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Physical Review A

Q representation of the atomic coherent states and the origin of fluctuations in superfluorescence

C. T. Lee

Published Dec 1, 1984• 23 citations
Publisher

Abstract

To demonstrate its attractiveness, the distribution function for the atomic coherent states called the $Q$ 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.

DOI
10.1103/physreva.30.3308

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The self-organization process of superfluorescence can be explained clearly basing on the thermodynamics. The present paper includes three parts: 1) the equations of superfluorescence radiation and heat transform cetion are derived, and their transient processes are analysed; 2) the possibility of the generation of superfluorescence is theoretically verified by the analysis of the linear stability of the thermo-dynamic Maxwell-Bloch equations, and the coherent process of superfluorescence is analysed by the temporal coherence function of the field of radiation obtained: 3) the effective initial tipping angle θ0 and the delay time τD are derived from the fluctuation in statistical thermodynamics. Numerical results are in good agreement with those accepted up to now.

Acta Physica Sinica•Jan 1, 1995•0 citations
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.
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 "superfluorescence equations" deduced in a previous paper are generalized to the case of finite temperature and translated into the atomic coherent-state representation of Arecchi et al. We get three coupled partial differential equations involving the quasiprobability distribution $Q(\ensuremath{\theta},t)$ of the atomic system and two suitable functions $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$ describing the stimulated processes which are relevant in oscillatory superfluorescence. Neglecting $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$, the equation for $Q(\ensuremath{\theta},t)$ reduces to the Fokker-Planck equation of Narducci et al. corrected by the contribution of thermal noise. The stationary solution is given by the unperturbed canonical state, which is shown to be the exact stationary solution of the single-mode model from which the superfluorescence equations have been deduced. These equations prescribe that $\mathfrak{M}(\ensuremath{\theta},t)$ always remain equal and opposite to $\mathcal{F}(\ensuremath{\theta},t)$, so that the superfluorescence equations reduce automatically to two equations. Eliminating $\mathcal{F}(\ensuremath{\theta},t)$ between the two remaining equations, one obtains a non-Markovian equation for $Q(\ensuremath{\theta},t)$ with derivatives of all orders in $\ensuremath{\theta}$. In the case of pure superfluorescence or super-radiance, such an equation reduces to a Markovian Fokker-Planck equation with derivatives of first and second order in $\ensuremath{\theta}$ which exhibits an interesting fourth-order contribution in the diffusion term. In fact, if one excludes a very small region around the north pole of the Bloch sphere, this term is much larger than the second-order contribution in the diffusion term of the equation of Narducci et al. In the case of oscillatory superfluorescence, the presence of memory effects enhances fluctuations, thus forbidding the existence of "classical" time evolutions.
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.
The super-radiant master equation (SME) of Bonifacio et al. is analyzed using the coherent-atomic-state representation. We have succeeded in deriving an exact Fokker-Planck equation for the density function corresponding to the reduced atomic density operator in the diagonal atomic-state representation. A solution to the Fokker-Planck equation has been provided in an elementary fashion for arbitrary atomic states which are sufficiently removed from the state of complete inversion at time zero. The general solution for arbitrary initial conditions (including the initial state of complete inversion) has been obtained using the method of eigen-function expansions and the final result expressed in terms of an integral over the initial density function. The moments of the collective atomic operators are also discussed.
The self-organization process of superfluorescence can be explained clearly basing on the thermodynamics. The present paper includes three parts: 1) the equations of superfluorescence radiation and heat transform cetion are derived, and their transient processes are analysed; 2) the possibility of the generation of superfluorescence is theoretically verified by the analysis of the linear stability of the thermo-dynamic Maxwell-Bloch equations, and the coherent process of superfluorescence is analysed by the temporal coherence function of the field of radiation obtained: 3) the effective initial tipping angle θ0 and the delay time τD are derived from the fluctuation in statistical thermodynamics. Numerical results are in good agreement with those accepted up to now.