Spectral quantum beating in mixed frequency/time-domain coherent multidimensional spectroscopy

Article abstract of DOI:10.1021/jp071929+

Coherent multidimensional spectroscopy performed in the mixed frequency/time domain exhibits both temporal and spectral quantum beating when two quantum states are simultaneously excited. The excitation of both quantum states can occur because either the spectral width of the states or the excitation pulse exceeds the frequency separation of the quantum states. The quantum beating appears as a line that broadens and splits into two peaks and then recombines as the time delay between excitation pulses increases. The splitting depends on the spectral width of the excitation pulses. We observe the spectral quantum beating between the two nearly degenerate asymmetric carbonyl stretch modes in a nickel tricarbonyl chelate using the nonrephasing, ground state bleaching coherence pathway in triply vibrationally enhanced four-wave mixing as the time delay between the first two excitation pulses changes.

Full text of DOI:10.1021/jp071929+

J. Phys. Chem. A 2007, 111, 6999-7005
6999
Spectral Quantum Beating in Mixed Frequency/Time-Domain Coherent Multidimensional
Spectroscopy
Andrei V. Pakoulev, Mark A. Rickard, Nathan A. Mathew, Kathryn M. Kornau, and
John C. Wright*
Department of Chemistry, UniVersity of Wisconsin, Madison, Wisconsin 53706
ReceiVed: March 9, 2007; In Final Form: May 17, 2007
Coherent multidimensional spectroscopy performed in the mixed frequency/time domain exhibits both temporal
and spectral quantum beating when two quantum states are simultaneously excited. The excitation of both
quantum states can occur because either the spectral width of the states or the excitation pulse exceeds the
frequency separation of the quantum states. The quantum beating appears as a line that broadens and splits
into two peaks and then recombines as the time delay between excitation pulses increases. The splitting
depends on the spectral width of the excitation pulses. We observe the spectral quantum beating between the
two nearly degenerate asymmetric carbonyl stretch modes in a nickel tricarbonyl chelate using the nonrephasing,
ground state bleaching coherence pathway in triply vibrationally enhanced four-wave mixing as the time
delay between the first two excitation pulses changes.
Introduction
signal enhancement dependence on the excitation pulse time
delays. This approach allows one to define a specific coherence
pathway and the specific quantum states involved in the
pathway. In particular, it allows one to isolate the coherence
transfer pathways that show the interstate couplings.14
In time-domain ultrafast spectroscopy, the bandwidth of
typical excitation pulses is sufficient to simultaneously excite
multiple coherences with different frequencies. The time-domain
output then exhibits quantum beating from the interference of
coherences with different frequencies, and Fourier transforma-
tion displays a frequency-domain spectrum of the quantum
In this paper, we report the frequency-domain manifestation
of quantum beating. The frequency-domain quantum beating
appears as a spectral line-splitting that oscillates as a function
of the time delay between the first and second excitation pulses
and has a period defined by the frequency difference between
the coherences. It resolves two closely spaced states that are
unresolved in conventional infrared spectra. The quantum
beating occurs only when two or more quantum states overlap
with the narrow excitation bandwidth, either because the width
of the states prevents their resolution or because the frequency
difference between the states is less than the excitation
bandwidth. We show how the temporal and spectral character-
istics of quantum beating change as the excitation pulse widths
change from the impulsive to the continuous wave limit. We
also show that the magnitude and the phase of the periodic
oscillation of the splitting are defined by the excitation pulses’
spectral width and the quantum states’ dephasing rates.
1-4
states. Time-domain coherent multidimensional spectroscopy
CMDS) uses a series of excitation pulses that excite a series
(
5-8
of quantum mechanical coherences, populations, or both. The
coherences form a temporally and spatially coherent array of
oscillators that emit a directional beam defined by phase
matching. Changing the time delays between the input pulses
creates temporal beating between coherences, and Fourier
transforming the coherences’ temporal dependence creates
multidimensional spectra over a range defined by the excitation
pulse bandwidth. Cross-peaks appear in the spectra when
quantum states are coupled by intra- or intermolecular interac-
tions. Phase coherence is required between the excitation pulses
over the entire measurement time.
