Vibrational Anharmonicity and Multilevel Vibrational Dephasing from Vibrational Echo Beats K

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Vibrational anharmonicity and multilevel vibrational dephasing from vibrational echo beats K. D. Rector, A. S. Kwok,a) C. Ferrante,b) and A. Tokmakoffc) Department of Chemistry, Stanford University, Stanford, California 94305 C. W. Rella Hansen Experimental Physics Laboratory, Stanford University, Stanford, California 94305 M. D. Fayerd) Department of Chemistry, Stanford University, Stanford, California 94305 ͑Received 22 January 1997; accepted 21 March 1997͒ Vibrational echo experiments were performed on the IR active CO stretching modes Ϫ1 (ϳ2000 cm ) of rhodium dicarbonylacetylacetonate ͓Rh͑CO͒2acac͔ and tungsten hexacarbonyl ͓W͑CO͒6͔ in dibutylphthalate and a mutant of myoglobin-CO ͑H64V-CO͒ in glycerol–water using ps IR pulses from a free electron laser. The echo decays display pronounced beats and are nonexponential. The beats and nonexponential decays arise because the bandwidths of the laser pulses exceed the vibrational anharmonicities, leading to the excitation and dephasing of a multilevel coherence. From the beat frequencies, the anharmonicities are determined to be 14.7, Ϫ1 13.5, and 25.4 cm , for W͑CO͒6,Rh͑CO͒2acac, and H64V-CO, respectively. From the components of the nonexponential decays, the vibrational dephasing at very low temperature of both the vϭ0–1 and vϭ1 – 2 transitions are determined. At the lowest temperatures, T2Ϸ2T1 , so the vϭ2 lifetimes are obtained for the three molecules. These are found to be signiﬁcantly shorter than the vϭ1 lifetimes. Although the vϭ1 lifetimes are similar for the three molecules, there is a wide variation in the vϭ2 lifetimes. © 1997 American Institute of Physics. ͓S0021-9606͑97͒02324-6͔

I. INTRODUCTION tional pump–probe spectroscopy.1,2 In this method, the vϭ0 1 transition is pumped with an intense ps IR pulse. The measurement of vibrational spectra using Fourier On a→ time scale short compared to the vibrational lifetime, a transform infrared ͑FTIR͒ spectroscopy provides a wealth of second IR probe, generally having a wide bandwidth, is information about the equilibrium structures of molecules passed through the sample and spectrally resolved. The and the effects of solvents on normal modes. However, im- vϭ1 2 transition appears as an absorption of the probe portant types of measurements, i.e., vibrational homoge- that is→ not present in the absence of the pump. This method neous linewidth and the vibrational anharmonicity, are either has been successfully applied,1,2 but it requires two ps IR difficult or impossible to perform with ordinary linear tech- pulses with different characteristics, i.e., wavelength and niques. Measurements of vibrational anharmonicities provide bandwidth. information about the shape of the vibrational potential sur- An alternative approach is to perform a vibrational face. Vibrational anharmonicities are not readily obtained us- echo3–6 using pulses having sufficiently wide bandwidth to ing a conventional FTIR because the vϭ0 2 transition is establish a multilevel coherence among the vϭ0, 1, and 2 → forbidden, and, therefore, the absorption at the frequency states.7 The multilevel coherence results in a vibrational echo corresponding to the vϭ0 2 transition is weak. In addi- decay with beats. The beats occur at the frequency of the → tion, the spectral region around a vϭ0 2 transition is fre- difference between the vϭ0 1 and vϭ1 2 transition en- → quently congested with combination bands, making it diffi- ergies, i.e., at the frequency→ of the vibrational→ anharmonic cult to identify the correct peak. A direct measurement of the splitting. This is not a conventional quantum beat. In a quan- vϭ1 2 transition is not usually feasible for the higher fre- tum beat, a state is coupled by the radiation field directly to → Ϫ quency modes (1000– 4000 cm 1) because the thermal two other states that fall within the bandwidth of the pulse. population at ambient temperatures of the vϭ1 state is down In the vibrational echo anharmonic beat ͑VEB͒ experiment, five to twenty factors of e compared to the vϭ0 state. the radiation field couples vϭ0 1 and then vϭ1 2. Another technique for measuring vibrational anharmo- There is no direct coupling between→vϭ0 2. → nicity involves the use of time resolved, two color vibra- In addition to measuring the vibrational→ anharmonicity through the echo decay beat frequency, the echo decay pro- a͒Permanent address: Department of Physics, Franklin and Marshall College, vides the homogeneous dephasing time ͑homogeneous line- Lancaster, PA 17604. width͒ of both the vϭ0–1 and vϭ1 – 2 transitions. The b͒Present address: Department of Chemistry, University of Munich, 80290 homogeneous linewidth provides information on the dynami- Munich Germany. c͒ cal interactions of a vibration with its environment. Vibra- Present address: Department of Chemistry, University of Chicago, Chi- 4,5 4,5 cago, IL 60637. tional echo experiments on liquids, glasses, and d͒Author to whom correspondence should be sent. proteins6,8 have shown that vibrational spectra may be

