We calculate the admittance of two types of Josephson weak links—the first is 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 superconducting reservoir and c is a constriction. The systems of the first type are analyzed on the basis of time-dependent Ginzburg-Landau equations derived by Gor'kov and Eliashberg for gapless superconductors with paramagnetic impurities. It is shown that the impedance $Z\left(\mathrm{\Omega }\right)$ has a maximum as a function of the frequency $\mathrm{\Omega }$, and the electric field $E_{\mathrm{\Omega }}$ is determined by two gauge-invariant quantities. One of them is the condensate momentum $Q_{\mathrm{\Omega }}$ and another is a 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 $E_{\text{Th}}=D/L^2\gg \mathrm{\Delta })$, we present a formula for admittance $Y$ valid frequencies $\mathrm{\Omega }$ and temperatures $T$ less than the Thouless energy $E_{\text{Th}}$ $\left(\hslash \mathrm{\Omega },T\ll E_{\text{Th}}\right)$ but arbitrary with respect to the energy gap $\mathrm{\Delta }$. It is shown that, at low temperatures, the absorption is absent $\left[\mathrm{Re}\left(Y\right)=0\right]$ if the frequency does not exceed the energy gap in the center of the constriction $(\mathrm{\Omega }<\mathrm{\Delta }cos{\phi }_0$, where $2{\phi }_0$ is the phase difference between the S reservoirs). The absorption gradually increases with increasing the difference $(\mathrm{\Omega }-\mathrm{\Delta }cos{\phi }_0)$ if $2{\phi }_0$ is less than the phase difference $2{\phi }_{\text{c}}$ corresponding to the critical Josephson current. In the interval $2{\phi }_{\text{c}}<2{\phi }_0<\pi$, 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 $\mathrm{\Omega }$, which is described by an analytical formula.

• Received 9 February 2017

DOI:https://doi.org/10.1103/PhysRevB.95.134518