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arXiv:2510.21636 · Physical Review Letters

Truncated Photon

Isak Cecil Onsager Rukan, Jan Gulla, Johannes Skaar

Published May 18, 2026• 1 citation
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Abstract

An elementary particle such as a photon cannot be cut in two pieces. Still it must be possible to truncate a photon with an optical shutter. The result is neither another photon nor a mix of a photon and a vacuum. Instead it is a superposition and mix of photon numbers up to infinity. This state is rather complicated, but nevertheless locally equivalent to a single photon or vacuum to the left and right, respectively, of a narrow transition region.

DOI
10.1103/94pm-hp34

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An elementary particle such as a photon cannot be cut in two pieces. Still it must be possible to truncate a photon with an optical shutter. The result is neither another photon nor a mix of a photon and a vacuum. Instead it is a superposition and mix of photon numbers up to infinity. This state is rather complicated, but nevertheless locally equivalent to a single photon or vacuum to the left and right, respectively, of a narrow transition region.
We propose a realizable experimental scheme to prepare superposition of the vacuum and one-photon states by truncating an input coherent state. The scheme is based on the quantum scissors device proposed by Pegg, Phillips, and Barnett [Phys. Rev. Lett. 81, 1604 (1998)] and uses photon-counting detectors, a single photon source, and linear optical elements. Realistic features of the photon counting and single-photon generation are taken into account and possible error sources are discussed together with their effect on the fidelity and efficiency of the truncation process. Wigner function and phase distribution of the generated states are given and discussed for the evaluation of the proposed scheme.
A Majorana field cannot be split into two spatial pieces, but a propagating Majorana can be truncated by a time-dependent shutter. We study this truncation for the two chiral components of a massless Majorana field in one spatial dimension, modeling the shutter as a smooth time-dependent SO(2)SO(2)SO(2) rotation of the two components. Although orthogonal at the level of the real field algebra, this rotation does not preserve the positive- and negative-frequency splitting that defines the Fock vacuum, so the in-vacuum becomes a fermionic squeezed state in outgoing modes. At any finite infrared regulator, a single incident Majorana is then a single outgoing quasiparticle on top of this squeezed vacuum. The squeezed background carries a universal soft response: its anomalous Bogoliubov kernel falls as the inverse sum of the two frequencies at low energy, so the anomalous Hilbert--Schmidt norm grows logarithmically with the infrared cutoff, with a coefficient fixed by the asymptotic shutter mismatch. The vacuum overlap exhibits a Fermi-edge type orthogonality exponent which, for the rotation-angle lift ΔθΔθΔθ with ∣Δθ∣<π|Δθ|<π∣Δθ∣<π, is exactly (Δθ/2π)2(Δθ/2π)^2(Δθ/2π)2, in analogy with the Fermi-edge singularity and the Kondo and Luttinger-liquid problems. We prove local equivalence non-perturbatively for the squeezed-vacuum factor under a compact-support hypothesis, and give an exact local pullback identity for the full one-quasiparticle state on the even CAR algebra; combined with wavepacket-localization assumptions, these yield the expected local vacuum and one-Majorana descriptions. We verify profile independence using a smooth switching family and contrast the temporal benchmark with a number-conserving beam-splitter benchmark whose entropy is bounded by log⁡2\log 2log2.
In a recent Physical Review Letter (accepted May 18, 2026), Rukan, Gulla, and Skaar demonstrate that truncating a photon with an optical shutter produces a superposition of photon number states extending to infinity, a result that challenges conventional quantum optical intuition. Here we show that this result is not an anomaly, but a direct and necessary consequence of the Al-Ani Fabric Theory. In this framework, the photon is not a point particle but a stable deformation knot—an intersection of propagating deformation waves in the spacetime fabric (the δ field). We demonstrate that: (1) The photon is a localized knot of the deformation field, not a fundamental particle. (2) The optical shutter acts as a discontinuity in the fabric's boundary conditions, not a measurement apparatus. Therefore, the knot does not collapse (Law 6 of the Al-Ani Mother Law), but scatters. (3) This scattering liberates the deformation field, allowing it to reorganize into all allowed harmonic modes of the new boundary condition. This reorganization is mathematically equivalent to a superposition of all possible photon number states. (4) The "infinite photon number" reported by Rukan et al. is not a paradox; it is the spectral decomposition of the scattered deformation wave packet. We derive the scattering cross-section directly from the mother equation ∂ₜδ = D∇²δ - Γδ + S and show that the resulting photon number distribution matches the "infinite spectrum" reported. This provides the first physical, geometric mechanism for the truncated photon effect, replacing mathematical formalism with causal geometry. This work not only explains the Rukan et al. result but also provides three falsifiable predictions: (1) the scattered state's entanglement entropy should follow the thread coherence model (ℓ_c = 15.3 m, Law 4), (2) a spectral cutoff at f_c = 7.73×10⁻¹⁵ Hz, and (3) independence from shutter closing time for all realistic shutters. Complete theoretical framework: DOI 10.5281/zenodo.20616527 (Al-Ani Mother Law)