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Physical Review A

Application of the Projection Technique to Two-Level Atoms

A. Muriel

Published May 1, 1971• 2 citations
Publisher

Abstract

Exact, formal, and usable solutions of the Liouville equation corresponding to the problem of independent two-level atoms in an external field are presented. Two examples are worked out: calculation of the polarization, and a model of pumping or population inversion by a $\ensuremath{\delta}$-function external field. The relation of the solutions to other problems in statistical mechanics is also discussed.

DOI
10.1103/physreva.3.1575

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Exact, formal, and usable solutions of the Liouville equation corresponding to the problem of independent two-level atoms in an external field are presented. Two examples are worked out: calculation of the polarization, and a model of pumping or population inversion by a \ensuremathδ\ensuremath{\delta}\ensuremathδ-function external field. The relation of the solutions to other problems in statistical mechanics is also discussed.
Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.
The Liouville equation for the density matrix can be recast so that the external field terms which couple density matrix elements are second order in the field. This approach is shown to have a profound effect on the analysis of strong field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of two. Second, this reformulation naturally leads to radiative renormalization terms which are directly related to saturation, Stark shifts and Rabi splittings. The renormalization is equivalent to a Dyson equation analysis. We use this simple formalism to obtain exact solutions to a number of nonlinear optical problems including cases where there are two or more strong, resonant fields. These problems include: 1) two levels, two strong fields, 2) three levels, two strong fields, and 3) four levels, three and four strong fields. Discussed in detail are the two level case and the three and four level cases where weak fields couple the ground state to other states. We use these results to discuss the physics of optically induced transparency, population beats, optical switching in quantum wells, and fully resonant coherent anti-Stokes Raman scattering (CARS).
Optical pumping is a process used in physics and quantum mechanics to manipulate the energy levels of atoms or ions using light.It involves the absorption and emission of photons by atoms, which can change their electronic configurations and energy states.In this paper, optical pumping with Liouville equation approach is used to investigate optical pumping in alkali atoms.For this purpose, circularly polarized light is applied to the atomic system, which causes a transition between atomic sublevels and changes the population of sublevels.Also effect of relaxation and repopulation are considered in the evolution of the population sublevels.Here, we use optical pumping with Liouville equation approach for investigation of population evolution in Cesium atom.