Experimental and The o ret i cal Studies of Quan tum Beats in Fluorescence

Transcription

1 Jour nal of the Ch nese Chem cal So c ety, 003, 50, Expermental an The o ret cal Stues of Quan tum Beats n Fluorescence C.. Chang a (, Cheng-Lang uang a (, Ch-Kung N a (,. L. Da b (, M. ayash c * (, K. K. Lang a (, Any Kung a (, I. C. Chen ( an S.. Ln a ( a In st tute of Atomc an Mo lec u lar Sc ences, Ac a e ma Snca, Tape, Ta wan, R.O.C. b De part ment of Chem s try an Bo chem s try, Un ver sty of Penn syl va na, Phl a el pha, Penn syl va na, U.S.A. c Cen ter for Con ense Mat ter Sc ences, Na tonal Ta wan Unversty, Ta pe, Ta wan, R.O.C. De part ment of Chem s try, Na tonal Tsng ua Un ver sty, snchu, Tawan, R.O.C. The ef fect of col l son on quan tum beat s stu e the o ret cally by ap ply ng the en sty ma trx metho. As an ap pl ca ton of ths the ory, we an a lyze the ex per men tally ob serve quan tum beat ap p ear ng n the tmeresolve flu o res cence pro fles of bacetyl. In or er to ex plan com pl cate flu o res cence beat ng pat terns, we pres ent a three-state moel an sev eral m por tant mo lec u lar prop er tes are mappe onto ths moel. We also n ves t gate how quan tum beat pat terns ap pear as a func ton of the num ber of the trp let states that are n volve n the n ter ac ton wth the sn glet state. Key wors: Bacetyl; Quan tum beat; Flu o res cence; Den sty ma trx metho; Three-state moel. IN TRO DUC TION In re cent years, quan tum beats ap pear ng n the m - cro-(or nano- sec on tme-resolve flu o res cence pro fles of mol e cules ex cte by co her ent pulse la sers have been ob - serve by many groups. -8 By an a lyz ng such quan tum beats, one can ob tan the m cro scopc prop er tes of the cou plngs among elec tronc, n par tc u lar, the sn glet-trplet cou plngs be cause of the tme-scale of the ex per ments. The m por tance of the ef fect of col l son on quan tum beats n the flu o res cence was frst ponte out by enke et al. an the o ret cally stu e. 9- For mol e cules be long ng to the n ter me ate case, the ef fects of collsonal ephasng may make an m por tant con tr bu ton to the tme-epenent be hav - or of the ex cte states. In the ex st ng the ory on the quan tum beats n the m cro-(or nano- sec on tme-resolve flu o res - cence pro fles, how ever, the ef fects of the col l son on the beat mo u la ton have only been par tally taken nto ac count. It has been shown that collsonal ephasng ue to elas tc scat ter ng can sg nf cantly change the quan tum beat pat tern. Quan tum beats n the flu o res cence of bacetyl have been ob serve by Chaken, Benson, Gurnck an Mc Don al. Most re cently there has been a re port of quan tum beats n bacetyl whch coul be an n ter est ng ex per men tal ex am ple for such a pro cess. In a pre v ous pa per, we have con frme the ex s tence of quan tum beats n bacetyl an re porte an ex - per men tal n ves t ga ton of the ef fect of col l sons. 3 We have pre sente two ex per men tal re sults an the cor re spon ng sm u la tons of f fer ent vbronc tran s tons wth f fer ent beat ng fre quen ces. We have foun one thng n com mon, that s, both of the Fou rer trans forme spec tra of the quan - tum beat ngs pres ent only one non-zero-frequency peak. Fg. shows the flu o res cence beat ng of G tran s ton. One can see that as the stag nant pres sure of the mol e cules n - creases, the beat ngs e cay faster. Note that the stag nant pres - sure means the pres sure of the mx ture of bacetyl an buffer gas mol e cules be fore eject ng from the noz zle, whch s pro - por tonal to the pres sure n the cham ber. In the pre v ous pa per, base on the fact that only one non-zero-frequency peak n the Fou rer spec tra ex sts, we have ap ple the en sty ma trx metho 4 wth the two-state moel to sm u late the ex per men tal re sults. Af ter a just ng the en ergy f fer ence an the elec tronc cou plng con stants be tween the sn glet state (S state an the trp let state (T state an the e cay con stants of these two states, we can ft the ex - per men tal beat ng qute well. We em pha sze that by con s - er ng the zeroth or er ba ss set rather than the mo lec u lar egenstates n the two-state moel, t s easy to map the ob - serve be hav or onto the mo lec u lar states gven by the the o - ret cal moel an can clearly un er stan how each elec tronc state con trb utes to the beat ng. In par tc u lar, we shoul stress here that the pres sure e pen ence of the pure ephasng con - Spe cal Is sue for the Sec on Worl we Ch nese The o ret cal an Com pu ta tonal Chem s try Con fer ence

