Quantum description of limit-cycle dynamics generated by a 2-mode Josephson junction circuit

We explore the dynamics of a driven-dissipative circuit where a Josephson junction is capacitively coupled to a linear oscillator. In the weak nonlinearity regime, Josephson potential can be approximated by a Kerr term enabling us to approximate the system as a Kerr oscillator bilinearly coupled to a linear oscillator. This system is previously shown (both theoretically and experimentally) to exhibit limit-cycle dynamics (frequency combs) in a strongly coupled, strongly driven, weakly nonlinear regime. [1,2] We calculate the system correlations in this regime in a semi-analytical way and show a computationally efficient method to characterize quantum correlations of the system. Although this method is favorable in terms of the computation cost, the linearization is not applicable in all regimes. We find the limits of the linearized approximation and then go beyond that to explore the full quantum dynamics of this system in the weakly-driven, strongly nonlinear regime numerically solving the master equation and show that the nonlinearity of the quantum system represses the formation of limit cycles in that regime.

[1]S.Khan, Phys. Rev. Lett. 120, 153601

[2] P.Lu, S.Khan, Phys. Rev. App. 15, 044031