[ ( )] + ρ VIII. NONLINEAR OPTICS -- QUANTUM PICTURE: 45 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 88

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1 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 88 VIII. NONLINEAR OPTICS -- QUANTUM PICTURE: 45 A QUANTUM MECHANICAL VIEW OF THE BASICS OF N ONLINEAR OPTICS 46 In what follows we draw on the discussion of the density operator in Reiew of Basic Quantum Mechanics: Dynamic Behaior 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 gien by P = Tr We take the total Hamiltonian of a particular system in the form ρ P [ VIII- ] H = H + H int + H random [ VIII- ] where H random is a Hamiltonian describing the random perturbations on the system by the thermal reseroir surrounding the system. Thus ρ t = i h ρ, H +H int [ ] + ρ t relax [ VIII-3 ] where ρ = i [ t h ρ,h random ] [ VIII-4 ] relax To find the nonlinear susceptibility, we make use of the following perturbation expansions: ρ= ρ ( ) +ρ ( ) +ρ ( ) +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 in Y. R. Shen's Principles of Nonlinear Optics, Wiley (984). R. Victor Jones, May 4, 88

2 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 89 P = P ( ) + + P ( ) +L [ VIII-5b ] P with P ( α) = Tr ρ α ( 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 ( ) t [ ]+ ρ = i h ρ ( ),H t relax [ VIII-6a ] ρ ( ) t ρ ( ) t ρ ( 3) t = i h = i h = i h {[ ρ ( ),H ]+ [ ρ ( ),H ]} + ρ () int t {[ ρ ( ),H ]+ [ ρ ( ),H ]} + ρ ( ) int t {[ ρ ( 3),H ]+ [ ρ ( ),H ]} + ρ ( 3) int t relax relax relax [ VIII-6b ] [ VIII-6c ] [ VIII-6d ] Following earlier considerations, it is reasonable to write n ρ n = Γ t n n n ρ n = Γ n relax n ρ n n [ VIII-7 ] Since the perturbing field is resolable into an appropriate set of Fourier components (either discrete or continuous), we may write R. Victor Jones, May 4, 89

3 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 9 H int = H int ( ω k ) exp ( i ω k t) [ VIII-8a ] k ρ= ρ( ω k ) exp ( i ω k t) [ VIII-8b ] k and resole Equations [ VIII-6 ] into a hierarchy of algebraic equations. member of that hierarchy becomes 47 () ρ n n { +Γ n n }= i h n ρ ( ) H int ω k ( ω k ) i ω n n ω k ( H int ( ω k ) ρ ) { } n = i h n H int ( ω k ) { n n } ( ) n ρ nn ρ The first [ VIII-9a ] or 48 () ρ n n ( ω k ) = i h n H int ( ω k ) The second member of the hierarchy becomes ρ n n ω k { ( n ω k )+Γ n n } = i h k i ω n ( ) n ρ nn ρ ( ) { n n }D ω n n ω k ;Γ n n. [ VIII-9b ] [ ] n ρ () ( ω k ),H int ( ω k ω k ) n [ VIII-a ] or ρ n n ω k = i h D ω () ( n n ω k ;Γ n n ) ρ n n ( ω k ) k n { n n H int ω k ω k n H int ( ω k ω k ) n ρ n n ω k } [ VIII-b ] 47 we presume in this deelopment that H int is a Hermitian operator. 48 Recall that D u ; w [ ] is the so called complex Lorentzian denominator. i( u ) + w R. Victor Jones, May 4, 9

4 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 9 Thus, we see that this expansion yields results akin to those embodied in Equations [VII-8 ] through [ VIII-3 ] in OCEF. TWO -PHOTON ABSORPTION Let us consider two-photon absorption by a single atom (or by a set of paired r k -states in a semiconductor as discussed Section 7 of IRM:ST). ω B ω ω A ω B ω A ω ω ω Two-Photon Absorption Process Tfrom earlier discussions, the appropriate interaction Hamiltonian is gien by H P = e r r E + ( R,t) = e r r E ( + ) r ( R,t )+ E r ( + ) r [ ( R,t )] [ VIII- ] r In the appendix to this section we establish -- iz., in Equation [ VIIIA- ] -- that the second approximation to the transition rate is gien by τ d d t f ( ) P i f t,t = π f H h int i + h f m f H int m m H int i ω i ω m δ( ω i ω f ) Thus for the two-photon process R. Victor Jones, May 4, 9

