Semiclassical theory of molecular nonlinear optical polarization

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1 Semcassca theory of moecuar nonnear optca poarzaton Mgue Ange Sepúveda a) and Shau Mukame Department of Chemstry, Unversty of Rochester, Rochester, New York Receved 27 February 1995; accepted 13 March 1995 The cacuaton of near and nonnear optca response as we as nonadabatc curve crossng processes depends on the tme evouton of the eectronc coherences off-dagona eements of the densty matrx. Unke ther dagona counterparts, the off-dagona eements do not have an obvous cassca mt. A semcassca approxmaton for the nonnear optca response functon, whch reveas the cassca orbt structure underyng the eectronc coherences between Born Oppenhemer surfaces, s deveoped. The resutng numerca propagaton, whch appes to arbtrary anharmonc potentas, s based on ntegratng the tme-dependent Schrödnger equaton usng the semcassca tme evouton operator, Van Veck propagator. Usng the present formasm t s possbe to descrbe semcasscay mutphoton processes nvovng severa Born Oppenhemer surfaces Amercan Insttute of Physcs. I. INTRODUCTION a Current address: Insttute for Fundamenta Chemstry, 34-4 Takano- Nshhrak-cho, Sakyo-ku, Kyoto 606, Japan. Nonnear optca spectroscopy s a rapdy deveopng fed that has aowed the study of propertes of moecuar systems n ways that are not possbe usng near technques. 1 3 Usng carefuy tmed and tuned sequences of utrashort aser puses, one can study the evouton of a system n mcroscopc moecuar tme scaes. Photon echo and hoe burnng technques aow the study of moecuar nteractons wth ther envronment n the qud or sod state, by competey emnatng nhomogeneous broadenng. 4,5 When nteractons of a chromophore wth the envronment are taken nto account, the numerca cacuaton of the tme evouton becomes a very hard probem. It cannot be easy computed quantum mechancay, athough there has recenty been some nterestng progress n that drecton. 6 If the nucear degrees of freedom of the chromophore are approxmated as harmonc oscators there are numerous methods to cacuate ts tme evouton, the nonnear poarzatons and optca susceptbtes. We can bascay dstngush two methodooges that attempt to address moecuar systems wth arbtrary potentas; frst are the methods based on the cassca smuaton of the nonnear response, 7 and second those based on the approxmate evauaton of Feynman path ntegras. 8 The former methods, usefu n the mt of strong dsspaton and ow resouton spectra, have the strength of handng effcenty arge number of degrees of freedom and reastc nteracton potentas. The methods based on Feynman path ntegras are more rgorous and can hande hgh resouton and ess damped dynamcs but have some strong mtatons reated to numerca convergence of oscatory ntegras and ntensve computer requrements. Semcassca technques for the treatment of moecuar systems offer a mdde ground between purey cassca moecuar dynamcs and exact Feynman paths. 9 They use cassca orbts as a reference zero order path and suppement t wth the approprate quantum phase that reproduces even extremey compcated quantum nterference. 10, The nonnear response functons, whch are necessary for descrbng the nonnear sgna, are most convenenty cacuated by propagatng the densty matrx n Louve space. 4,5 The densty matrx has severa advantages over the wave functon formasm: It s fuy tme ordered and s, therefore, cosey reated to experment. It further aows one to more naturay treat mxed states at fnte temperatures, and to emnate unnecessary bath degrees of freedom. In ths paper we combne the nonnear response functon formasm based on the evouton of the densty matrx wth semcassca technques and deveop a new method for cacuatng the nonnear poarzaton. Our method s capabe of descrbng asymptotcay the eectronc coherences that are greaty mportant n many spectroscopes, and aows the cassca nterpretaton of the dynamcs of moecuar systems under nonnear potentas. At the center of any semcassca theory n the tme doman s the Van Veck propagator, 12,13 ths s the asymptotc mt of the quantum propagator n coordnate representaton tme Green functon, xexpht/x as 0. The Van Veck propagator gves the probabty amptude to go from poston x to x n tme t. Asymptotcay ths transton matrx eement can be wrtten as the square root of the cassca probabty for ths event, tmes a phase gven by the cassca acton accumuated aong the orbt. Athough ths has been known for a ong tme, ts practca mpementaton has been very mted unt recenty. The numerca effort nvoved n fndng a cassca orbts subject to two vaue boundary condtons was the man obstace. In order to avod ths dffcuty one can represent the Van Veck formua usng the nta vaue representatons. 14 Fnay we note that the present formasm can be apped drecty to curve crossng and nonadabatc transtons. Ths probem s formay dentca to the nonnear response functon where the nonadabatc coupng pays the roe of the eectrc fed. The nonadabatc second-order rate constant s reated to the near response and the fourth-order correcton s reated to the thrd-order response. 15,16 In Sec. II we ntroduce the nonnar optca responses Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see J. Chem. Phys. 102 (23), 15 June /95/102(23)/9327/18/$ Amercan Insttute of Physcs 9327

2 9328 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton usng the densty matrx formasm. The forma semcassca mt s presented n Sec. III and a numerca mpementaton based on a two-trajectory approxmaton s deveoped n Sec. IV. The theory s used for the cacuaton of the thrd order response n Sec. V, and apped to a two-eectronc state system n Sec. VI. Fnay n Sec. VII, we sketch the most genera muttrajectory mpementaton of the semcassca response functon. II. LIOUVILLE SPACE PATHWAYS FOR THE DENSITY MATRIX Consder an ensembe of chromophores at equbrum. Before the nteracton wth the fed E(t), the state of the system s gven by the therma densty matrx (0) (). The tme evouton of the chromophore nteractng wth the fed s gven by the Louve Von Neuman equaton t H,, where H s the sum of the Hamtonan of the soated moecuar system H 0 and the dpoe fed moecue nteracton ˆ E(t): HH 0 ˆ Et. 2 We assume that the nteracton s weak enough for a perturbaton expanson of the densty matrx on the fed to be vad, (0) (1) (2). 