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

Physical Review A

Rotating-Wave Approximation and Spontaneous Emission

G. S. Agarwal

Published Nov 1, 1971• 82 citations
Publisher

Abstract

Spontaneous emission from two systems, namely, $N$ identical harmonic oscillators and $N$ identical two-level atoms, is studied without the use of the rotating-wave approximation. Certain new features of spontaneous emission, for instance, the dependence of the radiation rate on the initial dipole moment phase, are discussed.

DOI
10.1103/physreva.4.1778

Related papers

Spontaneous emission by two initially excited atoms

Kikuo Ujihara, Ho Trung Dung

We present an analysis of the spontaneous emission by two identical two-level atoms a distance R apart, both of which are initially in the upper state. This analysis applies for two interacting atoms emitting both in a free space and in a cavity structure. In this paper we use the Heisenberg picture. We work under the dipole approximation and the rotating-wave approximation. We assume that all the field modes are initially in the vacuum state and the two atoms A and B are in the upper state. We will concentrate on the motion of the product of the dipole operators of two atoms. Because we have two excitations at t=0, and because we work under rotating-wave approximation, we will always have two excitations, i.e., two excited atoms, or one excited atom plus one photon, or two unexcited atoms with two photons.

Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)•Jan 20, 2003•0 citations

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

S. Swain

The problem of a single two level atom in its excited state, placed in a cavity in which no photons are present initially, and interacting with N modes of the electromagnetic field is solved exactly in the electric dipole, rotating wave approximation. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time dependent wavefunction of the system is given. These quantities are evaluated analytically for N=1,2 and 3, and the time dependent properties of the system for these values of N are investigated and discussed.

Journal of physics. A, Proceedings of the Physical Society. General•Nov 1, 1972•36 citations

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

S. Swain

For Pt. I see ibid. vol.5, p.1587(1972). The problem considered is that of spontaneous emission by a single, two level atom which interacts with N modes of the electromagnetic field in the electric dipole, rotating wave approximation. The equations derived in Pt. I are solved numerically for N = 4,5,10,11,20,21 and 201 to find the eigenvalues and various time dependent properties of the system. The connection between this approach and conventional Wigner-Weisskopf theory is discussed.

Journal of physics. A, Proceedings of the Physical Society. General•Nov 1, 1972•12 citations

Rotating-Wave Approximation and Spontaneous Emission

G. S. Agarwal

This is an addendum to the author's previous paper [Phys. Rev. A 4, 1778 (1971)], where master equations describing the spontaneous emission from a collection of identical two-level atoms and oscillators were derived without the use of rotating-wave approximation. However, the terms corresponding to the frequency shifts were not adequately included. Here such terms are properly treated and the appropriate master equation is given. Explicit form of the frequency-shift terms is also given.

Physical Review A•Mar 1, 1973•48 citations

Interaction of N atoms with the radiation field in the restricted rotating-wave approximation. I. General theory

Miguel Orszag

A quantum model of N two-level atoms coupled by a single-mode radiation field under a restricted rotating-wave approximation is studied, assuming a dipole-interaction approximation and that the total number of excitations of the system is conserved. A solution is found for the case of spontaneous emission when the atoms are initially prepared in the state of complete inversion. This solution is valid for all times. A general solution of the model is also found for any initial configuration of the system, in the cases of both spontaneous and stimulated emission. This solution is in a closed form and valid for times tau < tau max .

Journal of Physics A Mathematical and General•Nov 1, 1979•8 citations

Spontaneous emission by two atoms with different resonance frequencies

Maurice T. Raiford

This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.

Physical Review A•Mar 1, 1974•11 citations
Spontaneous emission from two systems, namely, NNN identical harmonic oscillators and NNN identical two-level atoms, is studied without the use of the rotating-wave approximation. Certain new features of spontaneous emission, for instance, the dependence of the radiation rate on the initial dipole moment phase, are discussed.
We present an analysis of the spontaneous emission by two identical two-level atoms a distance R apart, both of which are initially in the upper state. This analysis applies for two interacting atoms emitting both in a free space and in a cavity structure. In this paper we use the Heisenberg picture. We work under the dipole approximation and the rotating-wave approximation. We assume that all the field modes are initially in the vacuum state and the two atoms A and B are in the upper state. We will concentrate on the motion of the product of the dipole operators of two atoms. Because we have two excitations at t=0, and because we work under rotating-wave approximation, we will always have two excitations, i.e., two excited atoms, or one excited atom plus one photon, or two unexcited atoms with two photons.
The problem of a single two level atom in its excited state, placed in a cavity in which no photons are present initially, and interacting with N modes of the electromagnetic field is solved exactly in the electric dipole, rotating wave approximation. Expressions for the energy eigenvalues and eigenvectors are derived, and an exact representation for the time dependent wavefunction of the system is given. These quantities are evaluated analytically for N=1,2 and 3, and the time dependent properties of the system for these values of N are investigated and discussed.
A quantum model of N two-level atoms coupled by a single-mode radiation field under a restricted rotating-wave approximation is studied, assuming a dipole-interaction approximation and that the total number of excitations of the system is conserved. A solution is found for the case of spontaneous emission when the atoms are initially prepared in the state of complete inversion. This solution is valid for all times. A general solution of the model is also found for any initial configuration of the system, in the cases of both spontaneous and stimulated emission. This solution is in a closed form and valid for times tau < tau max .
This paer is concerned with the spontaneous-emision rate of two two-level atoms which have different resonance frequencies and whose spatial separation is much smaller than the corresponding mean resonance wavelength. The total radiation rate of such an atom pair is calculated completely quantum mechanically as a function of time. Explicit results are given for various values of the frequeny difference and the following initial states of the atomic system: (a) both atoms in the excited state, (b) one taom in the excited state the other in the ground state, (c) a symmetric combination of those states, (d) an antisymmetric combination of those states, and product states where each atom is in an equal superposition of its ground and excited states, for which the expectation values of the dipole moments are nonzero and may be considered to be osicillating (e) in phase, (f) in opposite phae, and (g) in random phase. Oscillations are found to appear in the radiation rate in all cases when the frequency difference becomes larger than a certain critical value, viz., the mean half-width of the atomic resonance lines. The oscillations are especially pronounced for the initial atomic states (c), (d), (e), and (f), which exhibit superradiance and radiation trapping. For example, for atoms in the antisymmetric state (d), for which the radiation from two identical atoms is zero (completely self-trapped), a pair of nonidentical atoms begins radiating and even oscillates into the superradiant region at times.