We analyse theoretically the influence of inelastic processes on coherent excited states of a superconductor generated by the application of microwaves. These states were identified recently [Phys. Rev. Lett. 117, 047002 (2016)] as the source of resonance frequency shift in microwave superconducting micro-resonators at low temperatures where the quasiparticle density is negligible. As a convenient model we use a thin dirty superconducting strip in contact with a diffuse normal metallic reservoir through an opaque tunnel junction and calculate analytically both the spectral functions and the non-equilibrium distribution function in the presence of a monochromatic microwave signal. We demonstrate that in the quantum mode of depairing, when the frequency of the microwave signal is much larger than the normalised signal amplitude, but much less than the modulus of a superconductor order parameter, the change of the kinetic inductance is determined primarily by the change of the spectral functions. We propose an experimental system to measure the spectral functions as well as the non-equilibrium distribution function using a non-invasive tunnel-junction on a microwave-driven superconducting wire.
We analyse theoretically the influence of inelastic processes on coherent excited states of a superconductor generated by the application of microwaves. These states were identified recently [Phys. Rev. Lett. 117, 047002 (2016)] as the source of resonance frequency shift in microwave superconducting micro-resonators at low temperatures where the quasiparticle density is negligible. As a convenient model we use a thin dirty superconducting strip in contact with a diffuse normal metallic reservoir through an opaque tunnel junction and calculate analytically both the spectral functions and the non-equilibrium distribution function in the presence of a monochromatic microwave signal. We demonstrate that in the quantum mode of depairing, when the frequency of the microwave signal is much larger than the normalised signal amplitude, but much less than the modulus of a superconductor order parameter, the change of the kinetic inductance is determined primarily by the change of the spectral functions. We propose an experimental system to measure the spectral functions as well as the non-equilibrium distribution function using a non-invasive tunnel-junction on a microwave-driven superconducting wire.
Low temperature microwave superconducting resonators (SRs) are attractive candidates for producing quantum-sensitive, arrayable energy or power detectors for astrophysical and other precision measurement applications. Their readout uses a microwave probe signal with quanta of energy well below the threshold for pair-breaking in the superconductor. We have calculated the non-equilibrium quasiparticle and phonon distributions generated by the photons of the probe signal of a resonator operating well below its superconducting transition temperature T c as the absorbed probe power was changed using the coupled kinetic equations described by Chang and Scalapino. The calculations give insight into a rate equation estimate which suggests that the quasiparticle distributions can be driven far from their thermal equilibrium value for typical readout powers. From the driven quasiparticle distribution functions, the driven quasiparticle number densities and lifetimes were calculated. An effective temperature to describe the driven quasiparticles was defined. The non-equilibrium lifetimes were compared to the distribution-averaged thermal lifetimes at the effective temperature and good agreement was found typically within a few per cent. We used the non-equilibrium quasiparticle distribution to model a representative SR. The complex conductivity and hence the frequency dependence of the experimentally measured forward scattering parameter S 21 of the SR as a function of absorbed power were found. The non-equilibrium S 21 cannot be accurately modeled by a thermal distribution even at its own elevated temperature, having a higher quality factor in all cases studied, although for low absorbed powers the two effective temperatures are similar. From the non-equilibrium lifetimes and number densities we determined the achievable noise equivalent power (NEP) of the resonator used as a power detector as a function of absorbed microwave power. Simpler expressions to evaluate the effective quasiparticle temperature as a function of absorbed power have also been derived. We conclude that multiple photon absorption from the microwave probe increases the quasiparticle number above the thermal background and ultimately limits the achievable NEP of the resonator at temperatures well below T c .
Superconducting materials are useful for a wide range of applications, ranging from particle detectors to quantum computing. The presence of excitations (quasiparticles) in the superconductor can be a resource in some devices and a disturbance in others - in kinetic inductance detectors, the generation of quasiparticles by an incoming photon can be read out and used to calculate the energy of the photon, while the tunnelling of quasiparticles is a cause of decoherence in quantum computing. While early research mostly focused on superconductors close to the critical temperature, temperatures that are far below the critical temperatures are often used for current applications. At these low temperatures, the energy exchange between quasiparticles and phonons becomes slow, and even small disturbances may drive the energy distribution of the quasiparticles out of thermal equilibrium. In fact, experiments usually show a residual number of quasiparticles even at milikelvin temperatures, for which the thermal density of quasiparticles would be negligible. This thesis consists of two parts: In the first part, I model the energy distribution of quasiparticles in superconducting resonators: Quasiparticles absorb energy by absorbing photons and release it as phonons, which drives the superconductor out of thermal equilibrium. I solve a set of rate equations, both numerical and analytical, to calculate the energy distribution of the quasiparticles. For parameters typically used in kinetic inductance detectors made of aluminum, the microwave signal used to read out the resonator dominates the energy distribution of the quasiparticles, while their density is determined by other processes. At not too low temperatures, quasiparticles are mainly produced by thermal phonons, and my calculations reproduce recently measured quality factors of resonators at temperatures above 250 mK without free parameters. I show that even a very small number of photons of higher energy, which I ascribe to noise from the microwave generator, would dominate the quasiparticle density at lower temperatures. Under this assumption, my calculations also reproduce the quality factors at lower temperatures. In the second part of this thesis, I investigate the effects of spatial fluctuations of the superconducting energy gap on the dynamics of quasiparticles. Recently, it has been proposed that such fluctuations may lead to the localization of quasiparticles in states below the energy gap, which could cause a finite quasiparticle density even at very low temperatures. Using the Keldysh-Usadel formalism, I derive rates for the different processes (recombination/(de-) localization/generation) of quasiparticles in such systems. Using these rate equations I derive new upper limits for the quasiparticle density that could result from these localized states. These upper limits show that localization caused by weak magnetic impurities in typical aluminum systems does not lead to a relevant quasiparticle density. This contradicts previous theories, which only consider the recombination between two localized quasiparticles without examining their dynamics in detail.
