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IEEE Transactions on Applied Superconductivity

Comparative Study of Numerical Modeling Methods for a Flux-Pumped Josephson Parametric Amplifier

Seong Hyeon Park, Gahyun Choi, Bokyung Kim, Gyunghun Kim, Yong‐Ho Lee, Seungyong Hahn

Published Mar 3, 2025• 1 citation
Publisher

Abstract

Quantum-limited noise amplifiers based on superconducting Josephson junction elements are essential components for measuring low-power signals in a wide range of applications. The gain bandwidth of a flux-pumped Josephson parametric amplifier can be broadened by coupling junction elements with an impedance-matching network. Well-known modeling methods for these amplifiers typically assume ideal lumped-element circuits for passive components, but often overlook unexpected resonant modes arising from the bulk geometric features. To address this challenge, flux-pumped junction elements can be simulated using finite-element analysis (FEA) combined with the pumpistor modeling method, which accurately accounts for the geometric features of arbitrary impedance-matching circuits. Here, we present a benchmark study using a circuit topology to compare different modeling methods: (1) the coupled mode network method, (2) the harmonic balance method, (3) the analytical pumpistor method, and (4) the FEA-assisted pumpistor method. Our results highlight the importance of accurately reflecting geometric features to prevent unexpected resonant modes and avoid detuning from the target flux bias values.

