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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Circuit Quantum Electrodynamic Model of a Resonantly Phase-Matched Josephson Traveling Wave Parametric Amplifier

Michael Haider, Yongjie Yuan, J.H. Abundis-Patino, Johannes A. Russer, P. Russer, Christian Jirauschek

Published Jun 1, 2019• 1 citation
Publisher

Abstract

In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum nature of the device. A quantum mechanical treatment of the JTWPA was only given in a few recent papers, where [1] derives a Hamiltonian for a chain of unit cells without resonant phase-matching (RPM), and [2] solves the dynamics of the system with and without RPM, also partly considering noise and squeezed state generation. However, our approach differs from [2] in that we give an explicit solution to the resulting nonlinear wave equation for narrow-band signals, starting from discrete chain Hamiltonians with and without RPM. We focus on describing the dynamics of the device in a noiseless quantum setting and present solutions for the governing nonlinear wave equation in Fig. 1. The starting point of this work is the amplifier design as given in [3]. The mathematical description relies on the Hamiltonian of a discrete chain of unit cells, which was derived in [1], and which we extended by a RPM circuit [2]. As the size Δl of a single chain element is relatively small compared to the wavelength, the chain of discrete Josephson elements can be considered in terms of a continuous transmission line.

DOI
10.1109/cleoe-eqec.2019.8871821

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In recent years Josephson-junction-based parametric amplifiers have been developed for single-photon-level signals in the microwave spectral region, where they found applications in quantum information processing. Typically, Josephson traveling wave parametric amplifiers (JTWPA) have been studied based on classical circuit models. Although this approach shows good agreement with experimental results, it describes the system behavior in the framework of a classical theory, neglecting the quantum nature of the device. A quantum mechanical treatment of the JTWPA was only given in a few recent papers, where [1] derives a Hamiltonian for a chain of unit cells without resonant phase-matching (RPM), and [2] solves the dynamics of the system with and without RPM, also partly considering noise and squeezed state generation. However, our approach differs from [2] in that we give an explicit solution to the resulting nonlinear wave equation for narrow-band signals, starting from discrete chain Hamiltonians with and without RPM. We focus on describing the dynamics of the device in a noiseless quantum setting and present solutions for the governing nonlinear wave equation in Fig. 1. The starting point of this work is the amplifier design as given in [3]. The mathematical description relies on the Hamiltonian of a discrete chain of unit cells, which was derived in [1], and which we extended by a RPM circuit [2]. As the size Δl of a single chain element is relatively small compared to the wavelength, the chain of discrete Josephson elements can be considered in terms of a continuous transmission line.
A comprehensive numerical analysis of the dynamical response of Josephson Traveling Wave Parametric Amplifiers (JTWPA) under varying driving conditions is presented. The primary goal is to examine the effects of a current–phase relation (CPR) incorporating a second-harmonic contribution. The analysis further investigates how parameters such as pump intensity, signal frequency, and DC bias current influence the transition to chaos. The results reveal a wide range of dynamical regimes in JTWPAs, offering insights into optimizing device performance and stability. Notably, the shape of the CPR is shown to have a significant impact on the operational effectiveness and stability of JTWPAs. To obtain results closer to real devices, numerical modeling of JTWPAs implementing Resonant Phase Matching (RPM) circuitry has been performed. We have verified that the introduction of RPM allows a significant increase in the amplifier gain also in the case of non-sinusoidal CPR. These findings may have important implications for the development of next-generation JTWPAs, which are essential components in quantum computing and precision measurement applications.
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.
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.
This work presents the design, fabrication strategy, and numerical modeling of niobium-based Josephson traveling-wave parametric amplifiers (JTWPAs) for quantum microwave readout. These devices are designed as nonlinear transmission lines composed of more than two thousand Josephson junction unit cells, with periodically embedded resonators enabling phase matching. The use of Nb trilayer technology (Nb/AlO[Formula: see text]/Nb) with respect to Al-based devices ensures higher junction uniformity, lower losses, and higher resilience to stray magnetic fields. We developed a multi-scale modeling framework combining harmonic balance methods and time-domain simulations of nonlinear coupled differential equations to describe wave propagation, parametric amplification, and spectral dynamics. Numerical results predict broadband gain up to 20[Formula: see text]dB over multi-GHz bandwidths. These results demonstrate the potential of Nb-based JTWPAs as scalable components for quantum applications.
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.
Future progress and development in ultrasensitive detection and quantum-limited amplification of microwave signals in a wide frequency band is expected to be achieved by using new type architectures of parametric amplifiers based on chains of coupled Josephson junctions or superconducting quantum interference devices (SQUID). Over the past years the classical theory of the amplifiers in the framework of a nonlinear wave equation governing the node flux evolution along an array of coupled Josephson oscillators has been developed under continuum approximation. Although such amplifiers imply the quantum nature utilizing the Josephson effects there is no quantum approach to take into account quantum phenomena in Josephson traveling-wave parametric amplifiers (JTWPA). Here we formulate the Lagrangian approach to a one-dimensional chain of coupled Josephson junctions and take the first step to develop the quantum theory of JTWPAs by deriving an approximate quantum Hamiltonian of the chain.