Josephson's effect in the Schwarzschild background

TL;DR

This work presents a covariant framework for Josephson physics in static curved spacetimes, focusing on Schwarzschild geometry. By identifying two gauge-invariant structures—the condensate momentum $p_$ and the conserved current $j^$—and exploiting the timelike Killing field and Tolman–Ehrenfest redshift, the authors derive a redshifted AC Josephson law, a one-power lapse scaling for the DC critical current, and a gravity-insensitive DC interference pattern in a vertical SQUID to linear order. RF driving from infinity reveals that step positions are preserved in asymptotic variables while propagation phases translate lobes, separating redshift effects from geometric phases. The DC transport analysis via a covariant GL theory shows that $I_{c,infty}\,\simeq\,\alpha\,I_c^{\rm proper}$ and $P_infty\,=\,\alpha^2\,P^{\rm proper}$, with near-horizon and weak-field limits explored and extensions to rotating or analogue platforms discussed. Overall, the paper delivers an analytic, gauge- and coordinate-invariant dictionary linking local Josephson microphysics to asymptotic measurements in curved backgrounds, applicable from weak fields to near-horizon regimes.

Abstract

We develop a fully covariant, analytic framework for Josephson phenomena in static curved spacetimes and specialize it to the Schwarzschild exterior. The formulation rests on two invariant elements: the gauge-invariant condensate momentum that governs phase dynamics and the conserved current whose hypersurface flux encodes transport for an observer at infinity. Using the timelike Killing field to relate proper and asymptotic quantities, we derive a redshifted AC Josephson law in which the asymptotic phase-evolution rate is proportional to the difference of redshifted voltage drops, i.e. to $V_i^\infty \equiv α_i V_i^{\rm proper}$; equivalently, it depends on $α_i V_i^{\rm proper}$ for local control. Under RF drive specified at infinity, the Shapiro-step loci are invariant (expressed in asymptotic voltages) while propagation phases set any apparent lobe translation. For DC transport, a short-junction solution on a static slice yields the proper current-phase relation; mapping to asymptotic observables gives a single-power redshift scaling of critical currents, $I_{c,\infty}\propto αI_c^{\rm proper}$, whereas power scales as $P_\infty\propto α^2 P_{\rm proper}$. In a "vertical" dc-SQUID with junctions at different radii, gravity does not shift the DC interference pattern at linear order; it produces a small envelope deformation and an amplitude rescaling. Gravity does not alter the local Josephson microphysics; it reshapes the clocks and energy accounting that define measurements at infinity. The resulting predictions are gauge- and coordinate-invariant, operationally stated in terms of an experimenter who can control (proper vs. asymptotic bias), and remain analytic from the weak-field regime to the near-horizon limit.

Figures (7)

• Figure 1: AC redshift map. Josephson frequency observed at infinity, $\omega_{J\infty}$, versus radius $r/r_s$ under two bias conventions: with a fixed local (proper) bias, $\omega_{J\infty}\propto \alpha(r)=\sqrt{1-r_s/r}$ and is redshift-suppressed near the horizon; with a fixed asymptotic bias, $\omega_{J\infty}$ is independent of $\alpha$ and remains constant.
• Figure 2: Shapiro steps in asymptotic variables. Time-averaged Josephson current $\langle I\rangle/I_c$ versus asymptotic dc bias $V_\infty/(\hbar\Omega_\infty/2e)$. With both bias and drive defined at infinity, the step loci sit at integers $V_\infty = n \hbar\Omega_\infty/2e$ (vertical positions on the x-axis), independent of gravitational potential. The step heights are modulated by the propagation phase $\psi$ through Bessel coefficients $J_n(a)$ and the factor $\cos(n\psi)$; two choices of $\psi$ illustrate that only heights change, not positions.
• Figure 3: Lapse–radius profile and near-horizon scalings. In Schwarzschild, $\alpha(r)=\sqrt{1-r_s/r}$ governs the redshift. The DC critical current observed at infinity scales with a single power, $I_{c,\infty}/I_c^{\rm proper}=\alpha(r)$, while the power scales quadratically, $P_\infty/P_{\rm proper}=\alpha(r)^2$.
• Figure 4: Left panel: Near-horizon behavior. The Shapiro phase grows logarithmically as the horizon is approached: $\psi_{\rm Shapiro}(r)\propto -\ln(r-r_s)$ when plotted against $\ln[(r-r_s)/r_s]$ (solid). On the same panel, the DC critical current measured at infinity vanishes with a single power of the lapse, $I_{c,\infty}/I_c^{\rm proper}=\alpha(r)$ (dashed). The top axis shows the corresponding proper distance $\rho/r_s$ from the horizon. Right panel: Weak-field linearization check. Exact single-$\alpha$ scaling $\alpha(r)=\sqrt{1-r_s/r}$ (solid) against its first-order expansion $\alpha_{\rm lin}(r)=1-\frac{r_s}{2r}$ (dashed) over $3\le r/r_s\le 20$. The dotted curve shows the relative error $(\alpha_{\rm lin}-\alpha)/\alpha$ in percent.
• Figure 5: Vertical SQUID interference envelopes. DC pattern (solid) computed from two junctions at different radii with lapses $\alpha_1=0.72$ and $\alpha_2=0.85$ shows lobe centers at $\Phi_{\rm ext}/\Phi_0 \in \mathbb{Z}$ with lifted minima due to the asymmetry but no shift in the centers. Under RF drive, the $n=1$ Shapiro-step envelope (dashed) is translated by a propagation-phase difference $\Delta\psi=0.70\ \mathrm{rad}$; the translation is set by $\Delta\psi$, not by $\Delta\alpha$.