1 The equatons for the successve terms are then gven by 1 We thus obtan 0 t 0, 1 t 2 t 3 t t 2 d ;te 1 h.c. t d d1 1, 2 ;te 1 E 2 h.c. 2 t d 2 d , 2 ;te 1 E 2, 3 t 3 d2 d 3 t 2 d1 23 1, 2, 3 ;te 1 E 2 E 3 h.c. 3 t t d 3 d 2 2 d1 13 1, 2, 3 ;te 1 E 2 E 3 h.c. 7a 7b 7c 7d The symbo h.c. stands for the Hermtan conjugate of the prevous expresson. In Eqs. 7 we have ntroduced the Louve space generatng functons for the densty matrx of the nth order, k (n), whch are defned as foows 0 t H 0, 0 0, n t H 0, n ˆ Et,n1. 3a 3b 1 1 ;te H 0 t/ ˆ 1 0 e H 0 t/, 8a 22 1, 2 ;te H 0 t/ ˆ 2 ˆ 1 0 e H 0 t/, 8b Sovng the system of couped equatons, Eqs. 3 aows the determnaton of the poarzaton of the chromophore P(t). Drect substtuton of the densty matrx expanson then eads to the expanson of the poarzaton nduced n the chromophore: PtP 0 tp 1 tp 2 t 4 wth P n ttrˆ n t. 5 The poarzaton densty of a unform dstrbuton of chromophores s smpy obtaned by mutpyng the P (n) by N/V, the moar densty of chromophores. The frst term n the expanson Eq. 3a gves the tme evouton of the nta densty matrx subjected to the unperturbed Hamtonan. Snce the nta densty matrx represents the equbrum state of the system, (0) s tme ndependent. The succesve terms Eq. 3b can be soved by teraton usng the propagator for the unperturbed system, then n t t d n e H 0 t n / ˆ E n, n1 n e H 0 t n / , 2 ;te H 0 t/ ˆ 1 0 ˆ 2 e H 0 t/, 8c 23 1, 2, 3 ;te H 0 t/ ˆ 3 ˆ 2 ˆ 1 0 e H 0 t/, 13 1, 2, 3 ;te H 0 t/ ˆ 3 0 ˆ 1 ˆ 2 e H 0 t/. 8d 8e We have made use of the Hesenberg representaton of the transton dpoe moment ˆ (t): ˆ te H 0 t/ ˆ e H 0 t/. Notce that the second ntegras n Eqs. 7c and 7d are not tme ordered. It s preferabe to work wth tme-ordered expressons that can be easy nterpreted n terms of sequences of ght puses. We then recast Eqs.7c and 7d as t 2 d1 2 t 2 E 1 E 2, d 2 2 Qˆ j 1, 2 ;th.c. j1 9 10a J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

3 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 9329 FIG. 1. Doube-sded Feynman dagram for the two generatng functons contrbutng to the densty matrx term (2) and (3). Top: Second-order terms, Bottom: Thrd order terms. t 3 t 3 4 j1 3 d2 d 3 2 d1 Rˆ j 1, 2, 3 ;th.c. E 1 E 2 E 3. 10b Now the tme varabes 1, 2, 3 are ordered,.e., that The tme-ordered form s obtaned naturay when the Louve equaton for the densty matrx s soved. The operators Qˆ j and Rˆ j contan the compete nformaton necessary for the cacuaton of the nonnear response functons. These operators are defned n terms of the generatng functons ntroduced prevousy as Qˆ 1 1, 2 ;t 12 1, 2 ;t, Qˆ 2 1, 2 ;t 22 1, 2 ;t, Rˆ 1 1, 2, 3 ;t 13 2, 3, 1 ;t, Rˆ 2 1, 2, 3 ;t 13 1, 3, 2 ;t, Rˆ 3 1, 2, 3 ;t 13 1, 2, 3 ;t, a b 12a 12b 12c Rˆ 4 1, 2, ;t , 2, 3 ;t. 12d Each of these terms consttutes a pathway for the propagaton of the densty matrx n Louve space, 2 and can be represented by a doube-sded Feynman dagram Fg. 1. The generatng functons contan a the dynamca nformaton of the system, but a gven spectroscopc measurement seects ony part of that nformaton. Two knds of quantum nterference show up n the nonnear optca response functon. The frst s due to the bra and ket dynamcs tsef, severa quantum paths contrbute to ther dynamcs and the phase nterference between the paths create typca quantum effects. Second, because the tota response functons conssts of the sum of severa bra ket propagatons, or Feynman dagrams, an addtona nterference pattern s created. Each of the Feynman dagrams, or ways n whch ght puses nteract wth the system at a certan order, have a competey dfferent dynamca tme evouton. In the anguage used n prevous paragraphs, severa of the generatng functons contrbute to the same dpoe expectaton vaue. For exampe, the two generatng functons, 1 and 2 make the second-order term (2) and (2) (2) consequenty P (2). Ther Feynman dagrams aow one to vsuaze the two ways n whch two aser puses nteract va the densty matrx. In the case of 2 (2) both puses act on the ket sde whe n the 1 (2) one puse nteracts on the ket and at a dfferent tme a second puse acts on the bra sde of the densty matrx. The sum of both generatng functons provdes the second knd of nterference. A fuy quantum cacuaton of the generatng functons requres a smutaneous propagaton of the bra and the ket. In genera, t s not easy to cacuate the generatng functons exacty for reastc systems. However, sometmes the coupng of the chromophore wth an externa bath makes the system evove n tme casscay. Semcassca approaches for the cacuaton of these quanttes can then be both numercay and physcay very usefu. Semcassca equatons of moton for these generatng functons were deveoped n Ref. 15. The equatons were derved by assumng that the Louve generatng functon has a Gaussan form n phase space at a tmes. The equatons descrbe the evouton n terms of a snge reference trajectory whch represents both the bra and the ket. The actua densty matrx depends on the evouton n the vcnty of that trajectory. Ths nformaton s carred out by the second moments of the wave packet n phase space. The generatng functons may be obtaned by runnng a set of cassca trajectores aunched n the regon covered by the nta densty matrx. 17 The path foowed by the cassca orbts are determned not ony by the Hamtonan of the system H 0, but by the partcuar combnaton of propagators mpct n the dfferent generatng functons n Eqs. 8. A basc mtaton of ths procedure s the need to construct an effectve reference Hamtonan whch shoud reproduce both the bra and the ket evouton. Ths can be ony done rgorousy for harmonc systems. The resutng equatons of moton are exact, e.g., for a two eectronc eve system wth harmonc potentas that dffer n ther equbrum poston and frequency. For harmonc modes t s aso possbe to ncude dsspaton by coupng to a harmonc bath. 18 For anharmonc systems there s no genera way to construct ths reference Hamtonan and the procedure s, therefore, mted to short tmes and sma anharmonctes. In ths paper we deveop a new semcassca Green functon propagaton scheme that overcomes the mtatons of the reference Hamtonan approach and aows the asymptotc propagaton of bra and ket n genera anharmonc potentas. The reevant cassca orbts come n pars, one for J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

