We present a formalism for modelling parametric amplification by resonators subject to rate-limited nonlinearity of mixed reactive/dissipative character, with particular relevance to superconducting devices. The non-linearity is assumed to be characterised by a single state parameter, which responds to changes in the energy stored in the resonator with finite response time. We show how the operating point and small signal amplification behaviour of the pumped resonator can be calculated, characterised and optimised in terms of a set of three dimensionless parameters. The formalism is then illustrated with a simple, first-order, model nonlinearity and the implications for amplification via quasiparticle generation in a superconductor discussed. Throughout we describe how the parameters needed to characterise the device can be determined experimentally from steady-state measurements. A key result of this paper is that rate-limiting of a nonlinear mechanism does not preclude amplification, although it does limit the bandwidth over which it may be achieved.
We present a formalism for modelling parametric amplification by resonators subject to rate-limited nonlinearity of mixed reactive/dissipative character, with particular relevance to superconducting devices. The non-linearity is assumed to be characterised by a single state parameter, which responds to changes in the energy stored in the resonator with finite response time. We show how the operating point and small signal amplification behaviour of the pumped resonator can be calculated, characterised and optimised in terms of a set of three dimensionless parameters. The formalism is then illustrated with a simple, first-order, model nonlinearity and the implications for amplification via quasiparticle generation in a superconductor discussed. Throughout we describe how the parameters needed to characterise the device can be determined experimentally from steady-state measurements. A key result of this paper is that rate-limiting of a nonlinear mechanism does not preclude amplification, although it does limit the bandwidth over which it may be achieved.
Abstract The underlying nonlinear mechanisms behind the operation of travelling-wave parametric amplifiers (TWPAs) are important in determining their performance in terms of added noise, maximum gain, and bandwidth. We describe a method of characterising the underlying nonlinearity of a superconducting material in terms of its dissipative-reactive ratio and the response time of the underlying microscopic processes. We describe and calculate the different behaviour arising from the equilibrium supercurrent nonlinearity, which has low dissipation and fast response time, and the non-equilibrium heating nonlinearity, which has high dissipation and slow response time. We have fabricated TWPAs based on Al and Ti, and characterised their nonlinearities using our analysis. For both Al and Ti, the measured dissipative-reactive ratios and response times are quantitatively similar to predictions for the non-equilibrium heating nonlinearity. We were able to obtain more than 20 d B of peak power gain, although only over a narrow bandwidth of a few kilohertz. Our method of characterising the underlying nonlinearities could also be useful in the understanding and design of other superconducting nonlinear devices such as parametric up-converters, kinetic inductance Fourier transform spectrometers, and resonator parametric amplifiers.
Abstract Superconducting resonators and parametric amplifiers are important components in scientific systems such as kinetic inductance detector arrays, frequency-domain multiplexers for other superconducting bolometers, spin-ensemble based memories, and circuit quantum electrodynamics demonstrators. In this paper, we report microwave measurements of superconducting Ti, Nb, and NbN resonators and their use as parametric amplifiers. These half-wave resonators were fabricated under near identical sputtering and lithographic conditions to ensure a like-for-like comparison of material properties. We report a wide range of properties and behaviours in terms of transition temperatures, resistivities, rate-limiting nonlinear response times, nonlinear dissipation, signs of the nonlinear inductances and their dependences on temperature and resonance harmonic. We have successfully operated Nb and NbN resonators as high gain parametric amplifiers, achieving greater than 20 dB of power amplification. We have shown that for a half-wave resonator, amplification can be realised not only in the fundamental resonance but also in the higher harmonic resonances. Further, for materials with high transition temperatures, e.g. Nb and NbN, amplification can be achieved at ∼ 4 K, i.e. a temperature maintained by a pulse tube cooler. Finally, in materials systems that have very fast response times, e.g. NbN, we have found that a cross-harmonic type of amplification can be achieved by placing pump tone in a different resonant mode as the signal and the idler. This wide range of observations will have important implications on the design and application of superconducting parametric amplifiers.
