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

Symmetries in the optical Bloch equations and multiphoton processes

Bruno Zambon

Published Apr 19, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.63.053814

Related papers

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

Valery Alexandrovich Astapenko, M. I. Mutafyan

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.

Journal of Atomic Molecular Condensed Matter and Nano Physics•Dec 31, 2023•0 citations

Einstein's Coefficients and Rate Equations to Macroscopic Gain: Detailed Balance & Population Inversion for Atomic Physics Laser

Rajesh Vedala, Palak Kapoor, Harkirat Kaur

This paper presents a unified, step-by-step development of laser amplification from the microscopic viewpoint of photon-atom interactions. Starting with the quantization of light (E = hν), introducing Einstein's three coefficients for absorption and emission, derive the coupled rate equations, use detailed balance to recover Planck's law, connect the rates to quantum-mechanical dipole matrix elements, and culminate in the concept of population inversion and the physical picture of laser build-up. The goal of this presentation about understanding the Stimulated emission towards Laser concepts, an level 1 understanding prior to journey of understanding quantum computing.

•Aug 26, 2025•0 citations

Population Inversion, Saturation, and Laser Systems

Surendra Singh

This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.

CRC Press eBooks•Sep 23, 2026•0 citations

Driven two-level atoms as gain media for two or more photon amplification

Y. Zhu, S. Morin, Qilin Wu +3 more

It is well known that driven two-level atoms can exhibit optical amplification.1 This fact has led to the recent demonstration of single-photon lasing in cell and atomic-beam type ensembles of two-level atoms.2 We show that the same simple two-level system can also be utilized as a two- or more-photon amplifying medium. The underlying physics can be understood in terms of the atom field dressed-state picture. When driven by a field detuned from the atomic resonance, significant population inversion between the dressed-state doublets can be created. In the dressed picture, single-photon lasing occurs between inverted sublevels of adjacent dressed-state doublets. Transitions between nonadjacent doublets are still inverted, but require two or more photons. In the degenerate case, the multiphoton emission frequencies differ from the single-photon emission frequency and can hence be selectively enhanced using an optical cavity. Due to the peculiar structure of dressed levels, the multiphoton emissions are all connected through near resonant intermediate states. The multiphoton emissions can also be viewed as a stimulated hyper-Raman scattering of successively higher order.

Annual Meeting Optical Society of America•Jan 1, 1989•0 citations

Intrinsically Irreversible Multiphoton Laser Gain Mechanisms

Peter B. Sellin, Chung‐Chieh Yu, Jason R. Bochinski +1 more

Single-photon stimulated radiative interactions are universally symmetric relative to emission and absorption. We point out that certain multiphoton stimulated processes, whose completion is dependent on the spontaneous emission of at least one photon, do not behave analogously. The broken symmetry found in such cases invalidates simple correlations between population inversion and laser gain.

Physical Review Letters•Feb 24, 1997•22 citations

Quantum theory of driven multi-level systems

Aaron S. Manka, Lorenzo M. Narducci, Marlan O. Scully +3 more

We discuss a generalized version of the Scully-Lamb laser master equation for a collection of driven multilevel atoms in a lambda-type configuration. We are especially interested in the effect of the external driving field on the laser process with or without a population inversion between the lasing levels. We provide an exact solution of the photon probability distribution in steady state and discuss regimes of operation that can lead to sub-Poissonian statistics, as measured by the ratio between the variance and the average photon number. We solve the field master equation in the transient regime and analyze the effects of the external driving field on the linewidth of the emitted radiation. The largest reduction of the photon variance relative to the mean number corresponds to the injection into the cavity of atoms prepared in the nonlasing level. Injection of atoms in an incoherent mixture of the three energy states normally leads to super-Poissonian statistics.

Annual Meeting Optical Society of America•Jan 1, 1992•0 citations
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.
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.
This paper presents a unified, step-by-step development of laser amplification from the microscopic viewpoint of photon-atom interactions. Starting with the quantization of light (E = hν), introducing Einstein's three coefficients for absorption and emission, derive the coupled rate equations, use detailed balance to recover Planck's law, connect the rates to quantum-mechanical dipole matrix elements, and culminate in the concept of population inversion and the physical picture of laser build-up. The goal of this presentation about understanding the Stimulated emission towards Laser concepts, an level 1 understanding prior to journey of understanding quantum computing.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.
It is well known that driven two-level atoms can exhibit optical amplification.1 This fact has led to the recent demonstration of single-photon lasing in cell and atomic-beam type ensembles of two-level atoms.2 We show that the same simple two-level system can also be utilized as a two- or more-photon amplifying medium. The underlying physics can be understood in terms of the atom field dressed-state picture. When driven by a field detuned from the atomic resonance, significant population inversion between the dressed-state doublets can be created. In the dressed picture, single-photon lasing occurs between inverted sublevels of adjacent dressed-state doublets. Transitions between nonadjacent doublets are still inverted, but require two or more photons. In the degenerate case, the multiphoton emission frequencies differ from the single-photon emission frequency and can hence be selectively enhanced using an optical cavity. Due to the peculiar structure of dressed levels, the multiphoton emissions are all connected through near resonant intermediate states. The multiphoton emissions can also be viewed as a stimulated hyper-Raman scattering of successively higher order.
We discuss a generalized version of the Scully-Lamb laser master equation for a collection of driven multilevel atoms in a lambda-type configuration. We are especially interested in the effect of the external driving field on the laser process with or without a population inversion between the lasing levels. We provide an exact solution of the photon probability distribution in steady state and discuss regimes of operation that can lead to sub-Poissonian statistics, as measured by the ratio between the variance and the average photon number. We solve the field master equation in the transient regime and analyze the effects of the external driving field on the linewidth of the emitted radiation. The largest reduction of the photon variance relative to the mean number corresponds to the injection into the cavity of atoms prepared in the nonlasing level. Injection of atoms in an incoherent mixture of the three energy states normally leads to super-Poissonian statistics.