Dynamics of Quantum-Dynamics of Quantum-Degenerate Gases at Finite Degenerate Gases at Finite

TemperatureTemperature

Brian JacksonBrian Jackson

Inauguration meeting and Lev Pitaevskii’s Birthday: Trento, March 14-15

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 TGross-Pitaevskii equation:

Normalization condition:

a: s-wave scattering lengthm: 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 FormalismBose 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 FormalismGeneralized 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.)

),,( tf pr

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, 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

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:

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, 100404 (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, 180402 (2002).

JILA experimentExcitation scheme: modulate trap potential

m = 0

condensate

thermal cloud

= 1.95 T´ = 0.8

Drive frequencies

Solid symbols – maximum condensate amplitude

ENS experimentm = 0 mode in an

elongated trapExcitation 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).

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).

condensatethermal cloud

‘tophat’ excitation schemecollisions

experimenttheory

condensatethermal cloud

excite condensate onlycollisions

Fermi gasesMotivation: 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 gasesMotivation: 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

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, 273202 (2002).= relative velocity of colliding atoms

Collision cross-section:

Feshbach resonance:

Fermi gases

Jochim et al., PRL 89, 273202 (2002).= relative velocity of colliding atoms

Low k limit:

Fermi gasesFeshbach resonance:

Jochim et al., PRL 89, 273202 (2002).= relative velocity of colliding atoms

Unitarity limit:

Quadrupole collective modes:

In-phase modes:

L. Vichi, JLTP 121, 177 (2000)

Taking moments:

Taking moments:

• collisionless limit: ωτ » 1

• hydrodynamic limit: ωτ « 1

• intermediate regime: ωτ ~ 1

Solve set of equations for iR

Example: transverse breathing mode in a cigar-shaped trap

2R 0

)3/10(R

2)3/10( R 0

0

Low k limit:

Unitarity limit:

N=1.5105

=0.035

1)( 1

2)( 1

3)( 1

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…