5 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 9 π e4 τ h 4 f r E r ( + ) m m r E r ( + ) i δ( ω m ω i ω i ω f ) [ VIII- ] f m For two beam, two-photon processes τ π e4 h 4 f m f r E r ( + ) m m r r E + ω ω m + f r E r ( + ) m m r r ω ω m i E + i [ VIII-3a ] δ( ω +ω ω f ) and for one beam, two-photon processes π e4 f r r ( E + ) m m r E r + i δ ( ω h 4 m ω i ω beam ω f )[ VIII-3b ] f beam τ If we use once again the factorization used in Section VI of this lecture set -- iz. we write the initial state as i = i = { i, i } = A ( i ) F ( i ) where A ( i ) and F i are, respectiely, the initial electronic (atomic) and field (photon) states. Using closure on the intermediate photon states, we find for two beam, twophoton processes τ ( ) π e4 M (,) δ( ω h 4 +ω ω f ) Tr ρ E ( ) E ( ) E ( + ) E ( + ) [ ] [ VIII-4a ] where R. Victor Jones, May 4, 9

6 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 93 M (,) m A( f ) r A (m) ε ˆ ε ˆ A(m) r A (i) ω ω m + A( f ) r A (m) ε ˆ ε ˆ A(m) r A (i) ω ω m [ VIII-5a ] and for one beam, two-photon processes τ ( ) π e4 M (,) δ( ω h 4 ω f ) Tr ρ E ( ) E ( ) E ( + ) ( + ) [ E ] [ VIII-4b ] where M (,) A( f ) r A (m) ε ˆ ε ˆ A(m) r A (i) [ VIII-5b ] m ω ω m FOUR WAVE MIXING : Four Wae Mixing Spectroscopy: We explore here a particular set of four wae interactions. To that end, we consider an input field which consists of two plane waes = E α E α r, t [ ] k [ VIII-6 ] [ ( r ω t )] +c.c. k, ω exp i ( k r ω t ) +E α k, ω exp i where frequencies ω and ω lie in the isible part of the spectrum. As Equation [ VIII-3 ] of OCEF informs us, this input incident on a third-order nonlinear material, will directly generate a component of nonlinear polarization at a frequency ω -- iz. R. Victor Jones, May 4, 93

7 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 94 ( 3) (NL) Pα E β (3) ( r, t) = 3 χ αβγδ ω ;ω ; ω k, ω exp i ( k k ) r E γ k, ω E δ k, ω [ ] exp i ω [ t] [ VIII-7 ] which may be associated with the destruction of two photons at ω and the creation of photons at ω and ω. ω ω ω ω χ (3) ω Four wae mixing - direct process If, howeer, the material lacks a center of symmetry there is also an alternate path to obtain ω radiation. In a second-order material [ VIII-3 ] of OCEF informs us we can also hae a nonlinear polarization at ω ( ) (NL) Pα () ( r, t) = χ αβγ ( ω ;ω ) E β k k {, ω ω E γ k, ω () +χ αβγ ( ω ; ) E β k, ω E γ k, ω } exp i ( k k ) r [ ] exp i ω [ t] [ VIII-8 ] R. Victor Jones, May 4, 94

