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International Conference on Electromagnetics in Advanced Applications (ICEAA)

Quantum optics and the discrete-space discrete-time Jaynes-Cummings model

John M. Arnold

Published Sep 1, 2015• 0 citations
Publisher

Abstract

The Jaynes-Cummings model of early quantum optics is examined from the point of view of a discrete-space discrete-time formulation of the electromagnetic field (the FDTD formulation) excited by an elementary quantum Hertzian dipole associated to a single edge of the discrete-space lattice. The quantum Hertz dipole is a natural 2-level atomic system, and the quantum dynamics of the electromagnetic field in the discrete space surrounding the dipole, described by functional integrals over field space, is essentially a stochastic (Monte-Carlo) algorithm for solving the FDTD equations by searching for the field distribution about which the field action Lagrangian functional S is stationary.

DOI
10.1109/iceaa.2015.7297291

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The Jaynes-Cummings model of early quantum optics is examined from the point of view of a discrete-space discrete-time formulation of the electromagnetic field (the FDTD formulation) excited by an elementary quantum Hertzian dipole associated to a single edge of the discrete-space lattice. The quantum Hertz dipole is a natural 2-level atomic system, and the quantum dynamics of the electromagnetic field in the discrete space surrounding the dipole, described by functional integrals over field space, is essentially a stochastic (Monte-Carlo) algorithm for solving the FDTD equations by searching for the field distribution about which the field action Lagrangian functional S is stationary.
We have now reached a point where it is possible to consider more advanced and currently intensively developing topics in quantum optics. These particular topics have been chosen because of their importance in the development of modern laser physics, quantum atom optics, quantum information and computation. We start by considering the dynamics of an atom coupled in a free space to a multi-mode electromagnetic (EM) vacuum field. In free space an atom is coupled to many modes of the EM field and then an initial excitation in the system is irreversibly emitted from the atom into the field modes, never again to be reabsorbed by the atom, in contrast to a close system such as the Jaynes-Cummings model where an excitation was periodically exchanged between the atom and the field. The evolution of an excited atom coupled to a multi- mode EM field is a particular example of an open quantum system.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
New developments in the high techniques associated with quantum optics have made it possible to produce a high-Q cavity with an extremely small size comparable to the wavelength of atomic emission. In such cavities with so small size, many distinctive phenomena of atomic motion interacting with cavity field appear to be quite attractive. The Jaynes-Cummings model describes the interaction of a two-level atom with a single mode field. When two atoms enter the domain of the cavity field simultaneously, and interact with each other through the dipole-dipole coupling with the exchange of visual
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It is pointed out here that quantum phase properties, as well as dynamical properties, of light exhibit a sensitivity to the relative phase between the atomic dipole and light when the atom is prepared in a coherent superposition of its states in the two-level Jaynes-Cummings model interacting with a coherent state of light. The existence of the coherent-trapping of light is demonstrated by looking not only at the variance of the light phase about the coherent state value, and the number-phase uncertainty relation, but also at the factorization of the electric-field operator, and the photon-number distribution. We find also the factorization expression of the compound state of the atom-field system. We show once more that the phase properties of light based on the phase operator formalism of Pegg-Barnett and the coherence properties of light based on the coherence theory of Glauber are complementary in description of the optical quantum coherence.