Near extremal RN–AdS control of holographic Josephson transport

Abstract

We formulate a holographic weak-link construction in which Josephson transport is controlled by a near-extremal Reissner–Nordstrom–AdS background while the superconducting junction is carried by a dynamically distinct Abelian-Higgs probe sector. The RN–AdS geometry is supported by a homogeneous background Maxwell field \({\mathcal {B}}_\mu ,\) whereas the junction gauge field \(A_\mu \) and charged scalar \(\Psi \) enter the action with an overall factor \(\lambda ^2.\) After deriving the field equations, the limit \(\lambda \rightarrow 0\) leaves the nonlinear Maxwell–scalar equations intact on a fixed RN–AdS geometry while their stress tensor affects the metric only at \({\mathcal {O}}(\lambda ^2).\) This makes the fixed-background approximation a controlled leading-order truncation even for a finite spatial modulation of the probe chemical potential. The Josephson phase difference is defined from the gauge-invariant boundary data of the probe condensate, not from the RN charge itself. We identify the current-phase relation, critical current, coherence length, and small-phase stiffness as the principal transport quantities. In the SNS regime, the critical current and midpoint condensate probe the same spatial proximity scale, while higher harmonics diagnose enhanced transparency or departure from the opaque weak-link limit. The near-extremal RN–AdS throat introduces a radial scale distinct from the spatial proximity length. We derive its domain of validity and distinguish the AdS\(_2\) response coefficient from the Josephson current obtained after ultraviolet matching. For the canonical scalar mass \(m^2L^2=-2,\) the homogeneous normal probe throat violates the AdS\(_2\) stability bound at extremality for every real probe electric field. A real-index power law therefore requires a different stable reference sector and a separate derivation of its overlap with the junction response. The analysis provides a probe formulation, analytic proximity references, and infrared constraints; it does not constitute a numerical solution of the nonlinear RN–AdS junction.