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.
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.
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.
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.
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.
The "superfluorescence equations" deduced in a previous paper are generalized to the case of finite temperature and translated into the atomic coherent-state representation of Arecchi et al. We get three coupled partial differential equations involving the quasiprobability distribution $Q(\ensuremath{\theta},t)$ of the atomic system and two suitable functions $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$ describing the stimulated processes which are relevant in oscillatory superfluorescence. Neglecting $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$, the equation for $Q(\ensuremath{\theta},t)$ reduces to the Fokker-Planck equation of Narducci et al. corrected by the contribution of thermal noise. The stationary solution is given by the unperturbed canonical state, which is shown to be the exact stationary solution of the single-mode model from which the superfluorescence equations have been deduced. These equations prescribe that $\mathfrak{M}(\ensuremath{\theta},t)$ always remain equal and opposite to $\mathcal{F}(\ensuremath{\theta},t)$, so that the superfluorescence equations reduce automatically to two equations. Eliminating $\mathcal{F}(\ensuremath{\theta},t)$ between the two remaining equations, one obtains a non-Markovian equation for $Q(\ensuremath{\theta},t)$ with derivatives of all orders in $\ensuremath{\theta}$. In the case of pure superfluorescence or super-radiance, such an equation reduces to a Markovian Fokker-Planck equation with derivatives of first and second order in $\ensuremath{\theta}$ which exhibits an interesting fourth-order contribution in the diffusion term. In fact, if one excludes a very small region around the north pole of the Bloch sphere, this term is much larger than the second-order contribution in the diffusion term of the equation of Narducci et al. In the case of oscillatory superfluorescence, the presence of memory effects enhances fluctuations, thus forbidding the existence of "classical" time evolutions.
Recently a generalized master equation was derived that extends the Lindblad theory to highly non-Markovian quantum processes [H.-P. Breuer, Phys. Rev. A 75, 022103 (2007)]. We perform a stochastic unraveling of this master equation by considering n random state vectors that satisfy the corresponding stochastic differential equation for a piecewise deterministic process. As an application we consider a two-state system randomly coupled to an environment consisting of two energy bands with finite number of levels. Our numerical results are compared to results obtained from the time-convolutionless projection operator method using correlated projection superoperators and the exact solution of the Schr\"odinger equation for this system.
The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.