Long-time saturation of the Loschmidt echo in quantum chaotic billiards

TL;DR

This paper provides a quantitative analysis and explicit expression for the long-time saturation value of the Loschmidt echo in quantum chaotic billiards, supported by numerical calculations.

Contribution

It introduces a new analytical expression for the LE saturation value in quantum chaotic systems and validates it with numerical simulations.

Findings

Analytical expression for LE saturation value derived

Numerical results support the semiclassical theory

Long-time LE saturation inversely proportional to Hilbert space size

Abstract

The Loschmidt echo (LE) (or fidelity) quantifies the sensitivity of the time evolution of a quantum system with respect to a perturbation of the Hamiltonian. In a typical chaotic system the LE has been previously argued to exhibit a long-time saturation at a value inversely proportional to the effective size of the Hilbert space of the system. However, until now no quantitative results have been known and, in particular, no explicit expression for the proportionality constant has been proposed. In this paper we perform a quantitative analysis of the phenomenon of the LE saturation and provide the analytical expression for its long-time saturation value for a semiclassical particle in a two-dimensional chaotic billiard. We further perform extensive (fully quantum mechanical) numerical calculations of the LE saturation value and find the numerical results to support the semiclassical theory.