Quench dynamics of one-dimensional bosons in a commensurate periodic potential: A quantum kinetic equation approach

Results are presented for the dynamics arising due to a sudden quench of a boson interaction parameter with the simultaneous switching on of a commensurate periodic potential, the latter providing a source of nonlinearity that can cause inelastic scattering. A quantum kinetic equation is derived perturbatively in the periodic potential and solved within the leading order gradient expansion. A two-particle irreducible formalism is employed to construct the stress-momentum tensor and hence the conserved energy. The dynamics is studied in detail in the phase where the boson spectrum remains gapless. The periodic potential is found to give rise to multiparticle scattering processes that relaxes the boson distribution function. At long times the system is found to thermalize with a thermalization time that depends in a nonmonotonic way on the amount of energy injected into the system due to the quantum quench. This nonmonotonic behavior arises due to the competing effect of an increase of phase space for scattering on the one hand, and an enhancement of the orthogonality catastrophe on the other hand as the quench amplitude is increased. The approach to equilibrium is found to be purely exponential for large quench amplitudes, and more complex for smaller quench amplitudes.

Figure 1

The boson distribution function $n\left(q\right)$ generated by an interaction quench where $K_{\mathrm{neq}}=13.8$ and $K_{\mathrm{eq}}=3$ is $n\left(q\right)=\frac{1}{2}\left(\frac{K_{\mathrm{neq}}}{K_{\mathrm{eq}}}-1\right)\approx 1.797$ (continuous line). This is compared with the boson distribution at equilibrium at a nonzero temperature $1/[e^{u\left|q\right|/T_{\mathrm{eq}}}-1]$, where $T_{\mathrm{eq}}=2.23$ is the temperature associated with the energy injected into the system due to the quench (dashed line). At zero temperature equilibrium, $n\left(q\right)\equiv 0$.

Figure 2

The self-energies to leading order $\mathcal{O}\left(g^2\right)$ in the periodic potential. The solid lines are the boson Green's function $G$.

Figure 3

The distribution function $\Psi \left(q\right)=qF\left(q\right)$ plotted for increasing times for a quench where $K_{\mathrm{eq}}=3,K_{\mathrm{neq}}=13.8$. At the initial time, the quench generates the distribution given by the straight line $\Psi \left(q\right)=q\frac{K_{\mathrm{neq}}}{K_{\mathrm{eq}}}$. The distribution converges to the equilibrium distribution given by $\Psi \left(q\right)=q\coth \left(\frac{uq}{2T_{\mathrm{eq}}}\right)$.

Figure 4

At long times the distribution converges to ${\Psi }_{\mathrm{eq}}\left(q\right)=q\coth \left(\frac{uq}{2T_{\mathrm{eq}}}\right)$ (dashed line). The quenches and the time for which the distributions are shown correspond to $K_{\mathrm{eq}}=3,K_{\mathrm{neq}}=13.8,T=1162$ and $K_{\mathrm{eq}}=4,K_{\mathrm{neq}}=5.38,T=1072$. The final temperatures are, respectively, $T_{\mathrm{eq}}=2.22$ and $T_{\mathrm{eq}}=0.414$.

Figure 5

Comparison between the full model (dashed line) and the toy model (solid line) for $K_{\mathrm{eq}}=3$ and $K_{\mathrm{neq}}=13.8$. The differences are unobservable. The system thermalizes to $T_{\mathrm{eq}}$ (dot-dashed line).

Figure 6

Time evolution of the effective temperature for (bottom to top) $K_{\mathrm{neq}}=4.40,4.72,5.38,6.0,6.72,9.08,11.48,13.91$, and $17.72$ with $K_{\mathrm{eq}}=4$. For all quenches the system thermalizes to the equilibrium temperate $T_{\mathrm{eq}}$ (dashed line).

Figure 7

For small quenches, the relaxation times (given by the abscissa corresponding to the three black dots) decreases with increasing quench amplitude (dot-dashed line). The parameters are $K_{\mathrm{eq}}=4$ and $K_{\mathrm{neq}}=4.40,4.72$, and $5.38$ (bottom to top).

Figure 8

For large quenches, the relaxation time (given by the abscissa corresponding to the three black dots) increases with the quench amplitude (dot-dashed line). The parameters are $K_{\mathrm{eq}}=3$ and $K_{\mathrm{neq}}=8.39,11.3$, and $13.8$ (bottom to top).

Figure 9

The relaxation rate (inverse thermalization time) for quenches with different $K_{\mathrm{neq}}$ keeping $K_{\mathrm{eq}}$ fixed at $K_{\mathrm{eq}}=3$ and $K_{\mathrm{eq}}=4$. $[K_{\mathrm{neq}}-K_{\mathrm{eq}}]/K_{\mathrm{eq}}$ is proportional to the energy density injected due to the quench. The numerical results are compared with the analytic expression Eq. (109) obtained from perturbation theory.

Figure 10

Comparison between $T_{\mathrm{eff}}\left(T\right)$ obtained from the kinetic equation (solid line) with a relaxational ansazt $T_{\mathrm{eff}}^{\exp }\left(T\right)=T_{\mathrm{eq}}\left(1-e^{-\eta T}\right)$ (dashed lines). The parameters are $K_{\mathrm{eq}}=4$ and $K_{\mathrm{neq}}=4.4,4.72,5.38$, and $9.08$ (bottom to top). As the quench amplitude decreases, the deviation from a simple exponential relaxation becomes larger.