Three-slit interference with a which-path memory ancilla: A bright-dark state formulation

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In this paper, we study the effect of a which-path memory ancilla on the three-slit interference pattern within the framework of the bright–dark state description [Phys. Rev. Lett. 134, 133603 (2025)]. Whereas two slits give one bright mode, a photonic mode that couples to the detector atoms, and one dark mode, which does not couple to the detector atoms, three slits lead to one detector-coupled bright mode and a two-dimensional dark-mode subspace in the three-dimensional path space. We first discuss the classical three-slit interference pattern in terms of probability leakage into the dark subspace. We then study the von Neumann entropy S_D of the dark subspace and the coherence measures of the reduced photonic state obtained by tracing out the memory: dark-sector coherence C_D and bright–dark coherence C_{BD}. These quantities bring out the internal quantum structure of the two-dimensional dark subspace. We show that, whereas C_D is not by itself a basis-independent physical observable, C_{BD} is basis-invariant. We establish that, for two paths, C_{BD} is fixed entirely by the populations and the single pairwise coherence, whereas for multiple paths, M\geq 3, it acquires a genuinely multipath contribution generated by asymmetry among the pairwise which-path overlaps. Finally, we distinguish the part of the path-coherence loss that is recoverable through a suitable measurement of the memory from the irreducible coherence deficit imposed by an uncontrolled environment.

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