ls3_unit_8

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

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Unformatted text preview: 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 v P ( ) [ VIII-1 ] We take the total Hamiltonian of a particular system in the form H = H + 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 t = i h , H + H int ( ) [ ] + t relax [ VIII-3 ] where t relax = i h , H random [ ] [ VIII-4 ] To find the nonlinear susceptibility, we make use of the following perturbation expansions: = ( ) + 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) . THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 89 R. Victor Jones, May 4, 2000 89 v P = v P ( ) + v P 1 ( ) + v P 2 ( ) + L [ VIII-5b ] with v P ( ) = Tr ( ) 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: ( ) t = i h ( ) , H [ ] + ( ) t relax [ VIII-6a ] 1 ( ) t = i h 1 ( ) , H [ ] + ( ) , H int [ ] { } + 1 () t relax [ VIII-6b ] 2 ( ) t = i h 2 ( ) , H [ ] + 1 ( ) , H int [ ] { } + 2 ( ) t relax [ VIII-6c ] 3 ( ) t = i h 3 ( ) , H [ ] + 2 ( ) , H int [ ] { } + 3 ( ) t relax [ VIII-6d ] Following earlier considerations, it is reasonable to write n t relax n = n n n n = n n n n [ VIII-7 ] Since the perturbing field is resolvable into an appropriate set of Fourier components THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 89 R. Victor Jones, May 4, 2000 89 v P = v P ( ) + v P 1 ( ) + v P 2 ( ) + L [ VIII-5b ] with v P ( ) = Tr ( ) 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: ( ) t = i h ( ) , H [ ] + ( ) t relax [ VIII-6a ] 1 ( ) t = i h 1 ( ) , H [ ] + ( ) , H int [ ] { } + 1 () t relax [ VIII-6b ] 2 ( ) t = i h 2 ( ) , H [ ] + 1 ( ) , H int [ ] { } + 2 ( ) t relax [ VIII-6c ] 3 ( ) t = i h 3 ( )...
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This note was uploaded on 02/29/2012 for the course PHYSICS 216 taught by Professor Staff during the Fall '11 term at BU.

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

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