ANNALS OF OPTICS - XXV ENFMC - 2002 262 Coupled cavities interacting with two-level atom G. T. Nogueira 1 , J. A. Roversi 2 Instituto de Física “Gleb Wataghin” - Unicamp - Campinas, SP, Brasil 1 trevisan@ifi.unicamp.br 2 roversi@ifi.unicamp.br Abstract In this work, we study a system of two coupled cavities interacting with a two-level atom. We use a variable transformation that diagonalizes the Hamiltonian of the system composed by two coupled cavities to simplify the problem. Through this transformation, we can write the effective Hamiltonian of the whole system like that of an atom interacting with two independent effective mode of the electromagnetic field. Some results of the atomic inversion and the mean number of photons are showed and compared with the results obtained to the atom-field system with one cavity, using the Jaynes-Cummings model. The presence of the second cavity creates conditions of the generation of others non-classical states of the light from another previously established state. Introduction In the last years, the interest in the coupling between the fields of optical cavities has increased due to the possibility of this system being a way of transferring quantum information. An application in quantum computation would be a system of n cavities coupled through optical fibers, each one containing a two-level atom, simulating a network [1]. In this work, we study some dynamical aspects of a system composed by two coupled optical cavities, one of them containing a two-level atom, based in the Jaynes-Cummings model [2] [3] of the atom-field interaction and in the Zoubi et al.’s work [4] about coupled cavities. First of all, we study a diagonalization of the Hamiltonian of the two-coupled cavities system transforming the creation and annihilation operators of the fields in a new set of creation and annihilation operators. Written in terms of the new operators, the complete Hamiltonian of the atom-coupled cavities system takes the form of the system composed by an atom interacting with two independent modes of the radiation. We show that in the limit of strong coupling between the cavities (compared to the atom-field coupling, but weak coupling compared to the energy of the each separated cavity), the Hamiltonian in the interaction picture of this system takes the form of the interaction Hamiltonian of the Jaynes-Cummings with only one cavity [5]. Hamiltonian of the system There are in the literature proposals about studies of interaction between cavities, where the coupling between them is considered proportional to the overlapped cavity-fields if there is a change of photons by a non- linear way [5] (beam-splitter, optical fiber, etc.). Once the electromagnetic field in the second quantization approach is described by the creation and annihilation operators, the Hamiltonian of the two-coupled cavities system, for weak coupling compared to the ground state energy of the separated cavities (λ<<ω 1 e λ<<ω 2 ), is given by: ( ) ( ) ( ) † 2 1 2 † 1 2 † 2 2 1 † 1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ a a a a a a a a H C + λ + ω + ω =    , (1) where ω 1 and ω 2 are the frequencies of the cavities (1 and 2) and λ is the coupling constant that depends on the distance between the cavities. â i and â i † are the creation and annihilation operators correspondent to the cavity “i”. The first two terms of the right side of the above equation represent the Hamiltonian of the isolate cavities and the last one is the coupling term. On the other hand, the interaction between a two-level atom and a mono-mode field is described in the literature through the Jaynes-Cummings model. According to this model, in the Rotating Wave Approximation (RWA) the interaction Hamiltonian of this system is [3]: ( ) † 1 1 AC H gâ â σ σ - + = +  , (2) where g is the coupling constant between the atom and the field, σ + (σ - ) is the operator that takes the atom from the ground (excited) state to the excited (ground) state. Thus, the whole Hamiltonian for the system composed by two coupled cavities and a two-level atom, is written as: ˆ ˆ ˆ ˆ S C A AC H H H H = + + ,