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Dynamics of Quantum-Degenerate Gases at Finite Temperature

Dynamics of Quantum-Degenerate Gases at Finite Temperature. Brian Jackson. University of Trento, and INFM-BEC. Inauguration meeting and Lev Pitaevskii’s Birthday: Trento, March 14-15. In collaboration with:. Eugene Zaremba (Queen’s University, Canada)

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Dynamics of Quantum-Degenerate Gases at Finite Temperature

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

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

  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: a: s-wave scattering length m: atomic mass Normalization condition:

  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 But! Condensate now pushes on thermal cloud- the response of which leads to a damping and frequency shift of the mode

  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: ‘normal’ thermal cloud densities: ‘anomalous’ Dynamical Equations E. Zaremba, T. Nikuni, and A. Griffin, JLTP 116, 277 (1999)

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

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

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

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

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

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

  20. Applications Numerical simulations useful in understanding the following experiments, that studied collective modes at finite-T: • 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).

  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:

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

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

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

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

  27. Drive frequencies Solid symbols – maximum condensate amplitude

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

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

  30. ‘tophat’ excitation scheme condensate collisions thermal cloud

  31. experiment theory

  32. excite condensate only condensate collisions thermal cloud

  33. Motivation: Experiment by O’Hara et al., Science 298, 2179 (2002). • Cool 6Li atoms (50-50 mixture of 2 hyperfine states) to quantum degeneracy T « TF • Static B-field tuned close to Feshbach resonance, a~ -104 a0 • Observe anisotropic expansion of the cloud Fermi gases

  34. Motivation: Experiment by O’Hara et al., Science 298, 2179 (2002). • Cool 6Li atoms (50-50 mixture of 2 hyperfine states) to quantum degeneracy T « TF • Static B-field tuned close to Feshbach resonance, a~ -104 a0 • Observe anisotropic expansion of the cloud Fermi gases  • Hydrodynamic behaviour, implying either: • Gas is superfluid (BCS or BEC) • Gas is normal, but collisions are frequent

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

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

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

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

  39. Taking moments:

  40. Taking moments:

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

  42. Low k limit:

  43. N=1.5105 =0.035 Unitarity limit:

  44. 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…

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