Mathematical modelling of the impedance of a Josephson junction noise thermometer

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Abstract
Recent experimental work on noise thermometers consisting of a resistively shunted superconducting loop containing a Josephson junction (a resistive SQUID), has shown some novel behavior of the SQUID dc impedance as a function of rf power. We present a mathematical analysis of the intrinsic behavior of a resistive SQUID in the limit of negligible noise and negligible feedback to the rf circuit. A nonlinear, first-order differential equation is thought to be a reasonable descriptor of this system. Because the radio frequency driving the SQUID is much larger than the Josephson frequency, we are able to obtain a pair of equations in which no rf oscillating terms appear, and which are amenable to numerical solution. The dc impedance calculated from these equations has several, but not all, of the experimentally observed features.