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Physical Review A

Cavity quantum electrodynamics inside a hollow spherical cavity

W. Jhe, K. Jang

Published Feb 1, 1996• 28 citations
Publisher

Abstract

We present a general formalism for studying the cavity quantum electrodynamics for an atom or a radiating dipole inside a hollow spherical cavity. The atomic-energy-level shifts and the modified dipolar decay rates are calculated in specific cases. In particular, the energy shifts for an atom located at the center of an empty sphere (radius a) vary as 1/${\mathit{a}}^{4}$ for large ka and 1/${\mathit{a}}^{2}$ for small ka, where k is the wave number. We attribute these variations to the retarded Casimir-Polder interaction and the state-independent vacuum-field contribution, respectively. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.1126

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We present a general formalism for studying the cavity quantum electrodynamics for an atom or a radiating dipole inside a hollow spherical cavity. The atomic-energy-level shifts and the modified dipolar decay rates are calculated in specific cases. In particular, the energy shifts for an atom located at the center of an empty sphere (radius a) vary as 1/a4{\mathit{a}}^{4}a4 for large ka and 1/a2{\mathit{a}}^{2}a2 for small ka, where k is the wave number. We attribute these variations to the retarded Casimir-Polder interaction and the state-independent vacuum-field contribution, respectively. \textcopyright{} 1996 The American Physical Society.
We calculate the atomic energy-level shift and the modified dipolar radiation rate inside a hollow dielectric cylinder and outside a solid cylinder using a quantum-mechanical linear-response formalism in the dipole approximation. We first derive the electromagnetic fields scattered by the cylindrical surface for an oscillating dipole inside and outside the cylinder. When an atom is located on the axis of the cylindrical hollow, we obtain analytic expressions of the atomic level shifts in two limiting cases: when bbb (radius of hollow) is very small, the level shift is proportional to ${b}^{\ensuremath{-}2}$ which is associated with the kinetic-energy change of the atomic electron, whereas when bbb is very large, the shift is proportional to ${b}^{\ensuremath{-}4}$ which is identified as the retarded (Casimir-Polder) interaction energy. Moreover, we calculate the atomic potentials as a function of the position of atoms in the hollow region, which is important for the atom-guiding experiment. We also calculate the decay rates and find enhanced rates inside and outside the cylinder. In particular, we compare the radiative properties of an atom inside the hollow cylinder with those between two plates, and those near a cylinder with near a single surface.
We consider the situation where a two-level atom is placed in the vicinity of the center of a spherical cavity with a large numerical aperture. The vacuum field at the center of the cavity is actually equivalent to the one obtained in a microcavity, and both the dissipative and the reactive parts of the atom's spontaneous emission are significantly modified. Using an explicit calculation of the spatial dependence of the radiative relaxation rate and of the associated level shift, we show that for a weakly excitating light field, the atom can be attracted to the center of the cavity by vacuum-induced light shifts.
We calculate the energy-level shifts of an atom near a dielectric microsphere. With the quantum electrodynamic linear-response formalism and the exact field susceptibility satisfying the boundary conditions, the energy shifts of a ground-state atom are calculated numerically. From the numerical results, in particular, we are able to identify the short-range instantaneous van der Waals interaction energy and the long-range Casimir-Polder retarded interaction energy between two polarizable objects (atom and cavity).
This chapter focuses on cavity quantum electrodynamics. It presents the modifications of the spontaneous radiation rate in a cavity and the shifts of energy levels. The chapter focuses on experimental work, with an outline of a theoretical framework in which the experiments may be understood. In the laboratory, the study of atoms in cavities is a relatively young field in which much of the effort is still aimed at elucidating the basic physical principles at work and demonstrating the elementary modifications of energies and the rates of an atom in a cavity. However, the ability to adjust the electromagnetic spectrum of the vacuum is leading into a remarkable new domain of quantum physics in which atoms can decay from the ground state to an excited state, radiative corrections can be far larger than fine structure, and electromagnetic fields can be prepared with an exact number of photons. This realm of physics is now known as “cavity quantum electrodynamics.”
We investigate the quantum electrodynamic (QED) properties of an atomic electron close to the focus of a spherical mirror. We first show that the spontaneous emission and excited-state level shift of the atom can be fully suppressed with mirror-atom distances of many wavelengths. A three-dimensional theory predicts that the spectral density of vacuum fluctuations can indeed vanish within a volume ${\ensuremath{\lambda}}^{3}$ around the atom, with the use of a far-distant mirror covering only half of the atomic emission solid angle. The modification of these QED atomic properties is also computed as a function of the mirror size, and large effects are found for only moderate numerical apertures. We also evaluate the long-distance ground-state energy shift (Casimir-Polder shift) and find that it scales as $(\ensuremath{\lambda}/R){}^{2}$ at the focus of a hemispherical mirror of radius RRR, as opposed to the well-known $(\ensuremath{\lambda}/R){}^{4}$ scaling law for an atom at a distance RRR from an infinite plane mirror. Our results are relevant for investigations of QED effects as well as free-space coupling to single atoms using high-numerical-aperture lenses.
The energy-level shift of a ground-state atom in front of a nondispersive dielectric half-space is calculated by quantizing the electric field by means of a normal-mode expansion and applying second-order perturbation theory to the electric-dipole Hamiltonian $\mathit{\ensuremath{\mu}}\ensuremath{\cdot}\mathbf{E}.$ It is shown that the contributions to this shift coming from traveling and from evanescent waves can be combined into a single expression which lends itself readily to asymptotic analysis for large atom-surface separations, while in the opposite asymptotic regime when the atom is close to the surface the combined expression is less convenient. Employing a Green's-function formalism instead of the normal-mode expansion leads directly to the combined formula, and in that case it is advantageous to be able to apply the same transformation backwards and split the energy shift into a sum of distinct contributions corresponding to different physical processes. The analysis serves to shed light on common sources of error in the literature and paves the way for the study of more complicated models in cavity quantum electrodynamics.