Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Estudo sobre um modelo unidimensional para a emissão espontânea na aproximação semiclássica

Sı́lvio Crestana

Published Apr 20, 1983• 0 citations
Publisher

Abstract

Some generalizations are made onthe spontaneous emission by a plane of excited atoms, described by two leveI atom-model, in the ~1=1, ~m=1, transition and using the semiclassical radiation approximation -both discussed in the text.lnitially,theradiation rate of an infinite plane of excited atoms is investigated, using ~1=O, ~m=O, transition.It is shown that we can observe a limit solution depending on the coupling hetween field ànd matter.We simulate the effect of the emitter thickness,heretofore neglected, by using two infinite parallel planes of atoms, separated by a finite distance.Theauto-consistent set of eguation for the interaction between the radiant energy and matter was established, in the general case, when 61=1, 6m=1.Westill discussed two extreme situations: a) the planes are coincident and b) they are infinitely separated.We investigate, in some cases, by means of a computer , the influence of plane separation, the phase ~difference between electric dipoles and lnicial.energy•of each plane in the spontaneous emission behaviour when the field-matter coupling is weak.íNDICE Capítulo I

DOI
10.11606/d.54.1983.tde-25052015-161924

Related papers

Soluble one-dimensional semiclassical model for spontaneous emission

R. C. T. da Costa, Gustavo Perez Munguia

The radiation emitted by plane of excited two-level atoms is exactly calculated in the semiclassical approximation. The well-known hyperbolic-secant solution is obtained in the limit case of small plane densities.

Physical Review A•Nov 1, 1976•1 citation

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.

Physical Review A•Nov 1, 1971•62 citations

Radiative Transitions between Discrete Quantum States

Greg J. Tallents

Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.

Cambridge University Press eBooks•Feb 20, 2018•0 citations

Superradiance

Nicholas E. Rehler, Joseph H. Eberly

We introduce a straightforward quantum-electrodynamic approach to the problem of superradiant spontaneous emission from a system of two-level atoms, and also discuss two classical superradiant systems. The same physical concepts underlie our treatment of both the quantum and classical cases. Explicit expressions are found which describe the time evolution and directional character of the radiated power. Our approach applies to an arbitrary number of atoms which are confined to a region with linear dimensions which may be large compared with the mean wavelength of the emitted radiation. The radiative decay half-life of a large many-atom system is found to be much shorter than the natural lifetime of a single atom, but not as short as the half-life of a small atomic system with the same number of atoms. The far-field radiation pattern appropriate to the case of a circular cylinder of emitters which have been excited by a plane-wave pulse is considered in detail. Polar plots are presented of the radiation patterns produced when the atoms are excited by a plane-wave pulse. Exciting pulses traveling in directions both parallel and at an angle of 10\ifmmode^\circ\else\textdegree\fi{} to the axis of the cylinder are considered. A discussion and generalization of Dicke's "coherence-brightening" criterion is given.

Physical Review A•May 1, 1971•534 citations

Spontaneous Emission by a System of Identical Atoms

Demosthenes Dialetis

The spontaneous emission of a set of $N$ identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.

Physical Review A•Sep 1, 1970•42 citations

Excitation with Monochromatic Light of a Vibronically Broadened Molecular Band

Juan Alberto Beswick, R. Lefèbvre, A. M. Plumejeau

This paper studies the effect of a monochromatic electromagnetic field of arbitrary strength and duration on an isolated molecule presenting the energy level scheme which is the basis for current theories of radiationless processes in the so-called statistical limit: the system is assumed to possess bands of excited levels which are due to the dilution of well separated electric dipole allowed states into a dense set of forbidden states. The frequency of the light is set approximately equal to the transition frequency of one of these bands. Two models are set up, a quantized one with a n-photons one-mode field, and a semiclassical one with an oscillating electric field. They yield identical results. Two limiting cases, termed weak field and strong field can be obtained. In the weak field case, two different steps can be distinguished: first, the excitation of the dipole allowed state; second, the selective excitation of the molecular states at resonance with the field. The strong field case corresponds to a nonlinear response characterized by a damped nutation between ground and allowed states. After a long irradiation time the molecule is left in states which are distributed according to the level scheme corresponding to the ``molecule dressed with photons''. The spontaneous emission rate after the light is switched off is found not to depend on the particular form which is obtained for the initial state. This result, which is shown to hold for more general forms of the electric field in the semiclassical treatment, is a consequence of the analytical properties of the amplitudes.

