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 ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ 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 $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\perp}}$. The dipole oscillation relaxation rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ 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, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, the three methods are shown to give identical results. In the quasi-2D case, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\ne}0$ at zero temperature. In quasi-1D system, $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{T}^{3}$ if the Fermi energy ${E}_{F}$ lies below the critical value ${E}_{F}<3{\ensuremath{\omega}}_{\ensuremath{\perp}}/4$. Otherwise, unless the system is close to integrability, the rate $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}$ has a 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 $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$, with the inelastic rate $1/{\ensuremath{\tau}}_{\mathrm{in}}\ensuremath{\ne}0$ as the phase-space opens up at finite energy of excitation ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. While $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}{\ensuremath{\tau}}_{\mathrm{in}}$ in the hydrodynamic regime, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\ll}1/{\ensuremath{\tau}}_{\mathrm{in}}$, in the regime of rare collisions, ${\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremath{\gg}1/{\ensuremath{\tau}}_{\mathrm{in}}$, we obtain the opposite trend $1/{\ensuremath{\tau}}_{\ensuremath{\perp}}\ensuremath{\propto}1/{\ensuremath{\tau}}_{\mathrm{in}}$.