Large amplitude spin waves in ultra-cold gases

We discuss the theory of spin waves in non-degenerate ultra-cold gases, and compare various methods which can be used to obtain appropriate kinetic equations. We then study non-hydrodynamic-situations, where the amplitude of spin waves is sufficiently large to bring the system far from local equilibrium. The full position and momentum dependence of the distribution function must then be retained. In the first part of the article; we compare two general methods which can be used to derive a kinetic equation for a dilute gas of atoms (bosons or fermions) with two internal states (treated as a pseudo-spin 1/2). The collisional methods are in the spirit of Boltzmann's original derivation of his kinetic equation where, at each point of space, the effects of all sorts of possible binary collisions are added. We discuss two different versions of collisional methods, the Yvon-Snider approach and the S matrix approach. The second method uses the notion of mean field, which modifies the drift term of the kinetic equation, in the line of the Landau theory of transport in quantum liquids. For a dilute cold gas, it turns out that all these derivations lead to the same drift terms in the transport equation, but differ in the precise expression of the collision integral and in higher order gradient terms. In the second part of the article, the kinetic equation is applied to spin waves (or internal conversion) in trapped ultra-cold gases. Numerical simulations are used to illustrate the strongly non-hydrodynamic character of the spin waves recently observed with trapped Rb-87 atoms. The decay of the phenomenon, which takes place when the system relaxes back towards equilibrium, is also discussed, with a short comment on decoherence. In two appendices we calculate the Wigner transform of the interaction term in the S matrix method, to first order in gradients; Appendix A.1 treats the case of spin-independent interactions, Appendix A.2 that of spin-dependent interactions.