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Mechanisms of dephasing in femtosecond time-resolved coherent Raman scattering from molecules in liquids
journal of MOLECULAR
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 169-176
Mechanisms of Dephasing in Femtosecond Time-resolved Coherent Raman Scattering from Molecules in Liquids Y. Fujimura Department of Chemistry, Faculty of Science, Tohoku University, Sendal, 980-77, Japan Abstract The mechanisms of ultrafast dephasings appearing in the time-profile of coherent Raman scattering from molecules in liquids are examined. Two mechanisms are taken into account. One is associated with intermolecular interactions between molecules at different sites through heat bath modes. The other is associated with the interference between rovibrational Raman transitions. In the former mechanism, the time profile of the coherent Raman scattering is characterized by an exponential decay. This decay constant is called the intermolecular dephasing constant. For the latter mechanism, the time profile has an inhomogeneous nature and is characterized by a Gaussian form. The structure of the intermolecular dephasing constant is derived within the Markov approximation. We describe the relationship between the intermolecular dephasing constant and the intramolecular dephasing constant associated with spontaneous Raman transitions. Finally we analyze the ultrafast, sub-picosecond dephasings observed in coherent anti-Stokes Raman profiles of neat benzene liquid in terms of the rovibrational interference mechanism. I. Introduction In recent years, considerable attention has been giyen to ultrafast dynamics associated with molecular motions in liquids. Femtosecond time-resolved, nonlinear coherent Raman scattering spectroscopy is one method for clarifying the origin of such ultra_fast dynamics. 1-5 Ultrafast dynamics in liquids is reflected in the profiles of the time-resolved, nonlinear coherent Raman scattering spectroscopy. This method includes time-resolved coherent anti-Stokes Raman scattering(CARS), degenerate four-wave mixing, impulsive stimulated Raman scattering (ISRS),etc. When considering the molecular dephasing processes, differences in the creation of the molecular coherence between coherent and incoherent(spontaneous) Raman scattering should be noted. In the time-resolved, nonlinear, coherent scattering processes, each molecule in the liquid receives the photon wave vector transiently through spatial coherence of the photon field. Consequently, a transient intermolecular coherence is created. The transient macroscopic nonlinear polarization of the molecular ensemble is therefore given as the summation of the induced nonlinear polarization on each molecule. The output intensity of the coherent Raman scattering experiments is generally proportional to an ensemble average of the absolute square of the nonlinear polarization. 6,7 In the absence of long-range spatial correlation between molecules, the intensity can be expressed in terms of the absolute square of the ensemble-averaged nonlinear polarization. On the other hand, in incoherent Raman scattering processes, intramolecular rovibrational coherence of each molecule within the laser irradiation spot is created. The decay of rovibrational coherence is expressed by the ordinary intramolecular dephasing constant. This consists of the population decay constant and the pure dephasing constant. The former originates from an inelastic interaction between the molecule of interest and heat bath modes. The latter originates from an elastic interaction between the molecule and the heat bath modes. Thus, intermolecular phase dynamics from liquids is directly reflected in time-resolved, nonlinear, coherent Raman scattering profiles though the information content in both coherent and incoherent Raman scattering experiments is in principle the same. 8 It should be noted that the laser pump pulses create sets of intermolecular rovibrational coherence. For example, quantum beats appear in time profile of the CARS spectrum from a molecular mixture. 3, 9, 10 The beat frequency created is equal to the difference between frequencies of vibrational modes of molecules in the mixture. When many sets of intermolecular 0167-7322/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00850-0
170 rovibrational coherence are created, it can be expected that ultmfast dephasing will result from the interference between the sets of the rovibrational coherence. This phenomenon is one of the main origins of the inhomogeneous Gaussian component in the time profile. The time profile also has a photon polarization dependence. This polarization-dependent time profile has been observed in the sub-picosecond time regime in nonresonant CARS from benzene in liquid. 11 The present study uses a quantum statistical method to show how the time-profile of CARS from molecules in liquids is associated with the intermolecular dephasing mechanisms. Liouville space Feynman diagrams are used for the development of relevant transitions associated with pairs of molecules. 9-10 The structure of the intermolecular dephasing constant is clarified to identify the difference between the intermolecular dephasing constant and intramolecular dephasing constant in Sec.II. Section III presents the rovibrational interference mechanism. The sub-picosecond decay observed in CARS profile of neat benzene is explained in terms of this mechanism. II. Intensity of time-resolved C A R S Consider a time-resolved, electronically nonresonant CARS spectrum from a molecular liquid. In the CARS process, the laser pump pulses create a linear combination (that is the intermolecular rovibrational coherence) of Raman active rovibrational transitions between molecules at position rt and rm in the mixture. This stimulated Raman scattering process is carried out by two-coincident laser pulses(I, II) with central frequencies(wave vectors) mi(kI) and r By applying the third pulse with ml(km) to the liquid after time delay x, the time dependence of the intermolecular rovibrational coherence is detected through the measurement of the intensity of the scattered photon with ks.
