Phonon scattering from spatial relaxation of one-dimensional Bose gases

TL;DR

This paper develops a nonequilibrium quantum hydrodynamics framework to study how a spatial density modulation in a one-dimensional Bose gas relaxes and how its relaxation reveals the phonon scattering rate. Using a Keldysh 1PI approach and self-consistent Born resummation, the authors derive a time-dependent scattering rate $\gamma_{k,t}$ that couples to phonon fluctuations and converges to the equilibrium rate $\gamma_k$ at long times, with an algebraic approach $\gamma_{k,t}-\gamma_k \sim t^{-2/3}$. They show that the density relaxation after a weak periodic quench is governed by this evolving rate, and that the relaxation time is of order $\gamma_k^{-1}$; they further compute Beliaev and Landau regimes and demonstrate a concrete spectroscopic protocol to measure phonon lifetimes in experiments. The results provide a general framework to access phonon dynamics via the temporal evolution of local perturbations in quantum gases and establish a bridge between nonequilibrium relaxation and equilibrium scattering properties.

Abstract

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 $\smash{γ_k\propto k^{3/2}}$ of the system's phononic excitations. We show that the relaxation is generally governed by a nonequilibrium scattering rate $γ_{k,t}$ coupled to quantum fluctuations, which approaches its equilibrium value $γ_k$ only at long times. Numerical simulations of quantum kinetic equations reveal an algebraic convergence, $\smash{γ_{k,t} - γ_k \sim t^{-2/3}}$, 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.

Figures (5)

• Figure 1: Quench protocol and post-quench evolution of the spatial density. (a) Before the quench, the gas is homogeneneous, with a phonon distribution given by Eq. (\ref{['eq:npprequench']}). At $t=0$, a periodic potential is switched on, and switched off shortly after, at $t=\delta t$. (b) Right after the potential quench ($t=\delta t$), the density profile has acquired a periodic modulation, Eq. (\ref{['eq:density_a']}), and the phonon distribution is given by Eq. (\ref{['eq:pq_mom']}). (c) At later time, the density modulation exhibits both coherent oscillations and damping, cf. Eq. (\ref{['eq:rhofinal']}). The figure illustrates the effect of damping at times integer multiples of $2 \pi/\epsilon_{k_0}$.
• Figure 2: Phonon distribution $n_{k,t}$ at successive times following the periodic-potential quench, computed numerically from Eq. \ref{['eq:deltan']}. Time is expressed in units of $\tau\equiv 1/(2g\rho_0)$. At $t=0$, the distribution features a triangular peak centered at $k=k_0$, superimposed on a smooth thermal background at temperature $T_0$, Eq. \ref{['eq:pq_mom']}. In the long-time limit $t=\infty$, relaxation is complete and $n_{k,\infty}$ coincides with the thermal distribution \ref{['eq:BEinfinite']} at $T_f>T_0$. Parameters are chosen as $T_0/(2g\rho_0)=2.10^{-3}$, $\rho_0\xi=5$ (weakly interacting gas), $k_0\xi=3.10^{-2}$ and $T_f = T_0\sqrt{1+0.5}$. $\varepsilon$ is determined by inverting Eq. \ref{['eq:Tf']}, yielding $\varepsilon^2 = 0.24$.
• Figure 3: Deviation $|\delta n_{k,t}|\equiv |n_{k,t}-n_k|$ of the phonon distribution from its equilibrium value as a function of time, shown in (a) logarithmic and (b) log-log scales, for several momentum values (dips in the curves correspond to a change of sign of $\delta n_{k,t}$). The deviation decays exponentially at short times and algebraically at long times. Solid curves correspond to numerical solutions of Eq. \ref{['eq:deltan']}. In (a), dotted lines are exponential fits of the form $\exp(-1.5\gamma_k t),\, \exp(-1.7\gamma_k t),\ \exp(-1.9\gamma_k t)$ and $\exp(-2.1\gamma_k t)$ from top to bottom. In (b), dotted lines are algebraic fits of the form $t^{-\alpha}$, with $\alpha=0.593, 0.651$ and $0.666$ from smallest to highest $k\xi$ values. Parameters used in the numerics are the same as in Fig. \ref{['Fig_momt']}.
• Figure 4: (a) Nonequilibrium deviation $\delta\gamma_{k_0,t}$ of the scattering rate from its thermal value $\gamma_{k_0}$ as a function of time, shown for several probe momenta $k_0$. Solid lines correspond to numerical results from Eq. \ref{['eq:deltagamma']}, and dotted lines are power-law fits $\propto t^{-\alpha}$, with $\alpha=0.646, 0.662, 0.669$ and $0.671$ from top to bottom. (b) Effective scattering rate, Eq. \ref{['eq:gammakeff']}, as a function of time, for various $k_0$. At long time, $\gamma^\text{eff}_{k_0,t}\to \gamma_{k_0}$. Dotted curves are fits to a function of the form $1-\text{const}/t^\alpha$, using the same $\alpha$ values as in (a). The black dot marks the inflection point defining the relaxation time $t_\Sigma$. The final temperature is set to $T_f=T_0\sqrt{1+0.5}$ for all curves, which corresponds to $\varepsilon^2 = 9.10^{-5}, 9.10^{-4}, 9.10^{-3}, 6.10^{-2}$ and $2.10^{-1}$ from smallest to highest $k_0\xi$ values. Parameters used in the numerics are the same as in Fig. \ref{['Fig_momt']}.
• Figure 5: Relaxation time $t_\Sigma$ of the scattering rate $\gamma_{k_0,t}$ (blue dots), defined from the inflection point of the ${\gamma_{k_0,t}^\text{eff}}$ curves in Fig. \ref{['FIg_SEDecay']}(b). The black solid curve shows, for comparison, the inverse of the equilibrium scattering rate, $\gamma_k^{-1}$, while the red (green) dashed line indicates its Landau (Beliaev) limit, given by Eq. \ref{['eq:Landau']} [Eq.\ref{['eq:Beliaev']}]. Parameters used in the numerics are the same as in Fig. \ref{['Fig_momt']}.