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arXiv:2407.14109 · SIAM Journal on Applied Mathematics

Dynamic One-Photon Localization in a Discrete Model of Quantum Optics

Joseph Kraisler, Jeffrey Schenker, John C. Schotland

Published May 4, 2026• 0 citations
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Abstract

Abstract. We consider a recently proposed model for the propagation of one-photon states in a random medium of two-level atoms. We demonstrate the existence of Anderson localization of single-photon states in an energy band centered at the resonant energy of the atoms. Additionally, for a bosonic model of the atoms the results can be extended to multi-photon states.

DOI
10.1137/21m1428315

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Abstract. We consider a recently proposed model for the propagation of one-photon states in a random medium of two-level atoms. We demonstrate the existence of Anderson localization of single-photon states in an energy band centered at the resonant energy of the atoms. Additionally, for a bosonic model of the atoms the results can be extended to multi-photon states.
The propagation of an off-resonant, single-photon pulse in and into a medium of two-level atoms is considered. When the pulse is launched from within the medium, there are two propagation speeds, neither of which is the normal group velocity. For densities as small as 1011{}^{11}11 atoms/cm3{}^{3}3 and detunings of order 1011{}^{11}11 s${}^{\ensuremath{-}1}$, both propagation speeds approach one half the speed of light in vacuum. Moreover, rather remarkably, there are Rabi oscillations between the field and atomic excitation, even for this single-photon pulse. In contrast, for a pulse sent into the medium, the atom-field system remains adiabatically in a dressed state that propagates with the normal group velocity. In the limit that the index of refraction is approximately equal to unity, we obtain the Fresnel equations for the reflection and transmission coefficients based on this microscopic model. The transmission coefficient differs from the conventional result owing to the fact that we quantize the field in free space.
Abstract. We consider the quantum optics of a single photon interacting with a system of two-level atoms. The wave properties of this interacting system are determined by the spectral properties of a matrix Hamiltonian, involving a nonlocal partial differential operator, acting on photonic and atomic degrees of freedom. We study the spectral problem via a reduction to a spectral problem for a scalar nonlocal operator, which depends nonlinearly on the spectral parameter. We investigate two classes of solutions: Bound states are solutions that decay at infinity, while resonance states have locally finite energy and satisfy a non–self-adjoint outgoing radiation condition at infinity. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered these problems for atomic models with small, high-contrast inclusions. In this setting, we have derived asymptotic formulas for the resonances. Our results are illustrated with numerical computations.
Interplay between the effects of coherent radiation and localization of light is analysed. A system of two–level atoms is placed in a medium interacting with electromagnetic field. The matter–light interaction can result in the appearance of a band gap in the spectrum of polariton states. If an atom with a resonance frequency inside the gap is incorporated into such a medium, the atomic spontaneous emission is suppressed, which is termed the localization of light. However, a system of resonance atoms inside the gap can radiate due to their coherent interactions. The peculiarity of the coherent radiation by a system of atoms, under the localization of light for a single atom, is studied. 1 1 Localization of Light In some cases a medium may possess an electromagnetic band gap where a severe depressions of the photon density of states occurs. One such case
A model of the propagation of a quantized electromagnetic field in a one-dimensional photonic crystal that contains two-level atoms that interact with the field is proposed. The model separately considers the interaction of the quantized radiation with the quantum system and with the photonic crystal. The use of the model is exemplified by its application to the study of the decay of excited states of one- and two-level atoms placed in the photonic crystal. The possibility of the transformation of the front of a single photon by a finite dielectric is demonstrated.