The total Hamiltonian I2I is given as fl = I:t~ + fl~ + f l ~ ,
(l)
where I2IMBis the molecular Hamiltonian including the optically inactive heat bath modes, I2IRis the radiation field Hamiltonian, and I~IMR is the molecule-radiation-interaction Hamiltonian. Using the dipole approximation, and making use of the rotating-wave approximation, the interaction Hamiltonian is given as I21MR= Z
V e,
(2)
/=1 9t,,,,q ,v,,~ with
,
'
(-)
Ve = - 2.,2.., Mba Tba El b
exp(ikl "re)
a
9, ' (+) exp(-ikll 9 r e ) - Z Z M~b Tcb E.ll c
~~c M~cr Tdc •ll
exp(--ikiil ~ re).
b
(3)
g
The term M~tx denotes the matrix element of the electric-dipole moment between states ~t and ~, of the gth molecule. The operator ~
e
( -- I~t><~,l) is the transition operator from the ~, to the l.t
states. E denote the radiation field operators. The density operator for the total system ( ~ T )
is
satisfied with the equations of motion,
ih "~" [5T (t) = [V(t ),[5T ( t ) ] , (4) where the density operator with the overtilde denotes that it represents the density operator in the interaction picture. The dipole transition operator in the interaction picture is expressed as Q ( t ) = e x p I i ( I 2 I ~ + I21R)t/h] I2tMRexp[--i(121.~m+ I21p)t/h 1= f i g=l
Qe(t).
(5)
171 In the perturbative density-matrix method, the total density operator for the CARS process is given by expanding iST(t) in terms of the photon number (4) as (4) i OT (t)= { dt(8)} [V(tl), [V(t2) . . . . . IV(t7), [V(ts), OT(-~) .... ]]]1,
where { i d t ( 8 ' } = ( ~ ' ) S i d t i i d t 2 . . . .=.oo
-.oo
idt7idt
==.oo
8,
(6)
(7)
-.-oo
and OT (_oo) is the density operator for the total system at initial time t ---~-o.. This density operator is assumed to be given by
(8)
OT(__oo) = OM (-oo)O B (-oo) ~ R (...oo) , where 0M ("**) is the density operator of the molecules, 0B (--~
that of the heat bath and
(4) OR (.-o.) is that of the radiation field. The density operator for the scattering field 0s (t) is (4) derived from the total-system-density operator 0T (t) by tracing out the variables from both the molecules and the heat bath modes. The intensity of time-resolved CARS is defined in terms of the population of the scattered photons (W('r, Ak, Ws, t)) as 12 I(,, Ak) =
dadWs
Atdt~'~W(,, Ak, Ws, t), withAk = ki - kII + kiii - ks
(9)
The CARS intensity is expressed as I('t, Ak)--
ZZ Itm('l:)exp(iAk~ 9 (10) t=l m=l (t ~ m) After averaging Eq.(10) over the heat bath variables with the aid of the Liouville space Feynman diagram (shown in Fig. 1) within the factorization approximation, the main term for the CARS process is given as
oj" dt XXxX;X;XX;X a
b
c
d
a'
b'
c'
d'
At g m g g g m m m m X Pa Pa' Mdc Mcb Mba Ma'd' Md'c' Mc'b' Mb'a' t ,,(-) (t2)EllI)(t3)) ,,(+ ,,(+ '(t4)Eli ,,(- )0.6)) X I _.,dt 1 " ' " I t_oodt7 ( EIII ( EII ,, ((+) gm tm X ( EI )(ts) EI (t7)) Gaa:a,b,(t6-tT) Gba:a,b,(t5-t6) tm gm em em X Gba. a'c ,(t4-ts) Gca: a'c' (t3-t4) Gca: a' a' (t2-t3) Gda- a'd ,(tl-t2).
(11)
The term Pa denotes the population in the initial state of the molecule at site g. The term ..((El)(ti)
.. (+ EI )(tj))denotes ~ the incident radiation field correlation function.
The term
172 G ~ . es(ti-tj) represents the time-development matrix element of the intermolecular coherence between B ~ k and e ~ ~i from time ti to time ti in the presence of the molecule-heat bath interaction. This matrix element is d e f i n ~ by tm Gj~t: es(ti-tj)= [((I.tl~, X~I ~,m(ti-tj)I l.tl~, X~ ))]av. (12) where (3em(ti-tj), the propagator of the molecular pair whose constituents are located at sites g t~m
and m, is given as
gm
(3tm(ti-tj) = exp[-i f_.MB(ti-tj)],
trn
with I'~MB= [HMB, ]lb.
(13)
~al g site
t5 l
a
t4
~
t7
'"f
1.6
P
t
:
t
ti "la>
/ 7 m site
t2
.s/
1,3 t la'~'
Figure 1. A Liouville space Feynman diagram for CARS process. Each wavy line denotes the molecule-radiation field interaction. This diagram represents the time evolution of the intermolecular coherence between two molecules at g and m sites in the CARS process. Figure 1 shows the Liouville space Feynman diagram representing the time development of the intermolecular coherence between molecules at site g and m in the time-resolved CARS process. The two upper lines show the time development of intramolecular coherence of molecules at g. The two lower lines show the time development of intramolecular coherence of molecules at m, respectively. The upper and lower lines are connected through the incident laser fields(I, II, and III) indicated by wavy lines. In the ordinary pump-probe type time-resolved CARS experiments, the time duration from t3 to t4 corresponds to the pump-probe time x, and the intermolecular coherence time for pair molecules. In other words, it is the time dependence of the coherence between the Raman transition a <---~cat site g and a' <---~c'at site m which are detected in the CARS time profile.
