Lecture III
Trapped gases
in the classical regime
Bilbao 2004
Outline
Trapped gases in the dilute regime
d : interparticle length
l : de Broglie wavelength
l << d : collisions dominate (irreversibility)
l >> d : mean field dominate
To describe the gas : The Boltzmann equation
Kinetic
term
Confinement
term
Mean field
collisions
Mean field and dimensionality
Mean field energy
Thermal energy
où
For a « pure» condensate it remains only the contribution of the mean
field
Gross - Pitaevskii
PRA 66 033613 (2002)
Stationary solution of the BE in a box
Stationary solution:
volume of the box
l.h.s. OK, r.h.s:
Conservation of energy
elastic collisions
Exact solutions of the BE in a box
Choice of scattering
properties:
Maxwell’s like particle
Class of solutions:
Normalization
Tail
Gaussian
One can work out explicitly
M. Krook and T. T. Wu, PRL 36 1107 (1976)
Exact solutions of the BE in an isotropic harmonic potential
Relies on number of particle, energy and momentum conservation laws
No damping !
Stationary solution
One can readily generalize this
solution to the quantum
Boltzmann equation including
the bosonic or fermionic statistics.
L. Boltzmann, in Wissenschaftliche Abhandlungen, edited by F. Hasenorl (Barth, Leipzig, 1909), Vol. II, p. 83.
Two « classical »types of experiments:
thermal gas versus BEC
Time of flight:
time
Excitation modes:
monopole
quadrupole
time
Averages
Function of space
BE :
with
and
and velocity :
Collisional invariants
with
Number of particles
conserved.
Momentum
conservation
Energy
conservation
This is still valid for the quantum Boltzmann equation
Monopole mode
Harmonic and isotropic confinement
We obtain a closed set of linear equations
(1)
Linear only for
harmonic confinement
(2)
(3)
We readily obtain the conservation
of energy Eq. (1) + Eq. (3)
Valid for bosons or fermions.
Quadrupolar mode
Linear set of equations for
the averages
To solve we need further
approximations
1_ One relaxation time
2_ Gaussian ansatz similar to the
previous approach, but gives also
an estimate for the relaxation time
Only term affected by collisions
Test the accuracy by means
of a molecular dynamics (Bird)
Quadrupolar modes (results & experiments)
HD
Exp ENS
Acta Physica Polonica B 33 p 2213 (2002).
CL
Theory
PRA 60 4851 (1999).
Quadrupolar mode BEC / thermal cloud in
the hydrodynamic limit
Disk shape
Cigar shape
Application: spinning up a classical gas
Average methods combined with time relaxation aproach
well suited to quadratic potential
rotating anisotropy
Equilibrium
Angular momentum can be transferred only throught elastic
collisions. What is the typical time scale to transfer angular mometum
PRA 62 033607 (2000).
to the gas ?
Spinning up a classical gas (results)
Angular momentum (rotating anisotropy) :
Collisionless regime
Dissipation of angular momentum
(static anisotropy) :
with
Why it could be interested to spin up the
thermal gas
Collisionless gas in 1D
[1]
Equilibrium solution:
such that
We search for a solution of Eq. [1] of the form:
with
;
;
Can be easily integrated
We find an exact solution of Eq. [1].
Collisionless gas in 1D (results)
Modes :
By linearizing, oscillation frequency
, i.e. monopole mode.
time of flight:
Lost the information on the initial state
We probe the velocity distribution, it permits to measure
the temperature.
Time of flight of a collisionless gas in 2D and 3D
Equations :
Ellipticity :
Ellipticity
reflects
the isotropy
of the
velocity
distribution
temps
The opposite limit: hydrodynamic regime
We search for a solution of
the form:
Continuity equation :
Euler Equation + adiabaticity :
Time of flight in the hydrodynamic regime
Inversion of ellipticity at long
times i.e. similar behaviour as
for superfluid phases !
Necessity of a quantitative theorie
which links the elastic collision rate
to the evolution of ellipticity.
Time of flight from an anisotropic trap
Evolution of ellipticity as a function of
time for different collision rate
Scaling ansatz and approximations
BE with mean field in the time relaxation approach:
Scaling ansatz
Scaling form
for the relaxation
time
PRA 68 043608 (2003)
Equations for the scaling parameters
This approach permits to find all the known results in the collisionless
or hydrodynamic regime, it gives an interpolation from the collisionless
regime to the hydrodynamic regime.
Consistent with numerical simulations.
Recently generalized to include Fermi statistics EuroPhys. Lett. 67, 534 (2004)
Equations for the scaling parameters
Circle experimental points
Solid line theory of scaling
parameters with no adjustable
parameter
How to link t0 and the collision rate ?
Gaussian ansatz
Molecular dynamics (Bird method)
Ellipticity as a function of time (result of simulation)
fitted with the scaling laws with only one parameter t0
Deviation from the gaussian
anstaz in the hydrodynamic regime
Quadrupolar mode (2D)
One can also compare modes and time of flight