  We study the effect of frequent projective measurements on the dynamics of
quantum self-sustaining 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.
