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A. Theory of Vibrational Third-Order Nonlinear Spectroscopy

A. Theory of Vibrational Third-Order Nonlinear Spectroscopy

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283



obtained (46,76,77):

h



01



00 i



0



10



Dj



10 j



with the line shape function, g



2



exp



iω10



g



(4a)



, given by:



0



g



D



d



0



d 00 hυω10



00



υω10 0 i



(4b)



0



0



The corresponding double-sided Feynman diagram is shown in Fig. 2a.

The double-sided diagrams, which have been used for many years, (78)

have a very simple interpretation for spectroscopic or resonant processes

because of the applicability of the rotating wave approximation in that

case. The vertical lines are time lines with the earliest time at the bottom.

The left and right time lines display properties of the light fields as they

would appear on the ket and bra sides of the initial density operator 00

in a conventional expansion of the density matrix as a power series in the

fields. The indices of the density matrix element created by the interaction

are displayed horizontally. By convention the arrow pointing upward from

left to right signifies a positive wavevector, whereas upward from right

to left is a negative wave vector. Since the complex field is assumed to

have the form exp[iωt kr], a negative wave vector field implies a positive

frequency, which causes a transition from a lower to a higher energy state

on the bra time line or from a higher to lower energy state on the ket time

line. The last arrow refers to the generated field, which we will always

take as being emitted as the system undergoes a downward transition on

the ket side. Thus in the correlation function containing 01 t 10 0 00

the first interaction is on the left (ket) side of the initial density operator

and the wave vectors of the outgoing and input fields cancel. After the first

interaction the system is in a 1-0 coherence. The sign of the term is 1 n ,

where n is the number of interactions on the right (bra) side.

Within the same formalism, the response functions of the thirdorder Feynman diagrams in Fig. 2b can be written down immediately. For

example, the response function of the first (rephasing) diagram in Fig. 2b is:

R1 D h

Dj



01

10 j



4



3



10



e



iω10



1

3



00

1



2



01



0



10



Ð exp



2



i

3



i



υω10



0



d



0



0

1



Ci



υω10



0



2



d 0Ci



0



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0



υω10



0



d



0



5



284



Hamm and Hochstrasser



Figure 2 The double-sided Feynman diagrams, which have to be considered in

(a) a linear absorption experiment and (b) a nonlinear third-order experiment such

as photon echo, pump-probe, transient grating. The diagrams are arranged according

to the possible time orderings, as discussed in the text and illustrated in Fig. 4.



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Again with the help of the cumulant expansion (46) we obtain:

R1 D j



10 j



4



g



3



e



iω10

1



3



1



2



g



3



Ð exp Cg

2 Cg



3



g



2



1



1



g



2



6



with the same line shape function g as the one used for the linear absorption spectrum [Equation (4)]. By changing the absolute times 1 , 2 , and 3

to the time intervals t1 D 1 , t2 D 2

1 , and t3 D 3

2 corresponding

to the delay times that are under experimental control, we obtain the final

result for the response function:

R1 D R2 D j



10 j



e iω10 t3 t1 exp g t1 C g t2

g t3 C t2 C g t1 C t2 C t3

4



g t3



g t2 C t1

7



In order to calculate the response function of the Feynman diagram R3 , it is

further assumed that the transition frequency ω12 is anharmonically shifted

with respect to the ground states transition frequency so that, ω12 D ω01

. Another assumption that can be made (see later for a discussion of these

assumptions) is that the fluctuations between both level pairs are strictly

correlated υω12 D υω01 . This implies that only the harmonic part of the

potential surface is perturbed by the bath fluctuations and the anharmonicity

of the vibrator is unaffected. We then obtain for R3 :

j 10 j2 j 21 j2 e i ω10  t3 ω10 t1 Ð exp g t1 C g t2

g t2 C t1

g t3 C t2 C g t1 C t2 C t3



R3 D



g t3

8



The response functions for the nonrephasing Feynman diagram (see Fig. 2b)

are calculated accordingly:

