Abstract
Fourwave mixing measurements can reveal spectral and dynamics information that is hidden in linear spectra by the interactions among lightabsorbing molecules and with their environment. Coherent multidimensional optical spectroscopy is an important variant of fourwave mixing because it resolves a map of interactions and correlations between absorption bands. Previous coherent multidimensional optical spectroscopy measurements have used femtosecond pulses with great success, and it may seem that femtosecond pulses are necessary for such measurements. Here we present coherent twodimensional electronic spectra measured using incoherent light. The spectra of model molecular systems using broadband spectrally incoherent light are similar but not identical to those expected from measurements using femtosecond pulses. Specifically, the spectra show particular sensitivity to longlived intermediates such as photoisomers. The results will motivate the design of similar experiments in spectral ranges where femtosecond pulses are difficult to produce.
Introduction
Systems as diverse as atomic gases, molecules, proteins, polymers and semiconductor nanostructures have been studied using fourwave mixing spectroscopy^{1,2,3,4,5,6,7,8,9}. The measurements have led to new insights into energy transfer, peptide secondary structure, manybody interactions and other phenomena. Fourwave mixing measurements exploit the thirdorder nonlinear optical response of the material under study. Variants such as transient absorption use sequences of short laser pulses—often femtoseconds in duration—to probe or coherently control dynamics directly in the time domain^{10,11}; other variants such as coherent antiStokes Raman scattering use continuouswave (cw) laser excitation to probe spectral properties directly in the frequency domain^{1}; even other fourwave mixing measurements combine the two domains^{12,13}.
Coherent multidimensional optical spectroscopy is an important type of fourwave mixing measurement because it resolves correlations between excitation and emission frequencies of the signal as a function of a variable time delay^{6,9}. The spectroscopic technique—analogous to multidimensional NMR spectroscopy^{14}—has been performed in the infrared, visible and ultraviolet spectral regions^{15,16,17,18,19,20,21,22,23,24,25,26,27,28}. In the visible region, the technique is often called twodimensional electronic spectroscopy (2D ES), and it has been used to study important topics such as photosynthetic energy transfer^{29,30,31,32}.
As a consequence of how we think about timing using femtosecond laser pulses, coherent multidimensional optical spectroscopy seems to require coherent excitation. A first femtosecond excitation pulse (E_{A}) interacts with the sample to induce coherent superpositions between the ground and excited states; these polarizations oscillate at optical frequencies. After a delay τ_{1}, a second excitation pulse (E_{B}) interacts to induce either populations of the ground or excited states or lowfrequency coherent superpositions known as quantum beats inside the groundstate or excitedstate manifolds. After a delay τ_{2}, a third excitation pulse (E_{C}) interacts to induce new coherent superpositions between the ground and excited states that oscillate at optical frequencies. These polarization oscillations lead to emission of a coherent signal field (E_{sig}) during the third time period, τ_{3}, in the phasematched direction. Using a weak localoscillator field (E_{LO}) and a phasestable interferometer, the influence of the amplitude and phase of each excitation field on the emitted signal field can be measured as a function of all three time intervals. The oscillations at optical frequencies during τ_{1} and τ_{3} are usually presented in the Fourier domain as a 2D spectrum. Dynamics are often displayed parametrically by presenting a 2D spectrum for each τ_{2} value.
On the basis of the description, short laser pulses do seem necessary to measure coherent 2D spectra because the phase of the coherence needs to be preserved during τ_{1} and τ_{3}. Indeed, to date, all coherent multidimensional optical spectra have been measured using coherent laser excitation^{15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. The vast majority of measurements used femtosecond pulses and even small deviations from a perfectly linear spectral phase profile introduce artefacts^{33,34}.
Here we report coherent 2D optical spectra measured using quasicw incoherent light, I^{(4)} 2D ES, where I^{(4)} means four incoherent excitation fields. Using incoherent excitation in conjunction with timedomain interferometry, we observe that strong fourwave mixing signals can be generated in the phasematched direction; that an LO field can be used to extract relative phase and amplitude information; and that coherent oscillations at optical frequencies can be measured and then Fourier transformed to produce 2D spectra. The 2D spectra contain similar but not identical information as 2D spectra measured using femtosecond pulses. The 2D measurements using incoherent light are extensions of onedimensional (1D) fourwave mixing spectroscopy measurements using incoherent excitation, a field referred to as noisylight spectroscopy^{35,36,37,38,39,40}. The main feature of noisylight spectroscopy is that the time resolution is given by the coherence time of the light, and not the temporal envelope. In principle the noisy beams can be cw, although in practice they are often pulsed on the order of hundreds of nanoseconds. This is still essentially cw relative to femtosecond and picosecond material dynamics.
Results
Measurement using timeintegrated detection
We used broadbandamplified spontaneous emission (ASE) and a fourbeam interferometer to measure I^{(4)} 2D ES using timeintegrated detection; more details are presented in Methods. We chose to study the carbocyanine dye 1,1′diethyl2,2′tricarbocyanine perchlorate (DTP) in butanol as a model system.
One can gain significant intuition about I^{(4)} 2D ES by reviewing how a Michelson interferometer can be used to measure a linear absorption spectrum. Both I^{(4)} 2D ES and Fouriertransform linear absorption spectroscopy are coherent spectroscopy techniques that use spectrally incoherent light; both use multiplebeam interferometers; and both rely on timecorrelation functions as part of the signal detection method. Figure 1 depicts the two measurements. We present theoretical analysis in Methods and in ref. 40; below we provide physical descriptions.
