Tight bounds on recurrence time in closed quantum systems

TL;DR

This paper derives quantitative upper bounds on the recurrence time of isolated quantum systems, linking it to the Hamiltonian's properties and initial state coherence, advancing fundamental understanding of quantum recurrence phenomena.

Contribution

It establishes the first rigorous bounds on quantum recurrence time, relating it to escape time and Hamiltonian variance, and analyzes the effects of initial state coherence.

Findings

Upper bounds on recurrence time depend on Hilbert-space dimension and neighborhood size.

Recurrence time bounds are saturated for random Hamiltonians.

Initial state coherence influences recurrence behavior.

Abstract

The evolution of an isolated quantum system inevitably exhibits recurrence: the state returns to the vicinity of its initial condition after finite time. Despite its fundamental nature, a rigorous quantitative understanding of recurrence has been lacking. We establish upper bounds on the recurrence time, $t_{\mathrm{rec}} \lesssim t_{\mathrm{exit}}(\epsilon)(1/\epsilon)^d$, where $d$ is the Hilbert-space dimension, $\epsilon$ the neighborhood size, and $t_{\mathrm{exit}}(\epsilon)$ the escape time from this neighborhood. For pure states evolving under a Hamiltonian $H$, estimating $t_{\mathrm{exit}}$ is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state $\psi_t$ needs to depart from the $\epsilon$-vicinity of the initial state $\psi_0$. We provide a partial solution, showing that under mild assumptions $t_{\mathrm{exit}}(\epsilon) \approx \epsilon /\sqrt{ \Delta(H^2)}$, with $\Delta(H^2)$ the Hamiltonian variance in $\psi_0$. We show that our upper bound on $t_{\mathrm{rec}}$ is generically saturated for random Hamiltonians. Finally, we analyze the impact of coherence of the initial state in the eigenbasis of $H$ on recurrence behavior.