  We present a Hamiltonian-based, 3D theory in paraxial approximation for X-ray
amplified spontaneous emission (or superfluorescence) pumped by X-ray
free-electron laser. The seed field is included. The ensemble-averaged
Heisenberg equations become Maxwell-Bloch equations if factorization of
operator products is assumed and are adequate when the stimulated emission is
dominant. The spontaneous emission is accounted for by adding a random noise
term to the atomic coherence, the magnitude of which is uniquely determined
from the fact that the expectation value of the normally-ordered product
electric field operators associated with an atom does not factorize. The
modified Maxwell-Bloch equation we developed reproduces the results of the
previous 1D theory based on correlation functions.
