Lecture III Trapped gases in the classical regime

1. Lecture IIITrapped gases in the classical regime Bilbao 2004

2. Outline I-Boltzmann equation II-Method of averages III-Scaling factors method

3. I-Boltzmann equation

4. Trapped gases in the dilute regime Kinetic term Mean field collisions d : interparticle length l : de Broglie wavelength l << d : collisions dominate (irreversibility) l >> d : mean field dominate To describe the gas : The Boltzmann equation Confinement 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 Stationary solution: volume of the box l.h.s. OK, r.h.s: elastic collisions Conservation of energy

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

8. Exact solutions of the BE in an isotropic harmonic potential Relies on number of particle, energy and momentum conservation laws No damping ! One can readily generalize this solution to the quantum Boltzmann equation including the bosonic or fermionic statistics. Stationary solution L. Boltzmann, in Wissenschaftliche Abhandlungen, edited by F. Hasenorl (Barth, Leipzig, 1909), Vol. II, p. 83.

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

10. II-Method of averages

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

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 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.

14. 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 Test the accuracy by means of a molecular dynamics (Bird) Only term affected by collisions

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

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

17. Application: spinning up a classical gas rotating anisotropy Equilibrium Average methods combined with time relaxation aproach well suited to quadratic potential Angular momentum can be transferred only throught elastic collisions. What is the typical time scale to transfer angular mometum to the gas ? PRA 62 033607 (2000).

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

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

20. III-Scaling factors method

21. 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].

22. 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.

23. Time of flight of a collisionless gas in 2D and 3D Equations : Ellipticity : Ellipticity reflects the isotropy of the velocity distribution 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 Scaling form for the relaxation time PRA 68 043608 (2003)

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 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

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