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Modeling of 3D paraxial x-ray superfluorescence based on stochastic differential equations

Andrei Benediktovitch, Stasis Chuchurka, Aliaksei Halavanau, Špela Krušič, Nina Rohringer

Published Jan 1, 2022• 0 citations
Publisher

Abstract

Intense focused XFEL pulses can produce atoms in the core-hole state thus leading to collective x-ray emission (superfluorescence). We will present a 3 D description of paraxial x-ray superfluorescence based on rigorously derived stochastic differential equations.
DOI
10.1364/euvxray.2022.ew6a.2

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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.
Irradiation of a medium by short intense pulses from x-ray (XUV) free-electron lasers can result in saturated photoionization of inner electronic shells. As a result an inversion of populations between core levels appears. The resulting fluorescent radiation can be amplified during its propagation through the inverted medium and results in intense, quasi-transform-limited radiation bursts. While the optical counterpart of this phenomena, known as superfluorescence, was intensively investigated, a generalized treatment is needed in the x-ray (XUV) domain, where the dynamics of pumping and evolution due to fast decay processes play a crucial role. To provide a general theoretical approach, we start from the fundamental, quantized minimal coupling Hamiltonian of light-matter interaction and after a series of approximations arrive at a closed system of equations for the two-point correlation function of atomic coherences and the two-time correlation function of the emitted field. The obtained formalism enables us to investigate collective spontaneous emission in various regimes. It is extended consistently to include incoherent processes that are relevant in the x-ray (XUV) domain. These processes are introduced into the formalism by corresponding Lindblad superoperators. The connection to other approaches is discussed and numerical examples related to recent experiments are presented.
We propose an approach based on stochastic differential equations to describe superfluorescence in compact ensembles of multilevel emitters in the presence of various incoherent processes. This approach has a numerical complexity that does not depend on the number of emitters. The stochastic differential equations are derived directly from the quantum master equation. In this study, we present a series of numerical examples, comparing our solution to exact calculations and discussing the limits of applicability. For many relevant cases, the proposed stochastic differential equations provide accurate results and correctly capture quantum many-body correlation effects.