Quantum Statistical Theory of Superradiance. I

We discuss the cooperative decay of initial atomic excitation for a pencil-shaped active volume filled with two-level atoms. As long as the length of the sample is much smaller than a certain maximal cooperation length, the atom-field interaction producing the superradiant pulse can be treated in terms of the simplest possible laser model (single mode). The basic laser master equation turns out to be exactly solvable if specified for the superradiance limit which is characterized by two conditions: (i) The photons escape from the low-$Q$ cavity so fast that they cannot feed themselves back into atomic excitation to any appreciable amount (this no-feedback condition is equivalent to the above-mentioned requirement for the length of the sample). (ii) The incoherent atomic decay due to natural relaxation is so slow that the individual atomic dipoles do not dephase before engaging themselves cooperatively in the interaction with the electromagnetic field. The present paper presents the derivation and general discussion of the equations describing the statistical properties of atoms and field in a superradiant pulse. Analytical and numerical solutions will be presented in a subsequent paper.