ABSTRACT

E = iJ2 eky/2irnu;k/V (akefi* - a\e'iir) , k

where ak, a\ are the annihilation and creation operators of photons with the wave vector k, frequency ujk and polarization ek, satisfying the common boson commutation relations [ak, a}] = 8ki\ V stands for the cavity volume. The free field Hamiltonian has the form

k

Consider a collection of identical two-level atoms with the upper energy level |1) and the lower energy level |0). The free atomic Hamiltonian reads

where sZj — — |0j)(0j| and hcja is the energy separation between two levels. The electric dipole interaction of the cavity field with the atomic system is described by the Hamiltonian

where Ej is the electric field at the point fj which corresponds to the position of the j-th atom and dj is the electric dipole operator of this atom given by

Here s + i = | l j ) (0j | and s__, = |0 j ) ( l j | are the atomic transition operators and dj = e(lj\f\0j) is the matrix element of the electric dipole momentum (which are assumed to be the same for all the atoms). Note that the operators sZj, s±j obey the algebraic relations for the common Pauli matrices.