R4 D R5 D j 10 j4 e iω10

C g t3 C t2

R6 D



t3 Ct1



Ð exp g t1

g t1 C t2 C t3



g t2



g t3 C g t2 C t1

9



j 10 j2 j 21 j2 e i ω10  t3 Cω10 t1 Ð exp g t1

g t2

g t1 C t2 C t3

g t3 C g t2 C t1 C g t3 C t2



10



The remaining time orderings, for which there is no corresponding

process in a photon echo experiment on an electronic two-level system, are:

R7 D j

R8 D



j 21 j2 e iω10 t3 C2t2 Ct1 eit2 Ð exp Cg t1

g t2 C t1

g t3 C t2

g t1 C t2 C t3

10 j



2



j 10 j2 j 21 j2 e iω10

C g t3

g t2 C t1



t3 C2t2 Ct1



g t2 C g t3



ei t3 Ct2 Ð exp Cg t1

g t3 C t2

g t1 C t2 C t3



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11

g t2

12



286



Hamm and Hochstrasser



As expected, we find that the total response function 3lD1 Rl D 6lD4 Rl D

8

lD7 Rl D 0 (i.e., for each possible time ordering) vanishes exactly in the

harmonic case, defined by  D 0 and j 21 j2 D 2j 10 j2 . Furthermore, it can

be easily seen that in the case of a strict separation of time scales of homogeneous and inhomogeneous broadening, the line shape function becomes

g t D t/T2 C 2 t2 /2 and the total response function reduces exactly to

the result obtained within a Bloch picture (see, for example Refs. 52 and

75), e.g.,

3



Rl D 2j



10 j



4



e



iω10 t3 t1



Ð 1



eit3 Ð exp



t1 C t3 /T2



lD1



ð exp



2



t1



t3 2 /2



13



In particular, the time coordinate t2 disappears in Equation (13), which

emphasizes the fact that three-pulse photon echo experiments are necessary

to characterize effects that are beyond the Bloch picture.

Vibrational energy relaxation T1 is taken into account phenomenologically through the multiplicative factors:

exp



t1 /2 C t2 C t3 /2

T1



exp



t1 /2 C t2 C 3t3 /2

T1



for R1 , R2 , R4 , and R5



(14a)



and

for R3 and R6



(14b)



which assumes that vibrational relaxation from the D 2 to the D 1 level

is twice as fast as from the D 1 to the D 0 level and that it is constant

throughout the inhomogeneous distribution. Orientational relaxation for R1

to R6 is taken into account through the factor:

h



z



0



z t1



z t1



C t2



z t1



C t2 C t3 i D



2 1

2

C e

3 6 15



6Dt2



e



2D t1 Ct3



15

where h. . .i is an orientational average, z t is the projection of the transition dipole at t onto the laboratory z-axis, and D is the rotational diffusion coefficient. Thus, during the period t1 and t3 , where the system is

in a coherence, the four-wave mixing signal decays like the first-order

Legendre polynomial P1 cos  just as is found in a two-pulse echo (74).

However, during the period t2 , where the system is in a population state, it



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287



decays fractionally like P2 cos  . The formulas for other polarization conditions and for nonspherical rotors are easily obtained by standard methods

(79,80). Generally such simple diffusional models are not correct for small

molecules in solutions. Their orientational correlation functions are not

usually simple exponentials with parameters that depend on the molecular

size, but they also involve varying amounts of inertial motion, and there is

no general form for the intermediate region. Furthermore, there is in general

no relation between the decays of P1 and P2 such as occurs when there is

diffusive motion. However, if the orientational correlation functions could

be measured independently it would be straightforward to introduce them

in place of Equation (15) and then to solve the problem numerically.

