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Population relaxation and non-Markovian frequency fluctuations in third- and fifth-order Raman scattering
Chemical Physics 233 Ž1998. 267–285
Population relaxation and non-Markovian frequency fluctuations in third- and fifth-order Raman scattering Thomas Steffen, Koos Duppen
)
Ultrafast Laser and Spectroscopy Laboratory, Materials Science Center, UniÕersity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 22 October 1997
Abstract The third- and fifth-order Raman response is calculated for vibrational phase relaxation due to stochastic Gaussian frequency fluctuations. This model can be solved exactly using the cumulant expansion technique. It interpolates continuously between the homogeneous and inhomogeneous limit and allows for a description of non-Markovian dynamics arising from finite correlation timeŽs. of the frequency fluctuations. Model calculations demonstrate that temporally two-dimensional fifth-order Raman experiments are sensitive to the time scaleŽs. of these frequency fluctuations and also to the life times of the vibrational states. These processes, that cannot be distinguished in nonresonant third-order Raman scattering, can be unequivocally identified in fifth-order experiments. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The importance of vibrational excitations in chemical reactions has been recognized for many years w1x. Coupling of the molecular vibrations with a dynamical environment is responsible for the energy flow into and out of molecules, and for thermally activated processes w2x. From a fundamental point of view on chemistry, it is therefore essential to obtain direct experimental information on both the intramolecular vibrational dynamics and on the external degrees of freedom that act as a heat bath for the reacting system. Vibrational lineshape analysis is a very popular and powerful tool to investigate the local structure and dynamics in condensed phase systems w1,3x. The variation of pressure, temperature or solvent leads in general to changes in both the central frequency and the linewidth of a vibrational transition. These modifications reflect
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Corresponding author.
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 0 8 3 - 4
268
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
static and dynamic properties of the microscopic potential. A variety of models has been proposed to describe different kinds of intra- and intermolecular interactions that lead to vibrational population and phase relaxation w1,3–7x. These models can be tested by evaluating their capabilities to explain the observed infrared ŽIR. and Raman spectra. The conventional spectroscopic techniques have, however, a limited information content since they provide only indirect evidence for the mechanism and the time scaleŽs. of the line broadening w8,9x. In order to obtain additional information, Mukamel and co-workers w8,9x proposed to probe higher-order correlation functions by multiple pulse experiments. Examples for these techniques are the Raman echo and Raman pump–probe experiments, which are seventh-order nonlinear optical processes w10–12x, and temporally two-dimensional fifth-order Raman scattering. This latter technique has been performed impulsively on low-frequency inter-w13– 19x and intramolecular w20,21x modes, and nonimpulsively on high-frequency intramolecular modes w22,23x. By using these more advanced techniques, it is in principle possible to characterize the pure dephasing and to separate population and phase relaxation processes. The loss of macroscopic phase coherence is often modelled in the limits of fast or slow frequency modulation which give rise to homogeneous and inhomogeneous dephasing, respectively w1x. When the correlation time of the frequency fluctuations is comparable to the typical time scale of the experiment, these approximations break down and stochastic models have to be evoked w24–28x. The resulting non-Markovian dynamics were first discussed for spin-resonance experiments w24–27x, but electronic coherences in condensed phase systems have been investigated as well w29–34x. The Kubo–Anderson formalism was evoked to model vibrational phase relaxation in IR w35x and third-order Raman w36x spectra. Evidence for a finite correlation time of the frequency fluctuations in molecular liquids were obtained by time-resolved CARS w37–39x, impulsive stimulated scattering w40x and Raman line shape analysis w41–43x. The theory of electronically nonresonant nonimpulsive seventh-order experiments such as the Raman echo is well established since it closely resembles the resonant third-order response of a two-level system which is discussed in standard text books on nonlinear optical spectroscopy Že.g., w44x.. The fifth-order Raman techniques, which have been proposed w9x and demonstrated w13–23x recently, are related to the resonant second-order response which necessarily involves three levels w16x. Therefore, the quantum number dependence of the relaxation processes has to be considered explicitly in modelling these experiments w22,23,45x. The fifth-order Raman response can be observed only in the case of coupling constants with a nonlinear coordinate dependence w9x andror anharmonic potentials w46x. Up to now only a few models for population and phase relaxation in fifth-order experiments have been discussed. In their key paper Tanimura and Mukamel w9x employed a Brownian oscillator model, where the optically active oscillator is linearly coupled to a continuous distribution of optically dark bath oscillators. This model allows for a continuous change from underdamped to overdamped vibrations and yields in the limit of a white bath spectrum a level-independent population relaxation as the only decay process w45x. Khidekel and Mukamel w47x introduced a static Lorentzian distribution of oscillator frequencies as line broadening mechanism and demonstrated that this leads to a motional echo. Tominaga and Yoshihara w22,23x discussed the effects of partial correlation between the fluctuations of different vibrational levels. In our previous paper w45x we derived the third- and fifth-order response within the weak coupling limit for quantum-number-dependent population relaxation and pure dephasing rates, resulting from linear and quadratic coupling to a harmonic heat bath. In this paper we calculate the third- and fifth-order Raman response for random Gaussian frequency fluctuations that cause phase relaxation. This model, that can be solved exactly using the cumulant expansion technique w24–28x, interpolates continuously between homogeneous and inhomogeneous dephasing. The effect of a finite correlation time, which gives rise to non-Markovian dynamics, is discussed for fifth-order vibrational echoes, overtone dephasing, and impulsive excitation. Simulations of the fifth-order Raman echo show that this experiment is particularly sensitive to the time scale of the fluctuations. The predicted impulsive 2D Raman response illustrates that this experiment allows one to discriminate between pure dephasing and population relaxation processes, which is impossible in lower-order nonresonant experiments.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
269
2. Theory The third- and fifth-order Raman response functions have been calculated by several authors using time domain perturbation theory w9,45–47x. When processes involving hyperpolarizabilities can be neglected, the third- and fifth-order Raman response are given by w9,44,45x: R Ž3. Ž t 1 . s
i "
a Ž t 1 . , a Ž 0 . r Ž y` . :'
²
i "
J1 Ž t 1 . y J1) Ž t 1 . ,
Ž 1.
and: R Ž5. Ž t 2 , t 1 . s '
i
2
ž /² Ž ž /Ý
a t 2 q t 1 . , a Ž t 1 . , a Ž 0 . r Ž y` . :
" i
"
2
2
Qn Ž t 2 , t 1 . q Qn) Ž t 2 , t 1 . .
Ž 2.
ns1
Here, r Žy`. denotes the density matrix of the system in thermal equilibrium and a Žt . s U0y1 Žt . a U0 Žt . is the interaction picture representation of the polarizability. For a Hamiltonian H0 Ž t . that is explicitly time dependent, U0 Žt . is given by:
½
U0 Ž t . s exp y
i
t
5
H dt H Ž t. " 0 0
.
Ž 3.
The functions J1Žt 1 . and Qns1, 2 Žt 2 , t 1 . are easily obtained by writing out the commutators of Eqs. Ž1. and Ž2.. When these functions are developed in terms of the nuclear eigenstates < k : of the system, this yields: J1 Ž t 1 .
s ² a Ž t 1 . a Ž 0 . r Ž y` . : s Ý Pk² k
Q1 Ž t 2 , t 1 .
½
i
t1
5
½
d t H0 Ž t . a Ž q . exp y
H " 0
i
t1
5
s ² a Ž t 2 q t 1 . a Ž t 1 . a Ž 0 . r Ž y` . : i t 2q t 1 i s Ý Pk² k
½
=² m
½
i
5 ½
H
t1
5
d t H0 Ž t . a Ž q . exp y
H " 0
½
i
Ž 4.
d t H0 Ž t . < l :² l < a Ž q . < k : ,
H " 0
t 2q t 1
t1
5
d t H0 Ž t . < m:
H0
Ž 5.
5
d t H0 Ž t . < l :² l < a Ž q . < k : ,
H " 0
and: Q2 Ž t 2 , t 1 .
s y² a Ž t 1 . a Ž t 2 q t 1 . a Ž 0 . r Ž y` . : i t1 i s y Ý Pk² k
½
=² m
½
i
t 2q t 1
H " 0
5
H
5
½
½
d t H0 Ž t . a Ž q . exp y
i
t 2q t 1
H " 0
t1
H0
5
d t H0 Ž t . < m:
Ž 6.
5
dt H0 Ž t . < l :² l < a Ž q . < k : .
Here, the factor Pk denotes the occupation of state < k : in thermal equilibrium. The different terms on the right-hand side of Eqs. Ž1. and Ž2. represent the two and four Liouville space pathways that contribute to the third- and fifth-order Raman response, respectively. These pathways can be
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
270
depicted in the form of double-sided Feynman diagrams, which are shown in Fig. 1. Pure dephasing is now introduced by assuming that the energy difference Ek y El s " v k l between two states < k : and < l : undergoes random fluctuations due to coupling to a large heat bath. When the system states follow the heat bath adiabatically, the transition frequencies v k l Ž t . can be decomposed into their average values v k l and a fluctuating part d v k l Ž t .:
vkl Ž t . s vkl q dvkl Ž t . .
Ž 7.
Defining the transition matrix element a k l ' ² k < a < l :, Eqs. Ž4. – Ž6. can then be recast as:
k, l
Q1 Ž t 2 , t 1 . s
t1
¦ ½H
J1 Ž t 1 . s Ý Pk a k l a l k e i v k lt 1 exp i
Ý k, l, m
0
dt dvkl Ž t .
5;
,
Ž 8. t1
¦ ½H
Pk a k m a m l a l k e iŽ v k lt 1q v k mt 2 . exp i
t 2q t 1
dt dvkl Ž t . q i
0
dt dvkmŽ t .
Ht
1
5;
,
Ž 9.
5;
.
Ž 10 .
and: Q2 Ž t 2 , t 1 . s
Ý k, l, m
t1
¦ ½H
Pk a k m a m l a l k e iŽ v k lt 1q v m lt 2 . exp i
0
dt dvkl Ž t . q i
t 2q t 1
Ht
d t d vm l Ž t .
1
At his stage we have to specify the system Hamiltonian, which determines the eigenstates < k : and the transition frequencies v k l , the nature of the fluctuations d v k l Ž t ., and the coordinate dependence of the polarizability a Ž q . that is necessary to calculate the transition matrix elements a k l .
Fig. 1. Feynman diagrams depicting the Liouville space pathways of the third- and fifth-order Raman response. Ža. The third-order response, given by Eq. Ž1., can be captured by the two diagrams Ža.1. and Ža.2. that are the complex conjugate of each other. Žb. The fifth-order response, given by Eq. Ž2., involves four different diagrams. Diagrams Žb.1. and Žb.4. depict Q1Žt 2 , t 1 . and Q1) Žt 2 , t 1 ., while Žb.2. and Žb.3. represent Q 2) Žt 2 , t 1 . and Q2 Žt 2 , t 1 ., respectively.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
271
Fig. 2. Energy level diagram of a harmonic oscillator that undergoes random frequency fluctuations. For the description of pure dephasing only the energy difference between the involved states is important. For perfectly correlated fluctuations the spread of energies between states < k : and < k q2: is twice as large as that between < k : and < k q1:, and < k q1: and < k q2:.
