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JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 24 22 JUNE 2001

Wavelength selective modulation in femtosecond pump–probespectroscopy and its application to heme proteins

Florin Rosca, Anand T. N. Kumar, Dan Ionascu, Theodore Sjodin, Andrey A. Demidov,and Paul M. Championa)

Physics Department and Center for Interdisciplinary Research on Complex Systems,Northeastern University, Boston, Massachusetts 02115

~Received 13 December 2000; accepted 19 February 2001!

We demonstrate novel lock-in detection techniques, using wavelength selective modulation ofultrafast pump and probe laser pulses, to discriminate between vibrational coherence and electronicpopulation decay signals. The technique is particularly useful in extracting low frequencyoscillations from the monotonically decaying background, which often dominates the signal inresonant samples. The central idea behind the technique involves modulating the red and/or bluewings of the laser light spectrum at different frequencies,VR and VB , followed by a lock-indetection at the sum or difference frequency,VR6VB . The wavelength selective modulation anddetection discriminates against contributions to the pump–probe signal that arise from degenerateelectric field interventions~i.e., only field interactions involving different optical frequencies aredetected!. This technique can be applied to either the pump or the probe pulse to enhance theoff-diagonal terms of the pump induced density matrix, or to select the coherent components of thetwo-frequency polarizability. We apply this technique to a variety of heme-protein samples to revealthe presence of very low-frequency modes~;20 cm21). Such low-frequency modes are notobserved in standard pump–probe experiments due to the dominant signals from electronicpopulation decay associated with resonant conditions. Studies of the diatomic dissociation reactionof myoglobin ~MbNO→Mb1NO!, using wavelength selective modulation of the pump pulse,reveal the presence of an oscillatory signal corresponding to the 220 cm21 Fe–His mode. Thisobservation suggests that the spin selection rules involving the ferrous iron atom of the heme groupmay be relaxed in the NO complex. Mixed iron spin states associated with adiabatic coupling in theMbNO sample could explain the fast time scales and large amplitude that characterize the NOgeminate recombination. ©2001 American Institute of Physics.@DOI: 10.1063/1.1363673#

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I. INTRODUCTION

Femtosecond coherence spectroscopy~FCS! is a pump–probe technique that utilizes the bandwidth of femtoseclaser pulses to prepare and monitor coherent states in avariety of samples.1–10 This technique allows the real-timobservation of ultrafast processes. Following pump pulsecitation of a resonant two electronic level system, two typof processes can occur that we refer to as ‘‘field driven’’ a‘‘reaction driven.’’ In the first case the pump pulse excitthe Raman active modes of the sample without affectingstructure. In the reaction driven case, the excited electrostate rapidly decays into other electronic states due to nradiative surface crossing, and the system is left in a cohe‘‘product state.’’ The delayed probe pulse monitors tchange in the sample absorption induced by the pump pu

In this work we focus on the detection of time resolvlow-frequency vibrational modes in heme proteins. Theme proteins form a large class of biomolecules, whichinvolved in a variety of important biochemical reactions suas diatomic ligand storage and transport, enzyme cataland electron transport. The iron atom, placed at the centethe heme group~active site!, can reversibly bind diatomic

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molecules like NO, CO, and O2. Here, we study the ferricand ferrous unbound species as well as the NO bound sWe have developed wavelength selective modulation teniques to detect low-frequency vibrational modes coheredriven by a pump pulse resonant with the Soret absorpband of the heme chromophore. The low-frequency modetected11,12 in heme proteins are conceivably connectedthe energy transport between the active site and the probackbone,13–15 which is expected to be important to thphysiological functionality of the protein.

The signals generated using FCS under resonant cotions consist of damped oscillations corresponding to thebrational modes excited by the pump pulse, along withsuperimposed monotonically decaying background, usurelated to processes like electronic relaxation, ligand recobination ~for the samples involving photodissociation!, andvibrational cooling. Vibrational oscillations below 400 cm21

are accessible using pulses of;50 fs. The main goal of thepresent paper is to demonstrate experimental techniquesresolve the vibrational part of the signal from the monotocally decaying background. The motivation for this endeais the fact that very low-frequency modes~,40 cm21) arequite difficult to recover from the data by signal processiwhen they are superimposed on a strong~and often nonex-

4 © 2001 American Institute of Physics

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10885J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Femtosecond pump–probe spectroscopy

ponential! monotonically decaying background.Various data analysis methods are currently used to

tract the oscillatory components from the data.16 One popularmethod is a linear predictive singular value decomposit~LPSVD! algorithm17,18that uses decaying exponentials tothe monotonically decaying part of the signal and, atsame time, fits the oscillatory part of the signal using damcosine functions. An alternative way of fitting the expemental data is through a maximum entropy method~MEM!algorithm that fits the monotonic part of the signal usiexponentials with a continuous distribution of decay ratThe oscillatory part of the signal that is left after using MEto extract the background can be analyzed using LPSVDFourier transform algorithm. The power spectra generausing both methods are usually in excellent agreement. Hever, as the oscillatory frequency is lowered, both methof analysis begin to generate unsatisfactory results inpresence of a strong monotonic background. In this papeoffer an experimental solution to this problem.

