lecture iii trapped gases in the classical regime
DESCRIPTION
Lecture III Trapped gases in the classical regime. Bilbao 2004. Outline. I-Boltzmann equation. II-Method of averages. III-Scaling factors method. I-Boltzmann equation. Trapped gases in the dilute regime. Kinetic term. Mean field. collisions. d : interparticle length - PowerPoint PPT PresentationTRANSCRIPT
Lecture III
Trapped gases in the classical regime
Bilbao 2004
Outline
Trapped gases in the dilute regime
d : interparticle lengthde Broglie wavelength
<< d : collisions dominate (irreversibility) >> d : mean field dominate
To describe the gas : The Boltzmann equation
Confinementterm
collisionsMean fieldKinetic term
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
l.h.s. OK, r.h.s:
Conservation of energy elastic collisions
volume of the box
Stationary solution:
Exact solutions of the BE in a box
Class of solutions:
Normalization Tail
Maxwell’s like particleChoice of scatteringproperties:
M. Krook and T. T. Wu, PRL 36 1107 (1976)
GaussianOne can work out explicitly
Exact solutions of the BE in an isotropic harmonic potential
L. Boltzmann, in Wissenschaftliche Abhandlungen, edited by F. Hasenorl (Barth, Leipzig, 1909), Vol. II, p. 83.
No damping !
Relies on number of particle, energy and momentum conservation laws
One can readily generalize this solution to the quantum Boltzmann equation including the bosonic or fermionic statistics.
Stationary solution
Two « classical »types of experiments: thermal gas versus BEC
Time of flight:
Excitation modes:
time
monopole quadrupole
time
Averages
BE :
with
and
Function of space and velocity :
Collisional invariants
with
Number of particles conserved.
Momentumconservation
Energyconservation
This is still valid for the quantum Boltzmann equation
Monopole mode
Harmonic and isotropic confinement
Valid for bosons or fermions.
We obtain a closed set of linear equations
Linear only for harmonic confinement
We readily obtain the conservation of energy Eq. (1) + Eq. (3)
(3)
(2)
(1)
Quadrupolar mode
Linear set of equations forthe averages
Only term affected by collisions
To solve we need further approximations
1_ One relaxation time
2_ Gaussian ansatz similar to theprevious approach, but gives alsoan estimate for the relaxation time
Test the accuracy by means of a molecular dynamics (Bird)
Quadrupolar modes (results & experiments)
PRA 60 4851 (1999).
HD CL
Acta Physica Polonica B 33 p 2213 (2002).Exp ENS Theory
Quadrupolar mode BEC / thermal cloud in the hydrodynamic limit
Cigar shape
Disk shape
Application: spinning up a classical gas
Average methods combined with time relaxation aproach well suited to quadratic potential
rotating anisotropy
PRA 62 033607 (2000).
Equilibrium
Angular momentum can be transferred only throught elastic collisions. What is the typical time scale to transfer angular mometum to the gas ?
Angular momentum (rotating anisotropy) :
Dissipation of angular momentum (static anisotropy) :
with
Spinning up a classical gas (results)
Collisionless regime
Why it could be interested to spin up the thermal gas
Collisionless gas in 1D
Equilibrium solution: such that
[1]
We search for a solution of Eq. [1] of the form:
with ; ;
Can be easily integratedWe find an exact solution of Eq. [1].
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.
Collisionless gas in 1D (results)
Time of flight of a collisionless gas in 2D and 3D
Equations :
Ellipticity :
reflects the isotropy of the velocity distribution
Ellipticity
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 rateto 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
PRA 68 043608 (2003)
Scaling formfor the relaxationtime
Equations for the scaling parameters
Modes
Time of flight
This approach permits to find all the known results in the collisionlessor 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 adjustableparameter
How to link 0 and the collision rate ?
Ellipticity as a function of time (result of simulation)fitted with the scaling laws with only one parameter 0
Deviation from the gaussiananstaz in the hydrodynamic regime
Gaussian ansatz
Molecular dynamics (Bird method)
Quadrupolar mode (2D)
One can also compare modes and time of flight