Here, we consider a harmonic vibration with Ek s Ž k q 1r2. " v 0 so that the average transition frequencies are:
vkl s Ž k y l . v0 .
Ž 11 .
We further assume that the system is coupled to the radiation field via a coordinate-dependent polarizability:
a Ž q . s a1 q q a2 q 2 ,
Ž 12 .
where q denotes the vibrational coordinate of the optically active oscillator. The relevant Liouville space pathways, i.e. the nonvanishing combinations of k, l and m in the sums of Eqs. Ž8. – Ž10., and their relative amplitudes, determined by the matrix elements a k l , were calculated explicitly in Ref. w45x. Several statistical models have been discussed in literature in order to describe the fluctuation of the energy levels of the system w24–28x. Here, we assume that the modulations d v k l Ž t . of the vibrational transition frequencies are perfectly correlated and can be modelled as a stationary Gaussian processes, obeying w24–28x: dvkl Ž t . s Ž k y l . dv Ž t . ,
Ž 13 .
²d v Ž t . : s 0 ,
Ž 14 .
² d v Ž t 1 . d v Ž t 2 . : s ² d v Ž 0 . d v Ž t 2 y t 1 . : s D 2 exp yL < t 2 y t 1 < 4 ,
Ž 15 .
and:
where D and t C s Ly1 denote the root mean square and the correlation time of the frequency fluctuations. As a consequence of the perfect correlations the amplitude of the fluctuations of a two-quantum coherence such as < k :² k q 2 < is twice as large as that of a one-quantum coherence such as < k :² k q 1 < or < k q 1:² k q 2 <. This is shown schematically in Fig. 2. Within these assumption of harmonic nuclear motion ŽEq. Ž11.., non-Condon coordinate dependence of the polarizability ŽEq. Ž12.., and perfect correlation of the fluctuations ŽEq. Ž13.., the expressions for J1Žt 1 . and Qns1, 2 Žt 2 , t 1 . in Eqs. Ž8. – Ž10. can be further simplified. Using these results, the third- and fifth-order response functions Eqs. Ž1. and Ž2. are, in lowest order of the vibrational coordinate:
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
272
¦ ½
t1
R Ž3. Ž t 1 . s exp yi
dt dv Ž t .
H0
5;
sin v 0t 1 ,
Ž 16 .
and t1
¦½ H ¦½ H
R Ž5. Ž t 2 , t 1 . s exp yi
t 1q t 2
dt dv Ž t . qi
Ht
0
y exp yi
t1
¦ ½ ¦½
Ht
t1
H0
y exp yi
dt dv Ž t .
dt dv Ž t .
1
cos v 0 Ž t 1 y t 2 .
5;
cos v 0 Ž t 1 q t 2 .
5;
cos v 0t 1
t1
H0
5;
t 1q t 2
dt dv Ž t . yi
0
q exp yi
dt dv Ž t .
1
t 1q t 2
dt dv Ž t . yi
Ht
d t 2d v Ž t .
1
5;
cos v 0 Ž t 1 q 2t 2 . .
Ž 17 .
Here and in the remainder of the paper we have set the constant prefactors a 12rm v 0 in Eq. Ž16. and a 12a 2r2Ž m v 0 . 2 in Eq. Ž17. equal to one. Note that the fifth-order response function Eq. Ž17. does not comprise the term proportional to cos v 0 Ž2t 1 q t 2 ., that was found in Ref. w45x for level-dependent population relaxation. This term is absent here due to the destructive interference of different Liouville space pathways w45x. For a stationary Gaussian process the averages over the random fluctuations can be calculated exactly for all involved Liouville space pathways, since then all cumulants beyond the second order vanish w24–28x. For a single propagation time like in Eq. Ž16. one finds for the relaxation function f Ž1. Žt 1 . w24–28x: t1 t1 1 t1 f Ž1. Ž t 1 . ' exp yi d t d v Ž t . s exp y d t1 d t 2² d v Ž t 1 . d v Ž t 2 . : ' exp yg Ž1. Ž t 1 . 4 , 2 0 0 0 Ž 18 .
¦ ½
5; ½
H
H
H
5
where the line shape function g Ž1. Žt . is given by w24–28x:
D2
Ž ey Lt q Lt y 1 . . Ž 19 . L2 The relaxation functions appearing in the expression for R Ž5. Žt 2 , t 1 ., cf. Eq. Ž17., can be expressed in terms of g Ž1. Ž t . as well, as is shown in Appendix A. Using the results of Eqs. Ž18. and ŽA3. the third-order response function can be rewritten as w38,39x: g
Ž1.
Žt . s
R Ž3. Ž t 1 . s eyg
Ž1.
Žt 1 .
sin v 0t 1 ,
Ž 20 .
while the fifth-order signal is given by: R Ž5. Ž t 2 , t 1 . s ey2 g
Ž1.
y eyg
Žt 1 .y2 g Ž1.Žt 2 .qg Ž1.Žt 1qt 2 . Ž1.
Žt 1qt 2 .
yg Ž1.Žt 1 .
qe
y ey2 g
Ž1.
cos v 0 Ž t 1 y t 2 .
cos v 0 Ž t 1 q t 2 .
cos v 0t 1
Žt 2 .y2 g Ž1.Žt 1qt 2 .qg Ž1.Žt 1 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 21 .
These formulas describe the effect of Gaussian frequency fluctuations with an arbitrary correlation time t C on the optical response of a harmonic mode. In the next two sections we simulate the temporally two-dimensional fifth-order Raman signal for different ratios DrL and for different excitation conditions.
3. Nonimpulsive fifth-order Raman scattering
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
273
The fifth-order Raman response, given in Eq. Ž21., consists of four different terms, that carry different dynamic information. In impulsive experiments they all contribute simultaneously to the signal. This situation will be discussed in the next section. In this section we will first treat two non-impulsive fifth-order Raman experiments, in which the employed laser pulses are much longer than the vibrational period of the investigated mode. As a consequence the signal does not show any oscillations at frequency v 0 , but it is given by the relaxation function only, i.e. the appropriate prefactors of Eq. Ž21.. Using sequences of laser pulses with suitable frequency differences, it is possible to select only one of the Liouville space pathways depicted in Fig. 1. The two possible sequences of system states that give rise to fifth-order Raman echoes, are depicted in Fig. 3 in the form of double-sided Feynman diagrams. The experiment is initiated by two coincident laser pulses, which via stimulated Raman scattering induce a one-quantum transition. The resulting coherent superposition of the ground and the first vibrational state propagates for a period t 1 before a second pair of laser pulses induces a two-quantum transition from the ground state. After a propagation period t 2 a fifth pulse converts the resulting coherent superposition of the first and second vibrational state back to a population by a Stokes or anti-Stokes one-quantum transition. This nonimpulsive fifth-order Raman echo experiment involves laser pulses at three different wavelengths Ž . and has not been performed, yet. The signal is determined by the echo response function R Ž5. 5E t 1 , t 2 , given by the relaxation function of the first term in Eq. Ž21.: Ž1. R Ž5. Ž t 2 . y 2 g Ž1. Ž t 2 . q g Ž1. Ž t 1 q t 2 . 4 . 5E Ž t 2 , t 1 . s exp y2 g
Ž 22 .
This response function is identical to that of the well-known resonant photon echo sequence for stochastic frequency fluctuations w29–31,33,34x. The underlying microscopic process, however, is different: While in an ordinary photon echo experiment only two levels are involved, the system propagates in the fifth-order Raman echo during t 1 and t 2 in different coherences, <1:²0 < and <1:²2 <, respectively, as is shown in Fig. 3. Like the resonant two-pulse echo, the fifth-order Raman echo is sensitive to the processes that cause phase relaxation w9,13–17,45–47x: The phases acquired during the propagation times t 1 and t 2 have opposite sign and, therefore, phase coherence that is lost during t 1 can be recovered during t 2 , provided: Ž1. that the propagation times are shorter than the correlation time t C s Ly1 , and Ž2. that the fluctuations on the different levels are correlated. The simulated fifth-order echo response is shown in Fig. 4 for different ratios DrL. In the slow modulation limit, D 4 L, the frequency fluctuates very slowly on the time scale of the experiment. The function g Ž1. Ž t ., see Eq. Ž19., can then be approximated by g Ž1. Ž t . f 1r2 D 2 t 2 w24,28,31x, which yields a Gaussian decay of the third-order response due to the static distribution of vibrational frequencies, cf. Eq. Ž20.. The fifth-order
Fig. 3. Feynman diagrams depicting the Liouville space pathways of the fifth-order vibrational echo. It is assumed that the low-temperature limit " v r k B T 41 holds, implying that the system is initially in the ground state. In both diagrams initially a one-quantum coherence <1:²0 < and <0:²1 <, respectively, is induced that dephases during the propagation period t 1 . Then a two-quantum transition <0: ™ <2: prepares another one-quantum coherence, <1:²2 < and <2:²1 <, respectively, that propagates for a time t 2 . The phases acquired during t 1 and t 2 have opposite signs which in the inhomogeneous limit allows for rephasing. The left and right diagram apply for probing the anti-Stokes and Stokes fifth-order echo signal, respectively.
274
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
Ž . Ž . Fig. 4. Simulation of the fifth-order echo response function R Ž5. 5E t 1 , t 2 as function of the reduced times Dt 1 and Dt 2 for a D r L s100, Žb. 1.0 and Žc. 0.1. In the inhomogeneous limit an echo occurs at t 1 st 2 . In the intermediate regime the system behaves as if it was inhomogeneously broadened for times shorter than t C which for D r L s1 corresponds to reduced times Dt 1, 2 much smaller than 1. For Dt 1, 2 much larger than 1 the dynamics are rather homogeneous and the decays become exponential due to the loss of phase memory.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
275
response is in this limit given by: yŽ t 1yt 2 . R Ž5. 5E Ž t 2 , t 1 . s e
2
D 2 r2
.
Ž 23 .
The argument of the exponent becomes zero at t 1 s t 2 and the signal reaches a maximum. This shows up in the simulation of Fig. 4a as a clear echo signal along the diagonal. The decay of the signal intensity along the diagonal is determined by the finite correlation time t C s 1rL, that is contained in the exact expression Eq. Ž22.. In the opposite limit of fast modulation, D < L, the frequency fluctuates very fast on the time scale of the experiment which causes motional narrowing w24–28x. The function g Ž1. Ž t . can now be approximated by g Ž1. Ž t . f G ) t, where G ) s D 2rL denotes the decay constant of pure dephasing w24,28,31x. In this case the third-order response function decays exponentially with the decay constant G ) while the fifth-order response function reads: yG R Ž5. 5E Ž t 2 , t 1 . s e
)
Žt 1qt 2 .
.
Ž 24 .
