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

Two simple expressions for the spontaneous emission factor β

Martin P. van Exter, Gerard Nienhuis, J. P. Woerdman

Published Oct 1, 1996• 71 citations
Publisher

Abstract

In the literature two different simplified expressions exist for the spontaneous emission factor \ensuremath{\beta}, being the fraction of spontaneous emission radiated into a specific optical mode. These expressions are valid in two mutually exclusive regimes. By treating the coupling between one discrete cavity mode and the outside optical continuum quasi one dimensionally in the spirit of Fano [Phys. Rev. 124, 1866 (1961)], we extend the validity range of both expressions and show that they lead to essentially the same result. The relevance and limitation of this result are discussed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.3553

Related papers

Photon statistics of a cavity-QED laser: A comment on the laser–phase-transition analogy

Perry R. Rice, H. J. Carmichael

We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the ``system size'' and show that the concept of laser threshold is well-defined in the ``thermodynamic'' limit, ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.

Physical Review A•Nov 1, 1994•291 citations

Quantum theory of a one-dimensional optical cavity with output coupling. V. Spontaneous emission

Kikuo Ujihara

For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>

IEEE Journal of Quantum Electronics•Jul 1, 1988•9 citations

Spontaneous radiative decay of a continum

Dan Agassi

We obtain and discuss a closed-form expression for the time-dependent density matrix associated with spontaneous radiative decay from a continuum or a set of discrete states to a lower discrete state.Received 5 July 1983DOI:https://doi.org/10.1103/PhysRevA.30.2449©1984 American Physical Society

Physical Review A•Nov 1, 1984•43 citations

Quantum theory of spontaneous emission in a one-dimensional optical cavity with two-side output coupling

Xiao-Ping Feng, Kikuo Ujihara

A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).

Physical Review A•Mar 1, 1990•71 citations

Cavity-modified spontaneous emission: From Rabi oscillations to exponential decay

H. Gie\Sen, Jill D. Berger, Georg Mohs +2 more

We revisit the problem of spontaneous emission by a two-level system in a cavity, concentrating in particular on those aspects associated with the finite propagation time of light between the atom and the cavity boundaries. We find that spontaneous emission always occurs first at the free space rate and there is then an abrupt change in these dynamics when light reflected by the cavity mirrors returns to the atom, leading to interference effects that result either in an enhancement or a reduction of this rate or in a periodic exchange of excitation between the atom and the cavity field. From this point of view, reversible spontaneous emission in terms of single-mode ``vacuum Rabi oscillations'' can be interpreted as resulting from the interference between the multimode partial waves reflected at the cavity mirrors and the instantaneously radiated light. \textcopyright{} 1996 The American Physical Society.

Physical Review A•Apr 1, 1996•49 citations

Spontaneous emission in microcavities

R. Loudon, M.J. Adams

The transverse distributions of intensity inside and outside an optical cavity produced by spontaneous emission from atoms inside the cavity are reviewed. The mode properties inside a closed passive one-dimensional cavity are summarised and the spatial distributions of emitted light for a passive cavity with partially transmitting mirrors are discussed. A qualitative theory describes the effects of an amplifying medium inside the cavity, covering the regimes both below and above the lasing threshold. Expressions are derived for the angular spread of the output light in the presence of gain. A theory based on rigorous general relations between gain and noise provides more complete results for the spontaneous emission noise from an amplifying slab within an optical cavity.

IET Optoelectronics•Dec 10, 2007•1 citation
In the literature two different simplified expressions exist for the spontaneous emission factor \ensuremath{\beta}, being the fraction of spontaneous emission radiated into a specific optical mode. These expressions are valid in two mutually exclusive regimes. By treating the coupling between one discrete cavity mode and the outside optical continuum quasi one dimensionally in the spirit of Fano [Phys. Rev. 124, 1866 (1961)], we extend the validity range of both expressions and show that they lead to essentially the same result. The relevance and limitation of this result are discussed. \textcopyright{} 1996 The American Physical Society.
We develop a birth-death model for a cavity QED laser and analyze the dependence of the lasing threshold on the fraction, \ensuremath{\beta}, of spontaneous emission directed into the laser mode. We define threshold in terms of the Fano factor for the intracavity phonon number. We emphasize the role of \ensuremath{\beta} as a parameter characterizing the ``system size'' and show that the concept of laser threshold is well-defined in the ``thermodynamic'' limit, ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\rightarrow}\ensuremath{\infty}. An ideal cavity-QED laser operates far outside this limit and therefore, in contrast to a conventional laser, is not a threshold device. We present numerical results showing how this difference affects the noise properties of the device.
For pt.IV see J. Phys. Soc. Japan, vol.47, p.217-224 (1982). Spontaneous emission in a lossy one-dimensional cavity is analyzed under the perturbation approximation and the Wigner-Weisskopf approximation. The cavity loss due to output coupling is treated quantum mechanically by the use of a set of continuously distributed radiation modes. The enhancement and reduction factors in the spontaneous emission probability due to the presence of the cavity structure are derived. Interesting transients in spontaneous emission during a time of the order of the cavity decay time are predicted, which leads to an interpretation of the enhancement or the reduction in spontaneous emission in terms of feedback and self-interference effects. The detection of the emitted photon by a detector atom outside the cavity is also discussed.>
A quantum theory of spontaneous emission from an initially excited two-level atom in a one-dimensional optical cavity with output coupling from both sides is developed. Orthonormal mode functions with a continuous spectrum are employed, which are derived by imposing a periodic boundary condition on the whole space with a period much larger than the cavity length. The delay differential equation of the atomic state of Cook and Milonni [Phys. Rev. A 35, 5081 (1987)] is re-derived in a strict manner, where the reflectivity of the cavity mirrors is included naturally in the mode functions. An approximate solution at a single-resonant-mode limit shows the results of ``vacuum'' Rabi oscillation in an underdamped cavity and enhanced spontaneous emission rate in an overdamped cavity. For the latter case, it is found that in the optical range the spontaneous emission rate is enhanced by a factor F (finesse of the cavity).
We revisit the problem of spontaneous emission by a two-level system in a cavity, concentrating in particular on those aspects associated with the finite propagation time of light between the atom and the cavity boundaries. We find that spontaneous emission always occurs first at the free space rate and there is then an abrupt change in these dynamics when light reflected by the cavity mirrors returns to the atom, leading to interference effects that result either in an enhancement or a reduction of this rate or in a periodic exchange of excitation between the atom and the cavity field. From this point of view, reversible spontaneous emission in terms of single-mode ``vacuum Rabi oscillations'' can be interpreted as resulting from the interference between the multimode partial waves reflected at the cavity mirrors and the instantaneously radiated light. \textcopyright{} 1996 The American Physical Society.
The transverse distributions of intensity inside and outside an optical cavity produced by spontaneous emission from atoms inside the cavity are reviewed. The mode properties inside a closed passive one-dimensional cavity are summarised and the spatial distributions of emitted light for a passive cavity with partially transmitting mirrors are discussed. A qualitative theory describes the effects of an amplifying medium inside the cavity, covering the regimes both below and above the lasing threshold. Expressions are derived for the angular spread of the output light in the presence of gain. A theory based on rigorous general relations between gain and noise provides more complete results for the spontaneous emission noise from an amplifying slab within an optical cavity.