Diagrammatic evaluation of the density operator for nonlinear optical calculations

Volume 23, number 1

OPTICS COMMUNICATIONS

October 1977

DIAGRAMMATIC EVALUATION OF THE DENSITY OPERATOR FOR NONLINEAR OPTICAL CALCULATIONS ~ S.Y. YEE and T.K. GUSTAFSON ¢m Department o f Eleetrteal Engtneering and Computer Setenees and the Electronics Research Laboratory Um verst ty o f Cahforma, Berkeley 94 720, U S A

S.A.J. DRUET and J.-P.E. T A R A N ONERA, 92320 Ch~ttllon, France

Received 11 July 1977 Tune-ordered diagrammatic representations are shown to precisely define and to sunplify calculations of radiative perturbations to the density matrix. Nonlinear optical susceptibilities,here exemphfled by that of CARS, can be obtained by simple propagator rules. An interpretation of transient Raman scattering m terms of time-ordered contributions is also discussed.

Nonlinear optical polarization or susceptibility components have, to a great extent, been obtained from dens:ty operator calculations. In many cases the derkslty matrix elements are calculated by perturbation theory up to the desired order [ 1]. Such detailed calculations can be avoided by using a general expression for the density operator to the order in question [2] ; however, its apparent complexity has restricted :ts application. In this letter it is shown that th:s general solution, when expressed in terms o f propagators and represented diagrammatically, not only allows a precise identification o f m u l t l p h o t o n processed but also provides a simple means by which to readily calculate the associated susceptibilities. In general, the calculation o f any specified order of the density matrix involves the evaluation o f a number o f terms, each associated with a specific time-ordering of the various interactions that occur between the molecules and the radiation fields. This time-ordering is important; it arises since the density matrix is a statistical average o f the p r o d u c t o f the wave function and its complex conjugate, each o f which represents a possible evolution o f the molecular state. Thus a given sequence o f interactions between radiation modes and the wave function can have many different relative tlme-orderings with respect to the distract sequence o f interactions associated with the complex conjugate wave function. By means o f a diagrammatic representation, any time-ordered contribution to the density matrix can be precisely defined and readily wrxtten down. Diagrams have been utilized previously for the calculation o f nonl,near susceptibilities, but only in the limit o f negligible lifetime and coUisional broadening [3,4]. In this limit, the perturbations occuring on the wave function are strictly decoupled from (and can be calculated independently of) those occuring on its complex conjugate. Indeed, we show that all possible time-ordered diagrams for the given sequence o f Interactions on the complex and complex conjugate parts o f the density matrix then sum to provide the decoupled result. A clear representation o f the polarization, and density operator elemel,ts in general, can be based upon an extension o f the approaches used by Omont, Smith, and Cooper [5] for the calculation o f trans:tion rates in incoherent scattering, and by Bonch-Bruevich [6] for multi-photon transition probabilities. To do so, we begin with the Research sponsored in part by Army Research Office Grant - DAHC04-75-C0095, The National Aeronautics and Space Admlnlstranon Grant - NSG-2151, National Science Foundation Grant - ENG72-03860-A01 Present address: ONERA, Ch~tfllon and Ecole Polytechnxque, Palalseau, France.

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October 1977

following general perturbation solution of the density operator [2].

P(n)(t)=

2I,]~Dl

=0

Pf

... f

--~

- ~

K+(tq+l,tq)~

-

U(tq+ 1

×P }0)(llK-(to, t'l)U(t'l- tO) p[I 1 I--~J~ [ _vl K-(t'p,t;+l)U(~)+l

tq) K+(tl,to)U(tl t'p)

=

t0)ll)

.dtmdt'l...dt'n_ m,

at 1

(1)

