Quantum Diagonalization Method in the Tavis-Cummings Model
  To obtain the explicit form of evolution operator in the Tavis-Cummings model we must calculate the term ${e}^{-itg(S_{+}\otimes a+S_{-}\otimes a^{\dagger})}$ explicitly which is very hard. In this paper we try to make the quantum matrix $A\equiv S_{+}\otimes a+S_{-}\otimes a^{\dagger}$ 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 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 a noncommutative differential geometry.   However, a quantum diagonalization is not unique and is affected by some ambiguity arising from the noncommutativity of operators in quantum physics.   Our method may open a new point of view in Mathematical Physics or Quantum Physics. 
Kazuyuki Fujii, Kyoko Higashida, Ryosuke Kato, Tatsuo Suzuki, Yukako   Wada