Polynomial bases and Wigner coefficients for SU(3) ⊃ R3

“…The condition (17) is also the constraint useful in determining the expansion coefficients of the basis vector (16) expanded in terms of the basis vectors of U (3) ⊃ U (2) ⊃ U (1), which will be used in what follows. In the following, we only construct the highest weight state of SO(3) with M = L. Once the basis vector (16) for the highest weight state of SO(3) with M = L is known, the basis vector of SU (3) ⊃ SO(3) ⊃ SO(2) for any M can be expressed in the standard way as…”

Section: Canonical and Non-canonical Bases Of Su (3)mentioning

confidence: 99%

“…Similar non-orthogonal basis vectors of SU (3) ⊃ SO(3) ⊃ SO (2) have been constructed by Bargmann and Moshinsky [14], and further studied by Ališauskas [15]. In addition, Sharp et al have proposed the polynomial and stretched non-orthogonal bases [16,17]. Asherova and Smirnov have used projection operators expressed as polynomials in the SO(3) generators instead of integral operators in the group elements [18], for which the expressions are also complicated.…”

Section: Introductionmentioning

confidence: 96%

A new procedure for constructing basis vectors of SU(3)⊃SO(3)

et al. 2016

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A simple and effective algebraic angular momentum projection procedure for constructing basis vectors of SU (3) ⊃ SO(3) ⊃ SO(2) from the canonical U (3) ⊃ U (2) ⊃ U (1) basis vectors is outlined. The expansion coefficients are components of the null-space vectors of a projection matrix with, in general, four nonzero elements in each row, where the projection matrix is derived from known matrix elements of the U (3) generators in the canonical basis. The advantage of the new procedure lies in the fact that the Hill-Wheeler integral involved in the Elliott's projection operator method used previously is avoided, thereby achieving faster numerical calculations with improved accuracy. Selected analytical expressions of the expansion coefficients for the SU (3) irreps [n13, n23], or equally, (λ, µ) = (n13 − n23, n23) with λ and µ the SU (3) labels familiar from the Elliott model, are presented as examples for n23 ≤ 4. Explicit formulae for evaluating SO(3)-reduced matrix elements of SU (3) generators are derived. A general formula for evaluating the SU (3) ⊃ SO(3) Wigner coefficients is given, which is expressed in terms of the expansion coefficients and known U (3) ⊃ U (2) and U (2) ⊃ U (1) Wigner coefficients. Formulae for evaluating the elementary Wigner coefficients of SU (3) ⊃ SO(3), i. e., for the SU (3) coupling [n13, n23]⊗[1, 0], are explicitly given with some analytical examples shown to check the validity of the results. However, the Gram-Schmidt orthonormalization is still needed in order to provide orthonormalized basis vectors.

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“…The general correspondence between the occupational basis states |n 1 n 2 n 3 is found in [20] and given by …”

Section: The Evolutionmentioning

confidence: 99%

Quantum tomography of a system of three-level atoms

Klimov

Guise

2007

J. Phys. A: Math. Theor.

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We analyze the possibility of tomographic reconstruction of a system of three-level atoms in both non-degenerate and degenerate cases. In the non-degenerate case (when both transitions can be accessed independently) a complete reconstruction is possible. In the degenerate case (when both transitions are excited simultaneously) the complete reconstruction is achievable only for a single atom in the Ξ configuration. For multiple Ξ atoms, or even a single atom in the Λ configuration, only partial reconstruction is possible. Examples of one and two-atom cases are explicitly considered.

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Polynomial bases and Wigner coefficients for SU(3) ⊃ R3

Cited by 30 publications

References 6 publications

Algebraic Treatment of a Three-Oscillator System: Applications to Some Molecular Models

Algebraic Treatment of a Three-Oscillator System: Applications to Some Molecular Models

A new procedure for constructing basis vectors of SU(3)⊃SO(3)

Quantum tomography of a system of three-level atoms

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