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.
A theoretical model for the propagation of coherent light pulses in resonant media where a large number of atoms cooperate in spontaneous emission (superradiance) is discussed. The medium is supposedly free of inhomogeneous broadening, and no lossy relaxation mechanisms are taken into account. The conditions for the emission of all spontaneous radiation in the direction of pulse propagation are described. A semi-classical model is used and, at each point in the medium, the propagation field and the spontaneous damping field are considered separately. The equation governing the evolution of a superradiant system in the presence of an applied pulse is derived. A mechanism of energy transfer from the traveling pulse to the medium is described which includes superradiant spontaneous emission effects. As a result of the neglect of all lossy relaxation processes, all energy stored in the medium is eventually returned to the pulse by coherent spontaneous emission. Therefore, in this theory, all pulses travel through the medium without energy loss. Numerical results for square input pulses are discussed which show two possible types of propagation behavior: (a) Low input intensities produce a negative tail to the input (positive) pulse and sharpening of the tail with the advance in the medium; (b) higher input intensities produce, after sufficient penetration, a steady-state pulse envelope somewhat similar to the $2\ensuremath{\pi}$ pulses of self-induced transparency. The steady-state solutions of the equations of motion are found numerically and the pulse shapes are computed for different degrees of superradiance. In the limit of strong superradiance, very asymmetric pulse envelopes of large area are found. For negligible superradiance, approximately $2\ensuremath{\pi}$ hyperbolic-secant pulses are obtained. The effects of inhomogeneous broadening are briefly considered, and the consequences of important superradiant damping in photon-echo experiments are qualitatively discussed.
A theoretical model for superradiant damping effect in inhomogeneously broadened media and its application to photon-echo problems are discussed. Only optically thin systems are considered, and all nonradiative relaxation mechanisms are neglected. Numerical calculations are performed for a very wide square inhomogeneous line shape and square applied pulses. It is shown that excitation of the system becomes increasingly difficult as the superradiant lifetime ${\ensuremath{\tau}}_{s}$ becomes comparable with
T2∗. The energy reradiated after the end of the exciting pulse is considerable for ${\ensuremath{\tau}}_{s}\ensuremath{\approx}{T}_{2}^{*}$ and for some applied pulse areas. Two-pulse sequences produce multiple echoes as the microscopic polarizations spontaneously rephase. The first-echo amplitude maximum is found to occur for first-applied-pulse areas increasingly above $\frac{1}{2}\ensuremath{\pi}$ as ${\ensuremath{\tau}}_{s}$ decreases. For a given first-pulse area, the echo intensities increase initially as ${\ensuremath{\tau}}_{s}$ decreases (negligible damping) and finally decrease and vanish (important damping). The ratio of the second- to first-echo amplitudes is found to be proportional to $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$ for small superradiance levels and to decrease when ${\ensuremath{\tau}}_{s}$ approaches
T2∗. An approximate estimate of this ratio is derived for the region of linearity in $\frac{{T}_{2}^{*}}{{\ensuremath{\tau}}_{s}}$.
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.
We give a general nonperturbative treatment of cooperative emission in systems of
N two-level atoms, starting from first principles and including inhomogeneous broadening. In particular, we study superfluorescence, which is defined as the cooperative spontaneous emission, i.e., radiation rate proportional to
N2, from an atomic system initially excited with zero macroscopic dipole moment and a uniform population difference between the excited and the fundamental states. The atomic system is described by means of collective dipole operators. A fundamental justification is given for the existence of damped "quasimodes" of the mirrorless active volume. The damping of such modes is simply due to the propagation of the Maxwell field, which escapes from the active volume. A general atom-field master equation is derived for the system atoms plus field inside the active volume, described, respectively, in terms of collective dipole operators and quasimode operators. An important feature of this equation is that inhomogeneous broadening simply appears via a time-dependent atom-field coupling constant. In this paper we give a semiclassical treatment of such a master equation. For a pencil-shaped geometry of the active volume, generalized Maxwell-Bloch equations are derived for the envelopes of the radiation inside the active volume and polarization. Such equations take into account the two directions of propagation of the radiation and the inhomogeneous broadening. Suitably phrasing our initial condition in semiclassical terms, we find that propagation effects can be neglected at all times and the generalized Maxwell-Bloch equations reduce to a simple pendulum equation. On the basis of the discussion of the pendulum equation, we conclude that superfluorescence occurs when (i) the length
L of the active volume is much larger than a suitable threshold length
LT (this condition ensures that the dephasing atomic processes occur on a time scale much larger than the times characteristic of the cooperative emission); (ii) the length
L is smaller or of the same order of a suitable cooperation length
Lc (this condition ensures that cooperative spontaneous emission dominates stimulated processes, which give radiation proportional to
N). For $L\ensuremath{\ll}{L}_{c}$, one has a hyperbolic-secant superfluorescent pulse; for $L\ensuremath{\approx}{L}_{c}$, as one has in the recent experiments of Skribanowitz et al., one finds oscillations in the cooperative decay and in the radiation emission. Such oscillations are due to the contribution of stimulated processes. For $L\ensuremath{\gg}{L}_{c}$, this contribution increases. As a consequence one gets more oscillations in the radiated intensity, which becomes proportional to
N, so that superfluorescence effects disappear.