Multistability of a Josephson parametric amplifier coupled to a mechanical resonator

We study the dynamics of a Josephson parametric amplifier (JPA) coupled to a mechanical oscillator, as realized with a dc superconducting quantum interference device (SQUID) with an embedded movable arm. We analyze this system in the regime when the frequency of the mechanical oscillator is comparable in magnitude to the plasma oscillation of the SQUID. When the nanomechanical resonator is driven, it strongly affects the dynamics of the JPA. We show that this coupling can considerably modify the dynamics of the JPA and induce its multistability rather than common bistability. This analysis is relevant if one considers a JPA for detection of mechanical motion.