AV

where the ( t ) ' s and ( tp' ) ' s are the times o f the various Interactions between the molecules and the radiation fields, p}O) is the lnqtlal value o f the density matrix element at t = t O, In) is the elgenket of the free Hamlltonlan H 0 with elgenenergy hwn, i f ( t ) is the perturbation Hamtltonlan (we shall use the dipolar approximation and the Schrodlnger representation ~or H'(tq) = - p • E (tq) with p dipole m o m e n t operator and E(tq) Instantaneous values o f the electric field vector). The U(t t] ) are unit-step distributions (= 00 if t t < tl, = 1 if t 1 ~ t] ), which maintain causality 1 for the Interactions occurmg on the left and right hand sides o f p}I ), P represents the sum of all possible time-ordered permutations between the (t)'s and (t')'s subject to the constraints t I < t 2 . . . < t m and t !1 < t 2 < .. < t n-m imposed by the unit-step distributions. The product of terms on the left hand side of p}O) m eq. (1) describes the evolution of the wave function. Tlus evolution results from the m interactions occurmg at times tl,.. t ,... t m associated with Hamlltonlan H'(t ), + q q and from the propagations represented by the propagator K (t +1' to) which carries the wave function forward in q (0) . time between the two Interactions at t a and ta+ 1 The terms on the right hand side ofpz ' describe the stmultaneous e olutlon o ,f the complex conjugate wave function, K (t p , t p +.) being the associated propagator between the inter• , 1 actions at tp and tp+ 1 . When colhslons and other statistical parameters are neglected, one has ¢

V

"

--

K+(tz,tj) = ~ l n ) ( n l e x p {

lCOn(t'

~

,

--

t

t

t

J

X-(f, tl)=K+(ti,~) *.

tj)},

tl

Finally, AV denotes an average over the statistical parameters of the ensemble o f molecules (colhslons, thermal velocity, etc..). In general, an average must be taken over these statistical variables, then, the matrix elements of the evolution ' t '1+1) cannot be specified independently $1. However, for the tmpact approximaoperators K+(tq+l , tq) and K - ( t 1' tion, which is assumed from this point on, K+(t +1' t ) tufty depends on the ttme interval t +. - t and similarly f or K _( t ,, t ., 1); furthermore, for the period ~- o~/overlap .q of intervals t +1' t and t ,+1' t ,, it is q possible I q to specify the /* + q q 1 ,] averaged/product [K K ] AV" Therefore, for this time overlap, which begins at tq or t/ (whichever comes first), one can specify the matrix elements •

[(rlK+(r)lu) (v IK-(r)Is)] AV = ((rsl [/£(r)]

AvlUV)).

(2)

Further, one has K + ( r l + r2) = K + ( r l ) K + ( r 2 ) and similarly for K - Thus the products K+K - in eq. ( l ) can be written as products involving the Intervals between successive interactions rl, r2, e t c . . . , and since these intervals are independent and non overlapping, they can be averaged over independently. The statistically averaged matrix elements are written as (( rs IA(~-)l u v )) where A('r) IS [/£(~-)] AV and is familiar in the theory o f h n e broadening. If we further make the isolated hne approxunation, as discussed by Omont et al. [5], the most important matrix elements o f A are the diagonal terms of the form

((rs [ti(~-)luv)) : 6ru6sv

exp(-lCOrvr

dPrv'r),

:[:1 This discussion down to eq (3) follows rel [5] closely

(3)