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The integrated intensity of the echo homogeneous lines, the echo decay is a bi-exponential pulse is measured as a function of ␶. squared with beats. One exponential corresponds to the de- For the case in which the laser bandwidth is narrow with cay of the vϭ1 – 2 coherence and the other exponential cor- respect to the vibrational anharmonicity, such that the vibra- responds to the decay of the vϭ0 – 1 coherence. Therefore, tional coherence involves only the vϭ0 – 1 transition, the the VEB experiment allows both the homogeneous line- experiment is modeled well by a two level system. For a widths and the vibrational anharmonicity to be experimental Lorentzian homogeneous lineshape with linewidth observables. ⌫ϭ1/␲T2 , the echo decays as In this paper, VEB experiments on CO stretching modes of three molecules are presented: rhodiumdicarbonylacety- I͑␶͒ϭI0 exp͑Ϫ4␥␶͒, ͑1͒ lacetonate ͓Rh͑CO͒2acac͔ and tungsten hexacarbonyl ͓W͑CO͒ ͔ in dibutylphthalate ͑DBP͒, and CO bound to the 6 where ␥ϭ1/T for the vϭ0 – 1 transition. The homogeneous active site of a myoglobin protein in a 95:5 mixture of glyc- 2 dephasing time has contributions from the pure dephasing erol:water. The myoglobin protein is a mutant of wild type time, T* , and the vibrational lifetime, T , myoglobin in which the distal histidine ͑position 64͒ is re- 2 1 placed with a valine ͑H64V-CO͒.6 The protein pocket around 1 1 1 the CO in H64V is markedly different than either of the ϭ ϩ . ͑2͒ inorganic compounds. In the protein, the CO is bound to an ␲T2 ␲T2* 2␲T1 octahedral Fe which has four bonds to a large aromatic mac- rocycle. The last bond of the Fe is to the proximal histidine There can also be a contribution from orientational ͑position 93͒. relaxation.5 However, the experiments reported here were The vibrational echo experiments on Rh͑CO͒2acac are conducted in low temperature glassy solvents, so the contri- the first for this molecule. Vibrational echo studies of bution from orientational relaxation is negligible. The life- W͑CO͒6 in several liquid and glassy solvents have been time can be measured with a pump–probe experiment, and 3,5 reported, including the first observation of vibrational echo when combined with a vibrational echo measurement of 7 anharmonic beats on W͑CO͒6 in DBP. Further, extensive T2 , the pure dephasing contribution to the homogeneous vibrational echo studies and lifetime measurements of the linewidth can be determined. For the systems studied here, vϭ0 – 1 transition have been presented previously for the lifetime components of the homogeneous lines are only 6,8 Mb–CO and H64V-CO. slightly temperature dependent. The pure dephasing compo- In the following, Rh͑CO͒2acac data is presented as a nents are more strongly temperature dependent. function of the excitation frequency. As the frequency is de- When the laser bandwidth is similar to the anharmonic- creased, the beats become more pronounced. These results ity, short pulse excitation will create a three level coherence are compared to an approximate theoretical calculation, and involving the vϭ0, 1, and 2 vibrational levels. Previous the- the trends are found to be in accord. The vϭ0–1 and oretical work on a three level echo described an equally vϭ1 – 2 dephasing times and the anharmonicity are also ob- spaced system.9 The derivation of the VEB signals for three tained. For H64V-CO, the anharmonicity is much greater level systems has a large contribution from experiments and than in the two organometallic compounds. By tuning to theory studying coherent oscillations in semiconductor lower frequencies than the vϭ0 1 transition, beats are ob- structures.10,11 The echo signal can be described for an un- → served although the pulse bandwidth is insufficient to pro- equally spaced three-level system using a semiclassical dia- duce beats when centered on the vϭ0 1 line. Again, the grammatic perturbation theory treatment of the third-order → vϭ0–1 and vϭ1 – 2 dephasing times and the anharmonic- nonlinear polarizability.12,13 ͑See Appendix for details.͒ The ity are obtained. These results are compared to prior mea- three-level system is spaced by the frequencies ␻01 and surements on W͑CO͒6 in DBP. ␻12 , where ␻01ϭ␻12ϩ⌬, and ⌬Ӷ␻01 and ␻12 . ⌬ is the anharmonic vibrational energy splitting. The transition frequencies ␻ and ␻ lie within the bandwidth of the pulses. II. THE NATURE OF THE EXPERIMENT AND 01 12 For such a system, three independent resonant pathways PROCEDURES ͑diagrams͒ exist that result in rephasing and the generation of In a vibrational echo experiment, two IR pulses, tuned to the echo pulse. ͑See Fig. 6 in the Appendix.͒ In addition to the frequency of the molecular vibration of interest, are the two that describe a rephasing in a two level system,13 a crossed in the sample. The first pulse creates an ensemble of third diagram accounts for the possibility of rephasing the coherent superposition states that begin to dephase because vϭ1 – 2 coherence. As discussed in the appendix, for a finite of inhomogeneous broadening. A second pulse, delayed by pulse bandwidth, where the E-field amplitude differs at ␻01 7 time ␶, is incident on the sample at an angle ␪ with respect to and ␻12 , the decay is given by

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2 with the first pulse. The net result is the square of the Gauss- I͑␶͒ϭI͑0͒exp͑Ϫ2␥01␶͕͒͑E01•␮01͒ exp͑Ϫ2␥01␶͒ ian pulse is convolved with the Gaussian pulse, yielding the 2 ϩ͑E12•␮12͒ exp͑Ϫ2␥12␶͒Ϫ2͑E01•␮01͒ ͱ3/2 factor. The vibrational echo experiments were performed using ϫ͑E12•␮12͒exp͓Ϫ͑␥01ϩ␥12͒␶͔cos͑⌬␶ϩ␾͖͒. the Stanford Free Electron Laser ͑FEL͒. The FEL pulse train ͑3͒ consists of a macropulse having a duration of ϳ3 ms and repeating at 10 Hz. Within a macropulse is a series of mi- Here, E01 and E12 are amplitudes of the electric fields at the cropulses repeating at 11.8 MHz. Each transform limited respective transitions and ␮01 and ␮12 are the respective di- Gaussian micropulse has an energy of ϳ0.5 ␮J. The pulse pole transition moments which are constant. ␥01 and ␥12 are duration was ϳ1ps͑see below͒. The FEL frequency is ac- the corresponding homogeneous dephasing decay constants, tively stabilized to within 0.02% of the center frequency. and ⌬ is the vibrational anharmonic splitting frequency ͑the Both the autocorrelation and the spectrum were monitored beat frequency in the signal͒. I(0) contains all the factors continuously during experiments. that determine the strength of the signal but are not involved The experimental apparatus is a slightly modified ver- in either the time dependent decays or the wavelength depen- sion of one reported previously.6,8 Very briefly, the IR beam dence of the beats. The dephasing rates for the two transi- is split into two equal parts to become the two input pulses in tions, ␥01 and ␥12 , are phenomenological; no model has the vibrational echo sequence. To reduce sample heating been assumed for the coupling of these modes to the bath. while retaining high peak power, single micropulses are se- For the narrow bandwidth case (E12ϭ0), Eq. ͑1͒ is recov- lected from the macropulse at a reduced repetition rate using ered. The phase factor in Eq. ͑3͒ does not arise from the germanium acousto-optic modulators ͑AOM͒. One pulse is derivation, i.e., ␾ϭ0. As is standard in such derivations for selected at ϳ60 kHz, while the other is chopped to which analytical results can be obtained, the pulse is taken to ϳ30 kHz by a second AOM and sent through a motorized be a delta function in duration. Therefore, it has infinite optical delay line. The chopping is performed to do back- bandwidth, and the E-field amplitude is independent of fre- ground subtraction. The two beams are subsequently focused quency. The derivation was modified by specifying different to ϳ50 ␮m diameter spots in the sample using an off-axis E-fields at the two transition frequencies. However, it has paraboloidal mirror. The echo signal is subsequently focused been shown that inclusion of the finite bandwidth of the 14 onto an IR detector. The echo signal is acquired with a gated pulses leads to a phase factor that need not be zero. There- integrator and digitized by computer as the computer scans a fore, the phase factor was included, and it was found to be an stepper motor delay line. aid in fitting the data discussed below. The W͑CO͒6 and H64V samples were prepared as re- To compare Eq. ͑3͒ to data requires a convolution to ported previously.6,7 The Rh͑CO͒ acac sample was prepared account for the finite pulse duration. To extract an echo de- 2 as follows: Rh͑CO2͒acac ͑Aldrich, 99%͒ was added to DBP cay that is on the same time scale as the pulse duration re- ͑Aldrich, 99.9%͒ to a molarity of ϳ1ϫ10Ϫ3. The sample quires full consideration of the three time ordered interac- was placed in a custom built copper housing between CaF2 tions of the radiation fields with the vibrations. In these windows and an optical path of 400 ␮m. The samples were experiments, the echo decays are long compared to the pulse cooled using a He flow cryostat and the temperature was durations, so this procedure is unnecessary. However, the controlled to within 0.2 K. beat frequency is comparable to the pulse duration. There- fore, in the data, the beats appear with much less depth of modulation than they would have in the absence of a finite III. RESULTS AND DISCUSSION instrument response. To account for this, Eq. ͑3͒ is con- Figure 1͑a͒ displays data taken on Rh͑CO͒ acac in DBP volved with the echo time dependence that would be ob- 2 at 3.4 K using a frequency of 2004.0 cmϪ1. The center of the served for a sample with delta function response. This is vϭ0 1 transition is at 2010.1 cmϪ1. So this data is taken defined as the instrument response function. For finite instru- with→ the frequency tuned somewhat to the red of the line ment responses, the data are fit by center. The pulse bandwidth is 8 cmϪ1 FWHM, which corresponds to a 1.3 ps pulse duration. A fit to the data is also Ϫ1 3 S͑␶͒ϭT ͕T͓p͑ͱ 2␶͔͒xT͓I͑␶͔͖͒, ͑4͒ shown. The fit uses Eq. ͑3͒ convolved with a 1.6 ps FWHM instrument response. Figure 1͑b͒ shows the same data but where T and TϪ1 are Fourier transform and inverse Fourier withafittoEq.͑3͒that holds the cosine term constant at transform, respectively. I(t) is from Eq. ͑3͒, and p(␶) is the one, i.e., the decay kinetics are the same but there are no laser pulse envelope at the intensity level. In these experi- beats in the fitting function. The inset in Fig. 1͑b͒ are the ments, the pulse envelope is an essentially transform limited residuals, which contain only the beats. ⌬, the anharmonicity Gaussian. The factor of ͱ3/2 arises from the three interac- ͑the difference between the vϭ0 1 and the vϭ1 2 tran- tions of the two Gaussian pulses used in the vibrational echo sition frequencies͒ can be obtained→ from the fit in Fig.→ 1͑a͒ or experiment. This procedure is the convolution of the sample it can be read off directly from the inset in Fig. 1͑b͒. The response function with ͱ3/2 times the Gaussian pulse enve- results yield ⌬ϭ13.5Ϯ0.2 cmϪ1. lope. There are two interactions with the second pulse, which In the fit to Eq. ͑3͒, the fast component corresponds to gives rise to the pulse envelope squared. This is convolved the dephasing time of the vϭ1 – 2 vibrational coherence

