Explicit Form of the Evolution Operator of Tavis–Cummings Model : Three and Four Atoms Cases

In this letter the explicit form of evolution operator of the Tavis–Cummings model with three and four atoms is given. This is an important progress in quantum optics or mathematical physics. The purpose of this paper is to give an explicit form to the evolution operator of Tavis–Cummings model ([1]) with some atoms. This model is a very important one in Quantum Optics and has been studied widely, see [2] as general textbooks in quantum optics. We are studying a quantum computation and therefore want to study the model from this point of view, namely a quantum computation based on atoms of laser–cooled and trapped linearly in a cavity. We must in this model construct the controlled NOT gate or other controlled unitary gates to perform a quantum computation, see [3] as a general introduction to this subject. For that aim we need the explicit form of evolution operator of the models with one, two, three and four atoms (at least). As to the model of one atom or two atoms it is more or less known (see [4]), while as to the case of three and four atoms it has not been given as far as we know. Since we succeeded in finding the explicit form for three and four atoms cases we report it ([5]). E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] 1 The Tavis–Cummings model (with n–atoms) that we treat in this paper can be written as follows (we set h̄ = 1 for simplicity). H = ω1L ⊗ a†a+ ∆ 2 n ∑ i=1 σ (3) i ⊗ 1 + g n ∑