A quantum model for rf-SQUIDs based metamaterials enabling 3WM and 4WM Travelling Wave Parametric Amplification

TL;DR

This paper introduces a quantum Hamiltonian model for rf-SQUID-based metamaterials functioning as traveling wave parametric amplifiers, analyzing their nonlinearities and gain at the single-photon level for 3WM and 4WM regimes.

Contribution

It provides an analytical Hamiltonian framework for modeling rf-SQUID metamaterials in 3WM and 4WM regimes, including gain calculation and photon population dynamics.

Findings

Analytic expression for amplifier gain.

Photon population evolution in multimodal states.

Dependence of nonlinearities on circuit parameters.

Abstract

A quantum model for Josephson-based metamaterials working in the Three-Wave Mixing (3WM) and Four-Wave Mixing (4WM) regimes at the single-photon level is presented. The transmission line taken into account, namely Traveling Wave Josephson Parametric Amplifier (TWJPA), is a bipole composed by a chain of rf-SQUIDs which can be biased by a DC current or a magnetic field in order to activate the 3WM or 4WM nonlinearities. The model exploits a Hamiltonian approach to analytically determine the time evolution of the system both in the Heisenberg and interaction pictures. The former returns the analytic form of the gain of the amplifier, while the latter allows recovering the probability distributions vs. time of the photonic populations, for multimodal Fock and coherent input states. The dependence of the metamaterial's nonlinearities is presented in terms of circuit parameters in a lumped model framework while evaluating the effects of the experimental conditions on the model validity.