In linear absorption, coherent signals can be detected even when they are generated by incoherent sources. The incident field (E′) excites different molecules at different times, and it may seem that the total response—if measured as a superposition of induced signal fields—would average to zero. However, the detected signal arises because the field radiated by each molecule (E_{sig}, the free polarization decay) interferes destructively with the E′ that produced it. This reduces the intensity of the transmitted field () compared with the E′, and it occurs because the amplitude and phase of E_{sig} are intrinsically locked to E′. In other words, the sample reduces the amplitude of certain frequencies. These reductions will be revealed by the timecorrelation function operation central to the timeintegrated detection method using a Michelson interferometer, as depicted in Fig. 1a. A beam splitter produces two copies of E so that the two arms, E_{1} and E_{2}, are identical stochastic functions of time except for a relative interferometric delay η. The two beams recombine and the sum is detected by a photodiode. Detection of the cross terms at the intensity level is a timecorrelation function, . We depict this in Fig. 1b using wiggly lines to indicate the electric field amplitude as a function of the laboratory time variable t and angle brackets to indicate the timecorrelation function.
Individually, the electric fields depicted in Fig. 1b do not have obvious reference points for timing femtosecond processes because they are stochastic functions of time. We are able to extract meaningful information about the sample only by interferometric correlation. For each value of η, the backgroundsubtracted voltage provides the amplitude and sign of the correlation. We recorded the voltage as a function of η, shown in Fig. 1c, to yield the signal for the model system. Coherent oscillations due to the electronic transitions of the sample—which occur at optical frequencies—are evident. One could create the sample absorption spectrum presented in Fig. 2a by processing the Fourier transforms of two scans, one with the sample (shown in Fig. 1c) and the another without it (not shown).
A similar principle as above helps us to explain how the thirdorder nonlinear response is produced and detected with noisy light. As the thirdorder signal field is a stochastic function of time, we need to lock its amplitude and phase to the E′s. We accomplish this by producing multiple copies of the E′ using a phase mask in a fourbeam interferometer^{41}, Fig. 1d. The four fields are identical except for relative interferometric delays σ, τ, and ζ. In the BOX geometry, three of these fields, E_{A}, E_{B} and E_{C}, excite the molecules to produce a thirdorder signal field, E_{sig}, in the phasematched direction collinear with the localoscillator field E_{LO}. A photodiode detects the heterodyned signal. The measured intensity is proportional to the correlation between E_{LO} and E_{sig}, , see Fig. 1e, where we have introduced a fourth interferometric delay variable, φ, between E_{LO} and E_{sig}. This is done as a matter of explanatory convenience; the value of φ is determined in part by the experimentally controlled interferometric delays but can in principle depend on the material response as well. For each setting of φ, the (stochastic) E_{LO} acts as a temporal anchor on the laboratory time frame t against which the (stochastic) E_{sig} is correlated, similar to the Michelson interferometer. As the fields that produce E_{sig} are identical to E_{LO}, an interferometric scan over variable φ yields a 1D spectrum but is insufficient for producing a 2D spectrum. To make progress, we use knowledge of the fourwave mixing process to recast the timecorrelation function between E_{LO} and E_{sig} as the fourpoint timecorrelation function depicted in Fig. 1e, .
The fourpoint timecorrelation function contains the three interferometric delay variables under complete experimental control but is intractable. We employ the standard assumption of circular complex Gaussian statistics^{42} to obtain , where the twopoint timecorrelation functions are easier to evaluate and interpret. This expansion is depicted in Fig. 1e. The three interferometric variables remain in the twopoint timecorrelation functions, and a scan over any of the interferometric variables will lead to signal oscillations in a manner similar to the above example of linear absorption spectroscopy. Oscillations due to the electronic transitions of the sample at optical frequencies during σ and ζ are shown in the measured data presented in Fig. 1f.
A more rigorous description of I^{(4)} 2D ES relies on the formalism for nonlinear optical responses convolved with the details of how the optical fields are correlated. In standard descriptions of nonlinear optical spectroscopy, the signal field is expanded perturbatively in the light–matter interaction. For the thirdorder nonlinearity used here, there are multiple pathways for signal generation often called Liouville pathways^{2}. The key difference between femtosecond and noisylight 2D measurements is that in the former, the interferometric delays correspond directly to the interactiontime variables in the Liouville pathways. In the latter, the interferometric delays correspond only to the delay between identical time points along noisy electric fields. Any arbitrary point on the time profile of the noisy field can conceptually serve as such an ‘anchor point’, an idea closely related to the ‘fingerprint region’ concept in the noisylight literature^{35,36,37,38,39,40}. The asterisks in the schematics of the noisylight fields shown in Fig. 1b,e mark an arbitrarily chosen anchor point. The parameters η, σ, τ and ζ give experimental control over the delay between corresponding anchor points. The time at which any given field can act upon the sample is not under experimental control because of the quasicw nature of the noisy light. Therefore, all possible time orderings for the field–matter interactions have a role in producing the fourwave mixing signal. The nontrivial correspondence between the interactiontime variables in the Liouville pathways and the interferometric delays can be depicted and understood using a formalism cast in terms of factorized timecorrelation diagrams. See refs 39, 40 for a detailed description of this formalism. As depicted in Fig. 1e, two of the interferometric delay variables will respect the interaction times in a manner similar to femtosecond methods. However, the third will not; see Methods for more details.
We acquired the data presented in Fig. 1f by scanning σ and ζ from −80 to +80 fs with a step size of 0.8 fs at τ=0. Although almost all femtosecond 2D spectroscopy measurements use frequencyresolved detection, we performed this first measurement using timeintegrated detection to be consistent with the I^{(4)} 2D ES theory^{40} and nearly all 1D noisylight measurements, including refs 35, 36, 37, 38, 39. We highlight the signal as function of ζ with σ=0 in Fig. 1c. Using the projectionslice theorem and assuming negligible contribution from excitedstate absorption pathways, this trace is identical to a linear absorption measurement. The measured signal shows oscillations at optical frequencies corresponding to the electronic transitions as the interferometric delay ζ is scanned.
