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Maxwell-Bloch equation and the inverse scattering method

Ildar R. Gabitov, Vladimir Evgen'evich Zakharov, А. В. Михайлов

Theoretical and Mathematical PhysicsApr 1, 198581 citations

π/2 pulse shaping via analytic inversion of Bloch equations

Chang Jae Lee

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BNov 1, 19930 citations

Effect of relaxation on adiabatic following

Munir Hasan Nayfeh

A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.A solution is presented for the damped optical Bloch equations under the excitation of a smooth pulse by first deriving three independent third-order equations of the Bloch vector components. Each equation is reduced to quadratures by assuming that the logarithmic time derivative of the field amplitude is small compared to the Rabi frequency. This results in an approximate summation of the infinite-order time-dependent perturbation in the field amplitude. The relaxation-dependent induced damping of the population inversion is calculated. Also calculated are additional relaxation-dependent contributions to the intensity-dependent refractive index. The time-integrated intensity contribution tends to cause line asymmetry, which becomes, at later times, linear in ${\ensuremath{\gamma}}_{2}$ when ${\ensuremath{\gamma}}_{2}\ensuremath{\gg}{\ensuremath{\gamma}}_{1}$ and zero when $2{\ensuremath{\gamma}}_{2}={\ensuremath{\gamma}}_{1}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase-changing relaxations, respectively. The dependence of the spectral broadening on pulse length, pressure, and length of the sample is discussed.

Physical Review ASep 1, 19767 citations

On the solution of the Bloch equations

E. Arimondo, Davide Bertolini

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoAug 1, 19720 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.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:1105.4120American Journal of PhysicsSep 18, 20129 citations

Squeezing corrections to the Bloch equations

Mario Abundo, Luigi Accardi

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento BFeb 1, 19913 citations

Spectral Resolution and Inversion of the Bloch Equations with Relaxation

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