arXiv subjects

John C. Schotland

Publications and source records attributed to John C. Schotland.

At least 19 recordsLinked to original sources

arXiv2026

Reconstruction methods for inverse scattering problems with phaseless data

We investigate phaseless inverse scattering problem for the Schrödinger equation and develop reconstruction methods based on the inverse Born series (IBS). We consider three types of phaseless data: the far-field total field, the total field and the far-field scattered field. For phaseless total-field data, we extend the IBS framework and analyze its radius of convergence. In the far-field setting, we propose a Fourier-based reconstruction method that exploits the scattering reciprocity between incident and observation directions to recover Fourier coefficients of the scattering potential. For phaseless scattered-field data, we employ a polarization-based approach to recover phase information and enable IBS reconstruction. Numerical experiments are conducted to validate the proposed methods.

math-ph↗

arXiv2026

Inverse Scattering for Single Photons in Quantum Optics

We study inverse problems for a time-harmonic one-photon model describing the interaction of a single photon with a medium of stationary two-level atoms. After time-harmonic reduction, the unknown compactly supported atomic density appears as a frequency-dependent potential in a scattering equation for the half Laplacian. We prove high-frequency uniqueness results for three types of intensity data: source-driven measurements, renormalized far-field intensity measurements, and phaseless far-field measurements obtained from coherent superpositions of incident plane waves. In each case, the corresponding data, given at all sufficiently large frequencies, determine the atomic density uniquely; the source-driven result requires a geometric visibility condition on the source and observation sets.

math.AP↗

arXiv2026

One-photon communication in atomic media

We consider the problem of single-photon transmission through an atomic medium, using quantum channel fidelity to describe the resulting information loss. We find that the normalized fidelity decreases monotonically with coupling strength, establishing a performance bound for quantum communication through such media. Our results hold for several channel types and for deterministic and random media.

quant-ph↗

arXiv2026

Inverse Scattering for Dirac Equations Arising in Waveguide Arrays

We investigate inverse scattering problems for Dirac equations that arise as continuum models of waveguide arrays. We first establish the well-posedness of the forward models. For the associated inverse problems, we develop the inverse Born series and the reduced inverse Born series, providing analysis of convergence and rigorous error estimates. Numerical experiments are presented to validate the proposed algorithms and demonstrate their effectiveness.

math.NA↗

arXiv2026

Quantum field theory and inverse problems: Imaging with Entangled Photons

We consider the quantum field theory for a scalar model of the electromagnetic field interacting with a system of two-level atoms. In this setting, we show that it is possible to uniquely determine the density of atoms from measurements of the source to solution map for a system of nonlocal partial differential equations, which describe the scattering of a two-photon state from the atoms. The required measurements involve correlating the outputs of a point detector with an integrating detector, thereby exploiting information about the entanglement of the photons.

math.AP↗

arXiv2025

Band Structure and Dynamics of Single Photons in Atomic Lattices

We present a framework to investigate the collective properties of atomic lattices in one, two, and three dimensions. We analyze the single-photon band structure and associated atomic decay rates, revealing a fundamental dependence on dimensionality. One- and two-dimensional arrays are shown to be inherently radiative, exhibiting band gaps and decay rates that oscillate between superradiant and subradiant regimes, as a function of lattice spacing. In contrast, three-dimensional lattices are found to be fundamentally non-radiative due to the inhibition of spontaneous emission, with decay only at discrete Bragg resonances. Furthermore, we demonstrate that this structural difference dictates the system dynamics, which crosses over from dissipative decay in lower dimensions to coherent transport in three dimensions. Our results provide insight into cooperative effects in atomic arrays at the single-photon level.

quant-ph↗

arXiv2025

Learning the detector in optical tomography

We propose a method to reconstruct the optical absorption of a highly-scattering medium probed by diffuse light. The method consists of learning the optical detection system and then using this result to reconstruct the absorption. Our results are illustrated by numerical simulations.

physics.optics↗

arXiv2025

Optical theorem and generalized energy conservation for scattering of time-modulated waves

We introduce and study a generalized energy conservation relation for scattering of time-modulated waves, where conventional energy conservation does not hold. Based on this relation, we derive an optical theorem and compute the active power describing the power input or output due to scattering. Notably, the same system may be subject to energy gain, energy loss, or energy conservation depending on the frequency harmonics present in the wave field. Moreover, we show how the optical theorem derived herein may be used for image reconstruction based on measurements of the power. In particular, such measurements do not require information about the phase of the scattered field.

