zero and the φ independent rotational flow is precisely the V (x)in (5.4). Of course, (5.1) can be studied in its own right without embedding it intoan analogue framework but we find it useful to keep in mind where such an equationcomes from.

For the equation of motion (5.1) the reflection coefficient is related to the transmis-sion coefficient by

|R|2 = 1− ω − eV0

ω|T |2 (5.5)


and consequently that |R|2 > 1 is possible for sufficiently low frequencies ω < eV0.

The way we derived relation (5.5) was through the conserved Wronskian associatedwith (5.1). The action of the Wronskian functional

W [f ] := f ′∗f − f ∗f ′, (5.6)

where prime ′ denotes derivative with respect to x, is independent of x whenever fsatisfies the equation of motion (5.1) as can be easily demonstrated. When we chooseas an exact solution the ‘scattering solution’

f =

eikinx +Reikrx x→ −∞T eiktx x→ +∞

(5.7)

the conservation of W [f ] from −∞ to +∞ leads exactly to relation (5.5) independentlyof the form of the potential V (x) at finite x.

The allowed wave-numbers and those chosen for the scattering solution can beconcisely understood using a dispersion diagram as shown in Fig.5.1. We note thatthe scattering solution chooses only a subset of the set of all possible wave-numbersallowed by the dispersion relation

k2 =

ω2 x→ −∞(ω − eV0)2 x→ +∞

. (5.8)

in such a way that the solution can be interpreted as a scattering process: A localisedwave packet with momentum centered around kin traveling from −∞ converting intoa reflected wavepacket with momentum centered around kr which travels back to −∞and a transmitted wavepacket with momentum centered around kt which travels onto +∞. The direction of motion is given by the group velocity dω/dk of the packetand the momenta are determined by the dispersion relation. Explicitly solving thedispersion relation we have

kin = ω

kr = −ωkt = ω − eV0. (5.9)

Conserved inner product, normalisation

Associated with the 1+1 equation of motion (5.4) is the bilinear form

(f, g) = i

∫dx [f ∗ (∂t + ieV ) g − g (∂t − ieV ) f ∗] (5.10)


Figure 5.1: The linear dispersion relation (5.8) and the effective frequency Ω(x) shownin the two asymptotic regions, the green dash-dotted line at −∞ and the blue dashedline at +∞. We have shown the case where ω < eV0, specifically ω = 0.4 and eV0 = 0.7.Note that the intersections of the blue dashed line (the effective frequency at +∞) liebelow the k axis, allowing for a negative norm transmitted mode and consequentlysuperradiant scattering back to −∞.

which is conserved in time when f and g are solutions to (5.4). The norm of awavepacket (f, f) is conserved in time even after it splits into two oppositely mov-ing packets. A normalised fixed positive frequency wavepacket f at −∞ propagates tothe right where we can compute the norm to be

(f, f) =ω − eV (x)

ω. (5.11)

From this we understand such a wave packet to possess a negative norm in regionswhere ω < eV (x)1. Hence, when the initial wavepacket splits into two, the transmittedpart possesses negative norm necessarily implying that the reflected part has norm(which is equal to |R|2) larger than one.

We therefore understand that superradiance is tied to the existence of negative normmodes which, in this case, is tied to the sign of the effective frequency ω − eV0 in thetransmitted asymptotic region.

1Assuming the potential is such that a WKB approximation is valid for the packet and the notionof a spatially dependent wavenumber makes sense.


We also learn from (5.11) how to correctly normalise modes so that they form anorthonormal basis. In each asymptotic region the orthonormal modes making up theHilbert space of states are given by

eikx√|Ω|

=

eikx√|ω|, x = −∞

eikx√|ω−eV0|

, x = +∞, (5.12)

with Ω evaluated in the appropriate asymptotic region. The correctly normalised scat-tering solution therefore reads

f =

eikinx√

ω+ Reikrx√

ωx→ −∞

T eiktx√|ω−eV0|

x→ +∞. (5.13)

The effect of correctly normalising the modes in the scattering solution can equivalentlybe achieved by simply multiplying all transmission amplitudes obtained from the un-normalised scattering problem by a factor:

T normalised =

√|ω − eV0|

ωT unnormalised. (5.14)

This quantitative factor does not affect out qualitative results presented below which wecary out with the un-normalised solution, except when we discuss exact solutions andcan be safely ignored until that point. We choose to work in the un-normalised variablesfor the qualitative treatment since it renders the formulas and results more transparentuntil quantitative results are required at which point we append the relevant factors.

Exact solution

In what follows we will occasionally be interested in some exact solutions to the scatter-ing problem when higher derivative operators are appended to the equation of motion.In such cases exact solutions are very difficult to come by in the literature due tothe relative disinterest in higher order ODEs compared with traditional second orderequations of motion. Therefore we will find it convenient to make use of idealisedstepfunction potentials for which the exact solution follows directly upon imposingcontinuity conditions at the step and solving a linear equation. However, the use ofstep-functions introduces unphysical poles in the scattering amplitudes, as we will seeshortly in the non-dispersive case. These poles will also be found in the higher ordersolutions obtained with step-functions. Here in this short passage we show that thesepoles are indeed artifacts of the infinite slope in the non-dispersive case at the stepby solving exactly the scattering problem using smooth potentials and comparing the


scattering amplitudes. We argue similarly that the poles in the higher order waveequations can be cured analogously by using smooth potentials.

Let eV (x) = eV0Θ(x), where Θ is the Heaviside function, and f solving (5.1) begiven by (5.13). Then imposing the boundary conditions

limx→0+f(x) = limx→0−f(x) (5.15)

limx→0+f′(x) = limx→0−f

′(x) (5.16)

allows us to solve for (correctly normalised) R and T as

R =kin − ktkin + kt

T =

√eV0 − ω

ω

2kin

kin + kt(5.17)

which are singular when kin + kt = 0 which implies from (5.9) ω = eV0/2.

The equation of motion (5.1) is solvable exactly when eV (x) takes the form of asmooth tanh interpolation

eV (x) =eV0

2(1 + tanh (x/xp)) . (5.18)

where xp is a ‘thickness’ parameter of the interpolation. The squared transmissioncoefficient can be shown to be given exactly for this potential by

|T |2 =sinh(πωxp) sinh(π(eV0 − ω)xp)

sinh2(πxp(2ω − eV0))2 + cos2(π/2√

1− eV 20 x

2p

) . (5.19)

In Figure 5.2 we show the resolution of the singularities of the solution (5.17) bythe smooth model for various values of the ‘thickness’ parameter xp. We discuss thissmooth model and its (non-trivial) construction more completely in Appendix B.1.

5.2.2 Introducing dispersion

In real, ‘dirty’ physical systems the dispersion relation is often highly non-trivial as wehave seen in previous chapters on BEC and optical systems and in Section 2.5. In thischapter we will be concerned with polynomial modifications to the dispersion relation,which can be understood as low wavenumber approximations to a complicated physicaldispersion relations as a Taylor series in k gives a low k approximation to an arbitrarycontinuous function of k.


(a) (b)

(c)

‘(d)

Figure 5.2: The squared reflection and transmission coefficients for the stepfunctionmodel Rlin and Tlin compared with the squared reflection and transmission coefficientsfor the smooth model R and T for various values of the thickness parameter xp (see(5.18)) 5.2(a) xp = 2, 5.2(b) xp = 1, 5.2(c) xp = 1/2, 5.2(d) xp = 1/4 — note in 5.2(d)the smoothed result is not actually divergent, being regularised outside the plottedwindow. In all plots we take eV0 = 1 and plot up to ω = eV0 from where the scatteringis no longer superradiant.


Inspired by the quartic dispersion relation Ω2 = k2±k4/Λ2, where Λ is a dispersivemomentum scale and the ± notates super- and sub-luminal dispersion respectively, wepropose the following generalization of equation (5.1),

∓ 1

Λ2f ′′′′ + f ′′ + (ω − eΦ(x))2 f = 0, (5.20)

where f represents the fixed frequency mode after separation of the time dependencefrom the appropriate time dependent wave equation

∓ 1

Λ2f ′′′′ + f ′′ −

(d

dt− ieΦ(x)

)2

f = 0, (5.21)

and prime denotes derivative with respect to x. We choose the effective frequencyΩ(x) = ω − eΦ(x) to satisfy asymptotic relations similar to that in equation (5.2),

Ω2(x) =

ω2, for x→ −∞,(ω − eΦ0)2 , for x→ +∞,

(5.22)

where ω > 0 is understood as the conserved frequency of the mode and eΦ0 is apositive constant. In the asymptotic regions, the solutions of equation (5.20) are simpleexponentials exp (ikx) whose wavenumbers k satisfy the dispersion relations

∓ k4/Λ2 − k2 + ω2 = 0, x→ −∞, (5.23)

∓ k4/Λ2 − k2 + (ω − eΦ0)2 = 0, x→ +∞. (5.24)

In the subluminal case (lower sign), real solutions (ω, k) to the dispersion rela-tion correspond to the intersections of a lemniscate (figure-eight) and a straight line(see Figure. (5.3)) while in the superluminal case (upper sign) they correspond to theintersections of the quartic curve and a straight line, as shown in Figure (5.4).

In order to relate the asymptotic solutions at +∞ and −∞ without having to solvethe differential equation for all values of x, one needs a conserved quantity analogousto the Wronskian (5.6) for second order wave equations. Since our model is non-dissipative, we expect such a quantity to exist [130].

Conserved Wronskian, inner product

After some trivial algebra, it is possible to show that the quantity

Z[f ] = W1 +W2 ∓ Λ2W3, (5.25)

where

W1 = f ′′′∗f − f ∗f ′′′,W2 = f ′∗f ′′ − f ′′∗f ′, (5.26)

W3 = f ′∗f − f ∗f ′,


Figure 5.3: The dispersion relation for the subluminal case shown in terms of thedimensionless variables k/Λ and Ω/Λ where Ω = ω − eΦ0. The green dash-dotted linerepresents ω when x→ −∞ while the blue dashed line corresponds to the region x→∞. Here ω < eΦ0 is the fixed lab frequency satisfying the condition for superradiance.We have indicated in boldface those modes which are included in the scattering solution(5.36) kc = kin, kb = kr1 , kd = kr2 , kB = kt1 and kD = kt2 .

is conserved in space i.e. dZ/dx = 0 for any solution f of equation (5.20). However, itwill be more convenient to work with the scaled functional X defined by

X[f ] =Z[f ]

2iΛ2. (5.27)

This is because the action of X on a linear combination of ‘on shell’ plane waves(wavenumbers satisfying the dispersion relation) is very simple, given by

X

[∑n

Aneiknx

]=∑n

ΩdΩ

dkn|An|2. (5.28)

Th inner product in the dispersive theory is identical [30, 131] to the non-dispersiveone defined in (5.10). It is conserved when acting on solutions to equation (5.21) ascan be easily verified.


Figure 5.4: The dispersion relation for the superluminal case shown in terms of thedimensionless variables k/Λ and Ω/Λ. The blue dashed curve corresponds to the dis-persion relation at x → +∞ while the green dash-dotted curve corresponds to thedispersion at x→ −∞. We have chosen here a frequency in the superradiant intervalω < eΦ0.

Again, for wave-packets φ and ψ normalised at −∞, respectively peaked aroundthe single momenta k and k′, with fixed frequency ω and localised in a region whereΦ(x) is approximately constant, one can show that

(φ, ψ) =ω − eΦ(x)

ωδ(k − k′), (5.29)

independently of the dispersion. Therefore, modes for which ω − eΦ0 < 0 possessnegative norm at +∞.

Again we se that the correct normalisation of plane waves to render them orthonor-mal in their asymptotic region the effective frequency:

orthonormal mode =eikx√|Ω|

(5.30)

As already mentioned we remain with the un-normalised modes unless otherwise stated.


Quartic polynomials – In fact, the action of X in (5.28) can be shown to be

X

[∑i

Aieikix

]=∑i

|Ai|2ΩdΩ

dki+ “terms”, (5.31)

where “terms” are all proportional to

(ki + kj)(Λ2 − k2

i − k2j ), (5.32)

for pairs of roots. These terms vanish either when ki = −kj or when they sum toΛ2.

In general the roots of the polynomial z4 + a2z2 + a1z+ a0, satisfy the relations∑

n zn = 0 and∑

n z2n = −2a2 known as Vieta’s formulas.

The dispersion relation in the asymptotic regions considered above is

ω2 = k2 − k4

Λ2, (x→ −∞), (5.33)

(ω − eΦ0)2 = k2 − k4

Λ2, (x→ +∞). (5.34)

Hence the roots of the dispersion relation satisfy∑n

k2n = 2Λ2,

∑n

kn = 0, (5.35)

in both asymptotic regions. In this zero flow case, the roots come in pairs ofopposite sign. Either the two roots in any cross term are of opposite sign and thefirst factor of (5.32) vanishes or they are not and contribute half to the sum ofsquares i.e: they square sum to Λ2 and the second factor of (5.32) vanishes. Hence“terms” all vanish.

5.2.3 Subluminal scattering

Consider firstly the subluminal dispersion relation. In realistic scenarios, we do notexpect the dispersion relation ω(k) to have a maximum/minimum value above/belowwhich only imaginary solutions of the dispersion relation are possible2. Therefore,in order to guarantee that at least one real mode is available for the scattering, weassume that the dispersion parameter Λ is large compared to the interaction potentialeΦ0, i.e. we assume that eΦ0 < Λ/2.

2A notable exception to this expectation is the optical dispersion described in Section 2.4


Then the scattering process is captured succinctly in Figure 5.3 where the the green(Ω = ω) and blue (Ω = ω− eΦ0) lines represent the effective frequency when x→ −∞and x→ +∞, respectively. Note that there exist exactly four roots, corresponding tofour propagating degrees of freedom as one would expect from a fourth order differentialequation. We also note that, when ω < eΦ0, the blue dashed line is located below theΩ = 0 axis and the roots in the region x → +∞ inherit a relative sign betweentheir group and phase velocities with respect to the roots at x → +∞ (comparethe intersections of the green dash-dotted line (x → −∞) and the blue dashed line(x→ +∞) with the red curve).

We label the four roots associated with a frequency 0 < ω < eΦ0 as ka < kb < kc <kd when x→ −∞ and as kA < kB < kC < kD when x→ +∞ (they correspond to theintersections of the straight lines with the figure-eight in Figure. (5.3)). The followingtable summarises the character of these modes, where u and v notate right-movers andleft-movers respectively and blue boldface indicates those modes which are included inthe scattering solution.

Roots at x →−∞

a b c d

group velocity u v u vphase velocity v v u unorm + + + +

Roots at x →+∞

A B C D

group velocity v u v uphase velocity v v u unorm - - - -

We set up our scattering experiment in the standard way so that all incident wavesoriginate from x→ −∞, i.e. there is only one source of waves, located at x→ −∞ andno incoming signal from x→ +∞. Therefore, we impose the boundary condition thatthe modes A and C are unpopulated in the interaction region since they correspondto left-moving modes at x → +∞. Furthermore, we choose the incoming mode atx→ −∞ to be entirely composed of low-momentum c modes with no high-momentuma mode component. Hence there are only 5 distinct roots which are relevant for thescattering process (indicated in boldface in Figure 5.3) and we relabel them as kin := kc,the reflected modes kr1 := kb, kr2 := kd, and the transmitted modes kt1 := kB, kt2 := kD.

The scattering solution of equation (5.20) in the asymptotic limits is, therefore,given by

f →

eikinx +R1eikr1x +R2eikr2x, x→ −∞,T1eikt1x + T2eikt2x, x→ +∞

. (5.36)

In the asymptotic regions, it is possible to solve exactly the dispersion relation and


find the following explicit expressions for the roots,

kin = −kr1 =Λ√2

√1−

√1− 4ω2

Λ2,

kr2 =Λ√2

√1 +

√1− 4ω2

Λ2, (5.37)

kt1 = − Λ√2

√1−

√1− 4(ω − eΦ0)2

Λ2,

kt2 =Λ√2

√1 +

√1− 4(ω − eΦ0)2

Λ2.

Substituting equation (5.36) into the expression for the conserved generalised Wron-skian, equation (5.27), we find, after straightforward algebraic manipulation, the fol-lowing relation between the coefficients R1, R2, T1 and T2,

|R1|2 +

∣∣∣∣kr2kin∣∣∣∣ |R2|2 = 1 +

√Λ2 − 4(ω − eΦ0)2

Λ2 − 4ω2

(∣∣∣∣kt1kin∣∣∣∣ |T1|2 +

∣∣∣∣kt2kin∣∣∣∣ |T2|2

)> 1. (5.38)

This relation should be compared to the standard result for non-dispersive 1D scattering|R|2 = 1 − |T |2 and its generalization in the presence of an external potential |R|2 =1− (ω − eΦ0)|T |2/ω, as in equation (5.5).

As discussed above, in the linear dispersion case it is possible to achieve |R| > 1 forsufficiently low frequency scattering ω < eΦ0. For subluminal dispersion, the conclusionis similar. From expression (5.38) above, which is valid only when 0 < ω < eΦ0 < Λ/2,we conclude that the total reflection coefficient (i.e. the left hand side of equation (5.38))is always greater than one, characterizing superradiance.

Note that, in the general case without an exact solution we have no informationabout how this total reflection is distributed between the low and high wavenumberchannels represented by R1 and R2 respectively. However, by looking at the Λ seriesexpansion of the reflected and transmitted wavenumbers, we can draw some interestingconclusions about the regime Λ 1.

From equation (5.37), we obtain the following expansions for the relevant wavenum-


bers,

kin = −kr1 = ω +ω3

2Λ2+O(Λ−3),

kr2 = Λ− ω2

2Λ+O

(Λ−2

), (5.39)

kt1 = ω − eΦ0 +(ω − eΦ0)3

2Λ2+O(Λ−3),

kt2 = Λ− (ω − eΦ0)2

2Λ+O

(Λ−2

).

All possible modes of the system have the same energy, which is determined by theconserved lab frequency ω. The momentum, on the other hand, is determined by thewavenumber k and, therefore, is not the same for every mode.

From the expansions above, we note that kt1 , kr2 ∼ O(Λ) and kin, kr1 , kt1 ∼ O(Λ0).In particular, the difference between the high momentum modes kr2 and kt2 is smallkr2 − kt2 ∼ O (Λ−1). Intuitively, this means that the creation of a pair of these modes(kr2 , kt2) requires a negligible momentum change in the system even when Λ 1.Consequently, we expect mode production in these high momentum channels to be non-negligible. Moreover, regarding the low momentum channels, incident modes kin arebeing converted into kr1 modes and kt1 modes in the system, and should be unaffectedwhen Λ 1.

In the regime Λ 1, we therefore anticipate that the scattering consists of twoseparate kinds of balance, high and low momentum, in such a way that the Wronskianrelation (5.38) becomes equivalent to two different conservation relations, namely

|R1|2 ' 1 +

√Λ2 − 4(ω − eΦ0)2

Λ2 − 4ω2

∣∣∣∣kt1kin∣∣∣∣ |T1|2 > 1, (5.40)

for the low-momentum modes and∣∣∣∣kr2kin∣∣∣∣ |R2|2 '

√Λ2 − 4(ω − eΦ0)2

Λ2 − 4ω2

∣∣∣∣kt2kin∣∣∣∣ |T2|2, (5.41)

for the high-momentum modes. Note that, outside the regime Λ 1, the expressionscan not be expected to remain valid independently. In such a case, only the originalWronskian relation (5.38) is satisfied and we expect a mixing between all availablechannels.

This kind of separation of the Wronskian is confirmed below by using a stepfunctionmodel for eΦ(x) to compute the exact solution.


Exact step-function solution

We can lend weight to the argument for the decoupling of high and low momentumchannels by solving exactly the idealised problem in which the potential is chosen to bea step function, i.e. V (x) = eΦ0Θ(x). The boundary condition at the step is such thatthe solution and it first 3 derivatives are continuous there. Then the un-normalisedsolution is

R1 =(kin − kr2)(kin − kt1)(kin − kt2)(kr2 − kr1)(kr1 − kt1)(kr1 − kt2)

,

R2 =(kin − kr1)(kin − kt1)(kin − kt2)(kr1 − kr2)(kr2 − kt1)(kr2 − kt2)

, (5.42)

T1 =(kin − kr1)(kin − kr2)(kin − kt2)(kr1 − kt1)(kr2 − kt1)(kt1 − kt2)

,

T2 =(kin − kr1)(kin − kr2)(kin − kt1)(kr1 − kt2)(kr2 − kt2)(kt2 − kt1)

.

Note the poles at exactly the same frequency as in the non-dispersive solutions (5.17)which we assume are resolved in a smooth model3 (from Figure 5.3 we see that kt2 = kr2at the same frequency that kt1 = kr1 , that is, at ω = eΦ0/2).

