Stochastic modeling of paraxial x-ray superfluorescence

An approach to modeling the dynamics of x-ray amplified spontaneous emission (ASE) and superfluorescence (SF) -- collective x-ray fluorescence emission phenomenon initiated by intense X-ray Free Electron Laser pump pulse -- is developed based on Stochastic Partial Differential Equations. The equations are derived from the first principles, the approximations, derivation steps, and extensions specific to stimulated x-ray emission are performed. The resulting equations have a form of three-dimensional -- in paraxial approximation -- Maxwell-Bloch equations augmented with the noise terms for both field and atomic variables. The derived noise terms possess specific correlation properties that enable correct reconstruction of spontaneous emission. As a result, the developed formalism is uniformly suitable for the description of all stages of the stimulated x-ray emission: spontaneous emission, ASE, and superfluorescence. Our numerical scheme circumvents the problem of run-away trajectories which is typical for stochastic differential equations based on positive P-representations. Numerical examples illustrating multiple properties of the emitted field -- e.g., spatio-temporal coherence -- are presented. We expect that the developed formalism will form a solid base for interpreting stimulated x-ray emission spectroscopy data, modeling the X-ray Laser Oscillator (XLO), and describing other experiments that employ x-ray superfluorescence phenomena.