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

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

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

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

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

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

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

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

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

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

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!

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

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)

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

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.)

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

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

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

Numerical Methods B. Jackson and E. Zaremba, PRA 66, (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

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, (2002). Numerical simulations useful in understanding the following experiments, that studied collective modes at finite-T:

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)

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

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

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, (2002).

JILA experiment Excitation scheme: modulate trap potential m = 0

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

Drive frequencies Solid symbols – maximum condensate amplitude

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, (2002). Experiment: F. Chevy et al., PRL 88, (2002).

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, (2002). Experiment: F. Chevy et al., PRL 88, (2002).

condensate thermal cloud ‘tophat’ excitation scheme collisions

experiment theory

condensate thermal cloud excite condensate only collisions

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~ a 0 Observe anisotropic expansion of the cloud

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~ 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 

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

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

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

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

Taking moments:

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

Low k limit:

Unitarity limit: N=1.5  10 5 =0.035

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…