The Journal of Chemical Physics•Apr 15, 1972•7 citations
Some generalizations are made onthe spontaneous emission by a plane of excited atoms, described by two leveI atom-model, in the ~1=1, ~m=1, transition and using the semiclassical radiation approximation -both discussed in the text.lnitially,theradiation rate of an infinite plane of excited atoms is investigated, using ~1=O, ~m=O, transition.It is shown that we can observe a limit solution depending on the coupling hetween field ànd matter.We simulate the effect of the emitter thickness,heretofore neglected, by using two infinite parallel planes of atoms, separated by a finite distance.Theauto-consistent set of eguation for the interaction between the radiant energy and matter was established, in the general case, when 61=1, 6m=1.Westill discussed two extreme situations: a) the planes are coincident and b) they are infinitely separated.We investigate, in some cases, by means of a computer , the influence of plane separation, the phase ~difference between electric dipoles and lnicial.energy•of each plane in the spontaneous emission behaviour when the field-matter coupling is weak.íNDICE Capítulo I
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 NNN identical two-level atoms is investigated further. It is shown that the equation of motion for the p(<N)p(<N)p(<N)-atom density matrix is coupled to the (p+1p+1p+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 ttt 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 1N\frac{1}{N}N1​. 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 2N2N2N 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.
Radiation can change the energy state of matter via three optical processes: spontaneous emission, absorption and stimulated emission. In spontaneous emission, an excited atomic or molecular state decays to a lower energy state with the emission of a photon of energy equal to the difference in the quantum state energies. Photon absorption is the reverse process, with a photon causing excitation from a lower to an excited quantum state. Stimulated emission was first proposed by Einstein in a paper published in 1917. He attempted to balance spontaneous emission and photon absorption, but could not get the correct balance without the postulated process of stimulated emission (see Section 4.2). With stimulated emission, an excited quantum state is stimulated to decay to a lower quantum state by an impinging photon. The existing photon survives with another identical photon being created (so that energy is conserved). The rates at which radiative transitions occur can be calculated using quantum mechanics. The time-dependent Schrodinger equation is used with the system wavefunction comprising time-varying linear combinations of the two wavefunctions for the lower and upper quantum states. In semi-classical treatments, an atom is treated quantum mechanically with the radiation field treated classically. There is a perturbation in energy associated with, say, the electric field of the radiation interacting with the electric dipole of the atom, though other interactions can be significant if the electric dipole interaction energy is small. In absorption, for example, a differential equation representing the change in ‘weighting’ of the wavefunction from the lower to upper state is obtained. A ‘time constant’ for the speed of change represents the rate of the radiative process. We examine the quantum mechanical nature of radiation absorption in Section 10.1 before discussing (Section 10.3) another important parameter arising with radiative transitions between discrete bound quantum states: the lineshape function. Quantum Theory of the Atom–Radiation Interaction An expression for the Einstein B-coefficient for photo-absorption is derived using quantum mechanics in this section. Einstein determined the detailed balance relationship between the rates of spontaneous emission, photo-absorption and stimulated emission when light interacts with an atom by considering an atomic system of two energy levels which are in equilibrium with a black-body radiation field. We use a semi-classical treatment where the atom is treated quantum mechanically and the radiation field is treated as a classical oscillating electric field.
We introduce a straightforward quantum-electrodynamic approach to the problem of superradiant spontaneous emission from a system of two-level atoms, and also discuss two classical superradiant systems. The same physical concepts underlie our treatment of both the quantum and classical cases. Explicit expressions are found which describe the time evolution and directional character of the radiated power. Our approach applies to an arbitrary number of atoms which are confined to a region with linear dimensions which may be large compared with the mean wavelength of the emitted radiation. The radiative decay half-life of a large many-atom system is found to be much shorter than the natural lifetime of a single atom, but not as short as the half-life of a small atomic system with the same number of atoms. The far-field radiation pattern appropriate to the case of a circular cylinder of emitters which have been excited by a plane-wave pulse is considered in detail. Polar plots are presented of the radiation patterns produced when the atoms are excited by a plane-wave pulse. Exciting pulses traveling in directions both parallel and at an angle of 10\ifmmode^\circ\else\textdegree\fi{} to the axis of the cylinder are considered. A discussion and generalization of Dicke's "coherence-brightening" criterion is given.
The spontaneous emission of a set of NNN identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.
This paper studies the effect of a monochromatic electromagnetic field of arbitrary strength and duration on an isolated molecule presenting the energy level scheme which is the basis for current theories of radiationless processes in the so-called statistical limit: the system is assumed to possess bands of excited levels which are due to the dilution of well separated electric dipole allowed states into a dense set of forbidden states. The frequency of the light is set approximately equal to the transition frequency of one of these bands. Two models are set up, a quantized one with a n-photons one-mode field, and a semiclassical one with an oscillating electric field. They yield identical results. Two limiting cases, termed weak field and strong field can be obtained. In the weak field case, two different steps can be distinguished: first, the excitation of the dipole allowed state; second, the selective excitation of the molecular states at resonance with the field. The strong field case corresponds to a nonlinear response characterized by a damped nutation between ground and allowed states. After a long irradiation time the molecule is left in states which are distributed according to the level scheme corresponding to the ``molecule dressed with photons''. The spontaneous emission rate after the light is switched off is found not to depend on the particular form which is obtained for the initial state. This result, which is shown to hold for more general forms of the electric field in the semiclassical treatment, is a consequence of the analytical properties of the amplitudes.