4 9330 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton TABLE I. Left and rght operators for the thrd-order generatng functon (3), 1,2. S x x tlẋt,x t dt. 15 the tme evouton of the ket and one for the bra. By retanng the bra and ket tme evoutons separatey we obtan an mproved approxmaton that better mmcs the orgna quantum probem. III. SEMICLASSICAL LIMIT OF THE LIOUVILLE SPACE GENERATING FUNCTIONS A the fundamenta spectroscopc observabes can be derved from (n), and consequenty from the operators Qˆ j s Rˆ j s. Four-wave mxng spectroscopc meassurements drecty probe the nonnear responses R j s whe the functons Q j s are reated to three-wave spectroscopes. In genera t s not easy to cacuate the generatng functons exacty for reastc systems. However sometmes the coupng of the chromophore wth an externa bath makes the system evove n tme more casscay. Semcassca approaches for the cacuaton of these quanttes can then be both numercay and physcay very usefu. The passage to the asymptotc mt 0 s carred out by wrtng Eqs.8 n the coordnate representaton x n x r dx r dx x Û x x 0 x r x r Û r x r. 13 Both operators to the eft and rght of the nta (0) () are the tme evouton operator for the nta densty matrx that yeds the th generatng functon to nth order at tme t. The formuas for the eft and rght operators of the thrd-order term are gven n Tabe I. Left and rght operators obvousy depend on n, the perturbaton order, but we have omtted ths ndex from Eq. 13 for the sake of smpcty. The basc budng bock for semcasscs n the tme doman s the Van Veck propagator 9,12,13 m x t expht/x 0 Û r 2 e H 0 t/ Û 2 e H 0 (t 3 )/ ˆ e H 0 ( 3 2 )/ ˆ e H 0 ( 2 1 )/ ˆ e H 0 1 / Û r 1 e H 0 1 / ˆ e H 0 ( 2 1 )/ ˆ e H 0 (t 2 )/ Û 1 e H 0 (t 3 )/ ˆ e H 0 3 / 2 1/2 dp dx t exp 1/2 Sx t,x Equaton 14 s the asymptotc mt of the tmedependent Green functon. From the correspondence prncpe we know that the moduus square of ths eement s the cassca probabty of gong from x to x t as 0. The cassca probabty s gven by dp/dx t see Appendx A. The phase factor S(x t,x) s the cassca acton accumuated aong the cassca orbt aunched at x and arrvng at x t. Ths acton s easy cacuated by ntegratng the Lagrangan of the system aong the cassca orbt: The sum runs over a possbe cassca orbts satsfyng the gven boundary condtons x t,x at the specfc tme t. When there s ony one orbt n the sum then the semcassca transton probabty s dentca to the cassca, as s the case for the harmonc oscator. However, n genera, there w be more than one orbt n the sum, and n that case the probabty amptude w not be smpy the sum of cassca probabty amptudes but, due to the phases, there w be an nterference. Fnay the factor n the exponent s an nteger known as the Morse ndex not to be confused wth the dpoe moment operator ˆ ). It was ntroduced by Gutzwer to extend the vadty of Van Veck s orgna formua beyond caustcs. 13 The Morse ndex s normay cacuated by countng the number of tmes dx t /dp goes through zero aong the orbt. Usng the stabty anayss summarzed n Appendx A, ths s aso equvaent to detectng the number of tmes the stabty matrx eement M 21 changes sgn aong the orbt. The zero s countng procedure s appcabe when ( 2 H/p 2 )0, whch s generay the case. For other knds of Hamtonans a dfferent procedure s foowed for cacuatng the Morse ndex. 19 The semcassca formua Eq. 14 has been used to cacuate the near response of a mutdmensona nonnear system subject to an eectromagnetc fed. 20 Generay the Hamtonan H 0 w be the sum over severa Born Oppenhemer surfaces correspondng to adabatc eectronc states e : H 0 e H e Smary the dpoe moment operator s a sum over the dpoe coupngs between the dfferent eectronc states for a moecue wth no permament dpoe ˆ e ˆ j e j. j 17 Usng the above expressons for the Hamtonan, the dpoe moment and the semcassca Green functon we derve formuas for the eft rght propagators Û and Û r as the product of severa Van Veck propagators ntegrated over the ntermedate nucear coordnates: x Û x dx 1,...,dx k x e H 1 1 / x 1 x 1 ˆ 1, 2 x 2...x k1 ˆ k1, k x k x k e H k k / x, x r Û r x r dx 1,...,dx k x r e H 1 1 / x 1 x 1 ˆ 1, 2 x 2...x k1 ˆ k1, k x k x k e H k k / x r. 18a 18b J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

5 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 9331 The dpoe operator coupngs transform the generatng functons nto a sequenta chan of propagators on dfferent eectronc surfaces 1, 2,... In the Condon approxmaton, whch w be used n ths artce, the transton dpoe moment s assumed to sowy vary wth respect to the nucear degrees of freedom. Then the x k1 ˆ, j x k eements are constant functons wth respect to the nucear degrees of freedom. In ths approxmaton we w set a the ˆ, j 1. Integraton of Eqs. 18 over the ntermedate postons gves essentay the propabty amptude to go from x to x when propagatng for 1 on an eectronc state, 2 on another eectronc state, etc. In the non-condon approxmaton the expct coordnate dependence of the transton dpoe moments can st be taken nto account by neary expandng the dpoe functon about the cassca orbt runnng between nta and fna postons. The ntegraton of Eqs. 18 are st very dffcut because t nvoves the determnaton of k cassca orbts for every x,x par and for every set of tmes. Moreover momentum s not necessary preserved when passng from the path x x k to x k x k1 ; a the orbts are dsjonted n prncpe Fg. 2. However, n the asymptotc 0 mt we can cacuate the ntegras Eqs. 18a and 18b by statonary phase. Dong so we obtan 1/2 x Û x 2 2 1/2 S x x exp S x,x 2, 19a 1/2 x r Û r x r 2 2 1/2 S r x r x r exp S rx r,x r 2 r r, 19b where the actons S r (x r,x r ) and S (x,x ) are cacuated aong the statonary paths gong casscay from x to x on the eft and from x r to x r on the rght operators. Both statonary paths are sequences of cassca orbts subject to the condton of momentum and poston conservaton when swtchng between eectronc states. Ths condton s a drect consequence of the ast statonary phase approxmaton. Then the eft statonary path, for exampe, w start at x,go forward 1 n tme, swtch potenta surface stayng on the same spot n phase space, and contnues for 2, etc.... unt competng the sequence of evouton operators n Û. The ndces r and are the sums of the rght eft Morse ndces aong the cassca segments n the statonary path see Fg. 1. There are aso addtona phase shfts, r that arse from the statonary phase approxmaton and are a resut of constructng the fu semcassca propagator from severa shorter ones. We w dea n more deta wth the determnaton of