Using the exact expressions (5.37), we can compute the action of the functional Xon each mode and its coefficient appearing in the scattering solution4 (5.36) as

X[eikinx

]= ω +O

(Λ−1

)X[R1 eikr1x

]=

(eΦ)2ω

(2ω − eΦ)2+O

(Λ−1

)X[R2 eikr2x

]= −16 ω2(Λ− 2ω − 2eΦ0)

(2ω − eΦ)2+O

(Λ−1

)X[T1 eikt1x

]=

4(ω − eΦ0)ω2

(2ω − eΦ)2+O

(Λ−1

)(5.44)

X[T2 eikt2x

]= −16 ω2(Λ− 2ω − 2eΦ0)

(2ω − eΦ)2+O

(Λ−1

).

3We have not shown that the poles are resolved in the dispersive theory since an exact smoothsolution is only available in the non-dispersive theory but believe they should be nevertheless beresolved in this way.

4Note that, by (5.14) we have

X[T normalisede−ikx/

√Ω]

= X[T normalisedeikx/

√ω]

= X[T normalisedeikx

]/√ω (5.43)

so that we can use both the unnormalised solution and unnormalised coefficients to get the same resultas the properly normalised solution and coefficients.


(a) (b)

Figure 5.5: The squared reflection and transmission coefficients for the low momentumchannels |R1|2 and |T1|2 for the subluminally dispersive solutions to the stepfunctionmodel for eΦ0 = 1 as computed from (5.42) compared with the reflection and transmis-sion coefficients computed from the the non-dispersive stepfunction model where theseare the only scattering channels. In 5.5(a) the results are shown for Λ = 5 (significantdispersion) while in 5.5(b) we show the results for Λ = 50 (small deviation from lineardispersion).

Hence the action of the functional X on the two channels kr2 and kt2 is identical upto terms of O(1/Λ) meaning that these contributions cancel from equation (5.38) andhence equation (5.40) is satisfied up to O(1/Λ), implying superradiance in the lowmomentum channel alone. This also implies that (5.41) is satisfied up to O(1/Λ).Note that this result also shows that the non-dispersive scattering process is recoveredin the non-dispersive Λ→∞ limit.

In Figure 5.5 we plot the low momentum channel squared reflection and transmissioncoefficients for two different regimes, significant (small Λ) and slight (large Λ) dispersionand compare them with the result from the linear dispersive case. In Figure 5.6 onthe other hand we plot only the high momentum channel coefficients |R2|2 and |T2|2including as a reference the low momentum |R1|2 coefficient.

These plots provide us with an interesting result since they show that, using stepfunctions, the population of the new channels due to dispersion, kr2 and kt2 , do not goto zero when the dispersion is slight, confirming the expectations we had above abouta momentum balance favouring simultaneous production of kr2 and kt2 modes. In fact,each of these high momentum channels contributes a universally large amount which


(a) (b)

Figure 5.6: The reflection and transmission coefficients |R2|2 and |T2|2 for the dispersivestepfunction model for eΦ0 = 1 as computed from (5.42) in 5.6(a) for Λ = 4 while in5.6(b) for Λ = 50. In both cases we include for reference also the coefficient |R1|2,showing that the high momentum channels R2 and T2 are less occupied than the lowmomentum channels for low frequencies but much more populated for high frequencyup to the superradiance bound ω = eΦ0.

cancels out from the total reflection coefficient (5.38) in the limit of large Λ.

Another very interesting observation is that the presence of the additional channelscorresponding to the high momenta kr2 and kt2 in fact act to increase the total reflectionreceived back at −∞ even though we only send in one low momentum kin incident wave.This is shown in Figure 5.7 where we plot the result from linear dispersion |Rlin|2 alongwith two objects which one might consider representative of the total reflection: theparticle current given by the left-hand side of (5.38) |R1|2 + |kr2/kin||R2|2 in one subfigure and the total power of reflection given sums of square amplitudes |R1|2 + |R2|2.We plot these quantities for the generic values of eΦ0 = 1 and Λ = 2, noting that thecoefficient R2 unphysically grows without bound as Λ → +∞ further enhancing thisresult (perhaps unphysically) at higher Λ.

5.2.4 Superluminal scattering

Let us now turn our attention to the case of superluminal dispersion. Once again, wecan understand much of the scattering process from the dispersion diagram, shown inFigure (5.4). The main difference with the subluminal case is that now there are only


(a) (b)

Figure 5.7: The total reflected current (5.7(a)) and total reflected power (5.7(b)) forΛ = 2 and eΦ0 = 1 plotted against the squared reflection coefficient of the lineardispersive solution. Note that the presence of the additional channels acts to increasethe observed reflection at −∞, improving the magnitude of superradiance from thenon-dispersive result.

two real roots in either asymptotic region, corresponding to one left-moving and oneright-moving mode. The other two solutions to the fourth order equation are imaginaryroots, corresponding to exponentially decaying and exponentially growing modes.

Similarly to the subluminal case, we are interested in the scattering of an incidentlow-frequency (ω < eΦ0) wave originating from x→ −∞, which is converted into a re-flected left-moving mode, a transmitted right-moving mode and exponentially decayingmodes (as a boundary condition, we impose the fact that there can be no exponentiallygrowing modes in the asymptotic regions). Therefore, the solution of equation (5.20)corresponding to this scattering process is given by,

f →

eikinx +Reikrx + Ere

kerx, x→ −∞,T eiktx + Ete

−ketx, x→ +∞,(5.45)

where the coefficients R and T are related, respectively, to reflection and transmis-sion coefficients (see below), and the coefficients Er and Et are the coefficients of theexponential modes.

The wavenumbers k in equation (5.45) are obtained directly from the dispersion


relations, equations (5.23) and (5.24),

kin = −kr =Λ√2

√−1 +

√1 +

4ω2

Λ2,

ker =Λ√2

√1 +

√1 +

4ω2

Λ2, (5.46)

kt = − Λ√2

√−1 +

√1 +

4(ω − eΦ0)2

Λ2,

ket =Λ√2

√1 +

√1 +

4(ω − eΦ0)2

Λ2.

A conservation equation can be obtained by plugging equation (5.45) into equa-tion (5.28) and equating the generalised Wronskian at ±∞. One might expect a dis-tinction with the standard result due to the extra two exponentially decaying modesin the solution. However, it turns out that these extra contributions to the generalisedWronskian are also exponentially decaying and one obtains the following reflectioncoefficient (valid for 0 < ω < eΦ0),

|R|2 = 1 +

√Λ2 + 4(ω − eΦ0)2

Λ2 + 4ω2

∣∣∣∣ ktkin

∣∣∣∣ |T |2 > 1, (5.47)

which, similarly to the non-dispersive case, is always greater than one. Note also thatthe expression above reduces to the usual non-dispersive reflection coefficient (5.5) inthe limit Λ→∞.

In Figure 5.8 we plot the exact reflection and transmission coefficients for the step-function model and the non-dispersive result (the algebraic result (5.17) applies toboth scattering problems since the solution (5.17) was obtained independently of thescattering momenta) showing the agreement with the non-dispersive result.

5.3 Non-zero flow (or, curved geometry)

In this section we make the second generalisation anticipated in the introduction, thatis, to non-zero flows in the direction of scattering. This is equivalent to a generalisationto a non-flat background geometry in the analogue language.

The equation of motion we will use can be understood to arise again from an acousticgeometry of the form

ds2 = dt2 − (dx− v(x)dt)2 (5.48)

5.3. Non-zero flow (or, curved geometry) 159

(a) (b)

Figure 5.8: The reflection and transmission coefficients for the low momentum channels|R|2 and |T |2 for the superluminally dispersive solutions to the stepfunction modelfor eΦ0 = 1 as computed from (5.42) compared with the reflection and transmissioncoefficients computed from the the non-dispersive stepfunction model where these arethe only scattering channels. In 5.8(a) the results are shown for Λ = 1 (significantdispersion) while in 5.8(b) we show the results for Λ = 5 (‘small’ deviation from lineardispersion).

but now with the first component W of the flow velocity v = (W,V ) non-zero. Westill requite that both W and V be only functions of one component x of the spatialcoordinate x = (x, y).

The non-dispersive d’Alembertian equation associated to (5.48) is[(∂

∂t− ipV − ∂

∂xW

)(∂

∂t− ipV −W ∂

∂x

)− p2 +

∂2

∂x2

]fω,p(x) = 0 (5.49)

where p is the momentum eigenvalue in the y direction.

Below we will work with a slightly modified equation of motion without the p2 termin the interests of simplicity as we did above. This additional term acts as a massfor modes propagating only in the x direction, where the scattering takes place, andlater on we will discuss the implications of including such a mass term which has beenrecently discussed in [132, 133]. Such a mass term is also considered in [30] in the non-dispersive, zero flow context where it was shown that the window of frequencies wheresuperradiance occurs is not only bounded from above by eΦ0 but also from below byp2 as we saw in 2.1.4.


5.3.1 Equation of motion, dispersion

Specifying the modified dispersion relation in the co-moving frame of the fluid as thequartic one, the equation of motion (5.20) generalises, through (5.49) to

∓ 1

Λ2

d4f

dx4+d2f

dx2+

(ω − eΦ(x) + i

d

dxW (x)

)(ω − eΦ(x) + iW (x)

d

dx

)f = 0. (5.50)

Similarly to the zero-flow case, it is useful to write the equation above in the form

f ′′′′(x) + α(x)f ′′(x) + β(x)f ′(x) + γ(x)f = 0, (5.51)

where

α = ±Λ2(1−W 2(x)

)β = ∓Λ2 [2W (x)∂xW (x)− 2iW (x) (ω − eΦ(x))] (5.52)

γ = ±Λ2[(ω − eΦ(x))2 + i∂x (W (x)(ω − eΦ(x)))

].

Observe that these coefficients are not independent but satisfy the following relations,

γ − γ∗ = 2iIm(γ) = iIm(∂xβ) = ∂x

(β − β∗

2

), (5.53)

Re(β) =1

2(β + β∗) = ∂xα. (5.54)

We choose the functions W (x) and Φ(x) to be asymptotically constant

W (x) =

0, x→ −∞,W0, x→ +∞,

(5.55)

eΦ(x) =

0, x→ −∞,eΦ0, x→ +∞,

(5.56)

so that, at ±∞, any solution to (5.50) can be decomposed into plane waves whichsatisfy the dispersion relation

k2 ± k4

Λ2=

ω2, x→ −∞,(ω − eΦ0 − kW0)2 , x→ +∞,

(5.57)

where the ± stands for super- and sub-luminal dispersion. At this stage we will notspecify whether W0 > 1 or < 1 corresponding to supersonic and subsonic asymptoticflow speeds respectively. As previously, we work under the assumption that ω, eΦ0 Λin order for the wavenumbers to be real.


5.3.2 The Wronskian and inner product reconsidered

The generalised Wronskian found previously, equation (5.25), is not conserved whenthe flow W is non-zero. However, one can show that a modification of it, namely

Z[f ] = W1 +W2 + αW3 − i Im(β)|f |2, (5.58)

and the corresponding scaled quantity X[f ] = Z[f ]/(2iΛ2) are independent of x whenf satisfies the equation of motion (5.50).

The action of the functional X on a linear combination of ‘on-shell’ plane wavestakes precisely the same form as in the zero-flow case,

X

[∑n

Aneiknx

]=∑n

|An|2Ωdω

dkn, (5.59)

but now with Ω = ω − eΦ − kW . In particular, this relation holds in the asymptoticregions at ±∞. In the zero-flow case, it was the quantity ω − eΦ0 which entered theright hand side of equation (5.59) in place of Ω and the existence of superradiance wasrelated to the condition that ω − eΦ0 < 0. We can anticipate that the condition ofsuperradiance in non-zero flows will be favoured by modes for which Ω < 0.

The generalization of the inner product (5.10) to non-zero flows is given by

(φ, ψ) = i

∫dx [φ∗ (∂t − ieΦ +W∂x)ψ − ψ (∂t + ieΦ +W∂x)φ

∗] ,

which is conserved for solutions of the time-dependent version of the equation of motion(5.50) as long as we choose the quartic modification to the dispersion5 as we showin Appendix B.2. It is clear, therefore, that the norm of a localised wavepacket φnormalised at −∞ (that is, with the free frequency, ω) of momentum k is proportionalto the factor Ω evaluated at its position,

(φ, φ) =ω − eΦ(x)− kW (x)

ωδ(0), (5.60)

in corroboration of our expectations above on the role of the sign of effective frequencyΩ being the litmus test for superradiance. That is, the negative norm modes are thosefor which Ω < 0.

Wronskian trick – In fact, again, the action of X (5.59) on the the linear com-bination of plane waves also contains some “cross terms” which can be shown tobe

“cross term”=

[Ω(ki)

dω

dki+ Ω(kj)

dω

dkj+

1

Λ2(ki − kj)2(ki + kj)

]Re(AiA

∗je

i(ki−kj)x)

5It suffices that we modify by an even power of k for this conclusion to hold.


spoiling the nice result (5.59). In the zero flow case, we were able to find algebraicrelations amongst the roots and use a symmetry argument to show that the crossterms all vanish. No such algebraic result is available in this case but we know thatsuch terms must vanish because of the independent fact that dX[f ]/dx = 0.

5.3.3 Subluminal scattering

Let us consider first the case of a subluminal dispersion relation. The inclusion of aflow velocity introduces many new features to the analysis which both complicate thecomputations but also allow a description of new phenomena and physical processes.

As shown in the dispersion diagram in Figure 5.9, for fixed eΦ0/Λ < W0 < 1, thereare two distinct intervals of frequencies separated by a critical frequency6 ωcrit < eΦ0 inwhich we expect superradiance: For 0 < ω < ωcrit (region I) two propagating degrees offreedom are admitted, both right-moving in the lab frame; for ωcrit < ω < eΦ0 (regionII) there are four real roots of the dispersion relation corresponding to four propagatingdegrees of freedom, three right-moving and one left-moving (with respect to the lab-frame). Note that the requirement that W0 is not too small, specifically W0 > eΦ0/Λ, isnecessary in order to guarantee that the left-most root in region I (i.e. the intersectionof an ω = constant line with the red dispersion curve in Figure 5.9) has positive groupvelocity and hence defines a true transmitted mode7. If 0 < W0 < eΦ0/Λ, the situationis qualitatively identical as the zero flow case discussed in section 5.2.3.

On the other hand, if W0 > 1 (see the blue-dashed curve in Figure. 5.9), there isonly one possible regime: For all frequencies 0 < ω < eΦ0, two right-moving modes areadmitted.

An interesting fact is that, due to the absence entirely of left-moving modes at +∞when ω < ωcrit, this system is a model for the event horizon of an analogue blackhole with modified dispersion relations. However, the precise location of the horizon,besides being ω-dependent, is also rather ill-defined, relying on a global solution tothe equation of motion in the vicinity of a classical turning point. This region can bestudied by WKB methods and Hamilton–Jacobi theory [134], but it is not of specificinterest to us here.

According to the analysis above, the scattering of an incoming wave from −∞ willresult in transmission through two or three channels, depending on whether ω < ωcrit

6In terms of the other parameters of the problem, ωcrit is given by

ωcrit ' −[

2

3(1−W0))

]3/2Λ + eΦ0. (5.61)

7See the Appendix for further discussion on this peculiarity of the quartic dispersive model as wellas a discussion on the role of the asymptotic flow W (x→ −∞).


Figure 5.9: The subluminal dispersion curve in the lab frame at x → +∞ for twodifferent flow velocities eΦ0/Λ < W0 < 1 (red solid line) and W0 > 1 (blue dashedcurve). The coloured regions described in the text are associated with the red curve:region I (green), ω < ωcrit, region II (blue) ωcrit < ω < eΦ0, region III (red), non-superradiantly scattering region. We have plotted the dispersion for eΦ0 = 0.4 andΛ = 1. In the non-interacting region, since W (x)→ 0 there, the dispersion is describedby the green dash-dotted curve of Figure 5.3.

or not. An exact solution to the scattering problem can be decomposed as

f =

eikinx +R1eikr1x +R2eikr2x, x→ −∞,T1eikt1x + T2eikt2x, x→ +∞,

(5.62)

when 0 < ω < ωcrit (if W0 < 1) or 0 < ω < eΦ0 (if W0 > 1) and as

f =

eikinx +R1eikr1x +R2eikr2x, x→ −∞,T1eikt1x + T2eikt2x + T3eikt3x, x→ +∞,

(5.63)


when ωcrit < ω < eΦ0 (only possible if W0 < 1)8. The wavenumbers kin, kr1 and kr2 aregiven by equation (5.37) and labeled in Figure. 5.3, while the transmitted wavenumbersare not, in general, expressible as simple functions of the parameters. Plugging thesolutions (5.62) and (5.63) into the functional X of equation (5.59), we find, aftersome algebraic manipulation, the following relationship between the transmission andreflection coefficients,

|R1|2 +

∣∣∣∣kr2kin∣∣∣∣ |R2|2 = 1− Λ√

Λ2 − 4ω2

(∑n

vgnkin

(ω − eΦ0 − ktnW0) |Tn|2), (5.64)

where the sum is over all (2 or 3, depending on ω and W0) transmission channels.Here, vgn = vg(ktn) are the group velocities of the transmitted modes at +∞. The lefthand side of equation (5.64) can be interpreted as the total reflection coefficient for theincident modes.

Since the group velocities vgn of the transmitted modes are, by definition, alwayspositive, the sign of the contribution from each channel ktn to the right hand side of(5.64) is determined by the factor Ω(k) = ω − eΦ0 − kW0 evaluated at ktn , as weanticipated previously. When only two transmission channels are admitted, one of thetwo factors Ω(k) is strictly negative while in the case of three transmission channels,two of the three factors Ω(k) are strictly negative. Therefore, in both situations thereis one root which contributes an overall negative amount to the right hand side ofequation (5.64) and thus reduces the magnitude of the total reflection. Because ofthese troublesome modes, the right hand side is not strictly greater than 1 and wecannot straightforwardly conclude superradiance. In general, in order to fully answerthe question of superradiance, one would need to specify W (x) and Φ(x) in all spaceand solve for the coefficients Rn and Tn which we will do later on using a stepfunctionmodel for W and eΦ.

Large Λ approximation: W0 < 1

To better understand the relation between equation (5.64) and superradiance, we nowfocus on small deviations (Λ 1) from the linear dispersion case. For fixed W0 < 1, thesize of region I in Figure. 5.9 is zero since as Λ increases local minima of the dispersioncurve shown drops below the horizontal axis. The value of Λ at which this occurs is

Λ =eΦ0[

23(1−W0)

]2/3 , (5.65)

which is large when W0 is close to 1(see the Section 5.3.3 for more details of ωcrit). Wetherefore assume that the frequency ω lies in region II, where three transmission and

8Again the properly normalised solution involves plane waves divided by a factor of√|Ω| as in the

zero-flow case, the difference being that now Ω is also a function of k. Hence when we come to exactsolutions we multiply the un-normalised transmission amplitudes by a factor of

√|Ω|/ω.


two reflection channels are available (the wave function is described by equation (5.63)).As explained above, in the most general case one would need to solve the equation ofmotion for all x in order to conclude superradiance or not from equation (5.64).

Similarly to the zero flow case, by looking at the series expansions of the relevantwavenumbers, we can make useful predictions about the transmission coefficients inthis Λ 1 regime.

We start the analysis by solving equation (5.57) in the asymptotic region x→ +∞and expressing the obtained transmitted wavenumbers as power series in Λ 9,

kt1 =ω − eΦ0

1 +W+

1

2

(ω − eΦ0)3

(1 +W )4Λ−2 +O(Λ−4), (5.66)

kt2,3 = ±Λ√

1−W 20 +W0

ω − eΦ0

1−W 20

+O(Λ−1

). (5.67)

The series expansions of the incident wavenumber kin and of the reflected wavenumbers,kr1 and kr2 , are given, as previously, by equation (5.39). From these Λ expansions, wenote that, unless W0 ∼ 1, we have kt2,3 , kr2 ∼ O(Λ) and kin, kr1 ∼ O(Λ0). However,unlike in the zero-flow case, the difference between the high momentum transmittedmodes kt2,3 and the high momentum reflected modes kr2 is not negligible for Λ 1,being of order O (Λ) due to the difference of W0 from zero. Hence, the appearance ofsuch modes in our system requires a large momentum change. Therefore, we expect theconversion of incident modes kin into transmitted modes kt2 and kt3 and into reflectedmodes kr2 to be disfavored in our system in comparison with the low momentumtransmission/reflection channel involving kt1 and kr1 . In other words, we expect (andwill give evidence for below) that the coefficients R2 and T2 will be much smallerthan R1 and T1, and that the Wronskian condition (5.64) will include only the lowmomentum channel, i.e.

|R1|2 = 1− ω − eΦ0

ω(1 +W0)|T1|2. (5.68)

From this relation for the reflection coefficient, we would conclude that superradi-ance occurs for all frequencies in region II (ωcrit < ω < eΦ0) when Λ 1. Note thatthis conclusion certainly does not hold in the case of a general dispersive parameter. Ifthe condition Λ 1 is not satisfied, there can be a mixture between the high and lowmomentum channels. Consequently, as discussed before, the total reflection coefficientgiven by equation (5.64) is not necessary larger than one. In such a case, a definiteanswer about superradiance can only be obtained by solving the differential equationsat every spatial point x. The results of doing exactly this for a simplified stepfunctionmodel is given below in Section 5.3.5.

