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arXiv:2112.05512 · Physical Review Letters

Bright and Dark States of Light: The Quantum Origin of Classical Interference

Celso J. Villas-Bôas, C. E. Máximo, Paulo J. Paulino, Romain Bachelard, Gerhard Rempe

Published Apr 3, 2025• 19 citations
arXivHTMLPDF

Abstract

Classical theory asserts that several electromagnetic waves cannot interact with matter if they interfere destructively to zero, whereas quantum mechanics predicts a nontrivial light-matter dynamics even when the average electric field vanishes. Here, we show that in quantum optics, classical interference emerges from collective bright and dark states of light, i.e., particular cases of two-mode binomial states, which are entangled superpositions of multimode photon-number states. This makes it possible to explain wave interference using the particle description of light and the superposition principle for linear systems only. It also sheds new light on an old debate concerning the origin of complementarity.

DOI
10.1103/physrevlett.134.133603

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Classical theory asserts that several electromagnetic waves cannot interact with matter if they interfere destructively to zero, whereas quantum mechanics predicts a nontrivial light-matter dynamics even when the average electric field vanishes. Here, we show that in quantum optics, classical interference emerges from collective bright and dark states of light, i.e., particular cases of two-mode binomial states, which are entangled superpositions of multimode photon-number states. This makes it possible to explain wave interference using the particle description of light and the superposition principle for linear systems only. It also sheds new light on an old debate concerning the origin of complementarity.
Interference phenomena, such as those observed in Young’s double-slit experiment, are foundational to quantum mechanics, yet their interpretation continues to spark debate. Villas-Boas et al. [Phys. Rev. Lett. 134, 133603 (2025)] propose a quantum-optical framework attributing non-detection in regions of destructive interference to photons occupying ``perfectly dark'' states, which they claim do not interact with a two-level atom sensor due to vanishing coupling in the Jaynes-Cummings Hamiltonian. We argue that this interpretation is fundamentally flawed. Dark states, such as the out-of-phase coherent state ∣α,−α⟩ |\alpha, -\alpha\rangle ∣α,−α⟩, are readily detectable in alternative experimental setups, including dispersive coupling in cavity QED and photon-counting detectors like photomultiplier tubes or avalanche photodiodes, which are sensitive to the photon number operator. This detectability undermines the assertion that dark states are the intrinsic cause of non-detection in destructive interference regions. Instead, the quantum mechanical superposition principle, combined with the Born rule, fully accounts for interference patterns across all detection schemes, as evidenced by well-established experiments such as single-photon double-slit interference, Hong-Ou-Mandel two-photon interference, and cavity QED measurements. By overgeneralizing a detector-specific effect, the dark-state framework introduces an unnecessary and redundant construct, as the standard quantum mechanical formalism already provides a complete and experimentally validated explanation of interference phenomena.
Building upon the recently introduced particle interpretation of the double-slit experiment [Phys. Rev. Lett. 134, 133603 (2025)] which attributes interference phenomena to detector-coupled (bright) and detector-uncoupled (dark) states of light, we develop a continuous-mode extension of the bright- and dark-state framework. This extension addresses a conceptual distinction between interference and diffraction, that is, the transition from a finite set of discrete paths to a continuum of modes. Through the construction of a complete detector-oriented basis for single-slit diffraction, we demonstrate that the observed diffraction pattern arises from projection of the photon state onto a single bright mode by identifying the detectable and undetectable modes, with photons detected at intensity minima having zero probability, as they reside in modes spanning an infinite-dimensional dark subspace. Our approach thus provides a unified particle-based explanation of diffraction that connects quantum and classical wave optics, and reveals distinctive quantum signatures in higher-order correlations.
The physical interaction processes behind the emergence of dark and bright fringes registered by the detectors at the output of optical interferometers is explained. This knowledge should be helpful to interferometrists to make better physical interpretations of their data. The belief in mysterious “interference of single indivisible photon” will disappear once we recognize that the spatial or temporal energy re-distributions are generated by the physical transformation experienced by the detecting molecules drawing energy from all the light beams. The molecules could be photodetectors at the interferometer output, or the beam combining dielectric boundary. The superposition principle (SP), represented by the linear mathematical sum of two or more wave amplitudes, does not represent an observable phenomenon. The superposition effect (SE), represented by the non-linear square modulus of the joint dipolar stimulation of the detectors by all the superposed waves, is observable. We present two laboratory experiments to clarify these points. Both classical beam combiners and quantum detectors are capable of generating superposition fringes of intensity variations. The logic of “quantumness” of light is narrowly relevant only when a quantum detector deciphers the fringes; it is not valid for classical beam combiners. We will also discuss “entanglement” based on these experiments.
With the recently introduced particle interpretation of the double-slit experiment for light fields [Phys. Rev. Lett. 134, 13360 (2025)], all related interference phenomena can be reinterpreted in terms of light particle states that either couple (bright) or do not couple (dark) with detectors. Here, we apply this approach to multimode pulsed lasers, unifying the description of their generation inside optical cavities and propagation outside them, now relying solely on quantum mechanics, i.e., without invoking wave superposition to explain pulse shaping. Specifically, we demonstrate that multimode interference presents a dominance of particle dark states over the bright ones and mode-locked pulsed lasers consist of a continuous photon beam, with photons forming bright states during pulses and dark states in between. Additionally, we analyze mode-locked pulsed lasers, showing that the theoretical bright-to-dark state ratio matches the experimental pulse-to-interval duration ratio.
The interference of the beams of light led to the foundations of wave optics and this motivated founding fathers of quantum mechanics to arrive at the wave equation for quantum particles like electrons. With the advent of quantum theory of light by Planck, Dirac pondered over how the interference would arise with single photons as these fundamental entities could not be broken in two parts like light waves. He introduced the idea of the quantum amplitudes for the paths which needed to be summed up coherently before finding the intensity or the photon detection probability. I will present a very important application of the interference among several paths to understand in clear physical terms the phenomena of super radiance ie the emergence of enhanced radiation from a system initially prepared in a collective state. I will show how how a collective state leads to several paths along which photon can be emitted. A constructive interference of these paths leads to super radiance where as a destructive interference leads to sub radiance.