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Journal of the Optical Society of America B

One-photon electrodynamics in optical fiber with fluorophore systems II One-polariton propagation in matter and fibers from the one-photon correspondence principle

Rod W. C. Vance, Francois J. Ladouceur

Published Mar 15, 2007• 3 citations
Publisher

Abstract

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.

DOI
10.1364/josab.24.000942

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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.
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.
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.
We put forth an approach to obtain a quantum master equation for the propagation of light in nonlinear fiber optics by relying on simple quantum pictures of the processes (linear and nonlinear) occurring along propagation in an optical fiber. This equation is shown to be in excellent agreement with the classical Generalized Nonlinear Schrodinger Equation and predicts the effects of self-steepening and spontaneous Raman scattering. Last, we apply these results to the analysis of two cases of relevance in quantum technologies: single-photon frequency translation and spontaneous four-wave mixing.
Abstract Photons strongly coupled to material systems constitute a novel system for realizing non-linear optics at the level of individual photons and studying the dynamics of non-equilibrium quantum many-body system. We give a simple physical polariton-picture of the dynamics of photons coupled to a one-dimensional array of two-level atoms. This picture allows a fully analytical description of the dynamics in terms of polariton scattering inside the medium and reflections of the polaritons from the edge of the array. We show that inelastic collisions, previously identified in small systems, also occur in infinite systems and are related to the existence of multiple bands in the dispersion relation. The developed theory constitutes an effective field theory for the dynamics, which can be used for studies of non-linear optics and many-body dynamics. As a specific example we map the system to the Lieb–Liniger model and show that a so-called Tonks–Girardeau gas of photons is a stable eigenstate of the system in the limit of many emitters.