Universal Level Statistics of the Out-of-Time-Ordered Operator

The out-of-time-ordered correlator (OTOC) 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, $\hat{\Lambda}(t) = \ln \left( - \left[\hat{x}(t),\hat{p}(0) \right]^2 \right)/(2t)$, 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 Lyapunov exponents in a partitioned phase space. We provide 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 that we demonstrate at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference effects.