Characterization of Classical and Quantum Chaos With Out-of-Time-Ordered Correlator in Stadium Billiard

Out-of-time-ordered correlator (OTOC) has been gaining attention as an indicator of chaos in quantum systems due to its simple interpretation in semiclassical limit. In particular, its rate of exponential growth at $\hbar_{\rm eff} \to 0$ is closely related to the classical Lyapunov exponent. The Bunimovich stadium billiard is a seminal classically chaotic model in which a particle moves freely inside a two-dimensional stadium-shaped infinite potential well. We analyze this system and find that suitably defined OTOC in appropriate parameter regimes shows early-time exponential growth (up to the Ehrenfest-time scale) at a rate attributed to the Lyapunov exponent. We also show that the Wigner-Dyson level statistics related to the chaotic nature of the classical billiard can be extracted from the late-time-OTOC matrix. These findings make OTOC a unified tool for characterization of classical and quantum chaos.