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FIG. 2. ͑a͒ Rh͑CO͒2acac vibrational echo data taken at 3.4 K at laser frequencies, from top to bottom, 2020.2, 2012.1, 2004.0, and 1996.0 cmϪ1. The center of the vϭ0 1 transition is 2010.1 cmϪ1. The dephasing times FIG. 1. a Rh CO acac in DBP at 3.4 K and laser frequency of → ͑ ͒ ͑ ͒2 of the vϭ1 2 and vϭ0 1 transitions are 15 and 92 ps, respectively. ͑b͒ 2004.0 cmϪ1 and fit using Eqs. ͑3͒ and ͑4͒. The center of the vϭ0 1 → → Ϫ1 Ϫ1 → Calculated echo decays obtained using Eq. ͑3͒ for several ratios of the transition is at 2010.1 cm . The pulse bandwidth is 8 cm FWHM, which E-field amplitudes. These curves use lifetimes, beats frequencies, and corresponds to a 1.3 ps pulse duration. The fit to the data uses Eq. ͑3͒ phases determined from the Rh͑CO͒2acac data, and have been convolved convolved with a 1.6 ps FWHM instrument response and yields T2s of 15 with an 1.6 ps FWHM instrument response. Lines, from top to bottom in ͑b͒, and 92 ps for the vϭ1 2 and vϭ0 1 levels, respectively. ͑b͒ The same → → have ratios of the E fields at the vϭ0 1 and vϭ1 2 transitions of data but with a fit with Eq. ͑3͒ that holds the cos term constant at 1. The 99.5/0.5, 95/5, 67/33, and 20/80, respectively.→ The calculation→ presented in inset in b are the residuals. This is a method of displaying only the beats. ͑ ͒ ͑b͒ are qualitatively very similar to the data presented in ͑a͒. ⌬, the anharmonicity ͑the difference between the vϭ0 1 and the v → ϭ1 2 transition frequencies͒ can be obtained from the fit. The results → yield ⌬ϭ13.5Ϯ0.2 cmϪ1. tion is 2010.1 cmϪ1. The first data set is at 2020.2 cmϪ1.In this data set, there are no apparent beats. These data can be (T2(12)) and the slow component corresponds to the fit with a single exponential, Eq. ͑1͒. At this laser frequency, dephasing time of the vϭ0 – 1 vibrational coherence there is little or no overlap of the pulse bandwidth with the (T2(01)). As can be seen from both Figs. 1͑a͒ and 1͑b͒, the vϭ1 2 transition. The single exponential decay yields → Ϫ1 fits to the data are quite good. The decay constants yield ␥01 only. Tuning to lower energy, 2012.1 cm , there are homogeneous dephasing times of T2(12)ϭ15 ps and only very low amplitude beats, which are almost lost in the T2(01)ϭ92 ps, respectively. In fitting the data, the phase noise. However, the data cannot be fit well to a single expo- factor ␾ in Eq. ͑3͒ was varied in addition to the other param- nential decay. To obtain a good fit, Eqs. ͑3͒ and ͑4͒ are eters. While the data can be fit with ␾ϭ0, the fitting routine needed. This frequency is still to the blue of the line center of consistently returned values of ␾Ϸ␲/2. the vϭ0 1 transition, so the overlap of the pulse band- Equation ͑3͒ predicts that the magnitudes of the compo- width with→ the vϭ1 2 transition is small. The next data nents of decay and the amplitude of the beats are related to set, at 2004.0 cmϪ1,→ definitely display beats. In this case, the strengths of the E-fields at the two transition frequencies. there is significant overlap of the pulse bandwidth with the For a finite bandwidth pulse, as the excitation frequency is vϭ1 2 transition, and the beat amplitude and fast decay moved from around the peak of the vϭ0 1 transition to component→ magnitude increase markedly. Finally, the lower energies, the beats will become more→ pronounced, and 1996.0 cmϪ1 data set shows significant beats as there is now the component of the decay corresponding to the relaxation extensive overlap of the excitation bandwidth with the of the vϭ1 – 2 coherence will become larger. Figure 2͑a͒ vϭ1 2 transition. All data sets indicate that the dephasing → displays Rh͑CO͒2acac vibrational echo data taken at 3.4 K at times of the vϭ1–2 and vϭ0 – 1 transitions are 15 and 92 a variety of frequencies. The center of the vϭ0 1 transi- ps, respectively. There is a clear phase shift in the data of →

