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D. The Three-Pulse Photon Echo Experiment

D. The Three-Pulse Photon Echo Experiment

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292



Hamm and Hochstrasser



third light interaction. This photon echo is formed only under rephasing

conditions and thus gives rise to a t-integrated signal which exceeds that

in the nonrephasing direction. As delay time T increases, spectral diffusion

destroys the phase memory, and eventually the response functions 3lD1 Rl

and 6lD4 Rl become identical. In other words, the difference between the

rephasing and the nonrephasing signals is a measure of the time variation

of the inhomogeneous distribution.

The extraction of the required information from the data over the full

ranges of and T shall be explained in some detail here. In the experimental

setup used by us (31,41), the pulses k1 and k3 traversed computer controlled

optical delay stages, while pulse k2 was held fixed in time. Thus, one of the

delay stages controlled the delay time ’ between the peaks of pulses k1

and k2 , while the second one controlled delay time T’ between the peaks

of the pulses k2 and k3 (see Fig. 3). The three pulses can be time ordered

in six ways (see Fig. 4). The Feynman diagram for the six permutations

are given in Fig. 2. The permutations (123), corresponding to T0 , 0 > 0,

and (132), corresponding to T0 C 0 > 0 > T0 , give a rephased part of the

echo signal since interchanging pulses 2 and 3 does not change the signal

under the k1 C k2 C k3 direction. Likewise, the response functions for

the nonrephased parts of the echo signal [(213) and (312)] and those for

(321) and (231), which are neither rephasing nor nonrephasing and which

have no correspondence in a photon echo experiment on a electronic twolevel system, have the same responses. When the delay times are controlled

by variations in T0 and 0 , the six time orderings can be arranged in the

quadrants spanned by these time coordinates as shown in Fig. 4. It is readily

seen from Fig. 4 that independent continuous scanning of T0 and 0 in this

space over the complete time ranges encounters the unwanted permutations

(231) and (321), and there is no simple continuous scan mode that just

picks out the desired rephasing and nonrephasing Feynman diagrams. This

is why we use instead the coordinate set and T, where denotes the

positive or negative separation between the peaks of pulses k1 and k2 , and

T denotes the positive separation between the second and third pulses. Thus

T measures the separation between k3 and k2 if > 0, while if < 0 it

measures the separation between k3 and k1 . The coordinate transformation

between T0 , 0 and T, can be made either during the experiment by moving

only one stepping motor for > 0 and both for < 0 or by rearranging

the data afterwards in the computer. The former method was used by the

Fleming group (19) and the latter method by us.

If the laser pulses were infinitely short (υ-shaped), and T would be

identical to t1 and t2 [see Equation (16)] and the discontinuous coordinate



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293



Figure 4 Distribution of all six possible time orderings, labeled (ijk), onto the

four quadrants spanned by time coordinates 0 and T0 (shown for the k1 C k2 C k3

phase matching direction).



transformation would not cause any difficulties. However, in reality, when

the pulse width is finite, this transformation is somewhat arbitrary, since

it introduces a discontinuity at D 0 into a process, which, of course, is

perfectly continuous. This representation of the data is nevertheless chosen

throughout this paper [and also by other authors (19)], since the transformed

coordinate system emphasizes the equivalence between the set of time coordinates , T and t1 , t2 . The latter set of time variables is the one used when

representing nonlinear processes in the form of Feynman diagrams (see

Fig. 2), so that using the transformed coordinate system and T is physically intuitive. However, it should be kept in mind that these sets of time

coordinates are not identical, but they are connected by the convolution of

the response functions with the electric fields [i.e., Equation (16)].

The photon echo (rephasing) signal can be compared with the reverse

photon echo (nonrephasing) signal by scanning the delay time from negative to positive delay times or by collecting the signals in the k1 C k2 C

k3 and the Ck1 k2 C k3 phase-matching directions (13,14,16–18,31,41).

Both procedures switch from the sets of graphs R4 R6 to R1 R3 , and

both procedures are often used simultaneously. In the work on electronic



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Hamm and Hochstrasser



photons, (13,14,16–19,99,100) the asymmetry between the rephasing and

the nonrephasing signal usually has been measured with the help of the

so-called peak shift, which is defined as half the time between the peak of

the k1 C k2 C k3 signal for > 0 and the Ck1 k2 C k3 signal for < 0.