physics.optics↗

arXiv2025

Transport models for wave propagation in scattering media with nonlinear absorption

This work considers the propagation of high-frequency waves in highly-scattering media where physical absorption of a nonlinear nature occurs. Using the classical tools of the Wigner transform and multiscale analysis, we derive semilinear radiative transport models for the phase-space intensity and the diffusive limits of such transport models. As an application, we consider an inverse problem for the semilinear transport equation, where we reconstruct the absorption coefficients of the equation from a functional of its solution. We obtain a uniqueness result on the inverse problem.

math.AP↗

arXiv2024

One- and Two-Photon Localization in Quantum Optics

We consider the lattice analog of a recently proposed continuum model for the propagation of one- and two-photon states in a random medium. We find that there is localization of single photons in an energy band centered at the resonant energy of the atoms. Moreover, there is also localization of photons at arbitrarily large energies. For the case of two photons, there is localization in an energy band centered at twice the resonant frequency.

quant-ph↗

arXiv2024

Nonlinearity helps the convergence of the inverse Born series

In previous work of the authors, we investigated the Born and inverse Born series for a scalar wave equation with linear and nonlinear terms, the nonlinearity being cubic of Kerr type [8]. We reported conditions which guarantee convergence of the inverse Born series, enabling recovery of the coefficients of the linear and nonlinear terms. In this work, we show that if the coefficient of the linear term is known, an arbitrarily strong Kerr nonlinearity can be reconstructed, for sufficiently small data. Additionally, we show that similar convergence results hold for general polynomial nonlinearities. Our results are illustrated with numerical examples.

math.NA↗

arXiv2024

Dynamic One Photon Localization in a Discrete Model of Quantum Optics

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 multiple photon states.

quant-ph↗

arXiv2024

Computational inverse scattering with internal sources: a reproducing kernel Hilbert space approach

We present a method to reconstruct the dielectric susceptibility (scattering potential) of an inhomogeneous scattering medium, based on the solution to the inverse scattering problem with internal sources. We employ the theory of reproducing kernel Hilbert spaces, together with regularization to recover the susceptibility of two- and three-dimensional scattering media. Numerical examples illustrate the effectiveness of the proposed reconstruction method.

math.NA↗

arXiv2023

Nonlocal PDEs and quantum optics: band structure of periodic atomic sytems

We continue our study of the quantum optics of a single photon interacting with a system of two level atoms. In this work we investigate the case of a periodic arrangement of atoms. We provide a general structure theorem characterizing the band functions of this problem, which comprise the spectrum of the associated Hamiltonian. Additionally, we study atomic densities arising as periodically arranged scaled inclusions. For this family of examples, we obtain explicit asymptotic formulas for the band functions.

math-ph↗

arXiv2022

Born and inverse Born series for scattering problems with Kerr nonlinearities

We consider the Born and inverse Born series for scalar waves with a cubic nonlinearity of Kerr type. We find a recursive formula for the operators in the Born series and prove their boundedness. This result gives conditions which guarantee convergence of the Born series, and subsequently yields conditions which guarantee convergence of the inverse Born series. We also use fixed point theory to give alternate explicit conditions for convergence of the Born series. We illustrate our results with numerical experiments.

math.NA↗

arXiv2022

A fast, high-order numerical method for the simulation of single-excitation states in quantum optics

We consider the numerical solution of a nonlocal partial differential equation which models the process of collective spontaneous emission in a two-level atomic system containing a single photon. We reformulate the problem as an integro-differential equation for the atomic degrees of freedom, and describe an efficient solver for the case of a Gaussian atomic density. The problem of history dependence arising from the integral formulation is addressed using sum-of-exponentials history compression. We demonstrate the solver on two systems of physical interest: in the first, an initially-excited atom decays into a photon by spontaneous emission, and in the second, a photon pulse is used to an excite an atom, which then decays.

math.NA↗

arXiv2022

Kinetic equations for two-photon light in random media

We consider the propagation of light in a random medium of two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a two-photon state and show that at long times and large distances, the corresponding average probability densities can be determined from the solutions to a system of kinetic equations.

physics.optics↗

arXiv2022

Inverse Source Problem for Acoustically-Modulated Electromagnetic Waves

We propose a method to reconstruct the electrical current density from acoustically-modulated boundary measurements of time-harmonic electromagnetic fields. We show that the current can be uniquely reconstructed with Lipschitz stability. We also report numerical simulations to illustrate the analytical results.

math.AP↗