Tie Evolution of Matter States W. M. Hetherington February 15, 1 The Tie-Evolution Operat The tie-evolution of a wavefunction is deterined by the effect of a tie evolution operat through the relation Ψ r, t) = Ut)Ψ r, t), with U) = 1. 1) Using this in the Schrödinger equation i t Ψ r, t) = HΨ r, t) = i t Ut) ) Ψ r, ) = HUt)Ψ r, ), ) yields the differential equation f U i Ut) = HUt). 3) t If H is independent of tie, the solution is the siple tie dependent phase fact of a stationary state Ut) = e i ~ Ht. 4) Since the stationary state Ψ) is an eigenfunction of H with eigenvalue E, Ut)Ψ) = e i ~ Ht Ψ) = e i ~ Et Ψ). 5) When the Hailtonian is a general function of tie Ut) = U) i H dt )Ut )dt. 6) The solution to this integral equation can be obtained by interation. The first der approxiation can be obtained by substituting the zero der approxiation U) = 1 into the integral Ut) = 1 i Ht ) dt. 7) The second der approxiation is obtained by replacing Ut) in the integral with the first der expression Ut) = 1 i Ht )dt + i ) Ht )Ht )dt dt, 8) 1
and so fth Ut) = 1 i Ht 1 ) dt 1 + i ) + i ) 3 Ht 1 )Ht ) dt 1 dt Ht 1 )Ht )Ht 3 ) dt 1 dt dt 3 +, 9) When H = H o + V t), Ut) can be written in the f Ut) = U o t)u v t), where U o t) = e i ~ Hot. 1) Using this definition in the Schrodinger equation yields ) ) i t U o U v + U o t U v = HU o U v = H o U o U v + V t)u o U v. 11) Then,??) leads to siply Operating on the left with U o yields the relation H o U o U v + iu o t U v = H o U o U v + V t)u o U v, 1) iu o t U v = V t)u o U v. 13) i t U v = U V t)u ou v = V H t)u v, 14) where V H t) is the Heisenberg representation of the operat Vt). Solving the resulting integral equation by iteration yields U v t) = 1 i V H t 1 ) dt 1 + i ) + i ) 3 V H t 1 )V H t ) dt 1 dt V H t 1 )V H t )V H t 3 ) dt 1 dt dt 3 +. 15) When there is no applied tie-dependent electroagnetic field other perturbation, then the Hailtonian ay consist of two ters: one large operat H o f which the eigenstates can be found, and one sall operat V = constant operat considered to be a perturbation. Then the expression f U is again?? but the tie dependence of the integrands is particulary siple.
Expansion of Ψ in a basis of pseudo-stationary states Coonly, we know only the set of eigenstates φ l : l = 1,,... of an approxiate tie-independent Hailtonian H a, with H a φ l = ε l φ l. Even when the total Hailtonian has no tie dependence, Ψ will be a tie-dependent linear cobination of the basis functions. Suppose that H o = H a + H p, where H p represents a sall tie-independent perturbation, such as spin-bit coupling. Then Ψt) = k c k t)φ k = k c k t) k a, 16) where the set of coefficients c k is deterined by perturbation they a variational approach, and the subscript a on the index k indicates that the function k a is an eigenstate of H a. Using the concept of a vect space, the set k a is presued to be a coplete basis f the description of any state vect Ψ. An iptant general expression f the coefficients is c k t) = k a Ψt), 17) which is siply the projection of Ψ on to the k a axis. Now suppose that we know that at tie t = the syste is in a definite eigenstate of H a, that is Ψ) = l a. We also know that So, Ψt) = Ut)Ψ) = UΨ o. 18) c k t) = k Ψt) = k UΨ o = k a U l a = U kl. 19) The zero-der expression f c k t) arises fro the zero-der approxiation f U, that is Ut) = U o U v = U o. Then c k t) = k a U o l a = k a e i ~ H at l a = e i ~ ε lt k a