Frequency-domain CMDS methods such as doubly vibra-
tionally enhanced (DOVE) and triply vibrationally enhanced
(
TRIVE) four-wave mixing (FWM) measure the signal en-
9-13
hancements as a function of the excitation frequencies.
The
Experimental
excitation pulses are long, and their bandwidth is narrower than
the transitions, so any pulse typically excites only single
transitions. The accessible frequency range is defined only by
the tuning range of the excitation sources, and phase coherence
is required only during the time the excitation pulses are present.
Temporal information is contained in the transition line shapes.
Mixed frequency/time-domain DOVE- and TRIVE-FWM mea-
surements use pulses with intermediate spectral widths that are
The TRIVE-FWM experiments were performed using two
optical parametric amplifiers (OPAs) that were pumped by a
Ti:sapphire regenerative amplifier. The difference frequency
between the signal and idler of each OPA provided two tunable
infrared excitation beams, ω1 and ω2. The ω2 beam was split
to create a Bk 2 and a Bk 2′ beam. The excitation beams were focused
into a sample at angles in a box phase matching geometry such
that Bk 4 ) Bk 1 - Bk 2 + Bk 2′, where each k-vector is labeled according
to its frequency. The excitation pulses were 900 fs wide and
have an 18 cm fwhm. Delay lines adjusted the relative time
delays, τ21 ≡ τ2 - τ1 and τ2′1 ≡ τ2′ - τ1, between the Bk 1, Bk 2, and
Bk 2′ pulses. A monochromator with a HgCdTe2 detector measured
the FWM output beam at a frequency ωm. Five variables control
the TRIVE-FWM spectroscopy: ω1, ω2, ωm, τ21, and τ2′1.
Experiments acquire two-dimensional cross sections through this
comparable to the widths of the transitions and temporal widths
that are comparable to the transition lifetime.1
4-19
The multi-
-
1
dimensional spectra are measured in the frequency domain from
the signal enhancement dependence on the excitation frequen-
cies. The dynamics are measured in the time domain from the
*
To whom correspondence should be addressed. E-mail:
wright@chem.wisc.edu.
1
0.1021/jp071929+ CCC: $37.00 © 2007 American Chemical Society
Published on Web 07/12/2007

7000 J. Phys. Chem. A, Vol. 111, No. 30, 2007
Pakoulev et al.
stretch modes are labeled a1 and a2, and the symmetric carbonyl
stretch is labeled b. The two asymmetric stretches are nearly
degenerate and are not resolved in the infrared spectrum.
Figure 1 shows the Fourier transform infrared spectrum
(FTIR) of the NTC/C6D6 solution (black line) and pure C6D6
(
blue line) with a 200 µm path length. The two asymmetric
-1
stretches are centered near 1995 cm , and the symmetric stretch
appears at 2069 cm . A reasonable fit of the asymmetric stretch
region can be made with a single Gaussian peak centered at
-1
-
1
-1
1
995 cm and a fwhm of 18 cm (Figure 2a). The only
discrepancy is the small difference at the top of the peak. A
better fit requires two Gaussian peaks, one centered at 1990
-
1
-1
-1
cm with a fwhm of 13 cm and one at 1999 cm with a
-
1
fwhm of 12 cm (Figure 2b). The FTIR spectrum and fits,
although suggestive, do not definitively confirm that the
asymmetric stretches in NTC have different frequencies.
Theoretical Model of Quantum Beating. In TRIVE-FWM,
transitions occur from the ground state labeled g to two
vibrational modes labeled a and b, where a and b can represent
any two vibrational states. There are 12 coherence pathways
consisting of six time orderings (labeled I-VI) involving either
Figure 1. FTIR spectrum of Ni(CO)
pure C
3 3 6 6
(PPh ) in C D (black line) and
6
6
D .
five-dimensional space, typically by measuring the FWM
intensity as a function of two variables and fixed values of the
other three. For the experiments described in this paper, the
time delays were set to values that resulted in the following
temporal ordering of the excitation pulses: Bk 2′, Bk 2, and Bk 1. The
monochromator was tuned so ωm ) ω1. These experimental
parametric (labeled R) or nonparametric (labeled â) path-
ways.