The third-order polarization is obtained by convolution of the response

functions with the electric fields of the three laser pulses:

1



P3 t D



1



dt3

0



ð E2 t



0



t3



1



dt2



t2 e



Rl t1 , t2 , t3 Ð E3 t



dt1

0



iω t t3 t2



t3 e



iω t t3



l



E1 Ł t



t3



t2



t1 eiω t



t3 t2 t1



16



where the E1,2,3 are the field envelopes of the laser pulses with wave vectors

vector k1 , k2 , and k3 , respectively, and ω is the carrier frequency of the light

pulses. When the laser pulses are not overlapping in time, the time ordering

is the same in the whole integration area and the sum in Equation (16) runs

over those Feynman diagrams relevant for that particular time ordering

(l D 1, 2, 3, l D 4, 5, 6, or l D 8, 9, respectively). In the case of overlapping

laser pulses, the set of response functions is interchanged each time the

integration in Equation (16) switches the time ordering.

In a homodyne detection scheme, such as in the stimulated photon

echo experiments described in the next paragraph, the detector measures

the t-integrated intensity of the square of the third-order polarization

S T,



1



D



jP 3 T, , t j2 dt



(17)



0



where and T are the time delays between the peaks of pulse k1 and k2

and pulse k2 and k3 , respectively (see Section III.D).

B. Limitations of the Stochastic Model



The response function R2 , which develops on the vibrational ground state

after the second light interaction, is the same as R1 [see Equation (7)]. This

is a consequence of the stochastic ansatz in Equation (3), which implies that

the bath influences the vibrational frequency of the solute but excitation of



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288



Hamm and Hochstrasser



the solute does not perturb the bath. In optical spectroscopy there is a

Stokes shift which is due to the response of the bath molecules to the

changed charge distribution of the solute after electronic excitation (81).

This effect causes a nonzero imaginary part in the line shape function g t

and the response functions R1 and R2 to become different (46). Vibrational transitions might undergo dynamical spectral shifts analogous to the

Stokes shifts of electronic transitions. The nuclear motion dependence of

the dipole or higher electric multipoles could induce a dynamical vibrational shift that will influence the interpretations of vibrational dynamics

and cooling in general. Such dynamics is not included in, for example,

Redfield theory (82–86) of multiple vibrational level systems coupled to

bath. The usual electronic Stokes shift is caused by the response of the bath

molecules to the changed charge distribution of the solute after excitation

(87,88). For high-frequency vibrations the change in the static, averaged

electric moment would be sensed by the solvent. However, when the solvent

responds comparably or faster than the solute motion, both the curvature and

the absolute energy of the potential surface may be changed, and any shift

of a vibrational frequency will not simply record the change in free energy

computed for a change in electric moment. Clearly the dipole moment as a

function of the internuclear distance is already partly incorporated into the

potential of mean force, which determines the mean frequencies in the BornOppenheimer approximation. Nevertheless, the vibrational state dependence

of the dipole moment is well known (89–92) and changes of 0.03–0.2 D

between D 0 and D 1 have been reported. Although the dipole (or electric quadrupole for nearest neighbor effects) changes are much smaller than

for electronic transitions, the spectral resolution is improved in the infrared

by a similar factor; therefore, when such dynamical shifts are present they

might be observable. In summary, excitation of D 1 by a short pulse

should create a nonequilibrium distribution of solvent/solute configurations

because of the mixed mode anharmonicity. Both the negative stimulated

emission peak and the new (1 ! 2 transition) absorption in a IR-pump–IR

probe experiment could shift and change width in time, while the bleaching

peak should remain constant. It should be possible to measure shifts of less

than one twentieth of the bandwidth. In some cases, specific modes (93)

may dominate the anharmonicity while in others more complex solvent

motions may be involved. In any event, further experimental work is needed

to evaluate the assumption that R1 D R2 .

Another simplifying assumption included at this point in our model

so far is the strict correlation of the fluctuations between the 0-1 and the

1-2 level pairs υω12 D υω01 . This assumption is based on the notion that



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