D. The Three-Pulse Photon Echo Experiment
Tải bản đầy đủ - 0trang
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
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
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
Copyright © 2001 by Taylor & Francis Group, LLC
294
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
Copyright © 2001 by Taylor & Francis Group, LLC
Proteins and Peptides
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
Copyright © 2001 by Taylor & Francis Group, LLC