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IEEE Transactions on Applied Superconductivity

Modeling of Josephson Traveling Wave Parametric Amplifiers

Claudio Guarcello, Guerino Avallone, C. Barone, M. Borghesi, S. Capelli, Giovanni Carapella, Anna Paola Caricato, Iacopo Carusotto, Alessandro Cian, 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 Greco, Danilo Labranca, Angelo Leo, Carlo Ligi, Giovanni Maccarrone, Federica Mantegazzini, B. Margesin, Giuseppe Maruccio, C. Mauro, R. Mezzena, Anna Grazia Monteduro, Angelo Nucciotti, L. Oberto, Luca Origo, S. Pagano, Vincenzo Claudio Pierro, L. Piersanti, M. Rajteri, A. Rettaroli, Silvia Rizzato, Andrea Vinante, M. Zannoni

Published Oct 14, 2022• 16 citations
Publisher

Abstract

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.

DOI
10.1109/tasc.2022.3214751

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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.

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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.
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.
Josephson traveling-wave parametric amplifiers (TWPAs) enable the amplification of weak microwave signals close to the quantum limit with large bandwidth, which has a broad range of applications in superconducting quantum computing and in the operation of single-photon detectors. While the large bandwidth allows for their use in frequency-multiplexed detection architectures, an increased number of readout tones per amplifier puts more stringent requirements on the dynamic range to avoid saturation. Here, we characterize the undesired mixing processes between the different frequency-multiplexed tones applied to a Josephson TWPA, a phenomenon also known as intermodulation distortion. The effect becomes particularly significant when the amplifier is operated close to its saturation power. Furthermore, we demonstrate that intermodulation distortion can lead to significant crosstalk and reduction of fidelity for multiplexed readout of superconducting qubits. We suggest using large detunings between the pump and signal frequencies to mitigate crosstalk. Our work provides insights into the limitations of current Josephson TWPAs and highlights the importance of performing further research on these devices.
The advent of ultra-low noise microwave amplifiers revolutionized several research fields demanding quantum-limited technologies. Exploiting a theoretical bimodal description of a linear phase-preserving amplifier, in this contribution we analyze some of the intrinsic properties of a model architecture (i.e., an rf-SQUID based Josephson Traveling Wave Parametric Amplifier) in terms of amplification and noise generation for key case study input states (Fock and coherent). Furthermore, we present an analysis of the output signals generated by the parametric amplification mechanism when thermal noise fluctuations feed the device.
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.
As the need for scaling up quantum computers increases there is also an increased need for more compact adjacent circuitry surrounding the quantum processor. In the amplification of a qubit's readout-signal there is a need for quantum-limited amplifiers achieving a high signal-to-noise ratio. Today, it is common to use traveling-wave parametric amplifiers (TWPAs), which are currently powered by an external microwave pump. To further reduce the circuitry an option is to use an on-chip pump only requiring DC-wires to operate. This is possible due to the AC Josephson effect which can convert a DC voltage to an AC current. By connecting multiple Josephson junctions in series it is possible to increase the output power by utilizing their coherent oscillations. The purpose of this project was to simulate and characterize Josephson voltage controlled oscillators in terms of power and frequency-output, with the goal of providing the pump to a TWPA. This also included bandpass filters utilizing the Josephson inductance, to remove higher order harmonics of the oscillations. These bandpass filters were also characterized both in simulations and in measurements. To evaluate these components, a cryogenic testbed was implemented, and benchmarked, with a capacity of 12 parallel experiments. Simulations showed that it was possible to use Josephson junctions to produce stable oscillations which were filtered to give a pure 5 GHz signal with a -86.78 dBm power. When the corresponding device was measured it was not possible to observe the harmonic spectrum, likely due to noise in the DC-biasing system. The measured bandpass filters exhibited either high insertion loss or limited bandwidth as compared to design-specifications. A power-threshold was identified at approximately -90 dBm at which the bandpass filters showed severe non-linear effects, both in simulations and in measurements.
Superconducting traveling-wave parametric amplifiers (TWPAs) are increasingly used in various applications, including quantum computing, quantum sensing, and dark matter detection. However, one important characteristic of these amplifiers, gain compression, has not received much attention. As a result, there is a lack of comprehensive experimental exploration of this phenomenon in the existing literature. In this study, we present an experimental investigation of gain compression in a Josephson traveling-wave parametric amplifier based on a four-wave mixing process. We have implemented a novel setup to monitor the complex transmission of both the pump and signal tones, which allows us to simultaneously track pump depletion and signal amplification as functions of signal power and frequency across the entire bandwidth of the device. Our findings indicate that, while pump depletion occurs during gain compression, it is not the only mechanism involved in the saturation of a TWPA. Power-induced phase-matching processes also take place within the device. This study provides valuable insights for optimizing TWPAs for applications that require high total input power, such as multiplexed qubit readout or broadband photon emission.