the ndex ater. By conventon the paths for the eft and rght operators run forward n tme. Lookng back to the expresson for the generatng functons, Eq. 13, ts straghtforward to wrte down ts semcassca mt by substtutng the eft rght propagators by Eqs. 19. Our next concern woud be how to evauate ths expresson n practce. The smpest possbe approach s expaned n Sec. IV. It s based on the Gaussan wave packet method 21,22 and conssts of expandng the cassca dynamcs about a par of cassca orbts. We w then appy ths method to a two eve chromophore n Sec. VI. IV. SEMICLASSICAL EXPANSION OF THE GENERATING FUNCTION Gven the semcassca expressons for the eft and rght propagators Eqs. 19 t woud seem that the ony step necessary for the determnaton of the generatng functons s the ntegraton of Eq. 13. For four-wave mxng experments we need to evauate the thrd-order generatng functons (3) 1,2, 2 then x 3 x r 1 dx 2 r dx x 0 x r M 21 M r 21 1/2 exp S x,x S rx r,x r 2, 20 where r and r. The ntegraton s carred out over a possbe nta poston pars x,x r. We have made use of the tota stabty matrx aong the eft and rght propagatons, M and M r,as defned n Appendx A. The acton functons have a compcated dependence on nta and fna postons many due to the root search probem mentoned earer; for every set of nta fna condtons the approprate nta momenta must be determned n order to obtan the actons and phase shfts. In genera the cacuaton of ths ntegra seems numercay very chaengng. Nevertheess, severa approxmatons can be shown to gve extraordnary good resuts thanks to the ocazaton of the nta state. The functon x (0) ()x r n the ntegrand s often nonzero ony n a mted regon of confguraton space. Ths reduces the amount of computatona effort. Moreover, f the nta state s ocazed n a reatvey sma regon, one can expand the cassca actons and dynamcs about the center of the dstrbuton. Let us assume that the nta densty matrx s Gaussan: x 0 x r 2 expx 2 x r 2 2x x r. 21 For exampe, for a harmonc oscaton at equbrum the parameters and are gven by m 2 coth, m 2 snh, 22a 22b J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

6 9332 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton an anaytca dependence on the nta and fna coordnates: S x,x S p x x M x 2M x 2 21 M 22 x 2M 2 1 x 21 M x x, 25a 21 S r x r,x r S rp rx r x r M r r x 2M r x r 2 21 M r 22 r x 2M r 2 1 r x 21 M r x rx r. 25b 21 FIG. 2. Phase space dagram of two knds of paths contrbutng to the eft or rght propagators: An arbtrary quantum path for the eft rght propagator between x,x dash ne, and the statonary pont sod ne whch s smpy a casscay connected orbt jonnng x,x. where 1/kT. If the nta densty matrx s not Gaussan, ether because the eectronc surface s not harmonc or the nta ensembe s not n equbrum, one can aways decompose (0) () as a sum over Gaussans ocazed n confguraton space. The basc approxmaton for the two-orbt semcassca expanson s to assume that the dynamcs of the bra and the ket can be approxmated as a movng Gaussan aong the cassca orbt aunched at the center of the nta densty matrx. The center poston of the (0) () s x 0, x r 0 and the average momenta p 0, p r 0. From ths snge pont two cassca orbts (x,x r) depart and foow the dynamcs specfed by the eft and rght tme evoutons. Logcay any other cassca orbt whose nta condtons were taken n the nearby regon of phase space woud ead to a dfferent cassca path. Nevertheess for a short tme t s vad to represent every orbt departng from arbtrary nta postons x,x r and momenta p,p r and reachng x,x r by expandng around the two orbts that ntate ther dynamcs at the center of the nta densty matrx see Fg. 3. Upon nearzaton of the equatons of moton aong the gudng orbts we wrte p p M x x M 21 p M 12 x, p M 22 x, p r p rm r p r M r 12 x r, x r x rm r 21 p r M r 22 x r. 23a 23b 24a 24b The expanson coeffcents are the eements of the tota stabty matrces M and M r, as defned n Appendx A. The cassca actons n Eq. 20 can aso be expanded about the two orbts, or gudng trajectores, thus provdng The above expansons rey on the ocazaton of the nta state. Insertng them nto Eq. 20 and performng the ntegratons, the generatng functon transforms nto a Gaussan quadrature 1/2 x 3 x r h o 2 2 M det A 21 M r 21 1/2 where B A M 21 M 22 2M 21 exp 0 1 4B A 1 B, x x, M 21 M r 22 r 2M r x r x r, 27 0 S S r p x x p rx r x r M 2M 21 x x 2 M r r x 2M r x r r r., Upon reducton of Eq. 20 to Gaussan quadrature we have not ony expanded the actons quadratcay about the two gudng trajectores but aso assumed that the prefactor to the exponent s constant over the entre doman. Ths s certany a good approxmaton for a narrow Gaussan. The exponenta functon n Eq. 20 vares rapdy reatve to the prefactor whch represents the cassca probabty. Ths probabty s aso cacuated aong the two gudng orbts: M r and M n the prevous equatons are the tota stabty matrces aong the gudng orbts for the rght and eft propagaton, respectvey. Equaton 26 s smpe to nterpret and to mpement. A the nformaton requred to determne the generatng functon s contaned n the cassca dynamcs of two orbts aunched at the center of the nta densty matrx. The generatng functon gves the tme evouton of ths nta densty matrx under the eft rght propagators. The par of cas- J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

7 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 9333 sca orbts serve as a gudng phase space path for the dynamcs of the densty matrx; one orbt represents the evovng bra and the other represents the ket. As a consequence of the quadratc expansons of the cassca dynamcs and actons, the generatng functon s a Gaussan functon at a tmes centered at x,x r. Ths s precsey the mtaton of ths two-orbt mpementaton: The current expansons are exact for harmonc potentas but ony appy n genera anharmonc systems for reatvey short tmes. Later we w anayze ths method and ts mtatons wth an exampe; we w aso descrbe a generazaton of the semcassca propagaton that s not restrcted to short tmes. V. THE THIRD-ORDER RESPONSE FUNCTION Q 1 t 2,t 1 Trˆ 12 tt 2 t 1,tt 1 ;t, Q 2 t 2,t 1 Trˆ 22 tt 2 t 1,tt 1 ;t, R 1 t 3,t 2,t 1 Trˆ 13 tt 3 t 2,tt 3,tt 3 t 2 t 1 ;t, R 2 t 3,t 2,t 1 Trˆ 13 tt 3 t 2 t 1,tt 3,tt 3 t 2 ;t, R 3 t 3,t 2,t 1 32b 32c 32d 32e So far we have ntroduced the semcassca mt of the perturbatve terms n the expanson of the densty matrx for a system nteractng wth an externa eectromagnetc fed. The generatng functons prevousy cacuated are an ntermedate step for the determnaton of a physca observabe. The trace of the dpoe moment operator over the evovng densty matrx yeds the poarzaton of the chromophore under the nfuence of the eectrc fed. Now et us ntroduce the response functons S (1),S (2),S (3) : 2 P 0 ttrˆ 0, P 1 t 0 dt1 S 1 t 1 Ett 1, P 2 t 0 dt2 0 dt1 S 2 t 2,t 1 Ett 2 t 1 Ett 2, P 3 t 0 dt3 0 dt2 0 dt1 S 3 t 3,t 2,t 1 30a 30b 30c Ett 3 t 2 t 1 Ett 2 t 1 Ett 1. 