Non-equilibrium superconductivity in superconducting resonators

Abstract: 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 $T_N<sup>*$ to describe the driven $f(E)$. The lifetime was compared to the distribution averaged thermal lifetime at $T_N<sup>*$ 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 $T_{21}<sup>*$ having a higher quality-factor in all cases studied and for low $P_{abs}$ $T_{21}^*\sim T_N<sup>*$. 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 $T_N<sup>*$ as a function of $P_{abs}$ were derived which give a very good account of $T_N<sup>*$ 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.