15,16,18,19
For example, the parametric and nonparametric
2′
9
-2
1
pathways for time ordering VI are gg
8 ag 8 gg 98 bg (VIR)
2′
9
-2
1
and gg
8 ag 8 aa 98 (a + b)a (VIâ), respectively, where the
choices define a single coherence pathway that is labeled
pathway VIR.1
5,16
letters represent the diagonal and off-diagonal density matrix
elements for the coherences and populations of states g, a, b,
and (a + b) and the numbers above the arrows label the
frequencies of the three excitation beams. For the case presented
in this work, the second interaction can also access a different,
nearby vibrational state and create a zero quantum coherence
It corresponds to the nonrephasing, ground
state bleaching pathway in 2D-IR or transient grating experi-
ments. The first excitation creates a coherence between a
vibrational state and the ground state. The second excitation
pulse creates a ground state population grating. The third
excitation creates a second coherence between a vibrational state
2′
-2
1
(which can be different or the same as that in the first coherence)
by the pathway gg
98 a1g
8 a1a2 98 (a1 + b)a2. This paper
and the ground state. This last coherence forms a spatially and
temporally coherent array of oscillators that emits the directional
output beam.
uses TRIVE pathway VIR as a specific example of frequency-
domain quantum beating. Other pathways also exhibit quantum
beating. The bg density matrix element for pathway VIR is
The sample was the organometallic complex Ni(CO)3(PPh3)
∞
∞
∞
(
(
abbreviated as NTC). It was synthesized from a 10 mM Ni-
CO)2(PPh3)2 tetrahydrofuran solution that was converted to the
F (t, τ , τ , τ ) ∝ ú ∫ ∫ ∫ E(ω ,t -
bg
1
2
2′
2′ a
-
∞
-∞ -∞
τ )G (t - t )E(ω ,t - τ )G (t - t )E(ω ,t - τ )G (t -
NTC by oxidation in the air. After 3 h, the solvent was
evaporated under a nitrogen atmosphere, and the remaining solid
was dissolved in deuterated benzene. The oxidation formed also
NiO solid that was removed by filtration. NTC has tetrahedral
geometry and three carbonyl vibrational modes. The asymmetric
2′
ag
b
a
2
b
2
gg
c
b
1
c
1
bg
t ) dt dt dt (1)
c
a
b
c
where ú contains the sign and proportionality factors, E(ωi, t -
-
iω (t-τ )-((t-τ )2)/(σ 2)
i i i i
τi) ) e
is the excitation electric field, Gmn(t)
) peak at 1996 cm- with (a) a single Gaussian peak and (b) two Gaussian peaks. The black and green curves
1
Figure 2. Fit of the Ni(CO)
3
(PPh
3
represent the data and fits, respectively.

Spectral Quantum Beating
J. Phys. Chem. A, Vol. 111, No. 30, 2007 7001
Figure 3. The VIR and VIâ coherence pathways with states shown on the right side.
Figure 4. Schematic diagram indicating the positions of the peaks corresponding to each of the pathways sketched in Figure 3. (a) The positions
when the two asymmetric modes are degenerate and (b) the positions when they are split. The peak positions correspond to vibrational states with
-
1
-1
-1
energies a
1
) 1990 cm , a
2
) 1999 cm , and b ) 2069 cm
.
θ(t) e-
iωmnt-Γmn
t
is the molecular response involving the mn
reasons, a 2D TRIVE spectrum is much more sensitive to the
frequencies of the fundamental modes than a linear spec-
trum.
)
density matrix element (frequency ωmn and dephasing rate Γmn),
θ(t) ≡ 0 when t < 0 and θ(t) ≡ 1 when t g 0 and the integration
is performed over the three interaction times.20 Numbers label
the excitation field variables (ωi, τi, σi), and letters label the
time ordered interaction times: ta, tb, tc where ta < tb < tc. This
equation has two resonances when ω2′ ) ωag and ω1 ) ωbg.
