A simple model of the electromagnetic field coupled to matter as a system of coupled quantum harmonic oscillators (QHOs) with a photon number conserving Hamiltonian is specialized to a translationally invariant optical fiber and then equipped with a QHO model for a one-photon fluorophore source. In the narrowband fluorescence limit, this fluorophore model becomes the same as the classical model for a current loop antenna.
A simple model of the electromagnetic field coupled to matter as a system of coupled quantum harmonic oscillators (QHOs) with a photon number conserving Hamiltonian is specialized to a translationally invariant optical fiber and then equipped with a QHO model for a one-photon fluorophore source. In the narrowband fluorescence limit, this fluorophore model becomes the same as the classical model for a current loop antenna.
A system of coupled quantum harmonic oscillators (QHOs) whose Hamiltonian conserves photon number begets a one-photon correspondence principle (OPCoP), which allows solutions to the classical linear Maxwell equations for propagation in matter to be reinterpreted as a precise description of a one-photon state. We state, prove, and explore the OPCoP, whereby the vast extant body of linear, sourceless optical waveguide theory [Optical Waveguide Theory (Chapman & Hall, 1983)] can be applied to the exact analysis of one-photon propagation in optical fibers.
A system of coupled quantum harmonic oscillators whose Hamiltonian conserves photon number begets a one-photon correspondence principle (OPCoP), which allows solutions to the classical linear Maxwell equations for propagation in matter to be reinterpreted as precise descriptions of one-photon states. With the help of the OPCoP, we derive the linear classical Maxwell equations from the Schrödinger equation for one-polariton state evolution. The role of the matter's initial quantum state in setting the macroscopic medium parameters is made explicit. It is shown that most of the kinds of linear Maxwell equations possible follow from this model, thus showing that the vast extant body of linear, sourceless optical waveguide theory [Optical Waveguide Theory (Chapman and Hall, 1983)] can be applied to the exact analysis of one-photon propagation in optical fibers.
Introduction Quantum optics The atom is the most elementary constituent of any model that describes the quantum nature of light–matter interaction. Because atoms emit and absorb light at well-defined frequencies, nineteenth century scientists thought of them as collections of harmonically oscillating electric dipole moments or EHDs. In the language of modern physics, the latter represent dipolar transitions among the various quantum mechanical states of an atom. In a strict definition, the field of quantum optics deals with problems that not only require the quantization of matter but also of the electromagnetic field, with examples such as (i) generation of squeezed light or Fock states, (ii) strong coupling of an atom and a photon, (iii) entanglement of a photon with an atom and (iv) Casimir and van der Waals forces. There are also many other important topics that have been discussed within the quantum optics community but do not necessarily require a full quantum electrodynamic (QED) treatment. Examples are (i) cooling and trapping of atoms, (ii) precision spectroscopy and (iii) modification of spontaneous emission. The simple picture of a TLS as an EHD remains very insightful and valuable to this day. Indeed, much of what we discuss in this chapter has to do with the interplay between the quantum and classical mechanical characters of dipolar oscillators. For instance, the extinction cross-section of a TLS, given by 3λ 2 /2π, can be derived just as well using quantum mechanics [70] or classical optics [234]. Another example, albeit more subtle, concerns the spontaneous emission rate.