This procedure is perfectly adequate in the case of electronic transitions,

since there is always a peak shift >0 when there is inhomogeneity. The

peak shift occurs because the k1 C k2 C k3 signal is larger for a given

> 0 (rephasing condition) than it is for

(nonrephasing condition), so

that the k1 C k2 C k3 signal peaks at > 0, given the signal is continuous

at D 0. The same argument holds for the Ck1 k2 C k3 signal for < 0.

However, the discontinuous coordinate transformation described

above can introduce difficulties in the peak shift measurement, particularly

when vibrational energy relaxation T1 occurs on a similar timescale as

dephasing, which is often the case for vibrational transitions, but generally

not for electronic transitions. In that case, the peak shift can be zero at

the discontinuous point D 0, although the overall k1 C k2 C k3 and the

Ck1 k2 C k3 signals are not at all identical. Therefore, we have chosen

to use the normalized first moment of the k1 C k2 C k3 signal

M1 T D



1

1



ÐS



1

k1 Ck2 Ck3



T,



d



1



S



k1 Ck2 Ck3



T,



d



(19)



as a measure of the asymmetry between the rephasing and the nonrephasing

signal. If the signal S T, were symmetrical with respect to its peak (which

is commonly the case in optical photon echo experiments because of the fact

that the time resolution of even the fastest laser pulses instrument response

time is of the same order as electronic dephasing processes), the peak shift

and the first moment would be identical. However, neither the first moment

nor the peak shift have much fundamental physical meaning for vibrational

echoes other than they are convenient measures of the asymmetry of the

signal versus at a given value of T. They assess qualitatively whether

an inhomogeneous distribution still exists after time T, so the approach

of the photon echo signals to a symmetrical form centered at D 0, at

which point M1 D 0 mimics in some respects the evolution of the spectral

diffusion and gives an idea of the time scales of the spectral diffusion

processes. For electronic two-level systems that the peak shift follows the

energy gap fluctuation correlation function in certain limits (19,99,100).

E. Spectral Diffusion of Small Molecules in Water



The asymmetrical stretching mode of the linear azide ion N3 exhibits

a very strong infrared absorption

D 5 ð 10 18 cm2 and was therefore



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295



chosen as our first example of IR-stimulated echo spectroscopy. Our expectation was that N3 could be used as a probe of the dynamics of the solvent,

which in this case was water. Its energy relaxation rate (T1 D 2.3 ps), orientational diffusion constant D D 0.023 ps 1 , and anharmonicity /2 c D

25 cm 1 are known from pump-probe and anisotropy experiments (66).

Figure 5a shows the stimulated photon echo signal from azide ion in

the k1 C k2 C k3 and the Ck1 k2 C k3 directions, which were collected

by two independent IR detectors. The beam directions were arranged in

the so-called box configuration, (101) (see Fig. 3). The experiments show

clearly that neither the k1 C k2 C k3 nor the Ck1 k2 C k3 signals are

symmetrical with respect to D 0. Moreover, the asymmetry diminishes

with T and has mostly disappeared by T ³ 5 ps. These results show that

the vibrational coherence introduced by pulse 1 and the corresponding

phase information stored in the population created by pulse 2 continues

to be detectable after rephasing is induced by pulse 3, albeit in an everdecreasing manner, as T increases. The rephasing process that generates

the echo only occurs when there is some memory of the original inhomogeneous distribution of frequencies. In other words, the difference between

both signals is due to a delayed photon echo, which is formed only under

rephasing conditions ( > 0 for the k1 C k2 C k3 signal and < 0 for

the Ck1 k2 C k3 signal), so that these results clearly prove that a certain

amount of inhomogeneity is present for small T. The inhomogeneity is

diminished on the picosecond time scale by spectral diffusion processes.

Clearly, the asymmetrical stretching mode of the azide ion in water is not in

the motional narrowing limit, which is what is generally assumed for vibrational dephasing. The normalized first moment M1 T of the k1 C k2 C k3

signal (see Fig. 6) indicates that the inhomogeneity decays on at least two

time scales. Furthermore, the fact that M1 T is still finite at the longest

measured times implies the existence of a small inhomogeneous contribution, which is essentially static on the time scale of these experiments.

Consequently, the following model for the transition frequency correlation

function was used to fit the photon echo data (Fig. 5):

2



hυω10



υω10 0 i D



i 2 e



t/



i



C 0 2



(20)



iD1



A global fit was performed by applying the formalism outlined in

Section III. A [Equation (7)–(12), (14)–(17)], which connects the transition frequency fluctuation correlation function hυω10 υω10 0 i to the

three-pulse photon echo signal S T, and by varying the five parameters in the right-hand side of Equation (20). The fit and the resulting first



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