Silke Weinfurtner, Matt Visser and Stefano Liberati

1. Massive minimal coupled scalar field from a 2-component Bose-Einstein condensate presented at ESF COSLAB Network Conference August 28th - September 4th 2005 Smolenice, Slovakia by Silke Weinfurtner, Matt Visser and Stefano Liberati

2. Excitations in Bose-Einstein condensates: sound waves in a 2-component BEC Interpretation of massless and massive classical scalar fields in curved space-time. Dispersion relation for coupled sound waves in a 2-component BEC in the hydrodynamic limit Application as an Analogue Model for Quantum Gravity Phenomenology: Talk on Friday: Stefano Liberati (11:00) What I am going to talk about.

3. 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, 1015atoms/cm3 • very cold temperature, T1K Bose-Einstein condensation in theory • 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

4. UBB UAA UAB 2-component Bose-Einstein condensation. 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 Kinematics is given by 2 coupled Gross-Pitaevskii equation • many-body Hamiltonian • time-dependence via Heisenberg equation of motion • replacing field operators by classical fields

5. 2-component Bose-Einstein condensation. Gross-Pitaevskii equations Macroscopic wave functions

6. mass-densitymatrix background velocity • 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 2-component Bose-Einstein condensation. From the GPE to a pair of coupled wave equations Physical interpretations:

7. the external laser field modifies the interactions ~ ~ ~ ~ UBB UBB UAA UAA ~ UAB • 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: 2-component Bose-Einstein condensation. Fine tuning of the interactions via the external coupling field :

8. 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 2-component Bose-Einstein condensation. Beyond the hydrodynamic limit We are in the hydrodynamic limit if the wave length of the perturbations is much smaller then the healing length!

9. 2-component Bose-Einstein condensation. The role of different initial phases for the model contribution to mass term damping terms

10. 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.. The 2-component BEC as an Analogue Model for Gravity. Sound waves in a 1-component BEC can be treated as an Analogue Model for Gravity for massless particles. How to continue: • 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

11. B1 • The system is in an eigenstate, if: • the perturbed phases are in-phase • the perturbed phases are in anti-phase A1 Klein-Gordon equation for massive phonon modes. Decoupling the wave equation onto the two eigenstates

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

13. the densities and interactions within each condensate are equal • the mono-metricity condition must be which requires the fine tuning Klein-Gordon equation for massive phonon modes. The fine tuning for the decoupling the wave equations: The two speed of sounds are: Within this fine tuning the eigenfrequency of the anti-phase (massive) mode is:

14. 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. About the mass of the phonon mode..

15. the effective metric obtained by our calculations are the same one gets for a single BEC Klein-Gordon equation for massive phonon modes. 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!

16. Sound waves in a moving fluid. Supersonic and subsonic region… horizon fluid at rest  fluid velocity

17. 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.

18. Decoupled sound waves in a 2-component BEC in fluid at rest. high energetic perturbations low energetic perturbations fluid at rest fluid at rest

19. The first step towards an Analogue Model for QGP. Alternative route to obtain the dispersion relation 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. 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. How to continue: • 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

20. 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 wave equation for a uniform background at rest reduces to:

21. 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: Dispersion relation for high energy phonon modes. The dispersion relation is given by:

22. 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. • The external coupling is crucial in order to obtain a massive phonon mode. • 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. • For a uniform condensate at rest it is possible to calculate the dispersion relation without decoupling the phonon modes first. • In the hydrodynamic limit we can recover perfect special relativity with milder constraints, as for the physical acoustics. • 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). • This model is a suitable object to study Quantum Gravity Phenomenology.

23. Thank you for your attention.