# Mesoscopic Hamiltonian for Josephson travelling-wave parametric amplifiers

## TL;DR

This paper develops a comprehensive quantum and classical Hamiltonian framework for Josephson junction-based traveling-wave parametric amplifiers, enabling detailed analysis of four-wave mixing processes.

## Contribution

It introduces a mesoscopic Hamiltonian approach that generalizes classical coupled-mode equations for Josephson parametric amplification.

## Key findings

- Derivation of a quantum Hamiltonian for the process
- Connection between quantum and classical descriptions
- Validation of the theoretical model

## Abstract

We present a theory describing parametric amplification in a Josephson junction embedded transmission line. We will focus on the process of four-wave mixing under the assumption of an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled-mode equations as presented by O'Brien et al. [Phys. Rev. Lett. $\mathbf{113}:157001$] is shortly reviewed. Then a derivation of the full quantum theory is given using mesoscopic quantisation techniques in terms of discrete mode operators. This results in a Hamiltonian that describes the process of parametric amplification. We show that the coupled-mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.