  Using the methods of quantum trajectories we study numerically the phenomenon
of quantum synchronization in a quantum dissipative system with periodic
driving. Our results show that at small values of Planck constant $\hbar$ the
classical devil's staircase remains robust with respect to quantum fluctuations
while at large $\hbar$ values synchronization plateaus are destroyed. Quantum
synchronization in our model has close similarities with Shapiro steps in
Josephson junctions and it can be also realized in experiments with cold atoms.
