Loschmidt echo

Abstract

The Loschmidt echo is a measure of the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system. It allows to quantify the sensitivity of quantum evolution to perturbations. An initial quantum state |ψ0⟩ evolves during a time t under a Hamiltonian H1 reaching the state |ψt⟩. Aiming to recover the initial state |ψ0⟩ a new Hamiltonian −H2 is applied between t and 2t. Perfect recover of |ψ0⟩ would be achieved by choosing H2 to be equal to H1. This is not possible in realistic setups, and there always appears a difference between H2 and H1, leading to a non-perfect recovery of the initial state. The forward evolution between t and 2t under the Hamiltonian −H2 is equivalent to a backward evolution from t to 0 under H2, which embodies the notion of time-reversal. The Loschmidt echo studies are focused on the cases where the dynamics induced by the Hamiltonians H1 and H2 are non-trivial, or sufficiently complex (like that of a classically chaotic one-particle system or a many-body system).