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Fig. 2͑a͒ that should be noticed. The phase shift may be explained by a full theoretical analysis of this problem using finite bandwidth pulses.14 Figure 2͑b͒ shows calculated echo decays obtained using Eq. ͑3͒ for several ratios of the E-field amplitudes. These curves use dephasing times, beats frequencies, and phases determined from the Rh͑CO͒2acac data, and have been convolved with an 1.6 ps FWHM instrument response. Lines A, B, C, and D in Fig. 2͑b͒ have ratios of the vϭ0 1 and vϭ1 2 transition E-fields of 99.5/0.5, 95/5, 67/33,→ and 20/ → 80, respectively. In the calculations, ␮12ϭ&␮01 ,asisthe case for a harmonic oscillator. As the quotient of the two E fields decreases, the amplitudes of the beats increase, and there is an increased short time decay of the signal because the contribution of the ␥12 portion of the signal is increased. The magnitude of the beats in the simulation is also a function of the instrument response compared to the beat frequency. For all of the data presented in this paper, the instrument responses are in the 1 ps range. Since the beat frequencies are 2–5 ps, depending on the sample, the instrument response significantly decreases the observed magnitude of the beats. The calculation presented in Fig. 2͑b͒ are FIG. 3. Vibrational echo data on H64V in glycerol–water at 3.4 K with the laser tuned to 1957 cmϪ1. The center of the vϭ0 1 transition for this line qualitatively very similar to the data presented in Fig. 2͑a͒. Ϫ1 → Ϫ1 This demonstrates the basic validity of the description of the is 1969 cm . The excitation bandwidth was ϳ16 cm FWHM. The dephasing times of the vϭ1 2 and vϭ0 1 transitions are 20 and 65 ps, multilevel coherence and its frequency dependence. How- respectively. → → ever, as discussed above and in detail in the Appendix, the analytical expression given in Eq. ͑3͒ was derived for a delta function duration pulse, but with the standard type of deri- ture as 35 ps.15 Comparison of the line center echo decay and vation modified to include different E fields at the two tran- the pump–probe data demonstrate that at these low tempera- sition frequencies. A quantitative theoretical description of tures, the homogeneous dephasing time is approximately this problem cannot be obtained analytically because it will twice the lifetime (2T1), i.e., there is no significant pure include a finite pulse duration with a finite bandwidth. This dephasing. From the fit to the data, the decay constants, produces a complicated numerical problem that is under ␥12 and ␥01 , yield homogeneous dephasing times of 14 investigation. As shown above, Eq. ͑3͒ provides a good T2(12)ϭ20 ps and T2(01)ϭ65 ps, respectively. The fit also description of the results. Because it is analytical, it is very gives the beat frequency. The inset in Fig. 3 displays only the useful in data fitting. It gives all parameters correctly except beats, obtained in the same manner as described for Fig. the depth of the beats. 1͑b͒. The data are modulated with a 1.3 ps beat which cor- Previously, an excitation frequency dependent study of responds to an anharmonicity of 25.4Ϯ0.2 cmϪ1. This can 7 Ϫ1 vibrational beats on W͑CO͒6 in DBT was reported. These be compared to the ϳ26 cm anharmonicity reported for data display a frequency dependence which is substantially Mb-CO, which was measured using two color pump–probe different from the data and calculations presented in Fig. 2. experiments.2 While H64V-CO has a shift in frequency and a This difference probably arises because of the triple degen- 20% decrease in the rate of pure dephasing,6 within experi- eracy of the T1u CO transition of the W͑CO͒6. The triple mental error, replacing the distal histidine with a valine does degeneracy is undoubtedly broken in the glassy environment. not change the CO vibrational anharmonicity. This will lead to many more pathways for the interaction of These data on H64V-CO show that it is possible to use the radiation field with the system than are contained in the the VEB method to obtain vibrational anharmonicities which derivation of Eq. ͑3͒. The Rh͑CO͒2acac CO stretching mode are substantially larger than previously reported for on which the experiments were performed is nondegenerate, W͑CO͒6. Metal carbonyls with two or more equivalent car- and Eq. ͑3͒ provides a good description of the nondegenerate bonyls have very small anharmonicities. By using a pulse problem. duration of 670 fs FWHM and tuning to lower frequency VEB measurements can be made on a large variety of than the vϭ0 1 line center, it was possible to measure the systems. Figure 3 shows data taken on H64V in glycerol– moderate anharmonicity→ of H64V-CO. Currently, it is not water at 3.4 K with the laser tuned to 1957 cmϪ1. The center possible to produce significantly shorter pulses than 600 fs of the vϭ0 1 transition for this line is 1969 cmϪ1. The with the Stanford FEL. However, conventional laser/OPA excitation bandwidth→ for these data was ϳ16 cmϪ1 FWHM. systems can produce IR pulses which are substantially When the echo decay is measured on line center, the data are shorter. Therefore, using available technology, it should not a single exponential decay without beats. IR pump–probe be difficult to measure large anharmonicities of 100 cmϪ1 or experiments measured the vϭ0 1 lifetime at this tempera- more. →

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FIG. 5. Decay times T2(01)ϭ1/␥01 and T2(12)ϭ1/␥12 of W͑CO͒6 in DBT for temperatures between 10 and 150 K obtained from ﬁts to Eq. ͑3͒. The lack of temperature dependence of T2(12) at the lower temperatures shows that T2 is lifetime limited.

