Phonon scattering from spatial relaxation of one-dimensional Bose gases

We theoretically investigate the nonequilibrium relaxation of a spatial density modulation in a one-dimensional, weakly interacting Bose gas, and its connection to the equilibrium scattering rate $\smash{\gamma_k\propto k^{3/2}}$ of the system's phononic excitations. We show that the relaxation is generally governed by a nonequilibrium scattering rate $\gamma_{k,t}$ coupled to quantum fluctuations, which approaches its equilibrium value $\gamma_k$ only at long times. Numerical simulations of quantum kinetic equations reveal an algebraic convergence, $\smash{\gamma_{k,t} - \gamma_k \sim t^{-2/3}}$, confirmed by analytical predictions. More broadly, our results establish a theoretical framework for experimentally probing phonon dynamics through the temporal evolution of local perturbations in quantum gases.