  A paradigm for quantum synchronization is the quantum analog of the
Stuart--Landau oscillator, which corresponds to a van der Pol oscillator in the
limit of weak (i.e. vanishingly small) nonlinearity. Due to this limitation,
the quantum Stuart--Landau oscillator fails to capture interesting
nonlinearity-induced phenomena such as relaxation oscillations. To overcome
this deficiency we propose an alternative model which approximates the van der
Pol oscillator to finitely large nonlinearities while remaining numerically
tractable. This allows us to uncover interesting phenomena in the deep-quantum
strongly-nonlinear regime with no classical analog, such as the persistence of
amplitude death on resonance. We also report nonlinearity-induced position
correlations in reactively coupled quantum oscillators. Such coupled
oscillations become more and more correlated with increasing nonlinearity
before reaching some maximum. Again, this behavior is absent classically. We
also show how strong nonlinearity can enlarge the synchronization bandwidth in
both single and coupled oscillators. This effect can be harnessed to induce
mutual synchronization between two oscillators initially in amplitude death.