The data for W͑CO͒6 in DBP at 10 K demonstrates that T2Ϸ2T1 for both transitions. It is reasonable to assume that FIG. 4. Vibrational echo decay and fit W͑CO͒6 in dibutylphthalate at 10 K T2Ϸ2T1 for both transitions of Rh͑CO͒2acac in DBP and for and 1976.3 cmϪ1. The beats are separated by 2.3 ps, corresponding to a H64V-CO as well. These samples were studied at 3.4 K. It is vibrational anharmonic splitting of ⌬ϭ14.7 cmϪ1Ϯ0.3 cmϪ1. known that T2Ϸ2T1 for the vϭ0 1 transitions in both samples at this temperature.6,15 At these→ very low tempera- Figure 4 shows a vibrational echo decay for asymmetric tures, the samples are in the low temperature limit. The thermal fluctuations of the heat bath are insufficient to cause CO stretching mode of W͑CO͒6 in DBP at 10 K with the laser tuned to the center of the vϭ0 1 transition significant pure dephasing. Even electronic transitions of Ϫ1 7→ large molecules in low temperature glasses have very slow 1976.3 cm , as well as a fit using Eq. ͑3͒. The beats are 16 separated by 2.3 ps, corresponding to a vibrational anhar- pure dephasing times of ϳ1 ns. These are only measurable monic splitting of ⌬ϭ14.7 cmϪ1ϩϮ0.3 cmϪ1. This splitting because the electronic excited state lifetimes are long com- is in accord with the value of 15 cmϪ1Ϯ1cmϪ1subsequently pared to the vibrational lifetimes discussed here. The cou- obtained by Heilweil and co-workers from observation of the pling of vibrational transitions to the medium is much weaker than the coupling of electronic transitions as evi- vϭ1 2 and vϭ2 3 transitions of W͑CO͒6 in hexane using two→ color pump–probe→ experiments.1 denced by much smaller gas to solvent shifts in transition frequencies and much slower pure dephasing at elevated The experiments on W͑CO͒6 were performed as a function of temperature.7 From the fits to Eq. ͑3͒, the homoge- temperatures. In addition, there is no fundamental reason neous dephasing times for the vϭ0 1 and vϭ1 2 tran- why the pure dephasing of the vϭ1 – 2 transition should be sitions were obtained for temperatures→ between 10→ and 150 different than the vϭ0 – 1 pure dephasing. For these reasons, K. These are displayed in Fig. 5. Detailed analysis of the we will take the T2s of both transitions of all three samples temperature dependence of the vϭ0 – 1 dephasing demon- to be lifetime limited at the lowest temperature. strates that by 10 K the homogeneous linewidth is dominated The lifetimes, T1(1) and T1(2) of vϭ1 and vϭ2, re- 5 spectively, and dephasing times T2(01) and T2(12) of the by the vibrational lifetime, i.e., T2Ϸ2T1 . From the fit to Eq. ͑3͒ of the data at 10 K and using results from Ref. 7 for vϭ0–1 and vϭ1 – 2 transitions, respectively, are summa- rized in Table I. The approximate errors for T1(1) and the vϭ0 1 transition, the decay constants, ␥12 and ␥01 , → T2(01) is 3%. The approximate error for T2(12) is 5%. Also yield homogeneous dephasing times of T2(12)ϭ5.0 ps and reported in Table I are the ratios of lifetimes, RT T2(01)ϭ66 ps, respectively. Above 10 K, the vϭ0–1 ho- 1 1.3 5 ϭT (1)/T (2) and dephasing times, R ϭT (01)/T (12). mogeneous linewidth increases as a power law, T . How- 1 1 T2 2 2 ever, the temperature dependence of the vϭ1 – 2 dephasing The value of T1(1) is highly dependent on the choice of has a very different character. It is temperature independent the solute–solvent systems studied. The lifetime of up to ϳ90 K. This demonstrates that the vϭ1 – 2 dephasing W͑CO͒6 is solvent dependent. At room temperature, its life- is dominated by T1 with T2Ϸ2T1 since a pure dephasing time in CCl4 is ϳ700 ps, in CHCl3 is ϳ340 ps, in contribution to T2 always displays a temperature depen- 2-methylpentane is 150 ps. In the solvents CHCl3 and dence. Thus, at 10 K, both T2(01) and T2(12) are measures 2-methylpentane, the lifetimes display an inverted tempera- of the vibrational lifetimes of the two states. ture dependence,17–19 i.e., the lifetimes become faster as the

J. Chem. Phys., Vol. 106, No. 24, 22 June 1997

TABLE I. Experimental results. Summary of the lifetimes, T1(1) and T1(2) of vϭ1 and vϭ2, respectively, and dephasing times T2(01) and T2(12) of the vϭ0–1 and vϭ1 – 2 transitions, respectively. The approximate errors of T1(1) and T2(01) is 3%. The approximate error for T2(12) is 5%. The approximate error for T (2) is 10%. Also reported are the ratios of lifetimes, R ϭT (1)/T (2) and dephasing times, R 1 T1 1 1 T2 ϭT2(01)/T2(12).

Sample Anharmonicity T (01) T (12) R T (1) T (2) R 2 2 T2 1 1 T1 Ϫ1 W͑CO͒6 14.7 cm 66 ps 5 ps 13 33 ps 3 ps 11 Ϫ1 Rh͑CO͒2acac 13.5 cm 92 ps 15 ps 6 49 ps 9 ps 5 H64V-CO 25.4 cmϪ1 65 ps 20 ps 3.3 35 ps 14 ps 2.5

temperature is decreased. In 2-methylpentane, the lifetime at rate scales as the square of the matrix element. Therefore, 10 K is 100 ps. The change in lifetime with solvent shows when the initial state is 2, the ͱv factor squared yields 2, and that the solvent is intimately involved in the relaxation path- when the initial state is 1, the ͱv factor squared yields 1, way. The initial CO vibration relaxes into a combination of giving an R of 2. So from this consideration alone, the v solute and solvent modes plus a mode of the solvent con- ϭ1 lifetime should be twice as long as the vϭ2 lifetime. tinuum ͑solvent phonon͒ as necessary to conserve At low temperature, where there is no pure dephasing, 17–19 energy. As the complexity of the solvent increases, more T2 is determined by 2T1 . However, when looking at dephas- modes are available to provide a low-order pathway for the ing, if R is 2, the ratio, R ϭT (01)/T (12) should be 3. T1 T2 2 2 relaxation. In CCl4, a ﬁfth-order process is required that in- The dephasing rate is the average of the rates of population volves the annihilation of the initial excitation and the cre- decay out of the two levels involved. For dephasing of the ation of three solute–solvent vibrational excitations and a vϭ0 – 1 transition, 17–19 phonon. In CHCl3, the addition of the CH bending mode at ϳ1250 cmϪ1 reduces the order of the process to fourth- 1 ␥͑1͒ϩ␥͑0͒ 1 17–19 ϭ ϭ , ͑5͒ order, and the lifetime is reduced. 2-methylpentane has a T2͑01͒ 2 2T1͑1͒ wider variety of modes, and DBP an even wider variety. where ␥(0)ϭ1/T1(0) and ␥(1)ϭ1/T1(1). Since the rate out Presumably, these considerations are very similar for of the ground state is 0, the common result for a vϭ0–1 Rh͑CO͒2acac. vibrational echo of T2(01)ϭ2T1(1) is obtained. Rearrang- However, H64V-CO, other Mb-COs, and a large variety ing Eq. ͑5͒ yields of model heme-COs have been shown to relax in a different manner.20–22 The relaxation is essentially solvent 2 21,23,24 ␥͑1͒ϭ . ͑6͒ independent. Relaxation occurs through deposition of T2͑01͒ the CO vibrational energy into the enormous density of states The upper state dephasing is similar in that it is the average provided by the heme. The coupling has been shown to be of the rates out of the two levels as given in Eq. ͑7͒ via ␲-bond interactions. Back bonding from the metal–heme ␲ electron system into the CO ␲* antibonding molecular 1 ␥͑1͒ϩ␥͑2͒ orbital couples the CO vibration directly to the vast number ϭ . ͑7͒ T2͑12͒ 2 of heme vibrational states. An increase in the back bonding decreases the vibrational lifetime. Mb-CO has a lifetime of The vϭ2 lifetimes displayed in Table I were calculated from 16 ps. Replacing the distal histidine with a valine to form measured values of the vϭ1 lifetimes (␥(1)s) and the H64V-CO reduces the back bonding and slows the lifetime vϭ1 – 2 dephasing rates, using Eq. ͑7͒. The approximate er- to 35 ps. Therefore, by selecting the particular form of Mb, it ror for these calculated lifetimes is 10%. As stated above, the is possible to have a heme-CO system with a vϭ1 lifetime minimum ratio of the upper and lower dephasing rates is 3. that is similar to the simpler metal carbonyls. This facilitates This follows from the simpliﬁcation of Eq. ͑7͒ using the the comparison of the vϭ1 lifetimes and the vϭ2 lifetimes. Landau–Teller model for which ␥(2)ϭ2␥(1), A striking feature of Table I is the trend in the ratios 1 ␥͑1͒ϩ␥͑2͒ 3␥͑1͒ (RT )ofT1(1) to T1(2) for the three samples. In all cases, ϭ ϭ . ͑8͒ 1 T2͑12͒ 2 2 relaxation from vϭ2 is substantially faster than relaxation from vϭ1. In the most basic analysis, it would be expected Using the results of Eq. ͑6͒ in Eq. ͑8͒ yields that ratio R ϭT (1)/T (2) would be 2.25 This will occur if T1 1 1 1 3 ϭ . ͑9͒ the density of bath states and the strength of coupling to the T ͑12͒ T ͑01͒ bath for the relaxation process are independent of the initial 2 2 vibrational quantum number, v, and if no additional relax- Clearly, these factors of 2 and 3 alone do not account for the R and R listed in Table I, although they come very ation pathways become available for the higher v initial T1 T2 state. The action of a lowering operator contained in the close for H64V-CO. The vibrational relaxation in the two vibrational relaxation coupling matrix element for a metal carbonyls depends strongly on interactions with the v vϪ1 relaxation brings out a factor of ͱv. The relaxation solvent. The vϭ1 2 transition is shifted to lower energy → →

