A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from
N identical two-level atoms is investigated further. It is shown that the equation of motion for the
p(<N)-atom density matrix is coupled to the (
p+1)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time
t can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of
N1. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of
2N coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.
Coherent spontaneous emission from a system of
N two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number"
r is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
One-atom lasers are important because their governing equations can be solved exactly, even with a quantized field. We present a fully quantum-mechanical treatment of one-atom lasers modeled by quantum-optical master equations. These are solved numerically without any significant approximations. We show that laser action is possible with one atom, and that it might be achievable experimentally. Laser action is characterized by the dominance of stimulated emission over spontaneous emission. We use the one-atom laser model to investigate, without approximation, some interesting generic laser phenomena. Under certain conditions lasers produce intensity squeezed light, and then the laser linewidth increases with the pumping rate, in contrast with standard lasers. We also report ``self-quenching'' behavior: lasers with incoherent pumping out of the lower laser level turn off when the pumping is sufficiently fast because the coherence between the laser levels is destroyed.
Spontaneous emission from a system of
N 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〉.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.
The fluorescent spectra arising from the interaction of a three-level atom with a strong pump field and a weak signal field have been studied simultaneously. The atom consists of an upper excited state ‖2〉 and two lower ground states ‖3〉 and ‖1〉, which arise by removing the degeneracy of the ground state by applying internal or external fields. The laser field depletes the metastable state ‖3〉 by bringing the electrons into the excited state ‖2〉 from where the electrons emit photons and decay into the lower states through the transitions ‖2〉 → ‖3〉 and ‖2〉 → ‖1〉, which are described by the signal field. Using a classical description of the laser field, where in the model Hamiltonian the laser–atom interaction is treated classically while the free and interacting electron and signal fields are quantized, the decay process ‖2〉 ↔ ‖1〉 of the signal field is considered by evaluating the appropriate Green’s function of the system. The spectral function for the ‖2〉 ↔ ‖1〉 transition of the signal field describes one-photon, three-photon, and two-photon Raman processes, respectively. The one-photon spectra consist of the main peak at the signal frequency and a pair of sidebands, which are symmetrically located from the position of the main peak. The intensity of the main peak is positive while that of the sidebands is negative indicating that the signal is attenuated and is amplified at the corresponding frequencies, respectively. The three-photon and two-photon Raman spectra are described by a doublet, respectively, whose intensities are always negative, implying amplification of the signal field. The computed spectra are presented graphically and compared with those derived in a recent study, where the laser field is quantized and photon–photon correlations are taken into consideration in the limit of high photon densities of the laser field. A detailed discussion of both treatments is given for the processes under investigation.