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

Journal of Atomic Molecular Condensed Matter and Nano Physics

Bloch Equations Versus Balance Equation: From F. Bloch to A. Einstein

Valery Alexandrovich Astapenko, M. I. Mutafyan

Published Dec 31, 2023• 0 citations
Publisher

Abstract

We investigate the correspondence between balance equation for population inversion of Two-Level System (TLS) and equations for optical Bloch vector which describe the interaction of electromagnetic field and TLS. Based on an analytical treatment, the condition for such a correspondence is established and an expression is obtained for the rate of the radiative transition. Particularly, the problem of applicability of the perturbation theory when writing the balance equation, which also describes a strong change in the population inversion, is resolved.

DOI
10.26713/jamcnp.v10i1.1886

Related papers

Symmetries in the optical Bloch equations and multiphoton processes

Bruno Zambon

The optical Bloch equations exhibit symmetries that are directly related to the probabilities of the processes of absorption and stimulated emission. As a matter of fact, a picture similar to that holding for a two-level system can be recovered also in multilevel systems and, in particular, for any multiphoton process tuned between a given couple of levels. Thus, new quantities, such as the generalized Einstein B coefficients, directly related to the probabilities of the involved process, and atomic level indicators, whose inversion signals the existence of gain for that given process, can be defined. It is shown how these lead to valuable conceptual simplifications and to the disclosure of new insights into some popular problems of quantum optics. In particular, the issue of an efficient gain process to be realized within a multilevel atom-field configuration, a problem closely related to the widely investigated problem of lasing configurations without population inversion, can be viewed entirely as a rate problem and therefore analyzed with increased confidence.

Physical Review A•Apr 19, 2001•0 citations

Rate-equation and Bloch-equation Theories of Radiation Pressure on Two-level Atoms

A Al-Hilfy, R. Loudon

Theories of radiation pressure based on rate equations and on the optical Bloch equations are developed within a common framework that allows comparison of the two. The latter method is used to determine the time dependence of the mean momentum of a two-level atom subjected to monochromatic coherent light for general values of various broadening parameters and detuning. The expression for the corresponding radiation pressure force in the limit of steady-state atomic level populations includes previous results as special cases. The rate equation method provides a fully quantum-mechanical calculation of an expression obtained by Einstein for the rate of change of mean-square atomic momentum in thermal equilibrium.

Optica Acta International Journal of Optics•Sep 1, 1985•5 citations

Emergence of electromagnetically induced absorption in a perturbation solution of optical Bloch equations

Jelena Dimitrijević, D. Arsenović, B. M. Jelenkovic

We study phenomenon of electromagnetically induced absorption (EIA) in the Hanle configuration by applying a perturbative method to solve linear system of optical Bloch equations (OBEs) for the case of closed F g = 1 → F e = 2 transition. The method is applied assuming stationary case and a weak laser fields (Ω ≪ Γ, i.e., Rabi frequency small compared to spontaneous emission rate). This way, we calculate (both numerically and analytically) higher order corrections to density matrix. Odd corrections give contributions to optical coherences, while even corrections contribute to populations and Zeeman coherences. The method gives insight into mechanism of transfer of coherences and transfer of populations between Zeeman sublevels. We have found that the ground-state coherences (2nd correction to coherences) are crucial for the 4th correction to the change of populations which brings EIA type behavior of the Hanle spectrum. Using exact analytical expressions we further discuss further the role of decoherence of the ground-state sublevels when forming EIA.

Laser Physics•Apr 2, 2010•0 citations

A semiclassical optics derivation of Einstein’s rate equations

Robert Höppner, Eugenio Roldán, Germán J. de Valcárcel

We provide a semiclassical optics derivation of Einstein’s rate equations (ERE) for a two-level system illuminated by a broadband light field, setting a limit on their validity that depends on the spectral properties of the light field. Starting from the optical Bloch equations for individual atoms, the ensemble averaged atomic inversion is shown to follow ERE under two concurrent hypotheses: (i) the decorrelation of the inversion at a given time from the field at later times and (ii) a Markov approximation owing to the short correlation time of the light field. When the latter hypothesis is relaxed, we find effective Bloch equations for the ensemble average in which the atomic polarization decay rate is increased by an amount equal to the width of the light spectrum, which allows its adiabatic elimination for a large enough spectral width. The use of a phase-diffusion model of light allows us to check our results and hypotheses using numerical simulations of the corresponding stochastic differential equations. We take into account both light bandwidth and atomic linewidth, which allows us to discuss the differences existing between rate equations in the limits of large light bandwidth or large atomic linewidth. Only in the former case does one obtain ERE with the correct expression for Einstein’s B coefficient.

arXiv:1105.4120•American Journal of Physics•Sep 18, 2012•9 citations

The equation of two level system-intensive fields interaction and its analytic solution

Li C

In this paper, the general optical Bloch equation describing interaction between intensive fields and two level system is obtained without using rotating wave approximation. The first-order solution of this equation is obtained by iterative method and analytic formulas of transition probability as a function of time, perturbation frequency and strength are presented. These results are in agreement with those obtained by an iterative step-wise power series and numerical integration solution for the time dependent Schr(?)dinger equation. And these simplified and clear mathmatical formulas facilitate the application of physics.