One method of extracting the vibrational signal from tmonotonic background arises from the fact that thesetypes of signal are generated differently. A major componof the monotonic background is the population decay ofcited states, which does not require the spectral bandwavailable within the laser pulse to generate signal. If we csider a second order electric field interaction for each ofpump and probe pulses~which is the lowest order that cacontribute to the four wave mixing signal!, the two fieldinterventions involved in the pump~or probe! interactionwill have to involve identical frequency components frothe laser bandwidth in order to create~or monitor! an elec-tronic and/or vibrational population. On the other handvibrational coherence needs the spectral bandwidth of bthe pump and probe pulses in order to be observed. Tfields of different optical frequencies (v12v25v j ) areneeded within the pump~or probe! spectral bandwidth inorder to create~or detect! a coherence at frequencyv j . Wehave used the different electric field dependence of the polation and coherence terms to develop a technique thatferentiates between these two sources of signal. This tnique involves wavelength selective modulation of tspectral components of the pump and probe laser pulsedetailed presentation of the technique and its ability tosolve the oscillatory part of the signal from the monotonbackground is presented below.

II. MATERIALS AND METHODS

We use a self-mode-locked Ti:sapphire laser~VerdiPumped Mira 900, Coherent Inc.! to generate the pumpprobe laser pulses used in this work. The laser generfemtosecond pulses~45–100 fs! with a center wavelengthbetween 700 and 960 nm. When trying to detect lofrequency modes we use the longest pulses available~80–100 fs!. A 0.2 mm BBO crystal is used to double the IR ligin order to obtain blue pulses, resonant with the Soretsorption band of the heme proteins under study~420–435nm!. In all experiments discussed here, the pump and prlight have the same spectral content. The average poweblue light at the sample is 30 mW, which corresponds


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about 0.4 nJ per pulse pair. A pump–probe pulse pair affeless than 1% of the sample. We use a;3.5 inch spinningcell, rotating at;3500 rpm, to refresh the sample in thilluminated volume. Under these conditions, there isbuild-up of unligated species in the NO bound samples.

Potassium phosphate buffer~pH57.8, 0.1 M! is used todissolve the protein, and 2ml of 1 M dithionite solution isadded to 90ml of buffered sample to obtain the deoxy spcies. An additional 1ml of 1 M NaNO2 solution is added toprepare the NO adduct. The protoporphyrin IX~FePPIX! so-lution was prepared with the addition of 0.1 M2-methylimidazole~which coordinates to Fe! and 1.5% cetyltrimethyl ammonium bromide~CTAB!, to prevent sampleaggregation. The microperoxidase-8~MP8! solution was pre-pared with the addition of 1.5% detergent~CTAB!, to pre-vent sample aggregation. The octaethylporphyrin~OEP! wasdiluted in an aqueous sodium dodecyl sulphate~SDS! deter-gent micellar solution,19 with the addition of 2 M2-methylimidazole~which coordinates to Fe!. The H93G Mbsample was provided by Professor Doug Barrick~John Hop-kins University, Department of Biophysics! and was pre-pared according to standard procedures.20,21 The concentra-tion of protein is chosen so that the sample hasabsorbance of about 0.6 OD at the pump wavelength in amm cell.

The pump–probe setup11 is shown in Fig. 1 and involvesa grating pair to disperse the probe@Fig. 1~a!# or the pump@Fig. 1~b!# beams. The dispersed light is collimated by a leand is reflected back by a mirror. The distances betwthese three optical elements~grating–lens, lens–mirror! areequal to the focal length of the lens. This is a regular pushaping geometry22 in which the grating is used in a doublepass configuration. The returning beam makes a slight awith the incident one, such that it can be picked off on tedge of another mirror after the second reflection offgratings. The chirp is being compensated by a double prsetup for the undispersed beam and for the dispersed bthe position of the lens~L3! is varied.23 An acousto-opticmodulator~AOM! ~Neos Tecnologies! is used to modulatethe pump beam at 1.5 MHz. The pump–probe delay timevaried using a 1mm step translation stage~SM!.

In Fig. 1~a!, we show how the probe beam is dispersby the grating pair and modulated at selected wavelengThe AOM is placed on the other beam, the pump, whichnot dispersed. For the application presented here, achopper~CH10 made by Boston Electronics! ~FC! is placedon the dispersed beam in front of mirror M3 to modulate hof the probe’s spectrum at 900 Hz. Due to the fact thatfinal signal is detected at the working frequency of the fochopper, only the modulated part of the third-order polarition ~generated by the modulated part of the third fieldspectrum! will be detected in the final signal. Thus, thchopped part of the probe spectrum defines the spectraltent of the third field involved in signal generation in thfour wave mixing experiment. After the sample, the probeam is again dispersed by a grating. The modulated pathe probe spectrum is blocked, so that only the unmodulahalf of the probe spectrum is incident on the photodiode@seeFig. 1~a!#. The detected part of the probe spectrum defin


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10886 J. Chem. Phys., Vol. 114, No. 24, 22 June 2001 Rosca et al.

