Dynamics of Josephson junctions beyond the tunneling limit

TL;DR

The paper addresses how Josephson junction dynamics extend beyond the tunneling limit by deriving a generalized RCSJ model from a microscopic Keldysh action that retains all orders in tunneling. The authors obtain a time-nonlocal Langevin equation $\frac{\hbar C}{2e}\ddot{\varphi} + \mathcal{I}([\varphi];t) + \xi = I_b$, with $\mathcal{I}$ containing both supercurrent and nonlinear dissipative current, and then perform an adiabatic reduction to a local form $\mathcal{I}(\varphi,\dot{\varphi})$ while treating the Langevin current fluctuations $\xi$ as Gaussian under suitable conditions. A central result is a nonlinear fluctuation-dissipation theorem, $K_d(V) \simeq \frac{2T}{V} I_d(V)$ (and its Fourier components $\mathrm{Re}[\mathcal{K}_n(V)] = \frac{2T}{V}\mathrm{Re}[\mathcal{I}_n(V)]$), linking dissipative currents to current fluctuations beyond linear response. This framework provides a microscopic basis for nonreciprocal Josephson behavior (e.g., the diode effect) and accommodates higher harmonics and multiple Andreev reflections, with broad relevance to fast, transparent junctions and superconducting diode applications.

Abstract

The dynamics of the superconducting phase difference across a Josephson junction can be described within the resistively and capacitively shunted Josephson junction (RCSJ) model. Microscopic derivations of this model traditionally rely on the tunneling limit. Here, we present a derivation of a generalized version of the RCSJ model, which accounts for dissipative currents with nonlinear current-voltage characteristics as well as supercurrents with arbitrary current-phase relations. This requires a generalized fluctuation-dissipation theorem to describe the Langevin current, which we deduce along the lines of fluctuation theorems for mesoscopic conductors. Our work is motivated in particular by recent theories of the Josephson diode effect, which is not captured within the RCSJ model in the tunneling limit.

Figures (1)

• Figure 1: Nonlinear fluctuation-dissipation relation in junctions without [(a,c); black frame] and with magnetic adatom [(b,d), green frame]. We assume that the Kondo temperature is small compared to $\Delta$, so that Kondo correlations can be neglected. Panels (a) and (b) illustrate that the nonlinear features in $K_0(V)$ (solid line), which are especially rich in the presence of a magnetic adatom, are well reproduced by $2TI_0(V)/V$ (dashed line) over a voltage range that far transcends the linear regime of the current $I_0(V)$ (red, dotted line). Only at voltages of the order of half the temperature (here: $T = 0.5 \Delta$) the quantitative agreement breaks down, while qualitative features are still reproduced. This behavior can be systematically studied by considering the relative deviation $\abs{K_0(V)-2TI_0(V)/V}/K_0(V)$ for a variety of temperatures as shown in panels (c) and (d). One observes that the voltage at which the nonlinear fluctuation-dissipation relation is violated grows with increasing temperature. Model: The plain tunneling junction (a,c) is modeled by $h_{L,R} = \xi + \Delta \tau_1$, while the magnetic adatom junction (b,d) has $h_L = \xi + \Delta \tau_1$ and $h_R = h_L + (w\tau_3 - JS)\delta(\mathbf{r})$ with $\pi \nu_0 w = 1 = \pi \nu_0 J$ ($\nu_0$ is the normal state local density of states at the Fermi level). Parameters common to all panels: $\pi \nu_0 \vartheta = 0.5$, $\eta = 0.01\Delta$. This corresponds to normal state transparencies $\theta = 0.64$ [(a,b)] and $\theta=0.41$ [(c,d)]. We note that the peaks at $eV = 0$ in (c,d) are numerical artifacts originating from the truncation of the Floquet Green function approach at vanishing frequency (i.e., voltage). In fact, the equilibrium fluctuation-dissipation relation guarantees that the noise power exactly equals $2T \partial_V I_0(V)$ here.