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Physical Review Letters

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

C. T. Lee

Published Oct 8, 1979• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.43.1110

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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.

Physical Review A•Mar 1, 1983•50 citations
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.
Recent work is reviewed where dynamic diffraction coupling is examined in Superfluorescence SF with use of a one-way semiclassical model in which diffraction and transverse density variations are rigorously included. The Cs data are correctly simulated for the first time in conjunction with both an average tipping angle and quantum fluctuation (either longitudinal or transverse) at the initiating stage. This comprehensive review also encompasses rigorously the effects of pump dynamics on the SF evolution. The terminology is unified. Specification of certain pump beam initial conditions at a given frequency results in specific SF characteristics at another frequency as recently observed in CH3F and Ba. Pump dynamics are studied in both the average value of the tipping angle and quantum fluctuation at the initiation stage.
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.
To demonstrate its attractiveness, the distribution function for the atomic coherent states called the QQQ representation is used to describe the evolution of superfluorescence. An exact Fokker-Planck equation in the phase space is obtained. It indicates that the dynamics is almost completely classical, even in the very beginning. This implies that all the fluctuations in superfluorescence must originate from the initial distribution corresponding to a complete inversion. Therefore, in a sense, the initial tipping angle possesses physical reality and is not fictitious.
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.