Journal of Physics B Atomic and Molecular Physics

Spontaneous emission by a fully excited system of three identical atoms

Helen S. Freedhoff

Published Oct 14, 1986• 18 citations

Abstract

The Lehmberg-Agarwal master equation is used to study cooperative spontaneous emission by a system of three identical two-level atoms. A choice of basis states is described which allows an exact solution to be obtained for any geometrical configuration, any sample size, and any initial conditions of the atomic system, fully including both near-field and super-radiance effects.

DOI: 10.1088/0022-3700/19/19/017

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Journal of Physics B Atomic and Molecular Physics

Spontaneous emission by a fully excited system of three identical atoms

Helen S. Freedhoff

Published Oct 14, 1986• 18 citations

Publisher

Abstract

DOI: 10.1088/0022-3700/19/19/017

Related papers

Cooperative Spontaneous Emission from an Initially Fully Excited System of Three Identical Two‐level Atoms

Th. Richter

Abstract Using Lehmberg's master equation and taking into account dipole‐dipole near‐field interaction the time‐dependent intensity pattern, intensity – and angular correlations of the cooperative spontaneous radiation from an initially fully excited system of three identical two‐level atoms are calculated. The intensity pattern developes lobes in different directions at different times in such a way that contrary to the two‐atom system the spatial distribution of the total number of the emitted photons differs from that of the single‐atom radiation pattern. In particular three atoms placed in proper distances on a straight line emit preferentially their energy along this line. At the beginning of the emission process the probability to find two photons simultaneously in the same (or geometrical equivalent) direction is always highest but in the course of time, contrary to the two‐atom case, it may well happen that the probability of finding two photons in quite different directions becomes more favoured.Abstract Using Lehmberg's master equation and taking into account dipole‐dipole near‐field interaction the time‐dependent intensity pattern, intensity – and angular correlations of the cooperative spontaneous radiation from an initially fully excited system of three identical two‐level atoms are calculated. The intensity pattern developes lobes in different directions at different times in such a way that contrary to the two‐atom system the spatial distribution of the total number of the emitted photons differs from that of the single‐atom radiation pattern. In particular three atoms placed in proper distances on a straight line emit preferentially their energy along this line. At the beginning of the emission process the probability to find two photons simultaneously in the same (or geometrical equivalent) direction is always highest but in the course of time, contrary to the two‐atom case, it may well happen that the probability of finding two photons in quite different directions becomes more favoured.

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

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Wei Feng, Yong Li, Shi‐Yao Zhu

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Cooperative Spontaneous Emission from a Single‐quantum Excited Three‐atom System

Th. Richter

Abstract The total radiation rate of the cooperative spontaneous emission from an initially single‐quantum excited three‐atom system is calculated using Lehmberg's master equation and taking into account dipole‐dipole near field interaction. We find a strong influence of the precise configuration of the atomic dipoles and of the position of the initially excited atom on the temporal behaviour of the total radiation rate, especially if the atoms are separated by distances much smaller than the resonance wavelength.Abstract The total radiation rate of the cooperative spontaneous emission from an initially single‐quantum excited three‐atom system is calculated using Lehmberg's master equation and taking into account dipole‐dipole near field interaction. We find a strong influence of the precise configuration of the atomic dipoles and of the position of the initially excited atom on the temporal behaviour of the total radiation rate, especially if the atoms are separated by distances much smaller than the resonance wavelength.

Annalen der PhysikJan 1, 198112 citations

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

Ching Tsung Lee

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.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.

Physical Review AJul 1, 197715 citations

Spontaneous emission from a system of

N

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

|{z}_{0}〉

The cooperative spontaneous emission from a completely excited system of

N

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.

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.