On Faithful Matrix Representations of q-Deformed Models in Quantum Optics

Consider the q-deformed Lie algebra, tq:K^1,K^2q=1−qK^1K^2,K^3,K^1q=sK^3, K^1,K^4q=sK^4,K^3,K^2q=tK^3,K^2,K^4q=tK^4, and K^4,K^3q=rK^1, where r,s,t∈ℝ−0, subject to the physical properties: K^1 and K^2 are real diagonal operators, and K^3=K^4†, († is for Hermitian conjugation). The q-deformed Lie alg...

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Bibliographic Details
Published in:International journal of mathematics and mathematical sciences 2022-09, Vol.2022, p.1-8
Main Authors: Hanna, Latif A -M., Alazemi, Abdullah, Al-Dhafeeri, Anwar
Format: Article
Language:English
Subjects:
Algebra
Analysis
Conjugation
Deformation
Lie groups
Matrix representation
Operators (mathematics)
Optics
Oscillators
Partial differential equations
Physical properties
Quantum optics
Radiation
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Summary:Consider the q-deformed Lie algebra, tq:K^1,K^2q=1−qK^1K^2,K^3,K^1q=sK^3, K^1,K^4q=sK^4,K^3,K^2q=tK^3,K^2,K^4q=tK^4, and K^4,K^3q=rK^1, where r,s,t∈ℝ−0, subject to the physical properties: K^1 and K^2 are real diagonal operators, and K^3=K^4†, († is for Hermitian conjugation). The q-deformed Lie algebra, tq is introduced as a generalized model of the Tavis–Cummings model (Tavis and Cummings 1968, Bashir and Sebawe Abdalla 1995), namely, K^1,K^2=0,K^1,K^3=−2K^3,K^1,K^4=2K^4,K^2,K^3=K^3,K^2,K^4=K^4, and K^4,K^3=K^1, which is subject to the physical properties K^1 and K^2 are real diagonal operators, and K^3=K^4†. Faithful matrix representations of the least degree of tq are discussed, and conditions are given to guarantee the existence of the faithful representations.
ISSN:0161-1712
1687-0425
DOI:10.1155/2022/6737287