The signal decays single exponentially along both propagation times and there is no motional echo as is shown in Fig. 4c. This is due to the fact that in the fast modulation limit the loss of phase coherence is irreversible. In the intermediate modulation case, where D f L holds, the system dynamics change from inhomogeneous like behaviour at short times to homogeneous like behaviour at longer times. The simulation of Fig. 4b exhibits an initial Gaussian decay that becomes single exponential at times longer than t C . The exact expression of Eq. Ž22. has to be used in order to describe the change of the dynamics from short to long times. To our knowledge only one nonimpulsive temporally two-dimensional fifth-order Raman experiment has been performed yet: Tominaga and Yoshihara w22,23x reported results of overtone dephasing studies of the C–D stretch mode of neat CDCl 3 liquid. In this experiment the system propagates in a one-quantum coherence for a period t 1 and in a two-quantum coherence for a propagation period t 2 . The resulting coherence state after the time t 1 q t 2 is measured by a second-order Žanti-. Stokes transition. This process is depicted schematically in the Feynman diagrams of Fig. 5. Recovery of macroscopic phase is not possible in this experiment Žunless in case of fluctuations that are anti-correlated on the two involved transitions.. Because the experiment is performed nonimpulsively, the signal does not show any oscillations. It is governed by the overtone response Ž . Ž . function, R Ž5. OV t 1 , t 2 , which is given by the relaxation function of the last term in Eq. 21 : Ž1. R Ž5. Ž t 2 . y 2 g Ž1. Ž t 1 q t 2 . q g Ž1. Ž t 1 . 4 . OV Ž t 2 , t 1 . s exp y2 g
Ž 25 .
The simulated shape of the fifth-order overtone response function along t 2 for several values of t 1 is shown in Fig. 6 for DrL s 10, 1 and 0.1. In the first case the dynamics are in the slow modulation limit where the
Fig. 5. Feynman diagrams depicting the Liouville space pathways of fifth-order overtone dephasing. The left diagram applies for probing the second-order anti-Stokes signal while the right diagram describes the second-order Stokes signal.
276
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
response can be approximated by: 2
2 R Ž5. OV Ž t 2 , t 1 . s exp y Ž t 1 q 2 t 2 . D r2 4 .
Ž 26 .
When t 1 is changed the shape of the response changes significantly: For small t 1 the decay along t 2 is Gaussian, but as t 1 is increased the decay becomes faster. This is the result of the finite correlation time
Ž . Ž . Ž . Fig. 6. Simulation of the overtone dephasing response function R Ž5. OV t 2 , t 1 as function of the reduced time Dt 2 for a D r L s10, b D r L s1.0, Žc. D r L s 0.1. The fixed values of Dt 1 are 0 Ždotted., 1.0 Žlong dashed., 2.0 Ždashed dotted., 3.0 Žshort dashed., 4.0 Ždashed double dotted. and 5.0 Žsolid.. For discussion of details consult text.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
277
t C s 1rL, that is contained in the exact expression Eq. Ž25.. When either t 1 or t 2 become large, the slow modulation limit does not perfectly hold anymore and the decays tend to become exponential. Ž . In the opposite case of fast modulation the overtone response function R Ž5. OV t 1 , t 2 is given by: ) R Ž5. Ž t 1 q 4t 2 . 4 . Ž 27 . OV Ž t 2 , t 1 . s exp yG The exponential decay along the two time coordinates does not depend on the value of the other coordinate. Along t 2 the decay is always four times faster than the decay along t 1. Note that this result relies on the assumption of perfect correlation of the fluctuations of the different vibrational levels. The decay along t 2 is also four times faster than that of the third-order CARS response w22,45x. In the intermediate modulation regime the system behaves initially as if it were in the slow modulation limit. The response is Gaussian when t 1 and t 2 are small compared to the correlation time of the fluctuations tc . As the propagation times t 1 and t 2 are increased the decay becomes faster. At times much longer than 1rL the system can be described in the fast modulation limit. The decay does not longer change; it is single exponential. In their experiments on CDCl 3 Tominaga and Yoshihara w23x found that the decay along t 2 becomes faster as t 1 is increased. The time constants were two to three times smaller than those observed in third-order CARS experiment and in the fifth-order signal along t 1. The finite correlation time of the frequency fluctuations can qualitatively account for the observed faster decay along t 2 , as t 1 is increased, but the ratio of the decay constants along the two time axes is not correctly predicted w22,23x.
4. Impulsive fifth-order Raman scattering In this section simulations of the temporally two-dimensional fifth-order Raman response are presented for impulsive excitation, where the used laser pulses are much shorter than the period of the excited vibration. This experiment has been reported by a number of groups w13–17x who investigated low-frequency intermolecular motions. Very recently, Tokmakoff et al. w20,21x used sub-30 fs laser pulses and incorporated optical heterodyning, which allowed for the observation of high-frequency intramolecular modes as well. Under these conditions a large number of Liouville space pathways, containing different dynamic information, contribute to the signal, which makes the analysis of experimental results less straightforward. When a single mode is excited, the impulsive material response is determined by the full fifth-order response function, R Ž5. Žt 1 , t 2 ., given by Eq. Ž21.. Simulations of this function for different ratios DrL are shown in Fig. 7. In the slow modulation limit, L < D, the impulsive fifth-order Raman response function can be approximated by: R Ž5. Ž t 2 , t 1 . s eyŽ t 1yt 2 .
2
D 2 r2
yt 12 D 2 r2
qe
cos v 0 Ž t 1 y t 2 . y eyŽ t 1qt 2 . yŽ t 1q2 t 2 . 2 D 2 r2
cos v 0t 1 y e
2
D 2 r2
cos v 0 Ž t 1 q t 2 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 28 .
Most of the terms of the fifth-order response exhibit an enhanced decay along t 1 and t 2 due to the Žalmost. static frequency distribution of width D. This does not hold for the first term which reaches a maximum whenever t 1 s t 2 as discussed at the beginning of the previous section. The corresponding echo signal is clearly present along the diagonal in the simulation of Fig. 7a. Also, in this figure a prominent feature is visible around t 1 s 0 that does not decay along t 2 . It is due to the third term of Eq. Ž28. that involves a one-quantum coherence during the first and a population during the second propagation period. Since the population is not affected by the frequency fluctuations discussed here, the amplitude of this term does not change as function of t 2 . As will be discussed in detail below this allows for a separation of population and phase relaxation processes. In the opposite limit of fast modulation, L 4 D, the response can be approximated by: )
R Ž5. Ž t 2 , t 1 . s eyG
Žt 1qt 2 .
qe
yG
)
t1
Ž cos v 0 Ž t 1 y t 2 . y cos v 0 Ž t 1 q t 2 . .
cos v 0t 1 y eyG
)
Žt 1q4 t 2 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 29 .
278
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
Fig. 7. Absolute square of the simulated impulsive fifth-order response R Ž5. Žt 1 , t 2 . as function of the reduced times Dt 1 and Dt 2 for Ža. mainly inhomogeneous dephasing, i.e. D 4 L, Žb. distinct non-Markovian dynamics where D f L holds, and Žc. mainly homogeneous dephasing with D < L. The period of the oscillator, Tvib s 2 p r v 0 , is normalized to one. Consult text for details.
This result was derived previously w45x for Markovian phase relaxation rates G k l of coherent superpositions < k :² l < given by G k l s Ž k y l . 2G ) . It is evident that all terms of the fifth-order response decay monotonically as t 1 is increased. Since the phases fluctuate very fast on the timescale of the experiment, rephasing is not possible. As a consequence the simulated response shown in Fig. 7c does not show any echo.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
279
In the intermediate modulation regime, where L f D holds, the approximations of Eqs. Ž28. and Ž29. break down and the exact expression for R Ž5. Žt 1 , t 2 ., given in Eq. Ž21., has to be used. As shown in Fig. 7b, the system behaves as if it was inhomogeneously broadened when times t 1 and t 2 are small: There is a buildup of a feature at the diagonal t 1 s t 2 that decays rapidly due to the finite correlation time. This demonstrates that at times larger than t C the response approaches that of the homogeneous limit.
5. Population relaxation Up to now, we have considered only pure dephasing processes, but in general there are also decay channels due to population relaxation. For high-frequency intramolecular modes the population decay is often much slower than the phase relaxation, so that in third-order Raman experiments the former process can be neglected. This is probably not true for low-frequency intermolecular vibrations that are heavily damped. In general, these two relaxation processes cannot be discriminated by temporally one-dimensional techniques as depicted in Fig. 1a. In impulsive fifth-order Raman experiments the population relaxation cannot be neglected even for high-frequency intramolecular modes, since some of the contributing Liouville space pathways involve the propagation of a population < k :² k < as was mentioned in the previous section. It is then possible to separate population and phase relaxation due to the temporally two-dimensional character of this technique. In order to illustrate this new feature of the fifth-order Raman technique, we apply a model often used for the description of population relaxation in vibrational spectroscopy w45,48–50x. When the optically active vibration is linearly coupled to a harmonic heat bath, this induces in the limit of weak system–bath interaction only one-quantum transitions < k : ™ < k " 1:. It can be shown that the decay rate for the downward transition, g k ™ ky1 , then scales linearly with the quantum number k:
g k ™ ky1 s kg ,
Ž 30 .
where g ' g 1 ™ 0 is one half of the population decay rate from the first excited state to the ground state w45,48–50x. At finite temperature, there are next to downward also upward transitions. The upward transition rates g k ™ kq1 can be obtained from a detailed balancing condition. In thermal equilibrium the population n k of level k does not change, i.e.: d nk dt
s y Ž g k ™ ky1 q g k ™ kq1 . n k q g kq1 ™ k n kq1 q g ky1 ™ k n ky1 s 0 .
Ž 31 .
Starting from the ground state Ž k s 0. one has to consider only transitions from and to the first excited state which directly yields g 0 ™ 1 s g 1 ™ 0 n1rn 0 s g 1 ™ 0 h , where h ' expŽy" v 0rk B T . denotes the Boltzmann factor. Using this result the other rates g k ™ kq1 can be obtained successively:
g k ™ kq1 s Ž k q 1 . gh .
Ž 32 .
The total rate out of level k is now given by the sum of the upward and the downward transition: g k s g k ™ ky1 q g k ™ kq1. In Ref. w45x this decay of the population was incorporated by adding an imaginary part ig kr2 to the frequency v k , given in Eq. Ž11.. The resulting complex frequencies impose no major difficulty in deriving the third- and fifth-order response in the same way as described in Section 2. It should be noted, however, that in that approach the total number of oscillators, i.e. the sum of the diagonal elements < k :² k < is not conserved, since the feeding of the lower Žhigher. states due to the downward Župward. transitions is neglected. This approximation is valid in third order, where the system always propagates in coherent states < k :² k " 1 < w50x, but not in fifth order for those Liouville space pathways that comprise a diagonal state. In some of the Liouville space pathways a population is propagated during the second propagation time t 2 . When there is no level-dependent population relaxation, these paths yield the cos v 0t 1 term of Eqs. Ž17., Ž21.,
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
280
Ž28. and Ž29.. One of the corresponding Feynman diagrams is shown in Fig. 8a. Due to the population transfer described by Eq. Ž31. the amplitude of this diagram decreases, while additional diagrams occur that capture the system dynamics and the optical signal of the feeded states. In Fig. 8b–c two of these relaxation-induced diagrams are depicted that result from one-quantum upward and downward transitions, respectively. They are only the first step towards thermal equilibrium where the population of all levels is given by the Boltzmann distribution. For a self-consistent description of the redistribution of the population, it is necessary to solve the set of coupled linear differential equations, see Eq. Ž31., for all diagrams with the appropriate initial condition. In order to avoid the evaluation of an infinite set of coupled equations one has to introduce an effective upper level from where there are no upward transitions. The accuracy of this approximation is determined by the temperature and the chosen upper level. Here we consider the low-temperature limit, i.e. h < 1, where almost all population is initially in the ground state and thermally induced upwards transitions can be neglected. In this limit three different Liouville pathways contribute to the cos v 0t 1 term: The propagation of the ground-state population, that in the low-temperature limit does not decay along t 2 , is captured by diagrams Žb.1. and Žb.4. of Fig. 1 with k s 0, l s 1 and m s 0. The dynamics involving the diagonal state <1:²1 < are depicted in Fig. 1Žb.2. and Žb.3. for k s 0, l s 1 and m s 1. These pathways decay as function of t 2 with a rate constant g while the amplitude of the relaxation induced diagrams Žsee, e.g., Fig. 8c. rises with the same rate constant. When all these additional pathways are taken into account, the plateau that was found in Fig. 7 along t 2 for small t 1 , decays. The full fifth-order Raman response, that at low temperatures involves only the lowest three vibrational states, is now given by: g
R Ž5. Ž t 2 , t 1 . s ey 2 Žt 1q3t 2 . ey2 g
Ž1.
