Evidence of a Nonequilibrium Distribution of Quasiparticles in the Microwave Response of a Superconducting Aluminum Resonator

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.

Figure 1

(a) The measured microwave transmission ${|S_{21}|}^2$ as a function of frequency for sample $A$. The solid lines are taken for four different microwave readout powers ($P_{\text{read}}$) at a temperature of 64 mK. The dashed lines are taken at 349 mK. The same color coding applies. The arrows indicate increasing $P_{\text{read}}$. The inset shows the internal quality factor and resonant frequency as determined from the $S_{21}$ measurements as a function of $P_{\text{read}}$. (b),(c) The measured internal quality factor and resonant frequency as a function of temperature for various $P_{\text{read}}$ (the same legend applies). The arrows indicate increasing $P_{\text{read}}$. Simulation results are shown as lines. The inset is a representation of the effect of microwave absorption on the energy gap $\mathrm{\Delta }$. The temperature is normalized to the equilibrium $T_c$.

Figure 2

(a) The calculated quasiparticle distribution as a function of normalized energy. The two different panels are for temperatures of 120 and 320 mK. The insets show the same distributions on a log scale. (b) The phonon power flow from the film to the substrate as a function of normalized energy, for 120 mK (the inset shows the same lines on a log scale) and 320 mK. $dP$ is zero in thermal equilibrium.

Figure 3

(a) The real part of the complex conductivity, ${\sigma }_1$, as a function of temperature, calculated for three microwave readout powers at a frequency of 5.57 GHz. (b) The imaginary part of the conductivity, ${\sigma }_2$, as a function of temperature. (c) The calculated quasiparticle density as a function of temperature. (d) The quasiparticle recombination lifetime as function of temperature. (e) The difference of the energy gap for the driven distributions ($\mathrm{\Delta }$) compared to a thermal distribution (${\mathrm{\Delta }}_T$). The legend applies to all panels.

Figure 4

(a) The measured quasiparticle recombination time as obtained from a noise measurement for various microwave readout powers measured on sample $B$. (b) The measured internal quality factor and (c) resonant frequency. The legend applies to all panels.