Quantum diagonalization method in the Tavis-Commings model

To obtain the explicit form of evolution operator in the Tavis-Cummings model we must calculate the term e^{-itg(S+⊗a+S-⊗a†)} explicitly which is very hard. In this paper we try to make the quantum matrix A ≡ S₊⊗a+S_-⊗a^† diagonal to calculate e^-itgA and, moreover, to know a deep structure of the model. For the case of one, two and three atoms we give such a diagonalization which is the first nontrivial examples as far as we know, and reproduce the calculations of e^-itgA given in quant-ph/0404034. We also give a hint to an application to non-commutative differential geometry. However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the non-commutativity of operators in quantum physics. Our method may open a new point of view in mathematical or quantum physics.