Circuit quantum electrodynamic model of dissipative-dispersive Josephson traveling-wave parametric amplifiers

We present a quantum mechanical model for a four-wave mixing Josephson traveling-wave parametric amplifier including substrate losses and associated thermal fluctuations. Under the assumption of a strong undepleted classical pump tone, we derive an analytic solution for the bosonic annihilation operator of the weak signal photon field using temporal equations of motion in a reference timeframe, including chromatic dispersion. From this result, we can predict the asymmetric gain spectrum of a Josephson traveling-wave parametric amplifier due to non-zero substrate losses. We also predict the equivalent added input noise including quantum fluctuations as well as thermal noise contributions. Our results are in excellent agreement with recently published experimental data.