30d S (1) (t 1 ) s the near response of the system whch contros the near optca measurements. S (2) (t 1,t 2 ) and S (3) (t 1,t 2,t 3 ) are the second and thrd order nonnear response functons, respectvey, whch descrbe nonnear optca measurements. Comparson of equatons Eqs. 30 wth Eqs. 7 and Eqs. 10 determnes the response functons n terms of the generatng functons k (n) : S 1 t 1 Jt 1 c.c., 31a 2 S 2 t 2,t 1 Q j t 2,t 1 c.c., 31b 2 j1 3 S 3 t 3,t 2,t 1 4 j1 where Jt 1 Trˆ 1 tt 1 ;t, R j t 3,t 2,t 1 c.c., 31c 32a Trˆ 13 tt 3 t 2 t 1,tt 3 t 2,tt 3 ;t, R 4 t 3,t 2,t 1 Trˆ 23 tt 3 t 2 t 1,tt 3 t 2,tt 3 ;t. 32f 32g The tme varabes chosen for the response functons are the tme ntervas between nteractons t rather than the actua nteracton tmes. As we see n Eqs. 32 the functons J, Q j, and R j are obtaned through the tracng of the transton dpoe moment and the generatng functons. The tota thrdorder response s the sum of four traces, each of them s assocated to a partcuar pathway n Louve space symbozed by the doube-sded Feynman dagram, Fg. 1. The semcassca approxmaton to the traces Eqs. 32 can be obtaned by substtutng the generatng functons by the semcassca expresson derved n the prevous secton, Eq. 26. Athough n Sec. IV we derved the two-orbt semcassca expanson of the thrd-order generatng functons, (3), n fact that formua equay appes to the second and near order generatng functons; the ony change between the three cases s the number and sequence of quantum propagators n Û r,û. Consequenty the recpe for descrbng the evouton of the cassca orbts changes among the dfferent generatng functons due to the modfcaton of the tme evouton operators for the densty matrx. The nterpretaton of the nonnear response provdes a new physca nsght nto the dynamcs. Each J, R j, and Q j depend on the tme evouton of the densty matrx and n partcuar they ook for the survva probabty,.e., the probabty of fndng the system n the same regon of confguraton space after the eft rght propagaton. For ths reason the semcassca mt aows us to ook at the gudng orbts that carry the fux of quantum probabty amptude through phase space back to the nta doman and understand the dynamca mechansm for correatons n compcated nonnear systems. Snce the transton dpoe moment s set to unty the traces Eqs. 32 are smpy the traces of the generatng functons; n the case of the thrd-order terms R j t 1,t 2,t 3 dxx 3 x, j1,...,4, 1,2. 33 J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

8 9334 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton Usng Eq. 26 then m 0 R j t 3,t 2,t 1 2 sgn M 21 g 1/2 k 2 exp k 0 k 2 1 4k 2. r M 21 1/2 34 A the quanttes appearng n Eq. 34 are defned n Appendx B. The sgn functon refers to the sgn of the dependent varabe. R j ends up dependng ony on the nformaton of the two gudng cassca orbts. It s possbe to express the second- and thrd-order terms Q j and R j as mutpe tme correaton functons of the transton dpoe. By expcty wrtng the generatng functons n terms of the quantum propagators, one vsuazes them n terms of averages of physca quanttes ke the transton dpoe moment Q 1 t 2,t 1 ˆ tt 2 ˆ tˆ tt 2 t 1 0, 35a Q 2 t 2,t 1 ˆ tˆ tt 2 ˆ tt 2 t 1 0, R 1 t 3,t 2,t 1 ˆ tt 3 t 2 ˆ tt 3 ˆ tˆ tt 3 t 2 t 1 0, R 2 t 3,t 2,t 1 ˆ tt 3 t 2 t 1 ˆ tt 3 ˆ tˆ tt 3 t 2 0, R 3 t 3,t 2,t 1 ˆ tt 3 t 2 t 1 ˆ tt 3 t 2 ˆ tˆ tt 3 0, R 4 t 3,t 2,t 1 ˆ tˆ tt 3 ˆ tt 3 t 2 ˆ tt 3 t 2 t b 35c 35d 35e FIG. 3. Notaton used n the two-orbt semcassca expanson of the generatng functons. Sod nes are the centra gudng orbts and the dash nes are two arbtrary orbts n the neghborhood of the center of (0).. 35f Equaton 34 s exact for purey harmonc surfaces. Snce the semcassca propagators and the statonary phase approxmaton are exact n ths case, ths formua aso appes for the cacuaton of the second-order responses Q j. When the potenta surfaces are anharmonc, Eq. 34 yeds a good estmate of the dynamcs up to the tme when the nonnear dynamcs starts to create mutpe smutaneous cassca recurrences n the response functons R j. The recurrences n R j appear because at certan tme t there are cassca paths that take amptude from the nta densty matrx back to tsef. Ths must happen smutaneousy for the eft and rght propagaton. In the case of a harmonc oscator the dynamcs s very smpe: ony one orbt on the eft and one on the rght can brng amptude smutaneousy back to the nta densty matrx creatng a recurrence see Fg. 4a. However, for anharmonc systems more than one par of orbts can contrbute at the same tme to create a recurrence n the correaton functons. Therefore, by that tme, Eq. 34 woud not be vad because we have essentay used nearzaton of the cassca dynamcs about a snge orbt per propagator. Nonnear effects n the dynamcs requre nearzaton about more than one orbt, 20 Fg. 4b. VI. APPLICATION TO A TWO ELECTRONIC LEVEL SYSTEM After ntroducng the response functon formasm and ts semcassca mpementaton we are ready to appy ths machnery to a test mode. Let us consder a two eectronc eve moecuar system wth one nucear degree of freedom descrbed by the foowng Hamtonan H 0 h 1 xh 2 x22ˆ 12 Et1 2ˆ 21 Et x s the coordnate degree of freedom, e.g., an nternucear dstance, and j the dpoe transton moment coupng the Born Oppenhemer surfaces, j. In Eq. 36 h 1 and h 2 are the Hamtonans for the two eectronc surfaces. Athough the tensor nature of the fed system nteracton s cear we w not address t. For the sake of carty we assume that the fed s agned wth the dpoe moment of the chromophore, we can then wrte n scaar notaton the nteracton as ˆ j E(t). For the two-eve system consdered here the owest nonnear term n the poarzaton s the thrd-order one. P 2 (t) s dentcay zero. Substtuton of Eq. 36 n Eqs. 8e and 8d gves for the ower order nonnear contrbuton 13 te h 2 t 3 / e h 1 3 / 0 e h 1 1 / e h / e h 1 t 2 /, 23 te h 2 t 3 / e h / e h / 37a e h 1 1 / 0 e h 1 t/. 37b As expaned n a prevous secton, under the Condon approxmaton the dpoe transton moments are constant wth respect to nucear motons and we set ˆ j 1. Conse- TABLE II. Expct expressons for the eft and rght operators appearng n the thrd-order response. Û r 2 e h 1 t/ Û 2 e h 2 (t 3 )/ e h 1 ( 3 2 )/ e h 2 ( 2 1 )/ e h 1 1 / Û r 1 e h 1 1 / e h 2 ( 2 1 )/ e h 1 (t 2 )/ Û 1 e h 2 (t 3 )/ e h 1 3 / J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

