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