Interaction quenches in the one-dimensional Bose gas

The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.

Figure 1

Quench from a noninteracting initial state to arbitrary final interactions. Main panel: Local correlations $g_2$ and $g_3$ as functions of the coupling $\gamma$, calculated from the two truncated generalized Gibbs ensembles (GGE) (red dashed, blue solid) and from the grand canonical ensemble (GCE) (dot-dashed). The asymptotic behaviors are also shown (dotted). Inset: Density of quasimomenta, ${\rho }_{\text{LL}}^{\left(1\right)}\left(\lambda \right),{\rho }_{\text{LL}}^{\left(2\right)}\left(\lambda \right)$ in the two truncated GGE (red dashed, blue solid) and ${\rho }_{\text{LL}}^{\text{th}}\left(\lambda \right)$ in the GCE (black dot-dashed) for $\gamma =1.$

Figure 2

Quench to the TG regime ($\gamma =\infty$). Main panel: Equal time density-density correlation function. We compare GGE/saddle point (green solid) values with the large time result of a numerical solution of the dynamics of Ref. 41 (purple dot-dot-dashed). Inset: Momentum distribution function.