9 quenty the equatons for the generatng functons Eqs. 37 are smpy gven by products of tme evouton operators n ower and upper eectronc states. In Tabe II we have summarzed the eft and rght operators, Û and Û r, for ths system that yed the thrd-order response functon from the tme evouton of the nta densty matrx. The terms that determne the thrd-order poarzaton va Eq. 30 can be aso wrtten as R j t 1,t 2,t 3 dx dx dx r xû x x 0 x r x r Û r x, j1,...4, 1,2. 38 Û r and Û can be thought as the bra and ket tme evouton operators for the thrd order term. We have aready ponted out that expressng these operators as a product of Van Veck propagators, Eq. 14, unfortunatey eads to the generaton of a cascade of orbts. In Sec. III we aso ntroduced the noton of cacuatng by statonary phase approxmaton the cassca paths that take the system from nta to fna poston through the transtons between eectronc eves. The fna formua for R j Eq. 34 has a structure very smar to the Van Veck propagator, Eq. 14. As an exampe for the cacuaton of the ndex et us cacuate the expct form of x r Û 2 r x r. Substtuton of the Van Veck formua gves x r Û r 2 x r dx 1 x r e h 1 3 / x 1 x 1 e h 2 t 3 / x r M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 2 1/2 S T 2 S dx 1 2 S 1 x r x 1 1/2 2 S 2 x 1 x r 1/2 39a x r x r S T S 1 S 2, 1 2. x r x 1 1/2 2 S 2 x r x 1 1/2 2 1/2 S 1 S 2 2 x 1, 41a 41b 41c S T (x r,x r ) s the tota acton accumuated aong a cassca path gong from x r to x r preservng poston and momentum when passng from the eectronc surface 2 to 1 notce that the orbts run foward n tme. The ndex a tota Morse ndex s the sum of conjugate ponts found aong the two cassca orbts n the path 1 and 2. As mportant s the new phase shft emergng from the statonary phase approxmaton and due to the dscontnuty of the semcassca propagaton. Ths ndex arses when mposng the connectvty condton n the quantum paths jonnng x r and x r. can be cacuated by usng the stabty matrces of both cassca fragments, M (1) and M (2) see Appendx A. By defnton dp 1 dx 1M 1 dp r dx r, dp r dx rm 2 dp 1 dx 1. Then 0 f M 1 22 /M 1 21 M 2 /M f M 1 22 /M 1 21 M 2 /M exp S 1S b The actons are cacuated aong the cassca orbts jonng (x r,x 1 ) and (x 1,x r ) n tmes 3, t 3, respectvey. The two cassca orbts are dsconnected because they come from two separate roots to ndependent boundary condtons. Ths poses a probem because the use of the propagator Eq. 39b requres aunchng two separate manfods of orbts: one at x r wth arbtrary momentum and another at x 1 aso wth a possbe vaues of nta momentum. However, as 0 the man contrbuton to the ntegra Eq. 39b s a casscay connected orbt, whereby momentum and poston are preserved when swtchng propagators. We reach ths concuson by cacuatng the ntegra above by statonary phase: 1/2 x r Û 2 r x r 2 2 1/2 S T x r x r exp S Tx r,x r 2, 40 where we have made use of the foowng equates FIG. 4. Phase space dagram showng the nta densty matrx (0) () shaded area and the cassca eft and rght gudng orbts. In case a ony one famy of orbts serves as gude for the tme evouton. In case b two fames of orbts gude the dynamcs creatng a recurrence n the correaton functon. J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

10 9336 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton FIG. 7. Thrd order trace R 2 as a functon of t 3 for fxed t and t a.u.. FIG. 5. Mode two eve system: The ground state s a harmonc potenta, the excted state s a Morse potenta. Both curves are drawn as a functon of an nternucear dstance x. Ths new statonary phase approxmaton s very mportant because t avods reaunchng an x-manfod of orbts at x 1 to cacuate the second semcassca propagaton. We can then send a ray of orbts started at x r wth dfferent nta momentum and et them go through the eectronc surface change unt they reach ther fna destnaton at x. We now present a numerca exampe of the semcassca two-orbt method descrbed and compare the semcasscay cacuated thrd-order nonnear response wth ts quantum counterpart, cacuated usng a standard numerca agorthm. Consder a two eectronc eve system wth a harmonc ground state and a Morse excted state. The Hamtonans h 1,h 2 n Eq. 36 are then gven by h 1 x p2 2m 1 2 m2 x 2, 45a h 2 x p2 2m D e1exp e xx e 2 E ge, 45b where m1, 6.0 for the ground state harmonc oscator and D e 50, x e 1, for the frst excted state Morse potenta see Fg. 5. The eectronc energy dfference between the two surfaces E ge eads to an overa oscatory contrbuton to the traces R j, j1,...,4. To better apprecate the agreement between semcassca and quantum resuts n the foowng fgures we have made ths constant factor zero. The system starts at equbrum n the harmonc state wth kt10; k beng Botzman s constant and usng atomc unts 1. In Fgs. 6 9 we have potted the four traces that consttute the thrd-order response S (3) as a functon of t 3. The other two tme varabes t 1 and t 2 were fxed to 0.59 and 0.61 a.u., respectvey. The ast tme nterva t 3 descrbes the evouton of an eectronc coherence between the ground and excted states and by ettng t vary we see the agreement of the semcassca two-orbt approxmaton wth the quantum as a functon of tme. The quantum cacuatons were performed usng a FFT Fet Feck mpementaton of the Louve Von Neuman equaton. 23 The fgures show a reasonabe agreement; R 1, R 4, and R 2 happen to be better descrbed by the two-path expanson than R 3. In a cases the smpe semcassca approxmaton works amost FIG. 6. Thrd order trace R 1 as a functon of t 3 for fxed t and t a.u.. FIG. 8. Thrd order trace R 3 as a functon of t 3 for fxed t and t a.u.. J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

