John C. Schotland

We study single-photon scattering by atoms in arrays of one-way waveguides. We investigate both chiral and antichiral arrays, where the one-way waveguides are aligned in the same and opposite directions, respectively. In the chiral array, reciprocity is broken: one of the (spatial) dimensions is time-like, resulting in a light-cone feature of the scattered fields. In contrast, the antichiral array preserves reciprocity and exhibits scattering behavior typical of wave systems. In analogy with classical physical optics, we examine the geometrical optics, diffraction, and scattering regimes in the waveguide arrays. We illustrate our results using numerical simulations.

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 (IBS) and the reduced IBS, providing analysis of convergence and rigorous error estimates for the former. Numerical experiments are presented to validate the proposed algorithms.

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.

We continue our study, initiated in [E. O. Hiltunen et al., SIAM J. Math. Anal., 56 (2024), pp. 3802–3831], 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.

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 rotating wave approximation (RWA) plays a central role in the quantum dynamics of two-level systems. In this paper we derive corrections to the RWA using the renormalization group approach to asymptotic analysis. We study both the semiclassical and fully quantum Rabi models and compare our analytical results with numerical calculations in the ultrastrong coupling regime, where the atom-coupling is comparable to the driving frequency.

We introduce the hybrid broken-ray tomography (HBRT) for three-dimensional imaging of weakly scattering systems. The HBRT utilizes fluorescent contrast agents and combines the principles and advantages of the broken-ray tomography and the non-reciprocal broken-ray tomography introduced by us previously. The HBRT uses angularly resolved intensity measurements at the incident and fluorescence wavelengths to reconstruct the attenuation and scattering coefficients at the excitation wavelength anywhere within the sample, as well as the attenuation coefficient at the fluorescence wavelength and the contrast agent concentration in the regions of contrast agent accumulation. The principles of HBRT have been validated by Monte Carlo simulations.

We investigate the band structure and topological phases of one- and two-dimensional bipartite atomic lattices mediated by long-range dissipative radiative coupling. By deriving an effective non-Hermitian Hamiltonian for the single-excitation sector, we demonstrate that the low-energy dynamics of the system are governed by a Dirac equation with a complex Fermi velocity. We analyze the associated topological invariants for both the SSH and honeycomb models, utilizing synthetic gauge fields to break time-reversal symmetry in the latter. Finally, we explicitly verify the non-Hermitian bulk-edge correspondence by deriving analytical solutions for edge states localized at domain boundaries.

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.