Explicit Form of Solution of Two Atoms Tavis–Cummings Model

In this paper we consider the two atoms Tavis–Cummings model and give an explicit form to the solution of this model which will play a central role in quantum computation based on atoms of laser–cooled and trapped linearly in a cavity. We also present a problem of three atoms Tavis–Cummings model which is related to the construction of controlled–controlled NOT operation (gate) in quantum computation. E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] E-mail address : [email protected] 1 The purpose of this paper is to give an explicit form to the solution of Tavis–Cummings model ([1]) with one and two atoms. This model is a very important one in Quantum Optics and has been studied widely , see [2], [3] or [4] as general textbooks in quantum optics. See also recent papers [5], [6] and their references. We are studying a quantum computation and therefore want to study the model from this point of view, namely the quantum computation based on atoms of laser–cooled and trapped linearly in a cavity. We must in this model construct a controlled NOT gate or other controlled unitary gates to perform a quantum computation, see [7] as a general introduction to this subject. For that purpose we need the explicit form of solution of the models with one, two and three atoms. As for the model of one atom it is more or less well-known, and as for the case of two or three atoms it has not been given as far as we know. In this paper we give it for the case of two atoms, while we could not give it for the three atoms case, so we present it as a challenging problem. Anyway, let us start. The Tavis–Cummings model (with n–atoms) that we will 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 ∑ i=1 ( σ (+) i ⊗ a+ σ i ⊗ a† ) , (1) where ω is the frequency of radiation field, ∆ the energy difference of two level atoms, a and a† are annihilation and creation operators of the field, and g a coupling constant, and L = 2. Here σ (+) i , σ (−) i and σ (3) i are given as σ (s) i = 12 ⊗ · · · ⊗ 12 ⊗ σs ⊗ 12 ⊗ · · · ⊗ 12 (i− position) ∈ M(L,C) (2) where s is +, − and 3 respectively and σ+ = 