Acta Optica Sinica•Jan 1, 1983•0 citations

Departures From The Optical Bloch Equations

A. A. Villaeys

It is the aim of the present work to show some departures from the well-known optical Bloch equations usually obtained when the damping of a two-level system driven by a Glauber coherent mode of the radiation field is treated in the Markovian approximation. In particular, we show that for an arbitrary intensity we have a set of 4 n optical Bloch equations which can be reduced to the usual optical Bloch equations when the population of the mode becomes important. A criterion for the quasi-classical evolution can be deduced. Obviously, in both limits of an arbitrarily weak or strong excitation intensity the Mollow's results are obtained. In addition, we do not need here the introduction of a particular form like the plane wave used by Mollow for the corresponding classical field--as the problem is solved explicitly.

Optical Engineering•Jun 1, 1983•0 citations
We investigate the correspondence between balance equation for population inversion of Two-Level System (TLS) and equations for optical Bloch vector which describe the interaction of electromagnetic field and TLS. Based on an analytical treatment, the condition for such a correspondence is established and an expression is obtained for the rate of the radiative transition. Particularly, the problem of applicability of the perturbation theory when writing the balance equation, which also describes a strong change in the population inversion, is resolved.
The optical Bloch equations exhibit symmetries that are directly related to the probabilities of the processes of absorption and stimulated emission. As a matter of fact, a picture similar to that holding for a two-level system can be recovered also in multilevel systems and, in particular, for any multiphoton process tuned between a given couple of levels. Thus, new quantities, such as the generalized Einstein B coefficients, directly related to the probabilities of the involved process, and atomic level indicators, whose inversion signals the existence of gain for that given process, can be defined. It is shown how these lead to valuable conceptual simplifications and to the disclosure of new insights into some popular problems of quantum optics. In particular, the issue of an efficient gain process to be realized within a multilevel atom-field configuration, a problem closely related to the widely investigated problem of lasing configurations without population inversion, can be viewed entirely as a rate problem and therefore analyzed with increased confidence.
Theories of radiation pressure based on rate equations and on the optical Bloch equations are developed within a common framework that allows comparison of the two. The latter method is used to determine the time dependence of the mean momentum of a two-level atom subjected to monochromatic coherent light for general values of various broadening parameters and detuning. The expression for the corresponding radiation pressure force in the limit of steady-state atomic level populations includes previous results as special cases. The rate equation method provides a fully quantum-mechanical calculation of an expression obtained by Einstein for the rate of change of mean-square atomic momentum in thermal equilibrium.
We study phenomenon of electromagnetically induced absorption (EIA) in the Hanle configuration by applying a perturbative method to solve linear system of optical Bloch equations (OBEs) for the case of closed F g = 1 → F e = 2 transition. The method is applied assuming stationary case and a weak laser fields (Ω ≪ Γ, i.e., Rabi frequency small compared to spontaneous emission rate). This way, we calculate (both numerically and analytically) higher order corrections to density matrix. Odd corrections give contributions to optical coherences, while even corrections contribute to populations and Zeeman coherences. The method gives insight into mechanism of transfer of coherences and transfer of populations between Zeeman sublevels. We have found that the ground-state coherences (2nd correction to coherences) are crucial for the 4th correction to the change of populations which brings EIA type behavior of the Hanle spectrum. Using exact analytical expressions we further discuss further the role of decoherence of the ground-state sublevels when forming EIA.
We provide a semiclassical optics derivation of Einstein’s rate equations (ERE) for a two-level system illuminated by a broadband light field, setting a limit on their validity that depends on the spectral properties of the light field. Starting from the optical Bloch equations for individual atoms, the ensemble averaged atomic inversion is shown to follow ERE under two concurrent hypotheses: (i) the decorrelation of the inversion at a given time from the field at later times and (ii) a Markov approximation owing to the short correlation time of the light field. When the latter hypothesis is relaxed, we find effective Bloch equations for the ensemble average in which the atomic polarization decay rate is increased by an amount equal to the width of the light spectrum, which allows its adiabatic elimination for a large enough spectral width. The use of a phase-diffusion model of light allows us to check our results and hypotheses using numerical simulations of the corresponding stochastic differential equations. We take into account both light bandwidth and atomic linewidth, which allows us to discuss the differences existing between rate equations in the limits of large light bandwidth or large atomic linewidth. Only in the former case does one obtain ERE with the correct expression for Einstein’s B coefficient.
In this paper, the general optical Bloch equation describing interaction between intensive fields and two level system is obtained without using rotating wave approximation. The first-order solution of this equation is obtained by iterative method and analytic formulas of transition probability as a function of time, perturbation frequency and strength are presented. These results are in agreement with those obtained by an iterative step-wise power series and numerical integration solution for the time dependent Schr(?)dinger equation. And these simplified and clear mathmatical formulas facilitate the application of physics.
It is the aim of the present work to show some departures from the well-known optical Bloch equations usually obtained when the damping of a two-level system driven by a Glauber coherent mode of the radiation field is treated in the Markovian approximation. In particular, we show that for an arbitrary intensity we have a set of 4 n optical Bloch equations which can be reduced to the usual optical Bloch equations when the population of the mode becomes important. A criterion for the quasi-classical evolution can be deduced. Obviously, in both limits of an arbitrarily weak or strong excitation intensity the Mollow's results are obtained. In addition, we do not need here the introduction of a particular form like the plane wave used by Mollow for the corresponding classical field--as the problem is solved explicitly.