by Silke Weinfurtner, Matt Visser and Stefano Liberati Massive minimal coupled scalar field from a 2-component Bose-Einstein condensate ESF COSLAB Network Conference August 28th - September 4th 2005 Smolenice, Slovakia present ed 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. 87 Rb (Eric Cornell) or 23 Na (Wolfgang Ketterle) extremely low densities,  atoms / cm 3 very cold temperature, T  1  K 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

2-component Bose-Einstein condensation. Interactions in a coupled 2-component BEC low-energy elastic collisions within each species, U AA and U BB low-energy elastic collisions between the the two species, U AB 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 U AA U BB U AB

2-component Bose-Einstein condensation. Gross-Pitaevskii equations Macroscopic wave functions

2-component Bose-Einstein condensation. 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

2-component Bose-Einstein condensation. 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: U AA U BB U AB ~ ~ ~ U AA U BB ~ ~

2-component Bose-Einstein condensation. 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 2-component Bose-Einstein condensation.

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

Klein-Gordon equation for massive phonon modes. 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:

Klein-Gordon equation for massive phonon modes. 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

Klein-Gordon equation for massive phonon modes. 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

Klein-Gordon equation for massive phonon modes. the effective metric obtained by our calculations are the same one gets for a single BEC 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… horizon  fluid velocity fluid at rest 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 Decoupled sound waves in a 2-component BEC in fluid at rest. fluid at rest high energetic perturbationslow 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.