We have directly measured quasiparticle number fluctuations in a thin film superconducting Al resonator in thermal equilibrium. The spectrum of these fluctuations provides a measure of both the density and the lifetime of the quasiparticles. We observe that the quasiparticle density decreases exponentially with decreasing temperature, as theoretically predicted, but saturates below 160 mK to 25-55/μm(3). We show that this saturation is consistent with the measured saturation in the quasiparticle lifetime, which also explains similar observations in qubit decoherence times.
We have directly measured quasiparticle number fluctuations in a thin film superconducting Al resonator in thermal equilibrium. The spectrum of these fluctuations provides a measure of both the density and the lifetime of the quasiparticles. We observe that the quasiparticle density decreases exponentially with decreasing temperature, as theoretically predicted, but saturates below 160 mK to 25-55/μm(3). We show that this saturation is consistent with the measured saturation in the quasiparticle lifetime, which also explains similar observations in qubit decoherence times.
We measure temperature-dependent quasiparticle fluctuations in a small Al volume, embedded in a NbTiN superconducting microwave resonator. The resonator design allows for readout close to equilibrium. By placing the Al film on a membrane, we enhance the fluctuation level and separate quasiparticle effects from phonon effects. When lowering the temperature, the recombination time saturates and the fluctuation level reduces by a factor $\ensuremath{\sim}100$. From this we deduce that the number of free quasiparticles is still thermal. Therefore, the theoretical, inverse relation between the quasiparticle number and recombination time is invalid in this experiment. This is consistent with quasiparticle trapping, where on-trap recombination limits the observed quasiparticle lifetime.
We have measured the number of quasiparticles and their lifetime in aluminium superconducting microwave resonators. The number of excess quasiparticles below 160 mK decreases from 72 to 17 μm−3 with a 6 dB decrease of the microwave power. The quasiparticle lifetime increases accordingly from 1.4 to 3.5 ms. These properties of the superconductor were measured through the spectrum of correlated fluctuations in the quasiparticle system and condensate of the superconductor, which show up in the resonator amplitude and phase, respectively. Because uncorrelated noise sources vanish, fluctuations in the superconductor can be studied with a sensitivity close to the vacuum noise.
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 .
We present measurements of the quasiparticle relaxation time in Ta superconducting films shaped as planar resonators using the response of the complex conductivity to photon flux. These films have been implanted with magnetic (Mn) as well as non-magnetic atoms (Al). We find a significant decrease of the relaxation time with increasing impurity concentration for both cases, observing a reduction of the low temperature relaxation time, from 25 μs to 10 μs when implanting 100 ppm Mn. This indicates that impurities enhance quasiparticle relaxation in superconductors, regardless of their magnetic moment.
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.