9See Appendix B.3 for the details of the expansion for these roots and all other expansions appearingbelow.


Large Λ approximation: W0 > 1

Having analyzed the case of W0 < 1, let us now fix W0 > 1 and assume 0 < ω < eΦ0. Asdiscussed previously, there are two transmission and two reflection channels availablewhen W0 > 1 and the scattering problem is now described by equation (5.62). Since weare interested in the regime of large Λ, we calculate the wavenumber of the transmittedmodes up to next-to-leading order terms,

kt1 =ω − eΦ0

1 +W+

1

2

(ω − eΦ0)3

(1 +W )4Λ−2 +O(Λ−4),

kt2 = −ω − eΦ0

W − 1− 1

2

(ω − eΦ0)3

(W − 1)4Λ−2 +O(Λ−4). (5.69)

By direct substitution of these expressions into equation (5.64), one can straightfor-wardly determine the reflection coefficient in powers of Λ,

|R|2 +

∣∣∣∣kr2kin∣∣∣∣ |R2|2 = 1− ω − eΦ0

ω

(|T1|2 − |T2|2

)(5.70)

plus terms of O (Λ−2). Note that this second channel T2 is present even in the absenceof dispersion, being an upstream mode which is swept downstream by a super-sonicflow. From the equation above, it also becomes evident that the relation between thenorms |T1| and |T2| of the two transmission channels determines the occurrence or notof superradiance. We will show in Section 5.3.5 how the relative magnitudes of thesetwo transmission coefficients depend on the value of the asymptotic flow rate W0.

The critical case

An interesting situation to be analyzed is the critical case ω = ωcrit, which correspondsto the boundary between regions I and II in Figure. 5.9. In this scenario, the backgroundflow W0, when expanded in powers of Λ, relates 10to the critical frequency accordingto the following expression,

W0 = 1− 3

2(ωcrit − eΦ0)

23 Λ−

23 +O

(Λ−

43

). (5.71)

Note that the previous analysis leading to equation (5.68) relied on series expansions

(see equation (5.67)) which are not valid when W0 − 1 ∼ O(

Λ−23

). Therefore, in

order to analyze the possibility of superradiance in the critical case, we cannot useequation (5.68); instead, we have to start from the original Wronskian relation, equa-tion (5.64).

10Again, see Appendix B.3 for a derivation of these series.


Figure 5.10: The dispersion relation for the subluminal case in the region x→ +∞ atthe critical frequency. The blue line represents the effective frequency in the co-movingframe, Ω = ω − eΦ0 − kW0, and the red curve represents the dispersion relation atx → −∞ in the co-moving frame. Note that (ω − eΦ0)/Λ has to be sufficiently largein order to guarantee that kt2 be located in the top semi-plane. Otherwise, kt2 wouldcorrespond to a left-moving mode in the interacting region and would not be includedin a scattering solution.

In the critical regime, the dispersion relation (see Figure. 5.10) has three distinctsolutions. Two of these solutions, denoted by kt1 and kt2 , have positive group velocitiesin the lab frame and, therefore, are identified as transmitted modes. The other solution,denoted by k0, is a degenerate double root and, consequently, has a vanishing groupvelocity in the fluid frame. In order to obtain the reflection coefficient for the scatteringproblem, we first expand kt1 and kt2 as power series in Λ,

kt1 =ω − eΦ0

2− 3

8[−(ω − eΦ0)]

53 Λ−

23 +O

(Λ−

43

), (5.72)

kt2 = 2 [−(ω − eΦ0)]13 Λ

23 +O(1).

and then substitute the obtained expressions into equation (5.64). The final result isgiven by

|R1|2 +

∣∣∣∣kr2kin∣∣∣∣ |R2|2 = 1− ω − eΦ0

ω

(|T1|2 − 3 |T2|2

), (5.73)

plus terms of order O(

Λ−23

).

Observe again the importance of the relative sign of the norm in the two transmis-sion channels. Note, however, that the coefficient |T2| is related to the probability of


the initial incident mode (with wavenumber kin ≈ ω + O(Λ−1)) to be converted into

a transmitted mode of large wavenumber kt2 ≈ O(Λ23 ) which can only be balanced by

the high momentum reflected channel kr2 O (Λ). On the other hand, the coefficient |T1|is related to the probability of conversion into modes of wavenumber kt1 ≈ (ω−eΦ0)/2which is comparable to the low momentum kr1 channel. We, therefore, expect thesecond channel to be negligible in comparison with the first one, i.e. |T2| |T1|. Sinceωcrit < eΦ0, we also deduce that the right hand side of equation (5.73) would alwaysbe greater than one in the limit Λ 1. In other words, low-frequency waves in thecritical regime would be superradiantly scattered in our toy-model if small subluminalcorrections are added to the dispersion relation. Again, this dominance of |T1| over|T2| will be demonstrated for the stepfunction model in Section 5.3.5 corroborating theconclusion of superradiance in this case.

5.3.4 Superluminal scattering

We now turn to superluminal scattering in the presence of a flow which goes from sub- tosuper-sonic in a finite region. The superluminal case is quite different to the subluminalone since there does not exist any notion of a horizon or mode-independent blockingregion for high energy incident modes (the group velocity dω/dk is unbounded as afunction of k and only the low frequency modes which possess quasi-linear dispersionexperience a blocking region in such flows). We will follow the standard treatment [134]of analogue black holes with superluminal dispersion and analyze the transmission ofan incoming wave from −∞ through to +∞. In principle, one could also study thescattering of an upstream supersonically propagating mode from ∞ back to ±∞, butwe leave this analysis for a future study.

The relevant dispersion relation in the superluminal case is depicted in Figure. 5.11.Given a supersonic flow W0 > 1, there exists an interval of frequencies 0 < ω < ωcrit

(region I in Figure. 5.11) for which only two propagating modes are admitted, oneright-moving and one left-moving in the lab frame. For ωcrit < ω < eΦ0 (region IIin Figure. 5.11), however, there are four propagating modes, two transmitted right-movers and two left-movers. The third possibility is a subsonic flow W0 < 1 with0 < ω < eΦ0, for which there are always two propagating modes (see the blue dashedcurve in fig 5.11).

Let us discuss first the cases in which only two propagating modes are availableat x → +∞. Consider, therefore, the scattering of an incident wave from −∞ whosefrequency ω satisfies 0 < ω < ωcrit (if W0 > 1) or 0 < ω < eΦ0 (if W0 < 1). Notethat exactly one of these two propagating modes is a left-moving mode. Imposing theboundary condition that no wave generator is allowed at +∞, we conclude that noneof these left-moving modes can be present in the scattering solution. Consequently, thesolution of equation (5.50) corresponding to the scattering problem at hand is given


Figure 5.11: Superluminal dispersion: the lab frequency as a function of wavenumberω = ±

√k2 + k2/Λ2 + eΦ0 + kW when x → ∞ for two different flows, W0 > 1 (red,

solid) and W0 < 1 (blue, dashed). The intervals of frequency indicated refer to thered curve: region I (green, ω < ωcrit), region II (blue, ωcrit < ω < eΦ0) and region III(red, non-superradiant region). The intersection of horizontal straight lines with thered curve indicate propagating modes at that frequency. In both the green and blueregions we expect superradiant scattering in our model..

by,

f →

eikinx +Reikrx, x→ −∞,T1eiktx, x→ +∞,

(5.74)

plus negligible exponentially decaying channels. Note that the wavenumbers kin and krare the same as those which appear in equation (5.83) for the superluminal W = 0 case.In addition, the wavenumber kt1 represents the only available transmission channel (itcorresponds to the right-moving mode at x→ +∞).

On the other hand, if W0 > 1 and the frequency ω of the incident wave satisfies


ωcrit < ω < eΦ0 (region II in fig 5.11), there are, in principle, two extra propagatingchannels available (four in total, as discussed above). However, because of the boundarycondition imposed at x → +∞, only one extra transmission channel is included in ascattering solution (the other extra channel is always left-moving at x → +∞). Thescattering solution is then given by,

f →

eikinx +Reikrx, x→ −∞,T1eikt1x + T2eikt2x, x→∞,

(5.75)

where kin and kr are again given by equation (5.83) and kt1 and kt2 are the wavenumbersof the transmitted modes.

Using equation (5.59) to evaluate the functional X in both asymptotic regions, weobtain, similarly to the subluminal case, the following relation between the reflectionand transmission coefficients,

|R|2 = 1− Λ√Λ2 + 4ω2

(∑n

vgnkin

(ω − eΦ0 − ktnW0) |Tn|2), (5.76)

where the sum is over one or two transmission channels, depending on ω and whetherW0 > 1 or W0 < 1. Here, vg1 and vg2 are the group velocities of the transmittedmodes kt1 and kt2 , which are always positive by definition. Furthermore, it is possibleto show, for frequencies 0 < ω < eΦ0, that the factor (ω − eΦ0 − ktnW ) is alwaysnegative for modes kt1 and always positive for kt2 modes. Therefore, since only then = 1 transmission channel is available for frequencies lying in region I of Figure. 5.11,we conclude that the right hand side of equation (5.76) is greater than 1 and, therefore,the scattering is always superradiant. The situation for W0 < 1 and 0 < ω < ωcrit

is similar: Only the first transmission channel is available and superradiance alwaysoccurs. However, for frequencies located in region II, we cannot so easily concludesuperradiance since the extra transmission channel kt2 contributes an overall negativefactor in equation (5.76).

To obtain a conclusive answer, one would need to know the detailed structure ofW (x) and Φ(x) in the intermediate regime and solve the equations not only in theasymptotic regions but at every point x.

Large Λ approximation

In order to better understand the scattering of an incident wave whose frequency islocated in region II of Figure. 5.11 (which only occurs for W0 > 1) we shall considersmall deviations from linear dispersion, i.e. Λ 1. In such a case, we can expand thetwo transmission channels, kt1 and kt2 , in powers of Λ by solving perturbatively the


dispersion polynomial,

kt1 =ω − eΦ0

1 +W− 1

2

(ω − eΦ0)3

(1 +W )4Λ−2 +O(Λ−4), (5.77)

kt2 =ω − eΦ0

W − 1+

1

2

(ω − eΦ0)3

(W − 1)4Λ−2 +O(Λ−4), (5.78)

Substitute the obtained wavenumbers into equation (5.76) in order to determine thereflection coefficient for the scattering,

|R|2 = 1− ω − eΦ0

ω

(|T1|2 − |T2|2

)+O

(1/Λ2

). (5.79)

Since we assume ω < eΦ0 in region II, this reflection coefficient is larger than 1 whenever|T2| < |T1|. As predicted above, whether this condition is satisfied or not in a generalmodel would depend on the detailed structure of W (x) and Φ(x) in the intermediateregime [134] and the asymptotic value of W0.

From a practical point of view, in order to maximise the potential for superradiancein an experiment with a superluminally dispersive medium, one should choose theasymptotic flow W0 as small as possible while still being supersonic as this wouldminimise the size of region II and the extra positive-norm transmission channels therein.Choosing the flow as such to maximise the T1 channel is also consistent with ourintuition that scattering favors the channel which most closely matches the momentumof the reflected mode; in this case the wavenumber kt1 is closer to kr than kt2 is. Thisintuition will be confirmed below when we solve exactly the stepfunction model.

The critical case

Another interesting possibility that we now consider in detail is the critical regimeω = ωcrit. This situation corresponds to the boundary between regions I and II inFigure. 5.11 and is depicted, in the fluid frame, in Figure. 5.12. As we shown inAppendix B.3 the relation between the background flow W0 and the critical frequencyωcrit is given by the following expression,

W0 = 1 +3

2[−(ωcrit − eΦ0)]

23 Λ−

23 +O

(Λ−

43

)(5.80)

Note that, since W0 − 1 ∼ O(

Λ−23

), the approximate reflection coefficient (5.79)

obtained previously is not valid in the present case (check the denominators in equa-tion (5.90)). Consequently, we shall need different Λ expansions in order to obtain anappropriate expression for the reflection coefficient.

Like in the critical subluminal case, the dispersion relation has three distinct roots:the double root k0 (with vanishing group velocity in the lab frame), a right-moving


Figure 5.12: The dispersion relation in the region x→∞. The blue line represents theeffective frequency Ω = ω−eΦ0−kW of the critical frequency ωcrit associated with thefixed W0 > 1 at which the number of transmitted channels changes from one to two.

mode (with negative group velocity in the lab frame) and a transmitted mode (withpositive group velocity in the lab frame) whose wavenumber kt is given by

kt =ω − eΦ0

2− 3

8(ω − eΦ0)

53 Λ−

23 +O

(Λ−

43

). (5.81)

Applying as a boundary condition the fact that only right moving modes are allowedat +∞, we obtain, after substituting the relevant quantities into equation (5.76), thereflection coefficient for the scattering process,

|R|2 = 1− ω − eΦ0

ω|T |2 +O

(Λ−

23

). (5.82)

Since ωcrit < eΦ0, we conclude that the right hand side of the equation above is alwaysgreater than one when Λ 1. Hence, superradiance is expected to occur in thesuperluminal critical case for small deviations from the linear regime.


5.3.5 Some exact solutions for non-zero flow in the large Λapproximation

In this section we collect some exact solutions to the scattering problem in the variousscenarios (W0 > 1, W0 < 1, sub- super-luminal dispersion, different regions and thecritical case) presented above.

The purpose of this section is to try to get a handle on the actual spectrum ofthe superradiant modes and to try to start to study the quantitative robustness ofthe phenomenon to modifications to the dispersion relation as well as to address ourintuition from the general cases above on the relative magnitudes of the reflection andtransmission coefficients. Specifically, we wish to understand the relative magnitudesof |T1| and |T2| in equations (5.70), (5.73), and (5.79) in order to conclude not super-radiance respectively in the W0 > 1, and critical cases in subluminal scattering andthe W0 > 1 case for superluminal scattering for cases. We also give the exact solutionsin the other cases for completeness. Throughout we use exclusively the stepfunctionprofiles for the flow W and potential eΦ.

The algebraic solution to the scattering problem does not depend on the actualform of the scattering momenta, only on the number of channels and consequently onthe number of undetermined coefficients. Since we only have a handle on the scatteringmomenta as series in the dispersion parameter Λ the results here are to be consideredaccurate only for moderate to mild ‘amounts’ of dispersion such that a power series inΛ makes sense.

Subluminal models

When W0 > 1 and ω < eΦ0 the subluminal scattering involves two reflection and twotransmission channels and is described by the exact solution in (5.62) which we lookedat above reaching the relation (5.70). We were not able to make a general statement ofsuperradiance in such a case sue to the relative sign between the T1 and T2 contributionsto the right hand side of that relation. Here we show that, in the step function model,indeed, |T1| > |T2| for ‘low’ flows, implying superradiance in general there while forhigher flows |T2| can dominate leading to non-superradiant scattering.

The reflected channel momenta and the incoming momentum are expressed exactlyas

kin = −kr1 =Λ√2

√1−

√1− 4ω2

Λ2.

kr2 =Λ√2

√1 +

√1− 4ω2

Λ2. (5.83)


while the transmitted momenta are only expressible as the perturbative series

kt1 =ω − eΦ0

1 +W+

1

2

(ω − eΦ0)3

(1 +W )4Λ−2 +O(Λ−4), (5.84)

kt2 =ω − eΦ0

W − 1− 1

2

(ω − eΦ0)3

(W − 1)4Λ−2 +O(Λ−4). (5.85)

Imposing continuity in the first 3 derivatives and the function itself at x = 0 wecan solve for the reflection and transmission coefficients exactly as in the zero flowcase recovering exactly (5.42) which include the resonances at kt1 = −kin (both R1 andT1 →∞) and kt2 = −kin (bothR1 and T2 →∞). The properly normalised transmissioncoefficients have an additional factor of

√|Ω|/ω. Considering the problem at fixed ω

and eΦ0 these resonances occur for flows

W0 =eΦ0

ω, and W0 =

eΦ0

ω− 2. (5.86)

Note that we must have ω < eΦ0 for superradiance so these W0 lie in the intervals(1,+∞) and (−1,+∞) respectively depending on how small ω is. Observe that theseresonances are only possible due to the fact that the transmitted modes kti are negative,allowing for some factors in the denominators to vanish.

For generic values of ω and eΦ0 we plot these (properly normalised) coefficients inFig 5.13. Note that the perturbative expansion for kt1,2 breaks down for flows too closeto 1.

We see that the T1 transmission is always larger than T2 until the resonance inT2 starts to take effect. After the crossover due to the resonance the T2 coefficient isalways larger. Note also the superradiant reflection coefficients R1 and R2.

Superluminal models

For superluminal scattering at W0 > 1 the scattering solution only has one reflectioncoefficient

f →

eikinx +Reikrx, x→ −∞,T1eikt1x + T2eikt2x, x→∞,

(5.87)

and we have

kin = −kr =Λ√2

√−1 +

√1 +

4ω2

Λ2, (5.88)

and the perturbative series

kt1 =ω − eΦ0

1 +W− 1

2

(ω − eΦ0)3

(1 +W )4Λ−2 +O(Λ−4), (5.89)

kt2 =ω − eΦ0

W − 1+

1

2

(ω − eΦ0)3

(W − 1)4Λ−2 +O(Λ−4). (5.90)


Figure 5.13: Here we chose ω = 0.1, eΦ0 = 0.4 and plot the coefficients |R1,2| and|T1,2| as a function of asymptotic flow velocity W0 (R2 is only barely visible along thehorizontal axis). In these plots we choose Λ = 1

Again, this case was considered above where we reached the inconclusive relation (5.79)showing that the occurrence of superradiance depends on the relative magnitudes ofthe coefficients |T1| and |T2|. Here we show for the the stepfunction solution thatsimilarly to the subluminal exact solution above, we have |T1| > |T2| for ow flows untila resonance at higher flows in T2 leads to non-superradiant scattering.

The exact solution for the coefficients obtained by matching the first 2 derivativesand the function at x = 0 is

R = − (−kt2 + kin)(kin − kt1)(−kin − kt2)(−kin − kt1)

(5.91)

T1 = − 2(−kt2 + kin)kin

(kt1 − kt2)(−kin − kt1)(5.92)

T2 =2(kin − kt1)kin

(kt1 − kt2)(−kin − kt2)(5.93)

where we note again the poles which occur at the same place as in the subluminal case(in the term independent of Λ in the series).

This (normalised) solution is shown in Fig. 5.14 which is almost identical to thecorresponding W0 > 1 subluminal plot in Figure 5.13. Notice again that |T1| is greaterthan |T2| until the resonance in |T2| which occurs at W0 = eΦ0/ω. The resonance in


Figure 5.14: The reflection and transmission coefficients |R| and |T1,2| at fixed ω = 0.1eΦ0 = 0.4, Λ = 1 plotted as a function of asymptotic flow velocity.

|T1| can be made to be absent from this analysis if we choose eφ0/ω < 2. Also notethat after the resonance in |T2| we always have |T2| > |T1|.

5.3.6 Loose ends

Here we collect a few odds and ends which have been left open in the analysis above.

Minimal flow condition

A number of thorny issues arise when generalizing the discussion of superradiance tonon-zero flows in the subluminal case. One might have noticed that our model withnon-zero W0 < 1 flow and three transmission channels is not continuously connectedto the zero flow case in which there were only two transmission channels available.Consider, therefore, Figure 5.10 and the nature of the second transmitted mode kt2 inthe region x → +∞. If we follow this mode backwards in time (i.e. we move towardsthe region x→ −∞), the slope of the blue line Ω = ω− eΦ(x)− kW (x) in Figure 5.10will eventually decrease. As a result, depending on how W (x) behaves as we movetowards negative x, the group velocity of this root kt2 might increase until it becomesdivergent. This occurs when k/Λ = −1 and corresponds to a background velocityW (x) = −(ω − eΦ(x))/Λ. At this point it is not clear how the mode will behaveanymore since, somehow, the group velocity must jump discontinuously from +∞ to


−∞. In order to avoid this peculiarity, we impose the condition that our flow shouldnot be too small, i.e. W (x) > −(ω− eΦ(x))/Λ everywhere. We point out that it is notunnatural to assume such a condition when dealing with only slight deviations fromlinear dispersion (Λ → ∞). Moreover, if we wish this condition to be satisfied by allfrequencies in the band of interest 0 < ω < eΦ0, then we require W (x) > eΦ(x)/Λ. Inparticular, at x→ +∞, this inequality becomes W0 > eΦ0/Λ, which is always satisfiedsince we always work under the assumption that eΦ0 Λ.

