The elusive nature of the Lyapunov regime in the Loschmidt echo

Gregor Veble, Tomaž Prosen

Gregor Veble, Tomaž Prosen

Ignacio García-Mata, Diego A. Wisniacki

The Loschmidt echo (LE) is a measure of the stability and reversibility of quantum evolution under perturbations of the Hamiltonian. One of the expected and most relevant characteristics of this quantity for chaotic systems is an exponential decay with a perturbation-independent decay rate given by the classical Lyapunov exponent. However, a non-uniform decay—instead of the Lyapunov regime—has been reported in several systems. In this work, we find an analytical semiclassical expression for the averaged fidelity amplitude that can be directly related to the anomalous—unexpected—behaviour of the LE.

Fernando Cucchietti

Alexander Iomin

Chaotic dynamics of a nonlinear oscillator is considered in the semiclassical approximation. The Loschmidt echo as a measure of quantum stability to a time dependent variation is calculated. It is shown that an exponential decay of the Loschmidt echo is due to a Lyapunov exponent and it has a pure classical nature. The Lyapunov regime is observed for a time scale which is of the power law in semiclassical parameter.

Ignacio García-Mata, Diego A. Wisniacki

One of the fundamental manifestations of classical chaos is exponential sensitivity to initial conditions that is, two trajectories starting from nearly identical initial states diverge exponentially over time. This behavior is quantified by the Lyapunov exponents. Due to the unitary nature of quantum mechanics, such exponential divergence is elusive in quantum systems. As a result, several alternative quantities have been proposed and studied in recent years to capture analogous behavior. In this article, we present a pedagogical overview of three such quantities that have been the focus of intense research in recent years: the Loschmidt echo, out-of-time-order correlators (OTOCs), and Krylov complexity.