Post on 12-Feb-2016
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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…