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Journal of Experimental and Theoretical Physics

Basis of polarization-dressed states of an atom in an elliptically polarized resonant field

Прудников Олег Николаевич, A. V. Taǐchenachev, Anatolii M Tumaikin, Valeriy I. Yudin, Gerard Nienhuis

Published Dec 1, 2004• 7 citations
Publisher

Abstract

A basis of polarization-dressed states is proposed for atomic energy levels degenerate in the total angular momentum projections in the case of interaction with elliptically polarized light. It is shown that instead of selection rules for the magnetic quantum number, the interaction in this basis can be presented as the sum of direct transitions between corresponding pairs of polarization-dressed states of the upper and lower levels. The explicit form of the basis is derived for ten possible combinations of dipole transitions between energy levels with angular momenta J = 0, 1/2, 1, 3/2, and 2. The problem of Rabi oscillations in such a system is considered as an application.

DOI
10.1134/1.1854801

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Journal of Experimental and Theoretical Physics•Mar 1, 2004•0 citations
A basis of polarization-dressed states is proposed for atomic energy levels degenerate in the total angular momentum projections in the case of interaction with elliptically polarized light. It is shown that instead of selection rules for the magnetic quantum number, the interaction in this basis can be presented as the sum of direct transitions between corresponding pairs of polarization-dressed states of the upper and lower levels. The explicit form of the basis is derived for ten possible combinations of dipole transitions between energy levels with angular momenta J = 0, 1/2, 1, 3/2, and 2. The problem of Rabi oscillations in such a system is considered as an application.
Coherent population trapping (CPT), under the resonant interaction of elliptically polarized light with atoms having degenerated energy levels, is analytically treated without the use of a specific coordinate frame. Tensors describing dark states are presented in an explicit and invariant form for all atomic transitions, where CPT takes place. A correspondence between elliptical polarization vectors and pairs of unit spinors is established. It is shown that all dark states in the problem under consideration can be constructed from these spinors by taking multiple tensor products. In the case of an integer angular momentum, this general construction is reduced to a multiple tensor product of vectors.
The coherent population trapping (CPT) under resonance interaction of elliptically polarized light, with atoms having degenerated energy levels, is analytically treated in an invariant way, without using a specific coordinate frame. Tensors describing dark states are found in an explicit and invariant form for all atomic transitions, where CPT takes place. The correspondence between elliptical polarization vectors and pairs of unit spinors is established. It is shown that all dark states in the problem under consideration, can be constructed from these spinors, by taking multiple tensor products. In the case of an integer angular momentum, this general construction is reduced to a multiple tensor product of vectors. A general significance and some applications of these results are indicted.
An analysis is made of coherent population trapping as a result of resonant interaction of elliptically polarized light with atoms whose energy levels are degenerate with respect to the projection of the angular momentum and are coupled by a dipole transition. Explicit invariant expressions for dark states are obtained in tensor form for all transitions where population trapping occurs. A correspondence is established between the vector of the elliptic polarization and the pair of associated spinors. It is shown that all dark states can be constructed from these spinors by means of a multiple tensor product. For integer values of the angular momenta of the transitions these constructions reduce to spherical functions of a complex variable. As applications analytic expressions are obtained for the dark magneto-optic and geometric potentials, and the change in their profile with increasing angular momenta is analyzed.
The polarization of the resonance fluorescence of interacting atoms and the transfer of coherence, due to the exchange of excitation, are studied theoretically. The treatment is for the general case where both the excited and the ground states can have arbitrary angular momenta, except that they can be connected to each other by an allowed transition. Electrostatic dipole–dipole interaction and electromagnetic radiative interaction are considered simultaneously as the perturbations on the system. Heitler and Ma's generalized time-dependent perturbation method is used. The system is quantized with respect to the space-fixed z axis and a static magnetic field along the z axis may be present, so that the magnetic sublevels (m, m′, etc., for the excited state J, and α, β, etc., for the ground state J′) are not necessarily degenerate. The intensity of the scattered light of a given polarization, in a given direction, is expressed as a function of kR (where ℏck is the energy of excitation and R is the internuclear distance), and of the orientation of R with respect to z axis. The matrix elements (ℏ / 2)γmβ,αm′ and Vmβ,αm′, which cause resonant transfer of excitation between two atoms through radiative and electrostatic interaction, respectively, are also expressed as a function of kR, and of the orientation of R. For randomly oriented atoms, the polarization after being averaged over the random orientation of R becomes a function of kR. Numerical calculations have been carried out for S01 → 1P1 transition. The coherence is shown to be conserved upon resonant transfer of excitation when R is parallel to the quantization axis.
The unitary transformation from linearly to elliptically polarized states of photons is determined in a scheme of the second quantization (\textsection{}1). It is given by ${\mathbf{e}}_{l}={\mathbf{e}}_{1}cos\ensuremath{\phi}+i{\mathbf{e}}_{2}sin\ensuremath{\phi}$, ${\mathbf{e}}_{r}=i{\mathbf{e}}_{1}sin\ensuremath{\phi}+{\mathbf{e}}_{2}cos\ensuremath{\phi}$ where e1{\mathbf{e}}_{1}e1​ and e2{\mathbf{e}}_{2}e2​ are orthogonal unit vectors corresponding to linearly polarized states of photons and el{\mathbf{e}}_{l}el​ and er{\mathbf{e}}_{r}er​ are those corresponding to left and right elliptically polarized states of photons. A general formula for a probability of an emission of light from an atom is deduced on the basis of the Dirac electron theory so as to include up to both a magnetic dipole and an electric quadrupole term (\textsection{}2). Its general form is improved as compared with the hitherto obtained one. The general theory is applied to electric dipole and quadrupole and magnetic dipole spectral lines (\textsection{}3 and \textsection{}4). Polarization states of the Zeeman components corresponding to all combinations of magnetic and orbital magnetic quantum numbers of an atom are completely determined for an arbitrary given direction of an observation on the basis of quantum mechanics.
The dynamics of a two-level atom with optical transition J g → J e =J g + 1 under the action of elliptically polarized resonance radiation in the presence of a constant field directed along the radiation wave vector is studied in the approximation of a semiclassical description of the angular momentum orientation. It is shown that the atomic distribution over the angular momentum orientation exhibits a two-scale time evolution. At the first stage, after the beginning of irradiation, the angular momenta of atoms get oriented (over comparatively short time intervals) along the magnetic field, as well as in the opposite direction, depending on their initial orientation and the ellipticity of radiation. At the second (longer) stage, the redistribution of atoms takes place, as a result of which they are oriented predominantly in one of the above directions. The duration of the second stage is an exponential function of angular momentum J.