This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.
We introduce a straightforward quantum-electrodynamic approach to the problem of superradiant spontaneous emission from a system of two-level atoms, and also discuss two classical superradiant systems. The same physical concepts underlie our treatment of both the quantum and classical cases. Explicit expressions are found which describe the time evolution and directional character of the radiated power. Our approach applies to an arbitrary number of atoms which are confined to a region with linear dimensions which may be large compared with the mean wavelength of the emitted radiation. The radiative decay half-life of a large many-atom system is found to be much shorter than the natural lifetime of a single atom, but not as short as the half-life of a small atomic system with the same number of atoms. The far-field radiation pattern appropriate to the case of a circular cylinder of emitters which have been excited by a plane-wave pulse is considered in detail. Polar plots are presented of the radiation patterns produced when the atoms are excited by a plane-wave pulse. Exciting pulses traveling in directions both parallel and at an angle of 10\ifmmode^\circ\else\textdegree\fi{} to the axis of the cylinder are considered. A discussion and generalization of Dicke's "coherence-brightening" criterion is given.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms
A1,
A2, whose separation
r21 can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only
A1 inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators
O1,
O2. The atoms exhibit superradiant behavior, whereas
O1,
O2 tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that
A2 and
A2 radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
The interaction between electromagnetic radiation and two three-level atoms is investigated. The atom-atom separation is assumed to be smaller than the corresponding mean resonance wavelength. The total radiance rate of the atomic system is calculated as a function of time. Explicit results are given for several different initial states of the atomic system. Some of them exhibit superradiance, and some act as photon-trapping states. Instead of exponential decay, the oscillations appear in the time evolution of the radiation rate due to the correlation between atoms. The decay constants and coupling decay constants show strong effects on the emission rates. Completely differing from two identical double-level atoms, the cooperative effects always exist in the three-level atomic system except for the case in which the coupling decay constants ${\ensuremath{\gamma}}_{12}$ and ${\ensuremath{\gamma}}_{21}$ are zero.
The problem of the interaction of N atoms with a laser beam and vacuum modes can give rise to many interesting phenomena concerning the spontaneous emission of light and its propagation in the medium. The cooperative effects, for example, such as superadiance and subradiance, are effects related to the coherence created between the atoms when a photon is emitted spontaneously by a single excited atom. Superradiance can be defined as the enhancement of the spontaneous emission due to constructive interference of the scattered light. Its counterpart, subradiance, is the trapping of some remaining light due to destructive interference. In cold atoms, some previous theoretical works predict and characterize these two cooperative effects in a large and diluted atomic cloud, in the regime of low intensities and large detunings of the incident laser. The theoretical model is a coupled-dipole model for two-level atoms driven by a low-intensity field and in the scalar approach. The experiment consists in measuring the super- and subradiant decay rates from the temporal emitted intensity after the switch off of the incident laser in the steady state. Our experimental setup consists in a magneto-optical trap of rubidium 87 atoms at large resonant optical thicknesses. A probe beam excites the atoms close to the D2 line. The intensity emitted is detected by a single photon detector with no afterpulsing and a calibration procedure allows us to determine the resonant optical thickness of the cloud and its temperature. In this work, we report the experimental observation of super- and subradiance in a large cloud of cold atoms. For subradiance, the main result is the linear evolution of the characteristic time with the resonant optical thickness of the cloud and its independence of the detuning. For superradiance, we observe superradiance out of the forward direction. We verify the validity of our interpretations with the predictions of the coupled-dipole model. Finally, we discuss the interplay of subradiance and radiation trapping, as well as theoretical predictions for: a setup of a phased cloud, to control the subradiant amplitude emission; and temperature effects, where subradiance is shown to be robust in a large range of temperatures.
The present state of theoretical and experimental research on collective spontaneous emission (superradiance) in a system of radiators (atoms, molecules, or nuclei) is reviewed. The distinction between superradiance and the amplification of spontaneous emission is discussed. There is also a discussion of conditions under which the effect can be observed and of various theoretical methods for describing superradiance: the quantum single-mode and multimode models and the semiclassical approach. Theoretical papers on superradiance in systems with dimensions smaller than the wavelength of the radiation and also in extended systems are reviewed. It is shown that superradiance may occur in weakly amplifying media. A situation in which the superradiance is oscillatory is described. The possible use of superradiance to generate coherent emission in the x-ray and ? ranges is discussed. The superradiance accompanying Raman scattering of light in atomic and molecular media is studied. The theoretical results are compared with experimental observations of superradiance in the optical range. In an appendix, the nature of the phase transition in a system of radiators interacting through an electromagnetic field is discussed.