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 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.
The dynamics of the co-operative. spontaneous emission· from many· two-level atoms is investigated by introducing a specific form of quasi-probability distribution which apparently excludes the vacuum fluctuation of the radiation field but includes the ·fluctuation of the atomic dipole moment. It is shown that the equation of motion for the random variables associated with the atom-field coupled system can be written in a quite classical form. As a result, the formula is derived for computing the multi-time correlation functions of the radi ation field-essential to its coherence properties-which generalizes Bonifacio, Schwendimann and Haake's adiabatic corresponding theorem. § 1. Introduction. The co-operative spontaneous emission (so-called superradiance) from an as sembly of many two-level atoms first discussed by Dicke in 1954n has been studied by many authors with different methods, roughly speaking, classicall/> and quantum mechanically. 3>-5> The classical approach, though simple, has a special significance for the dynamics of a large number of atoms participating in phase, and has succeeded except when the population of the atoms is completely inverted. The failure of the simple classical treatment in the case of complete inversion is due to the fact that the effect of spontaneous emission is not considered at all, so that emission is impossible from such a state as with no dipole moment.2>.s> Therefore, if one takes into account adequately the quantum fluctuation of the tdtal dipole moment which exists even when the atoms are fully excited, one should be able to describe the spontaneous emission from a classical point of view. This idea was suggested by Senitzky6> in a rather intuitive argument. In the present paper an attempt is made to formulate the above problem from a quasi-probability point of view. To this end we make here a particular choice of the representation of the quantum density operators (or quasi-probability. distri bution functions abbreviated by QPDF hereafter) among others7l-Io> by which the quantum mechanical equation of motion may be transformed into a form in an . associated c-number phase space, i.e., the yvell-known P-representation 9> of Glauber IS taken for the field variables and the so-~alled Q~representation based on the
We investigate the statistical aspects of the cooperative emission of radiation by a system of many atoms. By solving the equation of motion for a suitably defined quasiprobability distribution which represents the density operator of the atoms, we find explicit expressions for the expectation values of products of atomic observables. The system is found to exhibit large quantum fluctuations for the most highly excited initial states of the atoms. For all others it exhibits nearly classical behavior. Previous numerical evaluations of the radiated intensity and its variance are found to be in good agreement with the present results.
Spontaneous emission from a system of
N identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state
∣z0〉.