Suppose that the complete Hamiltonian of a coupled system is parsed into two components
.
[ VA-1 ] The component
includes the Hamiltonians for the unperturbed material system, the free radiation field and interactions of the material system with available cavity modes. The component
is the Hamiltonian for the interactions which couple the material system to externally excited modes. As the first step in finding a fully quantal expression for the dielectric susceptibility, let us expand the state vector in the Schrödinger picture in terms of, presumably, known eigenkets of
-- viz. the dressed states of the unperturbed system --
.
[ VA-2 ] Following a now familiar track, we can use the Schrödinger equation of motion -- i.e.
[ VA-3 ] to obtain
.
[ VA-4 ] In turn, we obtain the following expansion for the time dependent expectation value of induced material system dipole moment:
[ VA-5 ] Differentiating this expression with respect to time and using Equation [ VA-4 ] we obtain
[ VA-6a ] Regrouping, we see that this expression can be written
[ VA-6b ] Since
is proportional to
, we see that, like magic,the first term vanishes!!! Hence,
.
[ VA-6c ] Differentiating this expression with respect to time and, again, using Equation [ VA-4 ] we obtain
.
[ VA-7 ] Our task is to now to attempt an interpretation this very nasty expression. To that end, we make use of Equation [ V-30 ] to write Equation [ VA-6c ] as
.
[ VA-8a ] Using the properties of the
and
operators (viz.
and
), this expression reduces to
[ VA-8b ] which may interpreted as a sum of a series of dipole moment components -- viz.
[ VA-8c ] Given this interpretation, we return to Equation [ VA-7 ] and use Equation [ V-6 ] to obtain
.
[ VA-9 ] Again using Equation [ V-30 ] and the properties of the
and
operators, we see that
[ VA-10a] which reduces to
[ VA-10b ] Substituting this expression and the expression in Equation [ V-33 ] into Equation [ VA-9 ] and, again, using the properties of the
and
operators it relatively straightforward to obtain
.
[ VA-11a ] Thus, in a kind of rotating field approximation, we get a set of driven harmonic oscillator equations of the form
.
[ VA-11b ] In the Weisskopf-Wigner approximation[1] -- i.e.
-- we can easily solve these equations and sum their results to obtain a standardized form for the frequency dependent of the dressed dielectric susceptibility of the system
[ VA-12 ]