The measured 2D data show that the intensitylevel signal oscillates at optical frequencies in all four quadrants, that beats between at least two frequencies are apparent, and the signal persists at long σ and ζ times along the line σ=−ζ. These observations are all consistent with the previous theoretical work^{40}. The persistent signals along σ=−ζ are in large part because of terms that account for the symmetry of the two nonconjugate fields, E_{A} and E_{C}; the terms originate from the second paircorrelator product depicted in Fig. 1e. In previous work^{40}, these were dubbed βtype unrestricted terms, and from the Bloch model it followed that the signals persist along σ=−ζ for the excitedstate lifetime. Here we stopped the scan at 80 fs. The consequence of this truncation appears in the resulting spectrum as apodization of the antidiagonal linewidth.
I^{(4)} 2D spectra from the timeintegrated data set
We produced the I^{(4)} 2D ES by Fourier transformation of the zeropadded data from two quadrants of the time–time data set. We did not make subcycle timing adjustments to each beam; thus, the socalled global phase was not corrected^{41,43}. Instead, we present magnitude I^{(4)} 2D ES displayed on linear colour scales. In femtosecond 2D ES, the direct relation between the interferometric delay setting and the interaction times of the Liouville pathways means that the rephasing (nonrephasing) pathways only generate signal as part of the rephasing (nonrephasing) pulsetiming sequence. In I^{(4)} 2D ES, because the light is always on, any Liouville pathway (rephasing, nonrephasing and twoquantum) can potentially contribute at any interferometric delay setting. In other words, although the ranges of the interferometric delays mimic certain types of femtosecond 2D ES measurements, the Liouville pathways are not restricted in the same manner. We therefore label Fig. 2b,c the ‘rephasinglike’ and ‘nonrephasinglike’ spectra, respectively. The rephasinglike spectrum originated from the quadrant of the time–time data set and the nonrephasinglike spectrum originated from the quadrant. We present the linear absorption spectrum of DTP in butanol and the spectrum of the incident light source in the solid and dashed lines, respectively, in Fig. 2a. The two lowestfrequency transitions of the sample are centred at 826 nm (363 THz) and 750 nm (400 THz).
Measurements and I^{(4)} 2D ES using frequencyresolved detection
Theoretical analysis, raw data, and scan and data treatment details are presented in Supplementary Notes 1–4, Supplementary Tables S1–S4 and Supplementary Figs S1–S15. Briefly, we used a spectrometer and chargecoupled device (CCD) camera to detect the heterodyne frequencyresolved signal using a noisylight version of spectral interferometry. This allowed rapid data acquisition but required a new theoretical analysis; ref. 40 and Supplementary Note 1 are relevant only to timeintegrated detection. In Supplementary Note 2 and Supplementary Tables S1–S4, we present analysis and results, respectively, for frequencyresolved detection of noisylight fourwave mixing signals. The analysis in Supplementary Note 3 and simulation results in Supplementary Figs S4 and S5 indicate that spectral interferometry methods similar to those used in femtosecond 2D ES should work, and indeed this is confirmed by experiments. Acquisition of a 2D spectrum at one τ value required about 2 min using frequencyresolved detection, whereas timeintegrated detection required about 10 h. Using this frequencyresolved detection method, we studied two complementary model systems to explore further the photoisomer signatures: DTP in butanol and anhydro11(4ethoxycarbonyl1piperazinyl)10,12ethylene3,3,3′,3′tetramethyl1,1′di(3sulfopropyl)4,5;4′,5′dibenzoindotricarbocyanine hydroxide (IR144) in a 1:4 mixture of methanol and ethylene glycol.
In Fig. 3, we present the I^{(4)} 2D ES of DTP in butanol at τ=100 fs at three different excitation energies. The lowestexcitation energy of 6 nJ per pulse is equivalent to the energy used in femtosecond 2D ES measurements. The spectra were ‘phased’ using the projectionslice theorem in the same manner used in femtosecond 2D ES measurements (see Supplementary Note 4). The spectra in Fig. 3 are total correlation spectra—the sum of the rephasinglike and nonrephasinglike components—which eliminate phasetwisted lineshapes in the same manner as femtosecond 2D ES measurements. The spectra show similar asymmetry as Fig. 2c; the abovediagonal cross peak is significantly brighter than the belowdiagonal cross peak.
In Fig. 4, we present measurements on IR144, a prototypical laser dye used in countless femtosecond spectroscopy measurements, for example, an initial report of femtosecond 2D ES^{15}. We show the linear absorption spectrum and the noisylight excitation spectrum in Fig. 4a. In Fig. 4b,c, we present I^{(4)} 2D ES of IR144 measured at two different pulse energies. The two I^{(4)} 2D ES differ in the amount of noise because of scatter. These spectra are total correlation spectra at τ=200 fs. In IR144, the belowdiagonal cross peak is significantly brighter than the abovediagonal cross peak, and this was independent of excitation spectrum (see Supplementary Figs S9 and S10).
Discussion
Previous noisylight measurements involved homodyned 1D fourwave mixing, but the most straightforward implementation of coherent 2D spectroscopy requires relative amplitude and phase information not measureable using homodyne detection. Recent theoretical work predicted that 2D spectra could be measured using incoherent light with heterodyne detection of a fourwave mixing signal^{40}. The first result reported here exactly follows the timeintegrated formulation described in ref. 40. We furthermore extend the previous theoretical work to frequencyresolved detection, which significantly decreases data acquisition time. The measurements confirm that coherent 2D optical spectroscopy measurements can be performed with spectrally incoherent excitation. The results mean that 2D spectroscopy requires a broadband light source but not necessarily femtosecond laser pulses, which are spectrally coherent. We describe the technique from the perspective of linear absorption spectroscopy—a measurement that can be performed with light of any spectral coherence—rather than from the perspective of femtosecond 2D spectroscopy. This qualitative description is useful, although it does not capture all of the important details underpinning the experiment. We envision one advantage of using incoherent excitation to measure 2D spectra is that users are not constrained by the costs and challenges associated with producing and controlling femtosecond laser pulses, which can be particularly difficult in the ultraviolet spectral region, for example. Another possible application is at even shorter wavelengths, such as 2D spectroscopy using Xrays^{44}, where it is also challenging to produce broadband transformlimited pulses. A third possible application is to apply the method described here directly to probe photochemical species such as photoisomers or triplet states, whose signatures can be weak in femtosecond 2D ES but strong in I^{(4)} 2D ES.
