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.
Cooperative emission from a collection of atoms prepared in an initial state of complete inversion has been studied extensively in the past. In this paper we study the cooperative emission in dense atomic systems where there is a large average number of atoms per cubic resonance wavelength. For systems with such large densities, effects of the induced dipole-dipole interaction among neighboring atoms become important. Recently it has been shown that such interatomic interaction in a dense atomic system can lead to novel nonlinear effects.1,2 The induced dipole-dipole interaction for two-level atoms appears as a nonlinear inversion dependent renormalization of the atomic resonance frequency which gives rise to a set of modified Maxwell-Bloch equations. We study the phenomenon of superfluorescence (SF) and amplified spontaneous emission (ASE) using the modified Maxwell-Bloch equations. Statistical effects of amplified quantum stochastic processes are analyzed in SF and ASE. Phase wave statistics are analyzed in the linear regime of coherent pulse buildup. Amplified effects in the nonlinear regime in the SF to ASE transition region are statistically analyzed. Dipole dephasing due to collisions is found to play an important role.
A superfluorescence effect in a V-type medium with initial Raman inversion, driven by a strong coherent field on one of the transitions, is studied analytically and numerically. It is shown that the collective spontaneous emission evolves in the presence of a minor population of an upper state of a superfluorescent transition. In the limit of the Rabi frequency of a driving field large compared to that of a superfluorescent pulse, the system of two equations for the fields and density-matrix equations for three states of the medium is reduced to Maxwell-Bloch equations for an effective two-level system. The shape of a superfluorescent pulse is perfectly described in terms of a slowly varying envelope function (which is a solution of the Maxwell-Bloch equations), filled with fast oscillations associated with Rabi floppings on the driving transition. In the average, i.e., after integrating over a period of Rabi oscillations, the upper superfluorescent state is populated less than other states, such that the phenomenon may be called ``superfluorescence without inversion'' by analogy with lasing without inversion.
In this paper we discuss the quantum features of cooperative radiation processes in a two-level system taking into account non-Markovian effects. If the initial condition is spatially uniform, the many-mode atom-field master equation which has been derived and semiclassically discussed in a previous paper, reduces to the well-known single-mode model laser master equation. Hence, using a suitable projection operator, we derive a generalized non-Markovian master equation which turns out to be correct already in the first Born approximation. This master equation incorporates the effects both of stimulated emission and absorption and of spontaneous emission. When stimulated processes can be neglected, it reduces to a previously known Markovian superradiance master equation, whereas in general it describes non-Markovian oscillations of the radiated intensity. Finally we derive a linear system of equations for the probability of occupation of the Dicke states which generalizes a previously discussed system, taking into account the contribution of stimulated processes.
We derive equations that describe the dynamics of three-level atoms with a cascade system of levels interacting with two resonant coherent fields in conditions where cooperative relaxation dominates over incoherent spontaneous emission. We calculate the temporal dynamics of the values of the atomic populations. The possibility of coherent population trapping in the presence of the cooperative decay is established. Finally, we calculate the averaged population of the intermediate level as a function of detuning for different values of the coupling constant.
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.