l a = e i ~ ε lt δ kl. ) So, the state of the syste does not change. The first der ter in U v contributes c k t) = k a U o i c k t) = i e i ~ ε lt k a H p l a c k t) = i e i ~ ε lt The probability of the state k a is c k t) = e i ~ H at H p e i ~ H at dt ) l a, 1) k a e i ~ Ha t H p e i ~ Ha t l a dt, ) e~ i ε k ε l )t dt = i ) ~ e i i εlt k a H p l a e~ i ε k ε l )t 1) ɛ k ɛ l ɛ k ɛ l ) k a H p l a )) ɛk ɛ l ) 1 cos t. 3), 4) which is just the first-der tie-independent perturbation result. This looks pathological as ɛ k ɛ l ), but it really is not since cos x = 1 x / +. The ost useful application of this expression is f the case of ɛ k ɛ l. Then c k t) = k a H p l a t. 5) 3
Ut) f the EM Field-Matter Interation It is custoary to consider the interaction between the electroagnetic field and atter to be weak relative to the interactions aong the atter particles theselves, and this is certainly the case f typical nonlinear optical phenoena. Therefe, we write H = H + H I t) + H F, 6) where H describes the atter, H F describes the field and H I t) describes the interaction. If H I t) =, then H has two independent ters, so the total wavefunction is a siple product of atter and field states, Ψt) = ψt)ξt). 7) We will assue that the Schrödinger equation f the atter has been solved and that the wavefunction f the atter is Ψt) = c l t)φ l = c l t) l, 8) l l which describes a wave packet coherent superposition of the stationary state solutions k. The field wavefunction is ξ = n k k, 9) k k where k ranges over odes of the radiation field that are of interest and k refers to all other odes. A coherent state function would be prefered, but the nuber state function akes the algebra sipler. The coherence of the radiation field can be handled at a later stage. The total Hailtonian is H = n e +n n 1 n e +n n p l q la rl, t) + l i j odes q i q j + 4πɛ r ij k ω k a k a k + 1 ). 3) Considering the interaction of the radiation field only with the electrons, the interaction energy operat is n e e H I = p l A r l, t) + A r ) n e e l, t) p l + A r l, t) A r l, t), 31) and the total Hailtonian is H = n e +n n p n e +n n l + l i j A tie-evolution operat is now odes q i q j + H I t) + 4πɛ r ij k ω k a k a k + 1 ), 3) H = H + H I t) + H F. 33) Ut) = U t)u v t), where U t) = e i ~ H +H F )t 34) and U v is defined by eq. 16. The operat V H t) = U t)h I t)u t) 35) ust now be carefully defined. First, specify H I precisely. The ter involving A will be igned as not being iptant f this discussion. Since we are wking in the Coulob gauge A =, and p A = i A = i A) + A = ia = A p 36) 4
as an operat relation. Therefe, n e H I = A r l, t) p l. 37) This is refered to as the oentu f of the interaction Hailtonian. The vect potential A is a siple ultiplicative operat as far as the atter states are concerned but contains annihilation and creation operats f the field states. So, consider only the atter operats f the oent. Using p l = i l as an operat on a atter state l chills the spine. Ftunately, the siple coutat relations x, p x = y, p y = z, p z = i, 38) x, p x = ip x, y, p y = ip y, z, p z = ip z 39) provide a way to replace p with r. We will need to evaluate integrals of the f k p l k. Using we find that r j, H = n e 1 r j, p l = i p j, 4) k p l k = i k r l, H k = i k r l H H r l k, 41) k p l k = i E k E k ) k r l k = iω k ω k) k r l k = iω k k k r l k. 4) Me thoughtful analyses yield the sae result. This replaceent of p with r will be used later. Notice that H I is a su of one-electron operats. Electrons interact with the field individually, and the results are additive. Without any loss in generality, the interaction of only a single particle will be considered.h I. Thus H I = A r, t) p. 43) Now, turn to the quantized field operat A. Fro an earlier discussion, A r, t) = odes j A j a j ˆɛ j e i k j r ω j t) + a j ˆɛ j e i k j r ω j t). 