Chaos in continuously monitored quantum systems: An optimal-path approach

Abstract

We predict that continuously monitored quantum dynamics can be chaotic. The optimal paths between past and future boundary conditions can diverge exponentially in time when there is time-dependent evolution and continuous weak monitoring. Optimal paths are defined by extremizing the global probability density to move between two boundary conditions, and are then expressed as solutions to a Hamiltonian dynamical system. We investigate the onset of chaos in pure-state qubit systems with optimal paths generated by a periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated in a scheme where two noncommuting observables of a qubit are continuously monitored, and one measurement strength is periodically modulated. The optimal quantum paths in this example bear similarities to the trajectories of the kicked rotor, or standard map, which is a paradigmatic example of classical chaos. We emphasize connections with the concept of resonance between integrable optimal paths and weak periodic perturbations, as well as our previous work on ``multipaths,'' and connect the optimal path chaos to instabilities in the underlying quantum trajectories.