Superfluorescence: Quantum-mechanical derivation of Maxwell-Bloch description with fluctuating field source

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 ${\tau }_D$, 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 ${\tau }_D=\left(\frac{{\tau }_R}{4}\right){\left[\ln {\left(2\mathrm{}\pi N\right)}^{\frac{1}{2}}\right]}^2$, where ${\tau }_R$ is the radiation time for collective decay and $N$ is the number of atoms. The corresponding effective initial tipping angle roughly equals $\frac{2}{{\mathrm{}\left(N\right)\mathrm{}}^{\frac{1}{2}}}$. A Fokker-Planck equation is derived to describe the statistics of the initial development of the tipping angle. The variance $\Delta {\tau }_D$ of the delay of the superfluorescence pulse satisfies approximately $\Delta {\tau }_D=\frac{2.3}{\ln N}$. A brief comparison with previous treatments of superfluorescence is given.