Higher order non-linear effects in a Josephson parametric amplifier

TL;DR

This paper derives a comprehensive quantum model for Josephson parametric amplifiers that includes all orders of non-linearity, revealing higher-order effects impacting amplifier performance and gain saturation.

Contribution

It introduces a quantum expression for JPA gain incorporating all non-linearity orders, extending beyond the typical cubic approximation.

Findings

Higher-order non-linear effects influence JPA gain.

Analysis of gain saturation for stronger signals.

Implications for optimizing JPA operation.

Abstract

Non-linearity of the current-phase relationship of a Josephson junction is the key resource for a Josephson parametric amplifier (JPA), the only device in which the quantum limit has so far been achieved at microwave frequencies. A standard approach to describe JPA takes into account only the lowest order (cubic) non-linearity resulting in a Duffing-like oscillator equation of motion or in a Kerr-type non-linearity term in the Hamiltonian. In this paper we derive the quantum expression for the gain of JPA including all orders of the Josephson junction non-linearity in the linear response regime. We then analyse gain saturation effect for stronger signals within semi-classical approach. Our results reveal non-linear effects of higher orders and their implications for operation of a JPA.