Equation of motion for the intermolecular coherence The equation of motion for the intermolecular coherence between two Raman transitions gm i ~---}jand i' <---~j' at site g and m ( Gji:i'j ,(t) ) is" d tm ~m }'m gm dt Gji:i'j'(t)---[il~ji:i'J'+ Fji'i'j':ji'i'j'(t)] Gji:i'j'(t) + ZZZZ (pp 'qq' )m (ji'ij ')
tm gm Fj,,i,j,: pq. p,q,(t)l Gpq: p,q,(t),
(14)
173 tm
t
t
m
m
where tiSji:i'j' = tiSj-l~i -(l~]j'-1~Si' ) is the frequency difference between two Raman transitions. tm
The real part of l"ji.i,j,:ji.i,j,(t) represents the intermolecular dephasing rate between two transitions, i <--,j and i' <--r The imaginary part represents the time-dependent frequency shift of tm
the transition. The real part of the term l"ji.i'j': pq.p,q,(t) represents the magnitude of the transfer between two intermolecular coherences, i<--->j and i' <--~j', and p~-cq and p'+->q' from two molecules. Structure o f the intermolecular vibrational d e p h a s i n g constant Within the Mark~v approximation, the intermolecular dephasing constant associated with the coherence decay between i+-->j and i' <--->j' vibrational Raman transitions is expressed in terms of the population decay constant from each state and intermolecular pure dephasing constant as 13 tm 1 e e t g t~rn(d) r'ji, i,j,:ji.i,j,-~(r'ii:ii + Fjj:jj + Fi,i,:i,i , + I"j.j,:j,j,)+ I"ji.i,j':ji.i'j', (15)
e where l"ii:ii is the population decay constant from state i of the molecule at site ~, and trn(d)
F ji.i'j':ji.i'j' the intermolecular pure dephasing constant. The population decay constant is defined as t 2g F i i : i i - h2 Z X X P i #42 ~(~tJmfB, iis) 9 (16) iB fs met and the intermolecular pure dephasing constant is defined as s 2~ r j i , i'j':ji,i'j' -- h2
i~B '~afB piBl(-')
- ( < j ' , iBI lqmBIj', f B > - < i ', iBI lqmBli ', fB>)P 5(C0iB.fB) 9 (17) Here lqeB(lqmB) represents the interaction Hamiltonian between the molecule at site e(m) and the heat bath mode. From the general structure of the intermolecular pure dephasing constant, the intermolecular dephasing constant for the transition i <--->jat e and i ,-->j at site m is expressed as ~m 1 t r'ji, ij:ji,ij -- 2 ( F i i : i i
~ m m + F j j . j j + Fii:ii + F j j : j j )
,
(18)
that is, there are no intramolecular and intermolecular pure dephasing terms. For Raman transitions i <--->jat site g and i <--->k(j ,:k)at site m, the dephasing constant is expressed as gm 1 e t m m gm(d) m ._ l"ji, ik'ji,ik -- 2(l"ii:ii + 1-'jj:jj + l"ii:ii + Fkk:kk) + l"ji, ik:ji.ik. (19) When the heat bath modes are different between sites g and m, these molecules are uncorrelated. In this case, the intermolecular dephasing constant consists of the sum of the intramolecular dephasing constants at the two sites, em e m Fji, ik: ji,ik = l"ji: ji + I ~ ki: ki. (20) When the molecules at different sites are correlated through heat bath modes, the intermolecular dephasing constant is not given by the sum of the intramolecular dephasing constants at the two sites. In this case it is given as em e m 1-'ji, ik:ji,ik = F j i : j i + I " k i : k i -
gm l'~ji, ik:ji,ik,
(21)
174 where I~ji,ik:ji,ik denotes the interference term between the two Raman transitions due to a common heat bath mode. At this poi.nt, a brief discussion about the site-dependence of the intermolecular dephasing constants is worthwhile. The population decay constant, and intramolecular dephasing constant are independent of the sites of the molecules in ordinary systems such as neat liquids. The magnitude of the intermolecular dephasing constant associated with i(--)j and i(---)k Raman transitions at different sites, on the other hand, do depend on the site when molecules are correlated through a common heat bath mode. The site-dependent intermolecular dephasing constant can be evaluated by using a multi-spherical-layer model. 9, l0 In this model the intermolecular dephasing constant is assumed to have the same magnitude within the same spherical layer. Under this assumption, the summation over sites ~ and m in Eq.(10) are carried out easily. 111. Intermolecular rovibratlonai interference m e c h a n i s m Okamoto and Yoshihara I l have reported a decay component of 0.39 ps in addition to a slow component of 2.4 ps in time-resolved CARS profiles of neat benzene at room temperature. This sub-pi~nd decay component is not attributed to the intermolecular vibrational dephasing effects discussed in the preceding section or to the effects of the pulsed lasers used. This is because the decay component has a Gaussian form and a photon polarization dependence. The slow component of 2.4 ps is identified with reorientational motions perpendicular to the C6 symmetry axis. Ultrafast intermolecular dynamics of benzene liquids has also been investigated by several groups using different nonlinear-coherent techniques. 14-18 The 0.39 ps dephasing time is explained in terms of interferences between rovibrational Raman transition amplitudes at different sites in this paper. To explain the origin of the 0.39 ps decay component on the basis of a rovibrational interference mechanism, we assume that the radius of the spot irradiated by the pump laser is much larger than an average correlation length between benzene molecules interacting through the heat bath modes. Further, the observation for the time-resolved CARS is performed in the direction of Ak=0. In this case, the molecules interact independently with the heat bath modes. The CARS intensity is proportional to the absolute square of the third-order nonlinear polarization, averaged over the heat bath mode and the distribution of rotational energy in the initial state. The rotational motion is treated quantum mechanically. For simplicity, rotational motions were treated within a free rotor approximation. This approximation is crude one in treating molecular librations in condensed phases and is only valid for high temperature limits in which the thermal energy is larger than an averaged librational bamer height. Photon polarization- dependence is taken into account by assuming that the geometry of benzene is that of a symmetric-top. Under the polarization condition [Z, X, Z, X] only anisotropic Raman transition contributes to the CARS process. The lime profile IZXZX(%)is given as 19 ZXZX
I
(x) oc I ~zxzx(%)12,
(22)
where
(2) X RJKN exp{-iB[N(N+ l)+2NJ]x}exp[-(i~fi+7 fi)x].
