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Journal of Experimental and Theoretical Physics

Analysis of correlations between atoms in terms of bivectors under coherent population trapping conditions

L. V. Il’ichev

Published Apr 1, 2006• 3 citations
Publisher

Abstract

Description of states of quantum systems in terms of bivectors (in the general case, polyvectors) is proposed as an alternative to vectors in a Hilbert space. This approach is developed for slow atoms under coherent population trapping conditions (in a dark state). The field of local two-dimensional dark subspaces is naturally associated with a local bivector field. An analysis in terms of bivectors is performed to recover the state of an atom or an atomic ensemble by using the condition for its being in some unknown dark state. This approach exposes natural correlations between atoms. The corresponding many-particle states belong to the class of states with positive partial transpose.

DOI
10.1134/s1063776106040066

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Description of states of quantum systems in terms of bivectors (in the general case, polyvectors) is proposed as an alternative to vectors in a Hilbert space. This approach is developed for slow atoms under coherent population trapping conditions (in a dark state). The field of local two-dimensional dark subspaces is naturally associated with a local bivector field. An analysis in terms of bivectors is performed to recover the state of an atom or an atomic ensemble by using the condition for its being in some unknown dark state. This approach exposes natural correlations between atoms. The corresponding many-particle states belong to the class of states with positive partial transpose.
The concept of geometric phase acquired by a quantum state during its evolution is generalized to the subspace of states of evolving quantum system. Physical motivation of this generalization comes from the effect of coherent population trapping. Under certain conditions, there exists a two-dimensional “dark” subspace of atomic ground states that does not interact with the external radiation. The structure of this “dark” subspace depends on the local field amplitude. We show that certain mathematical difficulties in finding the geometric phase for “dark” subspaces can be circumvented with the help of the orthogonal “bright” subspace.
The possibility of coherent population trapping in two electron states with aligned spins (orthosystem) is evidenced. From the analysis of a three-level atomic system containing two electrons, and driven by the two laser fields needed for coherent population trapping, a conceptually new kind of dark state appears. The properties of this trapping are physically interpreted in terms of a dark hole, instead of a dark two-electron state. This technique, among many other applications, offers the possibility of measuring, with subnatural resolution, some superposition-state matrix elements of the electron-electron correlation that due to their time dependent nature are inaccessible by standard measuring procedures.
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.
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.
A full quantum treatment of coherent population trapping (CPT) is given for a system of resonantly coupled atoms and electromagnetic field. We develop a regular analytical method of the construction of generalized dark states (GDS). It turns out that GDS do exist for all optical transitions FgtoFeF_g\\to F_eFg​toFe​, including bright transitions FtoF+1F\\to F+1FtoF+1 and F′′toF′′F''\\to F''F′′toF′′ with F′′F''F′′ a half-integer, for which the CPT effect is absent in a classical field.