2 63 J. Chn. Chem. Soc., Vol. 50, No. 3B, 003 Chang et al. stant s clearly men tone for the frst tme. In the pre v ous work, 3 we have men tone that for a case more com pl cate than the two-state case, t s f f cult to trans form the mo lec u lar egenstate re sults nto the zeroorer ba ss set re sults. That s, even n the Schrönger equa - ton ap proach one shoul use the zero-orer ba ss set for treat ng quan tum beats of com pl cate cases. In other wors, to an a lyze the ex per men tal quan tum beat re sults the zeroorer ba ss set s pre ferre an the en sty ma trx metho shoul be use when there s the col l son ef fect (or heat bath. In ths work we shall the o ret cally an a lyze more com - pl cate quan tum beat pat terns that are f fer ent from the two-state case re porte pre v ously. We wll ex pan the twostate moel to three an even more states cases. From the nu - mer cal cal cu la ton, one can see that when the num ber of states n creases, the beat ng s at ten u ate an the flu o res - cence pres ents sm ple ex po nen tal e cay. DEN SITY MA TRIX METOD Let us con ser a 3-state sys tem an use the zerothorer ba ss set. We as sume that ths sys tem con ssts of one sn glet state S an two trp let states T an U. The en ergy f - fer ences be tween S an T, S an U are an SU, re spec - tvely, an the cor re spon ng cou plngs an SU as shown n Fg.. In ths case, the equa tons of mo ton for the en sty ma tr ces can be wrt ten as ( ( t SU SU S TT ( ( t TU UT TU UT T TT UU ( ( t SU SU UT TU UT TU U UU ( ( TT ( t TU TU (-a (-b (-c (- ( ( UU ( UT UT, (-f t etc. By us ng the Laplace trans for ma ton, the above f fer en tal equa tons be comes p (0 Im( Im( SU S p (0 Im( Im( TT TT TU UT T TT p (0 Im( Im( p p p UU UU SU TU UT U UU (0 ( ( (0 ( ( (-a (-b (-c (- ( UU ( UT UT (-e (0 ( UT UT UT UT UT TT TU TU UT ( TT UU (.(-f Fg.. Ex per men tal quan tum beat re sults for the rovbronc state G 0 0 Fg.. A three-state moel. 0 00

3 The o ret cal Stues of Flu o res cence Quan tum Beats J. Chn. Chem. Soc., Vol. 50, No. 3B, ere we con ser TU = UT = 0 an UT TU 0. There fore, for ex am ple, the equa ton for can be ob tane by us ng Eqs. (- an (-e an by sub st tut ng Eqs. (3 an (4 nto Eq. (-a. Thus, we ob tan or Let W ( p Im[ ], etc., Eq. (6 can then be re - p wrt ten as Sm larly, we ob tan an ( p ( TT ( p ( UU ( p S (0 Im[ ( TT ] p Im[ SU ( UU ] p ( p S Im[ ] Im[ ] p p The for mal so lu ton for can be gven, n terms of the sec u - lar e ter m nant, by (0 Im[ ] p Im[ ] p ( p W ( p W ( p W ( p W ( p (0. UU S TT UU W ( p ( p W ( p W ( p W ( p T UT TT UT UU TT (0 W ( p W ( p ( p W ( p W ( p UT TT U UT UU UU (0. TT (3 (4 (5 (6 (7 (8 (9 (0 If we as sume that (0 =, TT (0 = UU (0 = 0 an S = T = U = an no tce that WUT ( p 0, then Eq. (0 re uces to or where It fol lows that (0 (0 (0 where, for ex am ple, W TT T UT UT UU UT U UT S T UT UT UT U UT Equa ton (4 can be re wrt ten as ( ( (3 (4 (5 (6 Carryng out the n verse Laplace trans for ma ton to Eq. (6 yels W p W W W W p W W p W W W W W p W W W W W p W W W 0 p W W UT UT W 0 p W W UT W p W W p W W W W W p W W ( p W ( p W D D ( p W W ( p W ( p W ( p W W ( p W W. ( p W ( p W ( p ( p W ( p W W ( p ( p. ( p [( p ][( p ]. 4 4 ( p [( p ][( ] p UT 0