FIG. 1. ~a! Description of the experimental setup usefor the frequency selective probe modulation. Thprobe beam is dispersed using a first grating~G1!, col-limated by a lens~L3!, and reflected back by the mirroM3. The fork chopper~FC! used to modulate parts othe probe spectrum is placed in front of M3. After thsample, the probe beam is dispersed again~using G2!and the unchopped part of it is detected with a phodiode ~P!. The signal is processed by two cascadlock-in amplifiers. The symbols used in the figure arlenses: L1–6, second harmonic generating crysSHG, beam splitter: BS, mirrors: M1–6, dispersiocompensator: DC, acousto-optic modulator: AOM, hawaveplate:l/2, quarter waveplate:l/4, diffraction grat-ings: G1 and G2, fork chopper: FC, stepping motoSM, sample cell: S, polarization analyzer: A, photodode: P.~b! Description of the experimental setup usefor the frequency selective pump modulation. Thpump beam is dispersed using a grating~G!. Two forkchoppers~FC1 and FC2! are used to modulate parts othe pump spectrum. The full spectral content of tprobe beam is detected with a photodiode~P!. The sec-ond lock-in amplifier is triggered by the sum of the twchopper frequencies~FC11FC2!.

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the spectral content of the fourth field. Because of this,detected signal will arise from the distinct spectral conteof the third-order polarization and the final electric field itervention.

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sample cell using a 2 in. lens. The pump beam is spatiablocked and selected against by a polarization analyzer,vided that the pump and probe beams have perpendicpolarization. The output of the photodiode is fed into the filock-in amplifier ~SRS 844! phase locked to the acousto


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optic modulator. The time constant on this first lock-in is sat 100ms so that the signal at 900 Hz will not be averagout. The output of the first lock-in is fed into a second o~SRS 850! that is phase locked to the 900 Hz fork choppThe time constant of this lock-in is set to 300 ms, whichcomparable to the waiting time of 200 ms spent by the stping motor~SM! at each delay point. The autocorrelationrun under the same conditions as the experiment in ordeaccount for the delay in the time response of the lock-in aprovide an accurate pump–probe zero delay. The seclock-in generates the final recorded signal. The motivatfor this setup will be addressed in more detail below.

When we apply the wavelength selective modulattechnique to the pump beam, the pump must be dispe@Fig. 1~b!#. Two fork choppers working at different frequencies (VR5900 Hz andVB5600 Hz) are used to modulatthe red and blue halves of the pump’s spectrum~e.g., see alsothe top right panel of Fig. 8!. The AOM is also placed on thepump beam. After the sample the pump beam is agblocked and the full spectrum of the probe is detected bphotodiode~P!. The output of the photodiode is fed into thfirst lock-in amplifier, with the time constant set at 100ms,phase locked to the AOM at 1.5 MHz. The output of the filock-in is fed into the second one, which is phase lockedthe sum of the two chopper frequencies (VSUM51.5 kHz!.To generate the sum frequency we use a frequency mixemix the two initial frequencies and an electronic filter~model3382, Krohn Hite Corp.! to select the sum from the initiafrequencies and their difference. In order to generate sigthat will be detected by the second lock-in~at VR1VB) thetwo pump field interventions have to be modulated atVR

andVB , respectively. The two fork choppers are placedthe dispersed pump beam’s spectrum such that the specomponents modulated atVR and VB do not overlap. Thetwo pump field interventions will consequently have distinspectral content.

III. WAVELENGTH SELECTIVE MODULATIONOF THE PROBE PULSE

The self-mode-locked Ti:sapphire laser produces pu(t550 fs! that can excite and probe vibrational coherenwithin its spectral bandwidth~;400 cm21). For biomol-ecules, the low-frequency modes~under 100 cm21) are ofspecial importance because, if they are more delocalithey may be involved in energy and information transpoFor example, in heme proteins, the doming mode ofheme is expected24,25 below 100 cm21.

We have previously discussed the methodology oftuned detection to selectively enhance the low- or hifrequency part of the measured signal.11,12,26,27In the detec-tion scheme for detuned detection the probe beamdispersed using a monochromator, and a photomultiptube detects a selected narrow bandwidth within the probspectrum. We have shown that detection close to the cawavelength of the probe pulse spectral distribution setively enhances the low-frequency modes~and the ‘‘zero fre-quency,’’ monotonic background signal!, while detection inthe wings of the spectral distribution enhances only the hfrequency modes.


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For the more commonly employed ‘‘open band’’ detetion, a photodiode is used to detect the full spectrum ofprobe beam and, once again, both the low-frequency pathe sample’s vibrational spectrum and the background sigare enhanced. In Fig. 2 we present examples of data geated using deoxy Mb with both detuned~upper panel! andopen band detection~lower panel!. As seen in the detunedmeasurements of Fig. 2, the high frequencies are seleover the decaying background, whereas in the open bmeasurements the low-frequency oscillations ride on a vstrong background signal. This can be intuitively understoif we think of the monotonically decaying background aszero frequency signal, which is enhanced along with the lofrequency modes. The lower the frequency of the modeare trying to detect, the harder it will be in the data analyto separate it from the background.

One of the novel aspects of the present work involvthe experimental discrimination between the backgroundthe low-frequency modes. In order to explain the techniqwe first need to discuss the key differences between thegin of the background and the oscillatory signals.

