Exact Relaxation in a Class of Nonequilibrium Quantum Lattice Systems

A reasonable physical intuition in the study of interacting quantum systems says that, independent of the initial state, the system will tend to equilibrate. In this work we introduce an experimentally accessible setting where relaxation to a steady state is exact, namely, for the Bose-Hubbard model quenched from a Mott quantum phase to the free strong superfluid regime. We rigorously prove that the evolving state locally relaxes to a steady state with maximum entropy constrained by second moments—thus maximizing the entanglement. Remarkably, for this to be true, no time average is necessary. Our argument includes a central limit theorem and exploits the finite speed of information transfer. We also show that for all periodic initial configurations (charge density waves) the system relaxes locally, and identify experimentally accessible signatures in optical lattices as well as implications for the foundations of statistical mechanics.

Figure 1

Intuitive picture of the relaxation process in the quenched Bose-Hubbard model: for any lattice site $i$ (or any block of sites) within a cone (dark gray) defined by the speed of information transfer excitations significantly contribute to the local mixing of the state at the site. Contributions from outside this cone (light gray) are exponentially suppressed. The incommensurate influence of the lattice sites in the cone gives rise to a relaxation to maximal entropy for large times $t$.

Figure 2

Second moments $⟨{\widehat{b}}_1^{†}{\widehat{b}}_{1+l}⟩$ (left) and $⟨{\widehat{b}}_2^{†}{\widehat{b}}_{2+l}⟩$ (right), $l=0$, 1, 5, 10, 15, 20, for an initial periodic configuration $|1,0,1,0\cdots ⟩$ on a lattice of size $N=300$. For larger times recurrences occur (not shown), while for $N\to \infty$, one has $\underset{t\to \infty }{lim}⟨{\widehat{b}}_i^{†}{\widehat{b}}_j⟩={\delta }_{i,j}/2$.