  We investigate synchronization effects in quantum self-sustained oscillators
theoretically using the micromaser as a model system. We use the probability
distribution for the relative phase as a tool for quantifying the emergence of
preferred phases when two micromasers are coupled together. Using perturbation
theory, we show that the behavior of the phase distribution is strongly
dependent on exactly how the oscillators are coupled. In the quantum regime
where photon occupation numbers are low we find that although synchronization
effects are rather weak, they are nevertheless significantly stronger than
expected from a semiclassical description of the phase dynamics. We also
compare the behavior of the phase distribution with the mutual information of
the two oscillators and show that they can behave in rather different ways.
