Number Fluctuations of Sparse Quasiparticles in a Superconductor

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 $25–55/\mu {\mathrm{m}}^3$. We show that this saturation is consistent with the measured saturation in the quasiparticle lifetime, which also explains similar observations in qubit decoherence times.

Figure 1

(a) Schematic of the process of generation (left) and recombination (right) of quasiparticles in a superconductor. The single arrow symbolizes a quasiparticle, the double arrow a Cooper pair, and the wavy arrow a phonon. (b) A microscope picture of the microwave resonator, 2 times reduced in length for visibility. The open ends set the half-wavelength resonance condition. The coplanar waveguide (CPW) throughline is used for the microwave excitation and readout (from contact 1 to 2) of the resonator. (c) The real and imaginary parts of the complex transmission $S_{21}$ as a function of frequency around the resonant frequency. The gray arrow indicates the direction of increasing frequency. We define a resonator amplitude $A$ and phase $\theta$ with respect to the resonance circle center as indicated. In red, the measured fluctuations in $S_{21}$ at the resonant frequency are shown. The fluctuations far off resonance are shown in blue, which are used as a calibration for system noise contributions.

Figure 2

(a) Corrected power spectral density of the resonator amplitude fluctuations as a function of frequency, calibrated for system noise contributions as depicted in (b). The spectra are plotted for six different temperatures. Note that the spectral density is temperature independent up to the roll-off frequency. The Lorentzian fits (dashed lines) show that the spectra can be described with a single time scale. (b) The system noise is subtracted from the noise spectrum on resonance. The system noise is obtained by taking a calibration measurement at a frequency far from the resonant frequency as indicated in Fig. 1c. (c) Part of the time trace of the resonator amplitude, filtered with a moving average filter with a time constant ${\tau }_r/2$. High energy impacts are observed every 20–30 s. Smaller impacts (inset) happen every 5–10 s. Parts with impacts larger than 5 times the standard deviation (0.43 eV) are rejected.

Figure 3

The quasiparticle lifetime, obtained from the roll-off frequency of the resonator amplitude noise spectrum (Fig. 2), as a function of temperature. The solid line is the lifetime, calculated from theory. Additionally, the lifetime is determined with a short pulse of light. The exponential decay of the excitation is fit, and the obtained decay time is shown. The measurements show consistently that the lifetime saturates to about 2.2 ms below 160 mK. The error bars represent statistical uncertainties obtained from the fitting procedure.

Figure 4

The number of quasiparticles as determined from the noise level as a function of temperature. The inset shows that the noise level ($S_A$) divided by the responsivity is almost temperature independent. The error bars indicate combined statistical uncertainties. Alternatively, the number of quasiparticles in the superconductor is determined from the measured quasiparticle lifetimes. The number of quasiparticles saturates at around 30 000–50 000 below 160 mK, where an exponential decrease is expected (solid line).