(23)
In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, ~ is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. ~(J)=0 if J<0 and ~(J)=l for all other values of J,
175
(2) R~r~N = [2(J+N)+ 1]
/
J
J +N
-K
K
1/2
c0il is the vibrational Raman transition frequency, and ~fi is the vibrational dephasing constant. Within the high temperature approximation, Eq.(22) is expressed as
I
('r -- ( - - - ~ ) 2 e x p [ - 2 7 f i l : ] {5+2exp(-8BkBTX2)}.
(24)
Here T denotes temperature, and ks the Boltzmann factor. This expression indicates that there are two decay components in the time-resolved CARS profile: one is Gaussian and the other is exponential. The Gaussian decay component originates from dephasings of the off-diagonal terms of the CARS intensity, Eq.(22).
,
2
.
:
_
o C ,.-4 rr
-1
0.O
2.0
,~.o e.o Time/ps
a.o
1o.o
Figure 2. CARS time profile calculated under the polarization condition [Z, X, Z, X]. The Stokes and probe polarizations are perpendicular and parallel to the pump one, respectively. The figure inset represents the time profile observed. 1l Figure 2 shows the CARS profile calculated using Eq.(24). The parameter set used is 8 T ~ 2 = 0 . 4 0 with units of t~ called the mean polarizability, t.0fi=992cm -1, B--O.19Ocm -l, kBT=206cm -1, and ~/fi=3.0cm -1. The value of ~fi was determined from the spontaneous Raman spectra of neat benzene. 20 The figure inset shows the observed profile for comparison. Note the excellent correspondence between the calculated CARS profile and the observed one in the sub-picosecond regime. This implies that the sub-picosecond decay component is due to the quantum mechanical interference between the anisotropic rotational Raman transition amplitudes whose transition frequencies are located within the band width of the pump pulse. The physical origin of this interference is a coherent excitation of rotational Raman transitions of molecules by the pump pulse. In other words, a rotational wavepacket in molecular ensemble is created as a result of the coherent excitation, the dissipation of the rotational wavepacket is reflected in CARS time profiles. Classically, an initial angular phase relation between rotors created by the pump pulse disappears as time develops because each rotor has its own mean angular frequency. McMorrow and Lotshaw 17 and Lotshaw et al. 18 have recently given the most detailed analyses of benzene liquid by using femtosecond Fourier-transform of the optical-heterodyne detected optical Kerr effect. They have observed two relaxation constants of 80 fs and 0.45 ps in addition to that of 2.3 ps. The dephasing time of 0.39 ps discussed in this paper corresponds to that of
176 0.45 ps mentioned above. This relaxation component has been suggested that it is associated with a relaxation of overdamped local collective modes. 17 In conclusion, two types of intermolecular dephasing mechanisms are consistent with ultrashort, time-resolved CARS profiles of molecules in liquids. The mechanisms were discussed in terms of the intermolecular vibrational and rovibrational coherence created in molecules at different sites. The intermolecular vibrational coherence decay mechanism is due to molecular interactions through the heat bath modes. In the ordinary case in which molecules interact independently with the heat bath, the intermolecular vibrational dephasing constant associated with two Raman transitions i,-->j and i4-~k(kg:j) is equal to the sum of the intramolecular dephasing constants relevant to the two transitions. In a correlated system, the magnitude of the intermolecular dephasing constant is less than the sum of the individual intramolecular dephasing constants. In a two-level molecular system, the intermolecular dephasing constant is given by a sum of the populations decay constants of the two states. The other type of the dephasing mechanisms discussed is due to the interferences between rovibrational Raman transitions. The time profile of the CARS spectrum is characterized by a Gaussian form and by a photon polarization dependence. This phenomenon has been observed by Okamoto and Yoshihara 11 in neat benzene and is explained in terms of the rovibrational interference mechanism. Finally, it is interesting to clarify the origin of the rapid decay component of 80 fs of benzene liquid and a relation between collective modes and a creation of a long range librational (rovibrational) coherence, l a, 19
Acknowledgments The author expresses his thank to Professor Yoshihara for his fruitful comments. This work was supported by the Ministry of Education, Science and Culture, Japan(Grant No. 0664(041), and a Grant-in-aid for Scientific Research in the Priority Area of "Non-equilibrium Processes in Solution" (02245203).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
S. Ruhman, B. Kohler, A.G. Joly, and K. A. Nelson, Chem. Phys. Lett., 141, 16 (198"0 D. McMorrow, W. T. Lotshaw, and G. A. Kenney Wallace, 1 EFF. J. Quantum Electron., 2 4 , 4 4 3 (1988). W. Zinth, R. Leonhardt, W. Holzaptdel, and W. Kaiser, I EFF. J. Quantum Electron., 24,455 (1987). K.A. Nelson and E. P. Ippen, Adv. Chem. Phys., 75, 1 (1988). S. Mukamel, Adv. Chem. Phys. 70, 165 (1988). N. Morita and T. Yajima, Phys. Rev., A30, 2525 (1984). S. Mukamel and E. Hanamura, Phys. Rev., A33, 1099 (1986). R.F. Loring and S. Mukamel, J. Chem. Phys., 83, 2116 (1985). M. Hayashi, Y. Nomura, and Y. Fujimura, J. Chem. Phys., 89, 32 (1988). M. Hayashi and Y. Fujimura, Chem. Phys., 140, 41 (1988). H. Okamoto and K. Yoshihara, J. Opt. Soc. Am., B7, 1702 (1990). M. Hayashi and Y. Fujimura, J. Opt. Soc. Am., 137, 1653 (1990). M. Hayashi, M. Sugawara, and Y. Fujimura, Phys. Rev., A43, 2416 (1991). T. Joo, M. A. Dugan and A. C. Albrecht, Chem. Phys. Lett., 177, 4(1991). T. Hattori and T. Kobayashi, J. Chem. Phys., 94, 3332 (1991). J. S. Friedman and C. Y. She, J. Chem. Phys., 99, 4960 (1993). D. McMorrow and W. T. Lotshaw, Chem. Phys. Lett., 2 0 1 , 3 6 9 (1993). W. T. Lotshaw, P. R. Staver, D. McMorrow, N. Thantu and J. S. Melinger, in Ultrafast Phenomena IX, Springer, p.91 (1994). M. Hayashi, Y. Fujimura, H. Okamoto, and K. Yoshihara, Chem. Phys. Lett., 196, 44 (1992). G.D. Patterson and J. E. Griffiths, J. Chem. Phys., 63, 2406 (1975).
