1. Lecture III Trapped gases in the classical regime Bilbao 2004

2. Outline

4. Trapped gases in the dilute regime d : interparticle length  de Broglie wavelength << d : collisions dominate (irreversibility) >> d : mean field dominate To describe the gas : The Boltzmann equation Confinement term collisionsMean field Kinetic term

5. 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)

6. 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:

7. Exact solutions of the BE in a box Class of solutions: NormalizationTail Maxwell’s like particle Choice of scattering properties: M. Krook and T. T. Wu, PRL 36 1107 (1976) Gaussian One can work out explicitly

8. 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

9. Two « classical »types of experiments: thermal gas versus BEC Time of flight: Excitation modes: time monopole quadrupole time

11. Averages BE : with and Function of space and velocity :

12. Collisional invariants with Number of particles conserved. Momentum conservation Energy conservation This is still valid for the quantum Boltzmann equation

13. 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)

14. Quadrupolar mode Linear set of equations for the averages Only term affected by collisions 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 Test the accuracy by means of a molecular dynamics (Bird)

15. Quadrupolar modes (results & experiments) PRA 60 4851 (1999). HD CL Acta Physica Polonica B 33 p 2213 (2002). Exp ENS Theory

16. Quadrupolar mode BEC / thermal cloud in the hydrodynamic limit Cigar shape Disk shape

17. 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 ?

18. Angular momentum (rotating anisotropy) : Dissipation of angular momentum (static anisotropy) : with Spinning up a classical gas (results) Collisionless regime

19. Why it could be interested to spin up the thermal gas

21. Collisionless gas in 1D Equilibrium solution: such that [1] We search for a solution of Eq. [1] of the form: with ;; Can be easily integrated We find an exact solution of Eq. [1].

22. 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)

23. Time of flight of a collisionless gas in 2D and 3D Equations : Ellipticity : reflects the isotropy of the velocity distribution Ellipticity temps

24. The opposite limit: hydrodynamic regime We search for a solution of the form: Continuity equation : Euler Equation + adiabaticity :

25. 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.

26. Time of flight from an anisotropic trap Evolution of ellipticity as a function of time for different collision rate

27. Scaling ansatz and approximations BE with mean field in the time relaxation approach: Scaling ansatz PRA 68 043608 (2003) Scaling form for the relaxation time

28. Equations for the scaling parameters Modes Time of flight 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)

29. Equations for the scaling parameters Circle experimental points Solid line theory of scaling parameters with no adjustable parameter

30. 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 gaussian anstaz in the hydrodynamic regime Gaussian ansatz Molecular dynamics (Bird method)

31. Quadrupolar mode (2D) One can also compare modes and time of flight