Žt 1 .y2 g Ž1.Žt 2 .qg Ž1.Žt 1 q t 2 .
cos v 0 Ž t 1 y t 2 .
g
y ey 2 Žt 1q t 2 . eyg
Ž1.
Žt 1 q t 2 .
cos v 0 Ž t 1 q t 2 .
g
q ey 2 Žt 1q2 t 2 . eyg
Ž1.
Žt 1 .
cos v 0t 1
g
y ey 2 Žt 1q2 t 2 . ey2 g
Ž1.
Žt 2 .y2 g Ž1.Žt 1 q t 2 .qg Ž1.Žt 1 .
cos v 0 Ž t 1 q 2t 2 .
g
q Ž 1 y ey gt 2 . ey 2 Ž2 t 1q t 2 . ey2 g
Ž1.
Žt 1 .qg Ž1.Žt 2 .y2 g Ž1.Žt 1 q t 2 .
cos v 0 Ž 2t 1 q t 2 . .
Ž 33 .
The first two terms describe Liouville space pathways where the system evolves in one-quantum coherences
Fig. 8. Additional Liouville space pathways due to population relaxation. The amplitude of the double-sided Feynman diagram Ža. decays as function of the propagation time t 2 due to thermally induced upward and downward transitions. The population transferred from the diagonal state < k q1:² k q1 < to < k q2:² k q2 < and < k :² k < gives rise to new, relaxation induced diagrams Žb. and Žc.. Note that the amplitude of these diagrams is determined by the kinetic equations for the populations n k ŽEq. Ž31.. and the matrix elements a k k .
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
281
Fig. 9. Simulation of the impulsive fifth-order Raman response as function of the reduced time t 2 r Tvib with t 1 s 0 ŽEq. Ž34... Here Tvib s 2 p r v 0 denotes the oscillation period. In all traces fast frequency modulation was assumed Ž D < L, D 2 r L f G ) . and the total linewidth of the third-order response G ) qg r2 was kept constant. Ža. In the absence of pure dephasing Ž g Ž1. Žt . s 0. the signal shows oscillations with period Tvib that decay two times faster than the third-order response. Žb. When pure dephasing and population relaxation are comparable Žshown for g r2 s G ) ., the signal is initially determined by oscillations due to the last term of Eq. Ž34.. At longer delays it decays exponentially with a rate constant g . Žc. When only pure dephasing occurs Žg s 0., the term proportional to cos v 0t 1 in Eq. Ž34. does not decay along t 2 yielding a plateau.
during t 1 and t 2 , while the third and fourth term correspond to a one-quantum coherence, followed by a population and a two-quantum coherence, respectively. The first and fourth term were already discussed in Section 3, in connection with the non-impulsive fifth-order Raman echo and overtone response functions. The fifth term, that comprises a two-quantum coherence during the first and a one-quantum coherence during the second propagation period, disappears for level-independent population relaxation due to perfect interference of the relevant Liouville space pathways w45x. It was therefore not present in Eqs. Ž17., Ž21., Ž28. and Ž29. where population relaxation was ignored.
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Because the five terms decay in different ways, it is now possible to separate population and phase relaxation processes. This is particularly evident for the fifth-order response along t 2 with t 1 s 0 1, where we find: R Ž5. Ž t 2 , t 1 s 0 . s eyg t 2 Ž 1 y ey4 g
Ž1.
Žt 2 .
cos 2 v 0t 2 . .
Ž 34 .
In order to demonstrate the sensitivity of the fifth-order response to the population relaxation, we have calculated the signal in the fast fluctuation limit, i.e. D < L, for different values of g while keeping the third-order line width gr2 q G ) constant. When there is no pure dephasing, the signal shows oscillations that fade out with the population relaxation rate g as is shown in Fig. 9a. In the other extreme case of only pure dephasing but no population relaxation, the initial oscillations, that originate from the term proportional to cos 2 v 0t 2 , are rapidly damped, while on a longer time scale there is a nondecaying plateau due to the cos v 0t 1 term, see Fig. 9c. When both pure dephasing and population relaxation processes are equally important, the signal of Fig. 9b is predicted: The initial oscillations are damped rapidly due to both mechanisms while on a longer time scale the decay is single exponential with the population decay rate g . This allows for an independent determination of the population relaxation time that cannot be extracted from the third-order Raman response.
6. Conclusions The nonresonant third- and fifth-order Raman response was calculated for pure dephasing due to random frequency fluctuations. The nuclear vibrations were modelled by a harmonic oscillator that is coupled to the radiation field via a polarizability with a nonlinear coordinate dependence. For fluctuations that are described by a stationary Gaussian process, the response functions can be solved exactly using the cumulant expansion technique. The obtained expressions for the response functions depend explicitly on the correlation time t C s 1rL of the frequency fluctuations and on the root mean square D of the average distribution. By changing the ratio DrL the results interpolate continuously from homogeneous dephasing Ž DrL < 1. via non-Markovian dynamics Ž DrL f 1. to inhomogeneous dephasing Ž DrL 4 1.. The fifth-order Raman response was discussed for two non-impulsive experiments. In temporally two-dimensional overtone dephasing w22,23x the phase relaxation of one- and two-quantum coherences is probed selectively. It was shown that the decays are Gaussian at early times. If at least one of the two propagation times is long compared to t C , the decays become faster and approach an exponential. The second type of non-impulsive fifth-order Raman experiment, which has not been performed, yet, selectively probes an echo-type of process carrying information on the time scaleŽs. of the frequency fluctuations. Model calculations were also performed for the impulsive fifth-order Raman response. In the case of dominant inhomogeneous broadening a motional echo was found at t 1 s t 2 while in the homogeneous limit the dephasing is irreversible and there is no echo. When the correlation time t C of the fluctuations is finite, the dynamics are again mainly inhomogeneous at times much shorter than t C and mainly homogeneous at times much longer than t C . At the intermediate times the resulting non-Markovian response cannot be approximated by one of these limits. The effect of Markovian level-dependent population relaxation was included in the treatment to demonstrate that the impulsive fifth-order Raman response allows in principle to separate population and phase relaxation
1
At t 1 s 0 hyperpolarizability contributions can contribute to the fifth-order response w45x. The resulting complications in the signal analysis can be avoided by either choosing appropriate polarization conditions w19x or setting the first delay time T1 equal to the pulse duration t P . In the latter case the expression of Eq. Ž34. is still a good approximation since in impulsive experiments the pulse duration is much smaller than the period of the excited mode.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
283
processes. This is impossible for third-order Raman experiments. Experimentally, this feature, that to our knowledge has not been discussed before, is best observed along t 2 at t 1 s 0.
Acknowledgements The investigations were supported by the Netherlands Foundations for Chemical Research ŽSON. and Physical Research ŽFOM. with financial aid from the Netherlands Organization for the Advancement of Science ŽNWO..
Appendix A. Stationary Gaussian frequency fluctuations in the fifth-order response The relaxation functions in the fifth-order Raman response due to the frequency fluctuations are all of the form, cf. Eq. Ž17.: t1
¦½
H0
¦½
H0
f 1Ž2. Ž t 2 , t 1 . s exp yi
dt Ž k yl . dv Ž t . q
t 2q t 1
Ht
d t Ž k y m. d v Ž t .
1
5;
,
Ž A1.
5;
.
Ž A2.
and:
f 2Ž2. Ž t 2 , t 1 . s exp yi
t1
dt Ž k yl . dv Ž t . q
t 2q t 1
Ht
dt Ž myl . dv Ž t .
1
Since for a stationary Gaussian process all cumulants beyond the second vanish w24–28x, Eq. Ž33. can be written as:
f 1Ž2.
Žt2 , t1.
1
t1
½ ¦H
s exp y
½
' exp y
2
1 2
dt Ž kyl . dv Ž t . q
0
g 1Ž2.
2
t 2q t 1
Ht
d t Ž k y m. d v Ž t .
1
;5
Ž A3.
5
Ž t 2 ,t 1 . .
The two-time line shape function g 1Ž2. Žt 2 , t 1 . can now be decomposed into a sum of the well-known one-time line shape functions g Ž1. Žt 1 ., given by Eq. Ž19.. From the definition of g 1Ž2. Žt 2 , t 1 . in the last equation we find: g 1Ž2. Ž t 2 , t 1 .
s Ž kyl.
2
t1
H0
d t1
t1
H0
d t 2 ² d v Ž t1 . d v Ž t 2 . : q Ž k y m .
t 2q t 1
Ht
d t 2 ² d v Ž t1 . d v Ž t 2 . : q 2 Ž k y l . Ž k y m .
1
2
t1
H0
2
t 2q t 1
Ht
d t1
d t1
1
t 2q t 1
Ht
d t 2 ² d v Ž t1 . d v Ž t 2 . :
1
2
' Ž k y l . A q Ž k y m. B q 2Ž k y m. Ž k y l . C .
Ž A4. The symbols A, B and C denote the double integrals in the first, second and third line of Eq. ŽA2., respectively. In Fig. 10 it is shown that 2 C s D y A y B holds where D denotes the double integral with both integrations running from 0 to t 2 q t 1. Using this equality we can rewrite g 1Ž2. Žt 2 , t 1 . as: g 1Ž2. Ž t 2 , t 1 . s Ž k y l . Ž m y l . A q Ž k y l . Ž l y m . B q Ž k y m . Ž k y l . D ,
Ž A5.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
284
X Fig. 10. Definition of the surface areas A, B, C, C and D of the two-dimensional integrals of Eqs. Ž17. and ŽA2.. For a stationary X Gaussian process one finds C s C and 2 C s D-A-B.
where all of the double integrals A, B and D can be expressed in terms of the one-time line shape function g Ž1. Žt ., defined in Eq. Ž18.. The final result therefore is: g 1Ž2. Ž t 2 , t 1 . s Ž k y l . Ž m y l . g Ž1. Ž t 1 . q Ž k y l . Ž l y m . g Ž1. Ž t 2 . q Ž k y m . Ž k y l . g Ž1. Ž t 1 q t 2 . .
Ž 40 . This procedure can also be followed f 2Ž2. Žt 2 , t 1 ., yielding finally Eq. Ž21..