J. Chem. Phys., Vol. 106, No. 24, 22 June 1997

Downloaded¬31¬Jul¬2002¬to¬171.64.123.74.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jsp 10034 Rector et al.: Vibrational dephasing from vibrational echo beats by ϳ14 cmϪ1 for both molecules. The relaxation pathway ͑3͒ provides a good description of the observables. It permits almost certainly includes a phonon mode of the glassy sol- extraction of the important physical parameters, i.e., the an- vent continuum. For a non-hydrogen bonding organic like harmonicity and the homogeneous linewidths of the v DBP, the continuum extends to ϳ200 cmϪ1.26 Therefore, if ϭ0 – 1 transition and the vϭ1 – 2 transition. The frequency the phonon mode is to the high energy side of the peak of the dependent Rh͑CO͒2acac data demonstrates how the ampli- density of states, a 14 cmϪ1 shift to lower energy could result tudes of the beats, and, therefore, the extent of the multilevel in an increase in the density of states, and therefore an in- coherence changes with the laser excitation frequency. Equa- crease in the vibrational relaxation rate for vϭ2. Also, an tion ͑3͒ can reproduce the general nature of the frequency increase in the strength of coupling results from the partici- dependence, but it does not provide a quantitative description pation of a different phonon. However, such a change is because the derivation of the analytical expression does not likely to only account for a portion of the RT for W͑CO͒6 include the finite duration of the excitation pulses. 1 At the very low temperatures of the experiments, the and Rh͑CO͒2acac. H64V-CO, which does not utilize the solvent in its relaxation would not have this effect. homogeneous dephasing times are determined by the vibra- A factor which may be the major contributor to the large tional lifetimes, T2Ϸ2T1 . The three molecules studied have similar vϭ1 lifetimes, but the decays of the multilevel co- RT values of the simple metal carbonyls is the opening of a 1 herences observed in the vibrational echo experiments show new relaxation pathway for relaxation from the vϭ2 state. that the vϭ2 lifetimes of all three molecules are signifi- The new pathway is direct relaxation from vϭ2tovϭ0. cantly faster and that they vary from one molecule to an- The vϭ2 0 transitions for the systems studied have ener- other. Possible mechanisms for the decrease in the lifetimes gies ϳ3950→ cmϪ1. It is possible for the numerous C–H tran- and the variations were discussed. sitions, which have energies in the 3300– 2900 cmϪ1 range, To observe the VEB signal, it is necessary to have the to participate in the vibrational relaxation. If the vϭ2 0 transform limited bandwidth of the excitation pulses compa- relaxation is a low order path, perhaps not involving a pho-→ rable to the anharmonic splitting. As demonstrated, the am- non because of the large number of C–H modes covering a plitude of the beats can be enhanced by tuning to lower fre- wide range of frequencies, it could substantially augment the quency so the that the pulse bandwidth can more effectively rate of relaxation out of the vϭ2 level. The experiment mea- overlap both the vϭ0 1 and vϭ1 2 transitions. How- sures the decay constant, ␥ , which is independent of the 2 ever, it is important to→ emphasize that→ the shortest possible pathway for leaving the vϭ2 level. Relaxation from pulse duration ͑widest bandwidth͒ is not always desirable in vϭ2 0 or from vϭ2 1 have the same effect on the de- a vibrational echo experiment. If the aim is to understand the cay→ of the coherence.→ While both W͑CO͒ and 6 dynamics of the a liquid, glass, or protein system by studying Rh͑CO͒ acac have the vϭ2 0 opened, its influence on 2 the pure dephasing of the vϭ0 – 1 transition, the multilevel R will depend on the details→ of the coupling matrix ele- T1 coherence may interfere by turning a single exponential de- ments and differences in the internal molecular modes that cay into a multiexponential decay with beats. At high tem- can participate in the relaxation. H64V-CO already has an peratures, where the echo decay time can be comparable to enormous density of coupled states provided by the heme, the beat period, it can be difficult to extract the true echo and therefore, the availability of an additional pathway ap- decay. Therefore, it is desirable to use a pulse duration that is parently makes less difference. appropriate to obtain the information that is being sought. The ratio R in Table I can have at least three contri- T1 butions: the factor of 2 which comes from the quantum num- ACKNOWLEDGMENTS ber dependence through the lowering operator, a change in the phonon density of states, and the additional vϭ2 0 The authors would like to thank Professor Alan pathway. While all three molecules have similar vϭ1 life-→ Schwettman and Professor Todd Smith, Physics Department, time, the experiments demonstrate that there can be a wide Stanford University, and their research groups at the Stan- range of rates for decay out of the vϭ2 level. This range ford Free Electron Laser Center whose efforts made these depends on the relative contribution of the three factors dis- experiments possible. We would also like to thank Professor cussed above. Dana D. Dlott and Professor Stephen G. Sligar, and Dr. Jef- frey R. Hill and Dr. Ellen Y. T. Chien for providing us with the H64V-CO sample used in these experiments. This re- IV. CONCLUDING REMARKS search was supported by the Office of Naval Research, ͑N00014-94-1-1024͒, by the National Science Foundation, In this paper, we have investigated multilevel vibrational Division of Materials Research ͑DMR93-22504͒, and by the coherences observed in vibrational echo experiments in three Office of Naval Research, ͑N00014-92-J-1227͒. systems, Rh͑CO͒2acac in dibutylphthalate, H64V in glycerol–water, and W͑CO͒ in dibutylphthalate. The beats 6 APPENDIX in the echo decay provide a direct measure of the vibrational anharmonicity. The observations in the three systems show We present a derivation of Eq. ͑3͒ in this Appendix. We that the VEB experiment can be of utility for measuring use a standard Markovian model in the low temperature vibrational anharmonicities provided the coherence decay is limit. This treatment is similar to that of Fourkas et al. where 9 sufficiently slow to permit observation of the beats. Equation the signal was derived assuming ␻12ϭ␻01 .

