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ls3_unit_8 - THE INTERACTION OF RADIATION AND MATTER...

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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 88 R. Victor Jones, May 4, 2000 88 VIII. N ONLINEAR O PTICS -- Q UANTUM P ICTURE : 45 A Q UANTUM M ECHANICAL V IEW OF THE B ASICS OF N ONLINEAR O PTICS 46 In what follows we draw on the discussion of the density operator in Review of Basic Quantum Mechanics: Dynamic Behavior of Quantum Systems, Section II of the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory (hereafter referred to as IRM:ST). The macroscopic polarization is given by v P = Tr ρ v P ( ) [ VIII-1 ] We take the total Hamiltonian of a particular system in the form H = H 0 + H int + H random [ VIII-2 ] where H random is a Hamiltonian describing the random perturbations on the system by the thermal reservoir surrounding the system. Thus ∂ρ t = i h ρ , H 0 + H int ( ) [ ] + ∂ρ t relax [ VIII-3 ] where ∂ρ t relax = i h ρ , H random [ ] [ VIII-4 ] To find the nonlinear susceptibility, we make use of the following perturbation expansions: ρ = ρ 0 ( ) + ρ 1 ( ) + ρ 2 ( ) + L [ VIII-5a ] 45 See Nonlinear Optics -- Classical Picture which is Section VII in the lecture set entitled On Classical Electromagnetic Fields (OCEF). 46 See, for example, Chapter 2 in Y. R. Shen's Principles of Nonlinear Optics , Wiley (1984) .
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 89 R. Victor Jones, May 4, 2000 89 v P = v P 0 ( ) + v P 1 ( ) + v P 2 ( ) + L [ VIII-5b ] with v P α ( ) = Tr ρ α ( ) v P ( ) [ VIII-5c ] By substituting these expansions into Equation [ VIII-3 ] and equating terms of like order in H int , we obtain the following hierarchy of equations: ∂ρ 0 ( ) t = i h ρ 0 ( ) , H 0 [ ] + ∂ρ 0 ( ) t relax [ VIII-6a ] ∂ρ 1 ( ) t = i h ρ 1 ( ) , H 0 [ ] + ρ 0 ( ) , H int [ ] { } + ∂ρ 1 () t relax [ VIII-6b ] ∂ρ 2 ( ) t = i h ρ 2 ( ) , H 0 [ ] + ρ 1 ( ) , H int [ ] { } + ∂ρ 2 ( ) t relax [ VIII-6c ] ∂ρ 3 ( ) t = i h ρ 3 ( ) , H 0 [ ] + ρ 2 ( ) , H int [ ] { } + ∂ρ 3 ( ) t relax [ VIII-6d ] ……………………………………………… Following earlier considerations, it is reasonable to write n ∂ρ t relax ʹஒ n = −Γ n ʹஒ n n ρ ʹஒ n = −Γ n ʹஒ n ρ n ʹஒ n [ VIII-7 ] Since the perturbing field is resolvable into an appropriate set of Fourier components (either discrete or continuous), we may write
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 90 R. Victor Jones, May 4, 2000 90 H int = H int ω k ( ) exp i ω k t ( ) k [ VIII-8a ] ρ = ρ ω k ( ) exp i ω k t ( ) k [ VIII-8b ] and resolve Equations [ VIII-6 ] into a hierarchy of algebraic equations. The first member of that hierarchy becomes 47 ρ n ʹஒ n 1 () ω k ( ) i ω n ʹஒ n −ω k ( ) + Γ n ʹஒ n { } = i h n ρ 0 ( ) H int ω k ( ) H int ω k ( ) ρ 0 ( ) { } ʹஒ n = i h n H int ω k ( ) ʹஒ n ρ nn 0 ( ) −ρ ʹஒ n ʹஒ n 0 ( ) { } [ VIII-9a ] or 48 ρ n ʹஒ n 1 () ω k ( ) = i h n H int ω k ( ) ʹஒ n ρ nn 0 ( ) −ρ ʹஒ n ʹஒ n 0 ( ) { } D ω n ʹஒ n −ω k ; Γ n ʹஒ n ( ) . [ VIII-9b ] The second member of the hierarchy becomes ρ n ʹஒ n 2 ( ) ω k ( ) i ω n ʹஒ n −ω k ( ) + Γ n ʹஒ n { } = i h n ρ 1 () ω ʹஒ k ( ) , H int ω k −ω ʹஒ k ( ) [ ] ʹஒ n ʹஒ k [ VIII-10a ] or ρ n ʹஒ n 2 ( ) ω k ( ) = i h D ω n ʹஒ n −ω k ; Γ n ʹஒ n ( ) ρ n ʹஒ ʹஒ n 1 () ω ʹஒ k ( ) ʹஒ ʹஒ n H int ω k −ω ʹஒ k ( ) ʹஒ n { ʹஒ ʹஒ n ʹஒ k
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