02 05 0 v 2 1 3 Fe b 20 03 How To Treat An N – Level System : A Proposal ∗ Kazuyuki FUJII † Department of Mathematical Sciences Yokohama City University Yokohama , 236 - 0027 Japan

In this paper we propose a Hamiltonian of the n–level system by making use of generalized Pauli matrices. An atom has usually many (finite or infinite) energy levels. However, to treat infinitely many ones at the same time is not realistic, so we treat an atom with finite energy levels. In the following we would like to consider an atom with n energy levels interacting with external (periodic) fields like cos(ωt+ φ) (an n–level system in our terminology). In Quantum Optics it is standard to consider a 2–level system, see for example [1] and [2]. The n–level system might be considered as a composite of 2–level systems. As for recent developments of the 2–level system including constructions of solutions in the strong coupling regime, see [3], [4], [5], [6], [13] and their references. See also [8] as a general introduction to two–level system, and [7], [10] as some applications to Quantum Optics. However we would like to treat an n–level system in a direct manner. Then we meet some troubles. First what is a Hamiltonian describing such a system ? As far as we This is prepared for the coming 11–th Numazu Meeting (6–8/March/2003) E-mail address : [email protected] 1 know such a Hamiltonian has not been known. As for some examples of n–level system see [12], [17], [18] and their references. In this paper we propose such a Hamiltonian. In the 2–level system the Hamiltonian is written by making use of Pauli matrices {σ1, σ3} (see the references above), so in the n– level system the Hamiltonian should be given by making use of generalized Pauli matrices {Σ1,Σ3}. We will write down it. We believe that it is a natural generalization Our real aim of this study is to apply some developments in this paper to the theory of qudits in Quantum Computation, see for example [11] and [15]. In it we must first of all construct all unitary elements in U(n) (a universality of 1–qudit). We expect that this work will become a starting point toward this direction . Let {σ1, σ2, σ3} be Pauli matrices and 12 a unit matrix : σ1 = 