J. Chem. Phys., Vol. 106, No. 24, 22 June 1997

There are three possible resonant quantum mechanical pathways that contribute to the vibrational echo signal ͑after tϭ0͒ when excitation bandwidth overlaps with three levels. These are shown in Figs. 6͑a͒ and 6͑b͒. Figure 6͑a͒ is the ladder diagram representation of the resonant pathways. The solid arrows denote interactions on the ket of the density matrix, while dotted arrows denote interactions on the bra of the density matrix. The signal is indicated with the dashed arrows. The first two pathways are the standard ones for the vibrational echo involving two levels. The third pathway involves excited-state absorption, and hence, interferes with the first two.9 The general polarization is a sum of the individual response functions integrated over the excitation fields

ϱ ϱ ϱ ϱ ͑3͒ ¯ ¯ ¯ P ϭ d␻G͑␻͒ dt3 dt2 dt1 E1*E2E3 ͵0 ͵0 ͵0 ͵0

ϫ Ri͑t1 ,t2 ,t3͒, ͑A1͒ ͚i where t1 ,t2 ,t3 are the times between each of the pulses in the vibrational echo sequence and Ri are the response functions. The distribution of transition frequencies, ␻,isre- ﬂected by the inhomogeneous distribution function, G( ). ␻ FIG. 6. ͑a͒ Latter diagram of the three pulse sequences in a VEB experi- For delta function pulses ment. The ﬁrst two diagrams are the two-level vibrational echo diagrams. ¯ The solid arrows denote interactions on the ket of the density matrix, while Eiϭ͉Ei͉␦͑tϪ␶i͒exp͑Ϫi␻t͒. ͑A2͒ dotted arrows denote interactions on the bra of the density matrix. The signal is indicated with the dashed arrows. ͑b͒ Another representation of the The response functions are given by three pulse sequences of in the VEB experiment. Time is taken to increase 4 moving up the parallel set of lines. R1ϭR2ϭ␮01 exp͑i␻01t1Ϫi␻01t3͒exp͑Ϫ␥01t1Ϫ␥01t3͒, ͑A3͒ 2 2 R3ϭϪ␮01␮12 exp͑i␻01t1Ϫi␻12t3͒exp͑Ϫ␥01t1Ϫ␥12t3͒, neous broadening is taken to be perfectly correlated in the A4 ͑ ͒ two transitions. Further, all transitions for a given distribu- where Iij(t)ϭexp(Ϫi␻ij tϪ␥ij t) and ␮ijϭ͉͗i͉␮͉j͉͘ and ␻ij tion have the same ␮ij and ␥ij . ϭ␻iϪ␻j . Note that the complete derivation depends also In the delta function pulse limit, the polarization is given on the t2 time, which is the time between the second and by the response function third interactions. In the echo experiment, the second pulse ϱ provides both the second and third interactions so the time ͑3͒ P ͑t3 ,t1͒ϭ G͑␦ij͒ ͚ Ri͑ti ,t3,␦͒ ͉E1*͉͉E2͉͉E3͉d␦ij , between them is zero for delta function pulses. Therefore, ͵0 ͫ i ͬ ͑A6͒ terms containing t2 in Eq. ͑A3͒ and ͑A4͒͑and henceforth͒ 0 have been dropped. substituting ␻ijϭ␻ijϩ␦ij yields Since for a purely harmonic system, ␮01ϭ&␮12 , the 4 third path will completely cancel the echo at tϭ0 and will R1ϭR2ϭ␮01 exp͓i␻01͑t1Ϫt3͔͒ cancel the echo signal at all time if ␻12ϭ␻01 and ␥01 ϫexp͓Ϫ␥ ͑t ϩt ͔͒exp͓i␦͑t Ϫt ͔͒, ͑A7͒ 9 01 1 3 1 3 ϭ␥12 . However, in real systems, the oscillator is not har- 2 2 monic, therefore ␻12Þ␻01 . As shown below, the beating in R3ϭϪ␮01␮12 exp͑i␻01t1Ϫi␻12t3Ϫ␥01t1Ϫ␥12t3͒ the data will arise from the difference between ␻12 and ␻01 ϫexp͓i␦͑t1Ϫt3͔͒. ͑A8͒ and the decay of the beats will be the average of the ␥01 and ␥12 times. Substituting Eqs. ͑A7͒ and ͑A8͒ into Eq. ͑A6͒ yields the To include inhomogeneous broadening, we take the dis- third-order polarization as ° tribution function to be a Gaussian centered at ␻ij : 2 2 Ϫ͑t3Ϫt1͒ ␴ P͑3͒͑t ,t ͒ϭ͉E*͉͉E ͉͉E ͉exp 1 Ϫ͑␻ Ϫ␻° ͒2 3 1 1 2 3 ͩ 2 ͪ ij ij G͑␻ij͒ϭ exp 2 ϭG͑␦ij͒, ͑A5͒ ͱ2␲␴ ͩ 2␴ ͪ 4 ϫ͕2␮01 exp͓i␻01͑t1Ϫt3͔͒ where ␦ij is the deviation from the center of the line. We will 2 2 ϫexp͓Ϫ␥01͑t1ϩt3͔͒Ϫ␮01␮12 further assume that the distribution function for the ␻01 transition is the same as the ␻12 transition. Also, the inhomoge- ϫexp͑i␻01t1Ϫi␻12t3Ϫ␥01t1Ϫ␥12t3͖͒. ͑A9͒

J. Chem. Phys., Vol. 106, No. 24, 22 June 1997

Downloaded¬31¬Jul¬2002¬to¬171.64.123.74.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jsp 10036 Rector et al.: Vibrational dephasing from vibrational echo beats

This shows the echo characteristic of being sharply peaked at 4 2 2 S͑t͒ϭ͑E01•␮01͒ ͓exp͑Ϫ2⌫01t͔͓͒4͑E01•␮01͒ ␮01 t1ϭt3 , if the distribution function is very broad. 2 2 The observed integrated signal is ϫexp͑Ϫ2⌫01t͒ϩ͑E12•␮12͒ ␮12 exp͑Ϫ2⌫12t͒ 2 2 Ϫ4͑E01•␮01͒ ͑E12•␮12͒ ␮01␮12 exp͑Ϫ⌫01t ϩϱ Ϫ⌫ t͒cos͑⌬t͔͒. ͑A13͒ 2 12 S͑t͒ϭ dt3͉P͑t1Ϫt3͉͒ . ͑A10͒ ͵Ϫϱ As can be seen from Figs. 6͑a͒ and 6͑b͒, all three pathways have the first two interactions involving only the vϭ0–1 transition. These interactions give rise to the first term. The Defining the intensity as the square of the absolute value of extra factors of ␮2 not multiplied by E2 give rise to the the E-field and substituting Eq. ͑A9͒ into Eq. ͑A10͒ yields signal.

