silke weinfurtner, matt visser and stefano liberati

bySilke Weinfurtner, Matt Visser and Stefano Liberati

Massive minimal coupled scalar field from a 2-componentBose-Einstein condensate

ESF COSLAB Network Conference

August 28th - September 4th 2005

Smolenice, Slovakia

presented at

Application as an Analogue Model for Quantum Gravity Phenomenology:Talk on Friday: Stefano Liberati (11:00)

Dispersion relation for coupled sound waves in a 2-component BEC in the hydrodynamic limit

Interpretation of massless and massive classical scalar fields in curved space-time.

What I am going to talk about.

Excitations in Bose-Einstein condensates: sound waves in a 2-component BEC

2-component Bose-Einstein condensation.

Bose-Einstein condensation in experiment

gas of bosons, e. g. 87Rb (Eric Cornell) or 23Na (Wolfgang Ketterle)

extremely low densities, 1015 atoms/cm3

very cold temperature, T1K

nearly all atoms occupy the ground state

non condensed atoms are neglected

microscopic system can be replaced by a classical mean-field, a macroscopic wave-function

Bose-Einstein condensation in theory


Interactions in a coupled 2-component BEC

low-energy elastic collisions within each species, UAA and UBB

low-energy elastic collisions between the the two species, UAB

transitions between the two species

many-body Hamiltonian

time-dependence via Heisenberg equation of motion

replacing field operators by classical fields

Kinematics is given by 2 coupled Gross-Pitaevskii equation

UAA

UBB

UAB


Gross-Pitaevskii equations

Macroscopic wave functions


From the GPE to a pair of coupled wave equations

Physical interpretations:

this equation represents kinematics of sound waves in the 2-component BEC

a small (in amplitude) perturbation in 2-component BEC results in pair of coupled sound waves

coupling matrix

this description holds for low and high energetic perturbations

interaction matrix + quantum pressure term

contains the modified interactions due to the external coupling

mass-density matrix background velocity


Fine tuning of the interactions via the external coupling field :

the external laser field modifies the interactions

the sign of can be positive or negative ( additional trapping frequency ), e.g it is possible to make the modified XX or XY interactions zero:

UAAUBB

UAB

~

~

~

UAAUBB

~ ~


Beyond the hydrodynamic limit

the quantum potential has to be taken into account

the quantum potential term (here in flat space-time) can be absorbed in the redefinition of the interaction matrix between the atoms (effective interaction matrix)

this term gets relevant at wave length comparable to the healing length

a change to momentum space shows the effective interaction is k-dependent

We are in the hydrodynamic limit if the wave length of the perturbations is much smaller then the healing length!

The role of different initial phases for the model

contribution to mass term

damping terms


The 2-component BEC as an Analogue Model for Gravity.

decoupling of the phonon modes on the level on the wave equation.

the two independent wave equations can be treated in the same way as a 1-component system

for each mode it is possible to assign a mass and space-time geometry

forcing the two space-times to be equal by adding a mono-metricity condition

Sound waves in a 1-component BEC can be treated as an Analogue Model for Gravity for massless particles.

The idea was to do the same with our 2-component BEC, hoping that we would get additional terms in the wave equation, which can be identified as the mass of the phonon-modes..

How to continue:

Klein-Gordon equation for massive phonon modes.

Decoupling the wave equation onto the two eigenstates

A1

B1

The system is in an eigenstate, if: the perturbed phases are in-phase the perturbed phases are in anti-

phase


The two decoupled wave equations can be written as two scalar fields in curved space-times:

in-phase mode

anti-phase mode

the in-phase mode represents a massless scalar field

the anti-phase mode represents a massive scalar field

the two effective metrics are different, due to different speeds of sound:


The fine tuning for the decoupling the wave equations:

The two speed of sounds are:

the mono-metricity condition must be which requires the fine tuning

Within this fine tuning the eigenfrequency of the anti-phase (massive) mode is:

the densities and interactions within each condensate are equal


About the mass of the phonon mode..

phonon mass is proportional to the laser-coupling , therefore you need a permanent coupling

it is possible to calculate the general expression for the mass of the phonon modes


the effective metric obtained by our calculations are the same one gets for a single BEC

€

gab ∝

−(c 2 − v02) −v0x −v0y −v0z

−v0x 1 0 0

−v0y 0 1 0

−v0z 0 0 1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

About the fine tuning in terms of possible space-times..

in principle the 2-component BEC Analogue Model is possible to reproduce all the configurations in the same way as in the simple BEC: e.g. Schwarzschild Black Hole, FRW and Minkowski space-time.

Note: For example, in the case of FRW where one changes the scattering length through an external potential, also the fine-tuning would have to be re-adjusted!

Supersonic and subsonic region…

ho

rizon

fluid velocity fluid at rest

€

c 2 − v02 < 0

€

c 2 − v02 > 0

€

rv 0 =

r 0

Sound waves in a moving fluid.

Dispersion relation for uniform condensate.

Changing into momentum space leads to the dispersion relation:

Note: The change to momentum space is only exact, if the densities are uniform and the background velocity is at rest ( Minkowski space-time ).

We recover perfect special relativity for the decoupled phonon modes in the hydrodynamic limit.

fluid at rest

€

rv 0 =

r 0

Decoupled sound waves in a 2-component BEC in fluid at rest.

fluid at rest

€

rv 0 =

r 0

high energetic perturbations low energetic perturbations

The first step towards an Analogue Model for QGP.

change the wave equation to position space

the dispersion relation

the modes have to fulfill the generalized Fresnel equation

in the hydrodynamic limit - for low energy - we want to recover special relativity

The 2-BEC Analogue Model presents a massive and massless scalar field. We also know from condensed matter physics, that for high energy modes the Lorentz invariance will be broken.

How to continue:

The idea is know to look at Minkowski space-time ( uniform density and zero background flow ) and calculate the dispersion relation for the two coupled modes in the hydrodynamic limit.

Alternative route to obtain the dispersion relation

Dispersion relation for high energy phonon modes.

The wave equation for a uniform background at rest reduces to:

for a uniform condensate is constant it is possible to introduce:

it is useful to introduce

after changing in momentum space we get the dispersion relation

the modes have to fulfill the generalized Fresnel equation

Dispersion relation for high energy phonon modes.

The dispersion relation is given by:

again, in the hydrodynamic limit we want to recover special relativity:

the following fine tuning is necessary to obtain LI in the hydrodynamic limit:

in terms of physical parameter the constraints are:

Conclusion and Outlook.

The kinematics for sound waves in a coupled 2-component BEC is analogue to a massive minimal coupled scalar field embedded in curved-space time.

For a uniform condensate at rest it is possible to calculate the dispersion relation without decoupling the phonon modes first.

The external coupling is crucial in order to obtain a massive phonon mode.

In the hydrodynamic limit we can recover perfect special relativity with milder constraints, as for the physical acoustics.

This model is a suitable object to study Quantum Gravity Phenomenology.

The transition rate can be used to tune the system.

For an arbitrary 2-component system the decoupling on the level of the wave equation (physical acoustics) puts strong tuning parameter onto the system.

The dispersion relation obtained from the two Klein-Gordon equations is Lorentz invariant, therefore we recovered perfect special relativity.

We know how we have do modify our theory for high energy modes (wave length comparable to the order of the healing length of the condensate).

Thank you for your attention.

silke weinfurtner, matt visser and stefano liberati

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