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Physical Review A

Comparison of the classical rate and the Einstein coefficient for spontaneous emission in a light-absorbing cavity

Vitaly V. Datsyuk, Saulius Juodkazis, Hiroaki Misawa

Published Aug 15, 2005• 4 citations
Publisher

Abstract

The rate of spontaneous electromagnetic emission in a cavity is the normalized output radiation rate in the classical electrodynamics and the Einstein $A$ coefficient in a quantum theory. We reveal the difference and similarity between the quantum and classical rates by calculations of the relative output power of the resonant spontaneous electric-dipole emission as a function of light absorption in a high-$Q$ cavity. We show that the relative resonant output power is equal to the relative width of a morphology-dependent resonance in the spectral dependence of the Einstein $A$ coefficient. The validity of the theory is demonstrated by numerical verification of an experimental result on a microsphere.

DOI
10.1103/physreva.72.025803

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The theory developed in the first two articles of this series, dealing with the interaction between the electromagnetic field in a cavity resonator and a number of two-level molecules, is generalized to include a Gaussian spread in the molecular frequency. The center of the molecular frequency distribution coincides with the cavity resonant frequency. There is a coherent driving field in the cavity at the same frequency, and cavity loss is taken into account.Using the formalism previously developed for a quantum-mechanical field in a lossy cavity, expressions are obtained by means of second-order perturbation theory for the expectation values of the field strength and field energy in the cavity, and of the power loss by the molecules. It is shown that the parts of the field energy resulting from induced and spontaneous emission, respectively, initially increase as the square of the time and approach steady-state values after (different, in general) transient periods, each of which is determined by two time constants: cavity relaxation time and inverse molecular frequency spread.It is also shown that both the induced and spontaneous emission power radiated by the molecules increase initially linearly with the time and approach steady-state values after transient periods. For the induced emission power, the transient period is determined by only one time constant, the inverse molecular frequency spread, while for the spontaneous emission power it is determined both by the inverse molecular frequency spread and the cavity relaxation time. The ratio of induced to spontaneous emission is initially $n$, and approaches a steady-state value $n{[\mathrm{exp}({r}^{2})(1\ensuremath{-}\mathrm{erf}r)]}^{\ensuremath{-}1},$ where $n$ is the driving field energy in units of the photon energy, and $r$ is the ratio of the cavity resonance width to molecular frequency spread. The seeming inconsistency of this value with the classical value of the ratio of the Einstein coefficients is discussed.

