1

Chaire Européenne du College de France (2004/2005)

Sandro Stringari

Lecture 47 Mar 05

Fluctuations of the order parameter

Previous Lecture. Equation for the order parameter.Gross-Pitaevskii theory. Healing length. Time dependent theory. Bogoliubov equations.

This Lecture.- Quantum fluctuations, BEC depletion and kinetic energy- Beyond mean field effects on collective oscillations- Thermal depletion. - Shift of critical temperature due to interactions

2

Fluctuations of order parameter

)(ˆ)()(ˆ 0 rrr Ψ+Ψ=Ψ δField operator

Total density of the system:

>ΨΨ<+Ψ>=ΨΨ=< ++ )(ˆ)(ˆ)()(ˆ)(ˆ)( 20 rrrrrrn δδ

0n nδ

Experiments based on imagingtechniques measure total density )()( 0 rnrn ≠

Physical origin of nδ

- Quantum fluctuations (small in very dilute Bose gases)- Thermal fluctuations (vanish at T=0)

Using plane wave representation of fieldoperator the many-body Hamiltonian

takes the form

)(ˆ)(ˆ)(ˆ)(ˆ2

)(ˆ)(ˆ2

ˆ 22

rrrrdrgrrdrm

H ∫∫ ΨΨΨΨ+Ψ∇Ψ−= +++h

h/1ˆ)(ˆ ripp

p

eV

ar ⋅∑=Ψ

∑∑ +−

++

+ +=p

ppqpqpppp

aaaaVgaa

mpH

2121ˆˆˆˆ

2ˆˆ

2

2

3

Quantum fluctuations in T=0 uniform Bose gas(brief summary of Bogoliubov theory)

Zero-th order term in H is obtained by keeping only termswith p=0 and using Bogoliubov replacement

Next approximation consists of keeping terms with two operatorswith + normalization condition

Hamiltonian can then be rewritten as:Naaaa

p pp =+∑ ≠++

000 ˆˆˆˆ

pp aa ˆ,+)

NNaa ≈≡=+000 ˆˆ

pp aa ˆ,+) 0≠p

2/gnNEH ==) (mean field energy)

)ˆˆˆˆ(21ˆˆ)

2(ˆ

2

0 pppppppaaaagnaagn

mpconstH −

+−

++≠

++++= ∑

4Hamiltonian is diagonalized by Bogoliubov transformation

+−−+= ppppp bvbua ˆˆˆ *

'' ,],[pppp bb δ=+

transforms particles into quasi-particles. Diagonalization is ensured by choice

2222

2

2)( cp

mpp +

=ε

2/122

21

)(22/,

±

+±=

pmcmpvu

ε maggnmc/4 2

2

hπ=

=

Results for u,v and coincide with predictionsof linearized time dependent Gross-Pitaevskii equation

h/εω =

Bogoliubov transformationreduces H to the form

bbpEH ˆˆ)( +∑+= ε pppgs

contains non trivial ultraviolet divergence. Its evaluation requires renormalization of effective potential)gsE

Physical observables can be expressed in terms of quasi-particle operators. Ground state is vacuum of quasi-particles 0=vacbp

)

Behaviour of :pn∞→

→pp 0

4/1

2/

pn

pmcn

p

p

→

→

is convergent in 3D. Gives number of atomsout of the condensate:

∑≠

=0p

pnNδ

2/13 )(38 na

NN

πδ

=

- Quantum depletion fixed by gas parameter. - Depletion should be small in order to apply Bogoliubobv theory.

Quantum depletionof the condensate

21

)(22/ˆˆ

222−

+=>==< +

pmcmpvaan pppp ε

5

1° Example: Momentum distribution ( T=0): )ˆˆ)(ˆˆ(ˆˆ ** +

−−−−++ ++= pppppppppp bvbubvbuaa

0≠p

because of two-body interactions0≠pn

6

Due to behaviour at large p, Bogoliubov theorypredicts divergence for kinetic energy in 3D.

