Equivalence of the effects on the complex conductivity of superconductor due to temperature change and external pair breaking

The responsivity of microwave kinetic inductance detector (MKID) is related to the derivative of the complex conductivity sigma with respect to the quasiparticle density n(qp). For thermal quasiparticles, this derivative d sigma/dn(qp) can be evaluated from s( T) and nqp( T) given by the Marttis-Bardeen theory (Phys. Rev. 14: 412, 1958). For excess quasiparticles due to external pair breaking, d sigma/dn(qp) can be evaluated by introducing an effective chemical potential mu* into the Marttis-Bardeens theory, as did by Owen and Scalapino ( Phys. Rev. Lett. 28: 1559, 1972). In this paper, we derive the expressions of d sigma/dn(qp) for both thermal and excess quasiparticles and show that they are approximately equal within the temperature and frequency range MKIDs operate. This equivalence validates the method of using bath temperature sweep to calibrate the responsivity of MKIDs. Response curves of an antenna-coupled submm MKID to bath temperature sweep and blackbody source temperature sweep are compared which demonstrates this equivalence.