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Physical Review A

Quantum theory of spontaneous emission in a one-dimensional optical cavity with two-side output coupling

Xiao-Ping Feng, Kikuo Ujihara

Published Mar 1, 1990• 71 citations
Publisher

Abstract

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).

DOI
10.1103/physreva.41.2668

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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.>
We revisit the problem of spontaneous emission by a two-level system in a cavity, concentrating in particular on those aspects associated with the finite propagation time of light between the atom and the cavity boundaries. We find that spontaneous emission always occurs first at the free space rate and there is then an abrupt change in these dynamics when light reflected by the cavity mirrors returns to the atom, leading to interference effects that result either in an enhancement or a reduction of this rate or in a periodic exchange of excitation between the atom and the cavity field. From this point of view, reversible spontaneous emission in terms of single-mode ``vacuum Rabi oscillations'' can be interpreted as resulting from the interference between the multimode partial waves reflected at the cavity mirrors and the instantaneously radiated light. \textcopyright{} 1996 The American Physical Society.
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.
We present two simple soluble quantum optical models in which an atomic system is coupled to a bath containing an infinite number of reservoir states. The total system is treated conservatively, but the atomic subsystem of interest exhibits decay. The first system consists of a single two-level atom coupled to an infinite number of modes of a one-dimensional cavity which are equally spaced in frequency. The coupling results in spontaneous emission by an excited atom into the modes of the cavity. The model also exhibits the subsequent re-excitation of the atom after the spontaneously emitted radiation is reflected from the ends of the cavity. The second is a single two-level atom coupled to a single mode of a damped optical cavity where the damping is provided by coupling to quantized oscillators in the walls of the cavity. The damped cavity alters the spontaneous emission rate from its free space value. The simplicity of both models makes them attractive as a method for introducing quantum mechanical damping phenomena.
Spontaneous emission in a planar microcavity by two identical and spatially separated two-level atoms, both of them initially excited, is considered under fourth-order perturbation approximation in atom-field coupling constants. A delay-differential equation with proper retardation times for the probability of both atoms in the upper state is derived and expressions for the emitted-light intensity and spectrum are given. It is numerically shown that, while two-atom cooperation is enhanced for small interatomic distances, the two-atom vacuum Rabi oscillation is somewhat suppressed compared to the single-atom oscillation level when the interatomic separation is close to the cavity transverse coherence length.