LIQUIDS ELSEVIER
Journal of Molecular Liquids, 65/66 (1995) 169-176
Mechanisms of Dephasing in Femtosecond Time-resolved Coherent Raman Scattering from Molecules in Liquids Y. Fujimura Department of Chemistry, Faculty of Science, Tohoku University, Sendal, 980-77, Japan Abstract The mechanisms of ultrafast dephasings appearing in the time-profile of coherent Raman scattering from molecules in liquids are examined. Two mechanisms are taken into account. One is associated with intermolecular interactions between molecules at different sites through heat bath modes. The other is associated with the interference between rovibrational Raman transitions. In the former mechanism, the time profile of the coherent Raman scattering is characterized by an exponential decay. This decay constant is called the intermolecular dephasing constant. For the latter mechanism, the time profile has an inhomogeneous nature and is characterized by a Gaussian form. The structure of the intermolecular dephasing constant is derived within the Markov approximation. We describe the relationship between the intermolecular dephasing constant and the intramolecular dephasing constant associated with spontaneous Raman transitions. Finally we analyze the ultrafast, sub-picosecond dephasings observed in coherent anti-Stokes Raman profiles of neat benzene liquid in terms of the rovibrational interference mechanism. I. Introduction In recent years, considerable attention has been giyen to ultrafast dynamics associated with molecular motions in liquids. Femtosecond time-resolved, nonlinear coherent Raman scattering spectroscopy is one method for clarifying the origin of such ultra_fast dynamics. 1-5 Ultrafast dynamics in liquids is reflected in the profiles of the time-resolved, nonlinear coherent Raman scattering spectroscopy. This method includes time-resolved coherent anti-Stokes Raman scattering(CARS), degenerate four-wave mixing, impulsive stimulated Raman scattering (ISRS),etc. When considering the molecular dephasing processes, differences in the creation of the molecular coherence between coherent and incoherent(spontaneous) Raman scattering should be noted. In the time-resolved, nonlinear, coherent scattering processes, each molecule in the liquid receives the photon wave vector transiently through spatial coherence of the photon field. Consequently, a transient intermolecular coherence is created. The transient macroscopic nonlinear polarization of the molecular ensemble is therefore given as the summation of the induced nonlinear polarization on each molecule. The output intensity of the coherent Raman scattering experiments is generally proportional to an ensemble average of the absolute square of the nonlinear polarization. 6,7 In the absence of long-range spatial correlation between molecules, the intensity can be expressed in terms of the absolute square of the ensemble-averaged nonlinear polarization. On the other hand, in incoherent Raman scattering processes, intramolecular rovibrational coherence of each molecule within the laser irradiation spot is created. The decay of rovibrational coherence is expressed by the ordinary intramolecular dephasing constant. This consists of the population decay constant and the pure dephasing constant. The former originates from an inelastic interaction between the molecule of interest and heat bath modes. The latter originates from an elastic interaction between the molecule and the heat bath modes. Thus, intermolecular phase dynamics from liquids is directly reflected in time-resolved, nonlinear, coherent Raman scattering profiles though the information content in both coherent and incoherent Raman scattering experiments is in principle the same. 8 It should be noted that the laser pump pulses create sets of intermolecular rovibrational coherence. For example, quantum beats appear in time profile of the CARS spectrum from a molecular mixture. 3, 9, 10 The beat frequency created is equal to the difference between frequencies of vibrational modes of molecules in the mixture. When many sets of intermolecular 0167-7322/95/$09.50 9 1995 Elsevier Science B.V. All rights reserved. SSDI 0167-7322 (95) 00850-0
170 rovibrational coherence are created, it can be expected that ultmfast dephasing will result from the interference between the sets of the rovibrational coherence. This phenomenon is one of the main origins of the inhomogeneous Gaussian component in the time profile. The time profile also has a photon polarization dependence. This polarization-dependent time profile has been observed in the sub-picosecond time regime in nonresonant CARS from benzene in liquid. 11 The present study uses a quantum statistical method to show how the time-profile of CARS from molecules in liquids is associated with the intermolecular dephasing mechanisms. Liouville space Feynman diagrams are used for the development of relevant transitions associated with pairs of molecules. 9-10 The structure of the intermolecular dephasing constant is clarified to identify the difference between the intermolecular dephasing constant and intramolecular dephasing constant in Sec.II. Section III presents the rovibrational interference mechanism. The sub-picosecond decay observed in CARS profile of neat benzene is explained in terms of this mechanism. II. Intensity of time-resolved C A R S Consider a time-resolved, electronically nonresonant CARS spectrum from a molecular liquid. In the CARS process, the laser pump pulses create a linear combination (that is the intermolecular rovibrational coherence) of Raman active rovibrational transitions between molecules at position rt and rm in the mixture. This stimulated Raman scattering process is carried out by two-coincident laser pulses(I, II) with central frequencies(wave vectors) mi(kI) and r By applying the third pulse with ml(km) to the liquid after time delay x, the time dependence of the intermolecular rovibrational coherence is detected through the measurement of the intensity of the scattered photon with ks.
