The quantum Loschmidt echo on flat tori

TL;DR

This paper derives explicit formulas for the Quantum Loschmidt Echo on flat tori, revealing how initial data and perturbations influence quantum sensitivity and decay in semiclassical regimes.

Contribution

It provides new explicit formulae for the Quantum Loschmidt Echo on flat tori and establishes criteria for non-decay based on initial data properties.

Findings

Explicit formulas involving defect measures and perturbations

Non-concentration criteria prevent macroscopic decay

Applications to coherent states and plane waves

Abstract

The Quantum Loschmidt Echo is a measurement of the sensitivity of a quantum system to perturbations of the Hamiltonian. In the case of the standard 2-torus, we derive some explicit formulae for this quantity in the transition regime where it is expected to decay in the semiclassical limit. The expression involves both a two-microlocal defect measure of the initial data and the form of the perturbation. As an application, we exhibit a non-concentration criterium on the sequence of initial data under which one does not observe a macroscopic decay of the Quantum Loschmidt Echo. We also apply our results to several examples of physically relevant initial data such as coherent states and plane waves.