Phase-Quadrature Response of Josephson Circuits to Nonequilibrium Thermal Gradients

This preprint presents a conservative effective field theory (EFT) for superconducting circuits subject to spatially varying nonequilibrium thermal gradients. The manuscript develops a minimal gauge-invariant framework in which the superconducting phase, the electromagnetic gauge field, and a coarse-grained nonequilibrium control scalar are treated as low-energy degrees of freedom. Within this framework, the leading symmetry-allowed coupling between the control-field gradient and the gauge-invariant phase gradient is derived and shown to produce a modified Josephson response. The central result is a predicted phase offset in Josephson weak links and SQUID-type architectures, together with a characteristic phase-quadrature signature that is distinct from ordinary heating-induced critical-current suppression. In particular, the proposed response appears in the cos⁡Δφ quadrature, whereas conventional thermal amplitude effects primarily enter through the sin⁡Δφ channel. The manuscript includes: a minimal gauge-invariant EFT construction for superconducting circuits under nonequilibrium thermal gradients derivation of the modified Josephson current-phase relation with an induced phase offset consistency checks including the uniform-field limit, Ward identities, and stability/positivity bounds order-of-magnitude detectability estimates for SQUID and Josephson-array platforms experimental discrimination strategies, including lock-in detection, gradient reversal, and arm-exchange controls, to distinguish genuine phase offsets from asymmetric heating artifacts This work is intended as a conservative, testable low-energy theory for phase-coherent superconducting circuits, rather than as a microscopic reformulation of superconductivity. The thermal-control scalar used in the analysis is treated as a coarse-grained operational field, with σ=ln⁡(T/T0) introduced only as a practical proxy for device-level estimates. Keywords: superconducting circuits, Josephson effect, SQUID, nonequilibrium thermal gradients, gauge invariance, effective field theory, quantum sensors