DOI
10.1109/tasc.2025.3546921

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Quantum-limited noise amplifiers based on superconducting Josephson junction elements are essential components for measuring low-power signals in a wide range of applications. The gain bandwidth of a flux-pumped Josephson parametric amplifier can be broadened by coupling junction elements with an impedance-matching network. Well-known modeling methods for these amplifiers typically assume ideal lumped-element circuits for passive components, but often overlook unexpected resonant modes arising from the bulk geometric features. To address this challenge, flux-pumped junction elements can be simulated using finite-element analysis (FEA) combined with the pumpistor modeling method, which accurately accounts for the geometric features of arbitrary impedance-matching circuits. Here, we present a benchmark study using a circuit topology to compare different modeling methods: (1) the coupled mode network method, (2) the harmonic balance method, (3) the analytical pumpistor method, and (4) the FEA-assisted pumpistor method. Our results highlight the importance of accurately reflecting geometric features to prevent unexpected resonant modes and avoid detuning from the target flux bias values.
Broadband quantum-limited amplifiers play a critical role in the single-shot readout of superconducting qubits, but a popular implementation, the traveling wave parametric amplifier, involves a complex design and fabrication process. Here, we present a simple design for a Josephson parametric amplifier, using a lumped element resonator comprising a superconducting quantum interference device whose useful bandwidth is enhanced with an on-chip impedance-matching circuit. We demonstrate a flux-coupling geometry that maximizes the coupling to the Josephson loop and minimizes spurious excitation of the amplifier resonant circuit. The amplifier, which operates in a flux-pumped mode, is demonstrated with a power gain of more than 20 dB over a bandwidth of about 300 MHz, where approximate noise measurements indicate quantum-limited performance. A procedure is given for optimizing the bandwidth for this kind of amplifier, using a linearized circuit simulation while minimizing non-linearities.
Abstract This work presents an analysis of a flux-pumped Josephson parametric amplifier (JPA) with a detailed superconducting quantum interference device (SQUID) model, using the harmonic balance (HB) method in Keysight’s Advanced Design System. The model is built using the generalized resistively shunted junction model of a Josephson junction and the quantization condition of a SQUID, incorporating the effects of the geometric inductance and parasitic parameters. This approach is used to simulate the behavior of an unmatched lumped element JPA and an impedance-matched JPA, showing the necessity of the detailed SQUID model for the simulation of a broadband JPA by comparison with the results based on a widely used, simplified model. Gain profiles obtained from the approach are consistent with existing experiments in the literature, with reasonable accuracy. Due to the HB method, the simulation is of high efficiency, making it feasible to sweep and optimize parameters. This provides a useful method for the analysis of Josephson parametric amplification and the design of a JPA.
Josephson parametric amplification is a tool of paramount importance in circuit-QED especially for the quantum-noise-limited single-shot read-out of superconducting qubits. We developed a Josephson parametric amplifier (JPA) based on a lumped-element LC resonator, in which the inductance L is composed by a geometric inductance and an array of 4 superconducting quantum interference devices (SQUIDs). We characterized the main figures of merit of the device, obtaining a −3 dB bandwidth BW = 15 MHz for a gain G = 21 dB and a 1 dB compression point P1dB = −115 dBm. The obtained results are promising for the future use of such JPA as the first stage of amplification for single-shot readout of superconducting qubits.
The development of large-scale superconducting quantum computers critically depends on the accurate readout of qubit states. To enable this, cryogenic parametric amplifiers have been proposed and developed, and have since found widespread use in small-signal detection applications. Among various amplifier architectures, Josephson traveling-wave parametric amplifiers (JTWPAs) are widely adopted due to their broad bandwidth, high saturation power, and compatibility with superconducting qubits. In this work, we present a design workflow of JTWPAs based on a co-simulation approach combining electromagnetic (EM) and circuit-level simulation. We apply this method to a uniform JTWPA, comparing our simulation results with experimental data. To mitigate the challenge of constructing lumped element circuit models usually used in conventional simulation method, we directly perform full EM analysis of the device to accurately determine its linear response. The extracted S parameters are then fed into a nonlinear harmonic balance simulator with Josephson junctions modeled as circuit elements. By doing so, the linear and gain properties of the JTWPA are extracted. Different factors that impact device performance including gain compression, phase matching, and fabrication imperfections are further investigated and modeled. A good agreement between simulation and measurements is identified, establishing the effectiveness of our approach in designing quantum parametric amplifiers.
Josephson traveling-wave parametric amplifiers (JTWPAs) are wideband, ultralow-noise amplifiers used to enable the readout of superconducting qubits. While individual JTWPAs have achieved high performance, behavior between devices is inconsistent due to wide manufacturing tolerances. Amplifier designs could be modified to improve resilience towards variations in amplifier components; however, existing device models often rely on analytical techniques that typically fail to incorporate component variations. To begin addressing this issue, a 1D numerical method for modeling JTWPAs is introduced in this work. The method treats the Josephson junctions and transmission lines in an amplifier as coupled subsystems and can easily incorporate arbitrary parameter variations. We discretize the transmission line subsystem with a finite element time domain method and the Josephson junction subsystem with a finite difference method, with leap-frog time marching used to evolve the system in time. We validate our method by comparing the computed gain to an analytical model for a traditional JTWPA architecture and one with resonant phase matching. We then use our method to demonstrate the impact of variations in Josephson junctions and phase-matching resonators on amplification. In future work, the method will be adjusted to incorporate additional amplifier architectures and extended to a 3D full-wave approach.
A Superconducting QUantum Interference Device (SQUID) modulated by a fast oscillating magnetic flux can be used as a parametric amplifier, providing gain with very little added noise. Here, we develop linearized models to describe the parametrically flux-pumped SQUID in terms of an impedance. An unpumped SQUID acts as an inductance, the Josephson inductance, whereas a flux-pumped SQUID develops an additional, parallel element which we have coined the “pumpistor.” Parametric gain can be understood as a result of a negative resistance of the pumpistor. In the degenerate case, the gain is sensitive to the relative phase between the pump and signal. In the nondegenerate case, gain is independent of this phase. We develop our models first for degenerate parametric pumping in the three-wave and four-wave cases, where the pump frequency is either twice or equal to the signal frequency, respectively. We then derive expressions for the nondegenerate case where the pump frequency is not a multiple of the signal frequency, for which it becomes necessary to consider idler tones that occur. For the nondegenerate three-wave case, we present an intuitive picture for a parametric amplifier containing a flux-pumped SQUID where current at the signal frequency depends upon the load impedance at an idler frequency. This understanding provides insight and readily testable predictions of circuits containing flux-pumped SQUIDs.