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October 1977

where corv= (cor - coy) and qbro is in general a complex constant describing collisional broadenings and lmeshifts. It is this approxtmatlon then that provides the usual damping term used in the density matrix time-evolution equations. If q~rv as zero, the matrix elements of A(r) factor into the matrix elements o f K ÷ and K - which can be written and averaged separately. With eq. (3) it is possible to express any individual time-ordered term of eq. (1) as a function of the transition frequencies, linewidths, llfetmaes and matrix elements. There are several advantages to be offered by using this approach over that of the density matrix differential equations: (i), it allows a direct evaluation of p(n) without calculating lower orders; (ii), it allows a diagrammatic visualization o f the processes contributing to the polarization and frees one of the necessity of using energy level diagrams, which are vague and have caused much confusion; (nl), one can investigate more general broadening processes and treat them more rigorously than with the phenomenologlcal approach usually taken with the density matrix. Any term o f e q . (1) can be associated with a diagram, according to the following roles' (1) the time axis is represented vertically; (11) the time evolution of the wave function (ket) is plotted to the left of this axis, and that o f the complex conjugate (bra) to the right; (Ill) the field interactions are represented by vertices in the standard way (fig. 1). The term in question is then unamblgously specified if: (a) the m interactions (or vertices) on the ket side are specified; (b) the relative time-ordering o f the vertices on the ket and bra sides is specified; (c) each vertex is specified as an emission or absorption operation. Once the diagram is specified, sample rules can be used for an evaluation o f the associated density matrix element. These rules can be easdy deduced from eq. (1) in the case o f interaction with monochromatic light waves. We shall do this taking zero initial values for off-diagonal components o f the density matrix, and assuming that the diagonal components are constant up to the first interaction; these Initial conditions apply to the usual cases of interest. Consider for instance the particular time-ordered case shown on fig. 2. The first interaction is taken to occur

IT/ME

I

TIME

H,=_p.~e*,:,,,~

-p.~-e

(b)

(a)

TIME

/ (c)

TIME

,j (d)

Fig 1. Standard representation of the radiative perturbations on the bra and ket (a) and (c) respectwely represent photon annlhdat]on and creatlon operators acting on the bra; (b) and (d) respectwely represent photon creation and anmhflatlon operators acting on the ket.

i

i

I

t=t~. ]H'/t'~)

I

-

t-t',/m/r)

,t:t o

U> f,, < I

Fig. 2 Diagrammatic representation of p (t) for a specific tmaeordered contribution of eq. (1)

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October 1977

¢

on t h e k e t side at t l and the second on the bra side at t 1 (1.e. t 1 > tl). Then in eq. (1),

[K+(tl '

to)K-(to'

tl)] AV = 1,

and

K+(t2' t l ) -H - ~' ( t l )-

ii)p)Ot ) (j[K-(tl, t'l )H'(t'l) -lh

can be written as . H'(t 1 )

K+(t2' t'l)K+(t'l '

tl) -~-

1])P~OI)(] ]K-(tl

H'(t'l) ' t'l) --lh

and for the approximations of eq. (3) and the desIgnanon of the states between interactions shown on fig. 2 and time interval r I , we have.

K+(t2,t'l)lr)(rlK+(rl )[r)(rl Letting

H'(t 1) be

H*(t'1 - 7-i)

~

H'(t'l)

I])P)°)(IIK-(rl)I])(f I _~

absorption this gives

K+(t2 , t'l)lr)(rl

- - P " gl exp{--lWl(t'l -- rl)}

2ih

[j) exp

{-(ICOrl + Cbrl)rl }p~O)(/I "H'(t'l)

With t O ~ _co to extract the steady state contribut:on, the mtegral over 71 [-f0o~ ... dr 1 ] gives: K+(t2 , t'l)lr)(r[ - P I J ) " Cl exp(-lC°lt'l)

2h(~--C°rl + idPrl +C°l)

