| Summary: | In this work, we investigate novel phase synchronisation features that occur in bistable oscillators, explored with trapped ions and oscillator-only systems, as well as in networks of spin-1 oscillators of varying size and geometry. We begin with two coupled trapped ions each driven by a two-quanta gain process whose dynamical states heavily influence the emergent relative phase preference. Large gain rates produce limit-cycle states where photon numbers can become large with relative phase distributions that are π-periodic with peaks at 0 and π, as extensively discussed in the literature. When the gain rate is low, however, the oscillators have very low photon occupation numbers which produces π-periodic distributions with peaks at π/2 and 3π/2. We find bistability between these limiting cases with a coexistence of limit-cycle and low-occupation states where the relative phase distribution can have π/2 periodicity. These results reveal that synchronisation manifests differently in quantum oscillators outside of the limit-cycle regime. Next, we investigate the origin of these features by proposing a minimal oscillator-only model that also exhibits bistability but with reduced complexity. Our model of two 321 oscillators is purely dissipative, with a two-photon gain balanced by single- and three-photon loss processes. Perturbation theory reveals that the values of π/2 and 3π/2 are due to the form of the number distribution that is produced by the two-quanta gain, unseen in thermal and van der Pol oscillators. Moving away from exploring bistability, we turn our attention to synchronisation in spin-1 oscillators which allows for the simulation of large networks. We derive an analytic form of the relative phase distribution of two spin-1 oscillators in a network that depends on only two complex values. The size and geometry of the network greatly affects the strength and form of synchronisation in the system. A strengthening of the synchronisation between next-nearest neighbours, compared to neighbours, is observed in chain and ring networks of three and four spins.
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