An analogous discontinuity exists in the x → −∞ region when we follow the highmomentum reflected mode kr2 backwards in time as it propagates from x → −∞towards the region of non-zero flow. This can be seen in Figure 5.3: As we movetowards the interacting region x→ +∞, the slope of the green dash-dotted line becomesnegative until, possibly, the intersection labelled kr2 crosses the ω axis and the groupvelocity jumps from −∞ to +∞. In order to avoid this unphysical jump, we assumeW (x) > (ω− eΦ(x))/Λ everywhere. In particular, at x→ −∞, this condition becomesW (−∞) > (ω− eΦ(−∞))/Λ. It is important to point out that we have conducted ouranalysis in this article without indulging this subtlety; the inequality is not satisfiedin any of our models, since we always assume eΦ(−∞) = 0 and W (−∞) = 0. Note,however, that choosing the flow velocity W (x) to satisfy this condition would imply thatthe kr2 modes are in fact not reflected modes and, therefore, should not be present in thescattering process because of the boundary conditions imposed. The main consequenceof these considerations is that, by requiring the minimal flow inequality in the non-interacting region, the second reflection coefficient R2 should be absent from the lefthand side of all transmission/reflection relations such as (5.64). Finally, we note thatthis same issue is discussed in Ref. [131] in the context of Hawking radiation. A similarinequality is needed for a Hawking signal to be present in their model.

Inertial motion superradiance

Throughout this paper, inspired by the usual condition for rotational superradiance(ω − mΩh < 0) in the black-hole case, we have analysed only scattering problems inwhich the frequency of the incident mode satisfies ω − eΦ0 < 0. However, by lookingat equation (5.60), one can see that the condition for negative-norm modes is givenby ω − eΦ0 − kW0 < 0 and, therefore, in principle an amplification of an incidentwave can also occur for ω − eΦ0 > 0 given that ω − eΦ0 − kW0 < 0. This kind ofsuperradiance is indeed possible and was analysed in ref. [29], where it was dubbedinertial motion superradiance and compared with rotational superradiance. In fact,inertial motion superradiance has long been known in the literature as the anomalousDoppler effect and the condition for negative norm modes is referred to as the Ginzburg-Frank condition [135]. Several phenomena in physics, like the Vavilov-Cherenkov effectand the Mach cones (which appear in supersonic airplanes) can be understood in termsof inertial motion superradiance [29].


Inclusion of mass

In reference [30], the authors consider the electrodynamic interpretation of superradi-ance which in the general case includes a mass term in the superradiant equation ofmotion

d2f

dx2+[(ω − eΦ0)− (p2 +m2)

]f = 0. (5.94)

Both a regular mass and the additional effective mass arising from the dimensionalreduction procedure have been included here. The authors then proceed with thegeneral analysis, including this mass term, concluding that superradiance indeed stilloccurs but the window of frequencies exhibiting superradiant scattering, which waspreviously ω ∈ S := (0, eΦ0) is now modified to the so-called ‘Klein region’√

p2 +m2 < ω < eΦ0 −√p2 +m2, (5.95)

which differs from S in general, particularly for low frequencies. It would be interestingto also include the effects of such a mass term in dispersive superradiance.

We have mentioned previously above and throughout this chapter that our modelequation for superradiance can also be thought of as arising from a dimensional re-duction in a different sense to the one of [30]. The role played by the electromagneticpotential eΦ(x) is played by a flow velocity in an additional dimension and the reductionformally introduces a mass term modification to for the remaining degrees of freedomin exactly the same way as above in (5.94) (see [132] for a discussion of this in thecontext of black hole radiance). This reduction could apply, for example, to the radialpart of a cylindrical surface wave scattering process on a cylindrically symmetric flowprofile. Therefore we can bootstrap the results of [30] to conclude that in such flowingfluid systems, the dimensionally reduced dynamics can still support superradiance inthe Klein region when dispersion is neglected.

In the article [132] the authors consider the effect of such a mass term on whilehole and black hole flow flow configurations in the presence of modified dispersion butwithout the superradiance inducing potential eΦ(x) that we have been considering inthis chapter. They study the so-called ‘undulation’, a zero-frequency but non-zeromomentum solution to the dispersion relation, normally present in white hole flows[64, 136]. This is show schematically in Figure 5.15 The effect of the mass is to introducesuch an undulation also in black hole flows.

Dispersion and dimensional reduction

When we introduced modified dispersion at the beginning of Section 5.2.2 we did soonly after dimensionally reducing to the 1+1 dimensional problem. Alternatively, wecould have introduced the modification in the higher dimensional theory, and only


Figure 5.15: The quartic superluminal dispersion relation for a massive excitation with-out the external potential (which was responsible for superradiance), the two additionalzero frequency roots are labeled as ku (for ’undulation’).

then dimensionally reduced. That is, we modified the dispersion by introducing higherpowers of k as

± k4 − k2 + (ω − kvx − pvy)2 − p2 = 0. (5.96)

where± refers to sub- and super-luminal dispersion respectively. On the other hand onecould introduce the modified dispersion before the dimensional reduction step leadingalso to quartic terms in p (the momentum in the direction which is integrated over)leading to

± k4 − k2 + (ω − kvx − pvy)2 ± p4 − p2 = 0. (5.97)

This second, more physically justified model has the interesting feature of the possibilityof an imaginary effective ‘mass’ in the subluminal case represented by the term p2−p4.For sufficiently large p this term becomes negative and may lead to some interestingeffects.


5.4 Conclusions summary and outlook

In this chapter we have investigated the qualitative robustness of superradiance againstmodifications to the dispersion relation. We have studied quartic polynomial modifi-cations both sub- and superluminal and concluded that 1D scattering of low frequencywaves can be superradiantly scattered. We have derived general formulas describingsuch dispersive scattering in both the simple case of zero flow in the direction of scat-tering and the more complicated case when the flow is allowed to propagate in thescattering direction. We have seen that the inclusion of such flows introduces an unex-pected complexity to the scattering process and we identified various regimes in whichsuperradiant scattering was possible and to what extent the non-dispersive results arerobust. We also began a quantitative analysis in this vein, using simplified stepfunctionpotentials to get a handle on the reflection and transmission spectra themselves.

In the two tables that follow we summarise the results of this chapter by stating ifand when superradiant scattering occurs for the various different scenarios. As we sawand as is evident from the table, the situation is far from straightforward.

For zero flow, the analysis was quite straight forward, since we were able to concludesuperradiance in the general case without the need to go to the large Λ approximation.An interesting quantitative result we obtained by moving to exact solution to thescattering by way of using an idealised stepfunction potential, was that in fact thelow momentum channel can be robust against the modifications to the dispersion forlarge Λ and, furthermore, the total reflection can be enhanced by the presence of theadditional degrees of freedom represented by the high momentum channels kr2 and kt2 .

Zero flow General case Low mom. channelSubluminal Always For large ΛSuperluminal Always Always

For the non-zero flow calculations, we found a much richer spectrum of scatteringphenomena which we have attempted to summarise in the following table.

Non-zero flow ω < ωcrit ω > ωcrit W0 > 1 Low mom. channelSubluminal For low W0 For high Λ For low W0 For high ΛSuperluminal Always For low W0 For low W0 N/A

We see that, in general, the superluminal scattering problem more naturally sup-ports superradiance, partly, it seems, due to the fact that for W0 < 1 it is still a twochannel scattering process as in the non-dispersive case with one transmitted and onereflected model. When W0 > 1 the scattering, although not always superradiant, hasno essential distinction from the non-dispersive case. The subluminal case is complex

5.4. Conclusions summary and outlook 181

with 4 channel scattering possible but we saw that the total reflection can be largerthan the reflection in the non-dispersive case.

The present work is intimately related to recent experimental realizations of ana-logue black holes, some of which have even studied the Hawking emission process. Inparticular, Ref. [23] exhibits the first detection of the classical analogue of Hawkingradiation using gravity waves in an open channel flow. One of the most importantlessons to be learned from that work is that even though vorticity and viscosity ef-fects cannot be completely removed from the experimental setup, they can be madeextremely small. In fact, they can be reduced to the point that the results predictedby the irrotational and inviscid theory match the results obtained experimentally withconsiderable accuracy. Based on this fact, together with our present results, we expectthat superradiance might occur (and be detected) in laboratory using gravity wavespropagating on water. This idea is currently being investigated by the authors.

5.4.1 What can we do next?

There are several ways in which one could generalise the theory we have presented inthis chapter. Firstly one could repeat the analysis for more realistic dispersion relationsand in this direction we are particularly interested in the dispersion relation for gravitywaves. The article [51] studies the gravity wave dispersion in detail, but not in thecontext of superradiance. It is given by

ω2(k) =

(gk +

σ

ρk3

)tanh (kh)

= ghk2 −(gh3

3− σh

ρ

)k4 +O

(k6), (5.98)

where g is the acceleration due to gravity and σ and ρ are respectively the surfacetension and density of the fluid. Hence our analysis above would apply to the case ofgravity waves where we identify

1

Λ2=

∣∣∣∣gh3

3− σh

ρ

∣∣∣∣ (5.99)

When capillary effects are sufficiently small (σ/ρ < gh2/3), the minus sign (sublumi-nal case) in equation (5.50) must be chosen; otherwise, for sufficiently large capillaryeffects, the plus sign (superluminal) is the one to be used. Additional effects due to amixing of sub- and superluminal dispersion might be present such a case and it wouldbe interesting to investigate further such a system particularly in the light of the pos-sibility of experimental detection directly of superradiant scattering in gravity waveexperiments (see next chapter 6).


Alternatively one could work with other physically interesting dispersion relationssuch as the optical one discussed in Chapter 4.

The generalisation of the presented theory in this chapter to higher order polynomialdispersion should be expected to be difficult based on the fact that one requires aconserved Wronskian to the higher order equation of motion. However it is possiblethat the form of the action of such a functional be universal in the sense that the resultshould always be of the form

X

[∑n

Aneiknx

]=∑n

|An|2Ωdω

dkn, (5.100)

for the correct generalised conserved quantityX. This extension can be directly checkedby looking for the appropriate generalisation for 5th or 6th degree equations of motion.

Another way in which to improve on the theory presented in this chapter would beto study the interplay between the dispersive Hawking mechanism and superradiance.In order to isolate the superradiance effect one could work with analogue geometrieswhich have vanishing surface gravity at the ‘horizon’. This can be achieved analytically,for example, with the flow profile

W (x) =

W0/2(1 + tanh(x)) x < 0

W0/2[1 + tanh

(2x0π

tan(πx2x0

))]0 < x < x0

W0 x > x0

..

which interpolates between 0 and f0, achieving f0 at finite x and having zero derivativethere.

A slightly more ambitious extension of this work and one in which the interplaybetween Hawking emission and superradiance could be addressed would a numericalinvestigation of the exact scattering solution for different and various flow profiles andprofiles for the potential eΦ(x). Such numerical work has been carried out for theHawking emission alone in many works in the literature the main works being [60, 118,129, 63, 127]. Such numerical studies would also be necessary to make predictions fora real superradiant spectrum in an actual experiment with a naturally occurring (andhence not idealised) flow profile.

Chapter 6

Experimental Black Holes and aBathtub Vortex

In this chapter we will be looking more closely at the experimental side of analoguegravity and in particular on the possibility of simulating the phenomena of superradi-ance known to occur for rotating black holes in a draining water vortex experiment. Werefer the reader back to 2.1.4 for a background discussion on superradiance in general.

The bridge between idealised theory and the real experiment we describe can betenuous at times. For this reason in this chapter we describe how we have adopteda mixed numerical, observational and theoretical approach to the experimental prob-lem. One of the major lessons learned from studying analogues is that many of thephenomena formulated on smooth Lorentz invariant spacetime manifolds are surpris-ingly robust. It is with these lessons in mind that we find ourselves looking beyondthe analogue model description of superradiance when it comes to a real experiment,often finding that an analogue description is not available or at best it is just a roughheuristic guide. Nevertheless, although the experiment we shall describe has not beenfully completed at this stage, we seriously expect some form of superradiance to bemanifested in our experiment and an observation of it to be feasible.

In the first section we will introduce the topic of experimental analogue gravity,and collect the material of Chapter 1. In the second we discuss the theory as it appliesto the draining vortex, the modifications and complications from the simple theorydiscussed in Section 2.3. In the third section we will describe an experimental setupand some measurements that have been made with it on the draining vortex systemwhile in the following section we briefly summarise the partial results obtained andlessons learned so far with the experiment. We conclude with a final section outliningthe future goals and plans for the experiment.

The work presented in this chapter will described in the forthcoming article [137].

184 Chapter 6. Experimental Black Holes and a Bathtub Vortex

6.1 Introduction

As introduced previously in 2.1.4, superradiance was first discovered for electromag-netic wave scattering in flat spacetime and only later discovered to be a more generalscattering phenomenon, including scattering from rotating Kerr black holes where itwas made ‘famous’ in the gravity community. In this way, indeed, superradiance hasplayed a special role for analogue gravity, the analogue having been discovered priorto the spacetime phenomenon. Of course the separation of ‘real’ from ‘analogy’ is onlysemantics as we have emphasised in previous chapters but the case of superradiance isunique in that the traditional order was reversed. Nonetheless, generating a superra-diant scattering process in the lab would be a simulation of the spacetime concept ofergo-region and is therefore of great interest.

In the same way that we have separated the phenomenon of ‘Hawking radiation’from the normal concepts of spacetime physics such as ‘horizon’, ‘Lorentz invariance’and ‘gravitational field’ superradiance does not rely on a geometric language for itsdefinition being simply that |R|2 > 1 for a (non-dispersive) scattering process where Ris the reflection coefficient for reflected waves from a scattering potential (see Chapter 2for the background discussion of superradiance in general).

A zeroth order attempt to generate superradiance experimentally would be to tryto simulate directly the equatorial Kerr geometry in a flowing fluid. We discussed inChapter 1 how this is difficult and perhaps a very unnatural thing to be trying to do to afluid. Instead, we can draw upon our lessons from studying superradiance in general andtry to find some fluid configuration which supports the necessary ingredients, namely,an acoustic regime at low wavenumbers and an ergoregion. The obvious candidateflowing fluid that comes to mind is the naturally occurring draining ‘bathtub’ vortex— the cylindrical symmetric flow from a draining tub . One huge experimental benefitof using the draining bathtub vortex is that, as mentioned, it is naturally occurring sothat one would not need to enter into any complex fluid manipulation techniques inorder to make the fluid flow in the correct way to generate the desired ergo-region.

6.2 The bathtub vortex theory

The physics of a draining vortex fluid is extremely complex and in fact is an active areaof research even in the fluid mechanics literature [138, 139]. This section is concernedwith the difficult problem of describing the background flow configuration and gravitywaves on top of such a vortex and to try to discover if at least some of the complex-ities can be smoothed over, allowing for description simple enough to provably admitsuperradiance.

As we described in Section 2.3, one such complication is the ubiquitous modifieddispersion relation which permeates discussions of horizons in analogue systems. In

6.2. The bathtub vortex theory 185

the previous section we demonstrated that superradiance is still possible for both sub-and super-luminal modifications to the linear dispersion relation.

Another major additional technical obstacle we must face in this endeavour is thefact that the (low frequency) surface wave propagation speed is not constant, owing tothe curved nature of the free surface. Thus we are confronted by the double edged swordof a non-constant flow rate and non-constant wave speed. In principle, furthermore, it isnot clear that we can have anything which resembles the appropriate acoustic geometryin a region large enough to allow for superradiant scattering since the analogy has beendemonstrated only for small deviations of the height of the fluid [50].

6.2.1 Vortex geometry with flat free surface

As a first approximation we might wish to model the draining vortex with a constantwave speed analogue, assuming in the process a constant depth of water above thebottom since

c2s =

√gh(r) (6.1)

where h(r) is the surface height as a function of r and g is the acceleration due togravity.

Of course this is a bad approximation near the center of the vortex where the free-surface becomes almost vertical and drops to the bottom of the tank. This kind ofmodel certainly is capable of predicting superradiance but it is almost as certainlyunphysical.

For example, in the Feynman Lectures on Physics [140] Feynman explicitly considersthe shape of the free surface of a draining vortex for ‘dry water’ (irrotational andincompressible).

The free surface h(r) can be shown very simply to be defined by

(h− h∞) =k

r2(6.2)

an example shown in Figure 6.1 However Feynman does not mention the flow velocitiesand in particular the surface flow velocities so we must look to additional theory inorder to find out if there can be an ergoregion in the flow at radii large enough for the‘slowly varying surface height’ approximation holds.

In the work [52] Unruh studies the problem in reverse, given a constant flow velocityand a bottom profile, what does the free surface look like? Alternatively one can providethe shape of the free surface and determine the bottom profile necessary to obtain theconstant flow velocity. Surface perturbations are shown to satisfy an equation of motionsimilar to that which we discussed in Section 2.3 of Chapter 2.


Figure 6.1: The free surface as given in (6.2) for ‘dry water’.

What we shall consider in our experiment is a flat bottom tank which drains nat-urally. Thus the free surface is determined entirely by the internal dynamics of thefluid itself and we have observed it to be extremely curved (see below in Section 6.5for a summary of some of our preliminary observations). Thus we would like to havea model which takes into account the curvature of the surface in such a way that wecan show that the surface wave speed is surpassed by the surface flow velocity in someregion where our approximations are valid.

6.2.2 Axisymmetrically draining fluid: curved free surface

The analysis of [50] and Section 2.3 can be applied to the case of a radially drainingcylindrically symmetric (‘vortex’) flow [141, 137].

In the general case an analogy does not exist for surface wave propagating on ageneral curved surface, being valid only when surface deformations are small withrespect to the wave length of the perturbations one is trying to describe and whenh′(r)/h 1 as we discussed in 2.3 and has been reported in [50]. However, theproblem of surface wave propagation is the subject of the following subsection. Firstlyhere, we discuss the background flow for the draining vortex.


This more physically justified approach explicitly includes the curved surface of thedraining vortex in the problem and we will describe how one might make take intoaccount the next to leading order corrections to the ‘slowly varying height’ approxima-tion.

The relevant coordinates are cylindrical polars (r, φ, z) and we are interested incylindrically symmetric time independent flows and in particular the height as a func-tion of radius h(r) and the radial and angular surface flow velocities vr(r) and vφ(r)as functions of r. To get these we will need to solve the full problem as a function of(r, φ, z) as we do below.

The equations of motion, again, are the continuity

∇ · v = 0, (6.3)

and the Euler equation,

dv

dt=∂v

∂t+ (v · ∇)v = −∇p

ρ− gz, (6.4)

where g, v and p are, respectively, the gravitational acceleration, the velocity and thepressure of the fluid. Note that we are assuming that the gravitational force is theonly external body force acting on the fluid and incompressibility (so that the densityis constant) and zero vorticity.

Besides the basic equations above, we need three boundary conditions already givenin Chapter 2:

1. vz|z=0 = 0;

2. The rate of change in the height of the fluid must be equal to the vertical velocityof the fluid at the surface,

vz|z=h =dh

dt

∣∣∣∣z=h

=∂h

∂t+(v‖∣∣z=h· ∇‖

)h; (6.5)

3. The pressure must be continuous at the interface.

We know that in the case of zero vorticity

vφ = B/r, (6.6)

where B is an integration constant, on general grounds1 and that the velocity potentialψ (where v = ∇ψ) can be expressed in terms of a single function ψ0 of r and φ as a

1The condition ∇× v = 0 implies three separate conditions, one for each component of the vectorrelation. We also assume the φ independence of all the velocities. The vanishing of the z componentimplies ∂zvφ = 0 so that vφ is only a function of r. Then the vanishing of the r component impliesvφ = r∂rvφ which can be solved to find vφ = B/r.


power series

ψ =∑n

zn

n!∇2n|| ψ0. (6.7)

where∇|| is the gradient operator restricted to the r, φ coordinates. The z independenceof ∂φψ = vφ = B/r this implies that ψ0 take the form ψ0 = ξ(r) +Bφ in order for theinfinite sum to collapse to the single z-independent term. It can be shown [50, 137]that the velocities vr and vz can be written as infinite sums of powers of the derivativeoperator

D2r :=

1

r

∂

∂r

(r∂

∂r

), (6.8)

acting on the radial part ξ(r) of the function ψ0:

vr =∑n

(−1)nz2n

(2n)!

