Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems. (17th November 2020)

• Record Type: Journal Article
• Authors: Leiner, David; Zeier, Robert; Glaser, Steffen J
• Is Part Of: Journal of physics. Volume 53:Number 49(2020)
• Publication Date: 2020-11-17
• Subjects: coupled spin systems -- Wigner functions -- tensor operators -- Clebsch–Gordan decomposition -- coefficients of fractional parentage -- representation theory; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ab93ff
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We study the representation and visualization of finite-dimensional, coupled quantum systems. To establish a generalized Wigner representation, multi-spin operators are decomposed into a symmetry-adapted tensor basis and are mapped to multiple spherical plots that are each assembled from linear combinations of spherical harmonics. We explicitly determine the corresponding symmetry-adapted tensor basis for up to six coupled spins 1/2 (qubits) using a first step that relies on a Clebsch–Gordan decomposition and a second step which is implemented with two different approaches based on explicit projection operators and coefficients of fractional parentage. The approach based on explicit projection operators is currently only applicable for up to four spins 1/2. The resulting generalized Wigner representation is illustrated with various examples for the cases of four to six coupled spins 1/2. We also treat the case of two coupled spins with arbitrary spin numbers (qudits) not necessarily equal to 1/2 and highlight a quantum system of a spin 1/2 coupled to a spin 1 (qutrit). Our work offers a much more detailed understanding of the symmetries appearing in coupled quantum systems.