Universal level statistics of the out-of-time-ordered operator

TL;DR

This paper introduces the Lyapunov operator as a spectral measure to connect quantum chaos indicators with classical Lyapunov exponents, demonstrating universal level statistics in chaotic quantum systems like the stadium billiard.

Contribution

It defines and analyzes the spectral statistics of the Lyapunov operator, establishing its connection to classical chaos and random matrix theory in quantum systems.

Findings

Lyapunov operator's level statistics follow Wigner-Dyson distribution in the bulk.

Spectral properties interpolate between classical Lyapunov exponents and quantum interference effects.

The approach provides a new tool for characterizing quantum chaos and quantum-to-classical transition.

Abstract

The out-of-time-ordered correlator has been proposed as an indicator of chaos in quantum systems due to its simple interpretation in the semiclassical limit. In particular, its rate of possible exponential growth at $\hbar \to 0$ is closely related to the classical Lyapunov exponent. Here we explore how this approach to quantum chaos relates to the random-matrix theoretical description. To do so, we introduce and study the level statistics of the logarithm of the out-of-time-ordered operator, $\hat{\Lambda}(t) = \ln \left( - \left[\hat{x}(t),\hat{p}_x(0) \right]^2 \right)/(2t)$, that we dub the "Lyapunovian" or "Lyapunov operator" for brevity. The Lyapunovian's level statistics is calculated explicitly for the quantum stadium billiard. It is shown that in the bulk of the filtered spectrum, this statistics perfectly aligns with the Wigner-Dyson distribution. One of the advantages of looking at the spectral statistics of this operator is that it has a well-defined semiclassical limit where it reduces to the matrix of uncorrelated classical finite-time Lyapunov exponents in a partitioned phase space. We provide a heuristic picture interpolating these two limits using Moyal quantum mechanics. Our results show that the Lyapunov operator may serve as a useful tool to characterize quantum chaos and in particular quantum-to-classical correspondence in chaotic systems, by connecting the semiclassical Lyapunov growth at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.