Eur. Phys. J. Plus (2026) 141: 230
https://doi.org/10.1140/epjp/s13360-026-07345-w

Regular Article

Antisymmetric dark state (II): effective potential, functional curvature, and refractive lensing

Independent Researchers, Murcia, Spain

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Received: 29 August 2025
Revised: 20 December 2025
Accepted: 20 January 2026
Published online: 4 March 2026

Abstract

We investigate the effective potential, spatial curvature, and refractive-lensing signatures generated by the antisymmetric dark-state (ADS) configuration introduced in our previous work. Building on insights from analogue-gravity models and analogue-horizon experiments in Bose–Einstein condensates, we formulate a purely quantum, spatial-only (absolute-time) framework in which geometry emerges from the internal coherence of the dark-state wavefunction, ${\psi }_{\text{dark}}$. In this setting, a single radial metric factor $A(r)\equiv g_{\mathit{rr}}^{(Q)}(r)$ is determined by the functional structure of ${\psi }_{\text{dark}}$, without invoking relativistic time warping or classical sources. The resulting geometric contribution to the specific effective potential, $U_Q(r)$, is regular and confining, enabling bound trajectories even at vanishing angular momentum. The Ricci scalar–computed on both the two-dimensional polar slice and the three-dimensional spherical geometry–is finite everywhere and tends to zero at large r. We further map A(r) to an effective refractive-index profile to analyse lensing phenomena; the predicted deflections are consistent with geometric confinement. All symbolic derivations, dynamical simulations, and figures are provided in the Supplementary material.