For simplicity consider a two-level atomic system with ground and excited states  and , respectively (using the Dirac bracket notation).  Let the energy difference between the states be  so that  is the transition frequency of the system.  Then the unperturbed Hamiltonian of the atom can be written as
Suppose the atom experiences an external classical electric field of frequency , given by
; e.g., a plane wave propagating in space.  Then under the dipole approximation the interaction Hamiltonian between the atom and the electric field can be expressed as
where  is the dipole moment operator of the atom.  The total Hamiltonian for the atom-light system is therefore   The atom does not have a dipole moment when it is in an energy eigenstate, so   This means that defining  allows the dipole operator to be written as
(with  denoting the complex conjugate).  The interaction Hamiltonian can then be shown to be
where  is the Rabi frequency and  is the counter-rotating frequency.  To see why the  terms are called counter-rotating consider a unitary transformation to the interaction or Dirac picture where the transformed Hamiltonian  is given by
where  is the detuning between the light field and the atom.