The spectra shown in Fig. 2 reflect characteristic features of the absorption spectrum of DTP. Diagonal peaks appear at 358 and 391 THz; these frequencies correspond to the two transitions observed in the linear absorption spectrum (slightly truncated by the excitation bandwidth and the measurement resolution). The lowerfrequency peak is the electronic S_{0}→S_{1} transition, whereas the higherfrequency peak is a bright vibronic transition. Cross peaks between the two transitions are evident, particularly in the nonrephasinglike spectrum (Fig. 2c). The cross peaks indicate that the two transitions share a common ground state; microscopically, the two transitions involve the same molecular orbitals (indeed, the same electronic states). Cross peaks are one of the most important features in 2D spectra because they directly reveal couplings, which are ultimately the source of electronic and vibrational dynamics. In this measurement, the cross peaks are due to strong coupling of a 37THz (1,220 cm^{−1}) vibrational mode to the main electronic transition^{45}. Such modes have also been shown to lead to cross peaks in femtosecond 2D ES studies of similar molecules^{22,41,46,47}. Our equipment cannot produce femtosecond pulses with more than about 10 nm of bandwidth at 800 nm, and therefore we were unfortunately not able to perform the equivalent 2D measurement using femtosecond pulses.
The diagonal peaks dominate the rephasinglike spectrum in Fig. 2b. The antidiagonal linewidths of the diagonal peaks are smaller than the diagonal linewidths, but this does not indicate inhomogeneous broadening as would be expected from femtosecond 2D spectroscopy. This is a significant difference between the femtosecond and noisylight methods. In femtosecond rephasing measurements, the ratio of the antidiagonal linewidth to the diagonal linewidth contains information about the relative contributions from homogeneous and inhomogeneous line broadening. In the noisylight spectrum performed here at τ=0, following the Bloch model^{40}, the antidiagonal linewidth yields the excitedstate lifetime because the group of terms that directly probe the lifetime decay accumulate signal strength by integrating over dephasing times in a constant manner during this experimental scanning scheme. Ideally, σ and ζ should be scanned to times beyond the excitedstate lifetime, which is not practical in this case because the excitedstate lifetime of DTP is about 30 ps (ref. 48), whereas the maximum scan range of the instrument is about 0.5 ps.
The value of the excitedstate lifetime is also important to determine the strength of certain subsets of signal pathways, see Methods. Although modest compared with many laser dyes, the 30ps timescale is sufficiently longer than the optical dephasing time of about 20 fs such that we expect steadystate terms (those of unrestricted topology in ref. 40) to be at least 1,000 times stronger than the dynamic terms (those of singly and doubly restricted topology in ref. 40); the timescale is also short enough compared with the excitation pulse duration (about 150 ns, see Supplementary Fig. S6) to ensure that the cw approximation is valid^{40}. The signals from βtype unrestricted terms could be eliminated to reveal the conventional inhomogeneous broadening information by either setting τ to a value larger than the dephasing time or to use two uncorrelated light sources, one producing E_{A} and E_{B} and the other producing E_{C} and E_{LO}.
The cross peak amplitudes in Figs. 2c and 3 represent a singular aspect of I^{(4)} 2D ES. Spectroscopists often use cross peaks in femtosecond 2D ES to probe the dynamics of energy transfer; one expects that the belowdiagonal cross peak—where the excitation energy is greater than the emission energy—should increase in amplitude as a function of waiting time as higherlying excited states decay to the lowerenergy state via intramolecular relaxation. Yet in the measurements of DTP, the abovediagonal cross peak is brighter than the belowdiagonal cross peak. On the other hand, the belowdiagonal cross peak is brighter than the abovediagonal cross peak in the measurement of IR144. The imbalance in both measurements persists at all values of τ, in multiple solvents and under a broad range of pulse spectrum excitation conditions, see Supplementary Figs S9 and S10. A reasonable explanation for the difference in the cross peak amplitudes of the two samples is that DTP photoisomerizes but IR144 does not. Indeed, the end groups used in the infrared (IR) family of laser dyes were developed to be bulky enough to prevent photoisomerization^{45}.
In many carbocyanine dyes including DTP, photoexcitation causes one or more bonds in the polymethine chain to isomerize. The photoproduct is blue shifted relative to the primary excitation of the alltrans ground state^{45,49}. Although the lifetime of the excited singlet state is on the order of picoseconds in DTP, photoisomers in carbocyanines can live for microseconds or even milliseconds at room temperature in solution^{45}. Probing photoisomerization in femtosecond spectroscopy typically requires significant data analysis of weak but longlived features^{50,51}. In contrast, in I^{(4)} 2D ES the powerful unrestricted terms integrate over the long lifetime, and thus lead to strong spectral features, indicating the presence of photochemical species. Thus, one big difference between I^{(4)} 2D ES and femtosecond 2D ES is that the amplitude of features in I^{(4)} 2D ES are proportional to the lifetime of species. This can be used to amplify spectral signatures of photochemical products or intermediates.
The linear absorption spectra show that the vibronic shoulder of IR144 is not as clearly resolved as the vibronic shoulder of DTP, and therefore the cross peaks in the I^{(4)} 2D ES of IR144 are less clearly resolved than the cross peaks in the I^{(4)} 2D ES of DTP. Nevertheless, the overall rectangular shape—consistent with numerous femtosecond 2D ES measurements of similar carbocyanine molecules where vibronic splittings do not exceed absorption linewidths^{22,25,28,41,46,47}—of the IR144 2D spectra indicate that both diagonal and cross peaks are present.