After the pump pulse interaction, vibrational coherenis induced in the sample, as manifested in the off-diagoterms of the second order density matrix. The delayed prpulse monitors the subsequent nonequilibrium responsthe medium, which includes vibrational and electronic pop

FIG. 2. FCS signal from deoxy Mb using detuned detection~upper panel!and open band detection~lower panel!. The circles are experimental datpoints and the solid line is the LPSVD fit. The insets show the absorpspectrum of deoxy Mb, the spectrum of the laser pulse~grayed!, and thedetection conditions. Two exponential decay times characterize the mtonic background of the FCS signal.


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lation decay in addition to vibrational coherences. Tpump–probe signals are readily analyzed in the wseparated pulse limit, where the pump and probe pulsestemporally separated.28 The generation of the third-order polarization and its subsequent detection have been studieding density matrix pathways26,27 and the doorway windowpicture.29,30 For the present analysis, it is convenient to epress the polarization in terms of the two frequenpolarizability,31,32a(v,v8), which reflects the nonstationarresponse of the medium induced by the pump pulse. Ifdenote the spectral envelope of the probe pulse electricasEb(v), the third-order polarization can be expressed a

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wheret is the pump–probe delay time. The final open badetected signal is obtained as the overlap of the final prfield with the third-order polarization

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The two-frequency polarizabilitya(v,v8) has the followingtypical form:

a~v,v8!5a0~v!d~v2v8!1(j

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where the summation is over all the vibrational frequencv j present in the system.a0(v)anda j (v) are functions thatdepend on the electron–nuclear coupling and the pumpduced displacement for the modev j as well as other properties that govern the line shape of the system. It is cfrom the above expressions that the oscillatory parts ofpump–probe signal arise from the off-diagonal parts~i.e.,vÞv8) of a(v,v8). These oscillatory signals are hence oserved only when the frequency components of the secprobe field in Eq.~2! are different from those of the first fielin Eq. ~1! that generated the third-order polarization. Moprecisely, the two optical frequencies involved in signal geeration must be separated by the vibrational mode frequev j . It is clear from Eq.~1! that the resulting third-ordepolarization consists of blue- and redshifted spectral futions, proportional toEb(v6v j ), which correspond to coherent anti-Stokes and Stokes Raman spectroscopy~CARSand CSRS resonances!26,27,33@here, we assume thata j (v) isslowly varying with respect toE(v)]. In contrast, the mono-tonically decaying~zero frequency! background signal isproportional to the ‘‘diagonal’’ parts ofa(v,v8), which areobserved only when the third and fourth field interactiohave the same optical frequency~i.e., v5v8).

Now, consider the case where a wavelength selecmodulation is applied to one half of the probe pulse spectron either the red or the blue side of the center~carrier! wave-length. The third order polarization detected by the lock-ingenerated by the modulated half of the spectrum, so thafinal signal in Eq.~2! can be obtained by monitoring thother half of the probe spectrum, as shown in Fig. 1~a!. InFig. 3, we show the case where the red half of the pr


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pulse spectrum is modulated and the Stokes and anti-Stshifted components of the modulated part of the polarizatare also shown. Since we selectively detect only themodulated part of the probe pulse spectrum, the overlapthe CARS~anti-Stokes! shifted polarization and the unmodulated half of the probe pulse spectrum determines the fisignal as described by Eq.~2!. It is clear that the overlap isnonzero only ifv jÞ0. In other words, the background (v j

50) components of the pump–probe signal can be expmentally eliminated using this approach. In order to copletely eliminate the background signal, no modulated prolight can be allowed to reach the photodiode. However, dto the finite resolution in selecting the detected portion oflight, it is sometimes necessary to let some of the modulaprobe light reach the detector. In this way we can be certhat the fields necessary for the detection of very lofrequency modes~spectrally very close to the modulatelight! are present. As a result, the low-frequency modes wsometimes ride on a small background signal.

One possible concern related to this technique is thattemporal shape of a pulse with a non-Gaussian spectraltribution might affect the measured signal. To study this psibility, we performed a series of control experiments in oder to define the conditions under which the ensuing sigcontains the expected oscillatory components withoutdistortions or artifacts. In Fig. 4 we present the signalstained using deoxy myoglobin (Mb21) under four differentconditions. As described earlier, half the spectral contenthe probe beam is chopped with a fork chopper. In the fipanel we present the signal obtained if a 0.2 nm specwindow of the unchopped portion of the probe light is dtected. This window is placed 165 cm21 from the cut madeby the fork chopper in the probe spectrum. The signal tdominates under these conditions is a damped 165 cm21

oscillation, symmetric around the zero delay time. Movi

FIG. 3. Graphic scheme depicting the spectral structure of the probeinterventions that generate the observed signal. The solid line is the mlated half of the probe spectrum~third field!. The dashed line represents thdetected third-order polarization, shifted from the carrier frequency byenergy of the observed vibrational mode. The circles represent the dethalf of probe pulse spectrum~fourth field!. The measured signal is generatein the overlapping spectral region between the modulated part of the thorder polarization and the fourth field.


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FIG. 4. Sequence of frequency selective probe modulation experiments on deoxy Mb under different detection conditions. Left panels present theratedFCS signals, whereas the right panels depict the detection conditions of the experiment. The grayed area is chopped by the fork chopper. In theseimentswe used 55 fs pump/probe pulses with a 434 nm carrier wavelength.

