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arXiv:2303.00853

Stochastic modeling of x-ray superfluorescence

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

Published Mar 1, 2023• 1 citation
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Abstract

An approach to modeling the dynamics of x-ray amplified spontaneous emission and superfluorescence -- the phenomenon of collective x-ray emission initiated by intense pulses of X-ray Free Electron Lasers -- is developed based on stochastic partial differential equations. The equations are derived from first principles, and the relevant approximations, derivation steps, and extensions specific to stimulated x-ray emission are presented. The resulting equations take the form of three-dimensional generalized Maxwell-Bloch equations augmented with noise terms for both field and atomic variables. The derived noise terms possess specific correlation properties that enable the correct reconstruction of spontaneous emission. Consequently, the developed theoretical formalism is universally suitable for describing all stages of stimulated x-ray emission: spontaneous emission, amplified spontaneous emission, and superfluorescence. We present numerical examples that illustrate various properties of the emitted field, including spatio-temporal coherence, spectral-angular and polarization characteristics. We anticipate that the proposed theoretical framework will establish a robust foundation for interpreting measurements in stimulated x-ray emission spectroscopy, modeling x-ray laser oscillators, and describing other experiments leveraging x-ray superfluorescence.

DOI
10.48550/arxiv.2303.00853

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An approach to modeling the dynamics of x-ray amplified spontaneous emission and superfluorescence -- the phenomenon of collective x-ray emission initiated by intense pulses of X-ray Free Electron Lasers -- is developed based on stochastic partial differential equations. The equations are derived from first principles, and the relevant approximations, derivation steps, and extensions specific to stimulated x-ray emission are presented. The resulting equations take the form of three-dimensional generalized Maxwell-Bloch equations augmented with noise terms for both field and atomic variables. The derived noise terms possess specific correlation properties that enable the correct reconstruction of spontaneous emission. Consequently, the developed theoretical formalism is universally suitable for describing all stages of stimulated x-ray emission: spontaneous emission, amplified spontaneous emission, and superfluorescence. We present numerical examples that illustrate various properties of the emitted field, including spatio-temporal coherence, spectral-angular and polarization characteristics. We anticipate that the proposed theoretical framework will establish a robust foundation for interpreting measurements in stimulated x-ray emission spectroscopy, modeling x-ray laser oscillators, and describing other experiments leveraging x-ray superfluorescence.
An approach to modeling the dynamics of x-ray amplified spontaneous emission and superfluorescence, the phenomenon of collective x-ray emission initiated by intense pulses of x-ray free-electron lasers, is developed based on stochastic partial differential equations. The equations are derived from first principles, and the relevant approximations, derivation steps, and extensions specific to stimulated x-ray emission are presented. The resulting equations take the form of three-dimensional generalized Maxwell-Bloch equations augmented with noise terms for both field and atomic variables. The derived noise terms possess specific correlation properties that enable the correct reconstruction of spontaneous emission. Consequently, the developed theoretical formalism is universally suitable for describing all stages of stimulated x-ray emission: spontaneous emission, amplified spontaneous emission, and superfluorescence. We present numerical examples that illustrate various properties of the emitted field, including spatiotemporal coherence and spectral-angular and polarization characteristics. We anticipate that the proposed theoretical framework will establish a robust foundation for interpreting measurements in stimulated x-ray emission spectroscopy, modeling x-ray laser oscillators, and describing other experiments leveraging x-ray superfluorescence.
A recently introduced theoretical framework for modeling the dynamics of x-ray amplified spontaneous emission is based on stochastic sampling of the density matrix of quantum emitters and the radiation field, similarly to other phase-space sampling techniques. While based on first principles and providing valuable theoretical insights, the original stochastic differential equations exhibit divergences and numerical instabilities. Here, we resolve this issue by accounting the stochastic components perturbatively. The refined formalism accurately reproduces the properties of spontaneous emission and proves universally applicable for describing all stages of collective x-ray emission in paraxial geometry, including spontaneous emission, amplified spontaneous emission, and the nonlinear regime. Through numerical examples, we analyze key features of superfluorescence in a one-dimensional approximation. Importantly, single realizations of the underlying stochastic equations can be fully interpreted as individual experimental observations of superfluorescence.
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.
The transition from the amplification of spontaneous emission to superfluorescence in a three-level and swept-gain medium excited by an x-ray free-electron-laser pulse is theoretically investigated. Given the specific timescale of an x-ray free-electron-laser pulse, we investigate the swept pumping process in detail, and our results show that the temporal structure of an x-ray free-electron-laser pulse plays a more critical role than its peak intensity does for producing population inversion. The typical watershed of two radiant regions depends on the optical depth of the gain medium for a given coherence time, namely, the particle number density and medium length are equally important. However, we find that medium length plays a more important role than particle density for making the forward-backward asymmetry. The transient gain length and the total medium length are identified as two important factors to observe length-induced backward transition. The present results suggest an application of parametric controls over a single-pass-amplified light source.