REPORTS

Femtosecond Phase-Coherent
Two-Dimensional Spectroscopy
Peifang Tian,1 Dorine Keusters,2 Yoshifumi Suzaki,3Warren S. Warren2
Femtosecond phase-coherent two-dimensional (2D) spectroscopy has been
experimentally demonstrated as the direct optical analog of 2D nuclear
magnetic resonance. An acousto-optic pulse shaper created a collinear
three-pulse sequence with well-controlled and variable interpulse delays
and phases, which interacted with a model atomic system of rubidium vapor.
The desired nonlinear polarization was selected by phase cycling (coadding
experimental results obtained with different interpulse phases). This method may enhance our ability to probe the femtosecond structural dynamics
of macromolecules.
Two of the central questions in physicalchemistry and biophysics are how the structures of complex molecular systems (suchas proteins and hydrogen-bonded liquids)
evolve, and how such dynamics are related
to important chemical and biological processes. Two-dimensional (2D) nuclear
magnetic resonance (NMR) can resolve thestructure of complex molecules with atomic
resolution, but its millisecond time scale is
too slow to follow ultrafast dynamics (tensof femtoseconds to picoseconds), which is
crucial to understanding many chemicaland biological processes (1–3). Both ultrafast electronic and vibrational 2D spectroscopy have recently shown promise for use
in probing the structures and dynamics ofsystems such as laser dyes, metal carbonyls, peptides, and proteins (3–16 ). However, these techniques use noncollinear pulses
to average over phase in a spatially coherent manner, rather than collinear phasecoherent pulses, which are pulses withwell-defined interpulse phases as in NMR(17 ). The latter approach, the direct opticalanalog of NMR, can provide advantages
over existing techniques for extracting
spectral information. We demonstrated experimentally that it is technically possible.
This direct optical analog of 2D NMR,
combined with current 2D spectroscopic
methods, can enhance capabilities to resolve the ultrafast structural dynamics ofcomplex systems.
In current ultrafast 2D experiments, amolecular system interacts with three ultrafast pulses traveling along different directions with wave vectors k1, k2, and k3,
respectively (Fig. 1A) (4–9). The first pulse
prepares the system in a particular coher1
Department of Electrical Engineering, 2Departmentof Chemistry, Center of Ultrafast Laser Applications,
Princeton University, Princeton, NJ 08544, USA. 3Department of Advanced Materials Science, Faculty ofEngineering, Kagawa University, Takamatsu, Kagawa
761– 0396, Japan.

ence, which oscillates during the period ␶1.
The second and third pulses interrupt thisoscillation and create a new coherence. The
complete polarization P is then detected at
a delay of ␶2 after the third pulse and as afunction of delay ␶1. A 2D function
P(␶1,␶2 ) is thus obtained and Fourier-transformed to create a 2D spectrum S(f1, f2 )(4–9). The experimental 2D spectral features ( peak positions, line shapes, and polarization dependence) reveal the propertiesof the molecular system (transition frequencies, coupling, anharmonicity, and interactions with the environment) fromwhich structural information can be deduced (3–16 ). Structural dynamics can bemonitored by taking a series of 2D spectraS(f1, f2 ) as a function of waiting time T
between the second and third pulses (4–12).
Different interactions between the molecular system and the three pulses generatemany peaks in the 2D spectrum S(f1, f2 )(17 ). Structural information can often be
obtained from just a few peaks caused byparticular (and often weak) third-order nonlinear polarizations, such as the subject of
this experiment: the photon echo signal
P(3)echo. Other peaks only give redundant information and congest the spectrum. In current ultrafast 2D spectroscopy, P(3)echo is isolated by measuring it in a unique direction
ks ⴝ k3 ⫹ k2 – k1 determined by thephase-matching condition (Fig. 1A) (4–9).
The phase information on P(3)echo is obtainedwith heterodyne detection. Phase matchinghas the advantage of spatial separation of
the desired and undesired signals.
However, P(3)echo also has a unique phasedependence on the input pulses. Therefore,
when collinear phase-coherent pulses interact
with the molecular system, P(3)echo can be selected by its phase dependence, leading to thedirect optical analog of 2D NMR: ultrafast
2D spectroscopy with phase cycling (17). In
this approach, phase information on P(3)echo isobtained by phase cycling (altering the inter-

