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Our dielectric JCM is a simplified version of Hopfield's model of dielectrics. The first to modify the Hopfield model to account for absorption were Huttner and Barnett
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7Tbere is another Kramers-Kronig dispersion relation that we do not mention explicitly here, where g I is given as an integral of gROver frequency.
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[33, 306,307]. Following their idea, we will now introduce in the dielectric JCM a coupling with a reservoir to model absorption [167]. Let the oscillators of the medium be coupled to a reservoir consisting of a continuum of harmonic oscillators. We will assume that this coupling strength V(w) has a maximum modulus that is much smaller than woo This allows us to make the rotating-wave approximation in the interaction term between the oscillators of the medium and those of the reservoir. We will also make a white-noise assumption that amounts to neglecting the frequency dependency of V (w) within the range over which it broadens the linewidth of the medium. This frequency range is given by 1I'1V(woW. So we are assuming that V(w) is approximately flat over a frequency interval of length 1I'1V(woW centered at woo That is nothing more than the regime where Fermi's golden rule holds. In this regime, the final result is independent of the particular form of V(w). Thus, for simplicity, we will adopt a Lorentzian shape for 1V (w) 12 , with V (w) given by
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+ i~ ,
(7.78)
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where Wo : ~ : /'i,. Figure 7.3 shows a plot of IV(wW in the golden rule regime we are considering.
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Figure 7.3 We are considering the regime where Fermi's golden rule holds. In this regime the actual shape of V (w) is irrelevant as long as it satisfies the broad requirements mentioned in the text. For simplicity, we take V(w) given by (7.78) so that W(w)1 2 is a Lorentzian. Then the golden rule regime is Wo /: :;. K, exemplified in this plot. All frequencies in the plot are in units of K. We used Wo = WOK, /:::;. = 10K. The inset shows an enlargement of the central part of the Lorentzian peak, where you can clearly see that V (w) is reasonably flat over the medium linewidth 1I'1V(woW.
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The new Hamiltonian is given by
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(1]) and Wk (1]) are the reservoir creation and annihilation operators. They where obey the usual commutation rules for the continuum
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(7.80) and commute with at, a, b}, and b It is important to stress that there is another j assumption implied in (7.79). This is the assumption that each atom has its own reservoirS and that the differences in their coupling strengths is negligible. We will see more about that in chapter 9 in the context of cavity damping. Substituting (7.1) and (7.2) in (7.79), we find that
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j=1 j=1
(at + a) (b} + bj )
(7.81) where 9j = -qjV 2;:0 sin
(:c Xj ).
(7.82)
As we have done in Section 7.1, we introduce the collective operators
Ih =
where
'E <Pkjbj,
(7.83)
(7.84) forj
k+ 1 andk < N, (7.85)
8Phonons or any other incoherent process in the dieleclric medium play the role of reservoir.
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cPkj = 0
for j > k
+ 1 and k =I N,
(7.86) (7.87)
L9}. j'=l
Like the b , they are annihilation operators obeying the commutation relations j (7.88) In terms of them, the "lossless" part of the Hamiltonian [i.e., the first three terms on the right-hand side of (7.81)] becomes
nwcata + nwo L BJBj + nGN (at + a) (B~ + BN) . j=l
(7.89)
The part of the Hamiltonian involving the reservoirs and the coupling with each atom of the medium [Le., the last two terms on the right-hand side of (7.81)] can be rewritten in terms of the collective operators substituting the bk by
bk = LcPjkBj. j=1
Then
(7.90)
N N N :LbkWk(ry) = LLcPjkBJWk(ry) k=lj=l k=l
N LBJt(ry), j=l
(7.91)
where
t(ry)
= LcPjkWk(ry). k=l
(7.92)
As you can easily verify, the t(ry) are annihilation operators; that is, (7.93) Moreover,just as we have shown in Section 7.1 for the