Heisenberg dipole operator

The result of using these identities as well as the Heisenberg definition of the time-dependenee of the dipole operator... 

In the Heisenberg representation a time-dependent dipole operator p(t) is generated from its value at some previous time t by a unitary transformation with the time-displacement operator exp — t )/h, so that... 

Here, the dipole moment operator at time t is related to that at the initial time through the usual Heisenberg transformation ... 

The ACF of the dipole moment operator of the fast mode may be written in the presence of Fermi resonances by aid of Eq. (10). Besides, the dipole moment operator at time t appearing in this equation is given by a Heisenberg equation involving the full Hamiltonian (225). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the real... 

Here, pDim(0) is the dipole moment operator of the dimer high frequency mode at initial time, whereas IDim(t) is the corresponding operator at time t, which is obtained by a Heisenberg transformation over the first one with the aid of the Hamiltonian HDav. 

Hence, the operators a in (67) also form a representation of the Weyl-Heisenberg algebra of the electric dipole photons. Employing this transformation (67) then gives the diagonal representation of the operator (63)... 

The preceding results lead to the conclusion that the radiation phase states (72) are dual to the conventional Fock number states nm). In turn, the operators (67) form the representation of the Weyl-Heisenberg algebra of the electric dipole photons dual to the operators am and [46]. 

Now consider the set of Stokes operators that can be obtained by canonical quantization of (132). On the other hand, the Stokes operators should by definition represent the complete set of independent Hermitian bilinear forms in the photon operators of creation and annihilation. It is clear that such a set is represented by the generators of the SU(3) subalgebra in the Weyl-Heisenberg algebra of electric dipole photons. The nine generators have the form [46]... 

Let us stress a very important difference between the representations of Stokes operators (137) and (157). If the former is valid only for the electric dipole photons, the latter describes an arbitrary multipole radiation with any X and j. The similarity in the operator structure and quantum phase properties is caused by the same number of degrees of freedom defining the representation of the SU(2) subalgebra in the Weyl-Heisenberg algebra. 

---