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2021 International Applied Computational Electromagnetics Society Symposium (ACES)

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

Michael Haider, Yongjie Yuan, Christian Jirauschek

Published Aug 1, 2021• 8 citations
Publisher

Abstract

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.

DOI
10.1109/aces53325.2021.00093

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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.
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 nonzero 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.
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 Ãnare the mode amplitudes in a co-rotating frame, ∆kTis the total phase mismatch, κndescribe the coupling due to four-wave-mixing, Γnare the damping factors and f~n{\tilde f_n}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 Gsas 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.
In this paper, we present models for the nonlinearity of different nonlinear elements for superconducting Josephson traveling-wave parametric amplifiers. Such devices are typically used as a first-stage readout amplifier for superconducting qubits in a cryogenic quantum computing environment. The nonlinear coefficients of different designs are evaluated as functions of an externally applied bias field, taking into account direct and external pumping schemes. Finally, we present results of the total gain for two different nonlinear transmission lines, one that is embedded with single Josephson junctions operating in four-wave mixing mode with direct pumping, and an externally pumped SNAIL-based amplifier operating in the three-wave mixing regime with a corresponding constant external flux bias. The gain is evaluated using a quantum circuit model including substrate losses and chromatic dispersion.
In this paper, we are constructing a circuit quantum electrodynamic model describing dissipative effects in a Josephson traveling-wave parametric amplifier (JTWPA). Losses and noise are treated using the quantum Langevin method. The total Hamiltonian of our model describes the coupling of the modes in a parametric four-wave-mixing scheme, as well as the interaction with the environment. Based on the Heisenberg equations of motion, we obtain the temporal evolution of the signal photon number and analyze the signal gain of the JTWPA.
Superconducting quantum computing is at the forefront of demonstrating quantum advantages for revolutionary applications in science and technology. One key component for the readout of qubit state information is given by superconducting traveling-wave parametric amplifiers (TWPAs) due to their low power dissipation and unprecedented quantum-limited noise performance. This work presents a black-box circuit quantum electrodynamic model for TWPAs with arbitrary superconducting nonlinear elements. A unified Hamiltonian description for directly pumped and flux-driven amplifiers is derived, and analytic results for the gain and the added quantum noise are presented, closely matching recent experimental realizations. We also offer an optimization strategy for nonlinear elements for large gain and low noise based on the proposed black-box quantum model. The results of this article will be helpful in the design and multitarget optimization of new and improved quantum-limited low-noise amplifiers.