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Results in Physics

Atomic population inversion in a two-level atom for shaped and chirped laser pulses: Exact solutions of Bloch equations with dephasing

Sofiane Grira, Nadia Boutabba, Hichem Eleuch

Published Jun 5, 2021• 3 citations
Publisher

Abstract

We perform a theoretical analysis of the Bloch equations of a two-level atom with dephasing. Exact analytical solutions with two different chirped and time-dependent detuned laser pulses are derived. The atomic population inversion as well as the coherence are explored for different initial populations. We show that the stationary population inversion can be controlled.

DOI
10.1016/j.rinp.2021.104419

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We perform a theoretical analysis of the Bloch equations of a two-level atom with dephasing. Exact analytical solutions with two different chirped and time-dependent detuned laser pulses are derived. The atomic population inversion as well as the coherence are explored for different initial populations. We show that the stationary population inversion can be controlled.
In this research study, we derive the exact solutions of the Bloch equations describing the dynamics of a two-level atom with dephasing. In the two-level atom, a strong laser pump couples a ground state to an upper excited state with a time-dependent Rabi-frequency. The exact solutions are given for the atomic population inversion and the real and imaginary parts of the coherence while the input pulse is an asymmetric hyperbolic cosine form. Additionally, the system is under a chirped detuning. The method of solving the Bloch equations analytically is a very tedious part of the research, and as far as we know, there are few exact solutions available in this field. Hence, our solutions might be of great interest to various research areas, including nuclear magnetic resonance, where analytical solutions to the Bloch equations play a major role in the study of the information on the state of the medium as determined by the NMR signals.
The excitation of a two-level system by a chirped laser pulse is analyzed using an analytical approach (in the perturbation-theory limit) that involves the modified rotating-wave approximation and the numerical solution to the Bloch equation. The dependence of the population of the upper energy level on the chirp is studied for various radiation intensities and pulse durations. An exact solution is compared with the result obtained using the modified rotating-wave approximation and the perturbation theory. It is demonstrated that, for a certain range of parameters, the excitation probability of a two-level system can be effectively controlled via a variation in the chirp.
A fast laser pulse with a q-deformed hyperbolic function shape is investigated in a Multi-level atomic system. Therefore, we first derive the exact solution of the Bloch equations describing a two-level atom excited by a q-deformed laser pulse with dephasing and time-dependent detuning. Next, we analyze the dynamic of the atomic population inversion at resonance and off-resonance of a Rubidium 87 three-level atom driven by a classical weak field and a strong q-deformed control laser. Finally, in order to get a deeper insight of the probe field's absorption and dispersion properties, we investigate the coherence's dependence on the q-deformation. Our work demonstrates that, the dynamic of the atomic system can be fully controlled through the manipulation of the asymmetry scaling parameter q of the q-deformed hyperbolic laser wave-form.
Doppler-free population inversion induced by coherent two-photon resonance is calculated analytically for a two-photon analog of the hyperbolic-secant pulse and numerically for a Gaussian, linearly chirped pulse. The numerical calculations are carried out for a two-photon resonance of a chirped ruby laser line with the $6{S}_{\frac{1}{2}}\ensuremath{\rightarrow}9{D}_{\frac{3}{2}}$ transition in Cs. Complete inversion is found when the fractional change in frequency during the pulse lies between 6 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}6}$ and 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$. The coherent two-photon resonances are described by optical Bloch equations which are derived from a multiple-time-scale perturbation theory.
The method of multiple scales is used to derive a solution of the damped optical Bloch equations of a two-level atomic system due to a strong pulsed field. The time dependence of the oscillations of the atomic inversion influenced by detuning and power broadening is found. The population inversion consists, in general, of three terms: a quasisteady term, quasisteady term that decays with time, and an oscillatory term that also decays with time. In the limit of constant fields, the solution of Torrey for damped systems and that of Rabi for undamped systems are recovered. For an adiabatic switching of the field, the solution for undamped systems reduces to that of Crisp in the adiabatic following limit. An equation describing the field envelope is derived for an arbitrary amount of detuning. At exact resonance, this equation reduces to a pendulum equation, in agreement with previous analyses.