4 634 J. Chn. Chem. Soc., Vol. 50, No. 3B, 003 Chang et al. where an t t D t D ( t Ae Be cost Ce cost (7 D 4 A B ( (8-a (,(8-b [ ][ ], 4 D ( 4 D D [ ][ ], D D( C D D [ ][ ]. D D( 4 (8-c (8- (8-e If SU, SU, an ap prox - mate form for Eq. (7 can be ob tane as ( t SU e [ ][ ] SU SU t cos e SU t e cos SU t. SU SU (9 Ap plyng Fou rer trans for ma ton to Eq. (9 leas to the beat - ng fre quen ces of the flu o res cence that wll ap pear ap prox - t, t mately at the po s tons of 0, an SU. Equa ton (9 may m ply that the 3-state case can be ap prox mate to be a com - po s ton of two -state cases. More over, when the en ergy f - fer ences be tween the sn glet an trp let states are larger than the cou plng con stants, the beat ng fre quen ces are om - nate by the en ergy f fer ences. For the two-state moel, one can con ser a sys tem shown n Fg. 3. In ths case, we ob tan an t ss t t ll sl Im( Im( s s sl ls s ss (0-a (0-b (0-c Equa ton (0 rep re sents the gen er al za ton of the two-state case. So lu ton of Eq. (0 wll pro ve the y nam cs n for ma - ton of ss(t an ll (t. Ths type of gen er al ze mas ter equa - tons has been em ploye by. ayash an hs co work ers to stuy the mag netc quench ng of mo lec u lar lu m nes cence. 5 RE SUL AND DIS C SION l l l l ll ( sl sl sl sl ( ss ll l l l '. sl l l In the pre v ous work, we have the o ret cally an a lyze two flu o res cence quan tum beats for a bacetyl mol e cule. The cor re spon ng elec tronc tran s tons are G an G , re spec tvely. From the Fast Fou rer Trans form (FFT spec tra, only one S T cou plng has been con s ere ue to one 0 peak pre sente. Now we an a lyze the quan - tum beats of G 0 0 tran s ton, whch s pre sente n Fg. 4. Frst for the -atm ata, the FFT spec trum shows that there are two 0 peaks, whch means that there are two trp let states, T an U, cou ple wth the S state. By us ng the -state cal cu la ton metho to ft each Fou rer spec tral peak of the beat, we can roughly es t mate the mo lec u lar pa ram e ters lke the en ergy f fer ences, the e cay con stants an the cou plng con stants be tween the S state an the cor re spon ng trp let Fg. 3. A gen er al ze two-state moel.

5 The o ret cal Stues of Flu o res cence Quan tum Beats J. Chn. Chem. Soc., Vol. 50, No. 3B, states. By us ng the 3-state cal cu la ton,.e. Eq. ( an ap ply - ng the fourth-orer Runge-Kutta metho, we can cal cu late (t an the cor re spon ng FFT spec tra nu mer cally. Sub - st tutng these mo lec u lar pa ram e ters nto Eq. ( an a just - ng them, we ob tan the cal cu la ton re sults of the quan tum beats an the FFT spec tra cor re spon ng to the ex per men tal ata. We ob tan the best ft f we aopt the mo lec u lar pa ram e - ters = 9.8 Mz, SU = 6.6 Mz, = 7. Mz an SU = 4.4 Mz an the ephasng con stants for S, T an U states are set to 0.56, 0.6 an.4 Mz, re spec tvely. Fg ure 5 shows com par sons of the cal cu late re sults wth the ex per men tal ata. In the sm u la tons, we have as sume that the cou plng con stant TU an all pure ephasng con stants are equal to zero. Now we go fur ther to sm u late the 4-atm an 9-atm cases by sm ply n creas ng the val ues. Fg ure 6 pres ents the pres sure e pen ence of the e cay con stants of each state. Ta - ble sum ma rzes the ft tng mo lec u lar pa ram e ters. One can Table. The Pressure Depenence of S, T an U of the Quantum Beat for the Rovbronc State G 0 0. The Fttng Energy Dfferences an SU an Couplng Constants an are 9.8 Mz, 6.6 Mz, 7. Mz an 4.4 Mz, Respectvely S (Mz T (Mz U (Mz atm atm atm no tce that when the en ergy gap be comes larger, the cou plng con stant be comes smaller, whch agrees well wth one s n tu - ton. One can no tce that as the num ber of T states n creases from two to three, the beat ng ex hb ts more com pl cate fea - tures. We may an tc pate that the beat ng fea ture s re late to the num ber of T states. Thus, we con ser some more com pl - cate cases. As sume that there are more than two T man fols an these two T man fols are cou ple wth S state but not wth other T states. ere we sm ply set the cou plng con stants of S state an each T state to be the same value: 37.6 Mz. The ephasng con stants for S, an T states are 0.67 Mz an 0.5 Mz, re spec tvely. Ap plyng Eq. (0 an us ng the same nu mer cal cal cu la ton metho men tone above, we have foun that when the num ber of T states n creases as 9, 5, 7 an 9, the beat ngs n the flu o res cence are ob v ously an n h late an the flu o res cence fea tures tens to ex hbt a sm ple fea ture as shown n Fg. 7. To ex am ne ths ten ency, we con ser a more ex treme case as shown n Fg. 8, n whch the T states are s trb ute wth the equal en ergy spac ng = 0.5 Mz an the num ber of T states as 3, 99 an 99. Fg ure 9 shows the cal cu late re sults. One can see that a sn gle ex po - nen tal e cay can be ob serve for the case n whch the num - ber of T states be comes 99. It shoul be note that the re cur rence tme can be eval u - ate as T ~. ( Fg. 4. Ex per men tal quan tum beat re sults for the rovbronc state G 0. 0 If = Mz, then one can see from Eq. ( that T = 6.9 s. That s, one can see a re cur rence re sult ng from the n ter fer - ence of a num ber of T states at T =.58 s n the pres ent case. To be able to ob serve such a sn gle ex po nen tal e cay, the flu o res cence mea sure ments shoul be per forme wthn the re cur rence tme. In a ton, a num ber of T states shoul be n volve n the n ter fer ence. In sum mary, by us ng the en sty ma trx metho an