∂

∂rD2nr ξ(r), (6.9)

vz = −∑n

(−1)nz2n+1

(2n+ 1)!D2nr ξ(r). (6.10)

Then the equations (6.3) and (6.4) reduce to the coupled equations∑n

(−1)nh2n

(2n+ 1)!

∂

∂rD2nr ξ(r) =

1

h

∫ h

0

vr(r, z)dz = −Ah∞rh

, (6.11)

and ∑n

(−1)nh2n

(2n)!

∂

∂rD2nr ξ(r) = − 1√

1 + h′2

√2g (h∞ − h)− B2

r2= vr|z=h . (6.12)

where h∞ is the asymptotic fluid height away from the vortex core and A is an inte-gration constant.

This pair of equations for the functions h(r) and ξ(r) is highly non-linear and ingeneral very difficult to solve without further approximations.

One quite crude approximation consists in keeping only the n = 0 terms in the sums(neglecting the D2

r derivatives) under which vr reduces from (6.9) to the z-independentform vr = ∂rξ(r) and the integral in (6.11) becomes trivial leaving

vr = −Ah∞rh

. (6.13)

Combining this with (6.12) we find the single equation for h

Ah∞rh

=1√

1 + h′2

√2g (h∞ − h)− B2

r2, (6.14)


which, further neglecting the derivative term h′(r), admits the solution for h(r)

h(r) =1

3

(h∞ −

B2

2gr2

)[1 + 2 cos

(θ

3

)], (6.15)

where

θ = cos−1

[1− 33

g

(Ah∞

2r

)2(h∞ −

B2

2gr2

)−3]. (6.16)

Therefore we have the desired solution for h(r) ((6.15)) and both the velocitiesvr (6.13) and vφ (6.6), albeit under a rough approximation which neglected higherderivatives of the quantities involved.

The ergoregion for this flow and surface profile, determined by the condition v2‖

∣∣∣z=h

=

v2r + v2

φ > c2s = gh (see Section 2.3 for the definition of g), corresponds to the interior

of the surface r = rE, where rE is given by

rE =B√gh∞

√3

2

√1 +

(3A

2B

)2

. (6.17)

It can be shown [142] that the horizon, where v2r = c2

s = gh is located at the point

rH =B√gh∞

[2

(A

B

) 13

sinh

(1

3cos−1

(B

A

))]− 32

. (6.18)

but that, somewhat surprisingly the height function h becomes singularly steep atprecisely this point. Hence the surface gravity

κH ≡1

2

d

dr

(c2 − v2

r

)∣∣rH

=1

2

(g + 2

A2h2∞

h3H

)dh

dr

∣∣∣∣rH

=∞. (6.19)

diverges at the horizon. Therefore we conclude that this model breaks down near thepredicted location of the horizon.

Another reason why the solution (6.15) might be considered unphysical is that theapproximation that the radial velocity vr be independent of height is known to be a badapproximation in draining vortices. We will see later that this is due to the presenceof so-called ‘Ekman pumping’ along the bottom boundary layer which carries the bulkof the flux of radial flow to the drain hole.

For illustration, including the first D2r terms in equations (6.11) and (6.12) results

in the set

∂ξ

∂r− h2

3!

∂

∂rD2rξ(r) = −Ah∞

rh, (6.20)

∂ξ

∂r− h2

2!

∂

∂rD2rξ(r) = − 1√

1 + h′2

√2g (h∞ − h)− B2

r2, (6.21)


A more complete analysis would include these corrections but, as we see here, thesemodifications result in a much more complicated set of coupled differential equationswhich should be solved in order to find h(r) and ξ(r).

6.2.3 Equation of motion for perturbations

Assuming that the radial flow is given by the form (6.13) and ignoring for a momentthe infinite surface gravity in this approximation we can show that superradiance ispredicted within this model.

With a flow profile and velocity field described by A, B and h(r) we can immediatelyuse the results of [50], given also in Section 2.3 above to write down the metric andwave equation

1√−g

∂µ(√−ggµν∂νδψ) = 0. (6.22)

For fixed frequency ω and angular eigenvalue m the equation of motion for radial partR(r) of the wave R(r)e−iωteimφ can be written as

d2R

dr2+ P (r)

dR

dr+Q(r)R = 0, (6.23)

for some complicated functions P and Q. We can use the results of Appendix A.1 tomanipulate this into the form

d2f

dr2+ Ω2(r)f = 0 (6.24)

similar to what we saw in Chapter 5 where r is a modified ‘tortoise-like’ radial coordi-nate, c∞ and cH are the wave speeds at +∞ the horizon respectively. The function Ωsatisfies the asymptotic relations

Ω2 =

ω2/c2∞ r → +∞ (r → +∞)(

ω − mBr2H

)2

/c2H r → −∞ (r → rH)

(6.25)

of the form required in order to give rise to superradiance [141, 137]. Using the re-sults presented in Section 2.1.4 we can conclude that, superradiance is present in thedraining vortex system as long as the approximations leading to the metric are valid.These are: small variation of the surface profile, long wavelength surface waves (lineardispersion). Moreover the results presented in Chapter 5 strongly suggest that thissuperradiance is robust against modifications to the dispersion relation. Such modi-fications are necessarily present in the system, the full dispersion relation for surface


waves given by

ω2(k) =

(gk +

σ

ρk3

)tanh (kh)

= ghk2 −(gh3

3− σh

ρ

)k4 +O

(k6), (6.26)

6.2.4 Non-linear fluid dynamics of a draining vortex

Here we collect some results concerning the non-linear dynamics of a draining watervortex and compare them with our observations and the linear (and first order correc-tions) material developed above.

This material is supposed to be a sobering reminder of the non-linear nature ofdraining fluid vortices and the difficulty of simulating nice acoustic geometries withthem.

The time independent and incompressible flow Navier Stokes equations are givenby

(u · ∇)u = −1

ρ∇p+ ν∇2u− 2Ω× u; ∇ · u = 0 (6.27)

where the various terms arise from competitive effects and are relevant in differentparts of a draining vortex.

The draining water vortex is surprisingly complex. The flow can be roughly dividedinto various boundary layers, the central core and the bulk flow, indicated in Figure 6.2These different regions are described by different approximation of the Navier Stokesequations.

Different regions of the flow

• Ekman layer

The Ekman layer exists on the bottom of the tank and consists mainly of a radialflow pattern. It is parameterized by the thickness δ =

√ν/Ω where ν is the

kinematic viscosity, Ω is the rotational velocity of the container and L is the sizeof the system.

The Ekman number (ratio betwen viscous and coriolis terms in (6.27))

Ek =ν

2ΩL2(6.28)

defines regions where this layer exists.


Figure 6.2: The observed flow profile reported in [138, 139] (graphic taken from thepaper [139]) with the Ekman upwelling region shown near the bottom of the tank nearthe vortex core. Note that in our experiments the central air core extended all the waydown to the bottom of the tank as shown schematically in fig 6.3.

• Surface boundary layer

It is the surface layer that we are most interested in due to the gravitationalanalogy for surface waves.

• Stewartson layer

The stewartson layer exists near the boundary of the circular container and isparametrized by LE

1/4k where Ek is the Ekman number.

Ekman Layer and pumping

In [138, 139] the flow profile in the bulk of a steady state draining vortex is investigatednumerically and experimentally. The authors observe a particular boundary layer effectknown as Ekman pumping whereby in certain parameter ranges the majority of theradial flow has its origin in the bottom Ekman layer of the draining fluid as shown inFig 6.2. This Ekman layer was observed to ‘well up’ nearer the center of the vortexgenerating a vertical flow which consequently convets into a strong downflow near thevortex core. In the so-called linear Ekman theory valid for small Rossby number2

the upwelling (or Ekman pumping) is zero when the vorticity vanishes. Hence thisupwelling is recognized as a non-linear effect due to the breakdown of the linear Ekmandescription and not necessarily because of the onset of vorticity near the core. Theobserved flow structure is diagrammatically shown in Fig. 6.2, which also labels the

2Recall that the Rossby number is defined as the ratio between the non-linear and Coriolis termsin the Navier Stokes equations and generally scales as R = U/(2ΩL2) where U is a typical velocity, Ωis the angular velocity and L is a typical scale of the system.

6.3. Method to experimentally determine superradiance 193

other boundary layers and the main bulk region which is shown to be in geostrophicbalance3 with a z independent velocity flow profile.

6.3 Method to experimentally determine superra-

diance

The traditional superradiant scattering process is in terms of the radial part of awave form. Generating a circular symmetric wave is subtle and measuring the circularamplitudes in all angular modes a complex data analysis task. From the theory side,ideally one would like to make a clean prediction for the superradiant (or not) spectrum.This is difficult for two reasons

• The background flow and profile is not sufficiently well understood theoreticallyto be able to conclude absolutely that superradiance should occur for the kinds ofdraining vortices that naturally occur in bathtubs – for example we are not surethat there is a horizon or ergo-region at all in these vortices (see Section 6.2).Hence we are forced to make certain (sometimes drastic) approximations andsimplifications in order to describe the background itself.

• More importantly (perhaps), the exact equation of motion for the propagationof perturbations on such background is difficult to obtain due to the complexnature of the excitations themselves on the highly curved free surface. Again weare forced to make approximations in order to obtain a tractable second orderwave equation for the perturbations with a domain of validity which is ratherunclear due to out lack of knowledge of the bulk background flow.

It is our expectation (hope!) that the approximations which lead to a tractableand superradiant predicting equation for the perturbations are valid in a region largeenough to include an ergo-region for long wavelength modes.

In thew initial stages of the experimental effort we have chosen to focus on thetheory described in Section 6.2 with the approximations described there using the flowvelocities vr and vφ proportional to 1/r — experimentally match the observed flows tothese functional forms by optimizing the parameters A and B (see (6.13) and (6.6)) tofit the data. However, due to the divergent surface gravity of the horizon in that model,we choose to abandon the derived form of the surface profile and instead measure thesurface directly in the lab.

That is, in order to overcome the manifold difficulties towards a clean theoreticalor clean experimental result, we use a combined numerical, observational and analyticmethod to determine superradiant scattering or not from the draining vortex:

3Recall that a rotating fluid is in geostrophic balance if the dominant balance is between Coriolisand pressure forces. In general, geostrophic balance implies a low Rossby number.


Set up stationary draining vortex

Measure free surface profile h(r)

Measure surface flow velocities A and B

Implement A, B and h(r) in a numerical routine simulating the scattering event computing R and T

In this way, the experiment combines an interesting mix of interdisciplinary physicswith an interesting mix of interdisciplinary methods.

6.4 The laboratory

In this section we describe the experimental procedure we followed to determine thebackground free surface profile and the radial and angular surface flow velocities. Wecarried out experiments in three separate phases using two different experimental se-tups. The second and third experiments were done one year after the first.

6.4.1 Experimental Setup

For this experiment we have used a tabletop sized plastic circular bucket of radius atthe base 30.1cm and slant height (see below) 27.7cm. The bucket walls were not verticalbut instead were slightly angled outwards such that the area of the circular horizontalcross sections of the bucket increased with vertical height. Water was pumped in froma piece of plastic tubing of flow radius of approximately 1cm and length approximately400cm which penetrated the surface of the water and was fixed to the bottom of thebucket with strong double sided tape such that the open end directed the water in anangular direction without a radial component giving the fluid angular momentum, seeFig. 6.3. The other end of the tube was fixed to a regular indoor house tap by forcingit over the tap, forming a watertight seal. To the open end of the tube was affixeda woven metal wire mesh to minimize turbulence at the inlet. In the bottom of thebucket a hole was cut in the centre of the bucket floor of diameter 2.9cm.

6.5. The results so far 195

Circular bucket

Free surface

Inflow pipe

(a)

Bucket wall

Inflow pipe

Flow direction

(b)

Figure 6.3: A cross-section (a) and top view (b) of the experimental setup. The circularmotion of the fluid flow is due to the non-zero azimuthal velocity component of inflowingwater.

6.4.2 Experimental procedure

By altering the inflow from the tap, a stationary vortex was obtained at various fluidheights. For each setting we took two independent data sets. There were 4 mainexperiments done with asymptotic fluid heights of approximately 180, 78, 63 and 25mmrespectively.

The first experimental goal was to obtain the free surface water profile. For thisdetection we focussed a two-dimensional light sheet from above, generated by a slideprojector shone through a narrow slit, onto a diameter of the cylindrically symmetricflow. We injected a small amount of Kalliroscope flakes into the water, recording thefree surface as illuminated by the light sheet. The images were taken using a digitalcamera.

The second experimental goal was to measure the angular and radial flow velocitieson the free surface. We placed a single particle (small piece of paper) on the surfaceof the water near the vortex and recorded the trajectory as it propagated towards thecentre of the vortex using a digital hand held camera fixed to an overhead structurepointing vertically down at the vortex.

We further applied flow visualization techniques in the bulk of the flow, to study indetail the bathtub vortex flow behaviour and its relevance for the analogy.

6.5 The results so far

The results so far collected are preliminary and served mainly as a testing ground tofine-tune our experimental setup and data collection methods.


The data analysis for both the sequence of images taken from the side and themovie from above were carried out in Matlab.

The data analysis from the photographs of the free surface and movies of the testparticles was carried out mainly by Silke Weinfurtner while the numerical code simu-lating the scattering process was written by Jason Penner.

Free surface profile

For each experimental run we recorded 10 profile images of the surface height function.In every image we applied several steps to determine the free surface profile 4 shownin Figure 6.4 .

These smooth fits were obtained from a non-linear regression optimization proce-dure on a three dimensional space of functions (each of the real number parametersshown were in fact given to a larger precision than two decimal places in the opti-mization). Note that the function space used in exprimental run. 4 differs from thoseused in the other three experiments in that a tanh function is used in place of thearctan shape. This was necessary in order to obtain a reasonable fit to the data but, asevidenced by a residuum data from that experiment (not shown here), even this spacewas not able to approximate the free surface to the same accuracy as was possiblefor the other three experiments. One possible explanation for this discrepancy is thatexp. 4, being the shallowest experiment we performed, may be described by a flowdynamics in which viscosity effects are non-negligible or at least play a more significantrole in the dynamics than in the other three experiments. An important remark isthat the functions that best fit the experimental data are different from the theoreticalprofile obtained, see eq. (6.15). This accounts for the fact that the fluid flow in theexperiments wasn’t completely irrotational and inviscid, as assumed in the theoreticalanalysis. Other contributing factor for this difference is that the slope of the heightprofile is not always negligible.

Our observations of the bulk flow

As a preliminary investigation we have performed a series of experiments observing thesurface flow velocities and free surface (described below) while also making some ob-servations of the bulk fluid dynamics by inserting colored dyes into the fluid at variousradii. We have observed a stagnant attractor region internally surrounding the vortex

4The water circuit in our experiments was not recycled, being allowed to drain into the sink afterflowing out the bottom of the tank, such that standard methods, i.e. illumination of free surface inwater with uniformly distributed fluorescent dye, could not be used. Instead, we worked with emergedKaliroscope flakes, and a more involved data analysis procedure to measure the water profile. Anyfurther analysis, such as studying the actual free surface excitations, the setup needs considerableimproved with this respect.

6.5. The results so far 197

Figure 6.4: The free surface data (in colored dots) superimposed with the result of anon-linear regression curve fitting procedure to determine an analytic form for the freesurface. The discrete data is fit here by a smooth functions used in our numerical codeto calculate superradiant scattering amplitudes.

and beneath the surface; the dyes coalesced in a torus shaped internal region and didnot dilute even over time scales of minutes while the fluid was in a stationary phasewith draining and inflow. This observation is schematically represented in fig 6.5. Sucha region is reminiscent of a similar stagnant region shown to exist in accretion disksaround rotating black holes [143]. This observation also shows that the approxima-tions leading to (6.15) are not accurate specifically the approximation that vr(r, z) isindependent of height z.

One particularly interesting feature of this non-draining region is the fact that thereappeared to be no ‘rotational’ component to the toroidal shaped flow. We observedthe dye to spiral inwards radially to a minimal radius leaving well separated flow linesvisible from above; the spiral flow lines did not mix and remained stable over timescales of 30 seconds and more.

In a future work we hope to investigate the manipulation of the size and characterof such a region by modifying the fluid flow. For example, it is our hypothesis thatthe intensity of the stagnant region is related to the relative radial flow due to Ek-


Free surface

Figure 6.5: Schematic diagram of the observed stagnant region in the bulk flow of ourdraining vortex.

man pumping and the radial flow present on the surface boundary layer. Hence byoptimizing the flow parameters to a region which disfavors a strong Ekman pumping,we hope to minimize the effects of the stagnant region. Note that in the theoreticalanalysis presented in this article above we have assumed a simplified laminar, vorticityfree internal flow structure neglecting viscosity.

Numerical simulations

We obtained the values of A and B (see equations (6.13) and (6.6) above) from measure-ments of the flow velocity by tracking test particles and using a particle velocimetrymethod. Subsequently we put these dat into a code which simulates the scatteringprocess. The output from the code is a value of T and R as a function of frequencyω. For low ω superradiant scattering is expected to occur in line with the generalunderstanding of superradiance (see 2.1.4).

A sample of the results from simulating the scattering process according to thedate collected from the first experiment is given in Figure 6.6 where we clearly seesuperradiant reflection coefficients. We note that these exact spectra are found using anon-dispersive theory described in Section 6.2. We include in Figure. 6.7 the true waterdispersion relation for water as well as the group velocity of surface modes showing theregime in which the non-dispersive result can be trusted. In Figure 6.8 we show theresults zoomed in on the non-dispersive window.

6.6. Conclusion and future directions 199

0 10 20 30 40 50 60 70 80 90 1001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

ω

|R|2

Reflection coefficient for superradiance

Exp.2, m=1Exp.2, m=2Exp.3, m=1Exp.32, m=2Exp.4, m=1Exp.4, m=2

2

33

44

2

Figure 6.6: The numerical output (the squared reflection coefficient |R|2) from a sim-ulation of the scattering process on the vortex neglecting dispersive effects. Thesesimulations were carried out with date collected from the experiment which includedthree different vortices differing in their asymptotic depths at large radii. These differ-ent vortices are labelled in this plot as exp 2 - 4 and we have also included the resultsfrom simulating the scattering of the first excited angular mode (m = 2) showing asmaller amount of superradiance as can be expected from (6.25). Note that the datafrom experiment 1 was of a lower quality (as can be seen in Figure 6.4 where the freesurface data seems to incorporate some systematic error (we fund that at this radiusduring experiment 1 there was a reflection of the light sheet from the bottom of thetank contaminating the image of the free surface) .

6.6 Conclusion and future directions

The series of experiments described above are ongoing at present. There have beentwo additional separate rounds of experimental testing performed on two differentprototypes for the full scale experiment.


Dispersion curveDispersion curve Shallow water limitShallow water limitk

00 5050 100100 150150 200200

omega

00

2020

4040

6060

8080

100100100

Divergence of dispersion from acoustic regime

(a)

Wave velocity Shallow water velocityk

0 50 100 150 200

velocity

0

0.1

0.2

0.3

0.4

0.5

0.6Group velocity of water waves

(b)

Figure 6.7: The true dispersion relation for water (given in (6.26)) compared to thelinear dispersion used in the numerical simulations in 6.7(a) while in 6.7(b) we showthe corresponding group velocities for surface waves. Note that the region in which thelinear dispersion is an accurate approximation is for approximately ω < 25 .

Of course, as we saw in Figure 6.6 our experiment can be said to have succeededin showing superradiance. However, there are good reasons to believe that both thetheory used and the experimental data obtained were too rough to completely trustthe result. At the very least, we can see in Figure. 6.7 that the results certainly shouldnot be trusted beyond about ω = 25s−1 and that even within this regime, the modelpredicts up to about 20% amplification which should be observable by the naked eyeand which we did not observe. Ultimately this is why we consider the results so faronly as point of principle that our methodology can be carried out. It is the goal nowto improve on the details of each step of the process in order to reach a conclusiveresult which can be shown to be trustable.

There are a number of different scientific projects which present themselves at thisstage of the experiment and we plan on pursuing as many as possible in the in nearfuture.

