Many advancements in optics have relied on the tight-binding approximation, which simplifies the description and prediction of complex system behaviors. This approximation describes the dynamics of the total light field by examining the coupling between the guided modes of individual single-mode substructures -- also known as coupled mode theory. However, the underlying assumption, that the guided modes of individual waveguides form an orthogonal basis, breaks down when waveguides are brought into close proximity or when larger arrays are considered. In this work, we systematically analyze the consequences of this non-orthogonality and show that it leads to a generalized eigenvalue problem involving an overlap matrix, causing a fundamental mismatch between the standard TB model and solutions of the paraxial wave equation. To resolve this issue, we introduce a modified TB framework based on the Löwdin orthogonalization, which constructs an orthonormal basis from the non-orthogonal guided modes while minimally altering their physical shape and preserving their symmetry properties. The resulting Löwdin-TB method restores the standard eigenvalue problem and yields excellent agreement with exact beam propagation simulations across a wide range of system sizes and waveguide separations. Furthermore, it captures important physical effects, such as enhanced long-range coupling and nontrivial hopping phases, that are absent in the standard approach.
Many advancements in optics have relied on the tight-binding approximation, which simplifies the description and prediction of complex system behaviors. This approximation describes the dynamics of the total light field by examining the coupling between the guided modes of individual single-mode substructures -- also known as coupled mode theory. However, the underlying assumption, that the guided modes of individual waveguides form an orthogonal basis, breaks down when waveguides are brought into close proximity or when larger arrays are considered. In this work, we systematically analyze the consequences of this non-orthogonality and show that it leads to a generalized eigenvalue problem involving an overlap matrix, causing a fundamental mismatch between the standard TB model and solutions of the paraxial wave equation. To resolve this issue, we introduce a modified TB framework based on the Löwdin orthogonalization, which constructs an orthonormal basis from the non-orthogonal guided modes while minimally altering their physical shape and preserving their symmetry properties. The resulting Löwdin-TB method restores the standard eigenvalue problem and yields excellent agreement with exact beam propagation simulations across a wide range of system sizes and waveguide separations. Furthermore, it captures important physical effects, such as enhanced long-range coupling and nontrivial hopping phases, that are absent in the standard approach.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
Recent advances are enabling quantum emitters to interact with photonic crystals, whose electromagnetic modes exhibit complex dispersion relations, spatial mode structure, and polarization textures. However, modeling light-matter behavior in these systems faces a persistent trade-off: electromagnetic approaches based on Maxwell-equation solvers provide realistic vectorial descriptions but are difficult to integrate with quantum many-body and non-perturbative methods, whereas simplified quantum-optical lattice models are tractable but typically rely on scalar and spatially independent light-matter couplings that miss essential features of these structured photonic environments. Here, we introduce a constructive framework to derive quantum-optical lattice descriptions that overcome this trade-off. Combining symmetry-constrained tight-binding constructions with numerically computed photonic band structures and field profiles, our method yields minimal, symmetry-enforced lattice Hamiltonians that reproduce the target photonic dispersion while retaining the mode-resolved (position- and polarization-dependent) structure of the light-matter coupling. We show that these models recover Green's-function-based emitter dynamics in the perturbative regime, while providing access to non-perturbative quantum dynamical simulations beyond emitter-only descriptions. As a proof of principle, we apply the framework to a two-dimensional photonic crystal and show that it captures polarization-dependent directional emission inaccessible to scalar models, while enabling the analysis of non-Markovian light-matter dynamics and entanglement. Our results provide a practical bridge between classical electromagnetic simulation tools and quantum-optical many-body and non-Markovian modeling in photonic crystal settings.
We present analytic solutions to the evolution in generalized tight-binding models, which consider complex first-neighbor couplings with equal amplitude and arbitrary phases. Our findings provide a powerful tool to efficiently calculate expectation values and correlations within the system, which are otherwise difficult to compute numerically. We apply our results to relevant examples in quantum light manipulation using -port linear couplers, describing the evolution of single (multi)-mode squeezing, single-photon added (subtracted) Gaussian states, and second-order site-to-site photon correlations. Significantly, our analytic results outperform standard numerical calculations. Our study paves the way for a comprehensive mathematical framework describing the spatial evolution of quantum states across a wide range of physical systems governed by the tight-binding model. Published by the American Physical Society 2024
We explore special features of quantum light-matter interactions inside structured waveguides due to their finite bandwidth, band edges, and nontrivial topological properties. We model the waveguides as either a tight-binding (TB) chain or a Su-Schrieffer-Heeger (SSH) chain. For unstructured waveguides with infinite bandwidth, the transmission and reflection coefficients of a side-coupled two-level emitter (2LE) are the same as the reflection and transmission coefficients of a direct-coupled 2LE. We show that this analogy breaks down for structured waveguides with finite bandwidth due to the appearance of Lamb shift only for the direct-coupled 2LE. We further predict a robust light-emitter coupling at zero collective decay width of a single giant 2LE (with two couplings at different points) near the band edges of the structured waveguides where topological features can be beneficial. Finally, we study single-photon dynamics in a heterojunction of a long TB and short SSH waveguide connected to a 2LE at the SSH end. We show the propagation of a photon from the excited emitter to the TB waveguide only when the SSH waveguide is in the topological phase. Thus, the heterojunction acts as a conditional propagation channel.
Waveguide quantum electrodynamics constitutes a modern paradigm for the interaction of light and matter, in which strong coupling, bath structure, and propagation delays can break the radiative conditions that quantum emitters typically encounter in free space. These characteristics intertwine the excitations of quantum emitters and guided radiation modes to form complex multiphoton dynamics. So far, combining the collective decay of the emitters with the non-Markovian effects induced by the modes has escaped a full solution and the detailed physics behind these systems remains unknown. Here we analyze such a collective non-Markovian decay in a minimal system of two excited emitters coupled to a one-dimensional single-band waveguide. We develop an exact solution for this system in terms of elementary functions that unveils hidden symmetries and predicts new forms of spontaneous decay. The collective non-Markovian dynamics, which are strongly dependent on the vacuum coupling and the detuning from the center of the band, show exotic features that can be characterized with a simple and readily available criterion. Our analytic methods shed light on the complexity of collective light-matter interactions and open up a pathway for understanding multiparticle open quantum systems. Published by the American Physical Society 2024
We present a Hamiltonian formulation of coupled mode theory for scenarios in which the coupled modes are associated with different "parent structures," such as two nearby waveguides. The relativistic nature of the photon leads to the complication that not any set of orthonormal modes can be used as a basis if the associated amplitudes are to satisfy canonical commutation relations. This difficulty is circumvented by the introduction of "dressed parent modes," which are in fact seen to be the "coupled modes" of the system. While an exact solution of the linear problem within the restricted basis of interest formally must be found before these modes can be constructed, in practice they can be constructed directly from the modes of the parent structures. The approach can be applied to periodic parent structures, such as photonic crystal waveguiding structures, as well as to simpler waveguides. We illustrate the accuracy of the various approximations employed by studying two sample systems in detail. We derive the linear coupled mode equations, and show how the approach can be immediately generalized from the linear regime to treat problems in nonlinear quantum optics.