8 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 95 if we hae driing fields E β k k, ω ω and E β k, ω. Such fields are, in fact, generate by the input field through the nonlinear polarization components ( ) (NL) Pα () = χ αβγ k k, ω E β ω ; k, ω E γ k, ω [ VIII-9a ] and ( ) (NL) Pα () = χ αβγ k, ω E β ω ;ω k, ω E γ k, ω [ VIII-9b ] ω ω ω χ () ω ω ω ω ω χ () () χ ω ω ω ω ω χ () We may sole for E β Four wae mixing - indirect processes k k,ω ω and E β k, ω from Equation [ VIII-3 ] in OCEF -- iz. Q δ αβ Q α Q β ω c ε or using the operator I ( k ) k k ε αβ ω E β k k = k k ( k ) Q, ω =µ ω NL P α { } ε ε αβ ( ω)i Q βδ ( ω c { )δ αδ } E δ Q, ω = ω c ε αβ defined in OCEF NL ( ω) P β ( Q, ω) [ VIII-a ] ( Q, ω) [ VIII-b ] R. Victor Jones, May 4, 95

9 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 96 If we define G αδ Q ; ω ε ε αβ { ( ω)i Q βδ ( ω c )δ αδ } [ VIII- ] we can, in principle, write E α Q, ω = ω c Q ; ω [ ] G αβ ε βδ NL ( ω) P δ ( Q, ω) [ VIII- ] Therefore, for a material lacking a center of symmetry, the complete nonlinear polarization may be written P α (NL) [ ] exp i ω ω ( r, t) = 3exp i ( k k ) r { (3) χ αβγδ ( ω ;ω ; ω ) () [ ] χ αη γ ω c [ t] E β ( ω ;ω )G ηλ + [ 3 ω c () ] χ αηδ ω ; G η λ k, ω E γ k k, ω ω k, ω E δ k, ω ε λφ ε λφ k, ω ω () χ φβδ ( ω ; ω ) ω χ φβ γ } () ω ;ω [ VIII-3 ] Notice the resonance in the second term when k k and ω are tuned to the wae number and frequency of coupled electromagnetic excitations or normal modes such as plasmons, polaritons and magnons. Degenerate Four Wae Mixing (DFWM) - Phase Conjugation: Consider now the reflected waes generated by the direct process -- i.e., Equation [ VIII-7 ] -- when the input field is the sum of two counter propagating pump fields, with [ k p,ω] and [ k p,ω], plus a complex wae E r in ( r, t) represented a range of wae numbers components { k },ω -- iz. l [ ] R. Victor Jones, May 4, 96

10 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 97 = E α + E α r, t [ ( )] + E α exp i ( k l ) l exp i k p r φ + + E α l l [ ] [ ] exp i k p r φ exp [ i ω t] + c.c. [ VIII-4 ] as shown below pump _ k p input signal nonlinear crystal pump - k p According to Equation [ VIII-7 ], reflected waes are generated by nonlinear polarizations components with [{ k },3ω], [{ k l p + k },ω], [{ k l p k }, ω], and l k { }, ω. The latter component is the so-called phase conjugate term and may l [ ] be written ( 3) (NL) Pα (3) ( r, t) = 3χ αβγδ ( ) ( ω;ω;ω) E β l E δ l + E γ k p, ω exp i φ + +φ * exp i k l r φ [ ] l which may be interpreted as the time reersed image of r [ ] exp[ i ω t] + c.c. E in r, t. [ VIII-5 ] TIME REVERSED IMAGING R. Victor Jones, May 4, 97

11 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 98 THE RAMAN EFFECT The Raman Nonlinearity: In our discussion of the dielectric susceptibility thus far we hae been implicitly assuming that the nuclei of system were clamped into fixed positions and hae focused on the nonlinearities associate with electron dynamics. This is too limited a iew. In general, we may expect that polarizability of, say, a molecule is a function of the nuclear positions and that the dipole moment can expressed p u ( s) α ({ }, E ) = p ( ) α ({ u ( s) }) + a u ( s) αβ ({ }) E β +L. [ VIII-6 ] u { ( s) } denotes a collection of nuclear positions where u ( s) is the displacement of the s nucleus from its nominal equilibrium position. Further, we may expand these terms in powers of the displacement -- iz. p α ({ u }) i p α ({ }) N + ( s) u p ( ) α u ( s) s β u β +L [ VIII-7a ] s= R. Victor Jones, May 4, 98