Experimental – After working with the circular bucket tank described above, wehave concluded that a different design is necessary in order to produce a smootherflow. One major problem with the circular bucket was that the inflow pipe penetratedthe free surface, generating turbulence and surface disturbances. Another issue wasthat the bucket was not completely transparent, making imaging of the free surface adifficult geometric problem. Finally, our data collection methods were crude: we used


2 4 6 8 10 12 14 16 18

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

ω

|R|2

Reflection coefficient for superradiance

Exp. 3Exp. 2Exp. 4

432

Figure 6.8: A zoomed in plot of some of the results of Figure 6.6 showing the behaviourin the region where we can trust the approximation which neglects dispersion

standard personal digital cameras to take images and movies, subsequently importingthem to a computer for analysis.

These difficulties have prompted us to come up with an improved design, shown inFigure 6.9, made of transparent material with long, isolates inflow channels allowingbetter flow characteristics and visualisation. We have also invested in some appropriatehardware, such an a dedicated computer for data analysis which streams data realtimefrom a high resolution black and white mounted camera. In the more distant futurewe plan on constructing a large-scale version of the improves experimental setup andbegin taking data fro it (expected to be running early in 2013).

Theory – It would be interesting to investigate the theoretical problem of wavespropagating on a highly curved surface such as the free surface profiles we found aboveand have observed in the lab. So far all theoretical analyses involve the assumption thatthe restoring force is the gravitational force. One can imagine that this approximationis poor when surface waves propagate into the vertical neck of the vortex core. In


Inflow

Inflow

Free surface

Top view

Side view

Figure 6.9: The new design for the tank of the draining bathtub vortex.

the neck, the waves propagate in fact vertically, fighting against the downflow ratherthan the radial flow we have been assuming. This is partly the source of some ofthe divergences we have seen in the linear theory. Furthermore, in the neck region,the gravitational force in not anymore the restoring force responsible for wave likebehaviour which most likely would be dominated by centrifugal forces and surfacetension likely giving rise to a highly modified dispersion relation. This at the very leastwould give a vastly different propagation velocity for long wavelength excitations andhence a different value for rH and hence profile of superradiance. The idea is that,thefurther the horizon site into the 1/r rotational velocity field, the higher the effectiveangular momentum of the horizon is and hence the larger the frequency below whichsuperradiance should occur is. The very definition of long wavelength excitation shouldalso change character in the neck region where the typical length scale is associatedwith the rotation rate and capillary effects. These two additional forces have beencompletely ignored in the theory above. It should be very interesting to calculate thedispersion relation for such waves and study the effect it has on superradiance.

Numerical – On the numerical side, the problem of including modified disper-sion relations comes hand in hand with solving difficult higher order ODEs. It hasbeen discussed in the literature [60, 118] some of the numerical difficulties of workingnumerically with such problems which include exponentially growing modes and thereduction of noise. We saw in Chapter 5 that, one cannot expect, in general, for thecontributions from the new, dispersive channels be small.

Conclusions and Outlook

This thesis has been about the (sometimes tortuous) path between theory and exper-iment in analogue gravity. The relative sophistication of the theoretical developmentsand experimental techniques and technologies varies markedly between the diverse ana-logue systems and we are often led to solve quite different problems when working onthe physics of different analogue models.

In Chapter 3 we looked at analogue models in BEC which have been widely studiedin the literature. The theories of dispersive Hawking radiation and DCE in BEC arewell developed and the most pressing issues are in practically observing the quantumemissions. In this light the idea that quantum correlations are the right observable hasarisen and several authors have looked at the structure of the quantum correlationspresent in a Hawking emission. We have looked at the related problem for DCE andshown that there exists a very particular signal buried in the Wightman function dueto the DCE emission. We showed how BECs can be used to simulate various DCEscenarios including emissions from inhomogeneous and finite size condensates whichare most similar to real BEC being currently produced experimentally.

In Chapter 4 we entered to discussion surrounding a series of optics experimentswhere a claim was made in the literature [91, 92] of an direct observation of Hawkingradiation. We modeled the experimental situation described in those articles with avery simple construction and showed how most of the pertinent features of the observedemission were reproduced in our model. In doing so we also highlighted the difficultyof making clean claims of the observation of QFT phenomena in dispersive analoguesystems since multiple and distinct physical mechanisms can simultaneously play arole.

In Chapter 5 we opened the discussion of the robustness of the phenomenon ofsuperradiance against quartic modifications to the dispersion relation, both sub- andsuperluminal showing the qualitative result that superradiance is possible in both staticand moving media. We also began the study of the quantitative robustness for low mo-mentum scattering processes, showing that the spectrum is roust when the dispersivemomentum scale is large. Finally, we also showed that the additional scattering chan-nels can in fact enhance the effects of superradiance making its detection more feasiblein real experiments.


In Chapter 6 we described an ongoing experimental project which aims to directlyor indirectly observe superradiant scattering from a common ‘bathtub’ draining watervortex. We discussed the difficulties faced when trying to confront the real physicalsystem with idealised theoretical work. In particular we observed that the analogycertainly breaks down near the center of the vortex and highlighted the main experi-mental task: to generate and measure a draining vortex which supports an analoguedescription (or modified dispersive version thereof) of an ergo-region sufficiently faraway from the highly non-linear region of the flow. The central core region is is poorlyunderstood both in terms of the existence of an analogue there as well as the basicfluid dynamics of that region.

Generalization vs Simulation — In the analogue gravity literature, the orig-inal goal of simulating Lorentz invariant physics seems to have been absorbed intothe wider, more far reaching goal of placing the Lorentz invariant results into a moregeneral context including violations of the linear dispersion relation. Traditionally thephilosophy of analogues has been that when we reproduce some phenomenon in ananalogue we are simulating the true effect of curved spacetime QFT. What has comeout more recently, under the guidance of Parentani (summarised in [118]) and his col-laborators but also stimulated by the extremely interesting experiments done by thegroup of Faccio et. al. (reported in [91, 92]), is the philosophy that the most importantthing we are doing is generalising some Lorentz invariant curved spacetime physics tothe Lorentz non-invariant, non-spacetime context. We are discovering that the plethoraof phenomena are really much more general processes in physics. What we are alsodiscovering, and this particularly applies to the lessons from the optics experiments,is that some of the phenomena might even be different aspects or different ways oflooking at the same underlying dispersive process. When dispersion is switched off,the effects appear to decouple, becoming quite distinct in their description and formu-lation. Here we cite two cases which point to this possibility: The Hawking processand superradiance both are represented as multi channel scattering processes in thepresence of dispersion as discussed in Chapter 5; the vacuum radiation associated withpropagating optical pulses as discussed in Chapter 4 connects the DCE and Hawkingprocess, where moving to a different frame of reference was seen to isolate one or theother aspect (Hawking or DCE) of the vacuum radiative process.

This vision for the unification of all the various quantum vacuum radiations indispersive theories is schematically shown in Figure 6.10. Such a unified perspective ofthe various quantum vacuum radiation effects is also presented in the recent article byDavies [144] but in the context of Lorentz invariant physics.

Outlook — Here we briefly summarise the status of the theoretical work as itpertains to analogue models for the three quantum vacuum radiations discussed in thisthesis:

• Dynamical Casimir effect – Here the theory is well developed and largely freeof complications. There exists a number of (perhaps) distinct versions of what


Hawking

Dynamical Casmir

Unruh

Superradiance

Quantum Vacuum

Radiation

Figure 6.10: The various quantum vacuum radiations seen as different corners of aunified phenomenon.

is referred to as DCE with the ‘moving or time dependent boundary condition’version currently receiving the most experimental attention. The spectrum iswell known in many scenarios and some exact analytical results available. Onedifficult theoretical challenge is the incorporation of finite size effects which weaddressed in Chapter ?? from within the confines of perturbation theory. TheDCE theory in dispersive systems is made straight forward by the fact that it canbe a ‘regular’ emission (as we have defined it) in some systems which sidestepsmany (but not all) of the transplanckian problems plaguing the Hawking emissionfor example.

• Superradiance – Non-dispersive superradiance, its field theoretical formulationand the quantum spontaneous version are well understood theoretically beingmature disciplines and playing a major role in the formulation of the black holethermodynamics. The generalisation to dispersive wave scattering is very new,initiated in this thesis and motivated by a desire to experimentally observe super-radiant scattering using (dispersive) gravity waves on a draining bathtub vortex.The connection with dispersive Hawking radiation will be interesting to investi-gate.

• Hawking radiation – Being one of the only predictions from semi-classical gravity,the non-dispersive Hawking emission has received wide interest from many com-munities interested in gravitational physics. Motivated by analogue models, the

study of dispersive Hawking radiation is also well developed for both sub- and su-perluminal modifications to the dispersion relation. Numerical simulations haveshown a remarkable robustness of the spectrum of produced particles and certainmore exotic configurations: multiple horizons, ‘time’ warps [145] and black holelasers [146] have also been thoroughly investigated.

On the experimental side, important progress has been made towards an observa-tion, direct or indirect of one or more of the quantum vacuum radiations. We summarisehere the status of some of the current, ongoing, and planned experiments in some wayrelated to analogue gravity:

• BEC — There are many groups around the world working on the experimentalside of BEC. Recently Steinhauser and collaborators have made a BEC blackhole. [147].

• Gravity waves — Already there have been important experimental results ob-tained concerning the dispersive Hawking effect by Weinfurtner et. al [23] aswell as Rousseaux [48]. The so called ‘hydrodynamical jump’ white hole has alsoreceived experimental attention [49]. Superradiant scattering is currently underinvestigation. It is fair to say that the experimental situation for gravity waveson water is very positive, being very simple to work with but, however, are notsuitable for an observation of the quantum spontaneous emissions.

• Nonlinear Optics — Experimental non-linear optical systems [93] perhaps cur-rently provide the best chance to directly observe the quantum spontaneousemissions and in fact have already yielded observational results in this direc-tion [91, 92]. Much experimental work needs to be done on these complex andhighly dispersive systems, the quantum processes within them and their abil-ity to support analogs of the various spacetime concepts such as horizon andergo-region. Due to the high complexity and sophistication of the technologiesnecessary to undertake these optics experiments, the groups doing experimentsare usually interested in many problems of which analogue gravity simulationsare just one.

In summary, it is fair to say that there is much work to be done, both theoreticallyand experimentally, but the current and recently generated interest in analogue gravityproblems by a diverse range of experimental groups, allows the community to start todelete the question mark from the title of the founding analogue gravity article [1]‘Experimental Black Hole Evaporation?’ as well as the related question marks forDCE and superradiant scattering.

Appendix A

Miscellaneous useful results

A.1 General method for getting rid of the friction

term

Frequently we wish to solve some differential equation with a friction term, a terminvolving the first derivative of the function. In this small appendix we demonstratehow one can in general reduce such an equation to one involving only 2nd derivativesand the function. Such a reduction often results in an equation of motion which canbe interpreted as a Schrodinger scattering problem in a particular potential — a classof problems with many known solutions or as an equation of motion such as (2.25)amenable to a Bogoliubov analysis. In this way one can select an appropriate solvedschrodinger potential which maps to the potential of the original frictional equation,even approximately. The idea here is that, the original equation is usually a toy model,or phenoomenological model for some phenomenon for which we have a choice of thepotentials which best match the physical system we are studying.

Consider the equation of motion

λf ′′ + µf ′ + νf = 0. (A.1)

Let the function g be defined as

f = ge−12

∫ x µ/λ. (A.2)

Then, [λg′′ +

(ν − λµ2/4− λµ′/2

)g]

e−12

∫ x µ/λ = 0. (A.3)

which is of the desired frictionless form.

A.2 Integrals

Here we collect some proper integrals which we use in the text∫ ∞0

un

sinh2(u)du =

πn

n(n+ 1), (A.4)∫ ∞

0

u2e−u2

du =

√π

4, (A.5)∫ ∞

0

u4e−u2

du =3√π

8, (A.6)∫ ∞

0

u3/2e−βu2−γudu =

3

4

√π(2β)−5/4eγ

2/(8β)D−5/2

(γ√2β

), (A.7)

where D is a parabolic cylinder function.

A.3 Gamma functions

Here we collect some facts about Gamma functions useful in the evaluation of certainBogoliubov coefficients. Firstly

Γ(1 + z) = zΓ(z), (A.8)

coupled with Γ(z) = Γ(z) and Euler’s reflection formula

Γ(1− z)Γ(z) =π

sin πz, (A.9)

gives rise to

|Γ(iA)|2 =π

A

1

sinhπA. (A.10)

Here we also used the definition of the hyperbolic function

isinh z = sin iz. (A.11)

Another important feature of the Gamma function is that it is nowhere vanishing.This can be seen from the relation A.9: π/sinπz is nowhere vanishing. Finally, someidentities involving Γ functions are given below.

Γ(nz − n) =Γ(nz)∏n

p=1(nz − p), (A.12)

Γ (nz − n/2) = Γ(nz − 1/2)× stuff. (A.13)

Appendix B

Superradiance Appendices

B.1 A pole in the scattering solution and its reso-

lution

Here we present a smooth model for an ergo-region which is exactly soluble and forwhich we write down exact expressions for the reflection and transmission coefficients.This model was discussed in Chapter 5. The purpose of this section is to demon-strate that the pole uncovered in the reflection and transmission amplitudes for a non-dispersive superradiant scattering process in 1 dimension with a step function potentialis an unphysical artifact of choosing the discontinuous step function.

B.1.1 The exact solution for modified and linear dispersion

The equation of motion we considered was

f ′′ + (ω − eΦ(x))2 f = 0, (B.1)

where the potential eΦ(x) was taken to be step function

eΦ(x) =

0 x < 0

eΦ0 x > 0.(B.2)

Solving for the momenta in the two regions we have we have

kin → ω,

kr → −ω,kt → −|ω − eΦ0|. (B.3)

210 Appendix B. Superradiance Appendices

Matching the solution across the step function up to third derivatives gives a set oflinear equations for the coefficients which has solution

R =kin − kt1kin + kt1

=eΦ0

2ω − eΦ0

,

T =2kin

kin + kt1=

2ω

2ω − eΦ0

, (B.4)

from which we confirm the standard and general result for the statement of superradi-ance

|R1|2 = 1− ω − eΦ0

ω|T1|2. (B.5)

We see that the general solution possesses a pole in both the reflected and trans-mitted amplitudes when the reflected momentum coincides with the transmitted one.This occurs when ω = eΦ0/2. In the next section we will show that such a pole isavoided by choosing a smooth potential Φ to replace the discontinuous step function.

B.1.2 Smooth model

Consider the potential

Φ(x) =

ω −

√ω2 − V (x) , x < xm

ω +√ω2 − V (x) , x > xm

, (B.6)

where

V (x) =Vf2

[1 + tanh(x/x0)] +Γ

cosh2(x/x0), (B.7)

and xm is a matching point, coinciding with the local maxima V (xm) of the functionV . Assume ω < eΦ0. Choosing

Vf = ω2 − (ω − eΦ0)2, (B.8)

and

Γ =(eΦ0)2

4, (B.9)

we find that V (xm) = ω2 and the function Φ is continuous and interpolates smoothlymonotonically between 0 and eΦ0 over a scale of the order of x0. This correspondsto the promised smooth transition to an ergo-region. This behaviour is shown inFig. B.1 where we also display the auxiliary potential V . In fact, it can be showngraphically (although algebraically it is somewhat of a mystery) that the function Φexactly corresponds to a tanh interpolation between 0 and eΦ0,

Φ(x) =eΦ0

2[1 + tanh(x/x0)] . (B.10)

B.1. A pole in the scattering solution and its resolution 211

Figure B.1: The behaviour of the potential Φ(x) and the auxiliary potential V (x) usedin the smooth ergo-region model described in the text. Here we have used the valuesω = 2, x0 = 1 and eΦ0 = 3. The

Crucially, the mode equation

d2f

dx2+ (ω − Φ(x))2f = 0, (B.11)

is written in terms of the potential V as

d2f

dx2+ (ω2 − V (x))f = 0, (B.12)

and it can be shown [148] that Eq. (B.12) with the potential V given by the so-calledRosen-Morse form (B.7) is exactly soluble in terms of Hypergeometric functions. Thetransmission coefficient is given exactly by

|T |2 =sinh(πωx0) sinh(π(eΦ0 − ω)x0)

sinh2(πeΦ0x0/2) + cos2(π/2√

1− (eΦ0)2x20)

(B.13)

By the general result B.5, we have that the reflection coefficient R is given by

|R|2 = 1− ω − eΦ0

ω|T |2 (B.14)


and we see that the special role played by ω = eΦ0/2 in the step function model now nolonger applies and we are left with a bounded (and superradiant) reflection coefficientfor all ω < eΦ0. This was shown in Chapter 5 in more detail.

This result shows that the pathological pole in the reflection and transmission am-plitudes for the step function model are an artifact of the unphysical discontinuousjump in the potential Φ.

An interesting observation is that when 2ω = eΦ0 (exactly where the pole waspresent in the step function model) the Rosen-Morse potential B.7 reduces to a sim-ple sech2 potential and the transmission coefficient becomes that for the sech2 modelaccordingly.

We note that this construction does not work for ω > eΦ0.

B.2 Conserved inner product with flow velocities

For a higher order equation of motion such as the one considered in Chapter 5 therestill exists a conserved inner product. This was given in Chapter 5 by

(φ1, φ2) = i

∫Σt

[φ∗1 (∂t − ieΦ +W∂x)φ2 − φ2 (∂t + ieΦ +W∂x)φ∗1] dx. (B.15)

Here we show that this is conserved when the dispersion function F 2 is an even function(so that its Taylor expansion is in powers of k2).

Firstly, it is clear that the potential eΦ(x) plays no role in the conservation since itcancels directly from the innerproduct.

Indeed we can write the inner product as being only over the dimension of interestin terms of the metric restricted to the (t, x) plane as

(φ1, φ2) = i

∫dx nαg

αβ (φ∗1∂βφ2 − φ∗2∂βφ∗1) := i

∫dx nαJ

α, (B.16)

so that the difference of of this quantity at two times isis given by the directed integralover the spacetime volume contained between the two time slices. By Stoke’s theoremthis is equal to the integral over the spacetime volume

(φ1, φ2)t2 − (φ1, φ2)t1 =

∫V

∇αJα =

∫V

∂αJα, (B.17)

since the determinant of the metric is trivial. The integrand is

∂αJα = ∂α

(gαβ (φ∗1∂βφ2 − φ2∂βφ

∗1))

= φ∗1[(∂t + ∂xv) (∂t + v∂x)− ∂2

x

]φ2 − φ2

[(∂t + ∂xv) (∂t + v∂x)− ∂2

x

]φ∗1

= φ∗1∂4xφ2 − φ2∂

4xφ∗1. (B.18)

B.3. Perturbation theory for the roots of the modified dispersionrelations 213

Note that this last line would be zero in the non-dispersive case. However, under theintegral this term does not contribute upon integration twice by parts whence∫

φ∗1∂4xφ2 − φ2∂

4xφ∗1 =

∫(∂2xφ∗1)(∂2

xφ2)− (∂2xφ2)(∂2

xφ∗1) = 0 (B.19)

This argument does not suffice when the dispersion function (squared) cannot bewritten as an expansion in even powers of the derivative operator since this last can-celation would then not be possible with an even splitting of the derivatives across thetwo fields.

B.3 Perturbation theory for the roots of the mod-

ified dispersion relations

The dispersion relation for the non-zero flow model in the asymptotic regions is givenby

k2 ∓ k4

Λ2=

ω2 x→ −∞(ω − eΦ0 − kW )2 x→∞

(B.20)

To solve these relations for k in terms of the other parameters of the problem it isbest to move to the dimensionless variable z = k/Λ. Furthermore we make use of thesmall parameters ε = ω/Λ and ε = (ω − eΦ0)/Λ which are small by the assumptionthat we are well outside of the region where the dispersion relation has no real roots(near the top of the figure 8) or alternatively in the physical regime of low frequencieswith respect to the dispersive scale. In terms of these variables the dispersion relationbecomes

z2 ∓ z4 =

ε2 x→ −∞(ε− zW )2 x→∞

. (B.21)

Restricting to the subluminal case we have in the ‘free’ region (where the potentialsand velocities vanish, the x → −∞ region) four roots expressed perturbatively up toO(ε3) as

z− = −1 +1

2ε2 (B.22)

z−s = −ε (B.23)

z+s = ε (B.24)

z+ = 1− 1

2ε2 (B.25)

where we have labeled the roots as either positive + or negative - with the extrasubscript ‘s’ to indicate the smaller (in absolute value) roots. It s the ‘s’ roots which


exist in a non-dispersive treatment, the other two roots appearing due to the modifiedquartic equation of motion and resulting quartic polynomial dispersion relation.

