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Physical Review A

Comments on some recent solutions of the superradiant master equation

Lorenzo M. Narducci, C. Alton Coulter, Charles M. Bowden

Published Feb 1, 1974• 9 citations
Publisher

Abstract

The results of Haake and Glauber concerning the solution of the superradiant master equation of Bonifacio et al., are compared with those of Degiorgio and Ghielmetti. This is accomplished by using the atomic coherent-state representation of Arecchi et al., which provides a convenient mathematical formalism for the comparison. The relationship of these results to the recently presented exact Fokker-Planck-equation representation of Narducci, Coulter, and Bowden is discussed. This representation is used to prove the equivalence of the main conclusions derived by Degiorgio and Ghielmetti to a special case of the results of Haake and Glauber. In addition, it is shown that the expectation values calculated by Haake and Glauber are particular solutions of a coupled hierarchy of equations of motion constructed from the exact Fokker-Planck equation, neglecting the noise and diffusion terms.

DOI
10.1103/physreva.9.999

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The results of Haake and Glauber concerning the solution of the superradiant master equation of Bonifacio et al., are compared with those of Degiorgio and Ghielmetti. This is accomplished by using the atomic coherent-state representation of Arecchi et al., which provides a convenient mathematical formalism for the comparison. The relationship of these results to the recently presented exact Fokker-Planck-equation representation of Narducci, Coulter, and Bowden is discussed. This representation is used to prove the equivalence of the main conclusions derived by Degiorgio and Ghielmetti to a special case of the results of Haake and Glauber. In addition, it is shown that the expectation values calculated by Haake and Glauber are particular solutions of a coupled hierarchy of equations of motion constructed from the exact Fokker-Planck equation, neglecting the noise and diffusion terms.
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.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
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.
The cooperative spontaneous emission from a system of NNN identical two-level atoms, excited arbitrarily and interacting with a single mode of radiation, is studied by using the Bonifacio-Schwendimann-Haake approach. The evolution of the probability distribution of the atomic system is considered as a Markov process described by a time-evolution matrix (TEM). The exact analytic expressions for the elements of this TEM are obtained by solving the superradiance master equation. It is proved that the TEM satisfies the normalization condition and the Chapman-Kolmogorov equation as required. The exact general formulas for the expectation values of various physical observables, expressed in terms of an arbitrary initial distribution, are obtained. Finally, these results are applied to a specific example, the superradiant initial state.
The cooperative spontaneous emission from a completely excited system of NNN identical two-level atoms is studied under the condition that the cooperation number is conserved. The master equation for such a system has been solved analytically. The exact expressions for the probability distribution and the expectation values of various observable physical quantities are presented. Through a power-series expansion of the exact photon-number expectation value and by using properties of Stirling numbers of the second kind, it is shown that our result agrees with that of Haake and Glauber asymptotically. As by-products, some new identities involving binomial coefficients and Stirling numbers of the second kind have also been obtained. The exact solution of the same master equation for arbitrary initial conditions will be presented in a subsequent publication.