The comparable equation for the VIâ pathway has resonances
when ω2′ ) ωag and ω1 ) ωa+b,a.
For this paper, the vibrational states can be any of the
fundamentals, overtones, and combination bands of the a1, a2,
and b vibrational states shown in Figure 3. For three vibrational
states, coherence pathways VIR and VIâ create 18 different
peaks in two-dimensional spectra of the intensity as a function
of ω1 and ω2. Figure 3 describes the state evolution for each
spectral feature, and Figure 4 shows a schematic two-
dimensional vibrational spectrum of these peaks. If states a1
and a2 are degenerate, the 2D ω1/ω2 spectrum for NTC will
resemble the schematic in Figure 4a. The alpha pathways are
parametric pathways that involve only fundamental modes and
are colored cyan; the beta pathways are nonparametric pathways
that involve overtones or combination bands and are colored
green. The 2D schematic spectrum changes if a1 and a2 are not
degenerate (Figure 4b). The frequency shifts create two coher-
ence pathways that interfere constructively or destructively at
the quantum level with other nearby pathways. For these
This paper is concerned with the quantum beating when the
two unresolved a1 and a2 vibrational states are excited with the
first excitation pulse (frequency ω2) as the excitation pulse width
varies from the impulsive to the continuous wave (cw) limit.
The two coherences that beat are probed at some later time, τ,
by the second and third excitation pulses. In order to obtain a
closed form solution, we consider that the first excitation results
from a square excitation pulse centered at τ2′ ) 0 with a full
width of 2s. For states a1 and a2, the polarization is then
s
E (ω ) e-iω2ta(e-(iωa1g+Γa1g)(τ-ta)
o
F(τ) ∝
∫
+
2
-
s
e-(iωa2g+Γa2g)(τ-ta)) dt (2)
a
Integrating F and assuming that τ > s yields eq 3 with ∆a,b )
ωa,b - ω2, where ω2 is the center frequency of the laser.
(
-iωa1g-Γa1g)τ
e
(ei∆
a1gs+Γa1g
s
-i∆a1gs-Γa1gs
F(τ) )
- e
) +
i∆ + Γ
a g
a g
1
1
e(-iωa2g-Γa2g)τ
(ei∆
a2gs+Γa2g
s
- e
-i∆a2gs-Γa2gs
) for τ > s (3)
i∆ + Γ
a g
a g
2
2

7002 J. Phys. Chem. A, Vol. 111, No. 30, 2007
Pakoulev et al.
Figure 6. Two-dimensional representation of the normalized intensity
as a function of the excitation frequency and the time delay (τ) between
the excitation and the measurement. The intensity is normalized to the
most intense feature at a given delay time. The dephasing rate
Figure 5. Two-dimensional representation of the intensity as a function
of the excitation frequency and the time delay (τ) between the excitation
and the measurement for a two-state system with no dephasing. The
excitation is a rectangular pulse (width of 2s) where (a) s ) 0.5 ps, (b)
s ) 2 ps, and (c) s ) 9.7 ps. The separation between the two states
corresponds to that in Table 1.
-
1
corresponds to 6 cm . Other parameters are identical to Figure 5.
The first two terms represent the individual resonances of the
two vibrational states broadened by the excitation pulse
bandwidth. The cos[(ωa g - ωa g)τ] in the last term is
The appendix shows the corresponding equation for arbitrary
times. In this simplified model, we assume that the next two
interactions are impulsive and transform F(τ) to the output
coherence, so the temporal and spectral behavior of F(τ) is
reflected in the output. The observed intensity for homodyne
2
1
responsible for the temporal quantum beating with a period T
) 2π/(ωa2g - ωa1g). Its amplitude is determined by the product
of the two sinc functions that depend on the detuning from each
resonance and the pulse width. It is this product that is
responsible for the spectral quantum beating described in this
paper. If τ ) (2n + 1)π/(ωa g - ωa g), cos[(ωa g - ωa g)τ] )
2
detection (integration of all output signal) is then I ∝ |Ftotal| .
