Bistability and Phase Synchronisation in Coupled Quantum SystemsTools Jessop, Matthew (2021) Bistability and Phase Synchronisation in Coupled Quantum Systems. PhD thesis, University of Nottingham.
AbstractIn 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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