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Physical Review Letters

Delay-Time Statistics and Inhomogeneous Line Broadening in Superfluorescence

Fritz Haake, Joseph W. Haus, Harald King, Guntram Schröder, Roy J. Glauber

Published Aug 18, 1980• 69 citations
Publisher

Abstract

A probability distribution $W(t)$ for the delay time $t$ of superfluorescent pulses is defined in terms of a functional integral, which is evaluated exactly. Although the linearized early-stage dynamics is used, full agreement is obtained with previous numerical and experimental results for the variance of $t$ and, more generally, for the shape of $W(t)$. The effect of inhomogeneous broadening of the atomic levels, showing that decrease of the dephasing time ${{T}_{2}}^{*}$ leads to substantial increase of the mean and variance of $t$, is also analyzed.

DOI
10.1103/physrevlett.45.558

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Physical Review A•Mar 1, 1983•50 citations
A probability distribution W(t)W(t)W(t) for the delay time ttt of superfluorescent pulses is defined in terms of a functional integral, which is evaluated exactly. Although the linearized early-stage dynamics is used, full agreement is obtained with previous numerical and experimental results for the variance of ttt and, more generally, for the shape of W(t)W(t)W(t). The effect of inhomogeneous broadening of the atomic levels, showing that decrease of the dephasing time T2∗{{T}_{2}}^{*}T2​∗ leads to substantial increase of the mean and variance of ttt, is also analyzed.
We discuss the delay-time statistics of superfluorescent pulses by introducing the concept of a passage time at which the field intensity reaches a specified value. Such passage times can be estimated accurately from the early-stage linear dynamics of the radiating system. The distribution function for the passage times, we show, can be expressed as a functional integral over the random zero point fluctuations of the atomic polarization. The functional integral can be evaluated exactly and reduced to an elementary closed form. Our theory applies both to the sharp and the broadened atomic line. The results we obtain agree with experimental data and numerical calculations already in existence for the unbroadened line. For the broadened line we predict substantial changes of the mean value and the variance of the passage time as the width of the atomic line increases. We also investigate the influence of the shape of the spectral distribution of atomic frequencies and show that the delay statistics depend significantly on the way in which the spectral density decreases in the wings of the atomic line.
We have developed a theoretical model that allows us to study the delay-time statistics of superfluorescent pulses for the case of collisional broadening with a Lorentzian line shape. Our model is based upon a more robust definition of delay time, based upon an energy rather than an intensity reference. We calculate the probability distribution of delay times in the initiation regime of superfluorescence, where the inversion is essentially constant. We find that, as the collisional dephasing rate is increased, the mean delay time increases and the distribution broadens. We compare our results with those for the case of inhomogeneous broadening with a Lorentzian line shape and find the unexpected result that the statistical behavior is identical.
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.
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.
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.