pulse phase shifts in a controlled manner).
Phase cycling has several advantages overphase matching. Pulses with specific controlled phases can alter the effective molecular evolution, thus coherently averaging away
specific interaction mechanisms or enhancing
signal intensity (1, 2, 17). The collinear approach also conveniently permits data acquisition in a “rotating frame” that dramaticallydecreases the number of required data points.
In addition, noncollinear phase matching
only works in extended systems (many absorbers with spatial distribution ⬎⬎1/⌬k,where ⌬k is the difference between the
wave vectors of the polarization and its
corresponding electric field), whereas thecollinear phase cycling is not constrainedby the size of the sample. Finally, the complexity of the phase-matching experiment
increases rapidly with the addition of extra
pulses coming from different directions.

Fig. 1. (A) Pulse sequence used in ultrafast 2D
spectroscopy with phase matching. Three excitation pulses, separated by delays ␶1 and T andtraveling in different directions k1, k2, and k3,
respectively, are focused onto the sample. The(3)photon echo signal Pecho
is emitted in a different direction ks ⴝ k3 ⫹ k2 – k1 and hence is(3)
spatially isolated. Pecho
is measured as a function of ␶1 and ␶2. (B) Schematic diagram of thelowest four energy levels of Rb atoms relevant
to our experiment: ground state 5S1/2, two
singly excited states 5P1/2 and 5P3/2, and one
doubly excited state 5D. (C) System I: D1 andD2 transitions share the same ground state(absorption wavelength ␭ at 794.7 and 780 nm,
respectively). (D) System II, with one ground(5S1/2 ), one singly (5P3/2 ) excited, and one
doubly (5D) excited state (corresponding to
two transitions at ␭ ⫽ 780 and 776 nm, respectively), is used to simulate anharmonicity.
(E) Pulse sequence used in ultrafast 2D spectroscopy with phase cycling. Three collinear
pulses, with phases ␾1, ␾2, and ␾3, are separated by delays of ␶1 and ␶2. This pulse sequence is equivalent to that of (A) by settingT ⫽ 0 and including a third pulse to detect thepolarization.

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Optical 2D spectroscopy with phase cycling was proposed in 1976 (1), and complex
phase-cycled sequences were demonstrated
with nanosecond resolution in 1981 (18). Recent developments in ultrafast pulse-shaping
technologies [such as the acousto-optic pulse
shaper (AOPS) and liquid-crystal– based
pulse shaper (LCPS)] and highly stable amplified ultrafast laser systems now make itpossible to create arbitrarily complex femtosecond pulse sequences (19–21). In fact, ultrafast 1D spectroscopy with phase cycling
was demonstrated by creating a phasecoherent pulse pair by using either an AOPS
or a complicated feedback loop (21, 22).
We generated a phase-coherent threepulse sequence using an AOPS and demonstrated ultrafast 2D spectroscopy using phasecycling on a model system: Rubidium (Rb)atomic vapor. Rb atoms can be used to model
two crucial properties: coupling and anharmonicity of a molecular system, which manifest themselves as cross peaks (f1 ⫽ ⫾ f2 ) on
a 2D spectrum (17). The lowest four energy
levels relevant to our experiment (Fig. 1B)
are the ground state 5S1/2, two singly excitedstates 5P1/2 and 5P3/2, and one doubly excitedstate 5D. The three relevant transitions are
D1: 5S1/2 3 5P1/2; D2: 5S1/2 3 5P3/2; and5P3/2 3 5D, with transition frequencies of fa,
fb, and fc, respectively. The transition 5P1/2
3 5D was not directly excited in our experiment because of its small transition dipole
moment. Depending on the input pulse spectrum, two systems can be obtained. System I(Fig. 1C) is a three-level “V” system, where
the two transitions D1 and D2 share the sameground state, allowing us to simulate coupling. System II (Fig. 1D) with one ground,
one singly excited, and one doubly excitedstate simulates anharmonicity. We can switch
between these two systems by controlling thefrequency content of the laser pulses.
Three collinear pulses with delays ␶1 and␶2 were used (Fig. 1E). The first and second
pulse are the excitation pulses and serve thesame role as the three pulses in Fig. 1A withT ⫽ 0. The phase and amplitude of the ultrafast oscillating polarization need to be measured as a function of ␶1 and ␶2, which isdifficult to achieve directly. Therefore, anadditional (third) pulse is added at delay ␶2
after the second pulse. This third pulse converts the polarization at delay ␶2 into a population of the excited states (18). This population reflects the amplitude and phase of thepolarization at ␶2 and is thus related to thethird-order polarization. Although the mostcommon method of magnetization detection
in NMR is nuclear induction (directly observing the oscillating magnetization), detection
directly analogous to our method is done inmagnetic resonance experiments such as
force microscopy. Hence, the complete polarization can be obtained by measuring the1554

