We comment on the work of Jodoin and Mandel concerning superradiance in an inhomogeneously broadened atomic system of small size. We argue that for any realistic geometry, when inhomogeneous broadening is small, spatial variation of the electric field plays a major role in determining superradiant behavior. When inhomogeneous broadening is large, we agree on the neglect of this spatial variation, but argue that all radiation-reaction effects are then small, and can be estimated simply.
The problem of superradiant emission from a small atomic system is treated under conditions in which the effects of inhomogeneous broadening are important, as in a solid. The treatment is semiclassical, with the assumption that all atoms see the same electromagnetic field. A set of coupled integrodifferential equations is obtained characterizing the state of each atom of resonant frequency \ensuremath{\omega} at time
t. It is found that, in general, atoms of different frequencies evolve differently in time, whether or not they start from the same initial state. The effect of the inhomogeneous broadening is generally to delay the onset and to reduce the amplitude of the superradiant pulse. Some effects of inhomogeneous broadening are manifest even when the inhomogeneous lifetime is much longer than the usual superradiance time. When the atomic system starts in the ground state and is then exposed to a coherent excitation pulse, superradiant effects can appear even when the inhomogeneous lifetime is very short compared with the usual superradiance time.
It is argued that superradiant processes are those in which the radiative response of an atom to an excitation is appreciably influenced by interactions with its neighbors, rather than processes in which the output intensity is proportional to the square of the number
N of atoms. On the basis of this criterion, optical free induction is a limiting response of an inhomogeneously broadened system in which cooperative effects become very small, but in which there is a strict
N2 dependence. In such a system, superradiance is manifest by a departure of the output from an
N2 dependence. Expressions are derived for the free induction response of an atomic system to an arbitrary excitation pulse, and are reduced to simple analytic forms in certain special cases. The free induction and superradiant responses are compared under conditions in which the inhomogeneous lifetime is shorter than the superradiance time $\frac{1}{N\ensuremath{\beta}}$, and an error in a previously published curve is corrected.
We discuss the problem of calculating the superradiant behavior in samples small compared to a radiation wavelength. Difficulties associated with the analysis of Dicke are considered in the framework of quantum electrodynamics and equivalently from a classical point of view. The evolution of superradiance in a small sphere is followed numerically with attention to spatial variation of the Bloch dipole density. Coherence is maintained except in the region ${(\mathrm{kR})}^{\ensuremath{-}1}|cos\ensuremath{\theta}|\ensuremath{\lesssim}2$ which is at the Bloch equator, where it suddenly disappears. Away from the equator, the spatial variations of dipole density oscillate with amplitude $\ensuremath{\sim}{(\mathrm{kR})}^{2}sin\ensuremath{\theta}$ and frequency $\ensuremath{\sim}{(\mathrm{kR})}^{\ensuremath{-}3}|cos\ensuremath{\theta}|$. All these results agree with the predictions of a theoretical discussion previously published.
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 have developed a novel method to describe superradiance and related cooperative and collective effects in a closed form. Using the method we derive a two-atom master equation in which any complexity of atomic levels, semiclassical coupling fields and quantum fluctuations in the fields can be included, at least in principle. As an example, we consider the dynamics of an initially inverted two-level system and show how even such in a simple system phenomena such as the initial radiation burst or broadening due to dipole-dipole interactions occur, but it is also possible to estimate the population of the subradiant state during the radiative decay. Finally, we find that correlation only, not entanglement is responsible for superradiance.
The theoretical foundations of the phenomenon known as superradiance still continue to attract considerable attention. Despite many valiant attempts at pedagogically clear presentations, the effect nevertheless still continues to generate some significant confusion. Part of the confusion arises from the fact that superradiance in a quantum field theory context is not the same as superradiance (superfluorescence) in some condensed matter contexts; part of the confusion arises from traditional but sometimes awkward normalization conventions, and part is due to sometimes unnecessary confusion between fluxes and probabilities. We shall argue that the key point underlying the effect is flux conservation (and, in the presence of dissipation, a controlled amount of flux nonconservation), and that attempting to phrase things in terms of reflection and transmission probabilities only works in the absence of superradiance. To help clarify the situation we present a simple exactly solvable toy model exhibiting both superradiance and damping.