The evolution of a small spherical system of many harmonic oscillators, initially in a coherent superradiant state, is considered. The dipole oscillators interact via their mutual fields, both longitudinal and transverse, including both radiation- and near-zone contributions. In principle, this interaction can dephase the vibration of oscillators and thereby decrease or eliminate the coherent superradiant contribution to the emitted radiation. It is shown, however, that if the system is sufficiently small, this dephasing makes only small corrections to the decay of the initial state. Thus the emission of a superradiant pulse and cooperative radiative decay are major processes in the system.
Recently, it has been suggested that the collective radiative decay of two point-like quantum emitters coupled to a waveguide, separated by a distance comparable to the coherence length of a spontaneously emitted photon, leads to an apparent "superradiance paradox" by which one cannot decide whether independent or collective emission occurs. Here we suggest an integrated optics platform to emulate the superradiance paradox, based on photon escape dynamics in waveguide lattices. Remarkably, Markovian decay dynamics and independent photon emission can be restored by frequent (Zeno-like) observation of the system.
When atoms are coupled to a common electromagnetic environment, the exchange of photons through dipole-dipole interactions leads to the emergence of cooperative effects. As a particular example, superradiance arises from spontaneous emission when this exchange leads to constructive interference of the emitted photons. Here, we introduce an integrated method for studying cooperative radiation in many-body systems. This method, which allows to study extended systems with arbitrarily large number of particles, can be formulated by an effective, nonlinear, two-atom master equation that describes the dynamics using a closed form which treats single- and many-body terms on an equal footing. We apply this method to a homogeneous gas of initially inverted two-level atoms, and demonstrate the appearance of both superradiance and subradiance, identifying a many-body coherence term as the source of these cooperative effects. We describe the many-body induced broadening -- which is analytically found to scale with the optical depth of the system -- and light shifts, and distinguish spontaneous effects from induced ones. In addition, we theoretically predict the time-dependence of subradiance, and the phase change of the radiated field during the cooperative decay.
The basic features of coherent spontaneous emission, including its dependence on the square of the number of participating emitters, are reviewed. Analogies with magnetic resonance phenomena are exploited, and we explain the need to go beyond such analogies in order to treat optical emission. Some recent results in the theory of optical superradiance are discussed. The radiative decay process is treated as a quantum-classical interpolation problem, and explicit solutions are given. In addition to its time dependence, we give the angular distribution of radiated power and the expected intensity fluctuations, and we estimate the maximum number of emitters which can cooperate in a superradiant event. Dicke's “coherence brightening” criterion is discussed from the point of view of the present paper. A “failure” of the criterion is exhibited and explained. Two of the purely quantum aspects of superradiance are isolated and discussed briefly.
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.