11 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 9337 FIG. 9. Thrd order trace R 4 as a functon of t 3 for fxed t and t a.u.. FIG.. Phase space portrat of the cassca gudng orbts for R 2. exacty at short tmes and t gets worse at onger tmes. As mentoned before, expandng the dynamcs about ony two cassca orbts takes the rsk of negectng contrbutons to the correaton functons from cassca paths that make a ong excurson far from our gudng orbts. Nevertheess ths method remans extremey usefu for reatvey short tme dynamcs; the dfference n computatona effort between the quantum and the two-path semcassca cacuatons s astonshng. The FFT cacuaton of the R j s took severa hours of CPU on a workstaton whe the semcassca two-path took seconds on the same machne. One of the nconvenences of the FFT method for ths partcuar probem s the necessty of propagatng the densty matrx usng two potenta surfaces wth very dfferent coordnate ntervas. In the present exampe the harmonc potenta s rather narrow n coordnates whe the Morse potenta s extended. Usng the semcassca approxmaton t s aso possbe to depct n phase space the cassca path foowed by the densty matrx for each of the terms R 1,...,R 4. Each term can be descrbed usng three pctures see Fgs : To the upper eft and ower rght s shown the eft and rght gudng orbts, respectvey, n phase space, n the ower eft the confguraton space trajectory descrbed by the densty matrx s aso potted by nterpoatng the coordnate excurson from the eft and rght cassca orbts. When the densty matrx s represented n the coordnate representaton (x,x r ) t appears as f ts tme evouton foows a twodmensona space; t woud seem dea to obtan equatons of moton for the evouton n (x,x r ) of the center of the movng densty matrx. However n reaty ths densty matrx s evovng n a four-dmensona space that contans the eft and rght momenta as we. Ths s precsey what s shown n Fgs ; the eft and rght gudng orbts expore the cassca dynamcs of the densty matrx n ts fu dmensonaty whe the projecton of the smutaneous eft rght postons gves the path descrbed by the densty matrx n the coordnate representaton. For the current exampe, these paths obvousy correspond to a reguar ntegrabe moton. We have used whte crces and the abes a, b, c to show the postons of the cassca gudng orbts after the t 1, t 2, and t 3 propagatons. Notce that the equatons of moton change abrupty at tmes t 1 and t 2, and so does the structure of the cassca gudng orbts. The rght gudng paths for the response functons R 1, R 2, R 3 are made of two cassca orbts, the frst one runnng on the excted Morse state and the second one on the ground harmonc potenta. On the other hand the eft gudng path runs aong the upper Morse potenta. Both gudng paths casscay descrbe an FIG. 10. Phase space portrat of the cassca gudng orbts for R 1. FIG. 12. Phase space portrat of the cassca gudng orbts for R 3. J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

12 9338 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton FIG. 13. Phase space portrat of the cassca gudng orbts for R 4. eectronc coherence created by the three fed nteractons. In the remanng response functon R 4 the three nteractons occur on the eft-hand sde of the densty matrx see Fg. 1. The phase space dagram of R 4 Fg. 13 then shows a eft gudng path wth three cassca orbts; the frst one runnng on the Morse potenta, the second on the harmonc ground state, and the thrd one agan on the upper Morse. The rght gudng orbt s smpy a statonary pont at the orgn because the ack of fed nteracton eaves the cassca orbt statonary at the mnmum of the harmonc potenta. The phase space pcture for the dynamcs of the densty matrx provdes a quck and nexpensve way to vsuaze and predct the response functons. Snce the nonnear response functons are gven by the traces of the tme evoved densty matrx, 1,2 (3), ony the confguraton space regon aong the x x r axs s mportant. When the densty matrx passes through ths axs t s expected to gve a recurson n the trace. Smary, the amptude of the trace shoud become neggbe far from the x x r axs. There s a second mportant factor for usng cassca phase space dagrams; f the cassca path (x,x r ) crosses the dagona x x r but the momentum of ether of the rght eft orbts s too hgh then the trace can be sma. Ths s easy understandabe: The cassca path n confguraton space eft-bottom pot n Fgs descrbes the moton of the center of the densty matrx ony, t does not te us about the overa average momentum of the movng dstrbuton. If the average momentum of the densty matrx s very hgh then 1,2 w (3) exhbt arge frnges and ts trace over the x x r may show neggbe resuts. To expore the nsght cassca mechancs gves nto the nonnear responses et us ook at the term R 4 for two dfferent sets of tmes t 1,t 2, agan as a functon of t 3. In Fg. 14 the traces R 4 for t , t 2 1.6, and t , t are shown. In the case of R 4 the trace s determned by the eft orbt because the rght orbt corresponds to the equbrum pont on the harmonc potenta. The cassca paths potted n the fgure have three segments, one for each of the t propagatons. The path of the frst run makes a very short excurson around the x 0, p 0 pont whe for the second run t s evdent that the cassca path descrbed durng the t 3 nterva s much arger. The rght orbt stays at x r 0 the harmonc statonary pont. Therefore ony when x s aso equa to 0, at the p axs, can the trace of R 4 be sgnfcant. There are ony two ponts per perod where there s an ntersecton between the Morse orbt and the p axs, and n both ponts the momentum of the gudng orbt s nonzero. As a resut the traces shoud be expected to be sma; aso the hgher the momentum at the crossng the smaer the trace. We can observe ths effect n the pot, the run for the second set of parameters s descrbed by a gudng orbt whch ntersects the x 0 condton wth a very arge momentum arge or sma momentum here means far or cose to the average nta momentum. In summary the two-orbt semcassca approxmaton to the propagaton of the densty matrx provdes an effcent way to vsuaze the regons of phase space expored by the evovng dstrbuton, and examne the eft and the rght sdes of the propagaton separatey. The dagrams n Fgs. 10 to 13 show then the cassca gudng orbts underyng the nonnear response contrbutons R 1, R 2, R 3, and R 4. The paths n confguraton space (x,x r ) ower-eft aow one to see when to expect an ncrease n the trace of the nonnear response. Smary the phase space projectons to the eft and rght aow one to read the average momentum of the densty matrx as t crosses the dagona x x r. The numerca effort nvoved n the two-orbt semcassca