population as a function of ␶1 and ␶2, which
can be done by measuring the fluorescencefrom or absorption by Rb atoms. Anotheralternative, which would probably be very
useful for infrared (IR) pulse sequences tostudy multilevel vibrational systems, is to
apply a final laser pulse at a wavelength (␭)
that is more strongly absorbed by a vibrationally excited molecule than by a groundstate molecule. For example, for a 2000 cm⫺1
vibrational frequency, we may go to the redend of the ultraviolet (UV) absorption spectrum and thus look at changes in UV-visible
fluorescence with phase cycling.
The total electric field E(t)can be written as
E(t) ⫽ [A(t ⫹ ␶1) e⫺i(2␲f0t ⫺ ␾1) ⫹ A(t)e⫺i(2␲f0t ⫺ ␾2)⫹ A(t ⫺ ␶2)e⫺i(2␲f0t ⫺ ␾3)]⫹ a complex conjugate where A(t) and f0 are the envelope andcentral frequency of each pulse, and ␾1, ␾2,and ␾3 are the phases of pulses 1, 2, and 3,
respectively.
In the system I case, the population ␳ of
5P1/2 and 5P3/2 after the three pulses is冘36

a i c i 共␾1,␾2,␾3)e i 2␲f 1␶1e i 2␲f 2 ␶2of other peaks in the 2D spectrum. For example, the peak at (0, 0) is caused by the linear
absorption of each individual pulse and does
not depend on pulse phase. To eliminate suchcongesting peaks, we take two sets of data for
two phase combinations ␾1,␾2,␾3 being XXX
and XXX, respectively, and record the corresponding total populations as ␳XXX (␶1,␶2 )and ␳XXX (␶1,␶2 ), where symbols X, Y, X, andY represent a phase of 0, ␲/2, ␲, and 3␲/2,
respectively. The amplitude of the photonecho peaks is doubled in the 2D spectrumof the two-step phase-cycled signal ␳2 ⫽
␳XXX – ␳XXX, whereas the DC peak disappears along with all of the peaks that do notdepend on the phase of the third pulse (17 ).
Populations for 16 different phase combinations of the three pulses can be recorded
and then combined correctly to select thephoton echo peaks while completely eliminating all of the other peaks. For example, thephoton echo peaks can be selected by thefollowing 16-step phase cycle (17)¯⫹ ␳X¯ XX¯ ⫺ ␳Y X Y ⫹␳16 ⫽ (␳XXX ⫺␳X XX¯ ⫺␳XXX

