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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Stationary spontaneous emission up to fourth-order approximation

Benjamin Fain

Published Aug 1, 1996• 1 citation
Publisher

Abstract

The intensity of spontaneous emission by atoms or molecules in a stationary state in contact with a thermal bath at temperature T, is derived up to the fourth-order approximation in electronic charge. The perturbative treatment of the intensity of the stationary spontaneous emission is shown to be valid only if the relaxation rate is much larger than w, the rate of spontaneous emission. A specific model gives the stationary spontaneous emission intensity for all values of and w.

DOI
10.1088/1355-5111/8/4/001

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The intensity of spontaneous emission by atoms or molecules in a stationary state in contact with a thermal bath at temperature T, is derived up to the fourth-order approximation in electronic charge. The perturbative treatment of the intensity of the stationary spontaneous emission is shown to be valid only if the relaxation rate is much larger than w, the rate of spontaneous emission. A specific model gives the stationary spontaneous emission intensity for all values of and w.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
To extend the capabilities of the existing spectroscopic methods of investigation and diagnostics of gaseous media, we consider a macroscopic effect of an increase in the rate of spontaneous emission, which results in the transition of a considerable part of excited molecules to lower-lying optically accessible states with light pulse emission along the direction of propagation of the exciting laser radiation. As an application of the effect of amplified spontaneous emission, we propose the population with its help of optically inaccessible electronic states of atoms and molecules.
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.
The system under consideration is a number of molecules contained in a resonant cavity and isolated from external influences. The molecules are assumed to have two energy levels, and the molecular frequencies have a Gaussian distribution centered at the cavity frequency; the initial states of the molecules are uncorrelated. The onset of correlation in the molecular behavior is studied by examining the field in the cavity and the power emitted by the molecules for effects depending on the square of the number of molecules, in a perturbation theory approach.It is shown that correlation effects manifest themselves in the fourth order interaction. Both the correlation energy in the field and the correlation power emitted by the molecules approach steady-state values after transient periods determined by the relaxation time of the cavity and the frequency spread of the molecules. A physical picture of the correlation effects, as being due to induced emission produced by the lowest order spontaneous and thermal emission, is investigated and found to be approximately correct. The ratio of correlation energy to lowest order spontaneous emission energy is derived. An analysis is made of the dependence of the results on the initial states of the molecules, and interpreted in terms of the physical picture. The effect of the presence of a number of cavity modes, rather than a single mode, within the frequency spread of the molecules is investigated under simplifying assumption, and is shown to multiply the correlation effects by the square of the number of modes.