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the detection window to a new position, 100 cm21 from thecut made by the fork chopper, changes the frequency ofobserved oscillation to 100 cm21, as seen in the seconpanel. Increasing the width of the detected spectral windto 1 nm damps the observed oscillation, as seen in the tpanel. The frequency of the damped symmetric oscillatiodetermined by the detuning between the edge of thechopper and the detection window, while the dampingdetermined by the width of the detection window. As seenthe fourth panel, the symmetric detuning oscillation cancompletely damped if the entire unmodulated half of tprobe spectrum is detected. The signal at negative time


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lays now shows just a monotonic decay, whereas at postime delays we observe the oscillatory signal characteristideoxy myoglobin. Thus, by chopping half of the probe sptrum and detecting the entire bandwidth of the unchoppspectral regions one can detect the coherent oscillationthe material and experimentally discriminate againstmonotonically decaying background. For all the data psented in this paper we have run symmetric scans and vfied that no oscillatory signals were detected for negattime delays. This is the best method to ensure that thetected signals are a measure of the sample response, utorted by the detection technique. Since a traditional autoc


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relation measurement sums photons containing fields ofsame wavelength, it yields no signal under these detecconditions.

IV. WAVELENGTH SELECTIVE MODULATIONOF THE PUMP PULSE

In a density matrix description of the signal the diagonelements represent the population terms, whereas thediagonal elements represent the coherence. In the prevsection, we discussed how the different frequency comnents of the probe pulse can be modulated and detectediscriminate against the population decay contribution tosignal. In the present section, we discuss how a similar chping scheme applied to the pump pulse can be used to scoherent oscillations based upon the optical frequenciethe fields needed for their generation. In this way, the signarising from two pump fields of identical frequencies~ex-pressed in the diagonal terms of the density matrix! can besuppressed.

As discussed earlier, two fork choppers are usedmodulate the red and blue sides of the pump pulse specat two different frequencies,VR andVB . This is schemati-cally depicted in the top panel of Fig. 5, where the shadregions correspond to the modulated blue and red side

FIG. 5. ~Upper panel! Description of the spectral modulation used on tpump beam. The red and blue parts of the spectrum are modulated aquenciesVR and VB. dv is the width of the unmodulated spectral regiobetween the two choppers.~Lower panel! Energy level diagrams describinthe creation of a vibrational coherence in the excited electronic state anpopulation of the first vibrational level of the excited electronic state. Telectronic states are labeled e~excited! and g~ground!, the vibrational statesare 0 and 1. The unmodulated gap width (dv) is compared to the energy othe excited state vibrational mode (v0). A similar picture can be easilyconstructed for ground state coherences.


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the pump pulse spectrum. The transmitted probe light iscident on a photodiode and, as described earlier, detectethe two cascaded lock-in amplifiers phase locked toAOM ~1.5 MHz! and to the sum frequency of the two forchoppers@see Fig. 1~b!#. In order to understand the mechnism behind the pump pulse modulation experiments,take a density matrix approach. When interacting with a telectronic level system, a femtosecond pump pulse canduce vibrational coherence in both the ground and thecited electronic states. The nonstationary state induced bypump pulse can be described using the second-order pebative expression for the doorway density matrix for tground and excited electronic states.34,35

In the Appendix, we consider a two-level system withsingle linearly coupled mode~frequencyv0), and present asimple analysis of the pump pulse modulation experimentthe excited state doorway density matrixre @Eq. ~A1!#. Sinceonly the signal modulated at the sum frequency (VR1VB) isdetected, only those terms of the second order density mawith one field component from the red and blue wings of tpump pulse spectrum@ER(t) and EB(t) in Eqs. ~A3a! and~A3b!# are relevant. It is shown in the Appendix that thelements of the density matrix (re)vv8 induced by the fieldcombinationsEREB andEBER will be suppressed in the signal detected at (VR1VB) if the condition u(v2v8)uv0

,dv is satisfied@see Fig. 5 and Eq.~A5!#. Here,dv is thefrequency gap determined by the distance between the blof the fork choppers that modulate the red and blue specparts of the pump pulse. In particular, the zero-frequen~background! signal, which corresponds to population decawill be suppressed as long asdv.0. This suggests how wecan use the capability of pump-wavelength modulationselectively eliminate the contributions of the backgroufrom the measured probe response. In the Appendix, wediscuss how electronic and vibrational dephasing procecan affect the resolution of the wavelength selective molation. While electronic dephasing does not affect the shaness of the frequency selection, vibrational dephasingaffect the resolution of the wavelength selective modulatiThus, the conditionv0,dv for suppression of thev0 fun-damental will not be sharp, but will exhibit a smooth behaior asdv is varied across the finite bandwidth of the vibrtional mode.