In terms of the problem parameters in the non-interacting region these roots are

k− = −Λ +1

2Λω2 (B.26)

k−s = −ω (B.27)

k+s = ω (B.28)

k+ = Λ− 1

2Λω2 (B.29)

which agrees, as it should, with the expansion of the exact solutions in the free region

k− = − Λ√2

√1 +

√1− 4ω2

Λ2(B.30)

k−s = − Λ√2

√1−

√1− 4ω2

Λ2(B.31)

k+s =Λ√2

√1−

√1− 4ω2

Λ2(B.32)

k+ =Λ√2

√1 +

√1− 4ω2

Λ2(B.33)

when expanding in powers of ε = ω/Λ.

In the interacting region, for W < 1 one finds

z− = −√

1−W 2 + εW

1−W 2+ε2

2

2W 2 + 1

(1−W 2)5/2(B.34)

z−s = −ε 1

1−W(B.35)

z+s = ε1

1 +W(B.36)

z+ =√

1−W 2 + εW

1−W 2− ε2

2

2W 2 + 1

(1−W 2)5/2. (B.37)

It is rather illuminating to re-write this expansion in terms of the variable λ =

B.3. Perturbation theory for the roots of the modified dispersionrelations 215

ε/(1−W 2)3/2 and q = z/√

1−W 2 as

q− = −1 +Wλ+ (2W 2 + 1)λ2

2(B.38)

q−s = −(1 +W )λ (B.39)

q+s = (1−W )λ (B.40)

q+ = 1 +Wλ− (2W 2 + 1)λ2

2(B.41)

In terms of these variables the dispersion relation reads

q2 − q4 = (1−W 2)λ2 − 2Wqλ (B.42)

where now λ is the small parameter. This representation shows us that the relevantsmall parameter governing the asymptotics is λ and not ε. This is also what is foundin the article [118] where a cubic dispersion relation is used.

In any case, in the physical variables the roots are approximated by the series

k− = −Λ√

1−W 2 +W (ω − eΦ0)

1−W 2+

(2W 2 + 1)(ω − eΦ0)2

2Λ(1−W 2)5/2(B.43)

k−s = −(ω − eΦ0)

1−W(B.44)

k+s =(ω − eΦ0)

1 +W(B.45)

k+ = Λ√

1−W 2 +W (ω − eΦ0)

1−W 2− (2W 2 + 1)(ω − eΦ0)2

2Λ(1−W 2)5/2(B.46)

B.3.1 Critical case

Let us consider the case of a subluminal dispersion. By taking the derivative of theexpression (B.20) with respect to k, we obtain the group velocity of the modes in thelab frame as a function of the wavenumber k,

vg(k) =∂ω

∂k= W0 ±

1− 2 k2

Λ2√1− k2

Λ2

= W0 +k − 2 k

3

Λ2

ω − eΦ0 − kW0

. (B.47)

In principle, the two equations vg(k) = 0 and the dispersion relation (B.20) are sufficientto obtain k0 and W0, the critical momentum of blocked upstream modes and the criticalflow velocity respectively, as functions of Λ, ε, and eΦ0. However, using the quarticdispersion relation we choose here, the algebra does not allow for a simple solution. On


the other had it is easy to obtain k0 as a function of W0 by straight forward substitutionWe have

k0 = (ω − eΦ0)3W0 +

√W 2

0 + 8

2(W 20 − 1)

. (B.48)

Plugging in the expression above into the condition vg(k0) = 0 we find

W0 =ω − eΦ0

2k0

+1

2

√ω − eΦ0

k20

− 8k20

Λ2+ 4 (B.49)

The wavenumber k0 also corresponds to a stationary (non-propagating) double rootof eq. (B.20); the other two solutions correspond to the propagating modes. They canbe obtained explicitly in terms of k0 and W0 (see fig. (5.10)),

kt1t2

= −k0 ± Λ

√1− 2k2

0

Λ2−W 2

0 . (B.50)

One can obtain explicit expressions for k0 and W0 by making the approximation

± (k − 1

2Λ2k3) ' ±

√k2 − k4/Λ2 = (ω − eΦ− kW ) . (B.51)

which is valid when k/Λ 1, which is accurate for the turning point when ω Λ.This approximation is accurate even for the largest roots up to a maximum error of afactor of

√2.

Using this approximation we find the results

k0 =[−Λ2(ω − eΦ0)

]1/3(B.52)

1−W0 =3

2

[−(ω − eΦ0)

Λ

]2/3

(B.53)

which are valid, as stated above, when ω,Φ0 Λ (note that, as usual, ω < eΦ0 inthese computations). Here we see that 1 −W0 and 1/k0 scale in the same way withΛ. This result was also found in [118] in the context of superluminal dispersion andHawking radiation where a turning point existed for the negative norm modes comingfrom within the supersonic flow region. Here the turning point exists for positive normmodes in the subsonic region.

B.4 Cylindrically symmetric dimensional reduction

The most general analogue metric for perturbations in a moving fluid written in cylin-drical coordinates is given by

g ∝

c2 − v2 vr r2vφvr −1 0r2vφ 0 −r2

(B.54)

B.4. Cylindrically symmetric dimensional reduction 217

where we have chosen the (+−−−) signature. Assuming homogeneity and a constantwave speed c we may take the constant of proportionality to be unity.

Hence the action is given by

S =1

2

∫dtdrdφ r

√ggµν∂µφ

∗∂νφ

=1

2

∫dtdrdφ r

[| (∂t + vr∂r + vθ∂θ)φ|2 − |∂rφ|2 −

∣∣∣∣1r∂θφ∣∣∣∣2]

(B.55)

Integration by parts is best done in the more abstract notation

S =1

2

∫d3x√g [∂µφ

∗(gµν∂νφ)] (B.56)

= −1

2

∫d3x√g [φ∗∇µ(gµν∂νφ)] (B.57)

which gives rise to the familiar field equation

∂µ (√ggµν∂νφ) = 0 (B.58)

Dividing through by√g we find

(∂t +1

r∂r (rvr) + vθ∂θ) (∂t + vr∂r + vθ∂θ)φ (B.59)

− 1

r∂r (r∂rφ)− 1

r2∂2θφ = 0 (B.60)

which can also be found by direct integration by parts of the explicit form of the action(B.55).

The radial version of the Fourier transform is the order 0 Hankel transform

H(f)(k) =: fk =

∫ ∞0

drf(r) rJ0(kr) (B.61)

where J0 is the order 0 Bessel J function. Using this we can show that, for example,the purely radial derivative term becomes

H(

1

r∂r (r∂rφ)

)= −k2H(φ) = −k2φk (B.62)

Proof : Using J ′ν(x) = Jν−1(x)− νJν(x)/x and J−ν = (−1)νJν we have

H(

1

r∂r (r∂rφ)

)=

∫ ∞0

drJ0(kr) ∂r (r∂rφ)

= [boundary term]−∫ ∞

0

dr (krJ1(kr)) ∂rφ

= [boundary term] + k

∫ ∞0

dr ∂r(rJ1(kr))φ

= −k2

∫ ∞0

dr rJ0(kr)φ = −k2φk (B.63)


However, in general the first order radial derivative terms do not enjoy such a niceanalogy with the plane wave case where first derivatives become −ik. In this radialcase, the “second derivative” operator (that operator which has as −k2 the eigenvaluewhen acting on the Bessel basis, which arises naturally in cylindrical coordinates asabove) does not simply factorise into two first order derivatives as in the plane geo-metric case where −d2/dx2 = (−id/dx)(−id/dx). For this reason, under the radialFourier transform (zeroth order Hankel transform) the wave equation does not becomean algebraic equation for the coefficients of the zeroth order transform. Instead thecoefficients become coupled to higher order Hankel transform (the Hankel transforminvolving higher order Bessel functions instead of the zeroth order ones) coefficients.We have, in regions where both vr and vθ are constant (where we usually read off thedispersion relation),

H(0)(∂t+1

r∂r (rvr) + vθ∂θ) (∂t + vr∂r + vθ∂θ)φ

= −k2H(0)φ+ (∂t + vθ∂θ)(−kvrH(1))φ+ (∂t + vθ∂θ)(kH(1) −H(0) 1

r)φ

+ (∂t + vθ∂θ)2H(0)φ. (B.64)

In the planar case, each of these two extra factors (proportional to the first order Hankeltransform H(1)) would be −ik which could be brought back into the squared factor torecover the perfect square form.

Another new feature of this field equation is that the angular velocity contributiondoes not factorise in the way we anticipated above in our toy model. Expanding inthe standard plane wave angular basis exp(imθ), as well as in a plane wave frequencybasis exp(−iωt) we do obtain a nice algebraic equation for the coefficients but it is ofa sightly difficult form involving a mass term in addition to the anticipated terms

1

r∂rr∂rf(r) +

[(ω + i

1

r∂rrvr −mvθ

)(ω + ivr∂r −mvθ

)+m2

r2

]f(r) = 0

Note that this equation is valid also in regions where vr and vθ have non-trivial rdependence. It is possible to show, however, that the presence of this mass term doesnot exclude the existence of superradiance. We will discuss this below in the contextof a simpler planar model.

This general equation of motion may also be written as

f ′′(r) + P (r)f ′(r) +Q(r)f(r) = 0 (B.65)

where

P (r) =1

r+ 2ivr(ω −mvθ)−

1

r∂r(rv

2r) (B.66)

Q(r) = (ω −mvθ)[(ω −mvθ) + i

(vrr

+ ∂r(vr))]

− imvr∂r(vθ) +m2

r2(B.67)

which can be put in the frictionless form by a suitable change of variables.

Appendix C

Optics Appendices

C.1 Exact homogeneous model for DCE – tanh(·)profile.

The equation of motion for the 1 + 1 dimensional electric field Φ with a time variableand momentum dependent refractive index n is

1

c2

(n2(t, k)φ,t

),t

= φ,xx, (C.1)

which can be written as∂2τφk + c2k2n2(τ, k)φk = 0, (C.2)

upon Fourier transformation and coordinate transformation

τ(t) =

∫ t dt′

n2(t′). (C.3)

We choose for the refractive index a tanh time variable perturbation on top of momen-tum dependent background value

n2(t, k) = n0(k)2 + 2η n0(k) tanh(t/t0) +O(η2), (C.4)

220 Appendix C. Optics Appendices

where η is a small parameter and n0(k) would be a model for the refractive indexsuch as the Sellmeier approximation. This choice is motivated by exact solubility ofthe model but also for its close approximation to the experimental Gaussian profilereported in [?] supplemented with the effects of pulse steepening whereby an initiallygaussian-like RIP develops a steep trailing edge and a shallow leading edge (mentionedagain below).

Then the equation of motion becomes

φk,ττ + c2k2(n2

0 + 2η n0 tanh(t(τ)/t0))φk = 0, (C.5)

which describes a time independent Schrodinger scattering problem

ψ,xx +2m

~2(E − V (x))ψ = 0, (C.6)

with V (x) = (V+ + V−)/2 + (V+ − V−)/2 tanh(x/x0) and

m =1

2c2k2~2, E = n2

0, V− = 2ηn0 and V+ = −2η n0. (C.7)

The Bogoliubov coefficient βk for our mode equation (C.5) is related to the transmissioncoefficient T for the scattering problem (C.6) by |βk|2 = 1/T − 1. From the literature[148] we find the transmission

T =sinh (πa+x0) sinh(πa−x0)

sinh2 (πx0(a+ + a−)/2); a± :=

√2m(E − V±)

~(C.8)

so that after some manipulation we find

|βk|2 =sinh2 (πx0(a+ − a−)/2)

sinh(πx0a+) sinh(πx0a−). (C.9)

In terms of the variables of our problem one has

a− = ck√n2

0 − 2ηn0 ' ck(n0 − η), a+ = ck√n2

0 + 2ηn0 ' ck (n0 + η) , (C.10)

so that

|βk|2 =sinh2 (ηπckt0)

sinh (πckt0(n0 + η)) sinh (πckt0(n0 − η)). (C.11)

By the smallness of η we may expand the sinh2 and identify the factors in thedenominator arriving at

|βk|2 'η2π2c2k2t20

sinh2 (πckt0n0)(C.12)

C.2. Homogeneous perturbation expansion for βk 221

To higher orders in η we have

|βk|2 ' η2 π2c2k2t20sinh2 (πckt0n0)

+ η4 π4c2k4t40sinh2(πckt0n0)

+O(η5) (C.13)

|βshk |2 ' η2 4π2c4k4t40n

20

sinh2(πckt0n0)− η3 16π2c6k6t60n

30

sinh2(πckt0n0)+ η4 16(15− π2)π2c8k8t80n

40

3 sinh2(πckt0n0)+O(η5)

(C.14)

where, interestingly, we see that there is no O(η3) term in the expansion of the tanhspectrum.

where we have also included the result we wrote down in Chapter 4 for the sech2

model for comparison.

C.2 Homogeneous perturbation expansion for βk

In Chapter 4 we have computed the Bogoliubov βk coefficient in two separate ways, oneexactly using known results for some specific potentials and one using a perturbativemethod. In the text we computed the various relevant features of the spectrum such asparticle flux and peak wavenumber of emission for the exact results. Here we computethe same quantities from the perturbative solution, comparing them with the exactcomputations. We will see that they agree closely justifying the perturbative methodas it applies to the RIP problem.

The peak wavenumber is

kpeak =1

ct0n0

' 4.79× 106m−1 (C.15)

and fluxes are

N1+1 = W × η2πt20c2

∫ ∞0

k2e−2c2t20n20k

2

dk (C.16)

=η2πt20c

2W

2√

2c3t30n30

∫ ∞0

u2e−u2

du (C.17)

=η2π√πW

8√

2ct0n30

(C.18)

' 1.12× 10−5 particles (C.19)


Perturbation theory Exact resultk m^(-1)

0 2#106 4#106 6#106 8#106 1#107 1.2#107

Flux

0

1#106

2#106

3#106

4#106

5#106Number flux as a function of wavenumber k

Figure C.1: The exact result for the sech2 model is given in green (lower curve); thisis flux against wavenumber where flux is calculated as k2|β|2 with β from 4.21. Thisshould be compared to the homogeneous perturbative calculation for a gaussian profilein red (upper curve) based on 4.37. In these figures we use t0 = 4×10−16s for the Exactsech2 result and t0 = 4.8× 10−16s for the perturbative result derived from a Gaussianprofile.

and

N3+1 = 4πVol× η2πt20c2

∫ ∞0

k4e−2c2t20n20k

2

dk (C.20)

=4η2π2t20c

2Vol

4√

2c5t50n50

∫ ∞0

u4e−u2

du (C.21)

=3η2π2

√πVol

8√

2c3t30n50

(C.22)

' 0.006 (shape A), 0.275 (shape B), particles (C.23)

which closely agree with the exact results derived for the exact model presented inChapter 4 (see that chapter for definitions of shape A and shape B).

In Fig. C.1 we compare the exact number spectrum as given by the exact analysisin 4.21 with the result obtained above using perturbation theory.

C.3 Perturbation theory with finite size effects

As described in Chapter 4 we wish to solve the equation of motion

φ,tt − c2n2(t, x)φ,xx = 0 (C.24)

C.3. Perturbation theory with finite size effects 223

when the function n2 is given by a background plus a small perturbation

n2(t, x) = n20 + ηf 2(t, x); η 1 (C.25)

under a perturbative assumption that φ(t, x) = φ(0)(t, x) + ηφ(1)(t, x) +O(η2).

We found that with a Gaussian RIP function

f 2(t, x) = e−x2/x20−t2/t20 , (C.26)

that

φ(0)(t, x) =

∫ ∞−∞

dk√2π

eikx−iωkt

n0

√2ωk

(C.27)

and

φ(1)k (t) =

1

ωk

∫ t

−∞dt′F (t′, k) sin ωk(t− t′). (C.28)

where

F (t, k) = e−t2/t20

∫dk√2π

e−iωkt

n0

√2ωk

c2k2[x0e−(k−k)2x20/4

](C.29)

and ω2k = c2n2

0k2.

The first order solution is therefore written as

φ(1)k =

1

ωk

∫ t

−∞dt′F (t′, k) sin ωk(t− t′)

=1

ωk

∫ t

−∞dt′

[e−t

′2/t20

∫dk√2π

c2k2x0e−iωkt

′

n0

√2ωk

e−(k−k)2x20/4

]sin ωk(t− t′)

=1

ωk

∫dk√2π

c2k2x0

n0

√2ωk

e−(k−k)2x20/4

[∫ t

dt′e−iωkt′−t′2/t20 sin ωk(t− t′)

](C.30)

The t′ integral is doable in terms of error functions and results in the asymptoticbehaviour for t→ +∞ as a sum of a pure positive and pure negative frequency piece

φ(1)k (t)→ 1

ωk

∫dk√2π

c2k2x0

n0

√2ωk

e−(k−k)2x20/4

[−i√πt02

e−t20(ωk−ωk)2/4e−iωkt

+i√πt02

e−t20(ωk+ωk)2/4eiωkt

]. (C.31)

Rearranging we have,

φ(1)k →

ic2x0t02

[I1k

e−iωkt

n0

√2ωk

+ I2k

eiωkt

n0

√2ωk

](C.32)


where

I1k =

1√ωk

∫dk√ωkk2e−

x204

(k−k)2e−t204c2n2

0(|k|−|k|)2 (C.33)

I2k =

1√ωk

∫dk√ωkk2e−

x204

(k−k)2e−t204c2n2

0(|k|+|k|)2 (C.34)

(C.35)

These k integrals I ik may be performed exactly using the following trick to eliminatethe cumbersome absolute values∫ ∞

−∞dkf(k) =

∫ ∞0

dk (f(k) + f(−k)) . (C.36)

We have

I1k =

1√ωk

∫ ∞0

dk√ωkk2e−

t204c2n2

0(k−k)2(

e−x204

(k−k)2 + e−x204

(k+k)2)

(C.37)

I2k =

1√ωk

∫ ∞0

dk√ωkk2e−

t204c2n2

0(k+k)2(

e−x204

(k−k)2 + e−x204

(k+k)2). (C.38)

Introducing dimensionless variables

K = t0cn0k/2, K = t0cn0k/2 (C.39)

and the dimensionless ratioξ =

x0

t0cn0

(C.40)

these integrals may be expressed as

I(1)k =

4

c2n20t0x0

ξ√K

∫ ∞0

dK K3/2e−(K−K)2[e−ξ

2(K−K)2 + e−ξ2(K+K)2

](C.41)

I(2)k =

4

c2n20t0x0

ξ√K

∫ ∞0

dK K3/2e−(K+K)2[e−ξ

2(K+K)2 + e−ξ2(K−K)2

]. (C.42)

in terms of which the Bogoliubov βk are

βk =ic2x0t0

2I(2)k (C.43)

Let us concentrate on the integrals. Using the experimentally reported values forthe parameters x0 and t0 (derived from the carrier wavelength of the RIP and theSellmeier dispersion relation) we find that

ξ ' 47.89 (C.44)

C.3. Perturbation theory with finite size effects 225

so we are in a parameter regime in which ξ2 1. We will exploit this limit. In thetrue ξ2 →∞ limit we have

ξ[e−ξ

2(K−K)2 + e−ξ2(K+K)2

]→√π δ(K − K) +

√π δ(K + K) (C.45)

which collapse the integrals giving the homogeneous result

βk → iηt0√πωkn2

0

e−ω2kt

20 as ξ →∞ (equivalently x0 →∞). (C.46)

This gave a spectrum which was peaked (in wavelength space) around 850nm whent0 = 4× 10−16s. Hence we expect this modified integral to be also peaked around thisvalue which in k space is at around 7.4× 106 m−1.

We will see below that this limit is in fact the physically relevant one for describingthe real RIP experiment and that the limit is essentially realised by the parameterchoices we have made to accurately describe the RIP experiment.

Parabolic cylindrical function solution

We need to manipulate the integrals I i a bit in order to use the result for the paraboliccylindrical functions:∫ ∞

0

xν−1e−βx2−γxdx = Γ(ν) (2β)−ν/2 exp

(γ2

8β

)D−ν

(γ√2β

). (C.47)

We are mostly concerned with βk so we will focus on the integral I(2)k . Expanding the

quadratics inside the integrals

I(2)k =

4

c2n20t0x0

ξ√K

e−(1+ξ2)K2

[∫ ∞0

K3/2e−(1+ξ2)K2−2(1+ξ2)KK

+

∫ ∞0

K3/2e−(1+ξ2)K2−2(1−ξ2)KK

](C.48)

so that

I(2)k =

4

c2n20t0x0

ξ√K

e−(1+ξ2)K2

Γ(5/2)(2(1 + ξ2)

)−5/4×[e(1+ξ2)K2/2D−5/2

(√2(1 + ξ2)K

)+ exp

((1− ξ2)2K2

2(1 + ξ2)

)D−5/2

(2(1− ξ2)K√

2(1 + ξ2)

)](C.49)

This solution has the ‘disadvantage’ that it diverges for small k as k−1/2. This is apeculiarity of the finite size computation and 1+1 dimensions. However when one moves


to the 3+1 dimensional interpretation of the flux using k2|βk|2 this divergence is cured.Note that this pole is integrable and hence for 1+1 dimensional flux computations doesnot represent a physical divergence of created particles.