4/1 p

Differently from total energy (kinetic + potential), whichcan be safely calculated in terms of scattering length, kinergy energy cannot be calculated in Bogoliubovtheory. Bogoliubov theory fails when . Kinetic energy depends on microscopic details of the force.

ap /h≈∞=∑

ppn

mp )(2

2

Behaviour of kinetic energy in a dilute Bose gas

Hard sphere Realistic microscopic potential

a aScattering length

gn21

gn21

mEnergy ( )ag

24 hπ=

gn21

≠gn21Kinetic energy

7

Quantum depletion in trapped Bose gas at T=0

32/13 )0(85))((

38)( anarnrdrn

NN

TFTFTFπ

πδ

== ∫

local density approximation

))((1)( 0 rVg

rn extTF −= µ

Effect is small in available configurationsCan become larger with Feshbach resonance

%1≈5/26/13 )(1.0)0(ho

TF aaNan =

Can one extract quantum depletion through measurement of density profile?

N=70 Helium droplet

total density

condensate

Quantum depletion arises from fluctuationsof the condensate. Separation in space between condensate and quantum depleted components is not possiblebecause they fully overlap.

8

2° example: density fluctuations

S(q)=1in T=0 idealBose gas

Density-density correlations arerelated to static structure factor >=< −qqqNS ρδρδ ˆˆ)(

By using Bogoliubov approach one can express density operatorIn terms of particle (and hence quasi-particle operators):

)ˆˆ)(()ˆˆ(ˆˆˆ qqqqqqkqkkq bbvuNaaNaa −+

−−++

+ ++=+==∑ρ

Using Bogoliubov results for u and v one finally finds (uniform gas)

∞→→→→

==qasqasmcq

qmqqS

102/

)(2)(

22 hh

εphonon regime

free particle regime22222 )2/()( cpmpp +=εBogoliubov dispersion law

Density fluctuations are strongly quenched in phonon regime

Result for S(q) coincides with predictions given by densityresponse function calculated in time-dependent GP theory

9

ideal gas

interacting gas

Response function

)),(),((),('' ωωπωχ −−= qSqSq

measured at MIT(Stamper-Kurn et al. 1999)

in situafterexpansion

- Result for structure factor can be generalized to trappedBEC gases (local density approximation).

- Structure factor has been measured with Bragg spectroscopy experiments (inelastic photon scattering)

Static structure factor

∫= ),()( ωω qSdqNS h

10

Beyond mean field effects:density profile and collective oscillations

Quantum correlations in interacting Bose gases modify equation ofstate with respect to mean field prediction)(nµ gnn =)(µ

Inclusion of quantum fluctuations in Bogoliubov theory(+ renormalization of coulpling constant) yieldsfirst correction to equation of state

)3321()( 3nagnnπ

µ +=Lee-Yang-Huang (1957) beyond mean field correction

New equation of state modifies both equilibrium and dynamic properties

0)()( µµ =+ rVn extEquilibrium density profile defined by

meanfield TFdensity

))((1)( 0 rVg

rn extTF −= µ3)(332)()()( arnrnrnrn TFTFTF π

−=

11

Beyond mean field effects: collective oscillations

Beyond mean field corrections in density profile are difficult to measure(would require 1% accuracy)

More promising possibility concerns the study of collective oscillations.

Change in equation of state change in frequency of hydrodynamic modes (easily measurable within 1% accuracy)

Corrections in collective frequencies due to beyond mean field affectseasily evaluable using perturbation theory in hydrodynamic equations

0))(21(

0)(

2 =++∇+∂∂

=∇+∂∂

extSS

S

Vnmvvt

m

nvnt

µ)

3321()( 3nagnnπ

µ +=

12

))((2

2

nn

nnt

m δµδ∂∂

∇∇=∂∂Linearized hydrodynamic equations:

In trapped gas beyond mean field effects affect both equilibrium density and chemical potential. One finds:

)(316)( 2/32

2/32 nnangnn TFTF δ

πδδω ∇−=∇∇+ ))((1)( 0 rV

grn extTF −= µ

Left hand side: HD equation in mean field regimeRight hand side: perturbative correction due to beyond mean field

Perturbative correction to collective frequencies is obtained bymultiplying HD eq. by mean field value and integrating by parts

δω*nδ

In uniform gas one recovers Belayev (1958) result for renormalized sound velocity

)161( 32 nagnmcπ

+=

13

Result for beyond mean field shift of collective frequencies:

∫∫ ∇

−=nndr

nnndr

ma TF

δδ

δδ

ωπωδω

*

2/3*2

2

2/3 )(

38

where are density variations and frequencies of hydrodynamics equations in mean field (Gross-Pitaevskii) regime.