If the blue and red modulated spectral regions~shaded inthe upper panel of Fig. 5! are separated by an unmodulatregion of widthdv ~determined by the distance between tblades of the two fork choppers!, the two fields that generatsignal atVR1VB will be energetically separated by at leadv. Two energy level diagrams are presented in the lowpanel of Fig. 5, associated with two possible pump inducprocesses: the creation of excited state coherence and plation. In this figure, we compare the energy of a vibrationmode (v0) to the spectral gap between the modulatedgions (dv). For dv50, when the fork choppers’ vaneleave no unmodulated spectral region between them, anybrational mode accessible within the pump pulse bandwwill be detected, since the necessary field frequenciesavailable from each modulated spectral region (VR andVB).As soon as the choppers are separated to leave an unm

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lated region between them~dv!, the availability of modu-lated fields is restricted. One can see that if the unmodulagap is less than the frequency of the vibrational mode (dv,v0), electric fields modulated atVR andVB are still avail-able to generate the coherence associated with that mHowever, if the unmodulated gap becomes larger thanmode frequency (dv.v0), the modulated fields can nlonger generate a coherence at this frequency. Processevolving pump induced population dynamics~needing twofields of the same color! will not be detected by this methoas long as the two choppers modulate distinct spectralgions and the detection is at the sum frequencyVR1VB .

Potentially, this approach can distinguish the coherresponse of an overdamped low-frequency mode frompopulation dynamics of electronic relaxation and coolingcan also select against signals generated through proclike sequential dissociation of a ligand, which may involthe initial preparation of populations and the subsequdynamics.9,36

V. APPLICATIONS TO HEME PROTEINS

A. Probe beam modulation

We have applied wavelength selective modulation teniques to heme protein samples in an attempt to more clereveal low-frequency modes in the absence of backgrouIn Fig. 6 we compare traditional open band measurementdeoxy Mb to the signals generated using wavelength setive modulation of the probe pulse@Fig. 1~a!#. As predicted,the signal generated using the selective modulation ofprobe pulse contains the oscillations in the absence of bground. On the other hand, the traditional open band sigcontains a strong, monotonically decaying backgroundperimposed with the oscillations. The absence of the motonic background signal in the data generated by wavelen

FIG. 6. Comparison between two experiments on deoxy Mb, one usinregular pump–probe setup with open band detection and the other usinfrequency selective probe modulation technique. The background signalmost completely removed and the oscillatory part of the signal~enlarged40 times in the inset! is clearly exposed when using the second techniqWe used 75 fs pump/probe pulses with a 442 nm center wavelength. Infrequency selective probe modulation experiment the red part of the pbeam is chopped and the blue part is detected with a photodiode.


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selective modulation of the probe beam facilitates the detion of low-frequency oscillatory signals with significantlimproved signal to noise.

An alternative way to enhance the detection of lofrequency modes vs background can in principle be achieby employing a detuned detection scheme11,12 ~see Fig. 2!.To optimize this scheme requires the availability of laspulses with bandwidth on the order of the mode frequen(v0) as well as high monochromator resolutio(!v0). However, under these conditions only a relativenarrow portion of the low-frequency vibrational spectrucan be enhanced. In contrast, the selective probe modulatechnique allows background free detection of a mubroader portion of the low frequency spectrum. Moreovthe detuned detection scheme uses only a small fractiothe probe laser light. Thus, as the resolution is increasedbetter enhancement of the oscillatory signal vs backgrounsignificantly smaller amount of light is available for detetion. As a result, the signal to noise in the detuned detecscheme is reduced as the signal to background ratio iscreased. In contrast, the wavelength selective modulatechnique detects half of the probe spectral content, whleads to an improvement in signal to background withoudecrease in the signal to noise ratio. In a detuned measment, it is straightforward to find the dependence of tsignal-to-noise ratio (SN) as a function of the oscillatory signal to background ratio (SB). Assuming statistical noise, thidependence is given by the following expression:

SN;expS 3V02ln 2

42

ln2 SB

8V02 ln 2

D ~4!

whereV05v0 /s and s is the full width at half maximum~FWHM! of the spectral content of the pulse intens@E2(v)#, and v0 is the energy of the specific mode to bdetected. This expression describes explicitly how the sigto noise decreases as the oscillatory signal to backgroratio is enhanced by detuning the detection window awfrom the carrier wavelength (vc). Plots of Eq.~4! are givenin Fig. 7 and demonstrate how this decrease becomes macute for lower frequency modes (v0,s, V0,1). As ex-pected, when the mode frequency begins to exceed the bwidth of the laser pulse (v0.s, V0.1), the signal alsodiminishes.

Since wavelength selective modulation does not suffedecrease in signal to noise as the signal to backgrounincreased, it is the preferred method for the detection of lofrequency modes. For example, when the time delay srange is increased and the pulses are broadened to;100 fs,wavelength selective modulation of the probe pulse allous to observe a mode near 20 cm21 in heme proteins thacannot be clearly detected using traditional measuremeOpen band measurements enhance low-frequency malong with the monotonic background, such that the lofrequency oscillations cannot be reliably extracted by danalysis from the strong background. The background foscillatory signal generated using the wavelength selecmodulation of the probe beam enables reliable detectionlow-frequency modes in FCS. In Figs. 8~A! and 8~B! we

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present the oscillatory part of the signals obtained by aping this technique to a variety of heme protein samples. Tbackground signal is decreased by at least an order of mnitude. The small amount of background present in thedata, due to the finite resolution in selecting the detecportion of the light, has been removed using the LPSValgorithm. The most notable feature37 of the signal generatedfrom ferrous myoglobin and ferrous protoporphyrin IX is thpresence of a long lived mode near 20 cm21.

