# ac properties of short Josephson weak links

## TL;DR

This paper investigates the admittance properties of short Josephson weak links, analyzing their frequency response and absorption characteristics using both Ginzburg-Landau and microscopic Green's function approaches.

## Contribution

It provides a comprehensive theoretical analysis of the admittance of two types of Josephson weak links, including formulas valid at various frequencies, temperatures, and phase differences.

## Key findings

- Impedance $Z(\Omega)$ has a maximum at a certain frequency.
- Absorption is absent below a threshold frequency related to the energy gap.
- Maximum admittance occurs near the critical temperature $T_c$.

## Abstract

The admittance of two types of Josephson weak links is calculated, i.e., of a one-dimensional superconducting wire with a local suppression of the order parameter, and the second is a short S-c-S structure, where S denotes a superconductor and c---a constriction. The systems of the first type are analyzed on the basis of time-dependent Ginzburg-Landau equations. We show that the impedance $Z(\Omega)$ has a maximum as a function of the frequency $\Omega$, and the electric field $E_{\Omega}$ is determined by two gauge-invariant quantities---the condensate momentum $Q_{\Omega}$ and the potential $\mu$ related to charge imbalance. The structures of the second type are studied on the basis of microscopic equations for quasiclassical Green's functions in the Keldysh technique. For short S-c-S contacts (the Thouless energy ${E_{\text{Th}} = D/L^{2} \gg \Delta}$) we present a formula for admittance $Y$ valid at frequencies $\Omega$ and temperatures $T$ less than the Thouless energy but arbitrary with respect to the energy gap $\Delta$. It is shown that, at low temperatures, the absorption is absent [${\mathrm{Re}(Y) = 0}$] if the frequency does not exceed the energy gap in the center of the constriction (${\Omega < \Delta \cos \varphi_{0}}$, where $2 \varphi_{0}$ is the phase difference between the S reservoirs). The absorption gradually increases with increasing the difference ${(\Omega - \Delta \cos \varphi_{0})}$ if $2 \varphi_{0}$ is less than the phase difference $2 \varphi_{\text{c}}$ corresponding to the critical Josephson current. In the interval ${2 \varphi_{\text{c}} < 2 \varphi_{0} < \pi}$, the absorption has a maximum. This interval of the phase difference is achievable in phase-biased Josephson junctions. Close to $T_{\text{c}}$ the admittance has a maximum at low $\Omega$ which is described by an analytical formula.