The hallmark of classical chaos is an exponential divergence of initially infinitesimally close trajectories - a phenomenon colloquially known as the “butterfly effect.” This exponential runaway of chaotic trajectories is quantitatively characterized by the Lyapunov exponent. Of great interest has been to understand how/if the butterfly effect and Lyapunov exponents generalize to quantum physics, where the notion of a trajectory does not exist. In this talk, I will discuss recent progress in resolving this fundamental challenge that is based on a newly introduced measure of quantum chaoticity – the out-of-time-ordered correlator or “Lyapunovian” – which enables to make a non-trivial connection between classical and quantum chaos in a variety of systems: from single-particle chaotic billiards to disordered condensed matter systems to models of black holes. I will illustrate the use of the Lyapunovian on a few standard examples that will be used to elucidate the nature of quantum chaotic dynamics, including suppression of the butterfly effect in quantum systems. I will conclude by formulating an intriguing conjecture connecting quasiclassical chaotic dynamics and statistics of energy levels in interacting many-body quantum systems.