44) Both creation a and annihilation a operats appear f all the polarizations and wave vects k. In a two-photon process, governed by the second ter in equation 16, any products of these operats would appear. We can restrict the contributions to A to only those operats which yield changes in the field which are easurable expected. Thus, f a two-photon absption of photons of odes ω 1, ˆɛ 1 ) and ω, ˆɛ ), we can use A r, t) = A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t) + A a ˆɛ e i k r ω t). 45) Siilarly, f su-frequency generation second haronic generation), photons of odes ω 1, ˆɛ 1 ) and ω, ˆɛ ) are annihilated, and one photon of ode ω 3, ˆɛ 3 ) is created. So, we can use A r, t) = A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t) + A a ˆɛ e i k r ω t) + A 3 a 3 ˆɛ 3 e i k 3 r ω 3 t). 46) All this can now be used to construct V H, V H t) = U t)h I t)u t) = e i ~ H +H F )t H I t)e i ~ H +H F )t. 47) 5
Continuing, V H t) = e ~ i H +H F )t A r, t) p e~ i H +H F )t. 48) We are now ready to ake use of the second der ter f U v t), U v t) = i ) V H t 1 )V H t ) dt 1 dt. 49) First, consider how this operat will be used. Fro??, Ψt) = Ut)ψ)ξ), 5) we see that we need to specify the initial state of the syste. Ordinarily, this would be given as a ξ a, where a is a stationary state of the atter Hailtonian and ξ a is a product of nuber states of the field. We will be looking f evidence of a particular final state f ξ f. Thus, we want to project Ψt) onto the f ξ f axis. Generally, Ψt) = l c l t) k a = Ut)ψ)ξ), 51) so, Using the product f f U yields Operating to the left with U, c f t) = fξ f Ψt) = fξ f Ut) aξ a = U fa. 5) fξ f U t)u v t) aξ a = fξ f e i ~ H +H F )t U v t) aξ a. 53) Using only the second der ter f U v, Continuing we arrive at the glious equation i ) e i ~ E f t fξ f U v t) aξ a. 54) e i ~ E f t fξ f U v t) aξ a. 55) i ) fξ f V H t 1 )V H t ) aξ a dt 1 dt, 56) fξ f e ~ i H +H F )t 1 A r, t 1 ) p A r, t ) p e~ i H +H F )t aξ a dt 1 dt. 57) If we use the third-der ter f U v, we arrive at the even e glious equation i ) 3 fξ f e i ~ H +H F )t 1 A r, t 1 ) p A r, t 3 ) p A r, t ) p e i ~ H +H F )t 3 aξ a dt 1 dt dt 3. 58) 6
And gliouser sill is the fourth-der equation i ) 4 3 Two-Photon Absption fξ f e ~ i H +H F )t 1 A r, t 3 ) p A r, t 4 ) p A r, t 1 ) p A r, t ) p e i ~ H +H F )t 4 aξ a dt 1 dt dt 3 dt 4. 59) To find the probability of finding the syste in state fξ f at tie t we need to take the absolute square of the glious equation. To evaluate i ) 1 fξ f A r, t 1 ) p A r, t ) p aξ a e~ i E at dt 1 dt, 6) we will insert a coplete set of states between the two operats fξ f A r, t 1 ) p jξ j jξ j A r, t ) p aξ a. 61) j Focusing on just the t integral, jξ j A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t ) + A a ˆɛ e i k r ω t ) p aξ a e~ i Ea t dt = jξ j A 1 a 1 ˆɛ 1 e i k 1 r ω 1 t ) p aξ a e i ~ E a t dt jξ j A a ˆɛ e i k r ω t ) p aξ a e i ~ E a t dt. 6) Thus, we have two different tie-derings: ω 1 followed by ω and vice versa. 7