In this paper, we present models for the nonlinearity of different nonlinear elements for superconducting Josephson traveling-wave parametric amplifiers. Such devices are typically used as a first-stage readout amplifier for superconducting qubits in a cryogenic quantum computing environment. The nonlinear coefficients of different designs are evaluated as functions of an externally applied bias field, taking into account direct and external pumping schemes. Finally, we present results of the total gain for two different nonlinear transmission lines, one that is embedded with single Josephson junctions operating in four-wave mixing mode with direct pumping, and an externally pumped SNAIL-based amplifier operating in the three-wave mixing regime with a corresponding constant external flux bias. The gain is evaluated using a quantum circuit model including substrate losses and chromatic dispersion.
In this paper, we present models for the nonlinearity of different nonlinear elements for superconducting Josephson traveling-wave parametric amplifiers. Such devices are typically used as a first-stage readout amplifier for superconducting qubits in a cryogenic quantum computing environment. The nonlinear coefficients of different designs are evaluated as functions of an externally applied bias field, taking into account direct and external pumping schemes. Finally, we present results of the total gain for two different nonlinear transmission lines, one that is embedded with single Josephson junctions operating in four-wave mixing mode with direct pumping, and an externally pumped SNAIL-based amplifier operating in the three-wave mixing regime with a corresponding constant external flux bias. The gain is evaluated using a quantum circuit model including substrate losses and chromatic dispersion.
Abstract A Josephson traveling-wave parametric amplifier (JTWPA) is indispensable for the high-fidelity readout of superconducting qubits, owing to its broadband gain and near-quantum-limited noise performance. Conventional JTWPAs implemented with 0-Josephson junctions (0-JTWPAs) avoid magnetic-flux bias but rely on Kerr nonlinearity, which leads to pump-power-dependent phase mismatch, gain ripple, and a limited dynamic range. More recently, three-wave-mixing (3WM) JTWPAs employing flux-biased rf-SQUIDs have been proposed to suppress Kerr effects. However, these designs require a significant external flux bias per unit cell, which complicates cryogenic hardware, introduces sensitivity to flux noise, and limits scalability in large arrays. In this study, we propose and analyze a π -junction-based JTWPA ( π -JTWPA) in which the intrinsic π phase shift of the junction effectively displaces the operating point of the current–phase relation. This approach aims to realize 3WM-type nonlinearity with reduced flux bias while preserving broadband amplification, thereby simplifying flux-bias distribution, enhancing robustness against flux noise, and providing a pathway toward scalable, multiplexed quantum readout architectures. In addition, we examine the potential applicability of the π -JTWPA in four-wave-mixing (4WM) operation.
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.
Josephson parametric amplifiers are an important part of a modern superconducting quantum computing platform and squeezed quantum states generation devices. Traveling wave and impedance-matched parametric amplifiers provide broad bandwidth for high-fidelity single-shot readout of multiple qubit superconducting circuits. Here, we present a quantum-limited 3-wave-mixing parametric amplifier based on superconducting nonlinear asymmetric inductive elements (SNAILs), whose useful bandwidth is enhanced with an on-chip two-section impedance-matching circuit based on microstrip transmission lines. The amplifier dynamic range is increased using an array of 67 SNAILs with 268 Josephson junctions, forming a nonlinear quarter-wave resonator. Operating in a current-pumped mode, we experimentally demonstrate an average gain of 17 dB across 300 MHz bandwidth, along with an average saturation power of –100 dBm, which can go as high as −97 dBm with quantum-limited noise performance. Moreover, the amplifier can be fabricated using a simple technology with just one e-beam lithography step.
We present an analytic approach for finding optimized design parameters for flux-driven traveling-wave parametric amplifiers with superconducting nonlinear asymmetric inductive elements. The parameter to be optimized, i.e., the gain coefficient, is extracted from a Hamiltonian quantum model for three-wave mixing flux-driven amplifiers. Analytic expressions for the nonlinear coefficients up to the third order in the current-phase relation are derived. The minimum potential energy is determined by finding the roots of a polynomial function. Candidates for optimal operation points are identified by minimizing the residual cross-phase modulation while at the same time maximizing the gain coefficient. Finally, the gain of a traveling-wave parametric amplifier implementation with optimized parameters is evaluated using a quantum model, including substrate losses and chromatic dispersion.
This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). \nThe unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. \nMoving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. \nA novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. \nA rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. \nKnowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.