# Phase synchronization in coupled bistable oscillators

## TL;DR

This paper presents a theoretical study of synchronization in quantum bistable oscillators, revealing how coupling induces various phase preferences depending on the dynamical state, with potential implications for quantum synchronization control.

## Contribution

It introduces a dissipative quantum oscillator model exhibiting bistability and analyzes how coupling leads to diverse phase synchronization behaviors.

## Key findings

- In the limit-cycle regime, phase peaks at 0 and π.
- In the low-occupation regime, phase peaks at π/2 and 3π/2.
- Intermediate regimes show π/2-periodic phase distributions.

## 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.