ωδ ,n

- Surface modes are unaffected by beyondmean field correction (insensitive to equation of state)

- Shift in m=0, l=0 mode (compression) in spherical trap:

and

02 =∇ nδ

22

53 Rrn −∝δ hoωω 5= 3)0(

12863 anπ

ωδω

=

Pitaevskii and Stringari, 1999Braaten and Pearson, 1999

14

1616968

21)(

)()0(12863

24

2

3

0

0

+−

+±=

=

±

±=

=

λλλλ

λπωδω

f

fanm

m

Result is easily generalizedto axially deformed trap:

⊥= ωωλ /z

Application to radial compression mode in cigar geometry

ENS experiment (2002):

Theory predicts

Compared to mean field value

510,2.5,26.1 === Nnmamaho µ55/26/13 102.2)/(1.0)0( −×== hoTF aaNan

⊥= ωω 007.2

⊥ω2

( , and ) 6/5=+f1<<λ ⊥= ωω 2 beyond mean field effect?

15

Quantum depletion is small in usual geometriesIt can be increased significantly by changing geometry of configurationExamples: - Low dimensional configurations (Lecture 5).

- Periodic potentials (Lecture 9) - Double well potential

Quantum depletion in double well potential

Hamiltonian in double well potential can be written in boson Hubbard form

)ˆˆˆˆ(2

)ˆˆˆˆˆˆˆˆ(4

abbabbbbaaaaEH JC ++++++ +−+=δ

tunneling between two wells

Eigenstates of are

(ground state) with energy

(excited state) with energy

)ˆˆˆˆ)(2/( abbaH Jsp++ +−= δ

vacba )ˆˆ(1++ +=ϕ 2/0 Jδε −=

2/1 Jδε +=vacba )ˆˆ(0++ −=ϕ

on site energy

16

Let us introduce operators and)ˆˆ(ˆ0 baa +=21 )ˆˆ(

21ˆ1 baa −=

Bogoliubov prescription then corresponds to setting NNaa ≈== +000 ˆˆ

By retaining only terms quadratic in and usingrelationship , the Hamiltonian takes the form

+11 ˆ,ˆ aa

Naaaa =+ ++1100 ˆˆˆˆ

Boson Hubbard Hamiltonian can be rewritten in the form)ˆˆ2ˆˆˆˆ(

8ˆˆ 11111111 aaaaaaEaaconstH C

J++++ ++++= δ

++= ββ ˆˆˆ1 vuaH is diagonalized by Bogoliubov transformation

with and , ±±= CJvu2/1

21

24/

+

J

NEε

δ)2/( CJJJ NE+= δδε

Hamiltonian takes diagonal form: ββε ˆˆ ++= JconstH

17

Excitation energy approaches :

“single particle” regime ( , Rabi frequency) when

“plasmon” regime ( ) when

JJ δε = NE JC /δ<<

)2/( CJJJ NE+= δδε

2/CJJ ENδε = NE JC /δ>>

Both regimes are described by Bogoliubov theoryprovided quantum depletion is small (they correspondto single particle and phonon regimes in uniform gas)

Quantum depletion is associated with occupation of state 1. One finds

21

24/ˆ 2

110 −+

=>==< +

J

CJ NEvaaNε

δδ

Condition of applicability of Bogoliubov theory ( ) isequivalent to condition (compatible with )

NN <<0δ

JC NE δ<< NE JC /δ>>

- Condition can be easily violated by reducing tunneling parameter, through increase of height of barrier.

- In the regime Bogoliubov theory is no longer valid and system jumps into BEC fragmented configuration (Lecture 1)

JC NE δ>>

18

Thermal depletion of the condensate (ideal gas)

)()()( 0 rnrnrn T+=Depletion in ideal Bose gas is due to thermal effect:

Calculation in harmonic trap yields ∫ −== ])/(1[)()( 300 CTTNrdrnTN

3/194.0 NkT hoC ωh=with

Differently from uniform gas BEC is visible in coordinate space because it is well separated from thermalcomponent (crucial feature of harmonic trapping)

22

ho

BT m

TkRω

≈Width of thermal component :

Width of condensate:ho

ho maR

ωh

=≈ 220

Ratio is large if3/120

2

NTTTk

RR

Cho

BT ≈≈ωh hokT ωh>

5/10 )/15( hoho aNaaR =

- Thermal component scarcely affected by interactions(thermal gas is very dilute)

- Condensate component strongly affected by interactionsIn Thomas-Fermi limit (a>0)

Role of interactions and BEC visibility in coordinate space19

6/1320

2

))0((1anT

TRR

C

T ≈and hence:

thermalBEC

Visibility in coordinate space isreduced with respect to ideal gas, but still since gas parameter is small.

