Conservation laws for Maxwell-Bloch equations applied to describe superfluorescence are studied. Modifications due to the initial boundary conditions are discussed. For initial polarization in the form of a constant tipping angle, conservation laws are integrated to give exact dependence of all field moments on the propagation length. Three regions of propagation are distinguished: the linear region, where the field energy is negligibly small, the region of rapid exponential growth, and the saturation region. The characteristic distance, where maximum radiation occurs, is explicitly evaluated; it defines the threshold length for superfluoresence. A small chirping, proportional to the square of the tipping angle, is found. The change of the pulse width with propagation length is estimated.
Conservation laws for Maxwell-Bloch equations applied to describe superfluorescence are studied. Modifications due to the initial boundary conditions are discussed. For initial polarization in the form of a constant tipping angle, conservation laws are integrated to give exact dependence of all field moments on the propagation length. Three regions of propagation are distinguished: the linear region, where the field energy is negligibly small, the region of rapid exponential growth, and the saturation region. The characteristic distance, where maximum radiation occurs, is explicitly evaluated; it defines the threshold length for superfluoresence. A small chirping, proportional to the square of the tipping angle, is found. The change of the pulse width with propagation length is estimated.
For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and N is the number of atoms. The corresponding effective initial tipping angle roughly equals (N)212. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.
The evolution of a superfluorescent pulse is studied in the region of large sample lengths.A solution to the Maxwell-Bloch semiclassical equations is found by using the inverse scattering method.It has the form of a onesoliton solution with varying parameters.Dependence on the pulse width and the retarded time is found without further approximations.These parameters are shown to be independent of the inhomogeneous broadening time.Their values are in good agreement with experiment.Asymmetry of the pulse is also investigated.
The self-organization process of superfluorescence can be explained clearly basing on the thermodynamics. The present paper includes three parts: 1) the equations of superfluorescence radiation and heat transform cetion are derived, and their transient processes are analysed; 2) the possibility of the generation of superfluorescence is theoretically verified by the analysis of the linear stability of the thermo-dynamic Maxwell-Bloch equations, and the coherent process of superfluorescence is analysed by the temporal coherence function of the field of radiation obtained: 3) the effective initial tipping angle θ0 and the delay time τD are derived from the fluctuation in statistical thermodynamics. Numerical results are in good agreement with those accepted up to now.
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.
Superfluorescence emission profiles are computed using one-way coupled Maxwell-Bloch equations. Transverse effects are included in the full three-spatial-dimension case as well as in the cylindrical-symmetry case. Initiating quantum fluctuations are approximated by a random polarization source with a completely random phase and root-mean-square tipping angle of N2, where N is the number of atoms in each volume element. These fluctuations reduce the tail of the output obtained with transverse effects alone. In fact, the fluctuations in output pulse shapes encompass the Cs data of Gibbs, Vrehen, and Hikspoors. The standard deviation for the delay time is found to be (12.5\ifmmode\pm\else\textpm\fi{}4)% for Fresnel number of 0.8 compared with the value (10\ifmmode\pm\else\textpm\fi{}2)% recently measured by Vrehen and der Weduwe, also for Cs. Inhomogeneous-broadening effects are also included in some simulations.
The fluctuating properties of superradiant and superfluorescent systems are investigated by supplementing the equations of motion for the expectation values of quantum observables with random force terms. In the case of a small atomic system the time evolution of the radiated intensity as well as the intensity fluctuations are computed. We also find the time dependence of the average dipole moment in the presence of chirping. For a pencil-shaped geometry, a generalized system of Maxwell–Bloch equations is analysed. The inhomogeneous broadening is then included. We compare the semiclassical equation for the probability distribution of the atomic variables with the quantum equations based both on normal and antinormal ordering. With the exception of the region in the vicinity of the poles of the Bloch sphere, a close correspondence between the semiclassical and quantum theories is established. The oscillatory superfluorescence is treated similarly to quantum analysis by employing a new method of Lugiato for open systems.