Spin dynamics in a polarized Bose gas.

At low temperature, the equation of motion' for the spin density m in nuclear polarized paramagnetic liquids (e.g. , liquid He, HeHe mixtures) and gases (e.g. , He, H i ) differs from the conventional Fick's law form Bm/t)t =D V' m of classical systems. In general, the equation becomes nonlinear and anisotropic and the transverse component has an oscillatory (wave) as well as a diffusive character. This wavelike behavior results from precession of the spins in the molecular field resulting from the exchange interaction introduced by the identical particle (anti) symmetrization. This prominent quantum effect in the spin dynamics of polarized systems has been confirmed experimentally. Specifically, standard spinecho experiments in liquid He (Ref. 3) and He-"He mixtures display spin-wave character and transverse spin oscillation modes (spin waves) were observed in liquid He (Ref. 5), HeHe mixtures, ' atomic hydrogen (H i ) gas, ' and He gas. ' In his derivation of the equation for the spin density in weakly polarized Fermi liquids, Leggett noted, without elaboration, that the transverse spin-diffusion relaxation time ~~ is the same as the longitudinal relaxation time ~~~. The equivalence of ~ll and vj for polarization M~O [M— = (n+ n)/(n++n —)], where n+ (n ) is the density of spin-up (-down) nuclei] may be justified using spin-rotation invariance alone (but, see below). On the other hand, Lhuillier and Laloe derived the spindynamical equation for nondegenerate Fermi and Bose gases by solving the kinetic equation developed by them. They found that the interference effect during the collisions (the identical spin-rotation effect) leads to a nonlinear spin hydrodynamic equation which has the same form as that derived by Leggett for Fermi liquids, with a change in sign of the exchange interaction term for Bose gases. Both the Leggett' and Lhuillier and Laloe equations contain a single difFusion coefficient [i.e., D~~ =Di, where Di (Di) is the longitudinal (transverse) diffusion coefficient, defined explicitly below]. Thus, the distinction between transverse and longitudinal diffusion was not emphasized. In 1985, Meyerovich" reasoned that, for degenerate polarized Fermi gases, ~~ ought to be different from ~~~ and that the equation for the spin density should contain two different diffusion coefficients, D~~ and D~. Jeon and Mullin' calculated 7~~ ~ and D~~ ~ for a dilute Fermi gas from the kinetic equation they presented' which is equivalent to the equation derived earlier by Silin. ' They showed that when T/TF (([(I+M)' ' —(I —M)' ']