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URSI General Assembly and Scientific Symposium

A Quantum Model of a Dissipative-Dispersive Josephson Traveling-Wave Parametric Amplifier Including Impedance-Mismatch-Induced Reflections

Michael Haider, Yongjie Yuan, Christian Jirauschek

Published Aug 19, 2023• 1 citation
Publisher

Abstract

A major challenge for the experimental realization of advanced superconducting traveling-wave parametric amplifiers is given by proper impedance matching of the input and output transmission lines and the active region with periodically embedded Josephson junction arrays. For a large impedance mismatch, non-negligible reflections occur at the interfaces, leading to a co-propagation of cavity-like modes. In this contribution, we want to assess the effects of improper impedance matching on the transmission spectrum of traveling-wave parametric amplifiers using a simple overlay cavity model. Our results can serve as a guideline for design limitations of Josephson-junction-embedded transmission lines.

DOI
10.23919/ursigass57860.2023.10265653

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A major challenge for the experimental realization of advanced superconducting traveling-wave parametric amplifiers is given by proper impedance matching of the input and output transmission lines and the active region with periodically embedded Josephson junction arrays. For a large impedance mismatch, non-negligible reflections occur at the interfaces, leading to a co-propagation of cavity-like modes. In this contribution, we want to assess the effects of improper impedance matching on the transmission spectrum of traveling-wave parametric amplifiers using a simple overlay cavity model. Our results can serve as a guideline for design limitations of Josephson-junction-embedded transmission lines.
The operating principle of traveling-wave parametric amplifiers is typically understood in terms of the standard coupled mode theory, which describes the evolution of forward propagating waves without any reflections, i.e., for perfect impedance matching. However, in practice, superconducting microwave amplifiers are unmatched nonlinear finite-length devices, where the reflecting waves undergo complex parametric processes, not described by the standard coupled mode theory. Here, we present an analytical solution for the TWPA gain, which includes the interaction of reflected waves. These reflections result in corrections to the well-known results of the standard coupled mode theory, which are obtained for both three-wave and four-wave mixing processes. Due to these reflections, the gain is enhanced and unwanted nonlinear phase modulations are suppressed. Predictions of the model are experimentally demonstrated on two types of unmatched TWPA, based on coplanar waveguides with a central wire consisting of (i) a high kinetic inductance superconductor, and (ii) an array of 2000 Josephson junctions.
In this contribution, we investigate different types of Josephson junction arrays that are embedded in coplanar microwave transmission lines, acting as traveling-wave parametric amplifiers. For the realization of traveling-wave parametric amplifiers, different Josephson junction array geometries have been proposed in the literature. Here, we investigate the amplifier’s Josephson nonlinearity along with the gain performance, following a general method to derive the nonlinear wave-mixing coefficients from the current-phase relation of different junction array topologies. These nonlinear coefficients are then incorporated into a corresponding Hamiltonian, which is used in order to solve the Heisenberg equations of motion of the photon number operator, determining the gain of the amplifier.
The paper investigates the interplay of nonlinear effects with the non-ideal transmission properties of a Josephson Travelling Wave Parametric Amplifier (JTWPA). Experimental characterization using a microwave and DC cryogenic setup reveals periodic transmission modulation linked to the tuning of the Josephson inductances embedded in the line. The effects of impedance matching and resonant parametric down-conversion (PDC) are observed on a broadband scale, and the impact of design parameters and fabrication issues on device performance is analyzed. These findings contribute to the optimizing techniques for JTWPA devices, advancing quantum-limited amplifiers.
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.
We propose a technique to overcome phase mismatch in Josephson-junction traveling wave parametric amplifiers in order to achieve high gain over a broad bandwidth. Using "resonant phase matching," we design a compact superconducting device consisting of a transmission line with subwavelength resonant inclusions that simultaneously achieves a gain of 20 dB, an instantaneous bandwidth of 3 GHz, and a saturation power of -98 dBm. Such an amplifier is well suited to cryogenic broadband microwave measurements such as the multiplexed readout of quantum coherent circuits based on superconducting, semiconducting, or nanomechanical elements, as well as traditional astronomical detectors.
We present a fishbone-like lumped-element artificial transmission line to overcome impedance mismatch in a reflection-type Josephson parametric amplifier between a nonlinear resonator and an external transmission line. Using this easily prepared architecture, we design and fabricate a broadband Josephson parametric amplifier, which has gain in an excess of 20 dB with a bandwidth of hundreds of MHz. Furthermore, by varying the working point of the device, the operating frequency of amplification can be tuned in a wide frequency range of 1 GHz while the amplifier operates in the mode of either three-wave mixing or four-wave mixing. Such a parametric amplifier is suitable for engineering applications of superconducting circuit quantum electrodynamics.