Quantum Zeno effect in self-sustaining systems: Suppressing phase diffusion via repeated measurements

We study the effect of frequent projective measurements on the dynamics of quantum selfsustaining systems, by considering the prototypical example of the quantum Van der Pol oscillator. Quantum fluctuations are responsible for phase diffusion which progressively blurs the semiclassical limit cycle dynamics and synchronization, either to an external driving, or between two coupled self-sustained oscillators. We show that by subjecting the system to repeated measurements of heterodyne type at an appropriate repetition frequency one can significantly suppress phase diffusion without spoiling the semiclassical dynamics. This quantum Zeno-like effect may be effective either in the case of one or two coupled van der Pol oscillators, and we discuss its possible implementation in the case of trapped ions.