Nonlinear impedance of a microwave-driven Josephson junction with noise

Abstract

The nonlinear impedance of a point Josephson junction is calculated under various conditions for the resistively shunted junction model in the presence of noise. The calculation proceeds by solving the Langevin equation for the mechanical problem of a Brownian particle in a tilted cosine potential in the presence of a strong ac force ignoring inertial effects. The exact solution of the infinite hierarchy of equations for the moments ͑expectation values of the Fourier components of the phase angle͒, which describe the dynamics of the junction, is expressed in terms of a matrix continued fraction. This solution allows one to evaluate the nonlinear response of the junction ͑nonlinear microwave impedance, for example͒ to an ac microwave current of arbitrary amplitude. Strong nonlinear effects in the resistance and the reactance are observed for large ac currents as is demonstrated by plotting the nonlinear response characteristics as a function of the model parameters. For weak ac currents and low noise strengths, our results agree closely with previously available linear response and nonlinear response noiseless solutions, respectively. Applications of the model to the interpretation of recent experimental data found in the literature for the nonlinear behavior of microwave impedance of superconducting weak links are discussed.

Quantum noise in a current-biased Josephson junction

“…This assumption is employed in almost all papers dealing with noise influence on Shapiro steps [13], on the impedance of microwave driven junctions [14,15] and other phenomena in Josephson junctions (more references are given in [16,17]). There are only few exceptions.…”

Noise-Assisted Microwave Up-conversion by Vortices in Thin-Film Superconductors with a dc-Biased Washboard Pinning Potential

“…Similarly to the problem of resistively shunted non-linear Josephson junction [22], in our case finite temperature overcritical biases can also lead to a sign change in the real part of the impedance. In the previous communication [21] it has been shown that such an anomalous impedance behavior is more pronounced for low temperatures and small ac amplitudes and is challenging with regard to the development of parametric amplifiers and improving efficiency of down converters operating with Abrikosov vortices.…”

Microwave Absorption by Vortices in Superconductors with a Washboard Pinning Potential

“…Whereas the general exact solution of the problem [25,27] has been obtained for non-zero temperature in terms of a matrix continued fraction [52], here we treat the problem analytically in terms of only elementary functions which allow a more intuitive description of the main effects. Solving the equation of motion for a vortex at T = 0, j 0 < j c , and j 1 → 0 in the general case, we also consider some important limiting cases of isotropic vortex viscosity and zero Hall constant provided it substantially helps us to elucidate the physical picture.…”