Non-equilibrium superconductivity in superconducting resonators
  We have calculated the non-equilibrium quasiparticle and phonon distributions $f(E)$, $n(\Omega)$, where $E$ and $\Omega$ are the quasiparticle and phonon energies respectively, generated by the photons of the probe signal of a low temperature superconducting resonator SR operating well-below its transition temperature $T_c$ as the absorbed probe power per unit volume $P_{abs}$ was changed. The calculations give insight into a rate equation estimate which suggests that the quasiparticle distributions can be driven far from the thermal equilibrium value for typical readout powers. From $f(E)$ the driven quasiparticle number density $N_{qp}$ and lifetime $\tau_r$ were calculated. Using $N_{qp}$ we defined an effective temperature $T_N^*$ to describe the driven $f(E)$. The lifetime was compared to the distribution averaged thermal lifetime at $T_N^*$ and good agreement was found typically within a few percent. We used $f(E)$ 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 $P_{abs}$ were found. The non-equilibrium $S_{21}$ cannot be accurately modeled by a thermal distribution at an elevated temperature $T_{21}^*$ having a higher quality-factor in all cases studied and for low $P_{abs}$ $T_{21}^*\sim T_N^*$.   Using $\tau_r$ and $N_{qp}$ we determined the achievable Noise Equivalent Power of the resonator used as a detector as a function of $P_{abs}$. Simpler expressions for $T_N^*$ as a function of $P_{abs}$ were derived which give a very good account of $T_N^*$ and also $N_{qp}$ and $\tau_r$. We conclude that multiple photon absorption from the probe increases the quasiparticle number above the thermal background and ultimately limits the achievable NEP of the resonator. 
D. J. Goldie and S. Withington
cond-mat.supr-con cond-mat.mes-hall