H'(t'l) p/(/0)(jl _ ~

This suffices to state the general roles permitting one to calculate the steady state contribution to p(t) corresponding to a given diagram. Beginning at t o and tracing up the diagram, one Introduces factors as follows (1) the initial density matrix element p(0) (2) the transition moment associated wit the first perturbation to p~O) (interaction at time t 1 with a field at COl): (rl-Pll) " t le - ~ It~2 (3) the propagation factor resulting from the evolution between t 1 and the tune of the next perturbation, Le. (1 for the case of fig. 2; the expression for this factor is gwen below, (4) the transition moment associated with the next interaction at t'1 and the corresponding propagation factor for the evolution between t 1 and the time of the following perturbation; this sequence is repeated up to the time t of interest; (5) multiply by In') (nl to obtain the density matrix operator contribution associated with element Onn,(t). Letting It)) and (ul be the ket and bra indicated on the diagram at any rime t" between two successive interactions, the propagation factor providing the evolution between the two mteracnons and to be used in (3) and (4) above, is calculated as -+ inverse of (1) the sum of the photon energies of all waves absorbed on the ket side and emitted on the bra side between the initial time t O and t", (ii) minus the sum of the photon energies of the waves emitted on the ket side and absorbed on the bra side, (ill) m i n u s the transition energy and damping as given by h(~vu - i ~ u ) . The + ( - ) sign is taken when the interaction preceeding t" has occured on the ket (bra) side o f the time axis. To illustrate the sunphclty and advantages offered by a diagrammatic approach, we consider the most important third order terms contributing to the Coherent Anti-Stokes Raman Scattering (CARS) susceptibility and those appropriate for the Raman Stokes susceptibility $2. :~2 Further consideranon of steady state susceptxbfllnes and transient coherent responses has been gwen In ref [7]. Transient and cw CARS is considered in detail by Druet et al [8] 4

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October 1977

The vibrational Raman-resonant CARS susceptibility has contributions from third order density matrix elements with m = 2 and m = 3. We shall only consider in the present discussion the contributions o f the molecules initially in the fundamental state la). The corresponding diagrams are shown in fig. 3. The density matrix element associated with each diagram of fig. 3 is easily written down from the rules specified above. Fig. 3a depicts a process at the end of which the molecules are returned to their initial state la} by the final anti-Stokes emission at 60 3 = +26o 1 co 2. Its contribution to the third order density operator at 60 3 is: p(B)(t) _~ 1

Ir'>o~Oa)
8h 3

l(I)r'a)

(4)

By exchanging the times o f the first two perturbations in fig. 3a (that is emitting 6o2 before absorbing col)' one obtains a contribution analogous to (4) with the vibrationally resonant term (col - 6°2 - coba + I@ba) but with ( - 6 o 2 -cora) in place of(co 1 - cora)" These two contributions could be termed a parametric process which returns the molecules to the ground state la} after the interaction. As indicated by the presence o f the qS's however, these parametric processes are lossy. Fig. 3b indicates the other basic third order process also providing a vibratyonally resonant polarization at frequency 6o3 but leaving the molecules in the vibrational state Ib} different from the initial state la) such that coba ~ col -- c°2" The density operator contribution from fig. 3b is seen to be. 0(3)(0_ -1 Ib)(bl-p

I - p " ~ 11r'} (r'l exp (-l(2co 1 - 602 )/} " t ~ l r ) ( r l - p ' e l l U O ~~x~ a a(0):a ,

(5 a)

(col - cora + iCbra)(col - 6°2 - coba + iqbba) (co3 --cobr' + irbbr')

8h 3

The last vlbratlonally resonant contribution is found once again by exchanging the times of the first two perturbations which results in a factor (-6o 2 - corafinstead of(co I - cora) In (5a). These two contributions correspond to a process where each molecule exchanges a vibrational quantum hcoba with the radiation fields. The four contributions associated with fig. 3 exhaust the vibratlonally Raman resonant contributions. Of these four contributions, the term associated with fig. 3a has two possible single p h o t o n resonances while the term associated with fig. 3b has the possibility o f one single p h o t o n resonance. F o r a given Raman line there is usually only

a33

_^S'-

*"

I I~? l_

•

~

_+I

Io4,

-/(o.I

<0.) (a)

.1

~.1

[ : Lo

J 2oj a .

cc

C:,l

(b)

Fig. 3 The main tune-ordered diagrammatic contributions to the CARS susceptlblhty to I

laser frequency, ,o 2 cy, to a = 2~o 1 - ~o2 = anti-Stokes frequency. (a) parametric process, (b) non-parametric process. =

=

Stokes frequen-

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October 1977

~--- co

60~.