The results in Figs 3 and 4 show that the signals are robust over a wide range of excitation energies. The peak irradiance used in many femtosecond experiments is about 10^{10 }W cm^{−2}. The lowestexcitation conditions in this work—5 nJ per pulse delivered in about 150 ns in an area <0.02 mm^{2}—correspond to a peak irradiance of about 10^{2} W cm^{−2}. Sunlight delivers about 0.1 W cm^{−2}. Thus, the lowestexcitation conditions correspond to about 1,000 suns, an excitation level achieved in highconcentration photovoltaic cells^{52}. As described in Methods, even lowerexcitation energies should generate detectable fourwave mixing signals.
We envison that these results will clarify and refocus the debate surrounding the interpretation of coherent oscillations in 2D ES recorded using femtosecond laser pulses. Several groups studied the interpretation of femtosecond 2D ES measurements of photosynthetic proteins and what those measurement reveal about the natural conditions for photosynthesis^{53,54,55,56} by focusing on the distinction between spectrally coherent and spectrally incoherent excitation conditions. We speculate that the significant distinction between excitation conditions is not the coherence of light but rather the fourwave mixing signal itself. The nonlinear response—measured using coherent or incoherent light—allows measurements of dynamics that are hidden by averaging in experiments that use steadystate signals and theories that model the linear response of lightharvesting complexes illuminated by sunlight. Thus, the key challenge is to understand how fourwave mixing signals can reveal variables that are often hidden by averaging over time and ensembles. In Supplementary Note 4 and Supplementary Figs S11–S15, we include preliminary measurements as a function of the waiting time, τ. In future work, we will further examine those results.
The results demonstrate that I^{(4)} 2D ES contains much the same as well as complementary information as femtosecond 2D ES. The measurements confirm the prediction^{40} that it is possible to create, correlate and measure coherent superpositions during multiple time periods—in the present work coherences oscillating at optical frequencies as a function of interferometric delays σ and ζ—using incoherent excitation light. The results show that I^{(4)} 2D ES is a sensitive probe of intermediates such as photoisomers that have long lifetimes and that coherent 2D optical spectroscopy measurements do not require femtosecondpulse excitation.
Methods
Experimental methods
The measurement apparatus for timeintegrated detection consisted of the light source described below, the same fourbeam interferometer that was used previously^{32,41}, and the detection scheme detailed below. Noisylight measurements require a light source that is spectrally incoherent, broadband and bright enough to generate nonlinear optical signals. For technical convenience, spatial coherence is also desirable. Few currently available sources simultaneously satisfy those requirements. We used the ASE from the optical cavity of an unseeded Ti:sapphire amplifier pumped by a frequencydoubled yttrium lithium fluoride laser. The thermal cavity produced pulses with duration of about 150 ns, a repetition rate of 5 KHz and satisfactory spatial coherence. We measured the ASE pulse duration using a 1nsrisetime photodiode; Supplementary Fig. S6 displays an example trace. Without modification, the biggest drawbacks to this light source are the lack of tunability away from 800 nm and the relatively narrow bandwidth of about 30 nm. We addressed the second drawback by inserting a spectral shaping filter (Alpine Research Optics SF6040) into the optical cavity to reduce gain narrowing; the ASE bandwidth more than tripled to about 100 nm as shown in Fig. 2a. The bandwidth corresponds to a coherence time of about 20 fs. The total output power was about 200 mW. This beam was sent into the interferometer setup where a 2D crosshatched phase mask produced four copies in the standard square (BOX) formation. Ultravioletfused silica wedge pairs inserted into three beams correspond to interferometric delays σ, τ and ζ, where E_{A}(t), E_{B}(t, σ), E_{C}(t, σ, τ) and E_{LO}(t, σ, τ, ζ). The interferometric delay variables (σ, τ, ζ) are analogous to the two common conventions in femtosecond 2D ES literature, (τ_{1}, τ_{2}, τ_{3}) and (τ, T, t). Different words have also been used to describe each time interval. The first interval has been called the coherence or excitation time, the second is the waiting or population time and the third is the emission or signal or detection time.
Each input beam was about 600 nJ per pulse before the sample without attenuation. This was well above the threshold needed to measure the phasematched homodyne thirdorder signal in the direction. The beam waist radius at the focus was ~75 μm. Timeintegrated detection involved a photodiode, a current preamplifier and data acquisition hardware for a personal computer. This equipment recorded a voltage value linearly related to the intensity of the heterodyned thirdorder nonlinear optical signal. One excitation beam was chopped at 500 Hz; the difference between the blocked and unblocked signals isolated the heterodyne signal, , from the homodyne contributions, and I_{LO}. There was minimal residual scatter. Unsubtracted fluorescence from the excitation beam that was chopped led to the small nonzero background in the timeintegrated data set shown in Fig. 1f. This nonzero background does not oscillate as a function of the interferometric delays and so it does not mar the I^{(4)} 2D ES. Frequencyresolved detection involved a spectrometer and CCD, and one beam was chopped at 50 Hz. This detection setup was constructed previously^{41} and optimized for the visible spectrum. The throughput of the setup from signal generation in the sample cell to detection at the CCD surface is >50% efficient at visible wavelengths. At 800 nm, the total efficiency of the same optics was <25%. At ~5 nJ per pulse, the electrical noise of the CCD was the largest source of noise. An optical apparatus optimized for 800 nm would increase efficiency significantly and would lead to detectable signals at excitation energies lower than 5 nJ per pulse.
We used the DTP (Exciton), IR144 (Exciton) and solvents (SigmaAldrich) without further purification. Solvation of DTP in butanol redshifted the absorption bands to improve the overlap of the sample and ASE spectra. The peak optical density in the 1mm path length sample cell was <0.3. A gear pump flowed the sample at 150 ml min^{−1} in the timeintegrated measurement and at 10 ml min^{−1} in the frequencydomain measurements to eliminate deleterious heating of the sample. The measurements were performed at room temperature.
