Semiclassical Phases of Charged Spin-1/2 Matter-Wave Interferometers in Gravitational-Wave Backgrounds

Abstract A matter wave propagating through curved spacetime accumulates a phase that encodes both geometry and gauge structure. We develop a semiclassical description of charged spin-$\tfrac{1}{2}$ matter-wave interferometry based on a WKB expansion of the general covariant Dirac equation. At leading order, and within the guided-arm approximation, the interferometric phase separates into three additive contributions: a dynamical phase associated with proper-time evolution, a spin phase arising from parallel transport in the local Lorentz frame, and an electromagnetic phase generated by curvature-induced perturbations of a background electromagnetic field. In a freely falling detector frame described by Fermi normal coordinates, the dynamical and spin channels are driven by the local gravitoelectric and gravitomagnetic tidal fields, while the electromagnetic channel follows from solving the detector-frame Maxwell equations for a uniform background magnetic field in the long-wavelength regime. For a weak gravitational wave, all three channels are controlled by the common tidal scale $\ddot h_{+,\times}\sim\Omega_{\rm gw}^2 h_0$, but enter through distinct couplings and geometry-dependent response kernels. Applied to an idealized square Mach--Zehnder interferometer, the dynamical channel reproduces the established tidal matter-wave response, and the spin channel is suppressed by the ratio of the Compton wavelength to the interferometer size. The electromagnetic channel is set by the applied magnetic flux rather than by the particle mass or spin: within the initial-value solution adopted here, its magnitude depends on the orientation of the background field relative to the wave, with the perpendicular configuration carrying an additional factor $c/v$, and the two orientations select different gravitational-wave polarizations.