# Phononic Josephson oscillation and self-trapping with two-phonon exchange interaction

## TL;DR

This paper introduces a bosonic Josephson junction in nonlinear mechanical resonators with two-phonon exchange, revealing regimes of oscillation and self-trapping, and showing transitions influenced by damping and system parameters.

## Contribution

It proposes a novel BJJ model with two-phonon exchange interaction and analyzes its dynamic regimes and transition conditions, expanding the understanding of phononic quantum phenomena.

## Key findings

- Demonstrates the existence of Josephson oscillation and self-trapping regimes.
- Shows transition between regimes driven by system parameters and damping.
- Provides a theoretical framework applicable to optical and microwave systems.

## Abstract

We propose a bosonic Josephson junction (BJJ) in two nonlinear mechanical resonator coupled through two-phonon exchange interaction induced by quadratic optomechanical couplings. The nonlinear dynamic equations and effective Hamiltonian are derived to describe behaviors of the BJJ. We show that the BJJ can work in two different dynamical regimes: Josephson oscillation and macroscopic self-trapping. The system can transfer from one regime to the other one when the self-interaction and asymmetric parameters exceed their critical values. We predict that a transition from Josephson oscillation to macroscopic self-trapping can be induced by the phonon damping in the asymmetric BJJs. Our results opens up a way to demonstrate BJJ with two-phonon exchange interaction and can be applied to other systems, such as the optical and microwave systems.