Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Canadian Journal of Physics

Semiclassical theory of superfluorescence

Andrzej Zardecki, Réjean Baribeau

Published Jun 1, 1977• 2 citations
Publisher

Abstract

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.

DOI
10.1139/p77-132

Related papers

Superradiant Intensity Fluctuations in Neoclassical Radiation Theory

Andrzej Zardecki

The fluctuation properties of a system of two level atoms are investigated by introducing a random precession term into the equations of motion of the atomic super Bloch vector. Starting from the assumptions of neoclassical radiation theory a general formula for the dipole field of a small system of atoms is derived. The intensity fluctuations are calculated in the narrowing limit for two extreme cases of very large and very small atomic to thermal energy ratios. The former case, describing a true superradiant emission, involves asymptotic solutions to the Fokker-Planck equation. The latter case, treated by perturbation theory, corresponds to spontaneous emission in the presence of incoherent pumping.

Canadian Journal of Physics•Dec 15, 1974•1 citation

Phase-space description of oscillatory superfluorescence

M. Gronchi, Luigi A. Lugiato

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.

Physical Review A•Feb 1, 1976•13 citations

Superfluorescence: Quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source

D. Polder, Martin F. H. Schuurmans, Quirin H. F. Vrehen

For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and $N$ is the number of atoms. The corresponding effective initial tipping angle roughly equals $\frac{2}{{(N)}^{\frac{1}{2}}}$. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.

Physical Review A•Mar 1, 1979•253 citations

Quantum statistics of oscillatory superfluorescence

M. Gronchi, Luigi A. Lugiato, P. Butera

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.

Physical Review A•Aug 1, 1978•4 citations

Fluctuations in superfluorescence

Fritz Haake, Harald King, Guntram Schröder +2 more

We show that certain properties of superfluorescent pulses radiated spontaneously by a system of many initially excited two-level atoms can be described by means of the classical Maxwell-Bloch equations. The quantum fluctuations which initiate the pulse are random initial polarizations. The radiated fields which arise from different initial polarization distributions vary significantly from pulse to pulse. Such variations represent quantum uncertainties amplified to the macroscopic level. Probability distributions are calculated for various parameters characterizing the shape of the output pulse as well as the mean and the variance of the radiated intensity.

Physical Review A•Nov 1, 1979•233 citations

Inverse scattering theory of superfluorescence

Heinz Steudel

The inverse scattering method enables a generic nonlinear theory of superfluorescence. In the semiclassical theory used here the process starts from a fluctuating polarization source. The source function is expanded with respect to an appropriate function system, and in this way it can be replaced by a set of normalized random numbers. The Gelfand-Levitan-Marchenko equations are solved approximately. Delay statistics and the pulse shape are studied, and the influence of inhomogeneous broadening is discussed. Results are compared with other theories as well as with experiment.

Quantum Optics Journal of the European Optical Society Part B•Oct 1, 1990•14 citations
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 fluctuation properties of a system of two level atoms are investigated by introducing a random precession term into the equations of motion of the atomic super Bloch vector. Starting from the assumptions of neoclassical radiation theory a general formula for the dipole field of a small system of atoms is derived. The intensity fluctuations are calculated in the narrowing limit for two extreme cases of very large and very small atomic to thermal energy ratios. The former case, describing a true superradiant emission, involves asymptotic solutions to the Fokker-Planck equation. The latter case, treated by perturbation theory, corresponds to spontaneous emission in the presence of incoherent pumping.
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.
For a pencil-shaped extended medium with Fresnel number equal to 1, we have quantum mechanically derived a description of the initiation of superfluorescence in terms of Maxwell-Bloch equations with a fluctuating source due to the zero-point fluctuations of the vacuum field. By the introduction of classical behavior, these equations are extended to include nonlinear behavior due to decreasing atomic inversion. The principal assumption in the derivation is that the main features of superfluorescence are governed by the interaction of atoms with field modes inside two small solid angles around the pencil axis. The delay ${\ensuremath{\tau}}_{D}$, defined as the time at which the mean-squared tipping angle of the collective Bloch vector attains the value 1, turns out to be given by ${\ensuremath{\tau}}_{D}=(\frac{{\ensuremath{\tau}}_{R}}{4}){[\mathrm{ln}{(2\ensuremath{\pi}N)}^{\frac{1}{2}}]}^{2}$, where ${\ensuremath{\tau}}_{R}$ is the radiation time for collective decay and NNN is the number of atoms. The corresponding effective initial tipping angle roughly equals 2(N)12\frac{2}{{(N)}^{\frac{1}{2}}}(N)21​2​. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}$ of the delay of the superfluorescence pulse satisfies approximately $\ensuremath{\Delta}{\ensuremath{\tau}}_{D}=\frac{2.3}{\mathrm{ln}N}$. A brief comparison with previous treatments of superfluorescence is given.
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 show that certain properties of superfluorescent pulses radiated spontaneously by a system of many initially excited two-level atoms can be described by means of the classical Maxwell-Bloch equations. The quantum fluctuations which initiate the pulse are random initial polarizations. The radiated fields which arise from different initial polarization distributions vary significantly from pulse to pulse. Such variations represent quantum uncertainties amplified to the macroscopic level. Probability distributions are calculated for various parameters characterizing the shape of the output pulse as well as the mean and the variance of the radiated intensity.
The inverse scattering method enables a generic nonlinear theory of superfluorescence. In the semiclassical theory used here the process starts from a fluctuating polarization source. The source function is expanded with respect to an appropriate function system, and in this way it can be replaced by a set of normalized random numbers. The Gelfand-Levitan-Marchenko equations are solved approximately. Delay statistics and the pulse shape are studied, and the influence of inhomogeneous broadening is discussed. Results are compared with other theories as well as with experiment.