Analytic results for a quantum quench from free to hard-core one-dimensional bosons

It is widely believed that the stationary properties after a quantum quench in integrable systems can be described by a generalized Gibbs ensemble (GGE), even if all of the analytical evidence is based on free theories in which the pre- and postquench modes are linearly related. In contrast, we consider the experimentally relevant quench of the one-dimensional Bose gas from zero to infinite interaction, in which the relation between modes is nonlinear, and consequently Wick's theorem does not hold. We provide exact analytical results for the time evolution of the dynamical density-density correlation function at any time after the quench and we prove that its stationary value is described by a GGE in which Wick's theorem is restored.

Figure 1

Equal-time density-density correlation function $\langle \widehat{\rho }(0,t)\widehat{\rho }(x,t)\rangle$. (a) As a function of the distance, we report the correlation for several different times. Equation (9) (solid line) is compared with the numerical results of Ref. [29] (points), showing an excellent agreement before the form-factor truncation and finite-size effects become important. (b) The same as in (a) for larger time. (c) $\langle \widehat{\rho }(0,t)\widehat{\rho }(x,t)\rangle$ for a few fixed $x$ as a function of time in logarithmic scale. Notice the highly oscillatory (and telescopic in $x$) behavior for very short time that is due to the presence of very high energy modes in the initial state. After reaching a global maximum, the correlation monotonically reaches the GGE value (the large-time plateau).

Figure 2

Large-time dynamical density-density correlation function $\langle \widehat{\rho }(0,t)\widehat{\rho }(x,t+\Delta t)\rangle$. (a) Subtracted correlation [i.e., $\langle \widehat{\rho }(0,t)\widehat{\rho }(x,t+\Delta t)\rangle -n{F}_0(x,\Delta t)$ to avoid the divergence at $x=\Delta t=0$] as a function of $\Delta t$ for $x=0$ and $nx=2$. The autocorrelation ($x=0$) does not depend on the elapsed time $t$, so the plot is valid for any $t>0$. The real part agrees perfectly with the numerical data in Ref. [29], but the imaginary part does not because of a different subtraction. (b) Full correlation as a function of $x$ for $n^2\Delta t=0.01$ and $n^2\Delta t=0.05$.