Fractionally Quantized Mean Recurrence Times in Repeatedly Measured Many-Body Systems

Recurrence in the dynamics of physical systems is an important phenomenon that has many far-reaching consequences. In classical physics, Poincare's recurrence theorem states that a complex system will return to its initial state within a finite time when left alone. This theorem has been extended to the quantum case and has been observed experimentally. In this work, we investigate quantum recurrence for a quantum system that interacts with periodic measurements. Specifically, we consider interacting spin systems where the measurements are performed on one spin. We ask the question: if the monitored spin is initially prepared in the upstate (for example), how long will it take to measure the spin for the first time in the upstate again? We show that the mean recurrence time is fractionally quantized and characterized by the number of dark states, which are eigenstates of the spin system where the monitored spin and the surrounding bath are not entangled. The mean recurrence time is invariant when changing the sampling rate, and this invariance is topologically protected by the quantized winding number. We also show how the mean recurrence time scales with the size of the system. Our results can potentially be observed in experiments with NV centers or quantum computers.