  We analyse the properties of the synchronisation transition in a many-body
system consisting of quantum van der Pol oscillators with all-to-all coupling
using a self-consistent mean-field method. We find that the synchronised state,
which the system can access for oscillator couplings above a critical value, is
characterised not just by a lower phase uncertainty than the corresponding
unsynchronised state, but also a higher number uncertainty. Just below the
critical coupling the system can evolve to the unsynchronised steady state via
a long-lived transient synchronised state. We investigate the way in which this
transient state eventually decays and show that the critical scaling of its
lifetime is consistent with a simple classical model.
