Unitary equilibrations: probability distribution of the Loschmidt echo

TL;DR

This paper investigates the probability distribution of the Loschmidt echo in closed quantum systems, revealing different equilibration regimes and a universal double-peaked distribution near criticality.

Contribution

It provides a detailed analysis of the full probability distribution of the Loschmidt echo after a quench, including exact results for a quasi-free system and identifying universal behaviors.

Findings

Gaussian distribution for small perturbations away from criticality

Universal double-peaked distribution near critical points

Different regimes of equilibration depending on initial state and quench

Abstract

Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly-solvable example of a quasi-free system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double peaked, "batman-hood" shaped, universal form.