6 636 J. Chn. Chem. Soc., Vol. 50, No. 3B, 003 Chang et al. Fg. 5. Com par son be tween ex per ment an the ory for G 0 0 ; (A stag nant pres sure atm; (B stag nant pres sure 4 atm; (C stag nant pres sure 9 atm.

7 The o ret cal Stues of Flu o res cence Quan tum Beats J. Chn. Chem. Soc., Vol. 50, No. 3B, Fg. 8. Ex treme cases for the gen er al ze two-state moel. Fg. 6. Pres sure e pen ence an ln ear ft tng of S, T ( an for G 0. 0 the zero-orer ba ss set con cept, the ex per men tal flu o res - cence quan tum beat re sults are well an a lyze. We have also pre sente that when more trp let states are taken nto ac count, the beat ng fea ture s mn she an sm ple ex po nen tal e - cay fea tures be come om nate. The e tale mech a nsm of raatonless tran s tons can be com pletely un er stoo only n ths way. Re ceve De cem ber 6, 00. Fg. 7. Ef fects of the num ber of T states on quan tum beat pat terns.

8 638 J. Chn. Chem. Soc., Vol. 50, No. 3B, 003 Chang et al. Fg. 9. The cal cu late florescence pro fles for ex treme cases of the gen er al ze two-state moel. REF ER ENCES. Chaken, J.; Benson, T.; Gurnck, M.; Mc Don al, J. D. Chem. Phys. Lett. 979, 6, 95; Chaken, J.; Gurnck, M.; Mc Don al, J. D. J. Chem. Phys. 98, 74, 06.. enke, W.; Selzle,. L.; ays, T. R.; Ln, S..; Schlag, E. W. Chem. Phys. Lett. 98, 77, (a Felker, P. M.; Lam bert, W. R.; Zewal, A.. Chem. Phys. Lett. 98, 89, 309. (b Felker, P. M.; Zewal, A.. b. 983, 0, 3. (c Lam bert, W. R.; Felker, P. M.; Zewal, A.

9 The o ret cal Stues of Flu o res cence Quan tum Beats J. Chn. Chem. Soc., Vol. 50, No. 3B, J. Chem. Phys. 984, 8, Sharfn, W.; Ivanco, M.; Wallace, S. C. J. Chem. Phys. 98, 76, 095; Ivanco, M.; ager, J.; Sharfn, W.; Wallace, S. C. b. 983, 78, Okajma, S.; Sagusa,.; Lm, E. C. J. Chem. Phys. 98, 76, van er Meer, B. J.; Jonkman,. Th.; ter orst, G. M.; Kommaneur, J. J. Chem. Phys. 98, 76, Watanabe,.; Tsuchya, S.; Koa, S. J. Phys. Chem. 983, 87, Brucat, P. J.; Zare, R. N. J. Chem. Phys. 983, 78, Kono,.; Fujmura, Y.; Ln, S.. J. Chem. Phys. 98, 75, Len, K. Chem. Phys. 980, 46, 79.. aberkorn, R.; Selzle,. L.; Detz, W.; Ln, S..; Schlag, E. W. Chem. Phys. 980, 5, Ntzan, A.; Jortner, J.; Rentzeps, P. M. Proc. Roy. Soc. 97, A37, Chang, C..; uang, C. L.; N, C. K.; Da,. L.; ayash, M.; Lang, K. K.; Kung, Any; Chen, I. C.; Ln, S.. Mo lec - u lar Phys cs. 00, 00, Ln, S..; Alen, R. G.; Islampour, R.; Ma,.; Vllaeys, A. A. Den sty Ma trx Metho an Femto secon Pro cesses (Worl Sc en tfc, Ln, S..; ayash, M.; Kono,.; ayash,.; Abe,.; Ikea, S. J. Chn. Chem. Soc. 997, 44, 03.