Results obtained on other samples containing Fe prporphyrin IX as the active site demonstrate that the 20 cm21

mode is sample dependent. A long-lived 23 cm21 mode~seeTable I! was observed in ferrous H93G myoglobin, a myglobin mutant for which histidine 93 is replaced by a glyci~which does not coordinate to the heme iron! and2-methylimidazole is ligated to the heme. In ferrous hemglobin, the mode is detected at 26 cm21 @see Fig. 8~A! andTable I#.

We have also studied several heme proteins that docontain protoporphyrin IX in the active site. For exampferrous cytochrome c, ferrous microperoxidase and ferroctaethylporphyrin do not show a low-frequency mode n20 cm21 @see Fig. 8~B! and Table I#. These results suggest38

that the;20 cm21 mode is protoporphyrin IX specific. Inthe NO bound species of ferrous myoglobin and hemoglothe 20–27 cm21 mode is observed with a much shorter liftime than in the unligated samples@see Fig. 8~A!#. This dif-ference in lifetime is significant, because for the ligatsamples the pump pulse triggers a photodissociation r

FIG. 7. Dependence of the signal-to-noise ratio (SN) for the detuned detec-tion scheme, as a function of the oscillatory signal to background ratio (SB)for five values ofV0 . Here,V05v0 /s /s, s is the bandwidth of the laselight, andv0 is the energy of the specific mode to be detected. The ploexpression@Eq. ~4!# describes explicitly how the signal to noise decreasesthe oscillatory signal-to-background ratio is enhanced by detuning thetection window away from the carrier wavelength (vc). The expressions forSN andSB used to derive Eq.~4! are:SN;(E•P(3)(V0))0.5;exp(2x2 ln 2)•exp(2(x2V0)

2ln 2) andSB;P(3)(V0)/E;exp(22 ln 2(V0222V0x)), where

x5(v2vc)/s is the detuning from the carrier wavelength normalizedthe bandwidth of the laser,E is the intensity of the electric field, andP(3)(V0) is the third-order polarization for the modev0 . The units arearbitrary since frequency independent scaling factors have not beencluded.


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tion. However, in both ligated and unligated samples,probe pulse interrogates the unligated species.

Special precautions were taken to check for the reliaity of the data showing the;20 cm21 oscillations. Samplesdisplaying a;20 cm21 oscillation, as well as samples thado not display it, were run ‘‘back to back’’~the sample cellswere switched and run under identical experimental contions! to ensure that the observed oscillatory signal is samdependent. Long symmetric scans were measured, andnegative time delay~probe before the pump! signal was ana-lyzed to check for the presence of symmetric damped oslations, similar to those presented in Fig. 4. No such signwere found. Moreover, the 26–27 cm21 mode in NO boundhemoglobin was detected using both the wavelength setive probe modulation technique and the standard open bdetection experiment. The monotonic part of the open bgenerated signal for HbNO is first fit using the MEM algrithm and the residual oscillatory signal is fit using thLPSVD algorithm, which distinctly reveals the 26–27 cm21

component in the power spectrum@dashed line in Fig. 8~A!#.This demonstrates definitively that the modes in the 20–cm21 region are properties of the studied samples and inpendent of the detection technique.

B. Pump beam modulation

In the case of wavelength selective probe beam modtion, only processes that need probe fields of different coto be detected are present in the final signal. The resultwavelength selective pump beam modulation are very silar to those obtained from probe beam modulation. Whphotostable samples are measured, the signal arisingpopulation terms~i.e., the absorption of a photon with twidentical pump field frequencies! generates a monotonicalldecaying background that is not passed through the seclock-in amplifier. In contrast, the vibrational coherences netwo different color pump fields to be generated and theymodulated atVR andVB so that the coherence signal passthrough the lock-in tuned toVR1VB .

It has previously been suggested39 that samples undergoing a ~nonadiabatic! photolysis reaction can lead to produstate vibrational coherences that are triggered byelectron–nuclear coupling forces that develop duringphotochemical reaction. In contrast to field driven cohences~which are created by pump fields of different colo!,the reaction driven coherences can, in principle, arise frpump induced electronic populations, which evolve rapidto the final electronic product state. Upon the absorptionthe pump photon, the system is projected onto an excstate potential and evolves, as the ligand dissociation taplace, towards the product state.9 Such reaction driven coherences, if they are generated by pump fields of the scolor, are not expected to appear in the pump beam modtion experiments. This is because the signals observed inwavelength selective pump beam modulation experimemust be generated from a pump field intervention from eof the two different spectral regions modulated atVR andVB by the fork choppers. Only this ‘‘off-diagonal’’ component of the pump induced signal will be modulated at t

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FIG. 8. ~A! Sequence of experimentusing frequency selective probe modulation on Mb and Hb. The left panelspresent the oscillatory part of the generated signal and the right panels shothe corresponding power spectra. Th20–27 cm21 oscillation extractedfrom the LPSVD fit is shown dis-placed from the data for the samplethat display this mode. A secondpower spectrum ~dashed line! forHbNO corresponds to the signal measured using open band detection. Thvibrational mode near 27 cm21 is acommon feature observed using bodetection conditions.~The mode at 42cm21 in the lower right panel isanomalously low in this data set and inormally found near 47 cm21.) ~B! Asimilar sequence of experiments oheme model compounds and cytochrome c.

eNOu

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eas

sum of the fork chopper frequencies (VR1VB) and be de-tected by the second lock-in amplifier.