␳Y¯ X Y ⫹ ␳Y X Y¯ ⫺ ␳Y¯ X Y¯ ) ⫹i⫽1where each term corresponds to a particularpolarization and thus to a unique peak in the
2D spectrum (17). The position, amplitude,
and phase dependence of each peak are (f1,f2 ), ai, and ci(␾1,␾2,␾3), respectively. The
photon echo signal includes two antidiagonal
peaks at (⌬fa, –⌬fa) and (⌬fb, –⌬fb) and twocross peaks (⌬fa, –⌬fb) and (⌬fb, –⌬fa) withthe same phase dependence ei(–␾1 ⫹ 2␾2 – ␾3)(17). ⌬fa ⫽ fa – fL and ⌬fb ⫽ fb – fL are thedifference frequencies between the transition
frequencies and the locked frequency fL of
the laser (23). Hence, the phase-cycling
scheme systematically reduces the oscillation
frequency and allows a reduction in the number of sampling points in the time domain(17, 21, 22).
The photon echo peaks can be understoodas follows. The first pulse excites coherences
D1 and D2 during ␶1. Two interactions withthe second pulse can reverse the phase of thepolarization, leading to an antidiagonal peakat (⌬fa, –⌬fa), or convert a D1 coherence into
a coherence at D2, resulting in a cross peak at(⌬fa, –⌬fb). Similarly, two other peaks appear
for those atoms oscillating at D2 during ␶1.
The phases that the photon echo peaks acquire during ␶1 are conjugate with those obtained during ␶2. For example, the antidiagonal peak (⌬fa, –⌬fa) has phases e i2␲⌬fa␶1 ande⫺i2␲⌬fa␶ 2. This rephasing property makes thephoton echo signal useful in probing the dynamics in the condensed phases. In general,cross peaks reflect the couplings between
chromophores and thus provide useful structural information (9).
The phase dependence of the photon echo
signal e i(⫺␾1 ⫹ 2␾2 ⫺ ␾3) is different from thosei(␳X¯ X Y¯ ⫺ ␳X¯ X Y ⫹ ␳ XXY ⫺ ␳ X X Y¯ ⫹ ␳YXX
⫺ ␳ Y X X¯ ⫺ ␳Y¯ X X ⫹ ␳Y¯ X X¯ )We experimentally realized this approach bysending an amplified pulse train (at 9 ␮J and

Fig. 2. (A) Pulse spectrum. (B to F) 2D spectra ofsystem I obtained from the 2D Fourier transformation of the ﬂuorescence signal (B) withoutphase cycling and with (C) 2-, (D) 4-, (E) 8-, and(F) 16-step phase cycling. Only photon echo
peaks are retained after a 16-step phase cycle, as
expected (⌬fa ⫽ –9.1 THz and ⌬fb ⫽ –2.1 THz).

