Josephson parametric amplifiers (JPA) are nonlinear devices that are used for quantum sensing and qubit readout in the microwave regime. While JPAs regularly operate near the quantum limit, their gain saturates for very small (few-photon) input power. In a previous work, we showed that the saturation power of JPAs is not limited by pump depletion, but instead by the fourth-order nonlinearity of Josephson junctions, the nonlinear circuit elements that enables amplification in JPAs. Here, we present a systematic study of the nonlinearities in JPAs, show which nonlinearities limit the saturation power, and present a strategy for optimizing the circuit parameters for achieving the best possible JPA. For concreteness, we focus on JPAs that are constructed around a Josephson ring modulator (JRM). We show that by tuning the external and shunt inductors, we should be able to take the best experimentally available JPAs and improve their saturation power by $\ensuremath{\approx}15\phantom{\rule{0.16em}{0ex}}\mathrm{dB}$. Finally, we argue that our methods and qualitative results are applicable to a broad range of cavity-based JPAs.
Josephson parametric amplifiers (JPA) are nonlinear devices that are used for quantum sensing and qubit readout in the microwave regime. While JPAs regularly operate near the quantum limit, their gain saturates for very small (few-photon) input power. In a previous work, we showed that the saturation power of JPAs is not limited by pump depletion, but instead by the fourth-order nonlinearity of Josephson junctions, the nonlinear circuit elements that enables amplification in JPAs. Here, we present a systematic study of the nonlinearities in JPAs, show which nonlinearities limit the saturation power, and present a strategy for optimizing the circuit parameters for achieving the best possible JPA. For concreteness, we focus on JPAs that are constructed around a Josephson ring modulator (JRM). We show that by tuning the external and shunt inductors, we should be able to take the best experimentally available JPAs and improve their saturation power by \ensuremath≈15dB. Finally, we argue that our methods and qualitative results are applicable to a broad range of cavity-based JPAs.
Circuit quantum-electrodynamics-based quantum information processing relies on low noise amplification for signal readout. In the realm of microwave superconducting circuits, this amplification is often achieved via Josephson parametric amplifiers (JPAs). In the past, these amplifiers exhibited low power added efficiency (PAE), which is roughly the fraction of pump power that is converted to output signal power. This is increasingly relevant because recent attempts to build high saturation power amplifiers achieve this at the cost of very low PAE, which in turn puts a high heat load on the cryostat and limits the number of these devices that a dilution refrigerator can host. Here, we numerically investigate upper bounds on PAE. We focus on a class of parametric amplifiers that consists of a capacitor shunted by a nonlinear inductive block. We first set a benchmark for this class of amplifiers by considering nonlinear blocks described by an arbitrary polynomial current-phase relation. Next, we identify that it is important for amplifiers with inductive blocks composed of repeating elements to have monotonic current-phase relations for each element in order to avoid exciting high-frequency modes. Using this design rule, we propose two circuit implementations for repeating elements in JPA inductive blocks. Finally, we investigate polynomial amplifier chains. We find that while amplifiers with higher gain have a lower PAE, regardless of the gain there is considerable room to improve as compared with state-of-the-art devices. For example, for a degenerate amplifier with a power gain of 20 dB, the PAE is $\ensuremath{\approx}0.1%$ for typical JPAs, 37.9% for our simpler circuit JPAs, 42.6% for our more complex circuit JPAs, 63.3% for our arbitrary polynomial amplifiers, and at least 98% for our amplifier chains.
One of the primary requirements for superconducting quantum information processors is high-fidelity, quantum non-demolition qubit readout. In superconducting circuits, we perform measurements with quantum-limited microwave amplifiers. These amplifiers are based on microwave cavities and powered by Josephson-junction based nonlinearities, which we drive to parametrically modulate the circuit and generate gain. However, cavity-based amplifiers generally suffer from a fixed gain-bandwidth product, a limited dynamic range, and the absence of directionality, and so they must be operated with external lossy microwave commercial components such as circulators. Many of these limitations can be circumvented by combining multiple parametric processes in a few-mode device, in particular by combining parametric drives which produce gain and those which produce gainless photon conversion. By combining balanced gain and conversion processes between a pair of modes, we have demonstrated transmission-only, phase-sensitive, gain-bandwidth-free amplification. We have also used measurements of transmon qubits with this mode of amplification to demonstrate its practical utility. More, by combining multiple instances of imbalanced gain and conversion processes among three modes, we have realized devices with non-reciprocal, transmission-only amplification, matched ports, and a large, gain-independent bandwidth. One of the biggest challenges in tuning multiple parametric processes is both inherent and dynamically-generated higher order nonlinearities in the device, which shift mode frequencies and parametric coupling strengths with pump power, preventing accurate balancing of the parametric processes. These can be controlled through engineering the Hamiltonian of the nonlinear element, in this thesis either a linearly-shunted Josephson Ring Modulator or SNAIL (Superconducting Nonlinear Asymmetric Inductive eLements), which for certain applied flux values can minimize or even change the sign of higher-order even nonlinearities. Additionally, we have explored a range of circuit parameters, allowing us to also control the relative size of odd-order nonlinearities. We have developed a series of prototypes which implement these concepts, and have achieved both high-dynamic range singly pumped amplifiers, and a promising platform for multi-parametric devices. We conclude by presenting recent progress towards implementing multiple parametric schemes that generates a practical directional amplifier, and discuss the remaining obstacles to a truly ideal parametric amplifiers.
