Physical Review A

Closed equations of motion for a system of N two-level atoms in the case of spontaneous emission

Gerhard Adam, JOSIP SEKE

Published Jun 1, 1981• 24 citations

Abstract

In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of $N$ identical two-level atoms. Three different observation levels of the atomic system are considered. First, exact closed integrodifferential equations for the expectation values of each observation level are obtained. Second, making the standard approximations---Born, Markov, and long-time approximations, we find that these equations reduce to closed differential equations for the expectation values. The quality of the Born approximation for different observation levels is also discussed. Moreover, in the obtained equations no kind of rotating-wave approximation was made.

DOI: 10.1103/physreva.23.3118

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Exact Equations of Motion and Numerical Results for Two-atom Spontaneous Emission in a Damped Cavity

JOSIP SEKE, G. Adam, Otto Hittmaır

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Closed equations of motion for expectation values of collective operators for spontaneous emission in the presence of external driving fields

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Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical SciencesDec 18, 19841 citation

Rotating-Wave Approximation and Spontaneous Emission

G. S. Agarwal

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Physical Review AMar 1, 197348 citations

Physical Review A

Closed equations of motion for a system of N two-level atoms in the case of spontaneous emission

Gerhard Adam, JOSIP SEKE

Published Jun 1, 1981• 24 citations

Publisher

Abstract

In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of

N

identical two-level atoms. Three different observation levels of the atomic system are considered. First, exact closed integrodifferential equations for the expectation values of each observation level are obtained. Second, making the standard approximations---Born, Markov, and long-time approximations, we find that these equations reduce to closed differential equations for the expectation values. The quality of the Born approximation for different observation levels is also discussed. Moreover, in the obtained equations no kind of rotating-wave approximation was made.

DOI: 10.1103/physreva.23.3118

Related papers

Exact Equations of Motion and Numerical Results for Two-atom Spontaneous Emission in a Damped Cavity

JOSIP SEKE, G. Adam, Otto Hittmaır

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Master-Equation Approach to Spontaneous Emission

G. S. Agarwal

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 $|{z}_{0}〉$ .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 $|{z}_{0}〉$.

Physical Review ANov 1, 1970232 citations

Master-Equation Approach to Spontaneous Emission. III. Many-Body Aspects of Emission from Two-Level Atoms and the Effect of Inhomogeneous Broadening

G. S. Agarwal

A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from $N$ identical two-level atoms is investigated further. It is shown that the equation of motion for the $p(<N)$ -atom density matrix is coupled to the ( $p+1$ )-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time $t$ can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of $\frac{1}{N}$ . The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of $2N$ coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from $N$ identical two-level atoms is investigated further. It is shown that the equation of motion for the $p(<N)$-atom density matrix is coupled to the ($p+1$)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time $t$ can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of $\frac{1}{N}$. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of $2N$ coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.

Physical Review ANov 1, 197162 citations

An exact analysis of spontaneous emission by a single two level atom in the rotating wave approximation II. Numerical results

S. Swain

Journal of physics. A, Proceedings of the Physical Society. GeneralNov 1, 197212 citations

Closed equations of motion for expectation values of collective operators for spontaneous emission in the presence of external driving fields

JOSIP SEKE

Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical SciencesDec 18, 19841 citation

Rotating-Wave Approximation and Spontaneous Emission

G. S. Agarwal

Physical Review AMar 1, 197348 citations

In this paper the equations of motion for open systems, derived recently by one of the present authors, are applied to the case of spontaneous emission from a system of

N

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}〉

A general theory of spontaneous emission was developed in Papers I and II of this series. In the present paper, the mastre equation describing spontaneous emission from

N

identical two-level atoms is investigated further. It is shown that the equation of motion for the

p(<N)

-atom density matrix is coupled to the (

p+1

)-atom density matrix. We first consider the initial excitation of the system to state $|{\ensuremath{\theta}}_{0},{\ensuremath{\phi}}_{0}〉$ (${\ensuremath{\theta}}_{0}<\ensuremath{\pi}$) and solve the hierarchy of equations so obtained in the "uncorrelated approximation." It is found that the presence of other atoms is equivalent to an external field which is determined from a self-consistent analysis. It is also found that in this approximation the state of the atom at time

t

can be described by a single parameter $\ensuremath{\theta}(t)$ ($\ensuremath{\theta}<\ensuremath{\pi}$), with ${sin}^{2}\frac{1}{2}\ensuremath{\theta}$ giving the probability that the atom is to be found in the excited state. Next, some improvements over the uncorrelated approximation are given. This is done by decoupling the equation of motion for higher-order mean values. It is found that the fluctuation $〈({S}_{i}^{Z}\ensuremath{-}〈{S}_{i}^{Z}〉)({S}_{j}^{Z}\ensuremath{-}〈{S}_{j}^{Z}〉)〉$ for any pair of atoms is of the order of

\frac{1}{N}

. The case when ${\ensuremath{\theta}}_{0}=\ensuremath{\pi}$ is considered separately and we obtain an expression for the radiation rate by making a "Hartree-Fock" type of approximation on the two-particle mean values. In this case, the behavior of the radiation rate is found to differ markedly from the "sech" behavior. It is then shown that the entire dynamics of two-level atoms emitting spontaneously can be described by a set of

2N

coupled first-order equations which clearly exhibit the type of nonlinearity (which is the analog of the van der Pol type of nonlinearity) for this problem and provide a better understanding of spontaneous emission. Finally in Sec. V, the theory is extended to include the effects of inhomogeneous broadening, and the functional dependence of the radiation rate on the atomic-line-shape factor is obtained.