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Population relaxation and non-Markovian frequency fluctuations in third- and fifth-order Raman scattering Thomas Steffen, Koos Duppen
)
Ultrafast Laser and Spectroscopy Laboratory, Materials Science Center, UniÕersity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Received 22 October 1997
Abstract The third- and fifth-order Raman response is calculated for vibrational phase relaxation due to stochastic Gaussian frequency fluctuations. This model can be solved exactly using the cumulant expansion technique. It interpolates continuously between the homogeneous and inhomogeneous limit and allows for a description of non-Markovian dynamics arising from finite correlation timeŽs. of the frequency fluctuations. Model calculations demonstrate that temporally two-dimensional fifth-order Raman experiments are sensitive to the time scaleŽs. of these frequency fluctuations and also to the life times of the vibrational states. These processes, that cannot be distinguished in nonresonant third-order Raman scattering, can be unequivocally identified in fifth-order experiments. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The importance of vibrational excitations in chemical reactions has been recognized for many years w1x. Coupling of the molecular vibrations with a dynamical environment is responsible for the energy flow into and out of molecules, and for thermally activated processes w2x. From a fundamental point of view on chemistry, it is therefore essential to obtain direct experimental information on both the intramolecular vibrational dynamics and on the external degrees of freedom that act as a heat bath for the reacting system. Vibrational lineshape analysis is a very popular and powerful tool to investigate the local structure and dynamics in condensed phase systems w1,3x. The variation of pressure, temperature or solvent leads in general to changes in both the central frequency and the linewidth of a vibrational transition. These modifications reflect
)
Corresponding author.
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 0 8 3 - 4
268
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
static and dynamic properties of the microscopic potential. A variety of models has been proposed to describe different kinds of intra- and intermolecular interactions that lead to vibrational population and phase relaxation w1,3–7x. These models can be tested by evaluating their capabilities to explain the observed infrared ŽIR. and Raman spectra. The conventional spectroscopic techniques have, however, a limited information content since they provide only indirect evidence for the mechanism and the time scaleŽs. of the line broadening w8,9x. In order to obtain additional information, Mukamel and co-workers w8,9x proposed to probe higher-order correlation functions by multiple pulse experiments. Examples for these techniques are the Raman echo and Raman pump–probe experiments, which are seventh-order nonlinear optical processes w10–12x, and temporally two-dimensional fifth-order Raman scattering. This latter technique has been performed impulsively on low-frequency inter-w13– 19x and intramolecular w20,21x modes, and nonimpulsively on high-frequency intramolecular modes w22,23x. By using these more advanced techniques, it is in principle possible to characterize the pure dephasing and to separate population and phase relaxation processes. The loss of macroscopic phase coherence is often modelled in the limits of fast or slow frequency modulation which give rise to homogeneous and inhomogeneous dephasing, respectively w1x. When the correlation time of the frequency fluctuations is comparable to the typical time scale of the experiment, these approximations break down and stochastic models have to be evoked w24–28x. The resulting non-Markovian dynamics were first discussed for spin-resonance experiments w24–27x, but electronic coherences in condensed phase systems have been investigated as well w29–34x. The Kubo–Anderson formalism was evoked to model vibrational phase relaxation in IR w35x and third-order Raman w36x spectra. Evidence for a finite correlation time of the frequency fluctuations in molecular liquids were obtained by time-resolved CARS w37–39x, impulsive stimulated scattering w40x and Raman line shape analysis w41–43x. The theory of electronically nonresonant nonimpulsive seventh-order experiments such as the Raman echo is well established since it closely resembles the resonant third-order response of a two-level system which is discussed in standard text books on nonlinear optical spectroscopy Že.g., w44x.. The fifth-order Raman techniques, which have been proposed w9x and demonstrated w13–23x recently, are related to the resonant second-order response which necessarily involves three levels w16x. Therefore, the quantum number dependence of the relaxation processes has to be considered explicitly in modelling these experiments w22,23,45x. The fifth-order Raman response can be observed only in the case of coupling constants with a nonlinear coordinate dependence w9x andror anharmonic potentials w46x. Up to now only a few models for population and phase relaxation in fifth-order experiments have been discussed. In their key paper Tanimura and Mukamel w9x employed a Brownian oscillator model, where the optically active oscillator is linearly coupled to a continuous distribution of optically dark bath oscillators. This model allows for a continuous change from underdamped to overdamped vibrations and yields in the limit of a white bath spectrum a level-independent population relaxation as the only decay process w45x. Khidekel and Mukamel w47x introduced a static Lorentzian distribution of oscillator frequencies as line broadening mechanism and demonstrated that this leads to a motional echo. Tominaga and Yoshihara w22,23x discussed the effects of partial correlation between the fluctuations of different vibrational levels. In our previous paper w45x we derived the third- and fifth-order response within the weak coupling limit for quantum-number-dependent population relaxation and pure dephasing rates, resulting from linear and quadratic coupling to a harmonic heat bath. In this paper we calculate the third- and fifth-order Raman response for random Gaussian frequency fluctuations that cause phase relaxation. This model, that can be solved exactly using the cumulant expansion technique w24–28x, interpolates continuously between homogeneous and inhomogeneous dephasing. The effect of a finite correlation time, which gives rise to non-Markovian dynamics, is discussed for fifth-order vibrational echoes, overtone dephasing, and impulsive excitation. Simulations of the fifth-order Raman echo show that this experiment is particularly sensitive to the time scale of the fluctuations. The predicted impulsive 2D Raman response illustrates that this experiment allows one to discriminate between pure dephasing and population relaxation processes, which is impossible in lower-order nonresonant experiments.
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269
2. Theory The third- and fifth-order Raman response functions have been calculated by several authors using time domain perturbation theory w9,45–47x. When processes involving hyperpolarizabilities can be neglected, the third- and fifth-order Raman response are given by w9,44,45x: R Ž3. Ž t 1 . s
i "
a Ž t 1 . , a Ž 0 . r Ž y` . :'
²
i "
J1 Ž t 1 . y J1) Ž t 1 . ,
Ž 1.
and: R Ž5. Ž t 2 , t 1 . s '
i
2
ž /² Ž ž /Ý
a t 2 q t 1 . , a Ž t 1 . , a Ž 0 . r Ž y` . :
" i
"
2
2
Qn Ž t 2 , t 1 . q Qn) Ž t 2 , t 1 . .
Ž 2.
ns1
Here, r Žy`. denotes the density matrix of the system in thermal equilibrium and a Žt . s U0y1 Žt . a U0 Žt . is the interaction picture representation of the polarizability. For a Hamiltonian H0 Ž t . that is explicitly time dependent, U0 Žt . is given by:
½
U0 Ž t . s exp y
i
t
5
H dt H Ž t. " 0 0
.
Ž 3.
The functions J1Žt 1 . and Qns1, 2 Žt 2 , t 1 . are easily obtained by writing out the commutators of Eqs. Ž1. and Ž2.. When these functions are developed in terms of the nuclear eigenstates < k : of the system, this yields: J1 Ž t 1 .
s ² a Ž t 1 . a Ž 0 . r Ž y` . : s Ý Pk² k
Q1 Ž t 2 , t 1 .
½
i
t1
5
½
d t H0 Ž t . a Ž q . exp y
H " 0
i
t1
5
s ² a Ž t 2 q t 1 . a Ž t 1 . a Ž 0 . r Ž y` . : i t 2q t 1 i s Ý Pk² k
½
=² m
½
i
5 ½
H
t1
5
d t H0 Ž t . a Ž q . exp y
H " 0
½
i
Ž 4.
d t H0 Ž t . < l :² l < a Ž q . < k : ,
H " 0
t 2q t 1
t1
5
d t H0 Ž t . < m:
H0
Ž 5.
5
d t H0 Ž t . < l :² l < a Ž q . < k : ,
H " 0
and: Q2 Ž t 2 , t 1 .
s y² a Ž t 1 . a Ž t 2 q t 1 . a Ž 0 . r Ž y` . : i t1 i s y Ý Pk² k
½
=² m
½
i
t 2q t 1
H " 0
5
H
5
½
½
d t H0 Ž t . a Ž q . exp y
i
t 2q t 1
H " 0
t1
H0
5
d t H0 Ž t . < m:
Ž 6.
5
dt H0 Ž t . < l :² l < a Ž q . < k : .
Here, the factor Pk denotes the occupation of state < k : in thermal equilibrium. The different terms on the right-hand side of Eqs. Ž1. and Ž2. represent the two and four Liouville space pathways that contribute to the third- and fifth-order Raman response, respectively. These pathways can be
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
270
depicted in the form of double-sided Feynman diagrams, which are shown in Fig. 1. Pure dephasing is now introduced by assuming that the energy difference Ek y El s " v k l between two states < k : and < l : undergoes random fluctuations due to coupling to a large heat bath. When the system states follow the heat bath adiabatically, the transition frequencies v k l Ž t . can be decomposed into their average values v k l and a fluctuating part d v k l Ž t .:
vkl Ž t . s vkl q dvkl Ž t . .
Ž 7.
Defining the transition matrix element a k l ' ² k < a < l :, Eqs. Ž4. – Ž6. can then be recast as:
k, l
Q1 Ž t 2 , t 1 . s
t1
¦ ½H
J1 Ž t 1 . s Ý Pk a k l a l k e i v k lt 1 exp i
Ý k, l, m
0
dt dvkl Ž t .
5;
,
Ž 8. t1
¦ ½H
Pk a k m a m l a l k e iŽ v k lt 1q v k mt 2 . exp i
t 2q t 1
dt dvkl Ž t . q i
0
dt dvkmŽ t .
Ht
1
5;
,
Ž 9.
5;
.
Ž 10 .
and: Q2 Ž t 2 , t 1 . s
Ý k, l, m
t1
¦ ½H
Pk a k m a m l a l k e iŽ v k lt 1q v m lt 2 . exp i
0
dt dvkl Ž t . q i
t 2q t 1
Ht
d t d vm l Ž t .
1
At his stage we have to specify the system Hamiltonian, which determines the eigenstates < k : and the transition frequencies v k l , the nature of the fluctuations d v k l Ž t ., and the coordinate dependence of the polarizability a Ž q . that is necessary to calculate the transition matrix elements a k l .
Fig. 1. Feynman diagrams depicting the Liouville space pathways of the third- and fifth-order Raman response. Ža. The third-order response, given by Eq. Ž1., can be captured by the two diagrams Ža.1. and Ža.2. that are the complex conjugate of each other. Žb. The fifth-order response, given by Eq. Ž2., involves four different diagrams. Diagrams Žb.1. and Žb.4. depict Q1Žt 2 , t 1 . and Q1) Žt 2 , t 1 ., while Žb.2. and Žb.3. represent Q 2) Žt 2 , t 1 . and Q2 Žt 2 , t 1 ., respectively.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
271
Fig. 2. Energy level diagram of a harmonic oscillator that undergoes random frequency fluctuations. For the description of pure dephasing only the energy difference between the involved states is important. For perfectly correlated fluctuations the spread of energies between states < k : and < k q2: is twice as large as that between < k : and < k q1:, and < k q1: and < k q2:.
Here, we consider a harmonic vibration with Ek s Ž k q 1r2. " v 0 so that the average transition frequencies are:
vkl s Ž k y l . v0 .
