Localized and Synchronous Measurement of Second and Third Order Nonlinearity in Superconductive Transmission Line Resonators

A method to locally observe the generation of nonlinearity of multiple orders inside the passband of Tl$_{2}$Ba$_{2}$CaCu$_{2}$O$_{8}$ and YBa$_{2}$Cu$_{3}$O$_{7}$ superconducting resonators has been developed. Nonlinear emission is stimulated locally in the vicinity of the probe by off-resonance, low frequency signals. By mixing of two very low frequency local currents and of one current in the passband, 2$^{nd}$ and 3$^{rd}$ order intermodulation distortion (IMD) occurs around resonance, permitting the determination of the currents associated with each order of nonlinearity. Synchronous measurement of different orders eliminates the ambiguity of emissions occurring on different time scales and at different skin depths. Comparison of synchronous 2$^{nd}$ and 3$^{rd}$ order IMD characterizes the extent of time reversal symmetry breaking (TRSB) in the superconductor. This technique is being used to perform spatially resolved TRSB studies, to examine the effect of doping on nonlinearity in Tl$_{2}$Ba$_{2}$CaCu$_{2}$O$_{8}$, and to search for a correlation between lithographic edge quality and 2$^{nd}$ and 3$^{rd}$ order nonlinearity.