At ~5 nJ per pulse, the diffraction efficiency was about 0.004% for either DTP or IR144, which is on par with femtosecond experiments. It is productive to describe the magnitude of the signal and the scaling to other samples from the standpoint of kinetics; using light that is ‘always on’, chromophores will be excited and then either produce signal or deexcite via another channel such as fluorescence or photoisomerization. These processes are ongoing at all times. The steadystate terms describe signals whereby the two excitation beams excite a fraction of the molecules, and the third beam scatters from this equilibrium transient grating. To address the scaling of the steadystate contribution to the signal in a simplified manner, we consider a simple rate equation model for a twolevel system; see for example, equation (7.4.6) in ref. 57. We find that the steadystate density of molecules in the excited state, N, scales as , where A is the absorption crosssection of the sample, B is the excitedstate lifetime and I is the laser intensity. The transient grating written into the sample will have an absorption modulation depth scaling as , and the grating efficiency will scale as . In the heterodynedetected case, the steadystate terms scale as (see for example, equation (48) in ref. 40).
Theoretical methods
To build intuition about the I^{(4)} 2D ES measurement, we describe linear absorption spectroscopy of a twolevel system (g and e) performed using a Michelson interferometer with quasicw incoherent excitation, as depicted in Fig. 1a–c. Linear absorption by a twolevel system can be described by the firstorder response function , where contains the frequency () and assumes exponential dephasing (). Times t_{1} and t_{2} correspond to the excitation and emission field–matter interactions, respectively.
The goal is to measure . The semiclassical initial field is , where is a complex stochastic function representing the random envelope and is the optical carrier frequency. Field interacts with the sample at time t_{1} to produce a polarization that radiates an electric field, E_{sig}, at a later time t_{2}. We must account for all possible interaction times. Mathematically, this is given by . Without a sample, E—the field entering the interferometer—is . When a sample is inserted, E contains two components, , where for convenience we assume . Field E is split into two parts, fields and . The timeintegrated signal at the photodiode is
The signal of interest will be contained in the cross term (I_{X})
Without a sample, the result is
where the integral over t is a correlation function that simplifies to , assuming a light source of infinite bandwidth and of any spectral phase profile. The timecorrelation function is depicted in Fig. 1b as brackets surrounding two wiggly lines representing the stochastic field envelopes. The result in equation (3) means that if the bandwidth were finite, oscillations at the carrier frequency would occur for a duration known as the coherence time.
When a sample is present, the crossterm intensity becomes
The signal from reproduces the temporal coherence profile of the excitation light (identical to equation (3)) and can be removed by a secondary measurement. The signal from leads to an oscillatory component containing the desired information, but this term is neglected in the lowdensity limit. The remaining terms from the expansion of equation (4) are the cross terms, which yield
Insertion results in
which can be simplified by the substitution and rearranged to yield
and then solved with the conjugate term to produce the signal expression
Incoherent excitation can indeed create coherent superpositions and allow measurement of dynamics much shorter than the pulse duration. In this case, the coherent superposition is between the ground and excited states, the dynamics are the optical dephasing and the pulse duration is cw. These oscillations are shown in Fig. 1c. Related descriptions of linear absorption can be found^{58}, and similar concepts regarding coherence are exploited in other methods such as whitelight interferometry and optical coherence tomography^{59}. We have used linear absorption to illustrate an important general principle: measuring coherent oscillations and dynamics using an interferometer does not always require pulses with short temporal envelopes.
In femtosecond 2D ES, it is customary to extract E_{sig} at the field level^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34}. In the example above, however, E_{sig} at the field level is not very useful because it is a stochastic function of time. We are able to extract meaningful information about the sample, in this case , only by interferometric correlation. In I^{(4)} 2D ES, a similar situation exists: E_{sig} is a stochastic function of time and information can be obtained only by interferometric correlation. Therefore, linear absorption spectroscopy is a helpful basis for understanding I^{(4)} 2D ES.
The I^{(4)} 2D ES theoretical analysis for timeintegrated detection was detailed previously for a Bloch fourlevel system^{40} and is outlined below. The analysis requires four fields and a large set of nonlinear response functions developed over three time periods between the field–matter interactions, . The key segment of the experimental apparatus is depicted in Fig. 1d. The measured intensitylevel signals involve fourpoint timecorrelation functions, which, under the standard assumption of circular complex Gaussian statistics, factorize into pairs of products of twopoint correlation functions, . The factorization is depicted in Fig. 1e. An example starting expression—meaning the equivalent of equation (6)—is
Examples of the final signal expressions are
where is the lifetime of excited state e_{1},
and
The first, equation (10), will lead to a diagonal peak at frequency coordinates , which does not depend on the value of τ, and the second example, equation (11), will lead to a cross peak at , which also does not depend on the value of τ. In these two example terms, the signal strength is related not only to the transition dipole moments, which is the same as femtosecond methods, but also to the excitedstate lifetime. The third example, equation (12), will lead to a diagonal peak at that oscillates coherently as a function of τ at frequency , a quantum beat.
The 128 expressions were categorized into three types based on the diagrammatic representation of the timecorrelation functions: unrestricted, singly restricted and doubly restricted. Signals from the unrestricted terms, including the first and second examples above, were predicted to be strong because they fully integrate signal over the excitedstate lifetime; such terms did not depend on τ. These signals are steady state, but they also contain information such as coupling or line broadening that is not directly measurable using linear spectroscopic methods. Signals from singly and doubly restricted terms do contain direct dynamics information during the waiting time τ, but they were expected to be weaker than signals because of unrestricted terms. This prediction depended on the ratio of the excitedstate lifetime to the optical dephasing time.
Finally, the method of coherent multidimensional spectroscopy using incoherent excitation was briefly used in NMR spectroscopy, and its history has been reviewed^{60}.
Additional information
How to cite this article: Turner, D. B. et al. Coherent multidimensional optical spectra measured using incoherent light. Nat. Commun. 4:2298 doi: 10.1038/ncomms3298 (2013)
References
 1.
Shen, Y. R. The Principles of Nonlinear Optics Wiley (1984).
 2.
