A systematic, rigorous, and complete investigation of the Bloch equations in time-harmonic driving classical field is performed. Our treatment is unique in that it takes full advantage of the partial fraction decomposition over real number field, which makes it possible to find and classify all analytic solutions. Torrey's analytic solution in the form of exponentially damped harmonic oscillations [Phys. Rev. {\bf 76}, 1059 (1949)] is found to dominate the parameter space, which justifies its use at numerous occasions in magnetic resonance and in quantum optics of atoms, molecules, and quantum dots. The unorthodox solutions of the Bloch equations, which do not have the form of exponentially damped harmonic oscillations, are confined to rather small detunings $δ^2\lesssim (γ-γ_t)^2/27$ and small field strengths $Ω^2\lesssim 8 (γ-γ_t)^2/27$ , where $γ$ and $γ_t$ describe decay rates of the excited state (the total population relaxation rate) and of the coherence, respectively. The unorthodox solutions being readily accessible experimentally are characterized by rather featureless time dependence.A systematic, rigorous, and complete investigation of the Bloch equations in time-harmonic driving classical field is performed. Our treatment is unique in that it takes full advantage of the partial fraction decomposition over real number field, which makes it possible to find and classify all analytic solutions. Torrey's analytic solution in the form of exponentially damped harmonic oscillations [Phys. Rev. {\bf 76}, 1059 (1949)] is found to dominate the parameter space, which justifies its use at numerous occasions in magnetic resonance and in quantum optics of atoms, molecules, and quantum dots. The unorthodox solutions of the Bloch equations, which do not have the form of exponentially damped harmonic oscillations, are confined to rather small detunings $δ^2\lesssim (γ-γ_t)^2/27$ and small field strengths $Ω^2\lesssim 8 (γ-γ_t)^2/27$, where $γ$ and $γ_t$ describe decay rates of the excited state (the total population relaxation rate) and of the coherence, respectively. The unorthodox solutions being readily accessible experimentally are characterized by rather featureless time dependence.

arXiv:1208.5736Aug 22, 20120 citations

QUANTUM MECHANICAL FORM OF THE DAMPED BLOCH EQUATIONS

Angelo Turrin

Jan 1, 19760 citations

Quantum Theory of Damped Oscillation

Shigemi Araki

Oct 1, 19830 citations

Laser gain and some of its consequences

Warren Nagourney

The laser gain equation is derived in two ways. First, the optical Bloch equations are derived, based upon a semiclassical treatment of a two-level atomic system and a radiation field. As a visualization aid, there follows a discussion of the formally identical classical Bloch equations, which describe the behavior of a magnetic moment in a combined static and rotating magnetic field. It is shown that laser gain is proportional to the imaginary part of the linear susceptibility, which is obtained from the semiclassical treatment. The second viewpoint uses a slightly modified form of Einstein’s A and B coefficients adapted to describing interactions between atoms and monochromatic fields. Various damping and broadening mechanisms are introduced in a reasonable but non-rigorous fashion, and the crucial distinction between homogeneous and inhomogeneous broadening is introduced.The laser gain equation is derived in two ways. First, the optical Bloch equations are derived, based upon a semiclassical treatment of a two-level atomic system and a radiation field. As a visualization aid, there follows a discussion of the formally identical classical Bloch equations, which describe the behavior of a magnetic moment in a combined static and rotating magnetic field. It is shown that laser gain is proportional to the imaginary part of the linear susceptibility, which is obtained from the semiclassical treatment. The second viewpoint uses a slightly modified form of Einstein’s A and B coefficients adapted to describing interactions between atoms and monochromatic fields. Various damping and broadening mechanisms are introduced in a reasonable but non-rigorous fashion, and the crucial distinction between homogeneous and inhomogeneous broadening is introduced.

Oxford University Press eBooksMay 8, 20140 citations

Quantum Theory of Damped Oscillation II

Shigemi Araki

Oct 1, 19850 citations

δ^2\lesssim (γ-γ_t)^2/27

and small field strengths

Ω^2\lesssim 8 (γ-γ_t)^2/27

, where

γ

and

γ_t

describe decay rates of the excited state (the total population relaxation rate) and of the coherence, respectively. The unorthodox solutions being readily accessible experimentally are characterized by rather featureless time dependence.