cacuaton of R j j1,...,4 s very sma. For ths reason t s rather easy to expore the behavor of the nonnear response functons n ther mutdmensona space t 1,t 2,t 3. One of the advantages of the nonnear optca response s that these functons provde a unfed pcture for many dfferent spectroscopc measurements. For exampe, a norma two-photon echo experment s descrbed by the functon R 2 (t 1,0,t 3 ) where t 1 s the tme dfference between the two ght puses and t 3 the tme nterva between the ast puse and the echo detecton. Aong the t 1 t 3 drecton t s possbe to emnate the nhomogenous broadenng that hnders the measurement of spectra nes n a norma near absorpton spectra. Usng our semcassca expanson we can now cacuate wth sma numerca effort R 2 (t 1,0,t 3 )as a functon of both t 1, t 3. Fgure 15 shows a contour pot of the absoute vaue of the nonnear response R 2 for our harmonc Morse mode, we may as we ca t the photon-echo response, as a functon of t 1 and t 3. The tme ntervas range from 0 to 6 a.u. Two cear knds of recurrences emerge n the response that we have abeed A and B. Departng from the orgn and foowng the t 1 t 3 dagona we frst observed an mmedate decay of the trace R 2 foowed by a sequenta seres of recurrences of type B,A,B,etc. Each of the recurrences can be dentfed wth partcuar cassca paths that nvove combnatons of perods of moton n the Morse and harmonc surfaces. The sgna, whch s an eectronc coherence, w depend on a combnaton of motons on both surfaces. Let us frst anayze recurrence type A wth the hep of the Feynman dagram shown at the bottom rght of Fg. 15. Durng the t 1 nterva,.e., after the nteracton wth frst puse, the bra sde of the densty matrx rght path has been excted to the upper Morse potenta whe the ket sde eft remans n the ground harmonc. The contour pot shows that J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see

13 M. A. Sepuveda and S. Mukame: Moecuar nonnear optca poarzaton 9339 FIG. 15. Contour pot of the nonnear response R 2 for a two photon echo measurement. FIG. 14. Comparson of two runs for R 4 (t 3,t 2,t 1 ) for fxed t 1,t 2 as a functon of t 3. In the upper two dagrams the phase space pcture (p,x )of the cassca path descrbed by the eft propagaton can be compared wth ther respectve correaton functons R 4. for t 3 0 and as t 1 ncreases the response frst decays and then rebuds n a seres of type A recurrences. The moment when ths happens for the frst tme concdes wth the frst perod of moton of the rght Morse orbt. Even f the t 1 moton contnues on forever, the eft gudng orbt remans statonary at x 0, p 0. Ths means that ony when the rght orbt goes through x r 0 wth sma or zero momentum there w be a recurrence A n the trace. The events A happen aong the t 3 0 axs perodcay at east for t 1 from 0 to 6.0) due to the perodc moton of the movng densty matrx aong the Morse orbt. Smar argument appes aong the t 1 0 axs. The same knd of recurrence type A appears aong the t 3 t 1 ne, e.g., at the abe A ocaton. The cassca reason for these recurrences s aso easy found by ookng at the phase space portrats of the eft and rght gudng orbts. Every one of these recurrences s due to the combnaton of an nteger number of perods aong the Morse orbt by the eft and rght gudng orbts. Because the Morse orbts have tme to compete a fu crcut n phase space they aways brng the densty matrx back to ts departure poston, thus amost competey rebudng the trace. In Fg. 16 we depct the cassca gudng orbts for the recurrence abeed B. Durng the tme nterva t 1 the dynamcs moves aong the t 3 0 axs and we observe the nta decay of the trace, by the end of haf-perod n the Morse orbt of the rght gudng path the densty matrx has moved aong the x r axs botton-eft dagram away from the mportant dagona x x r. As we mentoned, ony when the gudng path crosses ths dagona the trace buds up once more. Durng the t 3 propagaton the rght orbt descrbes a fu perod of moton n the harmonc potenta whe the eft orbt moves on the Morse excted state for haf a perod. As a resut there are two ntersectons of (x,x r ) wth the dagona. The frst fas to produce a recurrence n the response because the average momentum of the dstrbuton s extremey hgh n the rght orbt. The second ntersecton happens after one fu perod of the harmonc oscator n the rght orbt, and n ths case the average momentum of the densty matrx s zero same as the nta vaue. Ony at ths moment s there a fu revva of the trace. If the dynamcs contnue for onger t 3,.e., aong the horzonta ne n the R 2 contour pot, we see another coupe of dentca recurrences of type B whch are agan due to a combnaton of one fu harmonc and haf Morse perods. The cassca anayss dentfes recurrences A purey assocated wth the dynamcs on the Morse excted potenta. Ths s precsey what s to be expected for a photoabsorpton measurement and the trace of R 2 aong the t 1 0ort 3 0 axs s exacty that. Aong the dagona t 1 t 3 we have not ony the nformaton about the near absorpton spectrum recurrence type A but aso a new knd of recurrence type B whch s assocated to a combnaton of dynamcs n both the upper and ower eectronc states. As a coroary to ths anayss we reterate that ths smpe nterpretaton hods for the reatvey short tme scae shown n Fg. 15. For onger tmes the effect of the nonnear dynamcs n the Morse potenta ncreases and a smpe twoorbt expanson s not suffcent to descrbe the semcassca dynamcs. In that tme doman one has to use a semcassca mpementaton that makes use of the fu dynamcs n phase space, expandng the evouton of the densty matrx aong mutpe cassca orbts. In the foowng secton we descrbed such a method. However t s mportant to emphasze that the two-orbt expanson s much more powerfu than one may thnk superfcay; a possbe aternatve woud be to expand the upper Morse potenta quadratcay and approxmate t wth another harmonc potenta. Ths approxmaton s equvaent n our two-orbt pcture to havng gudng orbts that move on two harmonc surfaces; unfortunatey the frequences seen by the evovng orbts s aways constant due to the goba harmonc approxmaton to the potenta. On the other hand our two-orbt approxmaton adjusts the frequency of the upper harmonc ft constanty, makng t arger n the asymptotc sde of the Morse potenta and shorter cose to ts steep wa. J. Chem. Phys., Vo. 102, No. 23, 15 June 1995 Downoaded 07 Mar 2001 to Redstrbuton subject to AIP copyrght, see