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

Applied Physics Letters

Fluctuations in tipping angles and delay times of simple superfluorescence

C. T. Lee

Published Nov 1, 1982• 7 citations
Publisher

Abstract

It is shown that the essential statistical properties of the initial tipping angle θ0 can be obtained through a simple model and can be calculated directly from the super-radiance master equation. Thus, the more elaborate models and the tedious task of solving the master equation can all be avoided, as far as θ0 is concerned. As a byproduct, the statistical properties of the delay time first calculated by Degiorgio are rederived in a more rigorous and much simpler way.

DOI
10.1063/1.93707

Related papers

Superfluorescence pumped by a short pulse of arbitrary shape

C. T. Lee

Recognizing that the complete initial inversion required in superfluorescence (SF) can only be achieved in a finite time, we use two different master equations for two different stages of SF: one describing the initiation of SF during pumping and another one governing the free evolution of SF after pumping. The pumping process is treated as a Poisson process with varying rate to reflect a pump pulse of arbitrary shape. The exact solution to the first master equation is obtained and used as the initial condition for the second master equation. The statistical properties of the tipping angle and the delay time are then calculated explicity. It is shown that the effects of pumping process on these two quantities can be reduced to a parameter called effective head-start time (EHT). Two different expressions for this EHT have been derived: one with saturation effects and another one without.

Physical Review A•Jan 1, 1984•1 citation

Effects of Pumping Process on the Tipping Angle and Quantum Fluctuations in Superfluorescence

C. T. Lee

We propose a simple model to incorporate pumping process in the initial evolution of superfluorescence. The calculated initial tipping angle is closest, among various predictions, to the very recently measured value by Vrehen and Schuurmans. We also calculate the quantum fluctuations to account for the shot-to-shot variations in experimental results. Additional experiments with varying pumping duration are suggested to further test the validity of our model.

Physical Review Letters•Oct 8, 1979•4 citations

Direct Measurement of the Effective Initial Tipping Angle in Superfluorescence

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

We report the first direct measurement of the effective initial tipping angle ${\ensuremath{\theta}}_{0}$ in superfluorescence. A coherent pulse of small area $\ensuremath{\theta}$, resonant on the superfluorescence transition, is injected into a completely inverted sample of cesium atoms and the delay of the out-put pulse is measured as a function of $\ensuremath{\theta}$. The value of ${\ensuremath{\theta}}_{0}$ obtained agrees in essence with $\frac{2}{{(N)}^{\frac{1}{2}}}$. The latter value is predicted by recent theories which consider the quantum initiation of superfluorescence and which take propagation effects into account.

Physical Review Letters•Jan 22, 1979•77 citations

THERMODYNAMIC APPROACH FOR SUPERFLUORESCENCE

Eng Y

The self-organization process of superfluorescence can be explained clearly basing on the thermodynamics. The present paper includes three parts: 1) the equations of superfluorescence radiation and heat transform cetion are derived, and their transient processes are analysed; 2) the possibility of the generation of superfluorescence is theoretically verified by the analysis of the linear stability of the thermo-dynamic Maxwell-Bloch equations, and the coherent process of superfluorescence is analysed by the temporal coherence function of the field of radiation obtained: 3) the effective initial tipping angle θ0 and the delay time τD are derived from the fluctuation in statistical thermodynamics. Numerical results are in good agreement with those accepted up to now.

Acta Physica Sinica•Jan 1, 1995•0 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
It is shown that the essential statistical properties of the initial tipping angle θ0 can be obtained through a simple model and can be calculated directly from the super-radiance master equation. Thus, the more elaborate models and the tedious task of solving the master equation can all be avoided, as far as θ0 is concerned. As a byproduct, the statistical properties of the delay time first calculated by Degiorgio are rederived in a more rigorous and much simpler way.
Recognizing that the complete initial inversion required in superfluorescence (SF) can only be achieved in a finite time, we use two different master equations for two different stages of SF: one describing the initiation of SF during pumping and another one governing the free evolution of SF after pumping. The pumping process is treated as a Poisson process with varying rate to reflect a pump pulse of arbitrary shape. The exact solution to the first master equation is obtained and used as the initial condition for the second master equation. The statistical properties of the tipping angle and the delay time are then calculated explicity. It is shown that the effects of pumping process on these two quantities can be reduced to a parameter called effective head-start time (EHT). Two different expressions for this EHT have been derived: one with saturation effects and another one without.
We report the first direct measurement of the effective initial tipping angle ${\ensuremath{\theta}}_{0}$ in superfluorescence. A coherent pulse of small area $\ensuremath{\theta}$, resonant on the superfluorescence transition, is injected into a completely inverted sample of cesium atoms and the delay of the out-put pulse is measured as a function of $\ensuremath{\theta}$. The value of ${\ensuremath{\theta}}_{0}$ obtained agrees in essence with 2(N)12\frac{2}{{(N)}^{\frac{1}{2}}}(N)21​2​. The latter value is predicted by recent theories which consider the quantum initiation of superfluorescence and which take propagation effects into account.
The self-organization process of superfluorescence can be explained clearly basing on the thermodynamics. The present paper includes three parts: 1) the equations of superfluorescence radiation and heat transform cetion are derived, and their transient processes are analysed; 2) the possibility of the generation of superfluorescence is theoretically verified by the analysis of the linear stability of the thermo-dynamic Maxwell-Bloch equations, and the coherent process of superfluorescence is analysed by the temporal coherence function of the field of radiation obtained: 3) the effective initial tipping angle θ0 and the delay time τD are derived from the fluctuation in statistical thermodynamics. Numerical results are in good agreement with those accepted up to now.
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.