1. Dynamics of Quantum- Degenerate Gases at Finite Temperature Brian Jackson Inauguration meeting and Lev Pitaevskii’s Birthday: Trento, March 14-15 University of Trento, and INFM-BEC

2. In collaboration with: Eugene Zaremba (Queen’s University, Canada) Allan Griffin (University of Toronto, Canada) Jamie Williams (NIST, USA) Tetsuro Nikuni (Tokyo Univ. of Science, Japan) In Trento: Sandro Stringari Lev Pitaevskii Luciano Viverit

3. Bose-Einstein condensation: Cloud density vs. temperature Decreasing Temperature

4. Bose-Einstein condensation: Condensate fraction vs. temperature J. R. Ensher et al., Phys. Rev. Lett. 77, 4984 (1996)

5. Outline Bose-Einstein condensation at finite T collective modes ZNG theory and numerical methods applications: scissors, quadrupole, and transverse breathing modes Normal Fermi gases Collective modes in the unitarity limit Summary

6. Collective modes: zero T Condensate confined in magnetic trap, which can be approximated with the harmonic form:

7. Collective modes: zero T Change trap frequency: condensate undergoes undamped collective oscillations

8. Collective modes: zero T Gross-Pitaevskii equation: Normalization condition: a: s-wave scattering length m: atomic mass

9. Collective modes: finite T Finite temperature: Condensate now coexists with a noncondensed thermal cloud

10. Collective modes: finite T Change trap frequency: condensate now oscillates in the presence of the thermal cloud

11. Collective modes: finite T Condensate now pushes on thermal cloud- the response of which leads to a damping and frequency shift of the mode But!

12. Collective modes: finite T Change in trap frequency also excites collective oscillations of the thermal cloud, which can couple back to the condensate motion And

13. ZNG Formalism Bose broken symmetry: condensate wavefunction: condensate density: thermal cloud densities: ‘anomalous’ ‘normal’ Dynamical Equations E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999)

14. ZNG Formalism Generalized Gross-Pitaevskii equation: E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999) Popov approximation:

15. ZNG Formalism Boltzmann kinetic equation: E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999) Hartree-Fock excitations: moving in effective potential: phase space density: (semiclassical approx.)

16. ZNG Formalism E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999) Boltzmann kinetic equation:

17. ZNG Formalism E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999) Coupling: mean field coupling

18. ZNG Formalism E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999) Coupling: Collisional coupling (atom transfer)

19. Numerical Methods B. Jackson and E. Zaremba, PRA 66, 033606 (2002). Follow system dynamics in discrete time steps: 1.Solve GP equation for  with FFT split-operator method 2.Evolve Kinetic equation using N-body simulations: Collisionless dynamics – integrate Newton’s equations using a symplectic algorithm Collisions – included using Monte Carlo sampling 3.Include mean field coupling between condensate and thermal cloud

20. Applications Scissors modes (Oxford): O. M. Maragò et al., PRL 86, 3938 (2001). Quadrupole modes (JILA): D. S. Jin et al., PRL 78, 764 (1997). Transverse breathing mode (ENS): F. Chevy et al., PRL 88, 250402 (2002). Numerical simulations useful in understanding the following experiments, that studied collective modes at finite-T:

21. Scissors modes Excited by sudden rotation of the trap through a small angle at t = 0 Signature of superfluidity! D. Guéry-Odelin and S. Stringari, PRL 83, 4452 (1999) O. M. Maragò et al., PRL 84, 2056 (1999)

22. Scissors modes condensate frequency: with irrotational velocity field: thermal cloud frequencies:

24. Experiment: O. Maragò et al., PRL 86, 3938 (2001). Theory: B. Jackson and E. Zaremba., PRL 87, 100404 (2001).

25. m = 0 JILA experiment Experiment: D. S. Jin et al., PRL 78, 764 (1997). condensate: thermal cloud: Theory: B. Jackson and E. Zaremba., PRL 88, 180402 (2002).

26. JILA experiment Excitation scheme: modulate trap potential m = 0

27. condensate thermal cloud  = 1.95   T ´ = 0.8

28. Drive frequencies Solid symbols – maximum condensate amplitude

29. ENS experiment m = 0 mode in an elongated trap Excitation scheme: excites oscillations in both condensate and thermal cloud Theory: B. Jackson and E. Zaremba., PRL 89, 150402 (2002). Experiment: F. Chevy et al., PRL 88, 250402 (2002).

30. ENS experiment Condensate oscillates at Thermal cloud oscillates at Condensate and thermal cloud oscillate together with same amplitude at frequency m = 0 mode in an elongated trap Theory: B. Jackson and E. Zaremba., PRL 89, 150402 (2002). Experiment: F. Chevy et al., PRL 88, 250402 (2002).

31. condensate thermal cloud ‘tophat’ excitation scheme collisions

32. experiment theory

33. condensate thermal cloud excite condensate only collisions

34. Fermi gases Motivation: Experiment by O’Hara et al., Science 298, 2179 (2002). Cool 6 Li atoms (50-50 mixture of 2 hyperfine states) to quantum degeneracy T « T F Static B-field tuned close to Feshbach resonance, a~ -10 4 a 0 Observe anisotropic expansion of the cloud

35. Fermi gases Motivation: Experiment by O’Hara et al., Science 298, 2179 (2002). Cool 6 Li atoms (50-50 mixture of 2 hyperfine states) to quantum degeneracy T « T F Static B-field tuned close to Feshbach resonance, a~ -10 4 a 0 Observe anisotropic expansion of the cloud Hydrodynamic behaviour, implying either:  Gas is superfluid (BCS or BEC)  Gas is normal, but collisions are frequent 

36. Feshbach resonance: Fermi gases Jochim et al., PRL 89, 273202 (2002). = relative velocity of colliding atoms Collision cross- section:

37. Feshbach resonance: Fermi gases Jochim et al., PRL 89, 273202 (2002). = relative velocity of colliding atoms Low k limit:

38. Fermi gases Feshbach resonance: Jochim et al., PRL 89, 273202 (2002). = relative velocity of colliding atoms Unitarity limit:

39. Quadrupole collective modes: In-phase modes: L. Vichi, JLTP 121, 177 (2000)

40. Taking moments:

42. collisionless limit: ωτ » 1 hydrodynamic limit: ωτ « 1 intermediate regime: ωτ ~ 1 Solve set of equations for Example: transverse breathing mode in a cigar-shaped trap

43. Low k limit:

44. Unitarity limit: N=1.5  10 5 =0.035

45. Summary Bose condensates at finite temperatures:  studied damping and frequency shifts of various collective modes  Comparison with experiment shows good to excellent agreement, illustrating utility of scheme Normal Fermi gases:  relaxation times of collective modes  simulations  rotation, optical lattices, superfluid component…