Ž 11 .
We further assume that the system is coupled to the radiation field via a coordinate-dependent polarizability:
a Ž q . s a1 q q a2 q 2 ,
Ž 12 .
where q denotes the vibrational coordinate of the optically active oscillator. The relevant Liouville space pathways, i.e. the nonvanishing combinations of k, l and m in the sums of Eqs. Ž8. – Ž10., and their relative amplitudes, determined by the matrix elements a k l , were calculated explicitly in Ref. w45x. Several statistical models have been discussed in literature in order to describe the fluctuation of the energy levels of the system w24–28x. Here, we assume that the modulations d v k l Ž t . of the vibrational transition frequencies are perfectly correlated and can be modelled as a stationary Gaussian processes, obeying w24–28x: dvkl Ž t . s Ž k y l . dv Ž t . ,
Ž 13 .
²d v Ž t . : s 0 ,
Ž 14 .
² d v Ž t 1 . d v Ž t 2 . : s ² d v Ž 0 . d v Ž t 2 y t 1 . : s D 2 exp yL < t 2 y t 1 < 4 ,
Ž 15 .
and:
where D and t C s Ly1 denote the root mean square and the correlation time of the frequency fluctuations. As a consequence of the perfect correlations the amplitude of the fluctuations of a two-quantum coherence such as < k :² k q 2 < is twice as large as that of a one-quantum coherence such as < k :² k q 1 < or < k q 1:² k q 2 <. This is shown schematically in Fig. 2. Within these assumption of harmonic nuclear motion ŽEq. Ž11.., non-Condon coordinate dependence of the polarizability ŽEq. Ž12.., and perfect correlation of the fluctuations ŽEq. Ž13.., the expressions for J1Žt 1 . and Qns1, 2 Žt 2 , t 1 . in Eqs. Ž8. – Ž10. can be further simplified. Using these results, the third- and fifth-order response functions Eqs. Ž1. and Ž2. are, in lowest order of the vibrational coordinate:
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
272
¦ ½
t1
R Ž3. Ž t 1 . s exp yi
dt dv Ž t .
H0
5;
sin v 0t 1 ,
Ž 16 .
and t1
¦½ H ¦½ H
R Ž5. Ž t 2 , t 1 . s exp yi
t 1q t 2
dt dv Ž t . qi
Ht
0
y exp yi
t1
¦ ½ ¦½
Ht
t1
H0
y exp yi
dt dv Ž t .
dt dv Ž t .
1
cos v 0 Ž t 1 y t 2 .
5;
cos v 0 Ž t 1 q t 2 .
5;
cos v 0t 1
t1
H0
5;
t 1q t 2
dt dv Ž t . yi
0
q exp yi
dt dv Ž t .
1
t 1q t 2
dt dv Ž t . yi
Ht
d t 2d v Ž t .
1
5;
cos v 0 Ž t 1 q 2t 2 . .
Ž 17 .
Here and in the remainder of the paper we have set the constant prefactors a 12rm v 0 in Eq. Ž16. and a 12a 2r2Ž m v 0 . 2 in Eq. Ž17. equal to one. Note that the fifth-order response function Eq. Ž17. does not comprise the term proportional to cos v 0 Ž2t 1 q t 2 ., that was found in Ref. w45x for level-dependent population relaxation. This term is absent here due to the destructive interference of different Liouville space pathways w45x. For a stationary Gaussian process the averages over the random fluctuations can be calculated exactly for all involved Liouville space pathways, since then all cumulants beyond the second order vanish w24–28x. For a single propagation time like in Eq. Ž16. one finds for the relaxation function f Ž1. Žt 1 . w24–28x: t1 t1 1 t1 f Ž1. Ž t 1 . ' exp yi d t d v Ž t . s exp y d t1 d t 2² d v Ž t 1 . d v Ž t 2 . : ' exp yg Ž1. Ž t 1 . 4 , 2 0 0 0 Ž 18 .
¦ ½
5; ½
H
H
H
5
where the line shape function g Ž1. Žt . is given by w24–28x:
D2
Ž ey Lt q Lt y 1 . . Ž 19 . L2 The relaxation functions appearing in the expression for R Ž5. Žt 2 , t 1 ., cf. Eq. Ž17., can be expressed in terms of g Ž1. Ž t . as well, as is shown in Appendix A. Using the results of Eqs. Ž18. and ŽA3. the third-order response function can be rewritten as w38,39x: g
Ž1.
Žt . s
R Ž3. Ž t 1 . s eyg
Ž1.
Žt 1 .
sin v 0t 1 ,
Ž 20 .
while the fifth-order signal is given by: R Ž5. Ž t 2 , t 1 . s ey2 g
Ž1.
y eyg
Žt 1 .y2 g Ž1.Žt 2 .qg Ž1.Žt 1qt 2 . Ž1.
Žt 1qt 2 .
yg Ž1.Žt 1 .
qe
y ey2 g
Ž1.
cos v 0 Ž t 1 y t 2 .
cos v 0 Ž t 1 q t 2 .
cos v 0t 1
Žt 2 .y2 g Ž1.Žt 1qt 2 .qg Ž1.Žt 1 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 21 .
These formulas describe the effect of Gaussian frequency fluctuations with an arbitrary correlation time t C on the optical response of a harmonic mode. In the next two sections we simulate the temporally two-dimensional fifth-order Raman signal for different ratios DrL and for different excitation conditions.
3. Nonimpulsive fifth-order Raman scattering
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
273
The fifth-order Raman response, given in Eq. Ž21., consists of four different terms, that carry different dynamic information. In impulsive experiments they all contribute simultaneously to the signal. This situation will be discussed in the next section. In this section we will first treat two non-impulsive fifth-order Raman experiments, in which the employed laser pulses are much longer than the vibrational period of the investigated mode. As a consequence the signal does not show any oscillations at frequency v 0 , but it is given by the relaxation function only, i.e. the appropriate prefactors of Eq. Ž21.. Using sequences of laser pulses with suitable frequency differences, it is possible to select only one of the Liouville space pathways depicted in Fig. 1. The two possible sequences of system states that give rise to fifth-order Raman echoes, are depicted in Fig. 3 in the form of double-sided Feynman diagrams. The experiment is initiated by two coincident laser pulses, which via stimulated Raman scattering induce a one-quantum transition. The resulting coherent superposition of the ground and the first vibrational state propagates for a period t 1 before a second pair of laser pulses induces a two-quantum transition from the ground state. After a propagation period t 2 a fifth pulse converts the resulting coherent superposition of the first and second vibrational state back to a population by a Stokes or anti-Stokes one-quantum transition. This nonimpulsive fifth-order Raman echo experiment involves laser pulses at three different wavelengths Ž . and has not been performed, yet. The signal is determined by the echo response function R Ž5. 5E t 1 , t 2 , given by the relaxation function of the first term in Eq. Ž21.: Ž1. R Ž5. Ž t 2 . y 2 g Ž1. Ž t 2 . q g Ž1. Ž t 1 q t 2 . 4 . 5E Ž t 2 , t 1 . s exp y2 g
Ž 22 .
This response function is identical to that of the well-known resonant photon echo sequence for stochastic frequency fluctuations w29–31,33,34x. The underlying microscopic process, however, is different: While in an ordinary photon echo experiment only two levels are involved, the system propagates in the fifth-order Raman echo during t 1 and t 2 in different coherences, <1:²0 < and <1:²2 <, respectively, as is shown in Fig. 3. Like the resonant two-pulse echo, the fifth-order Raman echo is sensitive to the processes that cause phase relaxation w9,13–17,45–47x: The phases acquired during the propagation times t 1 and t 2 have opposite sign and, therefore, phase coherence that is lost during t 1 can be recovered during t 2 , provided: Ž1. that the propagation times are shorter than the correlation time t C s Ly1 , and Ž2. that the fluctuations on the different levels are correlated. The simulated fifth-order echo response is shown in Fig. 4 for different ratios DrL. In the slow modulation limit, D 4 L, the frequency fluctuates very slowly on the time scale of the experiment. The function g Ž1. Ž t ., see Eq. Ž19., can then be approximated by g Ž1. Ž t . f 1r2 D 2 t 2 w24,28,31x, which yields a Gaussian decay of the third-order response due to the static distribution of vibrational frequencies, cf. Eq. Ž20.. The fifth-order
Fig. 3. Feynman diagrams depicting the Liouville space pathways of the fifth-order vibrational echo. It is assumed that the low-temperature limit " v r k B T 41 holds, implying that the system is initially in the ground state. In both diagrams initially a one-quantum coherence <1:²0 < and <0:²1 <, respectively, is induced that dephases during the propagation period t 1 . Then a two-quantum transition <0: ™ <2: prepares another one-quantum coherence, <1:²2 < and <2:²1 <, respectively, that propagates for a time t 2 . The phases acquired during t 1 and t 2 have opposite signs which in the inhomogeneous limit allows for rephasing. The left and right diagram apply for probing the anti-Stokes and Stokes fifth-order echo signal, respectively.
274
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Ž . Ž . Fig. 4. Simulation of the fifth-order echo response function R Ž5. 5E t 1 , t 2 as function of the reduced times Dt 1 and Dt 2 for a D r L s100, Žb. 1.0 and Žc. 0.1. In the inhomogeneous limit an echo occurs at t 1 st 2 . In the intermediate regime the system behaves as if it was inhomogeneously broadened for times shorter than t C which for D r L s1 corresponds to reduced times Dt 1, 2 much smaller than 1. For Dt 1, 2 much larger than 1 the dynamics are rather homogeneous and the decays become exponential due to the loss of phase memory.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
275
response is in this limit given by: yŽ t 1yt 2 . R Ž5. 5E Ž t 2 , t 1 . s e
2
D 2 r2
.
Ž 23 .
The argument of the exponent becomes zero at t 1 s t 2 and the signal reaches a maximum. This shows up in the simulation of Fig. 4a as a clear echo signal along the diagonal. The decay of the signal intensity along the diagonal is determined by the finite correlation time t C s 1rL, that is contained in the exact expression Eq. Ž22.. In the opposite limit of fast modulation, D < L, the frequency fluctuates very fast on the time scale of the experiment which causes motional narrowing w24–28x. The function g Ž1. Ž t . can now be approximated by g Ž1. Ž t . f G ) t, where G ) s D 2rL denotes the decay constant of pure dephasing w24,28,31x. In this case the third-order response function decays exponentially with the decay constant G ) while the fifth-order response function reads: yG R Ž5. 5E Ž t 2 , t 1 . s e
)
Žt 1qt 2 .
.
Ž 24 .
