The intensity of radiation, stimulated by laser pulses of arbitrary shape, is derived up to fourth-order terms. It is shown that there is a linear fourth-order mechanism of emission that depends on linear susceptibilities of molecules. This linear mechanism reveals collective, super-radiant features. The intensity of laser-driven spontaneous emission (LDSE) is proportional to for samples smaller than , where is a wavelength of emitted radiation, and N is the number of molecules. For extended volumes , the intensity of LDSE is proportional to . Equations of motion describing collective radiation damping are derived.
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.
We present a numerical study of the spatial coherence of light that is radiated from a dilute ensemble of atoms. The spatial coherence is established as a result of the collective (cooperative) coupling of the atoms to the light and is qualitatively different from the coherence of a laser. Specifically, the coherence in collective spontaneous emission does not rely on population inversion and stimulated emission, is governed by antiphasing of the dipoles (subradiance), and the key figure of merit for the observed coherence is $\ensuremath{\approx}N/(L/\ensuremath{\lambda})$. Here,
N is the number of atoms in the ensemble,
L is the size of the sample, and $\ensuremath{\lambda}$ is the wavelength of the emitted light.
In order to study the cooperative emission of radiation from an extended system of
N two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled
j and
k via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the
z axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the
z dependence predominates, in the opposite case the (
x,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the
z dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the
z axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
We discuss from first principles the cooperative decay of a system of two-level atoms, initially prepared in an uncorrelated excited state with population inversion
N, and we give the conditions under which the superfluorescence effect occurs. Describing the atomic system in terms of collective variables, we derive a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes. The coupled equations of motion are solved with a self-consistent approach. It is found that the system goes through a nonexponential decay if the maximum length of the active volume is smaller than a "cooperation range" and larger than a "threshold length," in agreement with the one-mode theory. The radiation burst has a time width proportional to ${N}^{\ensuremath{-}1}$, and its intensity is proportional to
N2. Specializing to a pencil-shaped volume, we find that only two atomic modes need to be considered; in this case, the average emitted radiation is all condensed in the two diffraction patterns of the opposite axial modes.
Spontaneous decay is a phenomenon that has been described for the first time by Dirac, Wigner
and Weisskopf, at the early ages of the development of quantum electrodynamics. The theory is
applicable to a multi-body system only when atoms don’t interact during their decay. Dicke introduces
in his seminal paper the case where this assumption doesn’t hold.
The results he obtains not only tell that the system emits pulses, but the pulse duration now depends
on 1/N, where N is the number of participating atoms to the process. The pulses temporal profile
is described by a squared hyperbolic secant function. The resulting emission intensity scales as N^2.
The atomic system radiates more intensively than the ordinary fluorescent emission, this is the reason
why the process is called superradiance, or by a generalization to extended systems, superfluorescence.
The superradiant emission has preferential escape directions, depending on the atomic sample shape.
Superradiance is a collective process and it takes place when the group of excited atoms is left unper-
turbed during the formation of of the macroscopic dipole. A fundamental difference compared to the
spontaneous and stimulated emission processes is that all atoms act together as a single entity and it
isn’t possible to identify single atoms involved.
This remarkable behaviour is manifest when the atomic coherence of the system, while influenced by
environmental interactions that cause dephasing in atoms, persists for a sufficient interval of time.
Typical dephasing mechanisms, such as collisions in gases or lattice phonon interactions in crystals,
make its experimental observation difficult. A possible solution to this problem is the use of optical
crystals doped with rare-earths that are characterized by long and narrow levels lifetimes.