For clarity, we calculate the intensity when all dephasing rates
are zero. Then,
2
1
2
1
-
1, and eq 4 becomes
2
2
sin [(ω - ω )s] sin [(ω - ω )s]
a g
2
a g
2
1
2
I(τ, ω ) ∝
+
+
2
2
2
(
2n + 1)π
(ω
- ω2)
(ω - ω2)
a g
a g
1
2
I τ )
, ω2
∝
(
(ω - ω )
)
a g
a g
1
sin[(ω - ω )s] sin[(ω - ω )s]
2
a g
2
a g
2
1
2
sin[(ω - ω )s] sin[(ω - ω )s] 2
2
cos[(ω - ω )τ]
a g
a g
a g
2
a g
2
2
1
{
}
1
2
(ω
- ω )(ω - ω2)
a g 2 a g
1 2
-
(5)
(
(ω - ω2)
(ω - ω2)
)
a g
a g
2
(
4)
1

Spectral Quantum Beating
J. Phys. Chem. A, Vol. 111, No. 30, 2007 7003
Figure 7. Two-dimensional spectra of the log(FWM intensity) as a function of ω
1
and ω
2
when ω
m
1
) ω for time ordering VI and two sets of
delay times: (a) τ2′1 ) -6.0 ps, τ21 ) -5.0 ps and (b) τ2′1 ) -7.6 ps, τ21 ) -5.0 ps.
This function shows the spectral beating between the two
Results and Discussion
resonances. The intensity is zero if ω2 ) (ωa g - ωa g)/2, since
2
1
Figure 7 shows two-dimensional spectra of NTC for time
ordering VI and two sets of delay times. Both spectra show the
FWM intensity as a function of ω1 and ω2 when ωm ) ω1. The
left-hand 2D spectrum with τ2′1 ) -6.0 ps, τ21 ) -5.0 ps
the two sinc functions have the same amplitude at this point.
The intensity reaches a maximum at two other frequencies where
one or the other state is closer to resonance, so the two sinc
functions have different amplitudes. The splitting between the
maxima depends on the temporal width of the excitation pulse.
(
Figure 7a) has diagonal peaks at (ω1, ω2) ≈ (2000, 2000) and
-
1
(
2069, 2069) cm corresponding to creation of ag and bg
Figure 5 shows how the spectral and temporal behavior of
this two state system evolves as the excitation pulse width, s,
varies between impulsive and cw behavior. The two-dimensional
display shows the intensity as a function of the measurement
time, τ, and the excitation frequency, ω2. Short pulses have wide
bandwidths that excite both states equally, and temporal quantum
beats appear. Long pulses have narrow bandwidths that excite
primarily one state, so the temporal beating is weak, and the
states’ splitting is resolved. For intermediate pulse widths, the
relative amplitude of each state depends on the excitation
frequency’s detuning from each state, and temporal and spectral
coherences, respectively, and cross-peaks at (ω1, ω2) ≈ (2000,
2
069) corresponding to the creation of a bg and then an ag
-1
coherence and (2069, 2000) cm corresponding to creation of
an ag and then a bg coherence, all of which are broadened by
the presence of the other pathways. The transitions associated
with the a1 and a2 states are not resolved; neither is there any
resolution of the R and â transitions. This lack of resolution is
expected since the spectral separation of the peaks is small when
compared with the bandwidth of the excitation pulses.
The spectrum changes significantly, however, when τ2′1 is
increased to -7.6 ps (Figure 7b). This change corresponds to
an increase in the delay time between the first and second pulses
from 1.0 to 2.6 ps. There are two consequences. First, the upper
set of peaks is much stronger than the lower set of peaks. This
difference is caused because the bg coherence that results from
the first pulse in the upper set of peaks has a slower dephasing
rate than the ag coherence formed for the lower set of peaks.