The signal decays single exponentially along both propagation times and there is no motional echo as is shown in Fig. 4c. This is due to the fact that in the fast modulation limit the loss of phase coherence is irreversible. In the intermediate modulation case, where D f L holds, the system dynamics change from inhomogeneous like behaviour at short times to homogeneous like behaviour at longer times. The simulation of Fig. 4b exhibits an initial Gaussian decay that becomes single exponential at times longer than t C . The exact expression of Eq. Ž22. has to be used in order to describe the change of the dynamics from short to long times. To our knowledge only one nonimpulsive temporally two-dimensional fifth-order Raman experiment has been performed yet: Tominaga and Yoshihara w22,23x reported results of overtone dephasing studies of the C–D stretch mode of neat CDCl 3 liquid. In this experiment the system propagates in a one-quantum coherence for a period t 1 and in a two-quantum coherence for a propagation period t 2 . The resulting coherence state after the time t 1 q t 2 is measured by a second-order Žanti-. Stokes transition. This process is depicted schematically in the Feynman diagrams of Fig. 5. Recovery of macroscopic phase is not possible in this experiment Žunless in case of fluctuations that are anti-correlated on the two involved transitions.. Because the experiment is performed nonimpulsively, the signal does not show any oscillations. It is governed by the overtone response Ž . Ž . function, R Ž5. OV t 1 , t 2 , which is given by the relaxation function of the last term in Eq. 21 : Ž1. R Ž5. Ž t 2 . y 2 g Ž1. Ž t 1 q t 2 . q g Ž1. Ž t 1 . 4 . OV Ž t 2 , t 1 . s exp y2 g
Ž 25 .
The simulated shape of the fifth-order overtone response function along t 2 for several values of t 1 is shown in Fig. 6 for DrL s 10, 1 and 0.1. In the first case the dynamics are in the slow modulation limit where the
Fig. 5. Feynman diagrams depicting the Liouville space pathways of fifth-order overtone dephasing. The left diagram applies for probing the second-order anti-Stokes signal while the right diagram describes the second-order Stokes signal.
276
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
response can be approximated by: 2
2 R Ž5. OV Ž t 2 , t 1 . s exp y Ž t 1 q 2 t 2 . D r2 4 .
Ž 26 .
When t 1 is changed the shape of the response changes significantly: For small t 1 the decay along t 2 is Gaussian, but as t 1 is increased the decay becomes faster. This is the result of the finite correlation time
Ž . Ž . Ž . Fig. 6. Simulation of the overtone dephasing response function R Ž5. OV t 2 , t 1 as function of the reduced time Dt 2 for a D r L s10, b D r L s1.0, Žc. D r L s 0.1. The fixed values of Dt 1 are 0 Ždotted., 1.0 Žlong dashed., 2.0 Ždashed dotted., 3.0 Žshort dashed., 4.0 Ždashed double dotted. and 5.0 Žsolid.. For discussion of details consult text.
T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
277
t C s 1rL, that is contained in the exact expression Eq. Ž25.. When either t 1 or t 2 become large, the slow modulation limit does not perfectly hold anymore and the decays tend to become exponential. Ž . In the opposite case of fast modulation the overtone response function R Ž5. OV t 1 , t 2 is given by: ) R Ž5. Ž t 1 q 4t 2 . 4 . Ž 27 . OV Ž t 2 , t 1 . s exp yG The exponential decay along the two time coordinates does not depend on the value of the other coordinate. Along t 2 the decay is always four times faster than the decay along t 1. Note that this result relies on the assumption of perfect correlation of the fluctuations of the different vibrational levels. The decay along t 2 is also four times faster than that of the third-order CARS response w22,45x. In the intermediate modulation regime the system behaves initially as if it were in the slow modulation limit. The response is Gaussian when t 1 and t 2 are small compared to the correlation time of the fluctuations tc . As the propagation times t 1 and t 2 are increased the decay becomes faster. At times much longer than 1rL the system can be described in the fast modulation limit. The decay does not longer change; it is single exponential. In their experiments on CDCl 3 Tominaga and Yoshihara w23x found that the decay along t 2 becomes faster as t 1 is increased. The time constants were two to three times smaller than those observed in third-order CARS experiment and in the fifth-order signal along t 1. The finite correlation time of the frequency fluctuations can qualitatively account for the observed faster decay along t 2 , as t 1 is increased, but the ratio of the decay constants along the two time axes is not correctly predicted w22,23x.
4. Impulsive fifth-order Raman scattering In this section simulations of the temporally two-dimensional fifth-order Raman response are presented for impulsive excitation, where the used laser pulses are much shorter than the period of the excited vibration. This experiment has been reported by a number of groups w13–17x who investigated low-frequency intermolecular motions. Very recently, Tokmakoff et al. w20,21x used sub-30 fs laser pulses and incorporated optical heterodyning, which allowed for the observation of high-frequency intramolecular modes as well. Under these conditions a large number of Liouville space pathways, containing different dynamic information, contribute to the signal, which makes the analysis of experimental results less straightforward. When a single mode is excited, the impulsive material response is determined by the full fifth-order response function, R Ž5. Žt 1 , t 2 ., given by Eq. Ž21.. Simulations of this function for different ratios DrL are shown in Fig. 7. In the slow modulation limit, L < D, the impulsive fifth-order Raman response function can be approximated by: R Ž5. Ž t 2 , t 1 . s eyŽ t 1yt 2 .
2
D 2 r2
yt 12 D 2 r2
qe
cos v 0 Ž t 1 y t 2 . y eyŽ t 1qt 2 . yŽ t 1q2 t 2 . 2 D 2 r2
cos v 0t 1 y e
2
D 2 r2
cos v 0 Ž t 1 q t 2 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 28 .
Most of the terms of the fifth-order response exhibit an enhanced decay along t 1 and t 2 due to the Žalmost. static frequency distribution of width D. This does not hold for the first term which reaches a maximum whenever t 1 s t 2 as discussed at the beginning of the previous section. The corresponding echo signal is clearly present along the diagonal in the simulation of Fig. 7a. Also, in this figure a prominent feature is visible around t 1 s 0 that does not decay along t 2 . It is due to the third term of Eq. Ž28. that involves a one-quantum coherence during the first and a population during the second propagation period. Since the population is not affected by the frequency fluctuations discussed here, the amplitude of this term does not change as function of t 2 . As will be discussed in detail below this allows for a separation of population and phase relaxation processes. In the opposite limit of fast modulation, L 4 D, the response can be approximated by: )
R Ž5. Ž t 2 , t 1 . s eyG
Žt 1qt 2 .
qe
yG
)
t1
Ž cos v 0 Ž t 1 y t 2 . y cos v 0 Ž t 1 q t 2 . .
cos v 0t 1 y eyG
)
Žt 1q4 t 2 .
cos v 0 Ž t 1 q 2t 2 . .
Ž 29 .
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T. Steffen, K. Duppenr Chemical Physics 233 (1998) 267–285
Fig. 7. Absolute square of the simulated impulsive fifth-order response R Ž5. Žt 1 , t 2 . as function of the reduced times Dt 1 and Dt 2 for Ža. mainly inhomogeneous dephasing, i.e. D 4 L, Žb. distinct non-Markovian dynamics where D f L holds, and Žc. mainly homogeneous dephasing with D < L. The period of the oscillator, Tvib s 2 p r v 0 , is normalized to one. Consult text for details.
This result was derived previously w45x for Markovian phase relaxation rates G k l of coherent superpositions < k :² l < given by G k l s Ž k y l . 2G ) . It is evident that all terms of the fifth-order response decay monotonically as t 1 is increased. Since the phases fluctuate very fast on the timescale of the experiment, rephasing is not possible. As a consequence the simulated response shown in Fig. 7c does not show any echo.
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In the intermediate modulation regime, where L f D holds, the approximations of Eqs. Ž28. and Ž29. break down and the exact expression for R Ž5. Žt 1 , t 2 ., given in Eq. Ž21., has to be used. As shown in Fig. 7b, the system behaves as if it was inhomogeneously broadened when times t 1 and t 2 are small: There is a buildup of a feature at the diagonal t 1 s t 2 that decays rapidly due to the finite correlation time. This demonstrates that at times larger than t C the response approaches that of the homogeneous limit.
5. Population relaxation Up to now, we have considered only pure dephasing processes, but in general there are also decay channels due to population relaxation. For high-frequency intramolecular modes the population decay is often much slower than the phase relaxation, so that in third-order Raman experiments the former process can be neglected. This is probably not true for low-frequency intermolecular vibrations that are heavily damped. In general, these two relaxation processes cannot be discriminated by temporally one-dimensional techniques as depicted in Fig. 1a. In impulsive fifth-order Raman experiments the population relaxation cannot be neglected even for high-frequency intramolecular modes, since some of the contributing Liouville space pathways involve the propagation of a population < k :² k < as was mentioned in the previous section. It is then possible to separate population and phase relaxation due to the temporally two-dimensional character of this technique. In order to illustrate this new feature of the fifth-order Raman technique, we apply a model often used for the description of population relaxation in vibrational spectroscopy w45,48–50x. When the optically active vibration is linearly coupled to a harmonic heat bath, this induces in the limit of weak system–bath interaction only one-quantum transitions < k : ™ < k " 1:. It can be shown that the decay rate for the downward transition, g k ™ ky1 , then scales linearly with the quantum number k:
g k ™ ky1 s kg ,
Ž 30 .
where g ' g 1 ™ 0 is one half of the population decay rate from the first excited state to the ground state w45,48–50x. At finite temperature, there are next to downward also upward transitions. The upward transition rates g k ™ kq1 can be obtained from a detailed balancing condition. In thermal equilibrium the population n k of level k does not change, i.e.: d nk dt
s y Ž g k ™ ky1 q g k ™ kq1 . n k q g kq1 ™ k n kq1 q g ky1 ™ k n ky1 s 0 .
Ž 31 .
Starting from the ground state Ž k s 0. one has to consider only transitions from and to the first excited state which directly yields g 0 ™ 1 s g 1 ™ 0 n1rn 0 s g 1 ™ 0 h , where h ' expŽy" v 0rk B T . denotes the Boltzmann factor. Using this result the other rates g k ™ kq1 can be obtained successively:
g k ™ kq1 s Ž k q 1 . gh .
Ž 32 .
The total rate out of level k is now given by the sum of the upward and the downward transition: g k s g k ™ ky1 q g k ™ kq1. In Ref. w45x this decay of the population was incorporated by adding an imaginary part ig kr2 to the frequency v k , given in Eq. Ž11.. The resulting complex frequencies impose no major difficulty in deriving the third- and fifth-order response in the same way as described in Section 2. It should be noted, however, that in that approach the total number of oscillators, i.e. the sum of the diagonal elements < k :² k < is not conserved, since the feeding of the lower Žhigher. states due to the downward Župward. transitions is neglected. This approximation is valid in third order, where the system always propagates in coherent states < k :² k " 1 < w50x, but not in fifth order for those Liouville space pathways that comprise a diagonal state. In some of the Liouville space pathways a population is propagated during the second propagation time t 2 . When there is no level-dependent population relaxation, these paths yield the cos v 0t 1 term of Eqs. Ž17., Ž21.,
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Ž28. and Ž29.. One of the corresponding Feynman diagrams is shown in Fig. 8a. Due to the population transfer described by Eq. Ž31. the amplitude of this diagram decreases, while additional diagrams occur that capture the system dynamics and the optical signal of the feeded states. In Fig. 8b–c two of these relaxation-induced diagrams are depicted that result from one-quantum upward and downward transitions, respectively. They are only the first step towards thermal equilibrium where the population of all levels is given by the Boltzmann distribution. For a self-consistent description of the redistribution of the population, it is necessary to solve the set of coupled linear differential equations, see Eq. Ž31., for all diagrams with the appropriate initial condition. In order to avoid the evaluation of an infinite set of coupled equations one has to introduce an effective upper level from where there are no upward transitions. The accuracy of this approximation is determined by the temperature and the chosen upper level. Here we consider the low-temperature limit, i.e. h < 1, where almost all population is initially in the ground state and thermally induced upwards transitions can be neglected. In this limit three different Liouville pathways contribute to the cos v 0t 1 term: The propagation of the ground-state population, that in the low-temperature limit does not decay along t 2 , is captured by diagrams Žb.1. and Žb.4. of Fig. 1 with k s 0, l s 1 and m s 0. The dynamics involving the diagonal state <1:²1 < are depicted in Fig. 1Žb.2. and Žb.3. for k s 0, l s 1 and m s 1. These pathways decay as function of t 2 with a rate constant g while the amplitude of the relaxation induced diagrams Žsee, e.g., Fig. 8c. rises with the same rate constant. When all these additional pathways are taken into account, the plateau that was found in Fig. 7 along t 2 for small t 1 , decays. The full fifth-order Raman response, that at low temperatures involves only the lowest three vibrational states, is now given by: g
R Ž5. Ž t 2 , t 1 . s ey 2 Žt 1q3t 2 . ey2 g
Ž1.
