arXiv:2403.01597 · IEEE Transactions on Applied Superconductivity
Nonlinear Behavior of Josephson Traveling Wave Parametric Amplifiers
Claudio Guarcello, Felix Ahrens, Guerino Avallone, C. Barone, M. Borghesi, Luca Callegaro, Giovanni Carapella, Anna Paola Caricato, Iacopo Carusotto, Alessandro Cian, Alessandro D’Elia, Daniele Di Gioacchino, Emanuele Enrico, P. Falferi, Luca Fasolo, M. Faverzani, E. Ferri, Giovanni Filatrella, Claudio D. Gatti, Andrea Giachero, D. Giubertoni, V. Granata, Angelo Leo, Danilo Labranca, Carlo Ligi, Giovanni Maccarrone, Federica Mantegazzini, B. Margesin, Giuseppe Maruccio, R. Mezzena, Anna Grazia Monteduro, Roberto Moretti, Angelo Nucciotti, L. Oberto, Luca Origo, Alex Stephane Piedjou Komnang, S. Pagano, L. Piersanti, A. Rettaroli, Silvia Rizzato, S. Tocci, Andrea Vinante, M. Zannoni
Recent advancements in quantum technologies and advanced detection experiments have underscored the pressing need for the detection of exceedingly weak signals within the microwave frequency spectrum. Addressing this challenge, the Josephson Traveling Wave Parametric Amplifier (JTWPA) has been proposed as a cryogenic front-end amplifier capable of approaching the quantum noise limit while providing a relevant bandwidth. This research is centered on a comprehensive numerical investigation of the JTWPA, without resorting to simplifications regarding the nonlinearity of the essential components. Specifically, this study focuses on a thorough examination of the system, characterized by coupled nonlinear differential equations representing all components of the device. Proper input and output signals at the device's boundaries are considered. The analysis of the output signals undergoing the parametric amplification process involves a detailed exploration of phase-space dynamics and Fourier spectral analysis of the output voltage. This study is conducted while considering the parameters ruling the response of the device under pump and signal excitations. In addition to the expected signal amplification, the findings reveal that the nonlinear nature of the system can give rise to unforeseen phenomena, depending on the system's operational conditions, which include: the generation of pump tone harmonics, modulation of the signal gain, and incommensurate frequency generation—effects that are not easily accommodated by simplistic linearized approaches.
Recent advancements in quantum technologies and advanced detection experiments have underscored the pressing need for the detection of exceedingly weak signals within the microwave frequency spectrum. Addressing this challenge, the Josephson Traveling Wave Parametric Amplifier (JTWPA) has been proposed as a cryogenic front-end amplifier capable of approaching the quantum noise limit while providing a relevant bandwidth. This research is centered on a comprehensive numerical investigation of the JTWPA, without resorting to simplifications regarding the nonlinearity of the essential components. Specifically, this study focuses on a thorough examination of the system, characterized by coupled nonlinear differential equations representing all components of the device. Proper input and output signals at the device's boundaries are considered. The analysis of the output signals undergoing the parametric amplification process involves a detailed exploration of phase-space dynamics and Fourier spectral analysis of the output voltage. This study is conducted while considering the parameters ruling the response of the device under pump and signal excitations. In addition to the expected signal amplification, the findings reveal that the nonlinear nature of the system can give rise to unforeseen phenomena, depending on the system's operational conditions, which include: the generation of pump tone harmonics, modulation of the signal gain, and incommensurate frequency generation—effects that are not easily accommodated by simplistic linearized approaches.
The recent developments in quantum technologies, as well as advanced detection experiments, have raised the need to detect extremely weak signals in the microwave frequency spectrum. To this aim, the Josephson travelling wave parametric amplifier, a device capable of reaching the quantum noise limit while providing a wide bandwidth, has been proposed as a suitable cryogenic front-end amplifier. This work deals with the numerical study of a Josephson travelling wave parametric amplifier, without approximations regarding the nonlinearity of the key elements. In particular, we focus on the investigation of the system of coupled nonlinear differential equations representing all the cells of the Josephson travelling wave parametric amplifier, with proper input and output signals at the boundaries. The investigation of the output signals generated by the parametric amplification process explores the phase-space and the Fourier spectral analysis of the output voltage, as a function of the parameters describing the pump and signal tones that excite the device. Beside the expected behavior, i.e., the signal amplification, we show that, depending on the system operation, unwanted effects (such as pump tone harmonics, incommensurate frequency generation, and noise rise), which are not accounted for in simple linearized approaches, can be generated in the whole nonlinear system.
We tackled the numerical analysis of the dynamic response of a Josephson Traveling Wave Parametric Amplifier (JTWPA) by varying the driving parameters, with a focus on the pathways leading to chaotic behavior. By tuning the working conditions, we explore the broad spectrum of dynamical regimes accessible to JTWPAs, delineating the conditions under which transition to chaos occurs. Furthermore, we extend our investigation to junctions characterized by a non-sinusoidal current phase relation (CPR) and explore the consequences on the amplifier’s performance. Through the study of gain characteristics, Poincaré sections, and Fourier spectra, we provide an in-depth understanding of how CPR nonlinearity and nonsinusoidality influence the operational effectiveness and stability of JTWPAs. This investigation offers insights into optimizing the device for enhanced performance and robustness against chaotic disruptions, in order to establish a framework for predicting and controlling JTWPA behavior in practical applications. In fact, we identify the regions in the parameter space where the input signal is maximally amplified without excessive noise or undesired harmonics. This effort paves the way for the development of devices with tailored dynamic responses and for advancements in quantum computing and precision measurement technologies, where stability and high fidelity are of paramount importance.
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.
In this work, we develop a comprehensive numerical analysis of the dynamic response of a Josephson Traveling Wave Parametric Amplifier (JTWPA) by varying the driving parameters, with a focus on the pathways leading to chaotic behavior. By tuning the working conditions, we capture the broad spectrum of dynamical regimes accessible to JTWPAs, delineating the settings under which transition to chaos occurs. Furthermore, we extend our investigation to device formed by junctions characterized by a non--sinusoidal current phase relation (CPR) and exploring the impact of its shape on the amplifier's performance. Through the study of gain characteristics, Poincaré sections, and Fourier spectra, we provide an in-depth understanding of how non-linearity and CPR nonsinusoidality influence the JTWPAs' operational effectiveness and stability. This investigation offers insights into optimizing the device designs for enhanced performance and robustness against chaotic disruptions, in order to establish a framework for predicting and controlling JTWPA behavior in practical applications. This effort will pave the way for the development of devices with tailored dynamic responses and for advancements in quantum computing and precision measurement technologies, where stability and high fidelity are of paramount importance.
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 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.
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.