Abstract
We introduce a simple model system to study synchronization theoretically in quantum oscillators that are not just in limit-cycle states, but rather display a more complex bistable dynamics. Our oscillator model is purely dissipative, with a two-photon gain balanced by single- and three-photon loss processes. When the gain rate is low, loss processes dominate and the oscillator has a very low photon occupation number. In contrast, for large gain rates, the oscillator is driven into a limit-cycle state where photon numbers can become large. The bistability emerges between these limiting cases with a region of coexistence of limit-cycle and low-occupation states. Although an individual oscillator has no preferred phase, when two of them are coupled together a relative phase preference is generated which can indicate synchronization of the dynamics. We find that the form and strength of the relative phase preference varies widely depending on the dynamical states of the oscillators. In the limit-cycle regime, the phase distribution is $\pi$-periodic with peaks at $0$ and $\pi$, whilst in the low-occupation regime $\pi$-periodic phase distributions can be produced with peaks at $\pi/2$ and $3\pi/2$. Tuning the coupled system between these two regimes reveals a region where the relative phase distribution has $\pi/2$-periodicity.
Highlights
The past few years has seen rapid progress in engineering and probing the properties of nonlinear oscillators in the quantum regime [1,2,3,4]
For oscillators with low occupation numbers and no limit cycle, relative phases of π /2 and 3π /2 are preferred instead, a result which we argue can be understood as a result of the two-photon gain in this system
We have introduced a simple oscillator model with a twophoton gain process balanced by one- and three-photon losses that can be used to engineer a bistable oscillator state
Summary
The past few years has seen rapid progress in engineering and probing the properties of nonlinear oscillators in the quantum regime [1,2,3,4]. We instead explore how weak coupling generates synchronization, in the form of particular phase preferences, in a quantum oscillator which has a more complex bistable dynamics We do this by proposing a minimal model for a quantum oscillator that displays a limit-cycle state as well as a low occupationnumber state (in which the oscillator fluctuates about the origin) and can be tuned to a bistable regime in which both of these states coexist. We investigate in detail the phase synchronization that occurs when two of the bistable model oscillators are coupled via a weak photon exchange process This leads to a rich range of behavior in the relative phase distribution, with a different pattern of phase preferences emerging depending on the underlying dynamical states of the oscillators. IV and the Appendixes provide details about aspects of the calculations employed
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