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Manuscript Title: Recursive generation of Cartesian angular momentum coupling trees for SO(3).
Authors: B.S. Sherborne, G.E. Stedman
Program title: AMTREES
Catalogue identifier: ABRD_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 59(1990)417
Programming language: Pascal.
Computer: IBM PC.
Operating system: DOS 3.2.
RAM: 640K words
Keywords: General purpose, Rotation group, Angular momentum, Young's symmetrisers, Cartesian tensors, Jucys, Diagram techniques, Quantum theory of Angular momentum, Nonlinear optics, Polarisation, Clebsch-gordan.
Classification: 4.1.

Nature of problem:
When the geometry of a nonlinear optical experiment is altered, by changing the polarisation of one or more participating light beams or by space-time operations on the applied fields or the medium, the resulting change in the optical effect is restricted by symmetry considerations, and can be predicted from the knowlegde of a set of irreducible physical constants. As nonlinear optical processes of ever greater complexity become feasible and of interest, the construction of an efficient algorithm for determining these predictions is a useful goal, in view of the rapidly escalating difficulty of performing calculations when the number of vectors (polarisation or field) involved is greater than four.

Solution method:
As far as the photon polarisation and field orientation is concerned, the necessary group-theoretic analysis may be confined to SO(3) i.e. angular momentum coupling theory; the symmetries of the medium probably also require analysis of some subgroup of SO(3). Since the polarisation and field quantities are all vectors, we may usefully concentrate on the vector irreducible representation of SO(3), i.e. unit angular momentum, for all terminal angular momentum values. Various algorithms for reducing angular momentum coupling trees have been developed, some with possible computer implementation in mind. These would in the above context specify in an economical form the likely geometric restrictions on any optical process. Alternatively, or in addition to this, one may use tensor analysis in other or higher groups such as the unitary group in N dimensions, and then employ techniques for projection onto subspaces with definite irreducible character with respect to the orthogonal or symmetric groups to achieve a new elegence. As a result, the use of Cartesian tensor theory is also well established in the subject. Our analysis implements the computer manipulation of angular momentum coupling trees in two different forms.

Running time:
The most important parameter is the relevant tensorial rank, which is the total number of interactions with every photon and with any field summed over the amplitude of the process and its conjugate. For ranks up to 7, the run time for either algorithm is not important; above this value, for an IBM AT, runs take minutes (rank 8,9) or hours (rank 10).