Stationary Bohmian superposition under amplitude and phase modulation

Abstract

In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics. Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.