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Physical Review A

Quantum fluctuations in superfluorescence delay times

Quirin H. F. Vrehen, J. J. der Weduwe

Published Nov 1, 1981• 55 citations
Publisher

Abstract

Measurements are reported of the quantum fluctuations in superfluorescence delay times. The fluctuations depend strongly on the Fresnel number $F$ of the sample. For $F\ensuremath{\approx}1$ fair agreement with recent theories is obtained, while for larger $F$ the fluctuations are smaller. A qualitative explanation for this $F$ dependence is proposed. No correlation has been found between the delay times of pulses emitted from opposite ends of the same sample.

DOI
10.1103/physreva.24.2857

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Superfluorescence emission profiles are computed using one-way coupled Maxwell-Bloch equations. Transverse effects are included in the full three-spatial-dimension case as well as in the cylindrical-symmetry case. Initiating quantum fluctuations are approximated by a random polarization source with a completely random phase and root-mean-square tipping angle of $\frac{2}{\sqrt{N}}$, where $N$ is the number of atoms in each volume element. These fluctuations reduce the tail of the output obtained with transverse effects alone. In fact, the fluctuations in output pulse shapes encompass the Cs data of Gibbs, Vrehen, and Hikspoors. The standard deviation for the delay time is found to be (12.5\ifmmode\pm\else\textpm\fi{}4)% for Fresnel number of 0.8 compared with the value (10\ifmmode\pm\else\textpm\fi{}2)% recently measured by Vrehen and der Weduwe, also for Cs. Inhomogeneous-broadening effects are also included in some simulations.

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Technical Physics Letters•Jun 1, 2002•0 citations
Measurements are reported of the quantum fluctuations in superfluorescence delay times. The fluctuations depend strongly on the Fresnel number FFF of the sample. For F\ensuremath≈1F\ensuremath{\approx}1F\ensuremath≈1 fair agreement with recent theories is obtained, while for larger FFF the fluctuations are smaller. A qualitative explanation for this FFF dependence is proposed. No correlation has been found between the delay times of pulses emitted from opposite ends of the same sample.
Superfluorescence emission profiles are computed using one-way coupled Maxwell-Bloch equations. Transverse effects are included in the full three-spatial-dimension case as well as in the cylindrical-symmetry case. Initiating quantum fluctuations are approximated by a random polarization source with a completely random phase and root-mean-square tipping angle of 2N\frac{2}{\sqrt{N}}N​2​, where NNN is the number of atoms in each volume element. These fluctuations reduce the tail of the output obtained with transverse effects alone. In fact, the fluctuations in output pulse shapes encompass the Cs data of Gibbs, Vrehen, and Hikspoors. The standard deviation for the delay time is found to be (12.5\ifmmode\pm\else\textpm\fi{}4)% for Fresnel number of 0.8 compared with the value (10\ifmmode\pm\else\textpm\fi{}2)% recently measured by Vrehen and der Weduwe, also for Cs. Inhomogeneous-broadening effects are also included in some simulations.
Abstract : The interaction of a light beam with matter has varying degrees of dependence on the transverse profile of the beam. Two topics are considered here. The first topic, superfluorescence, involves the coherent emission generated by an initially inverted two-level medium. Calculations in the past have included either quantum fluctuations in the initial conditions or transverse effects, but never both. The calculations presented here include both transverse effects and quantum fluctuations. The results of the theoretical calculations are in good agreement with previous experimental results, both quantitatively in terms of time delay averages and distributions and also qualitatively in terms of pulse shapes. The second topic considered, optical bistability, involves devices which exhibit two stable output light intensities (a high and a low branch) over a limited range of input light intensities. Simulations performed here show the effect of diffraction, measured in terms of the Fresnel number, on how much of the device becomes bistable. It was found that for small Fresnel number cases, the entire device become bistable. For high Fresnel number cases, only part of the device becomes bistable while the rest of the device remains low transmitting. The results of an experiment performed are consistent with the simulations. (Author)
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.
We have formulated the quantum theory of formation of the superfluorescence pulse from a pencil-like amplifier. The coupled system of the linearized Bloch equations for two-level atoms and the three-dimensional wave equation for the electric field has been solved in the geometrical-optics approximation for Fresnel numbers $F \ensuremath{\ge} 1$. Several properties of the superfluorescence pulse are discussed. The spatial coherence function of the pulse has been found as a function of the Fresnel number. This function provides a precise formulation of the concept of "diffraction modes." The delay-time distribution function for various Fresnel numbers has been found. The results are consistent with experimental data and numerical simulations.
Correlation of the parameters (delay times, peak intensities) of counterpropagating superfluorescence pulses is studied within the framework of a one-dimensional theoretical model depending on the sharpness of boundaries of the excitation region. The concentration N 0 of the initially excited active centers is described by a smoothed profile (decaying to zero within the h − and h + wide intervals at the left-and right-hand ends of the sample, respectively), rather than by a rectangular profile. It is established that variation of the excitation region boundary widths h ± does not eliminate the observed correlations. Moreover, the correlations can significantly increase for the samples with a length of L = n λ/2, where n is any natural number and λ is the resonance light wavelength. The oscillating dependence of the parameters of counterpropagating superfluorescence pulses on the h ± values and a significant asymmetry of these pulses for h − ≠ h + are discussed.