Physical Review•Sep 15, 1960•19 citations
The rate of spontaneous electromagnetic emission in a cavity is the normalized output radiation rate in the classical electrodynamics and the Einstein AAA coefficient in a quantum theory. We reveal the difference and similarity between the quantum and classical rates by calculations of the relative output power of the resonant spontaneous electric-dipole emission as a function of light absorption in a high-QQQ cavity. We show that the relative resonant output power is equal to the relative width of a morphology-dependent resonance in the spectral dependence of the Einstein AAA coefficient. The validity of the theory is demonstrated by numerical verification of an experimental result on a microsphere.
The rates of spontaneous emission and stimulated emission and absorption for a two-level atom in a dielectric microsphere are calculated for weak E1 coupling. The A coefficient for spontaneous emission is proportional to the zero-point fluctuation of the electric field ℰij evaluated in the previous paper [ J. Opt. Soc. Am. B4, 1995 ( 1987)], which shows sharp resonances in frequency (Q > 104). It is verified that the ratio of the A and B coefficients is determined by the spectral density of thermal radiation. The level shift tends to repel the transition line away from the cavity resonance.
Starting from a quantum Maxwell equation for the vector potential operator, we study the spontaneous emission properties of a two-level atom interacting with the electromagnetic field in a lossless and inhomogeneous dielectric structure and express the spontaneous emission rate, the external quantum efficiency, and the spontaneous emission factor in terms of the classical Green function of the dielectric structure. Comparing these results with the corresponding classical results, we show that the above quantities can be calculated from the radiation field of a classical dipole. Based on this correspondence, a numerical algorithm is developed to calculate the modification of spontaneous emission in microcavities. The line shape of the emission spectrum of the light source and the decay of the cavity resonant modes are automatically taken into account in this algorithm. The approximate expressions for the spontaneous emission rate and the spontaneous emission factor are discussed.
The effects of light absorption and amplification on the transition rates of the electric-dipole emission of atoms or molecules embedded in micrometer-sized dielectric spheres are investigated in the framework of classical electrodynamics. The expression for the form factor of a resonant emission line is derived. Its dependence on the imaginary part of the light’s refractive index is investigated. It is discovered that the active sphere does not act as an optical resonator if light amplification is strong.
Abstract “Spontaneous emission of radiation” calculates the rate of spontaneous electric-dipole emission of a photon by an excited atom or molecule. The calculation proceeds by using basic quantum mechanics (i.e. not using the short cuts of Chapter 19); it uses quantum electrodynamics but is not, on that account, particularly difficult. A 2p–1s transition in hydrogen is used as exemplar; the radiation is elliptically polarized. The spectral line profile (lineshape function) is approximately Lorentzian, but has a high-frequency cut-off, needed to prevent the power radiated from diverging. A radiation-induced frequency shift is negligible. The width of the line profile agrees with the Einstein A-coefficient. A high-frequency cut-off is shown to apply similarly in the derivation of Golden Rule Number Two.
We consider the spontaneous emission in an absorbing medium by calculating the quantum flow of energy from an emitting atom. For this purpose, we extend the analysis by Power and Thirunamachandran of quantized Maxwell fields in the vicinity of an atom in the free space to a situation where the emitter is in a dielectric medium. We represent the quantum flow in terms of the non-equal time commutators involving electric and magnetic fields, as well as the local displacement field. The commutators are calculated applying a previously developed microscopic theory of quantization of radiation in dielectrics. Consequently, we derive from first principles the rate of spontaneous emission in an absorbing medium containing local field factors and other refractive contributions. The results support the previous introduction of the (virtual cavity) local field correction by Barnett et al on a phenomenological basis, as well as our previous approach in which the spontaneous emission is considered as a resonance energy transfer to the far-zone atoms. A distinctive feature of the present formalism is that it deals with the spontaneous emission directly by calculating the quantum flow from the emitting atom. Therefore, the method is free from divergences appearing in other approaches analysing the spontaneous decay in absorbing media.
The theory developed in the first two articles of this series, dealing with the interaction between the electromagnetic field in a cavity resonator and a number of two-level molecules, is generalized to include a Gaussian spread in the molecular frequency. The center of the molecular frequency distribution coincides with the cavity resonant frequency. There is a coherent driving field in the cavity at the same frequency, and cavity loss is taken into account.Using the formalism previously developed for a quantum-mechanical field in a lossy cavity, expressions are obtained by means of second-order perturbation theory for the expectation values of the field strength and field energy in the cavity, and of the power loss by the molecules. It is shown that the parts of the field energy resulting from induced and spontaneous emission, respectively, initially increase as the square of the time and approach steady-state values after (different, in general) transient periods, each of which is determined by two time constants: cavity relaxation time and inverse molecular frequency spread.It is also shown that both the induced and spontaneous emission power radiated by the molecules increase initially linearly with the time and approach steady-state values after transient periods. For the induced emission power, the transient period is determined by only one time constant, the inverse molecular frequency spread, while for the spontaneous emission power it is determined both by the inverse molecular frequency spread and the cavity relaxation time. The ratio of induced to spontaneous emission is initially nnn, and approaches a steady-state value $n{[\mathrm{exp}({r}^{2})(1\ensuremath{-}\mathrm{erf}r)]}^{\ensuremath{-}1},$ where nnn is the driving field energy in units of the photon energy, and rrr is the ratio of the cavity resonance width to molecular frequency spread. The seeming inconsistency of this value with the classical value of the ratio of the Einstein coefficients is discussed.