Including Substrate Losses and Noise in a Quantum Mechanical Model of a Josephson Traveling-Wave Parametric Amplifier

Substrate losses affect the performance of dispersive readout of superconducting qubits. We present a quantum mechanical model of a Josephson traveling-wave parametric amplifier, including substrate losses and the associated noise contributions. Losses and noise are introduced by coupling the system Hamiltonian to a bosonic bath, where the coupling constants can be related to the loss tangent of the transmission line substrate. Closed-form solutions for the temporal evolution of discrete mode operators in the presence of losses and noise are derived, under the assumption of a strong classical pump mode. A careful analysis of the loss-induced noise contributions can lead to the formulation of limits for the scaling of ultra-low noise qubit readout.