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Physical review. B, Condensed matter

Critique of the Bonifacio-Lugiato superfluorescence model

Irvin W. Kay, Bohdan Balko

Published Oct 1, 1993• 5 citations
Publisher

Abstract

This paper shows how the Bonifacio-Lugiato mean-field model of atomic superfluorescence, which is particularly attractive because it deals with a crystalline array of emitters, can be extended to the nuclear case. However, limits exist on the validity of certain approximations used in the development of the model. In particular, a numerical instability that increases without bound is shown to exist in the non-Markovian case, placing in doubt the application of the semiclassical theory derived for the non-Markovian case.

DOI
10.1103/physrevb.48.10011

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This paper shows how the Bonifacio-Lugiato mean-field model of atomic superfluorescence, which is particularly attractive because it deals with a crystalline array of emitters, can be extended to the nuclear case. However, limits exist on the validity of certain approximations used in the development of the model. In particular, a numerical instability that increases without bound is shown to exist in the non-Markovian case, placing in doubt the application of the semiclassical theory derived for the non-Markovian case.
Superfluorescence is treated in the Markovian master-equation approach. We show that we have to consider the complex coupling between any pair of atoms in order to reproduce superfluorescence signals with ringings as exhibited by experiments. We compare numerical results of our model with those obtained by a semiclassical approach of superfluorescence. The quantum treatment is found to be in agreement with the semiclassical approach only if the semiclassical tipping angle is taken equal to the square root of the number of radiating particles.
The fluctuating properties of superradiant and superfluorescent systems are investigated by supplementing the equations of motion for the expectation values of quantum observables with random force terms. In the case of a small atomic system the time evolution of the radiated intensity as well as the intensity fluctuations are computed. We also find the time dependence of the average dipole moment in the presence of chirping. For a pencil-shaped geometry, a generalized system of Maxwell–Bloch equations is analysed. The inhomogeneous broadening is then included. We compare the semiclassical equation for the probability distribution of the atomic variables with the quantum equations based both on normal and antinormal ordering. With the exception of the region in the vicinity of the poles of the Bloch sphere, a close correspondence between the semiclassical and quantum theories is established. The oscillatory superfluorescence is treated similarly to quantum analysis by employing a new method of Lugiato for open systems.
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.
A generalized master equation previously proposed to describe oscillatory superfluorescence is numerically solved in the case of strong ringing. When the atomic system is fully excited, anomalous fluctuations and large atom-field correlations make the time evolution of the mean quantities (e.g., number of photons, total population inversion) quantitatively different from those found in the semiclassical treatment. In particular, because of fluctuations, the minima of the radiation output are notably larger than zero. On the contrary, when the atomic system exhibits a sizable initial macroscopic polarization (superradiance), we find a remarkable non-Markoffian effect consisting of a sizable delay in the radiation emission. However, in such conditions all quantum effects disappear, as in the case of pure superradiance.
We investigate the emergence of cooperative radiation phenomena in ensembles of two-level atoms coupled to a lossy resonant cavity beyond the Markovian and mean-field approximations. By deriving a complete analytical solution for the two-emitter case and employing a numerically exact method for larger ensembles, we characterize the full transition from Markovian to non-Markovian collective dynamics for systems of up to 10310^3103 emitters. Our results reveal three distinct regimes: a Markovian phase exhibiting the standard superradiant burst, a non-Markovian phase featuring spontaneous superabsorption of the emitted field, and a critical regime marked by pulsed collective emission. We show that the critical spectral width separating these behaviors increases monotonically with the number of emitters, demonstrating that environmental memory effects can be enhanced by cooperativity. Finally, we find that the superradiant scaling of the peak intensity progressively degrades with increasing system size, approaching a subquadratic law in the limit of a perfect cavity. In this regime, spontaneous superabsorption emerges as a distinct manifestation of non-Markovian cooperativity.
We compare the experimental curves of Skribanowitz et al., showing superradiance from pumped HF gas, with the "single-mode" time-evolution formula of Bonifacio and Lugiato. All parameters are varied freely except the length LLL of the sample. It is found impossible to obtain even rough agreement. When ${\ensuremath{\tau}}_{c}$ is taken longer than Lc\frac{L}{c}cL​, no ringing is predicted; when shorter, the predicted ringing is many times faster than that observed.