Tight Bounds on Recurrence Time in Closed Quantum Systems

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_{rec}≲t_{exit}(ε)(1/ε)^{d}, where d is the Hilbert-space dimension, ε the neighborhood size, and t_{exit}(ε) the escape time from this neighborhood. For pure states evolving under a Hamiltonian H, estimating t_{exit} is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state ψ_{t} needs to depart from the ε vicinity of the initial state ψ_{0}. We provide a partial solution, showing that under mild assumptions t_{exit}(ε)≈ε/sqrt[Δ(H^{2})], with Δ(H^{2}) the Hamiltonian variance in ψ_{0}. We show that our upper bound on t_{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.