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arXiv:2203.02448 · Physical review. B./Physical review. B

Reflection-enhanced gain in traveling-wave parametric amplifiers

Samuel Kern, P. Neilinger, Evgeni V. Il’ichev, A. Sh. Sultanov, Matthias Schmelz, S. Linzen, J. Kunert, G. Oelsner, Ronny Stolz, Andrey V. Danilov, Sumedh Mahashabde, Aditya Jayaraman, Vladimir N Antonov, Sergey Kubatkin, M. Grajcar

Published May 22, 2023• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.107.174520

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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.
We extend the theory for a Josephson-junction traveling-wave parametric amplifier (TWPA) operating in the three-wave-mixing regime and we propose a scheme for achieving high gain. The continuous three-mode model [P.K. Tien, J. Appl. Phys. 29, 1347 (1958)] is, on one hand, extended to describe a discrete chain of Josephson junctions at high frequencies close to the spectral cutoff where there is no up-conversion. On the other hand, we also develop a continuous multimode theory for the small frequency limit where the frequency dispersion is close to linear. We find that in both cases the gain is significantly reduced compared to the prediction by the continuous three-mode model as the result of increasingly strong dispersion at the high frequencies and generation of up-converted modes at the small frequencies. The developed theory is in quantitative agreement with simulations of the full solution of the dynamical equations. To recover the high gain, we propose to engineer a chain with dispersive features to form a two-band frequency spectrum and to place the pump frequency within the upper band close to the spectral cutoff. We prove that there exists a sweet spot, where the signal and the pump are phase matched, while the up-conversion is inhibited. This results in a high gain, which grows exponentially with the length of the TWPA.
Traveling-wave parametric amplifiers may be fabricated from superconducting films that exhibit highly nonlinear kinetic inductance. The coplanar waveguide of such a microwave device, extending to a meter or more in length but compacted to reside on a chip of the order of a square centimeter, is engineered with periodic variations in its width. These width variations, or loadings, alter the dispersion characteristics of a nonlinear current propagating along the waveguide, changing its group velocity and modulation behavior. A strong pump and a small signal injected into one end of the waveguide mix parametrically in the presence of the nonlinear kinetic inductance. Engineered dispersion induces the favorable conditions of overall phase matching, leading to generation of idler products as well as signal amplification of wide bandwidth, high dynamic range, and low noise, making the device of particular use in quantum computing and photon detection. The authors present a theoretical framework in which signal gain may be calculated solely from loading design. This involves construction of a metamaterial band theory of the engineered dispersion, which is used as a basis to describe the mixing of nonlinear traveling waves.
Abstract Superconducting travelling-wave parametric amplifiers (TWPAs) play a vital role in a range of high-sensitivity applications. These devices can be realised using various superconducting materials, such as high kinetic inductance films or low-loss transmission lines embedded with discrete nonlinear elements like Josephson junctions (JJs) or superconducting quantum interference devices (SQUIDs), and can operate across different wave-mixing regimes. However, a unifying framework for quick assessment and thus efficiently evaluating the performance of these diverse TWPA architectures, particularly during the design phase, remains lacking. Most existing models are derived from first principles for specific TWPA designs and lack general applicability. While certain simulation program with integrated circuit emphasis (SPICE) tools can more accurately emulate TWPA behaviour post-design, they are typically computationally intensive, time-consuming, and offer limited physical insight; especially regarding key performance-determining factors such as phase matching. This, in turn, impedes the rapid identification of optimal TWPA configurations. In this work, we extend a previously introduced framework for kinetic-inductance (KI-) TWPAs and demonstrate its applicability to a broader class of χ ( 3 ) -type TWPA configurations, including bare JJ (JTWPA) and symmetric SQUID-based TWPAs, operating in all wave-mixing modes. This approach facilitates rapid design-space exploration prior to detailed optimisation using SPICE-based simulations. Our method accommodates a wide range of unit cell topologies and meta-material parameters without requiring ground-up derivations from first principles. We validate the framework by comparing it against representative models from the literature, including JJ-, KI-, and DC SQUID-based TWPA designs, and show that it reliably captures the first-order behaviour of their gain–bandwidth characteristics.
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.
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.
Superconducting parametric amplifiers have great promise for quantum-limited readout of superconducting qubits and detectors. Until recently, most superconducting parametric amplifiers had been based on resonant structures, limiting their bandwidth and dynamic range. Broadband traveling-wave parametric amplifiers based both on the nonlinear kinetic inductance of superconducting thin films and on Josephson junctions are in development. By modifying the dispersion property of the amplifier circuit, referred to as dispersion engineering, the gain can be greatly enhanced and the size can be reduced. We present two theoretical frameworks for analyzing and understanding such parametric amplifiers: 1) generalized coupled-mode equations and 2) a finite difference time domain (FDTD) model combined with a small signal analysis. We show how these analytical and numerical tools may be used to understand device performance.