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

Abstract

Consider the q -deformed Lie algebra, t q : K ^ 1 , K ^ 2 q = 1 − q K ^ 1 K ^ 2 , K ^ 3 , K ^ 1 q = s K ^ 3 , K ^ 1 , K ^ 4 q = s K ^ 4 , K ^ 3 , K ^ 2 q = t K ^ 3 , K ^ 2 , K ^ 4 q = t K ^ 4 , and K ^ 4 , K ^ 3 q = r K ^ 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, t q 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 = − 2 K ^ 3 , K ^ 1 , K ^ 4 = 2 K ^ 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 t q are discussed, and conditions are given to guarantee the existence of the faithful representations.