Thus far, we have developed the properties of the electromagnetic field at optical frequencies, based on Maxwell’s equations. These equations further give a classical macroscopic perspective on the coupling of the propagation media to the field, as presented in Chapter 2. The macroscopic properties of a medium are based on averaged microscopic properties. The microscopic energy structure of matter was presented in Chapter 3, covering gases, solids, and liquids by employing mostly quantum models. We now proceed to the next level of development, the dynamic description of the interaction between the optical field and the propagation medium as a function of the field frequency and propagation media variables (e.g., energy structure, temperature, and pressure). In this chapter, the classical electromagnetic field is coupled to discrete frequency oscillators via Newton’s equation of motion. This approach leads to the popular classical oscillator model, often presented in introductory books on lasers. The classical oscillator model is an incomplete theory and can be only a semiempirical model. In the next chapter, a more detailed and comprehensive approach, which also includes statistical and quantum mechanics, is used leading to robust semiclassical and quantum oscillator models. This chapter and the next are the basis for the applied models presented in Part II of this book. Classical electrodynamics is based on Maxwell’s equations, as given in Chapter 2, and the Lorentz force relation, as given below: . . . F = q[E + (v × B)]. (4.1) . . . These equations cover the classical description of the interaction of light and matter. The first term in Eq. 4.1 represents coupling of the electric field to the medium. As discussed in Chapter 2 (Section 2.2), the leading mechanism for this is the electric dipole moment. To see that this is the coupling mechanism in the first term, consider the potential function driving this force, F = −∇V(r) = −∇(−qr · E). The above expression contains the dipole moment, as defined in Chapter 2. The second term in Eq. 4.1 represents coupling of the magnetic field to the medium.
Techniques to controllably interface atoms with quantum optical fields form the basis for many applications in quantum information science. For example, photons are convenient to relay information over large quantum networks, while atoms naturally are physical systems that can process and store this information. Thus far, the available techniques to efficiently couple single photons with atomic media fall into one of the following, mostly independent, categories: i) cavity quantum electrodynamics (QED), where atomic interactions with light are enhanced via a high-finesse cavity, ii) coherent coupling with atomic ensembles exhibiting large optical depths, and iii) the use of fields tightly focused to dimensions smaller than or approaching the scattering cross-section of a single atom. Although remarkable achievements have been made with all of these approaches, a robust, scalable technique remains elusive. Recently, several groups have successfully demonstrated that cold atoms can be trapped near and interfaced with nanophotonic systems, such as hollow-core photonic crystal fibers [1] and tapered nanofibers [2–4]. The traps are well-characterized [2, 3, 5], and the nearly diffraction-limited transverse confinement of optical fields enables ∼ 10% coupling efficiency of a single atom to the fiber [2, 3]. This has already led to remarkable observations of strong light-matter interactions using relatively few atoms and low powers [1]. It has also been proposed that such techniques can be extended to highly configurable photonic crystal waveguides [6]. Together, these efforts raise the intriguing possibility for future nanophotonic systems with tremendous figures of merit, wherein atom-light interactions can be tailored nearly at will. Here, we discuss recent efforts to develop novel techniques to realize strong, controllable atom-photon and photon-photon interactions, which take full advantage of the parameter space afforded by nanophotonic interfaces and are not based upon the extension of existing techniques. As a specific example, we discuss a protocol to achieve “all-atomic” cavity QED [7], as briefly described below. First, we show that although the single-atom coupling to guided modes of a nanophotonic waveguide might be relatively weak, there exist collective modes of a trapped atomic ensemble whose coupling to light is enhanced by the square root of the atom number, √ NA. While collective effects are generally well-known, special consequences emerge in the nanofiber system when the atoms are trapped in a lattice. In particular, collective effects cause such a lattice to act as a near-perfect mirror for an incident field close to resonance. In analogy to cavity QED, we then demonstrate that two sets of atomic mirrors can form an effective cavity, which can greatly enhance the coupling of a single, specially chosen “impurity” atom (or a few impurity atoms) positioned inside. We introduce a novel quantum spin model to describe the atom-light coupling, which allows one to exactly map the atom-nanofiber interface onto the simple and elegant Jaynes-Cummings model of cavity QED [8]. A unique feature of our atomic mirrors compared to conventional cavities is that they have long relaxation times and are highly dispersive. Remarkably, even with very low mirror finesse (F ∼ 10 2 ), this property allows one to attain the “strong coupling” regime of cavity QED, where vacuum Rabi oscillations [9–12] occur between an excited impurity atom and a single “photon” stored in the cavity (or more precisely, in the atomic mirrors). Furthermore, as quantum mechanical objects, these atom mirrors can be used to store quantum information and transfer this information into propagating waveguide modes. These various features can be combined to realize all of the building blocks for scalable quantum information processing.