12 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 99 p α () u ({ ( i) }, E ) a u ( s) N αβ ({ }) E a ( ) β αβ E β + a ( R) s αβγ E β u γ +L [ VIII-7b ] s= The second term in Equation [ VIII-7b ] is the so called Raman nonlinearity. ( ) Since ( s) p u α u ( s) β ({ }) has the dimensions of charge, Equation [ VIII-7a ] is usually written where e αβ p α ({ u }) s p α N s ( s) u β +L. [ VIII-7a' ] + eαβ s= ( s) is the dynamic effectie charge tensor associated with the s nucleus. The nuclear displacements are a linear combination of the normal modes of the system = Γβ ( s; ν) q( ν) [ VIII-8 ] u β s 3N ν= so that we may recast Equations [ VIII-7a ] and [ VIII-7b ] in the form p α ({ u }) s p α 3N + eα ( ν) q( ν) +L [ VIII-9a ] ν= = a ( R) αβγ p α R N s= E β u γ s 3N R a αβ = ( ν) E β q( ν) [ VIII-9b ] ν= where e α ν N and a αβ R = e αβ ( s ) Γ β ( s; ν) s= s= N R ( ν) = a αβγ Γ γ ( s;ν). R. Victor Jones, May 4, 99

13 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE Quantum Theory of Raman Scattering The Raman nonlinearity proides a coupling between the photon field described by the Hamiltonian H rad = hω a k a k,σ + [ VIII-3 ] k, σ k, σ and the phonon field described by the Hamiltonian H ib = hω ν b ν b ν +. [ VIII-3 ] ν where b ν and b ν are, respectiely, the phonon creation and destruction operators. In the electric dipole approximation, the interaction Hamiltonian can be written 3N H int = ( R) a αβ ( ν) E α E β q( ν) [ VIII-3 ] ν= In earlier discussions we hae shown that q( ν) = h M ω ν ( b ν +b ν ) [ VIII-33a ] and π h ω E α = i k e ˆ V α ( k,σ) a k,σ a [ VIII-33b ] k, σ k, σ so that the complete interaction Hamiltonian may expressed as R. Victor Jones, May 4,

14 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE H int = 3N π h h ( R) V a M ω ν αβ ( ν) ( b ν +b ν ) ν= ω ˆ e k α ( k,σ) a k, σ a ω k, σ k k,σ k, σ ˆ a e β k, σ a k, σ ( k, σ ) To concentrate on the key issue, we write scattering Hamiltonian which is proportional to a a k, σ and which describes the scattering of a photon from the k, σ k, σ k,σ { } (initial) state to the { } (final) state H scat = π h h V M ω ν where a ( R) I,F; ν ( R) = a αβ ( ν) ˆ e α ( I) ˆ ( ω I ω F ) a ( R) ( I,F; ν) a F a I ( b ν +b ν ) [ VIII-35 ] e β ( F). We may now use the Fermi golden rule 49 to calculate the transition rate per "molecule" for inelastic Stokes scattering eents -- iz. for { n I,n F, n ν } { n I, n F +,n ν +} [ VIII-34 ] = π f H h scat i δ ω i ω f τ S = π h π h V h ω M ω ν I ω F a ( R) ( I,F; ν) n I,n F +, n ν + a F a I b ν n I, n F,n ν δ ( ωi ω F ω ν ) [ VIII-36a ] = 8π 3 τ S V h ω M ω ν I ω F a ( R) ( I,F; ν) n I ( n F +) ( n ν +) δ( ω I ω F ω ν ) [ VIII-36b ] 49 See the appendix to this section for a recapitulation of the theory of transition rates. R. Victor Jones, May 4,

