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IEEE Journal of Quantum Electronics

Quantum theory of a one-dimensional optical cavity with output coupling. V. Spontaneous emission

Kikuo Ujihara

Published Jul 1, 1988• 9 citations
Publisher

Abstract

For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>

DOI
10.1109/3.975

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For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>
Enhancement or suppression of spontaneous emission by the presence of a cavity has been a subject of numerous papers [1]. The effective number of radiation modes was estimated in terms of the cavity Q. Sachdev [2] has improved on this by an exact quantum Langevin treatment of a lossy, single-mode cavity and predicted a vacuum Rabi oscillation for high-Q cavities. Cook et al. [3] included retardation effects by incorporating multiple Fabry-Perot modes and treated the coupling loss by a heuristic introduction of the mirror reflectivity. The present author [4] has considered the same problem in perturbation and Wigner-Weisskopf approximations by use of a set of continuously distributed radiation modes which describes a cavity having output coupling.
A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).
A nonperturbative fully quantum-theoretical analysis describing the transient spontaneous emission of an initially excited two-level atom in a one-dimensional cavity with output coupling is presented. A delay-differential equation for the probability amplitude of the upper atomic state is strictly derived from the Schrodinger equation. With this equation, both the time evolution of the atomic state and the emission spectrum are calculated. Furthermore, the mode function makes the calculations of emitted fields both inside and outside the cavity possible. The calculated result of the vacuum Rabi oscillation in an underdamped cavity and the result of enhanced spontaneous emission rate in an overdamped cavity are presented. A retardation-governed revival of the upper atomic state is predicted. The calculated results also show the inhibition of spontaneous emission by detuning or by setting the atom at a node.>
The transverse distributions of intensity inside and outside an optical cavity produced by spontaneous emission from atoms inside the cavity are reviewed. The mode properties inside a closed passive one-dimensional cavity are summarised and the spatial distributions of emitted light for a passive cavity with partially transmitting mirrors are discussed. A qualitative theory describes the effects of an amplifying medium inside the cavity, covering the regimes both below and above the lasing threshold. Expressions are derived for the angular spread of the output light in the presence of gain. A theory based on rigorous general relations between gain and noise provides more complete results for the spontaneous emission noise from an amplifying slab within an optical cavity.
Spontaneous emission in a one-dimensional Fabry-P\'erot-type optical cavity is analyzed quantum mechanically. Via a monolayer dielectric cavity model, expressions describing the atomic-location dependence of the emission intensities and the spectra observed on both sides of the cavity are obtained. The probability amplitude for the upper atomic state is analyzed and its time-evolution equation is given. The probability amplitude is shown to be proportional to ``the electric field amplitude for a single emission event'' introduced classically by Huang et al. [Appl. Phys. Lett. 61, 2961 (1992)]. An expression for the emission spectrum applicable to a general one-dimensional cavity structure is obtained.
A detailed account of the calculation on the spontaneous emission rate in a very short optical cavity with plane-parallel dielectric mirrors is given. A cavity model consisting of a pair of plane dielectric slabs is used. Guided modes as well as outgoing modes are considered. The emission rate from a two-level atom is calculated under quantum-mechanical perturbation approximation. The enhancement factor for the rate is found to be of the order of unity except for extremely short cavities. The concept of quasimodes in a very short cavity and their effective area are discussed in relation to the enhancement factor. The coupling ratios into various quasimodes are derived.