In Fig. 4.6 we compare the first order finite size result C.43 with the first orderhomogeneous result 4.38.

Trying the second order term

Above and in Chapter 4 we saw that finite size effects are practically negligible atfirst order in perturbation theory in the specific RIP consided. Hence it would seemunnecessary to look for the next higher order term for φ in perturbation theory directlyand instead more reasonable to assume that the next higher order term φ(2) is also veryaccurately represented by the homogeneous approximation. Nevertheless here brieflywe write down the next higher order term in η for the perturbative expansion of thefinite size solution.

From C.24 and C.25 and the expansion φ = φ(0) + ηφ(1) + η2φ(2) we have

φ(2),tt − c2n2

0φ(2),xx = c2f 2φ(1)

,xx =: G(t, x) (C.50)

where again we see an an equation of the forced harmonic oscillator type which issoluble in terms of a simple retarded Green function. We have

φ(2)(t, x) = F−1

[1

ωk

∫ t

−∞F(G)(t′, k) sin ωk(t− t′)dt′

](t, x) (C.51)

where F is the Fourier operator. Now, the term F(G) is written in terms of the exact

solution we obtained above in terms of Cylindrical functions for φ(1)

kas

F(G)(t, k) =

∫dx√2π

e−ikxe−x2/x20−t2/t20

d

dx2F−1(φ

(1)

k(t))(x) (C.52)

(where the k is merely a labeling device and not a variable). Integrating by parts twicewe have

F(G)(t, k) =

∫dx√2π

e−ikx(

4x2

x40

+2ixk − 2

x20

− k2

)e−x

2/x20−t2/t20 F−1(φ(1)

k(t))(x)

(C.53)

= F(g(t) · F−1(φ

(1)

k(t)))

(k) (C.54)

=(F(g(t)) ∗ φ(1)

k (t))

(k) (C.55)

=

∫ ∞−∞F(g(t))(k′) · φ(1)

k′−k(t) dk′ (C.56)

where we have defined

g(t, x) :=

(4x2

x40

+2ixk − 2

x20

− k2

)e−x

2/x20−t2/t20 (C.57)

and used the convolution theorem for Fourier transforms of products of functions. Now,the first factor in the integrand is

F(g(t))(k′) =

∫dx√2π

(4x2

x40

+2ixk − 2

x20

− k2

)e−x

2/x20−t2/t20 (C.58)

= e−t2/t20

(− 4

x40

d2

dk2+

2ik

x20

id

dk− 2

x20

− k2

)∫dx√2π

e−ikxe−x2/x20 (C.59)

= e−t2/t20(−k2)

x0√2

e−x20k

2/4. (C.60)

Hence we see that the second order solution involves is a convolution of the first ordersolution over a Gaussian distribution.

This is rather messy but somewhere along the perturbative series one should startto see a non-isotropic Bogoliubov coefficient – that is, the second order solution (orperhaps only a higher order contribution) should be expressed not as a simple linearcombination of a positive and a negative frequency piece but as a function with a non-trivial Fourier spectrum (single particle decomposition). This intuition comes from thefact that the δ(k + k′) functional behaviour of the isotropic Bogoliubov coefficients isa direct result of the use of plane waves for the initial and final particle bases and thefact that the initial plane wave does not distort in space but only in time as the timedependent perturbation acts. Here, in the finite size situation, the initial plane wavewill distort spatially and temporally and we will be left with a kind of integral like

∫dx φin(t, x)e−ikx (C.61)

which will not in general evaluate to the delta function due to spatial distortion of theinitial condition (and ‘in’ state particle wave function) φin.

Beyond perturbation theory one would need to solve exactly the equation of motionto find such non-isotropic particle production. Intuitively, from Figure 4.5 one wouldexpect that in a 3+1 dimensional computation with one spatial dimension of the RIPsmall and one large with respect to the peak wavelength of emission, a non-isotropic

228 Appendix D. BEC Appendices

emission would produced.

Appendix D

BEC Appendices

D.1 Calculating the exact form of a Wightman func-

tion by contour inteation

In this appendix we calculate an integral given in the text (3.39), exactly. The integralrepresents the entanglement contribution to a two-point correlation function which weinvestigated in detail.

The integral we wish to calculate is∫ ∞0

I + I = 2Re

∫ ∞0

I. (D.1)

where

I =sin kx

xαkβ

∗ke−2ika2fη (D.2)

and

αk =2√AB

A+B

Γ(−iA)Γ(−iB)

Γ2 (−i(A+B)/2),

(D.3)

βk =−2√AB

B − AΓ(−iA)Γ(iB)

Γ2 (i(B − A)/2)(D.4)

and finallyA = kη0a

2i , B = kη0a

2f . (D.5)

ai, af and η0 being constants.

We would like to write the integral as an integral over the entire real line and notonly on (0,∞) in order to make use to the residue theorem and contour integration.

D.1. Calculating the exact form of a Wightman function bycontour inteation 229

Figure D.1: The contour γ ⊂ C in red used in the calculation of the integral∫I with

the first few poles ki0, kf0 and kf1 indicated.

However, upon inspection the integrand is in fact an odd function of real k and thecontour we wish to use will receive equal contributions from the negative real axis asthe positive, evaluating to zero for an odd function. Thus we make use of the signumfunction σ(z) = z/|z| which, crucially, changes the integrand into an even function ofk reducing to the identity on positive real numbers and minus the identity on negativereal numbers. This technique has the effect of isolating only the real part of thecontribution to the integral coming from I since the imaginary part is rendered oddby the signum function. Thus it will not be necessary to calculate the contributionfrom I but we will do so anyway as a check. Our plan is to use the semi-circle contourin the complex plane shown in figure D.1 and the residue theorem and show that thecontribution from the arc section of the contour converges to zero for large arcs, leavingonly the contribution from the real line. We have∫ ∞

0

I(k) + I(k) = 2Re

∫ ∞0

I(k) =

∫ ∞−∞

σ(k)I(k). (D.6)

Below we will refer to σI simply as I.

Explicitly we have

I = σ(k)e−2ika2fηΓ2(−iB)Γ(iA)Γ(−iA)[

Γ(− i

2(A+B)

)Γ(− i

2(B − A)

)]2 sin (kx)

x. (D.7)

This expression possesses the following two sets of poles in the lower half plane arising


from the simple poles of the Γ function on the negative integers:

− iB = −n ⇒ k = kfn := − in

η0a2f

, n ∈ N (Double poles) (D.8)

− iA = −n ⇒ k = kin := − in

η0a2i

, n ∈ N. (Simple poles) (D.9)

These poles are in the numerator since Γ is nonvanishing and hence does not contributea pole from the denominator. The simple poles of the Γ(iA) factor lie outside thecontour so we do not consider them further here. Below we make use of the followingimportant observations regarding Γ functions and the signum function:

σ(ir) = i, σ(−ir) = −i, r > 0 (D.10)

Res (Γ(z),−n) =(−1)n

n!, Res(Γ(λz),−n

λ) =

1

λ

(−1)n

n!, n ∈ N (D.11)

Res(Γ2(z),−n

)=: γn Res

(Γ2(λz),−n

λ

)=γnλ, n ∈ N. (D.12)

Here γn is sum of a pure fraction and a fraction multiplying the Euler constant whichconverges rather rapidly to 0 for large n. To calculate the residues we use the wellknown formula for an nth order pole

Res(f, z = a) := limz→adn−1

dzn−1((z − a)nf(z)) , (D.13)

so that for the simple poles at kun, we merely need to take a limit whereas the calculationof the residues of the double poles kfn require differentiation of the entire expressionI(k). Then,

Res(I, kin) = σ(k) e−2ika2fηΓ(−iB)2Γ(iA)[

Γ(− i

2(A+B)

)Γ(− i

2(B − A)

)]2 sin (kx)

x

∣∣∣∣∣k=kin

Res(Γ(−iA), kin)

= −e−2nT η

η0Γ(n)Γ2(−nT )

Γ2(−n

2(T + 1)

)Γ2(−n

2(T − 1)

) 1

xsinh

(nx

η0a2i

)×(

i

η0a2i

(−1)n

n!

).

(D.14)

For the double poles the calculation is slightly longer. We use the product rule toisolate the residue of Γ2,

Res(I, kfn) = limk→kfnd

dk

[(k − kfn)2I(k)

]= Θ(kfn, x, η)× Res

(Γ(−iB)2, kfn

)+ limk→kfn

[(k − kfn)2Γ(−iB)2 × d

dkΘ(k, x, η)

],

(D.15)

D.1. Calculating the exact form of a Wightman function bycontour inteation 231

where

Θ(k, x, η) := σ(k) e−2ika2fηΓ(iA)Γ(−iA)[

Γ(− i

2(A+B)

)Γ(− i

2(B − A)

)]2 sin (kx)

x. (D.16)

For reference note that

Θ(kfn, x, η) = Θne−2n η

η0

xsinh

(nx

η0a2f

)(D.17)

= −e−2n η

η0Γ(nT−1)Γ(−nT−1)

Γ2(−n

2(T−1 + 1)

)Γ2(−n

2(1− T−1)

) 1

xsinh

(nx

η0a2f

). (D.18)

It is possible to expand the Γ functions in the numerical coefficient Θn in terms ofelementary functions as

Θn =

n

8πTtan(πn2T

)∏n/2p=1

(p2 − n2

4T 2

)2

for n even

− T2nπ

(14− n2

4T 2

)2

cot(πn2T

)∏n/2+1/2p=1

((p+ 1

2)2 − n2

4T 2

)2

for n odd

. (D.19)

Note that since kfn is a double pole of both I and Γ2(−iB), the second term in D.15 isfinite as it should be. The first k derivative is

d

dkΘ(k, x, η)

∣∣∣∣k=kfn

= iΘ(kfn, x, η)

η0a2i

(Ψ(nT−1)−Ψ(−nT−1)

)− 2a2

fη + xcosh

(nxη0a2f

)sinh

(nxη0a2f

)η0(a2

f + a2i )Ψ

(−n

2(T−1 + 1)

)+ η0(a2

f − a2i )Ψ

(−n

2(1− T−1)

)],

(D.20)

where Ψ is the digamma function defined as Γ(z)′ = Ψ(z)Γ(z). Note that dσ(k)/dk =2δ(k) which evaluates to zero on the poles; δ(kfn) = 0. Hence we obtain the residue

Res(I, kfn) = iΘne−2n η

η01

xsinh

(nx

η0a2f

) γnη0a2

f

− 1

η20a

4f (n!)2

Kn − 2a2fη + x

cosh(

nxη0a2f

)sinh

(nxη0a2f

) ,

(D.21)where Kn stands for all the x and η independent terms inside the square bracket inD.20. The final result for the integral is therefore given by∫

γ

I = 2π∑n>0

[An

e−2nT η

η0

xsinh

(nx

η0a2i

)+ (Bn + η Cn)

e−2n η

η0

xsinh

(nx

η0a2f

)

+Dne−2n η

η0 cosh

(nx

η0a2f

)], (D.22)


where

An := − 1

η0a2i

(−1)n

n

Γ2(−nT )

Γ2(−n

2(T + 1)

)Γ2(−n

2(T − 1)

) , (D.23)

Bn := Θn

(γnη0a2

f

− Kn

η2oa

4f (n!)2

), (D.24)

Cn := − 2

η20a

2f

Θn

(n!)2, (D.25)

Dn =1

η20a

4f

Θn

(n!)2. (D.26)

Note that Kn has dimensions of time so that the dimensions of each term match.

It is also worthwhile to point out that the effect of the signum function is to con-tribute a factor of i in the I integral and a factor of −i in the I integral without whichthe two terms would not only be imaginary but in fact would contribute equally withthe opposite sign yielding the zero result anticipated above.

There remains only one last loose end and that is the fact that the integrand is notanalytic on the contour γ due to the non-differentiability of σ(z) at z = 0. We getaround this by modifying the contour around the origin into a small semi circle. Here wewill show that the lin integral over the semi circle vanished in the limit of infinitesimalradius. We temporarily return to the old notation in which I(k) = I(k)σ(k). We have

∫γ

I(z)σ(z) =

∫ 2π

π

dθ I(εeiθσ(εeiθ)

(D.27)

=−iε

∫ ε

−εdw f(w) (D.28)

=1

ε

∫ ε

−εdw Imf((w) (D.29)

≤ sup (Imf, [−ε, ε])2ε

ε(D.30)

= sup (Imf, [−ε, ε]) −→ 0 as ε→ 0, (D.31)

since Imf(z) smoothly goes to zero for z going to 0. Hence we may use the largesemi-circle contour with impunity.

D.2. Adiabatic limit of the entanglement contribution to theWightman function 233

D.2 Adiabatic limit of the entanglement contribu-

tion to the Wightman function

D.2.1 Preliminaries

The Wightman function under consideration is written

〈φ(η,x)φ(η,x′〉 =4π

2a2f

∫ ∞0

dk

[sin kx

x

(1 + 2|βk|2

)+ I + I∗

]. (D.32)

It is the contribution I that we are interested in. I is written in terms of the Bogoliubovcoefficients as

I = αkβ∗k

sin kx

xe−2ika2fη. (D.33)

=−4AB

B2 − A2

Γ2(−B)Γ(A)Γ(−A)[Γ(−1

2(A+B)

)Γ(−1

2(B − A)

)]2 sin (kx)

xe−2ika2fη (D.34)

where

A = ikη0a2i , B = ikη0a

2f . (D.35)

The adiabatic limit is the large η0k. Relabelling k = kη0 and dropping the tilde thiswill be the large k limit

It will be necessary to make use of the following asymptotic behaviour for Γ function

Γ(z) '√

2π

zzze−z, for |z| → ∞. (D.36)

We will insert the expansion D.36 into the Gamma functions in D.34 in steps: wecalculate the contributions from the z−1/2, zz and e−z separately an take their productat the end as well as the other extrenal factors not entering the assymptotic expansion.Firstly notice that the 2πs cancell on the top and bottom. The contribution from z−1/2

is1√−B

1√−B

1√A

1√−A[

1√−1/2(A+B)

1√−1/2(B−A)

]2 =1

4

(A+B)(B − A)

−iBA=i

4

B2 − A2

BA. (D.37)

From the exponential we have

e−(−B)−(−B)−A−(−A)

[e−(−1/2(A+B))−(−1/2(B−A))]2 =

e2B

eA+B+B−A = e0 = 1. (D.38)


Finally the zz terms

(−B)−B(−B)−BAA(−A)−A

[(−1/2(A+B))−1/2(A+B)(−1/2(B − A))−1/2(B−A)]2 (D.39)

=(−B)−2B(−1)−A

(−1/2)−(A+B)−(B−A)(A+B)−(A+B)(B − A)−(B−A)(D.40)

=(2B)−2B(−1)−A

(A+B)−(A+B)(B − A)−(B−A). (D.41)

Hence, the large k limit for the ratio of Γ functions is

Gammas ' i

4

B2 − A2

BA

(2B)−2B(−1)−A

(A+B)−(A+B)(B − A)−(B−A)(D.42)

=i

4

a4f − a4

i

a2i a

2f

(2ika2f )−2ika2f (−1)−ika

2i

[ik(a2i + a2

f )]−ik(a2i+a

2f )[ik(a2

f − a2i )]−ik(a2f−a

2i )

(D.43)

=i

4

a4f − a4

i

a2i a

2f

(ik)ik(−2a2f+a2i+a2f+a2f−a

2i )(2a2

f )−2ika2f (−1)−ika

2i

(a2i + a2

f )−ik(a2i+a

2f )(a2

f − a2i )−ik(a2f−a

2i )

(D.44)

=i

4

a4f − a4

i

a2i a

2f

(2a2f )−2ika2f (−1)−ika

2i

(a2i + a2

f )−ik(a2i+a

2f )(a2

f − a2i )−ik(a2f−a

2i )

(D.45)

=i

4

a4f − a4

i

a2i a

2f

(a2i + a2

f

a2i − a2

f

)ika2i(a4f − a4

i

4a4f

)ika2f

. (D.46)

=i

4

a4f − a4

i

a2i a

2f

(a2i + a2

f

a2f − a2

i

)ika2i(a4f − a4

i

4a4f

)ika2f

e−πka2i , (D.47)

where we have used the fact that ln(−1) = iπ. The full function I becomes

I ' −i

(a2i + a2

f

a2f − a2

i

)ika2i(a4f − a4

i

4a4f

)ika2fsin (kx)

xe−2ika2fηe−πka

2i , (D.48)

where we see the exponential damping for large k as expected physically. Taking thereal part we finally arrive at the desired result

I + I∗ ' 2

xsin (kx) sin k

(ln

(a2i + a2

f

a2f − a2

i

)a2i + ln

(a4f − a4

i

4a4f

)a2f − 2a2

fη

)e−πa

2i k.

(D.49)

This result was quoted in Chapter 3.

D.3. Glossary of basic concepts and approximations in the theoryof BEC 235

D.3 Glossary of basic concepts and approximations

in the theory of BEC

Below is a bullet pointed list of the various key concepts in the theory of BEC as itpertains to analogue gravity-ists.

• Dilute gas limit

This is defined as the regime in which three body interactions are neg-ligible. Algebraically we write

|a| << n−13 (D.50)

for the dilute gas limit. This is a fundamental assumption necessaryfor the whole picture of BEC. For example only in the dilute gas limitcan we say that the mean field function is the density. We are able toignore both the quantum and thermal depletion of the condensate.

• Hydrodynamic Limit

This is the limit in which the quantum pressure term is negligible. Thisterm is of quantum mechanical origin.

• Scattering length - the distance within which the atoms interact.

The scattering length is defined only for low energy collisions and de-pends on the physical intuition that due to de Broglie wave propertiesof particles, the very high frequency features of a scattering potentialare not visible to a slowly moving particle. One can approximate thepotential by a hard sphere in the low energy collisions the radius ofwhich is the scattering length. Essentially it is the distance a particleneeds to be from another in order to scatter.

• Healing length

Defined as

ξ =1√

8πna(D.51)

it characterises the length scale beyond which atoms can be thoughtof as not existing independently? It is a characteristic length scaleassociated with anisotropies in the density. A small healing length isnecessary for the hydrodynamic approximation.

Another interesting characterization of the healing length is in the typ-ical length scale that the ground state wave function goes from being


uniform to zero at the boundary. Essentially it is the solution to aconstrained Laplacian problem. (However, it should be noted that, inthe TF approximation the ground state wavefunction for a harmonicpotential is not of this flattened blob type but of inverted parabolatype with sharp edges.)

• Eikonal approximation - a high momentum approximation

• Hartree Fock ansatz

This is essentially the same as the dilute gas approximation. In the HFapproximation we assume that the full wavefunction of the many bodysystem is merely the tensor product of solutions to the GP equation.That is, if we solve the GP equation in a particular physical scenario asΨ(x) then the full many body wavefunction, in the HF approximationis

Φ(x1, . . . , xN) = Ψ(x1) . . .Ψ(xN) (D.52)

This approximation ignores the interactions among the elementary ex-citations and is not relevant to the case of perturbations around asolution where elementary excitations interact.

• Mean field theory

The mean field theory is applicable when the interatomic distance issmaller than the healing length. The mean field picture allows us totreat the condensate as a continuous field and not as a series of atoms.

• Thomas Fermi limit:

The thomas Fermi limit is that limit of the GP equation (that is, in-cluding interactions among the atoms) which neglects the quantumpressure (kinetic energy) term. The resulting TF solution is simply

Ψ(r) =√nTF (r) ; nTF (r) =

1

g(µ− Vext(r)) , (D.53)

since the full equation of motion is(−~2

2m∇2 + Vext + gΨ2

)Ψ = µΨ. (D.54)

There is a smooth transition between the non-interacting solution andthe TF solution given by the limits in the GP equation of the parameter

λTF = Na0

a(D.55)

D.3. Glossary of basic concepts and approximations in the theoryof BEC 237

where ao is the length scale associated with the geometric mean of theoscillator frequencies.

An important feature of the TF approximation is that it allows us toexpress the chemical potential in terms of the other variables as

µ =1

2~ω

(15Na

√mω

~

)2/5

. (D.56)

It is interesting to note that for a harmonic potential, the schrodingerequation would most naturally give the gaussian ground state for thewave function but we see that the TF ground state is an invertedparabola. We expect the TF approximation to require corrections nearthe sharp edge of the parabola. There exists solution matching procee-dures that show exactly this sort of smoothing of the edge in the regionwhere the potential can be approximated by a linear shape.

♥

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