The total Hamiltonian I2I is given as fl = I:t~ + fl~ + f l ~ ,
(l)
where I2IMBis the molecular Hamiltonian including the optically inactive heat bath modes, I2IRis the radiation field Hamiltonian, and I~IMR is the molecule-radiation-interaction Hamiltonian. Using the dipole approximation, and making use of the rotating-wave approximation, the interaction Hamiltonian is given as I21MR= Z
V e,
(2)
/=1 9t,,,,q ,v,,~ with
,
'
(-)
Ve = - 2.,2.., Mba Tba El b
exp(ikl "re)
a
9, ' (+) exp(-ikll 9 r e ) - Z Z M~b Tcb E.ll c
~~c M~cr Tdc •ll
exp(--ikiil ~ re).
b
(3)
g
The term M~tx denotes the matrix element of the electric-dipole moment between states ~t and ~, of the gth molecule. The operator ~
e
( -- I~t><~,l) is the transition operator from the ~, to the l.t
states. E denote the radiation field operators. The density operator for the total system ( ~ T )
is
satisfied with the equations of motion,
ih "~" [5T (t) = [V(t ),[5T ( t ) ] , (4) where the density operator with the overtilde denotes that it represents the density operator in the interaction picture. The dipole transition operator in the interaction picture is expressed as Q ( t ) = e x p I i ( I 2 I ~ + I21R)t/h] I2tMRexp[--i(121.~m+ I21p)t/h 1= f i g=l
Qe(t).
(5)
171 In the perturbative density-matrix method, the total density operator for the CARS process is given by expanding iST(t) in terms of the photon number (4) as (4) i OT (t)= { dt(8)} [V(tl), [V(t2) . . . . . IV(t7), [V(ts), OT(-~) .... ]]]1,
where { i d t ( 8 ' } = ( ~ ' ) S i d t i i d t 2 . . . .=.oo
-.oo
idt7idt
==.oo
8,
(6)
(7)
-.-oo
and OT (_oo) is the density operator for the total system at initial time t ---~-o.. This density operator is assumed to be given by
(8)
OT(__oo) = OM (-oo)O B (-oo) ~ R (...oo) , where 0M ("**) is the density operator of the molecules, 0B (--~
that of the heat bath and
(4) OR (.-o.) is that of the radiation field. The density operator for the scattering field 0s (t) is (4) derived from the total-system-density operator 0T (t) by tracing out the variables from both the molecules and the heat bath modes. The intensity of time-resolved CARS is defined in terms of the population of the scattered photons (W('r, Ak, Ws, t)) as 12 I(,, Ak) =
dadWs
Atdt~'~W(,, Ak, Ws, t), withAk = ki - kII + kiii - ks
(9)
The CARS intensity is expressed as I('t, Ak)--
ZZ Itm('l:)exp(iAk~ 9 (10) t=l m=l (t ~ m) After averaging Eq.(10) over the heat bath variables with the aid of the Liouville space Feynman diagram (shown in Fig. 1) within the factorization approximation, the main term for the CARS process is given as
oj" dt XXxX;X;XX;X a
b
c
d
a'
b'
c'
d'
At g m g g g m m m m X Pa Pa' Mdc Mcb Mba Ma'd' Md'c' Mc'b' Mb'a' t ,,(-) (t2)EllI)(t3)) ,,(+ ,,(+ '(t4)Eli ,,(- )0.6)) X I _.,dt 1 " ' " I t_oodt7 ( EIII ( EII ,, ((+) gm tm X ( EI )(ts) EI (t7)) Gaa:a,b,(t6-tT) Gba:a,b,(t5-t6) tm gm em em X Gba. a'c ,(t4-ts) Gca: a'c' (t3-t4) Gca: a' a' (t2-t3) Gda- a'd ,(tl-t2).
(11)
The term Pa denotes the population in the initial state of the molecule at site g. The term ..((El)(ti)
.. (+ EI )(tj))denotes ~ the incident radiation field correlation function.
The term
172 G ~ . es(ti-tj) represents the time-development matrix element of the intermolecular coherence between B ~ k and e ~ ~i from time ti to time ti in the presence of the molecule-heat bath interaction. This matrix element is d e f i n ~ by tm Gj~t: es(ti-tj)= [((I.tl~, X~I ~,m(ti-tj)I l.tl~, X~ ))]av. (12) where (3em(ti-tj), the propagator of the molecular pair whose constituents are located at sites g t~m
and m, is given as
gm
(3tm(ti-tj) = exp[-i f_.MB(ti-tj)],
trn
with I'~MB= [HMB, ]lb.
