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Additional resources for Quantum Optics (3rd Edition)
Introduction of the operators aˆ ν , aˆ ν† in Eq. 103) where ωνν = ∑ ωλ UνλUν∗ λ = ων∗ ν . 104) λ As expected, the nonmonochromatic photons are coupled to each other and have no ﬁxed energy in general. 105) as can be seen from Eq. 79). Using Eq. 101). The time-dependent nonmonochromatic modes Aν (r, t) are also called spatio-temporal modes (of the vector potential). Let the (excited) spatio-temporal modes Aν (r, t) be nonoverlapping pulselike wave packets,11 each of them can be described by a mid-frequency Ων and a slowly varying amplitude, i.
161) ˆ † Hˆ U. ˆ Hˆ = U 19) The extension to many atomic systems is straightforward. The uniˆ = ∏A U ˆ A , with U ˆ A from Eq. 155). 4 Dielectric background media The new Hamiltonian Hˆ expressed in terms of the new variables formally looks like the old minimal-coupling Hamiltonian Hˆ expressed in terms of the originally used variables. Expressing in Hˆ the new variables in terms of the original ones, we arrive at a multipolar-coupling Hamiltonian Hˆ = + 1 2 1 ˆ ˆ⊥ dr3 ε− 0 Π ( r) + P A ( r) 1 2 1 ∑ ma pˆ a + Q a a 1 0 2 ˆ ( r) + µ0−1 ∇ × A 2 ˆ [r A + s(rˆ a − rˆ A )]] ds s(rˆ a − rˆ A ) × [∇ × A 2 ˆ Coul .
C. ds s [(rˆ a − r A )× pˆ a ][∇× A 1 0 ˆ [r A + s(rˆ a − r A )]] ds s(rˆ a − r A ) × [∇ × A 2 . 233) differs from Eq. 150) in the ﬁrst term, which now describes the interaction of the polarization Pˆ A of the atomic system with the transformed transverse displacement ﬁeld ⊥ ˆ ⊥ (r) = [ε 0 Eˆ ⊥ (r) + Pˆ M D (r)] ⊥ ˆ (r) + Pˆ M ˆ⊥ = −Π (r) = ε 0 Eˆ ⊥ (r) + Pˆ ⊥ A ( r ) + PM ( r ) . 234) Note that all the primed quantities are deﬁned according to the deﬁnitions of the unprimed quantities except that therein the old variables fˆ (r, ω ), ˆf† (r, ω ) must be (formally) replaced by the new variables fˆ (r, ω ), ˆf † (r, ω ).