Nonlinear Response of Diffusive Superconductors to $ac$-electromagnetic Fields

TL;DR

This paper provides a comprehensive theoretical analysis of the nonlinear electromagnetic response of diffusive BCS superconductors, including harmonic generation and the Eliashberg effect, relevant for pump-probe experiments.

Contribution

It develops a generalized Usadel equation approach to analyze nonlinear responses up to third order, deriving expressions for Green's functions and physical observables in non-equilibrium superconductors.

Findings

Frequency and temperature dependence of the Eliashberg effect.

Behavior of second harmonic of the order parameter under irradiation.

Connection between down-conversion intensity and third harmonic generation.

Abstract

Motivated by the recent experimental progress in studying conventional and unconventional superconductors in a pump-probe setup, we perform a comprehensive theoretical analysis of the nonlinear response of a diffusive BCS conventional superconductor to the action of an alternating electromagnetic field using a generalized Usadel equation. We analyze the response up to the second order of the perturbation in the amplitude of the vector potential $\vec{A}$, the superconducting order parameter $\Delta$ and in the third order for the current $\vec{j}$. On the basis of this approach, we derive general expressions for the retarded (advanced) Green's functions, as well as the Keldysh function for an arbitrary number of harmonics of the incident field. Most importantly, we analyze the set of physical observables in a non-equilibrium superconductor, such as frequency and the temperature dependencies of the zero harmonic $\delta( \Delta)_0$ (Eliashberg effect), the second harmonic $\delta( \Delta)_{2\Omega}$, as well as the third harmonic for the electric current $j(3\Omega)$ under the action of a monochromatic irradiation. For the same set of parameters, we also analyze the behavior of the reflectivity and the down-conversion intensity of a thin superconducting film, discussed recently in the context of parametric amplification of superconductivity. We derive these quantities microscopically and show the connection of the down-conversion intensity to the third harmonic generation currents induced by the amplitude mode and the direct action of the electric field on the charge carriers.