Josephson traveling-wave parametric amplifiers (JTWPAs) achieve high parametric gain through four-wave-mixing with near quantum-limited noise performance and extremely low energy consumption. This makes them suitable for the detection of ultra-low-power microwave signals in the lowest temperature stage of a dilution refrigerator. Distributed circuit representations of JTWPAs have been studied in the framework of classical theory. A nonlinear wave equation describing the flux along a nonlinear transmission line was derived in [1]. Dispersion engineering was introduced in [2] to increase the amplifier’s gain and bandwidth by means of resonant phase-matching (RPM). In the classical treatment [1], dielectric loss is considered for the formulation of the wave equation, however, a closed-form solution was only given when dissipation was neglected. Our work focuses on the effects of noise and dissipation in the nonlinear transmission line. Our model differs from [1],[2], where we derive the telegrapher’s equations taking into account the loss and noise contributions to construct the current-flux-relation. In the super-conducting state, the dissipation due to the imperfect electric insulation in the substrate needs to be taken into account. It is represented by a parallel conductance and an additional noise source describing temporal fluctuations [3]. The flux along the transmission line is expanded into three discrete modes, representing pump, signal, and idler contributions. The equations of motion of the signal and idler modes result in\begin{equation*}{\partial _z}{\tilde A_{\text{s}}} = - \left( {{\Gamma _{\text{s}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{s}}} + {\text{i}}{\kappa _{\text{s}}}\tilde A_{\text{i}}^{\ast} + {\tilde f_{\text{s}}}\left( {i_{\text{s}}^{{\text{noise}}}} \right),\quad {\partial _z}{\tilde A_{\text{i}}} = - \left( {{\Gamma _{\text{i}}}/2 + {\text{i}}\Delta {k_{\text{T}}}/2} \right){\tilde A_{\text{i}}} + {\text{i}}{\kappa _{\text{i}}}\tilde A_{\text{s}}^{\ast} + {\tilde f_{\text{i}}}\left( {i_{\text{i}}^{{\text{noise}}}} \right),\tag{1}\end{equation*}where à n are the mode amplitudes in a co-rotating frame, ∆k T is the total phase mismatch, κ n describe the coupling due to four-wave-mixing, Γ n are the damping factors and ${\tilde f_n}$ represent thermal fluctuations on the transmission line, with n ∈ {s,i}. Pump-depletion was neglected when deriving the equations above. We define the signal power gain G s as the ratio of output to input power. Assuming no initial idler contribution at the input of the device, we obtain\begin{equation*}{G_{\text{S}}} = {\left| {\cosh (gz) + \frac{{ - {\Gamma _{\text{s}}} + {\Gamma _{\text{i}}} - 2{\text{i}}\Delta {k_{\text{T}}}}}{{4g}}\sinh (gz)} \right|^2}{{\text{e}}^{ - \frac{1}{2}\left( {{\Gamma _{\text{s}}} + {\Gamma _{\text{i}}}} \right)z}},\tag{2}\end{equation*}where g is the gain factor, which depends on the coupling strength, the pump amplitude and the total phase mismatch. The gain spectrum of the JTWPA setup from [4] was calculated according to our model with good agreement to the results in the literature. The gain spectrum of the JTWPA is plotted in Fig. 1 (a) without RPM and in Fig. 1 (b) with RPM, both for the case with and without dissipation.
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