Consequently, the bg coherence does not decay as quickly, so
the peaks that result from the bg coherence are stronger. Second,
the lower set of peaks at (ω , ω ) ≈ (2000, 2000) and (2069,
quantum beating occurs. If ω2 ) (ωa g - ωa g)/2, both states
2
1
contribute equally, and only temporal beating is observed. If τ
π/ωa g - ωa g, the contribution from both states depends on
)
2
1
the detuning from each one, and a spectral splitting occurs along
the ω2 axis. The spectral splitting appears periodically in time
and has the same period.
The case when the dephasing is not zero is significantly more
complex and appears in the appendix. The equation now
contains factors that depend on cos[(ωa g - ωa g)τ] and
2
1
1
2
-
1
sin[(ωa g - ωa g)τ], so the quantum beating occurs with the same
2000) cm split into two different sets of peaks in the ω2
2
1
period, but the phase is shifted by the pulse width and dephasing
rates. Figure 6 shows the changes in the spectral and temporal
behavior of the quantum beating for this case. In order to
compensate for the signal’s exponential decay, the intensity was
normalized to the brightest feature at each measurement time.
The spectral and temporal quantum beating become clearer with
this normalization. Note the dependence of the quantum
beating’s phase, the changes in the maximum and minimum
positions with the dephasing rate and the pulse width, and the
changes in the magnitude of the splitting with the pulse width.
For this case, the quantum beating is observed even for a long
pulse length because the nonzero dephasing rate has broadened
each transition so they spectrally overlap and interfere.
dimension. The lower row of peaks is located at ω ≈ 1980
2
-
1
-1
cm , and the upper row, at ω2 ≈ 2007 cm . These positions
do not correspond to any peaks in the FTIR data and indicate
the presence of frequency-domain quantum beating between a1g
and a2g coherences formed by the first excitation pulse. There
is no splitting or quantum beating for the upper set of peaks
since the first coherence is a single bg coherence.
Figure 8 is a two-dimensional display of the FWM intensity
as a function of τ2'1 and ω with fixed values of τ ) -5.0 ps
2
21
-
1
and ω1 ) ωm ) 2069 cm . It shows both the spectral and
temporal beating. Figure 8a shows the logarithm of the FWM
intensity; Figure 8b renormalizes the logarithm of the FWM
intensity to the maximum value at τ2′1 ) -7.0 ps; Figure 8c

7004 J. Phys. Chem. A, Vol. 111, No. 30, 2007
Pakoulev et al.
Figure 8. Two-dimensional representation of the log(FWM intensity) as a function of τ2′1 and ω
2
with a fixed value of ω
1
) 2069 cm-1. Parts a,
b, and c are taken under identical conditions. The data in b was obtained by resetting the signal amplification to maximize the FWM intensity at
τ
2′1 ) -7.0ps and enhance the signal at long delay times. The color bar again represents the log(FWM intensity). The FWM intensity has been
renormalized in c to the peak intensity at any given τ2′1 to observe how the spectral shapes and positions change with delay time. The color bar
represents the FWM intensity. Part d is a simulation of the renormalized data using the method described in the text and the parameters in Table
1
. It should be compared to part c.
TABLE 1: Parameter Values Used to Simulate
Experimental Data
renormalizes the data in a and b to the maximum of the FWM
intensity for each τ2′1 value. At τ2′1 ) -5.0 ps, there is a single
9 cm-
1
-
1
ω
a2,g-ω
broad peak at ω2 ) 2000 cm . The peak splits into two peaks
at τ2′1 ) -7.8 ps that then recombine at τ2'1 ) -8.4 ps. These
observations indicate the interference is not constant, but
periodic. The splitting is not symmetric. The higher energy peak
is stronger than the lower energy peak. A similar pattern of
splitting and recombination occurs with the R1, R2, R8, and
R9 pathways (data not shown).
a1,g
a
-1
σ
Γ
Γ
Γ
Γ
7.6 cm
6.5 cm
6.15 cm
2.25 cm
0.2 cm
-
1
a1,g
a2,g
b,g
-1
-1
1
-
g,g
a
Standard deviation of Gaussian excitation pulse.
