ls3_unit_8 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 88 R. Victor Jones, May 4, 2000 88 VIII. NONLINEAR OPTICS -- QUANTUM PICTURE: 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 ( 29 [ 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 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 ( 29 [ ] + t relax [ VIII-3 ] where t = i h ρ , H [ ] [ VIII-4 ] To find the nonlinear susceptibility, we make use of the following perturbation expansions: ρ= ρ 0 1 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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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 89 R. Victor Jones, May 4, 2000 89 v P = v 0 ( 29 + v 1 + v 2 + L [ VIII-5b ] with v α = Tr ρ α ( 29 v [ 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 [ ] + 0 t relax [ VIII-6a ] 1 t = i h ρ 1 , H 0 + ρ 0 , H int [ ] { } + 1 (29 t [ VIII-6b ] 2 t = i h ρ 2 , H 0 + ρ 1 , H int [ ] { } + 2 t [ VIII-6c ] 3 t = i h ρ 3 , H 0 + ρ 2 , H int [ ] { } + 3 t [ VIII-6d ] ……………………………………………… Following earlier considerations, it is reasonable to write n t n =-Γ n n n ρ n n n ρ n 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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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 90 R. Victor Jones, May 4, 2000 90 H int = H int ϖ k ( 29 exp - i ϖ k t ( 29 k [ VIII-8a ] ρ= ρ ϖ k ( 29 exp - i ϖ k t ( 29 k [ VIII-8b ] and resolve Equations [ VIII-6 ] into a hierarchy of algebraic equations. The first member of that hierarchy becomes 47 ρ n n 1 (29 ϖ k i ϖ n n k n n { } = i h n ρ 0 ( 29 H int ϖ k ( 29 - H int ϖ k ( 29 ρ 0 ( 29 { } n = i h n H int ϖ k ( 29 n ρ nn 0 ( 29 n n 0 ( 29 { } [ VIII-9a ] or 48 ρ n n 1 ϖ k = i h n H int ϖ k ( 29 n ρ nn 0 ( 29 n n 0 ( 29 { } D ϖ n n k ; Γ n n ( 29 . [ VIII-9b ] The second member of the hierarchy becomes ρ n n 2 ( 29 ϖ k ( 29 i ϖ n n k n n { } = i h n ρ 1 ϖ k , H int ϖ k [ ] n [ VIII-10a ] or ρ n n 2 ( 29 ϖ k ( 29 = i h D ϖ n n k ; Γ n n ( 29 ρ n n 1 ϖ ′′ n H int ϖ k n { n - n H int ϖ k n ρ n n 1 ( 29 ϖ k ( 29 } [ VIII-10b ] 47 we presume in this development that H int is a Hermitian operator.
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This note was uploaded on 01/31/2011 for the course PHYSICS 108 taught by Professor Staff during the Winter '08 term at UC Davis.

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ls3_unit_8 - THE INTERACTION OF RADIATION AND MATTER:...

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