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So each collective operator 13k has its own reservoir associated with the collective annihilation and creation operators Yk ( ry) and Yi (ry). Therefore, the total Hamiltonian is given in terms of these collective operators by
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(7.95) As in Section 7.1, we can separate (7.95) into one single damped oscillator, represented by the Nth collective atomic mode, that alone couples to the cavity mode, and N - 1 uncoupled damped oscillators. The latter are irrelevant here, for they do not affect the field dynamics. Just as we have done earlier in this chapter, we drop the subscript N from EN, YN(T/), and GN to simplify the notation. So we will be concerned with the following Hamiltonian: (7.96) where
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(7.98) (7.99)
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Unlike Huttner and Barnett [306, 307], we have a strictly microscopic model for the medium, where each constituent atom sits in the vacuum a given distance away from the others. The medium is discrete rather than continuous. From a practical point of view, however, there is no need to account for all this microscopic detail. So we take the macroscopic average of the physical variables in our theory. As we have done in Section 7.4, we expand the Dirac delta functions in (7.2) into Fourier sine series inside the cavity. Robinson's in'elevant spatial Fourier components are here all those with a spatial frequency higher than w e / c.
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Dielectric permittivity is a classical concept. Thus, the permittivity in our quantum theory should be the same that we obtain in classical electrodynamics. In this section we temporarily disregard the quantum nature of the radiation field to calculate the dielectric permittivity in our model. Later in this chapter this will be compared with the dielectric permittivity that appears in the Huttner-Barnett theory. We will denote the classical counterparts of a, E, and Y(T/) by a/../fi, f3/../fi, and Y(T/)/../fi, respectively. Making these replacements in (7.96) and using Poisson
brackets rather than commutators (see 2), we obtain the following equations of motion:
:t a = -iwea - iG (3 + (3*),
!(3 = -iwo(3 - iG (a + a*) - i
dry V(ry)Y(ry),
(7.101) (7.102)
!Y(rt) = -iry Y(ry) - iV*(ry) (3.
Assuming that the reservoir is not initially excited, we may take Y( ry) as vanishing at t = O. Then integrating (7.102), we find that
Y(ry, t) = -iV*(ry)
dt' e i (t'-t)71(3(t').
Substituting this formal result in (7.10 1), we get
:t(3 =
iwo(3 - iG (a
+ a*)
dry !V(ry) ,2
dt' e i (t'-t)71(3(t').
In the golden rule regime (wo ~ .Do ~ J<i.) considered here, the integrals on last term on the right-hand side of (7.104) can be done (Problem 7.1) and we find 9 that
= - (J<i.
+ iwo) (3 -
iG (a + a*).
The electric displacement D, m3gnetic field B, and polarization P are given by
rw; . ~x yT'Dsm (We)
(7.106) (7.107) (7.108)
rw; ~x -yr;Bcos (We)
2G' P=--where
j e . (We) , -xsm - x
(7.109) z To write down a closed system of differential equations for the real variables in (7.109) from the equations of motion (7.100), (7.102), and (7.105) for the complex variables a, (3, and Y(ry), we must enlarge (7.109) by including the imaginary part of (3,
a-a* 'D=a+a*, B= --.-, and X=(3+(3*.
(3 - (3*
9This is the famous Markovian approximation.
The physical meaning of to' is this. If we think the polarization P is produced by an oscillator of effective charge G'.fL in a region of size L, to' is proportional to the momentum of this oscillator. From (7.100), (1.102), (7.105), (7.109), and (7.110), we find that
dt'D = weB, -B = -we'D - 2GX , dt dt X = -KX +Woto',
d -to' = dt
-KtO' -
(7.111) (7.112) (7.113) (7.114)
WoX - 2G'D.
Differentiating (7.113) and using (7.114) to eliminate dtO' /dt. we obtain
dt2X = -K dt X +wo {-KtO' - WoX - 2G'D}.