# Non-degenerate parametric resonance in tunable superconducting cavity

## TL;DR

This paper presents a comprehensive theory of non-degenerate parametric resonance in tunable superconducting cavities, exploring nonlinear effects, amplification, noise squeezing, and frequency conversion with potential applications in quantum information processing.

## Contribution

It introduces a detailed theoretical framework for non-degenerate parametric resonance in superconducting cavities, including nonlinear dynamics, noise properties, and mode conversion analysis.

## Key findings

- Maximum gain values at the parametric instability threshold.
- Divergence in cavity response is linked to phase degeneracy, which can be lifted.
- Strong amplification leads to four-mode noise squeezing and enhanced signal-to-noise ratio.

## Abstract

We develop a theory for non-degenerate parametric resonance in a tunable superconducting cavity. We focus on nonlinear effects that are caused by nonlinear Josephson elements connected to the cavity. We analyze parametric amplification in a strong nonlinear regime at the parametric instability threshold, and calculate maximum gain values. Above the threshold, in the parametric oscillator regime the linear cavity response diverges at the oscillator frequency at all pump strengths. We show that this divergence is related to the continuous degeneracy of the free oscillator state with respect to the phase. Applying on-resonance input lifts the degeneracy and removes the divergence. We also investigate the quantum noise squeezing. It is shown that in the strong amplification regime the noise undergoes four-mode squeezing, and that in this regime the output signal to noise ratio can significantly exceed the input value. We also analyze the intermode frequency conversion and identify parameters at which full conversion is achieved.