Probing quantum chaos with the entropy of decoherent histories

Quantum chaos, a phenomenon that began to be studied in the last century, still does not have a rigorous understanding. By virtue of the correspondence principle, the properties of the system that lead to chaotic dynamics at the classical level must also be present in the underlying quantum system. In the classical case, the exponential divergence of nearby trajectories in time is described in terms of the Lyapunov exponent. However, in the quantum case, a similar description of chaos is strictly speaking impossible due to absence of trajectories. There are different approaches to remedy this situation, but the universal criterium of quantum chaos is absent. We propose the quantum chaos definition in the manner similar to classical one using decoherent histories as a quantum analog of trajectories. For this purpose we consider the model of open quantum kicked top interacting with environment, which is bosonic bath and illustrate this idea on it. Here environment plays the role of trajectory recording device. For kicked top model on classical level depending on the kick strength there is crossover between integrable and chaotic regimes. We show that for such a model the production of entropy of decoherent histories is radically different in the integrable and chaotic regimes. Thus, the entropy of an ensemble of quantum trajectories can be used as a signature of quantum chaos.