Žt 1 .y2 g Ž1.Žt 2 .qg Ž1.Žt 1 q t 2 .
cos v 0 Ž t 1 y t 2 .
g
y ey 2 Žt 1q t 2 . eyg
Ž1.
Žt 1 q t 2 .
cos v 0 Ž t 1 q t 2 .
g
q ey 2 Žt 1q2 t 2 . eyg
Ž1.
Žt 1 .
cos v 0t 1
g
y ey 2 Žt 1q2 t 2 . ey2 g
Ž1.
Žt 2 .y2 g Ž1.Žt 1 q t 2 .qg Ž1.Žt 1 .
cos v 0 Ž t 1 q 2t 2 .
g
q Ž 1 y ey gt 2 . ey 2 Ž2 t 1q t 2 . ey2 g
Ž1.
Žt 1 .qg Ž1.Žt 2 .y2 g Ž1.Žt 1 q t 2 .
cos v 0 Ž 2t 1 q t 2 . .
Ž 33 .
The first two terms describe Liouville space pathways where the system evolves in one-quantum coherences
Fig. 8. Additional Liouville space pathways due to population relaxation. The amplitude of the double-sided Feynman diagram Ža. decays as function of the propagation time t 2 due to thermally induced upward and downward transitions. The population transferred from the diagonal state < k q1:² k q1 < to < k q2:² k q2 < and < k :² k < gives rise to new, relaxation induced diagrams Žb. and Žc.. Note that the amplitude of these diagrams is determined by the kinetic equations for the populations n k ŽEq. Ž31.. and the matrix elements a k k .
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Fig. 9. Simulation of the impulsive fifth-order Raman response as function of the reduced time t 2 r Tvib with t 1 s 0 ŽEq. Ž34... Here Tvib s 2 p r v 0 denotes the oscillation period. In all traces fast frequency modulation was assumed Ž D < L, D 2 r L f G ) . and the total linewidth of the third-order response G ) qg r2 was kept constant. Ža. In the absence of pure dephasing Ž g Ž1. Žt . s 0. the signal shows oscillations with period Tvib that decay two times faster than the third-order response. Žb. When pure dephasing and population relaxation are comparable Žshown for g r2 s G ) ., the signal is initially determined by oscillations due to the last term of Eq. Ž34.. At longer delays it decays exponentially with a rate constant g . Žc. When only pure dephasing occurs Žg s 0., the term proportional to cos v 0t 1 in Eq. Ž34. does not decay along t 2 yielding a plateau.
during t 1 and t 2 , while the third and fourth term correspond to a one-quantum coherence, followed by a population and a two-quantum coherence, respectively. The first and fourth term were already discussed in Section 3, in connection with the non-impulsive fifth-order Raman echo and overtone response functions. The fifth term, that comprises a two-quantum coherence during the first and a one-quantum coherence during the second propagation period, disappears for level-independent population relaxation due to perfect interference of the relevant Liouville space pathways w45x. It was therefore not present in Eqs. Ž17., Ž21., Ž28. and Ž29. where population relaxation was ignored.
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Because the five terms decay in different ways, it is now possible to separate population and phase relaxation processes. This is particularly evident for the fifth-order response along t 2 with t 1 s 0 1, where we find: R Ž5. Ž t 2 , t 1 s 0 . s eyg t 2 Ž 1 y ey4 g
Ž1.
Žt 2 .
cos 2 v 0t 2 . .
Ž 34 .
In order to demonstrate the sensitivity of the fifth-order response to the population relaxation, we have calculated the signal in the fast fluctuation limit, i.e. D < L, for different values of g while keeping the third-order line width gr2 q G ) constant. When there is no pure dephasing, the signal shows oscillations that fade out with the population relaxation rate g as is shown in Fig. 9a. In the other extreme case of only pure dephasing but no population relaxation, the initial oscillations, that originate from the term proportional to cos 2 v 0t 2 , are rapidly damped, while on a longer time scale there is a nondecaying plateau due to the cos v 0t 1 term, see Fig. 9c. When both pure dephasing and population relaxation processes are equally important, the signal of Fig. 9b is predicted: The initial oscillations are damped rapidly due to both mechanisms while on a longer time scale the decay is single exponential with the population decay rate g . This allows for an independent determination of the population relaxation time that cannot be extracted from the third-order Raman response.
6. Conclusions The nonresonant third- and fifth-order Raman response was calculated for pure dephasing due to random frequency fluctuations. The nuclear vibrations were modelled by a harmonic oscillator that is coupled to the radiation field via a polarizability with a nonlinear coordinate dependence. For fluctuations that are described by a stationary Gaussian process, the response functions can be solved exactly using the cumulant expansion technique. The obtained expressions for the response functions depend explicitly on the correlation time t C s 1rL of the frequency fluctuations and on the root mean square D of the average distribution. By changing the ratio DrL the results interpolate continuously from homogeneous dephasing Ž DrL < 1. via non-Markovian dynamics Ž DrL f 1. to inhomogeneous dephasing Ž DrL 4 1.. The fifth-order Raman response was discussed for two non-impulsive experiments. In temporally two-dimensional overtone dephasing w22,23x the phase relaxation of one- and two-quantum coherences is probed selectively. It was shown that the decays are Gaussian at early times. If at least one of the two propagation times is long compared to t C , the decays become faster and approach an exponential. The second type of non-impulsive fifth-order Raman experiment, which has not been performed, yet, selectively probes an echo-type of process carrying information on the time scaleŽs. of the frequency fluctuations. Model calculations were also performed for the impulsive fifth-order Raman response. In the case of dominant inhomogeneous broadening a motional echo was found at t 1 s t 2 while in the homogeneous limit the dephasing is irreversible and there is no echo. When the correlation time t C of the fluctuations is finite, the dynamics are again mainly inhomogeneous at times much shorter than t C and mainly homogeneous at times much longer than t C . At the intermediate times the resulting non-Markovian response cannot be approximated by one of these limits. The effect of Markovian level-dependent population relaxation was included in the treatment to demonstrate that the impulsive fifth-order Raman response allows in principle to separate population and phase relaxation
1
At t 1 s 0 hyperpolarizability contributions can contribute to the fifth-order response w45x. The resulting complications in the signal analysis can be avoided by either choosing appropriate polarization conditions w19x or setting the first delay time T1 equal to the pulse duration t P . In the latter case the expression of Eq. Ž34. is still a good approximation since in impulsive experiments the pulse duration is much smaller than the period of the excited mode.
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processes. This is impossible for third-order Raman experiments. Experimentally, this feature, that to our knowledge has not been discussed before, is best observed along t 2 at t 1 s 0.
Acknowledgements The investigations were supported by the Netherlands Foundations for Chemical Research ŽSON. and Physical Research ŽFOM. with financial aid from the Netherlands Organization for the Advancement of Science ŽNWO..
Appendix A. Stationary Gaussian frequency fluctuations in the fifth-order response The relaxation functions in the fifth-order Raman response due to the frequency fluctuations are all of the form, cf. Eq. Ž17.: t1
¦½
H0
¦½
H0
f 1Ž2. Ž t 2 , t 1 . s exp yi
dt Ž k yl . dv Ž t . q
t 2q t 1
Ht
d t Ž k y m. d v Ž t .
1
5;
,
Ž A1.
5;
.
Ž A2.
and:
f 2Ž2. Ž t 2 , t 1 . s exp yi
t1
dt Ž k yl . dv Ž t . q
t 2q t 1
Ht
dt Ž myl . dv Ž t .
1
Since for a stationary Gaussian process all cumulants beyond the second vanish w24–28x, Eq. Ž33. can be written as:
f 1Ž2.
Žt2 , t1.
1
t1
½ ¦H
s exp y
½
' exp y
2
1 2
dt Ž kyl . dv Ž t . q
0
g 1Ž2.
2
t 2q t 1
Ht
d t Ž k y m. d v Ž t .
1
;5
Ž A3.
5
Ž t 2 ,t 1 . .
The two-time line shape function g 1Ž2. Žt 2 , t 1 . can now be decomposed into a sum of the well-known one-time line shape functions g Ž1. Žt 1 ., given by Eq. Ž19.. From the definition of g 1Ž2. Žt 2 , t 1 . in the last equation we find: g 1Ž2. Ž t 2 , t 1 .
s Ž kyl.
2
t1
H0
d t1
t1
H0
d t 2 ² d v Ž t1 . d v Ž t 2 . : q Ž k y m .
t 2q t 1
Ht
d t 2 ² d v Ž t1 . d v Ž t 2 . : q 2 Ž k y l . Ž k y m .
1
2
t1
H0
2
t 2q t 1
Ht
d t1
d t1
1
t 2q t 1
Ht
d t 2 ² d v Ž t1 . d v Ž t 2 . :
1
2
' Ž k y l . A q Ž k y m. B q 2Ž k y m. Ž k y l . C .
Ž A4. The symbols A, B and C denote the double integrals in the first, second and third line of Eq. ŽA2., respectively. In Fig. 10 it is shown that 2 C s D y A y B holds where D denotes the double integral with both integrations running from 0 to t 2 q t 1. Using this equality we can rewrite g 1Ž2. Žt 2 , t 1 . as: g 1Ž2. Ž t 2 , t 1 . s Ž k y l . Ž m y l . A q Ž k y l . Ž l y m . B q Ž k y m . Ž k y l . D ,
Ž A5.
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X Fig. 10. Definition of the surface areas A, B, C, C and D of the two-dimensional integrals of Eqs. Ž17. and ŽA2.. For a stationary X Gaussian process one finds C s C and 2 C s D-A-B.
where all of the double integrals A, B and D can be expressed in terms of the one-time line shape function g Ž1. Žt ., defined in Eq. Ž18.. The final result therefore is: g 1Ž2. Ž t 2 , t 1 . s Ž k y l . Ž m y l . g Ž1. Ž t 1 . q Ž k y l . Ž l y m . g Ž1. Ž t 2 . q Ž k y m . Ž k y l . g Ž1. Ž t 1 q t 2 . .
Ž 40 . This procedure can also be followed f 2Ž2. Žt 2 , t 1 ., yielding finally Eq. Ž21..
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