title: Origin of the exponential decay of the Loschmidt echo in integrable systems.
abstract: We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement.
authors.affiliation: Universite de Toulouse; UPS, Laboratoire de Physique Theorique (IRSAMC); F-31062 Toulouse, France and CNRS, LPT (IRSAMC), F-31062 Toulouse, France.
authors.forename: Rémy
authors.initials: R
authors.lastname: Dubertrand
authors.affiliation: Department of Mathematics and Information Sciences, Northumbria University, Newcastle Upon Tyne, NE1 8ST, United Kingdom and Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, D-01187 Dresden, Germany.
authors.forename: Arseni
authors.initials: A
authors.lastname: Goussev