15 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE and the transition rate per "molecule" for inelastic anti-stokes scattering eents -- iz. for { n I, n F, n ν } { n I, n F +, n ν } τ AS = π h = 8π3 V π h V h ω I ω F M ω ν ( a R )( I,F; ν) n I,n F +, n ν a F a I b ν n I, n F,n ν δ ( ωi +ω ν ω F ) h ω I ω F M ω ν a ( R) ( I,F; ν) n I ( n F +) n ν δ( ω I +ω ν ω F ) [ VIII-36c ] For a total number of "molecules" V ρ mol we may write the total time rate of change of photons of frequency ω F as d n F d t d n F d t S = 8π 3 ρ mol ω I ω F V AS h a R M ω ν ( I,F; ν) n I ( n F +) ( n ν +) δ( ω I ω F ω ν ) = 8π3 ρ mol ω I ω F V h a R M ω ν ( I,F; ν) n I ( n F +) n ν δ( ω I +ω ν ω F ) [ VIII-37a ] [ VIII-37b ] Stimulated Raman Effect: Simple model Suppose that a high power laser at ω I -- the pump -- propagates parallel to the z-axis and that the Raman medium occupies the half-space z >. The spatial rate of change of the Stokes photons at ω F -- the signal -- may be written 5 d n F z d z = G R n I z [ ] [ VIII-38 ] n F ( z) + 5 Assuming for the nonce that n ν. R. Victor Jones, May 4,

16 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 3 where G R = 8π 3 ρ mol ω I ω F cv γ ν h a ( R) I,F; ν M ω ν [ VIII-39 ] The rate constant γ ν is included this expression for gain to account for the broadening of a gien ibrational state. Each Stokes eent, of course, also depletes the incident beam so that d n I z d z = G R n I ( z) n F ( z)+ [ ] [ VIII-4 ] R. Victor Jones, May 4, 3

17 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 4 which means that n F ( z) + n I ( z) = n I ( ). [ VIII-4 ] Substituting this conseration condition into Equation [ VIII-38 ] we see that d n F z d z [ n F ( z) ] n F ( z) + = G R n I [ ] [ VIII-4a ] or that n I ( ) d n F z [ n F ( z) +] d n F z n F z [ ] = G + n R [ I ( ) ] d z [ VIII-4b ] Therefore n F ( z) = n I exp { G R [ + n I ( ) ] z } + exp G R + n I ( ) n I { [ ] z} [ VIII-43 ] This simple model predicts that Stokes photons initially increase linearly with distance { [ ] } -- and that there complete conersion at distances large compared to G R + n I i.e. lim z n F [ ] z [ VIII-44a ] ( z) = G R + n I lim z n F ( z ) = n I [ VIII-44b ] R. Victor Jones, May 4, 4

18 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 5 Coherent anti-stokes Raman Spectroscopy (CARS): There is an important four wae mixing process inoling the Raman nonlinearity. We may see how this comes about by realizing that Equation [ VIII-9 ] implies a potential energy contribution 3N R a αβ ν= ( ν) E α E β q ν [ VIII-45 so that inclusion of the Raman nonlinearity modifies the equation of motion for the ibrational normal modes -- iz. q ( t,ν) +ω ν q( t,ν)= M a ( R) αβ ( ν) E α E β [ VIII-46 ] When the normal mode is drien by a two (isible) wae input (see discussion of fourwae processes aboe), its dominant response will be at the difference frequency ω -- iz. R. Victor Jones, May 4, 5

19 THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 6 q ( t,ν) +ω ν q( t,ν) = M a R αβ ( ν) E α ω [ ]+c.c. E β ( ω ) exp i ( ω ) t [ VIII-47 ] so that q( t,ν) = M a αβ R ( ν) E α ω [ ] E β ( ω ) exp i ( ω ) t [ VIII-48 ] ω ν ω From Equation [ VIII-9b ] we see that excitation of a ibrational mode, through the Raman nonlinearity, generates a nonlinear polarization ( R) p α = M [ ] 3N ( R) ( R) a αβ ( ν) a γδ ( ν)e β ( ω ) E γ ( ω ) E δ ( ω ) exp i ( ω ) t [ VIII-49 ] ω ν ( ω ) ν= at the four wae mixing frequency ω. Thus, we see that the medium has an effectie third order susceptibility [ χ ( 3) αβγδ ( ω ) ] = χ ( 3 ) αβγδ eff + 3M a αβ R R ( ν) a γδ ( ν) [ VIII-5 ] ω ν ω which exhibits a resonance when ω equals ω ν. R. Victor Jones, May 4, 6