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Physical Review A

Calculation of analytic populations for a multilevel atom at low laser intensity

Geol Moon, Myung-Sun Heo, Seo Ro Shin, Heung‐Ryoul Noh, Wonho Jhe

Published Jul 21, 2008• 9 citations
Publisher

Abstract

We present an analytic calculation of the population of each magnetic sublevel of the $^{87}\mathrm{Rb}$ atom. Under the influence of a weak circularly polarized laser beam, the rate equations governing the internal dynamics of this atom were solved analytically and the results were compared with numerical calculations. As far as the condition of low intensity is satisfied, it is possible to obtain exact analytic populations regardless of the magnitude of energy spacings, transition lines, and laser polarization (circular or linear).

DOI
10.1103/physreva.78.015404

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We present an analytic calculation of the population of each magnetic sublevel of the 87Rb^{87}\mathrm{Rb}87Rb atom. Under the influence of a weak circularly polarized laser beam, the rate equations governing the internal dynamics of this atom were solved analytically and the results were compared with numerical calculations. As far as the condition of low intensity is satisfied, it is possible to obtain exact analytic populations regardless of the magnitude of energy spacings, transition lines, and laser polarization (circular or linear).
We present an analytical calculation of temporal evolution of populations for optically pumped atoms under the influence of weak, circularly polarized light. The differential equations for the populations of magnetic sublevels in the excited state, derived from rate equations, are expressed in the form of inhomogeneous second-order differential equations with constant coefficients. We present a general method of analytically solving these differential equations, and obtain explicit analytical forms of the populations of the ground state at the lowest order in the saturation parameter. The obtained populations can be used to calculate lineshapes in various laser spectroscopies, considering transit time relaxation.
We present an analytic calculation for linear susceptibility in pump-probe spectroscopy for a Doppler broadened atomic vapor. The analytic form of the population of each magnetic sublevel of a 87Rb^{87}\mathrm{Rb}87Rb atom, illuminated by a weak circularly polarized pump beam, was calculated for the lowest order of laser intensity and used in the calculation for the susceptibility of a counterpropagating (copropagating) probe beam. The obtained analytic results for the susceptibility can be used for calculating the absorption or the refractive index of the probe beam as functions of pump beam diameter and intensity.
We present analytical solutions for the populations of the D 1 transition line of 85 Rb atoms. The time evolution of the populations of the magnetic sublevels in the excited state at the weak intensity limit exhibits several kinds of homogeneous or inhomogeneous second-order differential equations. General solutions of the differential equations result in an analytical form of the population in each magnetic sublevel as a function of the elapsed time, laser intensity, and laser frequency detuning.
The evolution of the hyperfine structure magnetic sublevel populations of 87 Rb and 133 Cs atoms in resonant linearly polarized laser fields is theoretically analyzed. Analytical expressions are obtained for stationary populations of the magnetic sublevels of the ground state 5 2 S 1/2 F g = 1, F f = 2 in 87 Rb and 6 2 S 1/2 F g = 3, F f = 4 in 133 Cs under optical pumping at the F g ↔ F e = F g and F f ↔ F e = F f transitions of the D 2 lines ( n 2 S 1/2 n 2 P 3/2 ) versus the initial populations. The characteristic population stabilization time are obtained by numerical solution for the density matrix equations, depending on the intensity of the laser fields and the detuning of the optical resonance. It is shown that the alternation of the polarization direction of the laser field at the transition F f ↔ F e = F f almost completely transfers atoms to the lower sublevel of the "clock" M1 transition F g M = 0 ↔ F f M = 0 in rubidium and cesium frequency standards, which proportionally increases the signal of the recording system.
The interaction of high power laser radiation with the atoms in the external magnetic field leads to the new nonlinear effects due to modifications of populations of the states of the system 'atom plus field.' From one side it can be followed by new peculiarities in the effect of the coherent population trapping and from other -- the effects of the population modifications are important for explanation of regularities of nonlinear magneto-optical effects. The modification of population dynamics due to the parameters of the system -- radiation field intensities, detunings, polarizations, magnetic field strength and orientation, relaxation constants leads to the new features in polarization effects, resonant fluorescence and resonant Raman effects. The new features of the considered theory are that it is built for multilevel atoms and has not restrictions on the probe intensity compared to the pump intensity. The theory and method permit to investigate the populations modifications depending on modifications of the parameters of the system and to find the relation between population dynamics and new nonlinear resonant magneto-optical effects. The considered system consists of atomic gas of low concentration in the constant magnetic field and two monochromatic laser fields of different intensities and polarizations. Analytically the influence of magnetic field on population dynamics of three-level systems have been considered. For real multilevel atom the computer simulations show the dependence of population dynamics on relaxation constants and the differences of this dynamics for different magnetic sublevels of the excited states of the atom. The results agree with known theoretical and experimental works.
The evolution of the populations of the magnetic sublevels of the cesium atom (133Cs isotope) in resonant laser fields with linear polarization is analyzed using the equations for the density matrix. Analytic expressions are derived for stationary populations resulting from laser-induced optical transitions on the hyperfine structure components F g = 3 ↔ F e = 2, 3 and F f = 4 ↔ F e = 3, 4 of lines D1 (62S1/2 → 62P1/2) and D2 (62S1/2 → 62P3/2) depending on the initial values of the populations. The numerical solution of the evolution equations gives the characteristic times of stabilization of the steady regime as functions of laser field intensities and detuning from optical resonance. We determine the sequences of optical transitions increasing (by more than an order of magnitude) the population of the lower sublevel 62S1/2F g = 3 M = 0 of the “clock” microwave transition F g = 3 M = 0 ↔ F f = 4 M = 0 in the cesium frequency standard, which increases the signal intensity in the recording system by the same proportion.