Chaos and dynamical instability in a closed kicked system

Abstract

Quantum-classical correspondence plays an important role in understanding the emergence of classical chaos from underlying quantum mechanics. However, the transition from quantum to classical is not straightforward. Here we study a well-known closed-kicked spin system with a chaotic classical limit. The quantum dynamics takes the form of stroboscopic unitary kicks acting on a single spin system. By mapping it to a programmable quantum circuit, we show that NISQ devices can be a potential testbed for simulating quantum chaos. The results suggest that entanglement can be considered a signature of classical chaos even in the deep quantum regime. Extending the work to arbitrary spins and focusing on special Hamiltonian parameters, we then show that the system may acquire temporal periodicity. These temporal periodicities do not depend on the initial state. Throughout such periodic evolutions, no initial quantum state fully explores Hilbert space as either a state vector or phase space as a quasi-probability distribution despite the classical limit being chaotic. Because these state-independent temporal periodicities are present in all dimensions, their existence represents a universal violation of the correspondence principle. We also consider the stability of this periodic behavior as a function of the degree of chaos in the classical model. Our study suggests that even in the semi-classical regime, there are specific parameter values for which a quantum system never behaves classically or displays signatures of chaos.