Optical resonators store electromagnetic energy. The finite response time of optical resonators provides a feedback mechanism for controlling the dynamics of atomic and mechanical systems and to effectively exchange energy between light and matter. This chapter starts with a derivation of the reflection and transmission coefficients of a confocal optical cavity. The spectrum is characterized by multiple resonances and for most applications a single resonance can be singled out. This leads to the single-mode approximation. We derive the energy stored in the cavity and evaluate the fields of a cavity that is internally excited by a radiating dipole. We calculate the LDOS and derive an expression for cavity-enhanced emission (Purcell effect). We continue with an analysis of microsphere resonators with characteristic whispering-gallery modes and review the effective potential approach, which allows us to cast the problem in form of a Schr\“odinger equation, with parallels to quantum tunneling and radioactive decay. The next section is focused on deriving the cavity perturbation formula, which states that a change in energy is accompanied by a frequency shift. Having established a solid understanding of optical resonators we discuss the interplay of optical and mechanical degrees of freedom within the context of cavity optomechanics. We derive the optomechanical coupling rate and discuss the resolved sideband and the weak-retardation regimes.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The chapter discusses quantum emitters, exploring their fundamental mechanisms, properties, and applications. Beginning with two-level systems, we introduce the concept of extinction cross-section. To capture phenomena, such as fluorescence, the discussion extends to four-level systems and spontaneous as well as stimulated emission processes, crucial for understanding laser operation. We then examine the dependence of the quantum yield on the local environment. Single-photon emission is scrutinized in terms of the second-order autocorrelation function through both steady-state and time-dependent analyses, providing a comprehensive understanding of this essential feature of quantum emitters. The chapter further addresses the generation of indistinguishable single photons, a key requirement for quantum computing and secure communication. Various types of quantum emitters are then introduced, including fluorescent molecules, semiconductor quantum dots, and color centers in diamond, each with unique properties and applications. Finally, single molecules are presented as probes for localized fields, with an in-depth look at field distributions in a laser focus and sources of strongly localized fields.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

This chapter discusses light–matter interactions from a semiclassical point of view. By expanding the electromagnetic field into a Taylor series we derive the multipolar interaction potential and particle-field Hamiltonian. Then, using the Green function formalism, we calculate the fields of an oscillating dipole and, based on Poynting’s theorem, derive a general expression for the rate of energy dissipation in an arbitrary environment. This expression leads to the concept of local density of states (LDOS) and provides a direct link to spontaneous emission and atomic decay rates. The rate of energy dissipation of an oscillating dipole is also used to derive the absorption cross-section in terms of the polarizability. By accounting for radiation reaction and scattering losses, we obtain a compact expression for the dynamic polarizability. Dipole radiation and atomic decay rates can be enhanced via LDOS engineering, and the enhancement factor is referred to as the Purcell factor. We show that if the LDOS gets enhanced in a certain frequency range, it must be reduced in other frequency ranges, a feature described by the LDOS sum rule. After discussing the properties of a single dipole, we continue with analyzing the interaction between multiple dipoles. We derive the interaction potential and calculate the energy transfer rate between dipoles. For short distances we recover the famous Förster energy transfer formula. If the interaction energy becomes sufficiently large, we enter the regime of strong coupling, which gives rise to hybridized and delocalized modes, level splittings, and entanglement.

The chapter introduces the field of plasmonics, focusing on the unique optical properties of noble metals. We begin by describing noble metals as plasmas, introducing concepts such as electronic screening and the ponderomotive force. We then discuss the local optical response of noble metals through their frequency-dependent dielectric function. The chapter progresses to surface plasmons at plane interfaces, detailing the properties of surface plasmon polaritons (SPPs), methods for exciting SPPs, and their practical applications in sensors, particularly their sensitivity and utility in biochemical sensing. Next, we focus on plasmons supported by nanowires and nanoparticles, utilizing the quasistatic approximation and standing plasmon waves. We analyze plasmon resonances in complex nanostructures by introducing the concept of plasmon hybridization. These resonances play a crucial role in surface-enhanced Raman scattering (SERS), enabling the detection of low-concentration analytes. We further explore nonlinear plasmonics, which leads to phenomena such as harmonic generation and four-wave mixing. Finally, we address nonlocal plasmonics, discussing the impact of spatial dispersion and mesoscopic boundary conditions on plasmonic responses at very small scales, highlighting the quantum effects at the metal surface.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

This chapter reviews the main concepts of electromagnetic theory relevant for the understanding of this textbook. Based on Maxwell’s equations, we derive the wave equation and discuss homogeneous solutions, such as plane waves and evanescent waves. We derive the boundary conditions at interfaces between homogeneous media and the Fresnel reflection and transmission coefficients. We discuss energy conservation, causality, and reciprocity of electromagnetic fields. Point response functions are introduced (Green functions) in order to derive the inhomogeneous solution of the wave equation. The chapter concludes with the angular spectrum representation, a framework that allows arbitrary fields to be described as a superposition of plane and evanescent waves.

In practice, a radiating source is most commonly placed close to a surface or multiple interfaces. Examples are antennas mounted over ground or molecules placed on dielectric surfaces or waveguides. The topic has a long history dating back to Arnold Sommerfeld’s paper in 1909. To derive the fields of an arbitrary oriented dipole over a layered medium we have to find the corresponding Green function. We start by decomposing the free-space dyadic Green function into $s$ and $p$ polarized parts and then evaluate reflection and transmission for the two polarizations separately. Once the Green function of the layered reference system is found, we proceed to derive the radiated power and the far-fields of the dipole. We analyze the radiation patterns and the modes into which the energy is most effectively coupled. For dipoles over dielectric half-spaces, we find that evanescent field components couple predominantly into supercritical angles, giving rise to what is termed “forbidden light.” We discuss how recorded radiation patterns can be used to determine the orientation of the radiating dipole, such as the orientation of molecules fluorescing near a dielectric surface. The chapter concludes by reviewing the image dipole method and discussing its validity.