Mukamel, S. Principles of Nonlinear Optical Spectroscopy Oxford University Press (1995).
 3.
Shah, J. Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures SpringerVerlag (1999).
 4.
Boyd, R. W. Nonlinear Optics Academic Press (2003).
 5.
Hochstrasser, R. M. Twodimensional spectroscopy at infrared and optical frequencies. Proc. Natl Acad. Sci. USA 104, 14190–14196 (2007).
 6.
Cho, M. TwoDimensional Optical Spectroscopy CRC Press (2009).
 7.
Cheng, Y.C. & Fleming, G. R. Dynamics of light harvesting in photosynthesis. Ann. Rev. Phys. Chem. 60, 241–262 (2009).
 8.
Cundiff, S. T. et al. Optical twodimensional Fourier transform spectroscopy of semiconductors. Acc. Chem. Res. 42, 1423–1432 (2009).
 9.
Hamm, P. & Zanni, M. T. Concepts and Methods of 2D Infrared Spectroscopy Cambridge University Press (2011).
 10.
Fleming, G. R. Chemical Applications of Ultrafast Spectroscopy Oxford University Press (1986).
 11.
Zewail, A. H. Femtochemistry: Ultrafast Dynamics of the Chemical Bond World Scientific (1994).
 12.
Kukura, P., McCamant, D. W. & Mathies, R. A. Femtosecond Stimulated Raman Spectroscopy. Ann. Rev. Phys. Chem. 58, 461–488 (2007).
 13.
Pakoulev, A. V. et al. Multiply resonant coherent multidimensional spectroscopy: implications for materials science. J. Phys. Chem. Lett. 1, 822–828 (2010).
 14.
Ernst, R., Bodenhausen, G. & Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions Oxford Science Publications (1987).
 15.
Hybl, J. D., Albrecht, A. W., Gallagher Faeder, S. M. & Jonas, D. M. Twodimensonal electronic spectroscopy. Chem. Phys. Lett. 297, 307–313 (1998).
 16.
Hamm, P., Lim, M. & Hochstrasser, R. M. Structure of the amide I band of peptides measured by femtosecond nonlinearinfrared spectroscopy. J. Phys. Chem. B 102, 6123–6138 (1998).
 17.
Demirdöven, N., Khalil, M. & Tokmakoff, A. Correlated vibrational dynamics revealed by twodimensional infrared spectroscopy. Phys. Rev. Lett. 89, 237401 (2002).
 18.
Tian, P., Keusters, D., Suzaki, Y. & Warren, W. S. Femtosecond phasecoherent twodimensional spectroscopy. Science 300, 1553–1555 (2003).
 19.
Brixner, T. et al. Twodimensional spectroscopy of electronic couplings in photosynthesis. Nature 434, 625–628 (2005).
 20.
Fulmer, E. C., Ding, F., Mukherjee, P. & Zanni, M. T. Vibrational dynamics of ions in glass from fifthorder twodimensional infrared spectroscopy. Phys. Rev. Lett. 94, 067402 (2005).
 21.
Davis, J. A. et al. Noninterferometric twodimensional fouriertransform spectroscopy of multilevel systems. Phys. Rev. Lett. 100, 227401 (2008).
 22.
Prokhorenko, V. I., Halpin, A. & Miller, R. J. D. Coherentlycontrolled twodimensional photon echo electronic spectroscopy. Opt. Exp. 17, 9764–9779 (2009).
 23.
Stone, K. W. et al. Twoquantum 2D FT electronic spectroscopy of biexcitons in GaAs quantum wells. Science 324, 1169–1173 (2009).
 24.
Turner, D. B. & Nelson, K. A. Coherent measurements of highorder electronic correlations in quantum wells. Nature 466, 1089–1092 (2010).
 25.
Lewis, K. L. M. & Ogilvie, J. P. Probing photosynthetic energy and charge transfer with twodimensional electronic spectroscopy. J. Phys. Chem. Lett. 3, 503–510 (2012).
 26.
Wong, C. Y. et al. Electronic coherence lineshapes reveal hidden excitonic correlations in photosynthetic light harvesting. Nat. Chem. 4, 396–404 (2012).
 27.
Li, H., Bristow, A. D., Siemens, M. E., Moody, G. & Cundiff, S. T. Unraveling quantum pathways using optical 3D Fouriertransform spectroscopy. Nat. Commun. 4, 1390 (2013).
 28.
West, B. A., Molesky, B. P., Montoni, N. P. & Moran, A. M. Nonlinear optical signatures of ultraviolet lightinduced ring opening in αterpinene. New Journal of Physics 15, 025007 (2013).
 29.
Engel, G. S. et al. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–786 (2007).
 30.
Collini, E. et al. Coherently wired lightharvesting in photosynthetic marine algae at ambient temperature. Nature 463, 644–647 (2010).
 31.
Panitchayangkoon, G. et al. Longlived quantum coherence in photosynthetic complexes at physiological temperature. Proc. Natl Acad. Sci. USA 107, 12766–12770 (2010).
 32.
Turner, D. B. et al. Quantitative investigations of quantum coherence for a lightharvesting protein at conditions simulating photosynthesis. Phys. Chem. Chem. Phys. 14, 4857–4874 (2012).
 33.
Christensson, N. et al. Weakly chirped pulses in frequency resolved coherent spectroscopy. J. Chem. Phys. 132, 174508 (2010).
 34.
Tekavec, P. F., Myers, J. A., Lewis, K. L. M., Fuller, D. & Ogilvie, J. P. Effect of chirp on twodimensional fourier transform electronic spectra. Opt. Exp. 18, 11015–11024 (2010).
 35.
Morita, N. & Yajima, T. Ultrahightimeresolution coherent transient spectroscopy with incoherent light. Phys. Rev. A 30, 2525–2536 (1984).
 36.
Beach, R. & Hartmann, S. R. Incoherent photon echoes. Phys. Rev. Lett. 53, 663–666 (1984).
 37.
