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arXiv:2608.23535

Bound States in Exactly Solvable Polaritonic Models

Sam Johar, Ryogo Katahira, Joseph Kraisler

math-phmath.SPquant-phSubmitted Aug 24, 2026• Announced Aug 25, 2026
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Abstract

We study a one-dimensional scalar model for an effective-mass quasiparticle field coupled to a continuum of two-level atoms with at most one excitation present. We prove a general theorem on the existence of bound states under sign-definite spatially localized perturbations of constant atomic density and analyze the bound states for several exactly solvable examples.

Comments
15 pages, 4 figures

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We study a one-dimensional scalar model for an effective-mass quasiparticle field coupled to a continuum of two-level atoms with at most one excitation present. We prove a general theorem on the existence of bound states under sign-definite spatially localized perturbations of constant atomic density and analyze the bound states for several exactly solvable examples.
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We describe a new type of spatially periodic structure (lattice models): a polaritonic crystal formed by a two-dimensional lattice of trapped two-level atoms interacting with the electromagnetic field in a cavity (or in a one-dimensional array of tunnelling-coupled microcavities), which allows polaritons to be fully localised. Using a one-dimensional polaritonic crystal as an example, we analyse conditions for quantum degeneracy of a lower-polariton gas and those for quantum optical information recording and storage.
Quantum electrodynamics of two identical two-level atoms placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied within the framework of the dipole resonance (rotating-wave) approximation. For finite interatomic distances a polariton-atom bound state lying within the polaritonic gap is shown to be split into a doublet due to an effective atom-atom interaction. Evaluating the splitting as a function of the interatomic separation, we estimate a width of a polariton-impurity band when a system of spatially correlated atoms is present. The results obtained for the case of two atoms are then generalized to the case of an atomic chain (a one-dimensional atomic lattice). The equations describing a propagation of a band polariton are derived and the expressions for the band polariton spectrum and its effective mass are obtained. The model under consideration has quite a general structure and is applied to the case of photonic-band-gap materials as well. Numerical calculations are presented for SiC semiconductors.
Quantum electrodynamics of a single two-level atom placed within a frequency dispersive medium whose polariton spectrum has a gap due to photon coupling to a medium excitation (exciton, optical phonon, etc.) is studied. Despite a nonlocal effective polariton-polariton coupling, the system is proven to be integrable and is diagonalized exactly by means of the Bethe-ansatz technique. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits some unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons propagate within the gap, while a soliton with an odd number of polaritons is pinned to the atom and forms a many-polariton-atom bound state, in which the radiation and the medium polarization are localized in the vicinity of the atom. The model under consideration has a quite general structure and can also be applied to the case of photonic-band-gap materials. \textcopyright{} 1996 The American Physical Society.
A system of polaritons interacting with a two-level atom placed within a frequency dispersive medium is proven to be integrable and diagonalized exactly by the Bethe ansatz method, despite a nonlocal effective polariton-polariton coupling. Its spectrum consists of bound many-polariton complexes (quantum solitons) and exhibits unusual features due to the existence of the polaritonic gap. Only solitons containing an even number of polaritons (``even'' solitons) propagate within the gap, while an ``odd'' soliton is pinned to the atom and forms a many-polariton--atom bound state.
Abstract Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann’s Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier V ( x ) = − V 0 [ exp ( 2 ∣ x ∣ / a ) − 1 ] which for E < V 0 has a continuum of non-square-integrable, definite-parity, degenerate states. In this continuum, we show a surprising presence of discrete energy, square-integrable, definite-parity, non-degenerate states. For E > V 0 , there is again a continuum of complex scattering solutions ψ ( x ) whose real and imaginary parts though solutions of Schrödinger equation yet their parities cannot be ascertained as Cψ ( x ) is also a solution where C is an arbitrary complex non-real number.