Superconducting parametric amplifiers (SPAs) are critical components for ultralow-noise qubit readout in quantum computing, addressing the critical challenge of amplifying weak quantum signals without introducing noise that degrades coherence and computational fidelity. Unlike classical amplifiers, SPAs can achieve or closely approach quantum-limited performance, specifically the Standard Quantum Limit (SQL) of half a photon of added noise for phase-preserving amplification. The core principle of SPAs relies on parametric amplification, where energy is transferred from a strong pump tone to a weak input signal through non-dissipative nonlinear mixing processes. This is enabled by intrinsic nonlinearities in superconducting materials, primarily kinetic inductance in thin films (e.g., NbTiN, Al) and, more significantly, the Josephson effect in Josephson junctions. These nonlinear elements facilitate frequency mixing (three-wave or four-wave mixing) and can operate in phase-preserving or phase-sensitive amplification modes, with the latter allowing for noise squeezing below the SQL. This chapter emphasizes the significant role of resonator design in determining critical SPA performance metrics such as gain, bandwidth, and noise characteristics. It details both lumped-element (LC) and distributed-element (coplanar waveguide, CPW) resonators, discussing their unique properties, suitability for different frequency ranges, and the importance of achieving high-quality factors (Q) for efficient energy storage and minimal loss. A practical design and simulation of a meandered quarterwavelength CPW resonator coupled to a feed line is presented, illustrating how precise control over geometric parameters optimizes resonant frequency, coupling strength, and quality factor for high-fidelity qubit state discrimination.
We consider the effects of quasiparticle heating on the dynamical behaviour of superconducting microresonators operating with high readout powers. It is seen that resonance-curve distortion and hysteretic switching are an inevitable consequence of the forms of the quasiparticle heating and cooling functions. The model uses a diagrammatic representation of dynamical behaviour to gain an insight into how the instantaneous operating point moves in response to step changes in readout power, signal power, bath temperature, and frequency. The coupling quality factor has a marked effect on non-linear behaviour. The model is used to calculate the waveforms generated when a device switches between hysteretic states. The work suggests that measured response times need not be representative of intrinsic quasiparticle relaxation rates.
We discuss how reactive and dissipative nonlinearities affect the intrinsic response of superconducting thin-film resonators. We explain how most, if not all, of the complex phenomena commonly seen can be described by a model in which the underlying resonance is a single-pole Lorentzian, but whose centre frequency and quality factor change as external parameters, such as readout power and frequency, are varied. What is seen during a vector-network-analyser measurement is series of samples taken from an ideal Lorentzian that is shifting and spreading as the readout frequency is changed. According to this model, it is perfectly proper to refer to, and measure, the resonant frequency and quality factor of the underlying resonance, even though the swept-frequency curves appear highly distorted and hysteretic. In those cases where the resonance curve is highly distorted, the specific shape of the trajectory in the Argand plane gives valuable insights into the second-order physical processes present. We discuss the formulation and consequences of this approach in the case of nonlinear kinetic inductance, two-level-system loss, quasiparticle generation, and a generic model based on a power-law form. The generic model captures the key features of specific dissipative nonlinearities, but additionally leads to insights into how general dissipative processes create characteristic forms in the Argand plane. We provide detailed formulations in each case, and indicate how they lead to the wide variety of phenomena commonly seen in experimental data. We also explain how the properties of the underlying resonance can be extracted from this data. Overall, our paper provides a self-contained compendium of behaviour that will help practitioners interpret and determine important parameters from distorted swept-frequency measurements.
Nonlinear microwave resonators are a versatile tool in quantum information processing, enabling parametric amplification, continuous variable quantum computing, and engineered mode interactions. Many of these applications especially benefit from cubic nonlinearities enabling three-wave mixing; at the same time, they are limited by quartic nonlinearities giving rise to undesired Kerr effects. A recurrent challenge is therefore to engineer resonators with a finite cubic nonlinearity while suppressing quartic terms. Here, we investigate a superconducting resonator hosting two weak links fabricated from an aluminum-capped indium arsenide nanowire. We characterize the Kerr nonlinearity as a function of magnetic flux and gate bias, showing that it can be tuned to zero with either control parameter. Furthermore, we experimentally demonstrate three-wave mixing in a semiconductor-superconductor hybrid device, establishing nonzero cubic nonlinearity. An effective model based on Andreev bound states qualitatively captures the observed trends. Our results validate semiconductor-superconductor hybrid devices as a promising platform for tunable nonlinear superconducting circuits, with applications in parametric amplification, quantum control of bosonic modes, and engineering interactions between microwave modes.