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Progress of Theoretical Physics

Spontaneous Emission in Coaxial Cylindrical Cavities

K. Oshiro, K. Kakazu

Published Sep 1, 1997• 2 citations
Publisher

Abstract

The conditions for enhancement and inhibition of the spontaneous emission rate per second by an atom inside coaxial cavities are investigated in the electric-dipole approximation. A complete set of vector mode functions satisfying the boundary conditions is obtained utilizing field decomposition, which is needed to quantize the electromagnetic field. The behavior of the emission rate is generally quite complicated; there are sharp peaks at many resonance points, and the minimal values approach unity as the radius of the cavities increases. Emission is usually inhibited completely if the radius is very small. However, under certain conditions, one being a small radius, there is the possibility of very large emission due to the TEM mode.

DOI
10.1143/ptp.98.533

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Physical Review A•Nov 1, 1990•16 citations
The conditions for enhancement and inhibition of the spontaneous emission rate per second by an atom inside coaxial cavities are investigated in the electric-dipole approximation. A complete set of vector mode functions satisfying the boundary conditions is obtained utilizing field decomposition, which is needed to quantize the electromagnetic field. The behavior of the emission rate is generally quite complicated; there are sharp peaks at many resonance points, and the minimal values approach unity as the radius of the cavities increases. Emission is usually inhibited completely if the radius is very small. However, under certain conditions, one being a small radius, there is the possibility of very large emission due to the TEM mode.
We present a quantization procedure for the electromagnetic field in a circular cylindrical cavity. A complete set of vector mode functions satisfying the boundary conditions is obtained utilizing field decomposition. The quantized electromagnetic Hamiltonian is then derived. The conditions for enhancement and inhibition of the spontaneous emission rate per second by an atom inside the cavities are presented in the electric-dipole approximation. Emission is inhibited completely if the cavities are very small. Generally, the behavior is quite complicated; there are many sharp peaks at resonance points and minimal values approach that in free space as the radius of the cavities becomes large.
The normal modes of electromagnetic fields satisfying three different boundary conditions are constructed systematically. For each of these cases, cavity quantization of the electromagnetic fields is carried out explicitly. It is possible, using normal modes, to perform a unitary transformation of the minimal-coupling Hamiltonian of an atom inside the cavity to a multipolar form. It is shown that the boundary condition plays an important role in the transition rates for the spontaneous emission by the atom. Some explicit calculations are carried out for several different boundary conditions. These results become consistent with those for free space in the large-box limit.
This thesis is concerned with the theory of spontaneous emission in the context of cavity QED and the guiding of atoms by laser light in confined geometries. We focus on three different geometrical shapes of waveguide, namely the parallel plate guide and two kinds of cylindrical waveguide: one with a rectangular cross section and another with a circular cross section. These waveguides are assumed to be characterised by two distinct properties. Firstly, all three types of guide have perfect conductor boundary conditions, which means that the guide walls are infinitely conducting and, consequently, lossless. Secondly, the guides are assumed to have subwavelength dimensions (less than the atom electric dipole transition wavelength #lambda#). The first consequence of this is that the spontaneous emission process is possible only via a few cavity modes. For each type of structure, we examine the confinement of electromagnetic fields within it and proceed to quantise these fields by following the standard procedure, incorporating the boundary conditions at the guide walls. This allows the position-dependent spontaneous emission rate to be evaluated for an electric dipole located at an arbitrary point within the guide. Asymptotic limits of the spontaneous emission rate are then derived which serve as useful checks of the calculations. For the rectangular guide, we are able to recover the results appropriate for the parallel-plate case when side a of the rectangular cross section becomes large, while side b is kept fixed. For the circular guide, we are able to recover the results appropriate for the free space case when the radius of the guide becomes large. The second part of the thesis deals with the theory of the motion of atoms in the spatially varying light