1 S. M. Arrivo, T. P. Dougherty, W. T. Grubbs, and E. J. Heilweil, Chem. 3 8 4 4 Phys. Lett. 235, 247 ͑1995͒. S͑t͒ϭI ͓4␮01 exp͑Ϫ4␥01t͒ϩ␮01␮12 exp͑Ϫ2␥01t 2 J. C. Owrutsky, M. Li, B. Locke, and R. M. Hochstrasser, J. Phys. Chem. 99, 4842 ͑1995͒. 6 2 3 Ϫ2␥12t͒Ϫ2␮01␮12 exp͑Ϫ3␥01tϪ␥12t͒ D. Zimdars, A. Tokmakoff, S. Chen, S. R. Greenfield, and M. D. Fayer, Phys. Rev. Lett. 70, 2718 ͑1993͒. 4 ϫ͕exp͓i͑␻01Ϫ␻12͒t͔ϩexp͓Ϫi͑␻01Ϫ␻12͒t͔͖, A. Tokmakoff, D. Zimdars, R. S. Urdahl, R. S. Francis, A. S. Kwok, and M. D. Fayer, J. Phys. Chem. 99, 13310 ͑1995͒. ͑A11͒ 5 A. Tokmakoff and M. D. Fayer, J. Chem. Phys. 102, 2810 ͑1995͒. 6 K. D. Rector, C. W. Rella, A. S. Kwok, J. R. Hill, S. G. Sligar, E. Y. T. Chien, D. D. Dlott, and M. D. Fayer, J. Phys. Chem. B 101, 1468 ͑1997͒. which simplifies to 7 A. Tokmakoff, A. S. Kwok, R. S. Urdahl, R. S. Francis, and M. D. Fayer, Chem. Phys. Lett. 234, 289 ͑1995͒. 8 C. W. Rella, K. Rector, A. Kwok, J. R. Hill, H. A. Schwettman, D. D. Dlott, and M. D. Fayer, J. Phys. Chem. 100, 15620 ͑1996͒. 3 8 9 S͑t͒ϭI ͓exp͑Ϫ2␥01t͔͓͒␮01 exp͑Ϫ2␥01t͒ J. T. Fourkas, H. Kawashima, and K. A. Nelson, J. Chem. Phys. 103, 4393 ͑1995͒. 4 4 6 2 10 ϩ␮01␮12 exp͑Ϫ2␥12t͒ϩ␮01␮12 X. Zhu, M. S. Hybertsen, P. B. Littlewood, and M. C. Nuss, Phys. Rev. B 50, 11915 ͑1994͒. 11 ϫexp͑Ϫ␥01tϪ␥12t͒cos͑⌬t͔͒. ͑A12͒ S. T. Cundiff, Phys. Rev. A 49, 3114 ͑1994͒. 12 S. Mukamel and R. F. Loring, J. Opt. Soc. B 3, 595 ͑1986͒. 13 Y. J. Yan and S. Mukamel, J. Chem. Phys. 94, 179 ͑1991͒. This derivation assumed delta function pulses with infinite 14 J. Fourkas ͑private communication͒. 15 bandwidth. Therefore, the E fields have no frequency depen- K. D. Rector and M. D. Fayer ͑unpublished͒. 16 L. R. Narasimhan, K. A. Littau, D. W. Pack, Y. S. Bai, A. Elschner, and dence. There is no frequency dependence in Eq. ͑A12͒.In M. D. Fayer, Chem. Rev. 90, 439 ͑1990͒. particular, the depth of the modulation occurring at fre- 17 A. Tokmakoff, B. Sauter, and M. D. Fayer, J. Chem. Phys. 100, 9035 quency, ⌬, is independent of the laser frequency. The depth ͑1994͒. 18 V. M. Kenkre, A. Tokmakoff, and M. D. Fayer, J. Chem. Phys. 101, depends only on the values of the ␮01 and ␮12 , which are 10618 ͑1994͒. constants. This is in contradiction to the data in Fig. 2͑a͒. 19 P. Moore, A. Tokmakoff, T. Keyes, and M. D. Fayer, J. Chem. Phys. 103, The use of a delta function duration pulse makes it possible 3325 ͑1995͒. 20 to obtain an analytical expression. In the experiments, the J. R. Hill, D. D. Dlott, M. D. Fayer, C. W. Rella, M. M. Rosenblatt, K. S. Suslick, and C. J. Ziegler, J. Phys. Chem. 100, 218 ͑1996͒. laser pulses have finite durations and, therefore, finite band- 21 J. R. Hill, M. M. Rosenblatt, C. J. Ziegler, K. S. Suslick, D. D. Dlott, C. widths. The finite bandwidth results in different driving E W. Rella, and M. D. Fayer, J. Phys. Chem. 100, 18023 ͑1996͒. 22 fields, E01 and E12 , at the two transition frequencies. When D. D. Dlott, M. D. Fayer, J. R. Hill, C. W. Rella, K. S. Suslick, and C. J. the frequency is changed, the values of E and E change, Ziegler, J. Am. Chem. Soc. 118, 7853 ͑1996͒. 01 12 23 A. Ansari, J. Beredzen, D. Braunstein, B. R. Cowen, H. Frauenfelder, M. giving rise to the frequency dependence displayed in Fig. K. Hong, I. E. T. Iben, J. B. Johnson, P. Ormos, T. Sauke, R. Schroll, A. 2͑a͒. For a two level system, the coherence is produced by Schulte, P. J. Steinback, J. Vittitow, and R. D. Young, Biophys. Chem. 26, the coupling of the radiation field to the transition dipole as 337 ͑1987͒. 24 (E ␮). To include the effect of finite bandwidth pulses and D. Ivanov, J. T. Sage, M. Keim, J. R. Powell, S. A. Asher, and P. M. • Champion, J. Am. Chem. Soc. 116, 4139 ͑1994͒. yet preserve the analytical expression, we associate Eij with 25 L. Landau and E. Teller, Phys. Z. Sov. 10,34͑1936͒. 26 ␮ij where appropriate. This results in Eq. ͑A13͒, Y. J. Chang and E. W. Castner, Jr., J. Phys. Chem. 100, 3330 ͑1996͒.

J. Chem. Phys., Vol. 106, No. 24, 22 June 1997

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