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SIAM Journal on Mathematical Analysis

Nonlocal Partial Differential Equations and Quantum Optics: Bound States and Resonances

Erik Orvehed Hiltunen, Joseph Kraisler, John C. Schotland, Michael I. Weinstein

Published Jun 3, 2024• 3 citations
Publisher

Abstract

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.

DOI
10.1137/23m158142x

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Quarterly of Applied Mathematics•Sep 26, 2022•1 citation
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.
We consider the quantum optics of a single photon interacting with a system of two level atoms. This leads to the study of a nonlinear eigenproblem for a system of nonlocal partial differential equations. Two classes of solutions to these equations are studied. Bound states correspond to negative eigenvalues and resonances to eigenvalues with positive real parts. 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 the eigenproblem for atomic models with small high contrast inclusions. In this setting, we have derived asymptotic formulas for the eigenvalues. Our results are illustrated with numerical computations.
We study a one-dimensional single-excitation model of a scalar photon field coupled to a continuum of two-level atoms with quadratic density. The resulting matrix Hamiltonian has an unbounded spatially varying coupling and is not semibounded. We prove that it is essentially self-adjoint and determine its spectrum explicitly. The continuous spectrum is the half-line (−∞,Ω](-\infty,Ω](−∞,Ω], where ΩΩΩ is the atomic resonance frequency, while the spectrum above ΩΩΩ consists of a strictly increasing sequence of simple eigenvalues. These eigenvalues are characterized by the interlaced zeros of the Airy function and its derivative and satisfy \begin{align*} ω_n = \left(\frac{3πg n}{4}\right)^{1/2} + \fracΩ{4} + O(n^{-1/2}). \end{align*} The analysis is based on a nonlinear spectral reduction that yields a confining massless relativistic oscillator above ΩΩΩ and an inverted oscillator below ΩΩΩ.
Two-photon bound states in the continuum provide a stringent setting in which quantum interference must suppress radiative decay despite photon--photon interactions. Here, we prove a no-go theorem showing that a single nonlinear mode linearly coupled to a noninteracting bosonic continuum cannot support an interaction-active two-photon bound state in the continuum, provided the local spectral density vanishes only at isolated energies. Apart from this regularity condition, the result is independent of the continuum dispersion and the frequency-dependent coupling. Although such an exact bound state is forbidden, long-lived two-photon quasi-bound states remain possible. For a giant Kerr cavity nonlocally coupled to a waveguide, we derive the asymptotic scaling of the two-photon decay rate with the coupling-point separation from weak nonlinearity to the hard-core limit. We further identify a regime in which the two-photon resonance is substantially longer lived than the single-photon excitation. Our work establishes the limits of exact multiphoton confinement while opening a route to engineering long-lived interacting few-photon states.
The Hermitian properties of the Hamiltonian are related not only to the operator itself but also to the functions on which it acts. Hermitian Hamiltonians operate on functions in the L 2 Hilbert space which correspond to boundary conditions which vanish at infinity. In this chapter, in order to move into the non-Hermitian domain, we will impose on the solutions to the time-independent Schrödinger equation (TISE) different boundary conditions which lead to solutions which can be associated with different types of the complex poles of the scattering matrix. These solutions will contain information which was not available within the scope of functions in L 2 . By imposing outgoing boundary conditions on the eigenfunctions of the timeindependent Hamiltonian complex eigenvalues, E res = ε − (i/2)Г, are obtained. These complex energies are associated with decaying resonance states which were discussed in the previous two chapters. The bound states (if they exist) appear as real eigenvalues since they result from exactly such outgoing boundary conditions which appear under the threshold energy. When incoming boundary conditions are imposed two kind of solution are obtained. One type of solution is the complex conjugates of the decay resonance solutions mentioned above. In scattering theory text books (see Taylor for example) the physical resonance solutions are associated with the poles of the scattering matrix which are embedded in the lower half of the complex energy plane. However, in nuclear physics the complex poles embedded in the upper half of the complex energy plane, so-called virtual states , are denoted as capture resonances .
[ANGLÈS] In this thesis we tried to investigate the non-linear effects introduced by a generalized interaction of a two-level atom coupled to two photons. Scattering matrices were used to calculate the non-linear effects introduced and the dependence of outgoing frequencies of quantum light on these effects was found. It was seen as quite opposed to typical classical theories, that the non-linear susceptibility for an atom dictates very precise output frequencies from it. As a next step this work probes into quantum computing in which the two-photon bound state problem was mathematically established to show that a single two-level atom cannot be used in realization of non-linear logical operations in quantum computations. In the end a possible intuitive solution, involving dissimilar decay rates in three level systems, was thought of and shown that it can be effectively applied to quantum logic.
We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.