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Physical Review A

Exact solution of the superradiance master equation. II. Arbitrary initial excitation

Ching Tsung Lee

Published Jul 1, 1977• 15 citations
Publisher

Abstract

The cooperative spontaneous emission from a system of $N$ 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.

DOI
10.1103/physreva.16.301

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Physical Review A•Nov 1, 1976•9 citations
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.
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​〉.
In a previous investigation, a general theory for spontaneous emission from a system of NNN identical atoms or molecules was developed. This theory was based on the master equation recently derived in another paper. In that paper, the master equation relating to spontaneous emission from a system of harmonic oscillators was also derived. In the present investigation, the normally ordered correlation functions for the oscillator system are calculated and these are then used to calculate the radiation-field correlation functions in the far zone. These correlation functions are compared for two different modes of excitation, viz., (i) when each of the oscillators is excited initially to a Fock state, and (ii) when each of the oscillators is excited to some coherent state. It is found that the even-order (2n2n2n) correlation functions for the second mode of excitation (superradiant excitation) are of order N2n{N}^{2n}N2n higher than those for the first mode of excitation. It is also shown that the photoelectron counting distribution for the superradiant excitation is Poissonian. Finally, the non-Markoffian effects in the spontaneous emission are studied in detail and their connection with exact results is described.
In this paper we discuss the quantum features of cooperative radiation processes in a two-level system taking into account non-Markovian effects. If the initial condition is spatially uniform, the many-mode atom-field master equation which has been derived and semiclassically discussed in a previous paper, reduces to the well-known single-mode model laser master equation. Hence, using a suitable projection operator, we derive a generalized non-Markovian master equation which turns out to be correct already in the first Born approximation. This master equation incorporates the effects both of stimulated emission and absorption and of spontaneous emission. When stimulated processes can be neglected, it reduces to a previously known Markovian superradiance master equation, whereas in general it describes non-Markovian oscillations of the radiated intensity. Finally we derive a linear system of equations for the probability of occupation of the Dicke states which generalizes a previously discussed system, taking into account the contribution of stimulated processes.
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.
A system of NNN identical two-level atoms, distributed over an extended region, completely excited initially, and interacting with all modes of radiation field is considered. A rate equation is proposed to describe the evolution of such a system and the transition from initial nondirectional emission to eventual directional emission is reinvestigated. An analytical expression is obtained for the critical point of this transition.