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a 20-kHz repetition rate, generated by a
Coherent RegA 9000 Ti:sapphire regenerative amplifier) into an AOPS. For each
input pulse, a phase-coherent three-pulse
sequence was then generated, with thepulse spectrum (Fig. 2A) ␭ ⱕ 776 nm
blocked to obtain system I (20, 21). The
pulse sequence was then focused into a spot
size of ⬃100 ␮m with 20 nJ per pulse anda pulse duration of 70 fs in a Rb vapor cell
(2-mm path length, ⬃100°C). The fluorescence perpendicular to the input beam wascollected by a photomultiplier tube (PMT)(Hamamatsu R928). The signal from the
PMT was amplified and then measured by aboxcar integrator (Stanford Research
SR250). Each data point was an average of3000 laser shots, although much faster updating would be possible with a more sophisticated arbitrary waveform generator.
Data with a 16-step phase cycle werecollected as follows. We fixed the positionand phase of pulse 2 (␾2 ⫽ 0°) and scanned
␶1,␶2 (from 0.14 to 2.94 ps with a step size of
46 fs), ␾1, and ␾3, respectively. At each valueof (␶1,␶2 ), the total fluorescence for all 16
different combinations of the phases ␾1,
␾2 ⫽ 0, and ␾3 was recorded as ␳i(␶1,␶2 ),where the index i ⫽ 1,2,. . .,16 represents a
unique phase combination (24). Figure 2, B
to F, show the spectra obtained without phasecycling (Fig. 2B) and with 2-, 4-(␳4 ⫽ ␳2 ⫺
␳X̄XX ⫹ ␳X̄XX̄), 8-(␳8 ⫽ ␳4 ⫺ ␳YXY ⫹ ␳ȲXY⫹ ␳YXȲ ⫺ ␳ȲXȲ), and 16-step phase-cycled
signals, respectively (Fig. 2, C to F), where
about half of the extra peaks are removed
with each extra step. The spectrum with the
16-step phase cycle clearly shows all photonecho peaks with ⌬fa ⫽ –9.1 THz and ⌬fb ⫽–2.1 THz. The cross peaks arise because theD1 and D2 transitions share the same groundstate. The results here further imply thatstructural information such as coupling can
be clearly revealed by the phase-cycled
spectrum.
Next, system II was obtained by blocking
the spectrum near 794.7 nm. Because ⌬fb ⫽⌬fc, this three-level system can be treated as

Fig. 3. 2D spectra of system II obtained fromthe 2D Fourier transformation of the ﬂuorescence signal (A) without phase cycling and (B)with a two-step phase cycle (⌬fb ⫽ –3.7 THz
and ⌬fc ⫽ –1.6 THz). Spurious peaks along f2 ⫽–f1 indicate that this measurement is noisy.

a model for an anharmonic ladder. When thepulse sequence with phase combination XXX
interacts with system II, many peaks appear
in the 2D spectrum S(f1, f2 ); only peaks withthe same phase dependence ei(⫺␾1 ⫹ 2␾2 ⫺ ␾3)such as (⌬fb, –⌬fb) and (⌬fb, –⌬fc) would
remain after a 16-step phase cycle, where ⌬fb⫽ –3.7 THz and ⌬fc ⫽ –1.6 THz. Here thepulse energy was increased to ⬃100 nJ toreveal the weak peak (⌬fb, –⌬fc). The spectra
obtained from ␳XXX (Fig. 3A) and ␳2 ⫽ ␳XXX
⫺ ␳X̄XX (Fig. 3B) both show peak (⌬fb, –⌬fc),
as expected. The six circled peaks are causedby fifth-order polarizations and cascaded
third-order signals. Two of these peaks [(⌬fc,–⌬fb) and (⌬fc, –⌬fc)] have the same phasedependence as the photon echo and would
remain in the 2D spectrum after a 16-stepphase cycle. However, their amplitudes increase faster with the intensity of the input
pulses than that of (⌬fb, –⌬fc) caused by athird-order polarization. This property may
be used to eliminate or reduce the higher
order peaks.
We demonstrated femtosecond phasecoherent 2D spectroscopy with a simple pulse
sequence and on a model system. Because of
the nonidealities of the AOPS, spurious pulses
were generated along with the main pulses.
Some spurious pulses overlapped with the main
pulses in the time domain, which induced an
amplitude modulation with the same phase dependence as the desired signal, such as P(3)echo.
This problem may be solved by precompensating the spurious pulses or by replacing AOPS
with LCPS. Furthermore, the total delay
achievable was often constrained by either the
AOPS or LCPS to about tens of picoseconds. In
order to apply the phase-cycling method tomacromolecules such as proteins, we needhigh-quality phase-coherent sequences withmultiple (three or more) pulses in the mid-IR
and with arbitrary polarizations. As many as
hundreds of pulses can be generated with an
AOPS (21). Recently, linear polarized midIR shaped pulses have been demonstrated
by optical parametric amplification (25).
This method can be extended to make twopulse sequences with orthogonal linear polarization, which can then be combined toyield an arbitrarily (but well-controlled)polarized phase-coherent pulse sequence(26 ). Therefore, complex molecular systems may soon be studied with phase cycling. Furthermore, ultrafast laser spectroscopy provides an opportunity to combine
both phase cycling and phase matching(27 ), which may result in further simplification of the pulse sequence and allow us todetect more complicated processes.
References and Notes

