Thermalization of dipole oscillations in confined systems by rare collisions

Abstract

We study the relaxation of the center-of-mass, or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency ${\omega }_{\perp }$ fixed the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate, $1/{\tau }_{in}\ll {\omega }_{\perp }$. The dipole oscillations relaxation rate, $1/{\tau }_{\perp }$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation and applying the memory function formalism. As long as anharmonicity is weak, $1/{\tau }_{\perp }\ll 1/{\tau }_{in}$ the three methods are shown to give identical results. In quasi-2D case $1/{\tau }_{\perp }\ne 0$ at zero temperature. In quasi-1D system $1/{\tau }_{\perp }\propto T^3$ if the Fermi energy, $E_F$ lies below the critical value, $E_F<3{\omega }_{\perp }/4$. Otherwise, unless the system is close to integrability, the rate $1/{\tau }_{\perp }$ has the temperature dependence similar to that in quasi-2D. In all cases the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/{\tau }_{\perp }\propto 1/{\tau }_{in}$, with the inelastic rate $1/{\tau }_{in}\ne 0$ as the phase-space opens up at finite energy of excitation, $\hslash {\omega }_{\perp }$. While $1/{\tau }_{\perp }\propto {\tau }_{in}$ in the hydrodynamic regime, ${\omega }_{\perp }\ll 1/{\tau }_{in}$, in the regime of rare collisions, ${\omega }_{\perp }\gg 1/{\tau }_{in}$, we obtain the opposite trend $1/{\tau }_{\perp }\propto 1/{\tau }_{in}$.

34 pages, 10 figures

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