Transcript 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) 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
But!
Condensate now pushes on thermal cloud- the response of which leads to a damping and frequency shift of the mode
Collective modes: finite T
And Change in trap frequency also excites collective oscillations of the thermal cloud , which can couple back to the condensate motion
ZNG Formalism
Bose broken symmetry: condensate wavefunction: condensate density: thermal cloud densities:
‘normal’ ‘anomalous’ Dynamical Equations
E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
ZNG Formalism
Generalized Gross-Pitaevskii equation:
Popov approximation: E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
ZNG Formalism
Boltzmann kinetic equation: Hartree-Fock excitations: moving in effective potential: phase space density: (semiclassical approx.)
f
(
r
,
p
,
t
) E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
ZNG Formalism
Boltzmann kinetic equation:
E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
Coupling:
ZNG Formalism
mean field coupling E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
Coupling:
ZNG Formalism
Collisional coupling (atom transfer) E. Zaremba, T. Nikuni, and A. Griffin, JLTP
116
, 277 (1999)
Numerical Methods
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 B. Jackson and E. Zaremba, PRA
66
, 033606 (2002).
Applications
Numerical simulations useful in understanding the following experiments, that studied collective modes at finite-T:
• • •
Scissors modes
(Oxford): PRL
86
, 3938 (2001).
O. M. Maragò
et al
.,
Quadrupole modes
(JILA):
78
, 764 (1997).
D. S. Jin
et al.
, PRL
Transverse breathing mode
(ENS): F. Chevy
al.
, PRL
88
, 250402 (2002).
et
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).
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).
JILA experiment
Excitation scheme:
modulate trap potential
m = 0
= 1.95
T ´ = 0.8
condensate thermal cloud
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 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
Experiment:
F. Chevy
et al.
, PRL
88
, 250402 (2002).
Theory:
B. Jackson and E. Zaremba
.
, PRL
89
, 150402 (2002).
‘tophat’ excitation scheme collisions
condensate thermal cloud
experiment theory
excite condensate only collisions
condensate thermal 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~ -10 4 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~ -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
Fermi gases
Feshbach resonance:
Collision cross section:
Jochim et al., PRL 89, 273202 (2002).
= relative velocity of colliding atoms
Fermi gases
Feshbach resonance:
Low
k
limit:
Jochim et al., PRL 89, 273202 (2002).
= relative velocity of colliding atoms
Fermi gases
Feshbach resonance:
Unitarity limit:
Jochim et al., PRL 89, 273202 (2002).
= relative velocity of colliding atoms
Quadrupole collective modes: In-phase modes:
L. Vichi, JLTP
121
, 177 (2000)
Taking moments:
Taking moments:
Solve set of equations for
R
i
Example:
transverse breathing mode in a cigar-shaped trap • collisionless limit:
ωτ » 1
R
2 • hydrodynamic limit:
ωτ « 1
R
( 10 / • intermediate regime:
ωτ ~ 1
3 ) ( 10 / 3 )
R
2 0 0 0
Low
k
limit:
Unitarity limit: N=1.5
10 5 =0.035
( ) 1 3 ( ) 1 2 ( ) 1 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…