The quantum beating occurs in TRIVE because the two
different coherences excited by the first pulse have different
phases when the second pulse interacts and can constructively
or destructively interfere. Quantum beating is not seen between
the coherences that result after the third pulse because the
experiment does not temporally resolve the output coherence.
Consequently, beating is not observed along the ω1 axis, even
when the a1g and a2g coherences are excited. Figure 8d shows
a simulation of the temporal and spectral quantum beating using
a numerical integration of eq 1 assuming Gaussian shaped
pulses. Note that the splitting in the simulations is not symmetric.
The asymmetry is due to differences in the dephasing rates for
the two coherences. The higher energy peak has a slightly slower
dephasing rate, and consequently, its intensity is higher. The
frequency difference and the dephasing rates for the simulation
are summarized in Table 1. Their values match those obtained
by fitting the line shapes of the FTIR spectra. The frequency
difference of 9 cm would correspond to a quantum beating
period of 3.7 ps. If the excitation pulse were impulsive, the
intensity minimum would occur at a 1.85 ps delay (correspond-
ing to τ2′1 ) -6.85 ps). Instead, Figure 8 shows it appears at
2.75 ps (τ2′1) -7.75 ps). This shift is matched by the simulation
that shows that the difference is caused by the phase shift in
-
1

Spectral Quantum Beating
J. Phys. Chem. A, Vol. 111, No. 30, 2007 7005
F ) sin(∆ s) cosh(Γ s)[-
the quantum beating from the finite widths of the excitation
pulses and dephasing rates.
4
a
a2
a
2
(
∆ ∆ + Γ Γ ) sin(ωa1,a2τ) +
a
a
a
a
1
2
1
2
Conclusions
(∆ Γ - ∆ Γ ) cos(ωa1,a2τ)]
a a a a
2 1 1 2
Mixed frequency/time-domain coherent multidimensional
spectra exhibit a time- dependent spectral splitting between
unresolved quantum states that result from quantum beating
between quantum states that are simultaneously excited, either
because they overlap spectrally or because the excitation pulse
bandwidth is broader than their frequency difference. The
quantum beating creates a line shape that depends on the delay
time between excitation pulses and a splitting that depends on
the excitation pulse bandwidth. The spectral quantum beating
allows the mixed frequency/time-domain methods such as
DOVE and TRIVE-FWM to resolve quantum states that are
not resolved in conventional one-dimensional spectra.
F ) cos(∆ s) sinh(Γ s)[(∆ ∆ + Γ Γ ) cos(ωa1,a2τ) +
4
b
a
a
a
a
a
a
2
2
2
1
2
1
(∆ Γ - ∆ Γ ) sin(ω τ)]
a a a a
a1,a
2 1 1 2 2
F ) sin(∆ s) cosh(Γ s)
5
a1
a1
F ) sin(∆ s) cosh(Γ s)[(∆ ∆ + Γ Γ ) cos(ωa1,a2τ) +
6a
a
a
a
a
a
a
2
2
2
1
2
1
(∆ Γ - ∆ Γ ) sin(ωa1,a2τ)]
a a a a
2 1 1 2
F ) cos(∆ s) sinh(Γ s)[(∆ ∆ + Γ Γ ) sin(ω τ) +
6
b
a
a
a
a
a
a
a1,a
2
2
2
1
2
1
2
(
∆ Γ - ∆ Γ ) cos(ωa1,a2τ)]
a a a a
1 2 2 1
Acknowledgment. This work was supported by the National
Science Foundation under Grant CHE-0650431. Acknowledg-
ment is also made to the donors of the Petroleum Research Fund
of the American Chemical Society for their support of this
research. This work was supported in part by a fellowship from
Merck Research Laboratories. A.V.P. and M.A.R. contributed
equally to this reported research.
References and Notes
(
1) Hack, E.; Huber, J. R. Int. ReV. Phys. Chem. 1991, 10, 287.
2) Haroche, S.; Paisner, J. A.; Schawlow, A. L. Phys. ReV. Lett. 1973,
(
30, 948.
