On Faithful Matrix Representations of Polynomially q-deformed Tavis-Cummings Model

Abstract

Consider the polynomially q-deformed Lie algebra, ℓpq : [K+,K−]q = P(K0), [K0,K+]q = G(K+),[K−,K0]q =G(K−),[K◦,K+]q = F (K+),[K−,K◦]q = F (K−),[K0,K◦]q = (1 −q)K0K◦, whoseHamiltonianis H = ω1K0+(ω1 +ω2)K◦+ λ(t) K−eiϕ +K+e−iϕ , (λ is a time-dependent coupling
parameter). The polynomially q-deformed Lie algebra, ℓpq is introduced as a generalized model of the Tavis-Cummings model in [31], namely: [K1,K2] = K3,[K3,K1] = 2K1,[K3,K2] = −2K2,[K4,K1] = −K1,[K4,K2] = K2 and [K3,K4] = 0, subject to the physical properties K3 and K4 are real diagonal operators andλ(t) K2eiϕ +K1e−iϕ .Faithful matrix representations of least degree of ℓpq are discussed and conditions are given to guarantee the existence of the faithful representations. The Tavis-Cummings model is itself an extension of the model of the double quan
tized harmonic oscillator generated by K+,K− and K0, in
[13]