(13)
~al g site
t5 l
a
t4
~
t7
'"f
1.6
P
t
:
t
ti "la>
/ 7 m site
t2
.s/
1,3 t la'~'
Figure 1. A Liouville space Feynman diagram for CARS process. Each wavy line denotes the molecule-radiation field interaction. This diagram represents the time evolution of the intermolecular coherence between two molecules at g and m sites in the CARS process. Figure 1 shows the Liouville space Feynman diagram representing the time development of the intermolecular coherence between molecules at site g and m in the time-resolved CARS process. The two upper lines show the time development of intramolecular coherence of molecules at g. The two lower lines show the time development of intramolecular coherence of molecules at m, respectively. The upper and lower lines are connected through the incident laser fields(I, II, and III) indicated by wavy lines. In the ordinary pump-probe type time-resolved CARS experiments, the time duration from t3 to t4 corresponds to the pump-probe time x, and the intermolecular coherence time for pair molecules. In other words, it is the time dependence of the coherence between the Raman transition a <---~cat site g and a' <---~c'at site m which are detected in the CARS time profile.
Equation of motion for the intermolecular coherence The equation of motion for the intermolecular coherence between two Raman transitions gm i ~---}jand i' <---~j' at site g and m ( Gji:i'j ,(t) ) is" d tm ~m }'m gm dt Gji:i'j'(t)---[il~ji:i'J'+ Fji'i'j':ji'i'j'(t)] Gji:i'j'(t) + ZZZZ (pp 'qq' )m (ji'ij ')
tm gm Fj,,i,j,: pq. p,q,(t)l Gpq: p,q,(t),
(14)
173 tm
t
t
m
m
where tiSji:i'j' = tiSj-l~i -(l~]j'-1~Si' ) is the frequency difference between two Raman transitions. tm
The real part of l"ji.i,j,:ji.i,j,(t) represents the intermolecular dephasing rate between two transitions, i <--,j and i' <--r The imaginary part represents the time-dependent frequency shift of tm
the transition. The real part of the term l"ji.i'j': pq.p,q,(t) represents the magnitude of the transfer between two intermolecular coherences, i<--->j and i' <--~j', and p~-cq and p'+->q' from two molecules. Structure o f the intermolecular vibrational d e p h a s i n g constant Within the Mark~v approximation, the intermolecular dephasing constant associated with the coherence decay between i+-->j and i' <--->j' vibrational Raman transitions is expressed in terms of the population decay constant from each state and intermolecular pure dephasing constant as 13 tm 1 e e t g t~rn(d) r'ji, i,j,:ji.i,j,-~(r'ii:ii + Fjj:jj + Fi,i,:i,i , + I"j.j,:j,j,)+ I"ji.i,j':ji.i'j', (15)
e where l"ii:ii is the population decay constant from state i of the molecule at site ~, and trn(d)
F ji.i'j':ji.i'j' the intermolecular pure dephasing constant. The population decay constant is defined as t 2g F i i : i i - h2 Z X X P i #
i~B '~afB piBl(
- ( < j ' , iBI lqmBIj', f B > - < i ', iBI lqmBli ', fB>)P 5(C0iB.fB) 9 (17) Here lqeB(lqmB) represents the interaction Hamiltonian between the molecule at site e(m) and the heat bath mode. From the general structure of the intermolecular pure dephasing constant, the intermolecular dephasing constant for the transition i <--->jat e and i ,-->j at site m is expressed as ~m 1 t r'ji, ij:ji,ij -- 2 ( F i i : i i
~ m m + F j j . j j + Fii:ii + F j j : j j )
,
(18)
that is, there are no intramolecular and intermolecular pure dephasing terms. For Raman transitions i <--->jat site g and i <--->k(j ,:k)at site m, the dephasing constant is expressed as gm 1 e t m m gm(d) m ._ l"ji, ik'ji,ik -- 2(l"ii:ii + 1-'jj:jj + l"ii:ii + Fkk:kk) + l"ji, ik:ji.ik. (19) When the heat bath modes are different between sites g and m, these molecules are uncorrelated. In this case, the intermolecular dephasing constant consists of the sum of the intramolecular dephasing constants at the two sites, em e m Fji, ik: ji,ik = l"ji: ji + I ~ ki: ki. (20) When the molecules at different sites are correlated through heat bath modes, the intermolecular dephasing constant is not given by the sum of the intramolecular dephasing constants at the two sites. In this case it is given as em e m 1-'ji, ik:ji,ik = F j i : j i + I " k i : k i -
gm l'~ji, ik:ji,ik,
(21)
174 where I~ji,ik:ji,ik denotes the interference term between the two Raman transitions due to a common heat bath mode. At this poi.nt, a brief discussion about the site-dependence of the intermolecular dephasing constants is worthwhile. The population decay constant, and intramolecular dephasing constant are independent of the sites of the molecules in ordinary systems such as neat liquids. The magnitude of the intermolecular dephasing constant associated with i(--)j and i(---)k Raman transitions at different sites, on the other hand, do depend on the site when molecules are correlated through a common heat bath mode. The site-dependent intermolecular dephasing constant can be evaluated by using a multi-spherical-layer model. 9, l0 In this model the intermolecular dephasing constant is assumed to have the same magnitude within the same spherical layer. Under this assumption, the summation over sites ~ and m in Eq.(10) are carried out easily. 111. Intermolecular rovibratlonai interference m e c h a n i s m Okamoto and Yoshihara I l have reported a decay component of 0.39 ps in addition to a slow component of 2.4 ps in time-resolved CARS profiles of neat benzene at room temperature. This sub-pi~nd decay component is not attributed to the intermolecular vibrational dephasing effects discussed in the preceding section or to the effects of the pulsed lasers used. This is because the decay component has a Gaussian form and a photon polarization dependence. The slow component of 2.4 ps is identified with reorientational motions perpendicular to the C6 symmetry axis. Ultrafast intermolecular dynamics of benzene liquids has also been investigated by several groups using different nonlinear-coherent techniques. 14-18 The 0.39 ps dephasing time is explained in terms of interferences between rovibrational Raman transition amplitudes at different sites in this paper. To explain the origin of the 0.39 ps decay component on the basis of a rovibrational interference mechanism, we assume that the radius of the spot irradiated by the pump laser is much larger than an average correlation length between benzene molecules interacting through the heat bath modes. Further, the observation for the time-resolved CARS is performed in the direction of Ak=0. In this case, the molecules interact independently with the heat bath modes. The CARS intensity is proportional to the absolute square of the third-order nonlinear polarization, averaged over the heat bath mode and the distribution of rotational energy in the initial state. The rotational motion is treated quantum mechanically. For simplicity, rotational motions were treated within a free rotor approximation. This approximation is crude one in treating molecular librations in condensed phases and is only valid for high temperature limits in which the thermal energy is larger than an averaged librational bamer height. Photon polarization- dependence is taken into account by assuming that the geometry of benzene is that of a symmetric-top. Under the polarization condition [Z, X, Z, X] only anisotropic Raman transition contributes to the CARS process. The lime profile IZXZX(%)is given as 19 ZXZX
I
(x) oc I ~zxzx(%)12,
(22)
where
(2) X RJKN exp{-iB[N(N+ l)+2NJ]x}exp[-(i~fi+7 fi)x].