fields inside the three different shapes of waveguide. In each structure we examine the forces that act on the atom and solve the equation of motion leading to the atom dynamics when an electric dipole within the guide is subject to an excited p-polarised cavity mode. The characteristics of the atomic motion in the guide in the presence of a cavity mode are then explored. We show that under these conditions the atom becomes subject to a transverse dipole potential and a dissipative force, both of which vary across the guide and are also functions of the atom velocity along the mode propagation direction. The dipole potential is responsible for the transverse trapping (and hence the channelling) of atoms at specific regions of the cross section, while the dissipative force controls the longitudinal motion of the channelled atoms. The conditions facilitating atom guiding are explored using typical parameters for sodium atoms in guides of subwavelength dimensions. The roles of the van der Waals and Casimir-Polder potentials on the atomic motion in waveguides is clarified theoretically in the context of the parallel-plate guide and we show that such potentials are effective only at relatively short distances from the guide walls. Finally, we show that atomic motion in a hollow circular guide is subject to a quantised torque arising from the orbital angular momentum effects associated with any excited waveguide mode of order l > 0. We find that an appropriately detuned atom immersed in such a mode may be trapped radially in a vibrational state and be made to exhibit novel rotational effects associated with the light torque, including a rotational frequency shift. (author)
It is well known that spontaneous emission is not an intrinsic atomic property, but rather results from the coupling of the atom to the vacuum modes of the electromagnetic field. As such, it can be modified by tailoring the electromagnetic environment into which the atom can radiate. This was already realized by Purcell, who noted that the spontaneous emission rate can be enhanced if the atom placed inside a cavity is resonant with one of the cavity is resonant with one of the cavity modes, and by Kleppner, who discussed the opposite case of inhibited spontaneous emission. It has also been recognized that spontaneous emission need not be an irreversible process. Indeed, a system consisting of a single atom coupled to a single mode of the electromagnetic field undergoes a periodic exchange of excitation between the atom and the field. This periodic exchange remains dominant as long as the strength of the coupling between the atom and a cavity mode is itself dominant. 23 refs., 6 figs.
Recently, we observed enhancement and inhibition of visible spontaneous emission by atoms in a confocal resonator.1 In these experiments, the partial emission rate of the atom into the resonator modes is modified by a factor of 1/(1 − R), where R is the mirror reflectivity. The change in the total spontaneous emission rate is proportional to the solid angle ΔΩ subtended by the mirrors; the change was small (+1.6%, −0.5%) because ΔΩ ≈ 0.04 sr was small. Currently we are conducting experiments in which atoms are placed in a concentric resonator of large solid angle ΔΩ ≈ 3.6 sr and finesse of ~10. We estimate that a 30% decrease and 300% increase should be observable in the total spontaneous emission rate, when the resonator is tuned off or on-resonance, respectively. Results of this new experiment are presented. In addition to studying spontaneous processes, one can also consider the stimulated emission which occurs when a cavity field builds up which contains more than one photon. We have formulated a model which describes both the spontaneous and stimulated emission by the atom into the cavity and clarifies the relationship between enhanced spontaneous emission and stimulated emission.
Enhanced and inhibited spontaneous emission of two close atoms in an overdamped cavity is studied taking into account the dipole-dipole interaction between them. The system is described in the minimal-coupling scheme and the advantages of this choice for our problem are discussed. The dynamics is investigated using both quantum Langevin equations and Master equation techniques. This last approach allows one to describe the behavior of the atoms plus field system as a one-step process. The case in which both atoms are initially in their excited state and the mode is in its vacuum state is discussed in some detail. It is shown that, different from what occurs to two close atoms in free space, in an overdamped cavity dipolar interaction may modify the decay rate. In particular, we find that the two transitions the atomic system performs to reach its equilibrium configuration can have different rates, and that for a suitable choice of the detuning between the cavity mode and the unperturbed atomic frequency, one of the transitions occurs at an enhanced rate while the second is inhibited.