Microwave parametric amplifiers based on Josephson junctions have become indispensable components of many quantum information experiments. One key limitation which has not been well predicted by theory is the gain saturation behavior which limits the amplifier's ability to process large amplitude signals. The typical explanation for this behavior in phase-preserving amplifiers based on three-wave mixing, such as the Josephson Parametric Converter, is pump depletion, in which the consumption of pump photons to produce amplification results in a reduction in gain. However, in this work, we present experimental data and theoretical calculations showing that the fourth-order Kerr nonlinearities inherent in Josephson junctions are the dominant factor. The Kerr-based theory has the unusual property of causing saturation to both lower and higher gains, depending on bias conditions. This work presents an efficient methodology for optimizing device performance in the presence of Kerr nonlinearities while retaining device tunability and points to the necessity of controlling higher-order Hamiltonian terms to make further improvements in parametric devices.
The Josephson ring modulator (JRM) is a device, based on Josephson tunnel junctions, capable of performing nondegenerate mixing in the microwave regime without losses. The generic scattering matrix of the device is calculated by solving coupled quantum Langevin equations. Its form shows that the device can achieve quantum-limited noise performance both as an amplifier and a mixer. Fundamental limitations on simultaneous optimization of performance metrics like gain, bandwidth, and dynamic range (including the effect of pump depletion) are discussed. We also present three possible integrations of the JRM as the active medium in a different electromagnetic environment. The resulting circuits, named Josephson parametric converters (JPC), are discussed in detail, and experimental data on their dynamic range are found to be in good agreement with theoretical predictions. We also discuss future prospects and requisite optimization of JPC as a preamplifier for qubit readout applications.
Nonlinearity of the current-phase relationship of a Josephson junction is the key resource for a Josephson parametric amplifier (JPA) as well as for a Josephson traveling-wave parametric amplifier, the only devices in which the quantum limit for added noise has so far been approached at microwave frequencies. A standard approach to describe JPA takes into account only the lowest order (cubic) nonlinearity resulting in a Duffing-like oscillator equation of motion or in a Kerr-type nonlinearity term in the Hamiltonian. In this paper we derive the quantum expression for the gain of JPA including all orders of the Josephson junction nonlinearity in the linear response regime. We then analyze gain saturation effect for stronger signals within a semiclassical approach. Our results reveal nonlinear effects of higher orders and their implications for operation of a JPA.
Superconducting quantum circuits are a leading platform for building a large-scale quantum processor [1]. A crucial component necessary for the high-fidelity readout of qubits in this framework, is an ultra-low-noise microwave amplifier. Thus, over the past two decades, superconducting Josephson parametric amplifiers (JPAs) have been developed that achieve near quantum-limited noise performance, an order of magnitude lower than competing microwave amplifiers based on semiconductor transistors [2]. Nevertheless, JPAs suffer from drawbacks such as limited bandwidth and compression power that makes them difficult to use in an eventual large-scale quantum processor. Thus, an ongoing theme in JPA development is to optimize the circuit to improve these specifications without sacrificing other desirable properties such as the noise performance, ease of construction and operation, etc. Here, we review some results of optimizing a specific JPA circuit called the SNAIL parametric amplifier [3]–[5], shown in Fig. 1, where the acronym SNAIL refers to the Superconducting Nonlinear Asymmetric Inductive eLement. The SNAIL is a composite element made from conventional Al/AlOx/Al Josephson junctions, which has a nonlinear current-phase relation that can be approximated as a nonlinear inductance. By applying a strong microwave frequency pump to modulate the nonlinear inductance, parametric amplification can be realized. By device and circuit engineering we have demonstrated two orders of magnitude improvement in compression power (Fig. 2) in such devices without sacrificing other desirable characteristics.