~:oo

~

co z

~, + ~ q ~ , , 5 x

/

-

~ o

*, (e ~ , - oo

a°l

l

3

+1

[~,\t


k=k o

6O4

l a ) jao. o l,~ ( a l

l a ; > )o.{~ °l°~ <~1

(b) big 4 Diagrammatic contributions to the CARS susceptibility obtained by (a) interchanging the times of the third and second perturbations m diagram o f fig 3b, (b) interchanging the tunes of the third and first perturbations in diagram o f fig 3b

one of the resonant enhancement permitted because of selection rules. If the enhancement is provided by the field at frequency col only, these two terms give the main contribution to the resonant CARS susceptibility [9] (in the case that all the molecules are initially in state la)). By exchanging the thne of the third perturbation with that of any of the first two perturbations In fig. 3b one gets two other time-ordered diagrams (fig. 4). The terms associated with these diagrams are given in eqs. (5b, 5c) P (3)(0

- 1 Ib ) (b I - P " E~ Ir) (rl 8h 3

p(3)(t)

-

-1 Ib)(bl 8~ 3

- @°1

P " ~1 [a)P~ )(al - P " ~l Ir') (r'l exp {-l(2co 1 -- c o 2 ) t )

-cora + ldPra)(2C°l

("Or/"+ ldPrr')(co3 --

(5b)

C°br' + ldPbr')

p'e~[r)(rl -p'F. 1 [a)p~)(al - p " E11r')(r'l exp(-I(2Co I (O31 - coar' + IOear')(2col Wrr' + lqbrr')(co3 - cob/ + l~br')

- co2)t} (5C)

They possess one possible single photon resonance and no vibrational resonances so that they are generally small and contribute to the XN R of CARS $ 3. Nonetheless, their contribution can be significant m Resonant CARS, under certain conditions, as they result m a shift of the electronic resonance In the contribution (5a) depicted by fig 3b. This shift IS apparent m the case of no damping, since then the three tune-ordered dtagrams of figs. 3b, 4a, 4b can be combined to obtain the following contribution in place of (5a)" p(3)(t ) _

1 Ib)(bl - p .e~lr)(rJ - p . 8/i3

t 1 la)P(a°a)(al--p • ~llr')(r'l exp(-t(2co I - co2)t }

(COl _ coar,)(col _ 002 _ coba) ( _ c o l +

°°ra )

(6)

Indeed, contribution (6) comes from a non-time-ordered process as it carl also be obtained by diagrammatically evaluating the evolution of the wave function [10] and that of its complex conjugate independently and simply multiplying the results together. However, it is only in the case of no damping :~4 that the evolutions of the wave $3 In four wave mlxlhg (co4 = co 1 + o92 - 093) there are a total of 48 density matrix elements proportional to the ground state population alone, which contribute to the susceptibility [7,8]. In CARS they reduce to 24 the assomated diagrams can be referred from the four f u n d a m e n t a l diagrams o f tlgs 3 , 4 by m e a n s o f time-ordering, vertex exchanges, and for each diagram a reflection of the vernces from the ket (bra) to the bra (ket) with respect to the tune axis :~4 This result stdl holds provided the initial state has an infinite h f e t u n e and q~nn' + e°n'n" = °°nn' for any n, n ' , n"