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Proc. Natl. Acad. Sci. U.S.A. 98, 11265 (2001).
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Chem. A, in press.
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15. S. Mukamel, Annu. Rev. Phys. Chem. 51, 691 (2000).
16. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford Univ. Press, Oxford, 1995).
17. D. Keusters, H.-S. Tan, W. S. Warren, J. Phys. Chem. A
103, 10369 (1999).
18. W. S. Warren, A. H. Zewail, J. Chem. Phys. 75, 5956
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20. W. G. Yang, D. Keusters, D. Goswami, W. S. Warren,
Opt. Lett. 23, 1843 (1998).
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N.F. Scherer, Eds. (Springer, Berlin, 2001), pp. 165–
167.
22. N. F. Scherer et al., J. Chem. Phys. 95, 1487 (1991).
23. The locked frequency is f L ⫽ f 0 – C0␦T, where C0⫽
4.34 MHz/ps is determined by the AOPS. ␦T is thedelay between the arrival times of the radiofrequency wave controlling the AOPS and the laser pulse at
the AOPS. Hence, ␦T can be adjusted to shift theentire photon echo spectrum. The choice of f 0 is
arbitrary and affects ␦T accordingly. However, thelocked frequency remains the same with a different
choice of f 0. In our experiment, we adjusted ␦T andthen directly measured its effect on the peak positions ⌬fa and ⌬fb.
24. Other polarization signals such as the peaks with aphase dependence of ei(␾1 ⫺ ␾3) can be selected by adifferent 16-step phase cycle as
共␳ XXX ⫺ ␳X X X̄ ⫺ ␳X̄ X X ⫹ ␳X̄ X X̄ ⫹ ␳ Y X Y ⫺ ␳Ȳ X Y ⫺ ␳ Y X Ȳ⫹ ␳Ȳ X Ȳ ) ⫹ i(⫺ ␳X̄ XȲ ⫹ ␳X̄ X Y ⫺ ␳ X X Y ⫹ ␳ X X Ȳ ⫹
␳ Y X X ⫺ ␳ Y X X̄ ⫺ ␳ Ȳ X X ⫹ ␳Ȳ X X̄ )Hence, the 16-step phase cycle is a general approachhere. For a particular polarization signal, a smaller
number of phase cycles may be adequate. For exam(3) . If symbols
ple, a 10-step phase cycle can select Pecho
Q, R, S, and T represent a phase of 2␲/5, 4␲/5, 6␲/5,(3) is selected by
and 8␲/5, respectively; Pecho␳10⫽(␳ XX ⫺␳ XX̄ ⫹ e ⫺i4␲/5␳QX ⫹ e i␲/5␳QX̄ ⫹ e i2␲/5␳RX ⫹e⫺i3␲/5␳ RX̄ ⫹e⫺i2␲/5␳SX)⫹ei3␲/5␳SX̄ ⫹ e i4␲/5 ␳TX ⫹e⫺i␲/5␳T X̄
25. H.-S. Tan, W. S. Warren, in preparation.
26. H.-S. Tan, personal communication.
27. W. P. de Boeij, M. S. Pshenichnikov, D. A. Wiersma,
Chem. Phys. 233, 287 (1998).
28. We acknowledge the ﬁnancial support from the Ofﬁce of Naval Research and the New Jersey Commission on Science and Technology. We thank H.-S. Tanfor pointing out the 10-step phase cycle to select(3) and L. Xu for his assistance in the experiment.
Pecho
13 February 2003; accepted 18 April 2003

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