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Physical Review A

Coherent excitation of a degenerate two-level system by an elliptically polarized laser pulse

Nikolay V. Vitanov, Z. Kis, Bruce W. Shore

Published Dec 30, 2003• 19 citations
Publisher

Abstract

The elaborate linkage pattern of the interaction Hamiltonian descriptive of pulsed laser interaction of elliptically polarized light with a degenerate two-level system can, for coherent excitation, be reduced to sets of independent pairs of coupled two-state equations by means of the Morris-Shore transformation [Phys. Rev. A 27, 906 (1983)]. By extending the earlier defining work, which dealt only with time-independent interactions, to consider various pulse shapes, one can obtain exact analytic solutions to various multilevel linkages. We use these to find the conditions on the interaction parameters (e.g., the ellipticity) leading to various population transfer schemes, for example, to achieve either population inversion or orientation. We illustrate these with the M and W linkages amongst magnetic sublevels of $J=1$ or $J=2$ excited by elliptically polarized light.

DOI
10.1103/physreva.68.063414

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The elaborate linkage pattern of the interaction Hamiltonian descriptive of pulsed laser interaction of elliptically polarized light with a degenerate two-level system can, for coherent excitation, be reduced to sets of independent pairs of coupled two-state equations by means of the Morris-Shore transformation [Phys. Rev. A 27, 906 (1983)]. By extending the earlier defining work, which dealt only with time-independent interactions, to consider various pulse shapes, one can obtain exact analytic solutions to various multilevel linkages. We use these to find the conditions on the interaction parameters (e.g., the ellipticity) leading to various population transfer schemes, for example, to achieve either population inversion or orientation. We illustrate these with the M and W linkages amongst magnetic sublevels of J=1J=1J=1 or J=2J=2J=2 excited by elliptically polarized light.
Examining the dynamics of a degenerate two-level system excited by coherent resonant light, we exhibit a transformation of basis states which reduces the problem to a set of independent nondegenerate one- and two-state systems. More generally, reduction of multilevel Hamiltonians requires only that the states can be collected into two sets; within each set the states are degenerate, unlinked by Hamiltonian matrix elements. The remaining Hamiltonian linkages must share the same time dependence, but are otherwise arbitrary. We apply this method to reduction of the angular momentum degeneracy of a two-level atom excited by elliptically polarized light and to multiple-laser resonant excitation of multiple levels into a common level.
Descriptions of coherent excitation of multi-state quantum systems model changes through coupled ordinary differential equations that become more cumbersome as the number of involved quantum states increases. It is always useful to find simplifications of the equations, ideally involving exact analytic solutions. The ultimate simplification is to a set of independent pairs of quantum states. Such sets of independent two-state linkages occur naturally when we describe the excitation between two sets of degenerate Zeeman sublevels by a field that is linearly or circularly polarized. The Morris–Shore (MS) transformation, a Hilbert-space coordinate change, generalizes this procedure to replace the description of an elaborate linkage pattern of an -state Hamiltonian in the rotating-wave approximation by a set of independent two-state systems – a basis of paired bright states and excited states supplemented with dark states and spectator states. The three-state lambda-linkage found in stimulated Raman transitions is a simple example. Both bright and dark states have found application in various procedures for manipulating quantum states, either to achieve population transfer between states or to create coherent superpositions; the dark and spectator states coincide with particular dressed (or adiabatic) states. The present review describes the MS transformation, noting its historical background, and discusses examples of its use.
The technique of stimulated Raman adiabatic passage has become an established procedure for producing complete population inversion in atoms or molecules via application of Stokes and pump pulses in a counterintuitive sequence. We discuss here some of the new and important phenomena that arise when some of the states involved have nonzero angular momentum, the field polarization directions have no simplifying symmetries, and Zeeman splitting lifts the magnetic sublevel degeneracy. We consider the effects of the linkage structure of the Hamiltonian, the effects of various polarization directions, and the effects of various choices for carrier frequencies, as expressed by detunings from single-photon resonance. In particular, we consider an example with angular momentum J=0\ensuremath{\leftrightarrows}J=1\ensuremath{\leftrightarrows}J=2 in which nine magnetic sublevels occur. We show that under suitable conditions it is possible, simply through the appropriate tuning of pump and Stokes carrier frequencies, to place the entire population into any specified single magnetic sublevel.
We present an experimental investigation of ultrafast laser-driven Rabi oscillations of atomic rubidium. Because the broadband spectrum of an ultrafast laser pulse simultaneously couples all of the electronic hyperfine transitions between the excited and ground states, the complex excitation linkages involved with the D1{D}_{1}D1​ or D2{D}_{2}D2​ transition are energy degenerate. By application of the Morris-Shore transformation, this study shows that the considered multistate system is reduced to a set of independent two-state systems and dark states. In experiments performed by ultrafast laser interactions of atomic rubidium in the strong interaction regime, we demonstrate that the ultrafast dynamics of the considered system are governed by no more than two decoupled Rabi oscillations when it interacts with ultrafast laser pulses of any polarization state. We further show the implications of this result with regard to possible control of photoelectron polarizations.
Propagation of elliptically polarized light pulses in a medium of two-level atoms with degenerate energy levels under coherent population trapping has been studied within the density-matrix formalism and reduced Maxwell’s equation. It has been shown that ellipticity pulses and the orientation angle of the polarization ellipse move with a deceleration. The analytical expression for the group velocity of an arbitrary J → J dark transition ( J is an integer) has been derived. In addition, a previously unknown effect of forced phase modulation under the variation of the spatial orientation of the polarization ellipse has been discovered. This phase modulation includes a pilot pulse passing through the medium at the speed of light in vacuum and a slow pulse moving synchronously with the pulse of the orientation angle.
A simple method for measuring the parameters of a completely or partially coherent superposition of multiple atomic states is proposed. The method, which can be applied in principle to a superposition of arbitrary energy states, is described in the case of a superposition of magnetic sublevels of a degenerate ground level. It is based upon coupling the sublevels involved in the superposition to the sublevels of a degenerate excited level by using a single elliptically polarized laser pulse. Such an interaction maps the parameters of the initial superposition onto the populations of the excited sublevels. By measuring the subsequent fluorescence from the excited sublevels for several suitably chosen laser polarizations one can deduce unambiguously the parameters of the initial superposition.