Volume 23, number 1

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October 1977

function and of its complex conjugate are decoupled, so that the time-ordering of the vertices on the ket side with respect to those on the bra side is irrelevant. It is only in this limit that previous diagrammatic evaluations o f polarization components are valid [3,4]. However, as observed from eq. (6), the individual time-ordered contributions are lost. This combination of the individual tlme-orderings can be demonstrated from eq. (1) with the condition that eq. (2) can be factored into the matrix elements o f K ÷ and K - and is thus independent of the time-ordering. One can then sum all the mdwidual terms m P [ U ( t - t m ) U ( t m - t m _ l ) . . "] for any specified m to obtain a single term [U(t - t m ) U(t m - t m -1 ) ' ' " U ( t l - to)] [(U(t - t'n_ m _1 ) U(t'n_ m -1 - t'n-m-2 ) ' ' " U(t'l - tO)] which is factored into the two independent products giving respectively the time-ordering o f the Interactions on wave function and the time-ordering of those on the complex conjugate. The diagrams of figs. 3b, 4a, 4b can also be used to describe the Raman effect, by replacing the emission of 601 on the bra side by an absorption and the absorption of 6o3 by a Stokes emission at co2. All three diagrams contribute to the Raman susceptlbdlty; however, only the term associated with fig. 3b is important offelectronlc resonance. A parametric diagram analogous to that of fig. 3a is not possible here as Raman scattering always involves energy exchange between the molecules and the radiation fields. The analysis of these three diagrams for steady-state spontaneous Raman scattering reduces to that of Omont [5] The transient behaviour subsequent to a sudden termination o f the laser field at w 1 (bat not of the spontaneous Stokes field at 602) can also be obtained from eq. (1) and is easily visualized from the diagrams. The Raman processes for the tlme-ordermgs depicted in fig. 4 results, after the two absorptions at COl, in a perturbation of the density matrix element Orr,, which decays with a rate q~rr' when the laser field is shut off. In the case of one intermediate state, [r) ~- Ir') and q~rr is thus the inverse of the lifettme o f state Ir). When Orr is next perturbed by the Stokes field, three transients result: (1) the termination of the steady state initiates a natural response at frequency COrb with decay rate dPrb proportional to the propagator 1/(60 2 -- 6orb + i~rb); (2) The decaying population factor Prr generates two further transients proportional to 1/[co 2 - 6orb + i(dPrb -(brr)] " one is driven by 602 and is hence at that frequency with a decay rate ~rr' the other Is the natural response and thus at frequency 6orb and decay rate d~rb. The terms assocmted with these diagrams correspond to the Hot Luminescence terms of Shen [ 1 1 ]. The remaining diagram (fig. 3b), as is easily seen, can only result in a natural response at frequency 6orb w i t h decay rate "brb and corresponds to the transient resonant Raman term of Shen [ 11 ]. In conclusion, it is felt that many non-linear optical effects can be viewed from the diagrammatic representation of the density matrix. The temporal perturbation evolution is unambiguously deYmed and polarization or population changes are easily and directly evaluated. Although m the present paper steady-state polarization components have been specifically addressed, the analysis of electronically enhanced transient coherent birefringence [7] has demonstrated similar sunphcity and clarity.

References [1] N Bloembergen, Nonlinear optics (W.A. BenJamin, N Y., 1965). [2] C.P. Shchter, Principles of magnetic resonance (Harper and Row, N Y., 1q63) p. 134. [3 ] A. Yarw, Quantum electromcs (John Wiley and Sons, N.Y., 1975), 2nd ed., p 557 [4] J.F. Ward, Reviews Mod. Phys. 37 (1965) 1. [5] A Omont, E.W. Smith and J. Cooper, The astrophysical Journal 175 (1972) 185 [6] A M Bonch-Bruevlch and Y A. Khodovol, Usp. Flz Nauk 85 (Jan 1965) 3-64 [7] S.Y. Yee and T.K. Gustafson, submitted to Phys. Rev. [8] S. Druet, B. Attal, T K Gustafson, J.P. Taran, to be pubhshed [9] P.D. Maker and R.W Terhune, Phys. Rev. A 137 (1965) 801 [ 10] A. Messiah, M~canlque quantlque, tome 2 (Dunod, 1964) p 618. [11] YR. Shen, Phys. Rev 139 (1974) 622