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

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