Quantum Master Equation for a Lossy Josephson Traveling-Wave Parametric Amplifier

Abstract

In this paper, we present a theoretical framework for investigating losses and thermal fluctuations in a Josephson traveling-wave parametric amplifier (JTWPA). Our model is based on a discrete-mode Hamiltonian which includes a four-wave-mixing process and system-reservoir interactions. From this Hamiltonian, we derive a quantum master equation that describes the lossy Josephson-junction-embedded transmission line. The resulting equation of motion for the reduced density operator is applied for evaluating the average photon number in the signal mode. We present an analytic solution for the case of a dispersionless transmission line, where the phase-modulation effects can be neglected, and evaluate the photon number spectrum of a JTWPA structure with parameters from the literature. The quantum master equation is especially advantageous for a direct treatment of the expectation values of the photon number, in comparison to other strategies.