We address parametric amplifiers and kinetic inductance detectors, using concepts of the microscopic theory of superconductivity, and focusing on the interaction of microwave radiation with the superconducting condensate. This interaction was identified in recent experiments as the source of the apparent dissipation in microwave superconducting microresonators at low temperatures. Since the evaluation of the performance of practical devices based only on changes in kinetic inductance is not sufficiently informative about the underlying physical processes, we design an experiment with a tunnel measurement of a microwave-driven superconducting wire, in which the tunneling process is not affected by the microwaves. We conclude that such an experiment is feasible with current technology, but is unfortunately difficult to incorporate into standard superconducting resonators optimized for performance in applications. Nevertheless, given the limits of the commonly used phenomenological theories, such an experiment will provide the groundwork for further optimization of the performance.
Many superconducting devices rely on the finite gap in the excitation spectrum of a superconductor: thanks to this gap, at temperatures much smaller than the critical one the number of excitations (quasiparticles) that can impact the device’s behavior is exponentially small. Nevertheless, experiments at low temperature usually find a finite, non-negligible density of quasiparticles whose origin has been attributed to various non-equilibrium phenomena. Here, we investigate the role of photons with energy exceeding the pair-breaking threshold 2\Delta 2 Δ as a possible source for these quasiparticles in superconducting resonators. Modeling the interacting system of quasiparticles, phonons, sub-gap and pair-breaking photons using a kinetic equation approach, we find analytical expressions for the quasiparticles’ density and their energy distribution. Applying our theory to measurements of quality factor as function of temperature and for various readout powers, we find they could be explained by assuming a small number of photons above the pair-breaking threshold. We also show that frequency shift data can give evidence of quasiparticle heating.
I report measurements of two different types of superconducting devices illuminated by 780 nm light, one of the wavelengths needed in a proposed atom-superconductor hybrid quantum system. I illuminated a thin-film Al lumped-element resonator and observed the resonator quality factor and resonance frequency as a function of illumination intensity, microwave power, and temperature. The resonator was mounted in a 3d aluminum cavity. The variation in optically-induced loss due to microwave power was similar to the behavior expected for loss from a distribution of two-level systems. Although this behavior may suggest the presence of optically activated two-level systems, I found that the loss is better explained by the presence of nonequilibrium quasiparticles generated by the illumination and excited by the microwave drive. I described a model of the system where optical absorption creates an effective source of phonons and solved the coupled quasiparticle-phonon rate equations. I found good agreement between the simulation and the measured resonator quality factor and frequency shift as a function of temperature, microwave power, and optical illumination. I fabricated a transmon qubit and studied the qubit transition frequency and relaxation time as a function of illumination intensity and temperature. The qubit was mounted in a 3d aluminum cavity and coupled to the cavity forming a Jaynes-Cummings system. Qubit relaxation showed non-exponential behavior that I fit to a quasiparticle fluctuation model with two characteristic times. The transition frequency and both characteristic times decreased with increasing illumination intensity. For comparison, I described a nonequilibrium quasiparticle model for the expected frequency shift and relaxation time due to quasiparticle tunneling through the Josephson junction. While the quasiparticle simulation predicted the general qualitative behavior of the frequency shift and relaxation time, there were some significant discrepancies with the data. This suggests the model needs to be extended, for example by including a different gap in the two superconductor layers forming the junction, and by taking into account other possible sources of loss and decoherence.
In this thesis I describe work on developing theoretical and numerical models of supercon- ducting thin-film microwave resonators. Superconducting resonators are used in a variety of applications, one of which is as kinetic inductance detectors (KIDs). KIDs are ultra-low noise, highly sensitive, multiplexable detectors, with uses in a wide variety of fields including astrophysics, medical imaging, and particle physics. Resonators are also crucial for supercon- ducting qubit readout, superconducting mixers and parametric amplifiers, and as multiplexers for other devices. The results described in this thesis apply to all thin film resonators, but are primarily described in the context of KIDs. I develop models of how constant absorbed power affects superconducting thin films, driving them out of thermal equilibrium with the bath. Using the Chang & Scalapino equations, I calculate numerically the steady-state quasiparticle and phonon distributions for various sources of power (sub-gap and above-gap photons and phonons, single frequency and broadband). Many new results emerge, a few of which are: the quasiparticle heating effects of microwave power, explaining the experimentally measured saturation of resonator performance; the frequency dependence of the quasiparticle generation efficiency in the sub-mm wavelength range; and the importance of phonon trapping in thin films at low temperatures. I then use these nonequilibrium results in higher-level device models. Starting from a generic framework for resonators, I present and implement a model of a KID with variable thermal isolation. I calculate the effects of quasiparticle heating on both the large-signal and small-signal behaviour, highlighting the effects of electrothermal feedback caused by readout power. Electrothermal feedback is shown to be able to increase or decrease the magnitude and bandwidth of the responsivity and noise, depending on the operating point. Finally, I propose an experimental measurement of quasiparticle heating effects in a new device – a four-port ring resonator. In such a device, unlike in the standard two-port resonator, the device can be heated and read out independently. I develop detailed models for the thermal and electrical behaviour of these devices, and suggest a scheme by which the quasiparticle heating effects of microwave power, predicted in this thesis, can be measured.