Furthermore shapes of thermalAnd condensate components are different (bimodal distribution)

0RRT >

20

Visibility of BEC in momentum space

In superfluid helium information on BEC comes frommeasurement of momentum distribution. Data are availablefrom inelastic neutron scattering at high momentum and energy transfer where impulse approximation (IA) holdsand dynamic structure factor is proportional to n(p):

)()22

)((),(22

pnmp

mqpdpqSIA +

+−= ∫ ωδω h

)(~)()( 0 pnpNpn += δ

BEC quantum depletion at T=0

In practice corrections to IA should be included (final state interactions)

21

∫=

−=

=

),,()(

)2

(

)(),(

2

YppndpdpYJ

mq

qmY

YJqmqS

yxyx

hω

ω

scaling variable

At high momenta and energy transfer dynamic structure factorscales according to

Scaling functionmeasured in superfluidhelium (Sokol 1996).Delta peak due BECis smoothed out byfinal state interactionsand instrumental resolution

22

Measurement of momentum distribution in trapped Bose gases

In Bragg scattering experiments photons scatter inelastically fromatoms transferring or absorbing large momentum and energy(measured at MIT, Stenger et al. 1999). These measurements probe dynamic structure factor propotional,at high momentum transfer, to momentum distribution.

Width of measured signal is proportional to width of n(p)

Experiment at MIT has provenuncertainty limit RP ∆≈∆ /h

after 3ms

mPq /∆∝∆ν

thermal

BEC

23

Visibility of BEC in momentum space: trapped Bose gas

Width of thermal componentis scarcely affected by interactions:

TmkP BT ≈2

IDEAL GAS: width of condensatehom

RP ωh

h≈≈ 2

0

220

3/120

2

NTTTk

PP

Cho

BT ≈≈ωh

Ratio coincides with ratio 20

2 / RRT(in harmonic oscillator momentum and coordinate variables play symmetric role)

ROLE OF INTERACTIONS: width of condensate in momentum space reduced by interactions

00 RP h≈

Ratio strongly enhanced by interactions3/26/1320

2

))0(( NanTT

PP

C

T ≈

Higher visibility of BEC in momentum than in coordinate space

25

Effect of interactions on critical temperature

- Many body effect (due to correlations) relevant in uniformsystems (constant density). Even sign of the effect is not trivialRecenty work (Baym et al, 1999; Arnold, Moore 2001, Kashurnikovet al. 2001) predicts positive shift, proportional to 3/1an

- Mean field effect relevant in non uniform systems (constant N)(example: harmonic trapping)

- repulsive interactions, although small, tend to expand the gas, therebyreducing average density and hence the value of critical temperature.

Calculation of mean field effect:Just above critical temperature one can use Hartree Fock-Theory(HF theory can be developed also below , in the presenceof the condensate, not discussed here)

CT

26

Hartree-Fock theory describes the gas as a system of statisticallyindependent s.p. excitations governed by the Hamiltonian ( )

gnVm

H extsp 22

22

++∇−=h

Shift of CT

CTT ≥

Factor 2 arises from exchange term(absent in the condensate)

In semi-classical approximation one can write

1]),([exp1

)2( 3 −−= ∫ µεβπ pr

drdpNTh

If )(2

),(2

rVmppr ho+=ε

ideal gas

BEC starts at 0=µ

3/10 94.0 NTkNN hoCBT ωh=⇒=

ideal gas result

If BEC starts at )0(2gn=µ)(2)(2

),(2

rgnrVmppr ho ++=ε

In uniform gas (and hence ) does notdepend on interaction and one recovers ideal gas result

µε −),( prTN

27

In non uniform gas one can expand

Yielding, after straightforward integration,

δµµ

δ∂∂

+∂∂

+= TC

TTT

NTTNNN

C0 )0(2gn=δµ

6/10 43.1 N

aa

TT

hoC

C −=δ

Giorgini et al, 1996

Mean field negative shift:small, but measurable effect(Orsay; Gerbier et al 2003):

Many-body effect gives higherorder corrections in harmonictrap.

28

This lecture. Fluctuation of the order parameter.Quantum fluctuations and BEC depletion.Thermal depletion. Shift of critical temperature due to interactions

Next lecture. BEC in low dimensions. Theorems on long range order. Algebraic decay in low D. Mean field and beyond mean field. Collective oscillations in 1D gas.