Asaka, S., Nakatsuka, H., Fujiwara, M. & Matsuoka, M. Accumulated photon echoes with incoherent light in Nd^{3+}doped silicate glass. Phys. Rev. A 29, 2286–2289 (1984).
 38.
Tomita, M. & Matsuoka, M. Ultrafast pumpprobe measurement using intensity correlation of incoherent light. J. Opt. Soc. Am. B 3, 560–564 (1986).
 39.
Ulness, D. J. On the role of classical field time correlations in noisy light spectroscopy: color locking and a spectral filter analogy. J. Phys. Chem. A 107, 8111–8123 (2003).
 40.
Turner, D. B. et al. Twodimensional electronic spectroscopy using incoherent light: theoretical analysis. J. Phys. Chem. A doi:10.1021/jp310477y (2012).
 41.
Turner, D. B., Wilk, K. E., Curmi, P. M. G. & Scholes, G. D. Comparison of electronic and vibrational coherence measured by twodimensional electronic spectroscopy. J. Phys. Chem. Lett. 2, 1904–1911 (2011).
 42.
Goodman, J. W. Statistical Optics Wiley (1985).
 43.
Anna, J. M., Ostroumov, E. E., Maghlaoui, K., Barber, J. & Scholes, G. D. Twodimensional electronic spectroscopy reveals ultrafast downhill energy transfer in photosystem i trimers of the Cyanobacterium Thermosynechococcus elongates. J. Phys. Chem. Lett. 3, 3677–3684 (2012).
 44.
Schweigert, I. V. & Mukamel, S. Coherent ultrafast corehole correlation spectroscopy: xray analogues of multidimensional NMR. Phys. Rev. Lett. 99, 163001 (2007).
 45.
Meyer, Y. H., Pittman, M. & Plaza, P. Transient absorption of symmetrical carbocyanines. J. Photochem. Photobio. A: Chem. 114, 1–21 (1998).
 46.
Egorova, D. Detection of electronic and vibrational coherences in molecular systems by 2D electronic photon echo spectroscopy. Chem. Phys. 347, 166–176 (2008).
 47.
Christensson, N. et al. High frequency vibrational modulations in twodimensional electronic spectra and their resemblance to electronic coherence signatures. J. Phys. Chem. B 115, 5383–5391 (2011).
 48.
Benicewicz, P. K., Roberts, J. P. & Taylor, A. J. The absorption characteristics and recovery dynamics of the tricarbocyanine dye DTP. Opt. Commun. 86, 393–396 (1991).
 49.
Mishra, A. et al. Cyanines during the 1990, s: a review. Chem. Rev. 100, 1973–2011 (2000).
 50.
Sundström, V. & Gillbro, T. Viscositydependent isomerization yields in some cyanine dyes. a picosecond laser spectroscopy study. J. Phys. Chem. 86, 1788–1794 (1982).
 51.
Guarin, C. A., VillabonaMonsalve, J. P., LópezArteaga, R. & Peon, J. Dynamics of the higher lying excited states of cyanine dyes. An ultrafast fluorescence study. J. Phys. Chem. B 117, 7352–7362 (2013).
 52.
Baker, K. A., Karp, J. H., Trembley, E. J., Hallas, J. M. & Ford, J. E. Reactive selftracking solar concentrators: concept, design, and initial material characterization. App. Opt. 51, 1086–1094 (2012).
 53.
Mancal, T. & Valkunas, L. Exciton dynamics in photosynthetic complexes: excitation by coherent and incoherent light. N. J. Phys. 12, 065044 (2010).
 54.
Fassioli, F., OlayaCastro, A. & Scholes, G. D. Coherent energy transfer under incoherent excitation. J. Phys. Chem. Lett. 3, 3136–3142 (2012).
 55.
Brumer, P. & Shapiro, M. Molecular response in one photon absorption via natural light vs. pulsed laser excitation. Proc. Natl Acad. Sci. USA 109, 19575–19578 (2012).
 56.
Shaprio, M. Quantum dynamics experienced by a single molecular eigenstate excited by incoherent light. Phys. Rev. Lett. 110, 153003 (2013).
 57.
Verdeyen, J. T. Laser Electronics Prentice Hall (1995).
 58.
Perkins, W. D. Fourier transforminfrared spectroscopy. J. Chem. Ed. 63, A5–A10 (1986).
 59.
Huang, D. et al. Optical coherence tomography. Science 254, 1178–1181 (1991).
 60.
Blümich, B White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy. Prog. NMR Spectrosc. 19, 331–417 (1987).
Acknowledgements
The work in Toronto was supported by the DARPA QuBE program and the National Science and Engineering Research Council of Canada. We thank Alexej Jerschow for bringing ref. 60 to our attention.
Author information
Affiliations
Department of Chemistry, Institute for Optical Sciences and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George Street, Toronto, Ontario, Canada, M5S 3H6
 Daniel B. Turner
 , Paul C. Arpin
 , Scott D. McClure
 & Gregory D. Scholes
Department of Chemistry, Concordia College, Moorhead, Minnesota 56562, USA
 Darin J. Ulness
Authors
Search for Daniel B. Turner in:
Search for Paul C. Arpin in:
Search for Scott D. McClure in:
Search for Darin J. Ulness in:
Search for Gregory D. Scholes in:
Contributions
D.B.T. and D.J.U. conceived the experiment. D.B.T. and D.J.U. performed the theoretical analysis. D.B.T., P.C.A. and S.D.M. designed the experimental implementation, performed the measurements and analysed the data. All authors discussed the results. D.B.T. drafted the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Gregory D. Scholes.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary Figures S1S15, Supplementary Tables S1S4, Supplementary Notes 14 and Supplementary References
Rights and permissions
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

Electronic Couplings in (Bio) Chemical Processes
Topics in Current Chemistry (2018)

Energy transfer pathways in semiconducting carbon nanotubes revealed using twodimensional whitelight spectroscopy
Nature Communications (2015)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.