This paper augments previous theoretical descriptions of superradiance in an extended optically thick medium by giving expressions which limit the length and density of a high-gain system which can superradiate as a whole, or equivalently, the shortness of its output radiation pulses. Limits arising from experimental conditions such as finite inversion time, finite decay and dephasing times, finite transit time (cooperation length), feedback, diffraction, and Fresnel number not approximately one, are discussed. Modifications to the simple analytical expressions for the output radiation of the superradiant system due to each of these effects are described in detail. Detailed computer results are also given.
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.
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.
An analysis is made of the deexcitation kinetics of strongly excited extended systems of initially uncorrelated two-level radiators. The possibility of partially cooperative states of the system is predicted when the radiation intensity acquires values intermediate between the spontaneous decay intensity and the Dicke superradiance intensity. The conditions for the onset of cooperative processes are obtained. The threshold for the oscillatory emission conditions is determined and the oscillation frequency is found. Modulation of the emission intensity is predicted with different modulation frequencies before and after the superradiant pulse. The modulation frequencies are determined. Formulas are derived for the superradiant pulse delay times and intensities. The possibility of weakly anisotropic cooperative emission is predicted.
Superradiance is an extensive cooperative spontaneous emission phenomenon, exhibited by some atomic collective-spin states. However, distinct initial states differ in their decay dynamics. Dicke states
∣j,m⟩ with distinct numbers of excitations
n=m+j, driven by vacuum fluctuations, have their peak emission intensity shifted in time. Rotating Dicke states relative to the decay axis introduces an interesting parity structure that affects the pulse profile and photon correlations. Squeezed-bath prepared states undergo a squeezing-controlled crossover to the rotated Dicke states, making the emission character dependent on the amount of squeezing transferred from light to the atomic state. For semiclassical states with a macroscopic dipole moment, like the atomic coherent state, the emission intensity depends on their polarization. We present detailed results on the superradiant dynamics of a representative selection of states expanded in Dicke states to highlight the initial state as an independent dynamical control parameter. For large-
N, we are able to predict fairly accurately the pulse profile in each case using the mean-field approximation, an approach based on the Fokker--Planck equation. We also present comparative results on the intensity correlation function, quantify the coherent and incoherent contributions of the emission, and contrast between small and large ensembles.
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.