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Physical Review A

Relation between the delay time and the tipping angle for superradiance

Xi‐an Mao, Jin-Hong Chen

Published Dec 1, 1999• 2 citations
Publisher

Abstract

In optical superradiance studies, data were reported supporting a relation between the delay time $({\ensuremath{\tau}}_{D})$ and the tipping angle $({\ensuremath{\delta}}_{0}),$ ${\ensuremath{\tau}}_{D}=({\ensuremath{\tau}}_{R}/4)[\mathrm{ln}({\ensuremath{\delta}}_{0}/2\ensuremath{\pi}){]}^{2},$ where ${\ensuremath{\tau}}_{R}$ is the radiation damping time constant. We found that the same data also satisfy another relation, ${\ensuremath{\tau}}_{D}={\ensuremath{\tau}}_{R}\mathrm{ln}[\mathrm{cot}({\ensuremath{\delta}}_{0}/2)].$ The derivation of the two relations is reexamined. It is found that the former relation does not have sound theoretical background, and could lead to considerable errors when it is used to interpret experimental results.

DOI
10.1103/physreva.60.5140

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•Jan 1, 1978•1 citation
In optical superradiance studies, data were reported supporting a relation between the delay time (\ensuremathτD)({\ensuremath{\tau}}_{D})(\ensuremathτD​) and the tipping angle (\ensuremathδ0),({\ensuremath{\delta}}_{0}),(\ensuremathδ0​), \ensuremathτD=(\ensuremathτR/4)[ln(\ensuremathδ0/2\ensuremathπ)]2,{\ensuremath{\tau}}_{D}=({\ensuremath{\tau}}_{R}/4)[\mathrm{ln}({\ensuremath{\delta}}_{0}/2\ensuremath{\pi}){]}^{2},\ensuremathτD​=(\ensuremathτR​/4)[ln(\ensuremathδ0​/2\ensuremathπ)]2, where \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​ is the radiation damping time constant. We found that the same data also satisfy another relation, \ensuremathτD=\ensuremathτRln[cot(\ensuremathδ0/2)].{\ensuremath{\tau}}_{D}={\ensuremath{\tau}}_{R}\mathrm{ln}[\mathrm{cot}({\ensuremath{\delta}}_{0}/2)].\ensuremathτD​=\ensuremathτR​ln[cot(\ensuremathδ0​/2)]. The derivation of the two relations is reexamined. It is found that the former relation does not have sound theoretical background, and could lead to considerable errors when it is used to interpret experimental results.
According to Dicke's theory of superradiance, the relation between the delay time and the tipping angle for superradiant pulses can be expressed by , where is the radiation damping time constant. This relation is verified by nuclear magnetic resonance (NMR) experiments on a normal water sample. The radiation damping time and the delay time are independently measured, and the tipping angle is evaluated. The tipping angle can be as big as several degrees if the spin inversion is not perfect. It can also be as small as rad, when the residual coherence in the transverse plane is destroyed by a strong field gradient pulse.
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.
The authors report on an experimental study of intensity and delay fluctuations for a two-level system by analysing stochastic distribution of the polarisation of superradiant pulses emitted on the transition 6p 2 P 1 /2 to 6s 2 S 1/2 of rubidium atoms. They develop a theoretical interpretation of their results. Finally, use of these properties of polarisation fluctuations has permitted one to study the very beginning of the superradiant emission and to give a new estimate of the so-called 'tipping angle'.