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.
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.
In the superconducting state, the presence of a finite gap in the excitation spectrum implies that the number of excitations (quasiparticles) is exponentially small at temperatures well below the critical temperature. Conversely, minute perturbations can significantly impact both the distribution in energy and the number of quasiparticles. Typically, the interaction with the electromagnetic environment is the main perturbation source driving quasiparticles out of thermal equilibrium, while a phonon bath is responsible for restoration of equilibrium. Here we derive approximate analytical solutions for the quasiparticle distribution function in superconducting resonators and explore the impact of nonequilibrium on two measurable quantities: the resonator's quality factor and its resonance frequency. Applying our results to experimental data, we conclude that while at intermediate temperatures there is clear evidence for the nonequilibrium effects due to heating of the quasiparticles by photons, the low-temperature measurements are not explained by this mechanism.
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.
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 .
Understanding the detailed behaviour of superconducting pair breaking photon detectors such as Kinetic Inductance Detectors (KIDs) requires knowledge of the nonequilibrium quasiparticle energy distributions. We have previously calculated the steady state distributions resulting from uniform absorption of monochromatic sub gap and above gap frequency radiation by thin films. In this work, we use the same methods to calculate the effect of illumination by broadband sources, such as thermal radiation from astrophysical phenomena or from the readout system. Absorption of photons at multiple above gap frequencies is shown to leave unchanged the structure of the quasiparticle energy distribution close to the superconducting gap. Hence for typical absorbed powers, we find the effects of absorption of broadband pair breaking radiation can simply be considered as the sum of the effects of absorption of many monochromatic sources. Distribution averaged quantities, like quasiparticle generation efficiency η, match exactly a weighted average over the bandwidth of the source of calculations assuming a monochromatic source. For sub gap frequencies, however, distributing the absorbed power across multiple frequencies does change the low energy quasiparticle distribution. For moderate and high absorbed powers, this results in a significantly larger η–a higher number of excess quasiparticles for a broadband source compared to a monochromatic source of equal total absorbed power. Typically in KIDs the microwave power absorbed has a very narrow bandwidth, but in devices with broad resonance characteristics (low quality factors), this increase in η may be measurable.
Abstract Understanding the detailed behaviour of superconducting pair breaking photon detectors such as Kinetic Inductance Detectors (KIDs) requires knowledge of the nonequilibrium quasiparticle energy distributions. We have previously calculated the steady state distributions resulting from uniform absorption of monochromatic sub gap and above gap frequency radiation by thin films. In this work, we use the same methods to calculate the effect of illumination by broadband sources, such as thermal radiation from astrophysical phenomena or from the readout system. Absorption of photons at multiple above gap frequencies is shown to leave unchanged the structure of the quasiparticle energy distribution close to the superconducting gap. Hence for typical absorbed powers, we find the effects of absorption of broadband pair breaking radiation can simply be considered as the sum of the effects of absorption of many monochromatic sources. Distribution averaged quantities, like quasiparticle generation efficiency η , match exactly a weighted average over the bandwidth of the source of calculations assuming a monochromatic source. For sub gap frequencies, however, distributing the absorbed power across multiple frequencies does change the low energy quasiparticle distribution. For moderate and high absorbed powers, this results in a significantly larger η –a higher number of excess quasiparticles for a broadband source compared to a monochromatic source of equal total absorbed power. Typically in KIDs the microwave power absorbed has a very narrow bandwidth, but in devices with broad resonance characteristics (low quality factors), this increase in η may be measurable.
In a superconductor, absorption of photons with an energy below the superconducting gap leads to redistribution of quasiparticles over energy and thus induces a strong nonequilibrium quasiparticle energy distribution. We have measured the electrodynamic response, quality factor, and resonant frequency of a superconducting aluminium microwave resonator as a function of microwave power and temperature. Below 200 mK, both the quality factor and resonant frequency decrease with increasing microwave power, consistent with the creation of excess quasiparticles due to microwave absorption. Counterintuitively, above 200 mK, the quality factor and resonant frequency increase with increasing power. We demonstrate that the effect can only be understood by a nonthermal quasiparticle distribution.