(23)
In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, ~ is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. ~(J)=0 if J<0 and ~(J)=l for all other values of J,
175
(2) R~r~N = [2(J+N)+ 1]
/
J
J +N
-K
K
1/2
c0il is the vibrational Raman transition frequency, and ~fi is the vibrational dephasing constant. Within the high temperature approximation, Eq.(22) is expressed as
I
('r -- ( - - - ~ ) 2 e x p [ - 2 7 f i l : ] {5+2exp(-8BkBTX2)}.
(24)
Here T denotes temperature, and ks the Boltzmann factor. This expression indicates that there are two decay components in the time-resolved CARS profile: one is Gaussian and the other is exponential. The Gaussian decay component originates from dephasings of the off-diagonal terms of the CARS intensity, Eq.(22).
,
2
.
:
_
o C ,.-4 rr
-1
0.O
2.0
,~.o e.o Time/ps
a.o
1o.o
Figure 2. CARS time profile calculated under the polarization condition [Z, X, Z, X]. The Stokes and probe polarizations are perpendicular and parallel to the pump one, respectively. The figure inset represents the time profile observed. 1l Figure 2 shows the CARS profile calculated using Eq.(24). The parameter set used is 8 T ~ 2 = 0 . 4 0 with units of t~ called the mean polarizability, t.0fi=992cm -1, B--O.19Ocm -l, kBT=206cm -1, and ~/fi=3.0cm -1. The value of ~fi was determined from the spontaneous Raman spectra of neat benzene. 20 The figure inset shows the observed profile for comparison. Note the excellent correspondence between the calculated CARS profile and the observed one in the sub-picosecond regime. This implies that the sub-picosecond decay component is due to the quantum mechanical interference between the anisotropic rotational Raman transition amplitudes whose transition frequencies are located within the band width of the pump pulse. The physical origin of this interference is a coherent excitation of rotational Raman transitions of molecules by the pump pulse. In other words, a rotational wavepacket in molecular ensemble is created as a result of the coherent excitation, the dissipation of the rotational wavepacket is reflected in CARS time profiles. Classically, an initial angular phase relation between rotors created by the pump pulse disappears as time develops because each rotor has its own mean angular frequency. McMorrow and Lotshaw 17 and Lotshaw et al. 18 have recently given the most detailed analyses of benzene liquid by using femtosecond Fourier-transform of the optical-heterodyne detected optical Kerr effect. They have observed two relaxation constants of 80 fs and 0.45 ps in addition to that of 2.3 ps. The dephasing time of 0.39 ps discussed in this paper corresponds to that of
176 0.45 ps mentioned above. This relaxation component has been suggested that it is associated with a relaxation of overdamped local collective modes. 17 In conclusion, two types of intermolecular dephasing mechanisms are consistent with ultrashort, time-resolved CARS profiles of molecules in liquids. The mechanisms were discussed in terms of the intermolecular vibrational and rovibrational coherence created in molecules at different sites. The intermolecular vibrational coherence decay mechanism is due to molecular interactions through the heat bath modes. In the ordinary case in which molecules interact independently with the heat bath, the intermolecular vibrational dephasing constant associated with two Raman transitions i,-->j and i4-~k(kg:j) is equal to the sum of the intramolecular dephasing constants relevant to the two transitions. In a correlated system, the magnitude of the intermolecular dephasing constant is less than the sum of the individual intramolecular dephasing constants. In a two-level molecular system, the intermolecular dephasing constant is given by a sum of the populations decay constants of the two states. The other type of the dephasing mechanisms discussed is due to the interferences between rovibrational Raman transitions. The time profile of the CARS spectrum is characterized by a Gaussian form and by a photon polarization dependence. This phenomenon has been observed by Okamoto and Yoshihara 11 in neat benzene and is explained in terms of the rovibrational interference mechanism. Finally, it is interesting to clarify the origin of the rapid decay component of 80 fs of benzene liquid and a relation between collective modes and a creation of a long range librational (rovibrational) coherence, l a, 19
Acknowledgments The author expresses his thank to Professor Yoshihara for his fruitful comments. This work was supported by the Ministry of Education, Science and Culture, Japan(Grant No. 0664(041), and a Grant-in-aid for Scientific Research in the Priority Area of "Non-equilibrium Processes in Solution" (02245203).
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