(3) Bitto, H.; Huber, J. R. Acc. Chem. Res. 1992, 25, 65.
(
4) Carter, R. T.; Huber, J. R. Chem. Soc. ReV. 2000, 29, 305.
Appendix
(5) Wright, J. C.; Condon, N. J.; Murdoch, K. M.; Besemann, D. M.;
Meyer, K. A. J. Phys. Chem. A 2003, 107, 8166.
Equation 3 was derived from eq 2 under the assumption that
τ > s. The more general equation is
(
6) Jonas, D. M. Annu. ReV. Phys. Chem. 2003, 54, 425.
7) Mukamel, S. Annu. ReV. Phys. Chem. 2000, 51, 691.
(
e-iωa,bτ-Γa,bτ
i∆a,b + Γa,b
(8) Cho, M. Two Dimensional Vibrational Spectroscopy. In AdVances
in Multi-Photon Processes and Spectroscopy, 1st ed.; Lin, S. H.; Villaeys,
A. A.; Fujimura, Y., Eds.; World Scientific: Singapore 1999; Vol. 12, pp
1.
a,bτ-Γa,b
τ
-i∆a,bs-Γa,bs
F(τ) )
[(e-iω
- e
) θ(s - τ) +
ei∆a,bs+Γa,bs - e-i∆a,bs-Γa,bs) θ(τ - s)] θ(τ + s) (3′)
(
(9) Condon, N. J.; Wright, J. C. J. Phys. Chem. A 2005, 109, 721.
Equation 4 was derived from eq 3 if the dephasing rates were
zero. The more general equation is
(10) Murdoch, K. M.; Condon, N. J.; Zhao, W.; Besemann, D. M.;
Meyer, K. A.; Wright, J. C. Chem. Phys. Lett. 2001, 335, 349.
(11) Zhao, W.; Wright, J. C. Phys. ReV. Lett. 2000, 84, 1411.
I(τ, ω ) ∝ F + F [F (F + F ) + F (F + F )]
2
1
2
3
4a
4b
5
6a
6b
(12) Zhao, W.; Wright, J. C. J. Am. Chem. Soc. 1999, 121, 10994.
(4′)
(13) Zhao, W.; Wright, J. C. Phys. ReV. Lett. 1999, 83, 1950.
(
14) Rickard, M. A.; Pakoulev, A. V.; Kornau, K.; Mathew, N. A.;
where
Wright, J. C. J. Phys. Chem. A 2006, 110, 11384.
-
2Γa1τ
(15) Pakoulev, A. V.; Rickard, M. A.; Meyers, K. A.; Kornu, K.;
Mathew, N. A.; Thompson, D. C.; Wright, J. C. J. Phys. Chem. A 2006,
110, 3352.
e
F1 )
[cosh(2Γ s) - cos(2∆ s)] +
a a
1 1
2
2
∆a1 + Γa1
(16) Meyer, K. A.; Thompson, D. E.; Wright, J. C. J. Phys. Chem. A
-
2Γa2τ
e
2004, 108, 11485.
[cosh(2Γ s) - cos(2∆ s)]
a a
2 2
2
2
(17) Meyer, K. A. Frequency-Scanned Ultrafast Spectroscopic Tech-
∆a2 + Γa2
niques Applied to Infrared Four-Wave Mixing Spectroscopy; Ph. D.
Dissertation. University of Wisconsin-Madison, 2004.
e-
(Γa1+Γa2)τ
4
(
18) Besemann, D. M.; Meyer, K. A.; Wright, J. C. J. Phys. Chem. B
F2 )
2
2
2
2
2
004, 108, 10493.
(
∆a1 + Γ )(∆ + Γ )
a2
a a
1 2
(
19) Meyer, K. A.; Wright, J. C. Chem. Phys. Lett. 2003, 381, 642.
20) Mukamel, S. Principles of Nonlinear Optical Spectroscopy, 1st ed.;
(
F ) cos(∆ s) sinh(Γ s)
3
a
a
1
1
Oxford University Press: New York, 1995.