The oscillatory signals obtained by applying the wavlength selective pump beam modulation technique tobound myoglobin are presented in Fig. 9. The small resid


-

al

monotonic background~that has been subtracted from thpresented data! is probably related to the vibrational damping which affects the resolution of the technique~see thediscussion in the Appendix!. For the data presented in thfirst panel, the vanes of the two choppers were placed


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close to one another as possible, without being overlap~see the inset in the top right panel of Fig. 9!. If there wasoverlap, the spectral region modulated by both choppwould contain optical frequencies modulated atVR1VB ,leading to the detection of background signal, as in the sdard experiments. If a small gap is left between the chopvanes to optimally suppress the detection of backgroundnal, this will also select against the detection of the;20cm21 mode. Under typical pump beam modulation contions, the full vibrational spectrum~above;20 cm21) al-lowed by our temporal resolution is observed. The obsertion of oscillatory signals when using wavelength selectpump beam modulation implies that two field interventiohaving different colors trigger the detected vibrational cohences. The need for different optical frequencies in geneing the vibrational coherences in MbNO is further exempfied in the lower panels of Fig. 9. If the vanes of the two fochoppers are moved to leave between them a spectral re~dv! that is not modulated~see the insets in the lower righpanels of Fig. 9!, vibrational modes of frequency less thadv can no longer be detected atVR1VB . This is due to thefact that the two pump field frequencies, each from a specregion modulated by one of the fork choppers, cannotcloser to one another than the width of the unmodulaspectral region. It can be seen that, as the two vanesmoved further apart, more and more low frequency modisappear from the FCS signal~see the power spectra!. Thewavelength selective pump beam modulation techniquetherefore also be a useful tool to filter and isolate the moof interest. It is especially noteworthy that the Fe–His mowhich is Raman inactive in MbNO, also disappears asunmodulated gap between the fork choppers exceedscm21.

VI. DISCUSSION

We have developed novel wavelength selective modtion techniques that can be used to distinguish between

TABLE I. The frequency and lifetime for the 20 cm21 mode in hemesystems. The frequency~v in cm21), period of oscillation~T in ps!, lifetimeg21 in ps! and line width~y in cm21) of the;20 cm21 mode are displayedfor the studied heme proteins. The estimated uncertainty for the frequen;2 cm21. The associated lifetime uncertainty is about 25%.

Sample v (cm21) T ~ps! g (cm21) g21 ~ps!

MyoglobinDeoxy myoglobin 20 1.66 4.2 2.5NO bound myoglobin 20 1.66 13.2 0.8Deoxy H93G myoglobin 23 1.45 5.9 1.8Met myoglobin ¯

HemoglobinDeoxy hemoglobin 26 1.28 6.6 1.6NO bound hemoglobin 26 1.28 13.5 0.8

Model compoundsFerrous FePPIX12Me2Im 20 1.66 5.3 2Ferrous OEP12ME–Im ¯

CytochromesFerrous cytochrome C ¯

Ferrous octapeptide ¯


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coherence and population terms in the signals generatepump–probe femtosecond spectroscopy. Within a denmatrix description, these techniques are sensitive to thediagonal terms~coherences!, and discriminate against the dagonal terms~populations!, which contribute to the signalOne motivation for developing these techniques is to obtcoherent signals, free from the population induced baground decay that is often present in resonant samples.work described here presents an experimental solution toproblem of separating the oscillatory part of the signal frothe monotonically decaying background, a task that was pviously addressed in the data analysis.

Using these new experimental methods we are ableobserve modes having frequencies as low as 20 cm21 insome heme proteins. We also observed this mode in atoporphyrin IX model compound, where a detergent micerather than the protein backbone, surrounds the heme.porphyrin macrocycle is a ringed structure containing 20 cbon atoms bound to an inner core of 4 nitrogens. Offouter macrocycle, there are eight sites where substitugroups can be bound (R1– R8). Iron protoporphyrin IX~FePPIX! is an asymmetric porphyrin with two vinyl groupbound at R2 and R4, and two propionic acid groups at R6

and R7. Horse heart myoglobin contains a FePPIX actisite, which is covalently bound to the protein solely byhistidine~His93! coordinated to the iron. The presence of t20 cm21 mode in the FePPIX model compound, which hno amino acid backbone surrounding it, indicates thatobserved mode is localized at the heme. The mode neacm21 appears only in samples containing iron protoporphrin IX and its frequency is sample dependent. It is notewthy that most normal mode calculations24,25 do not predictmodes of such a low-frequency to be localized at the heThis is because these calculations are made consideringa bare porphyrin macrocycle~porphine!, without the sub-stituent groups present in FePPIX. Since the normal mcalculations24 predict the doming mode to be around 5cm21 ~the doming motion involves primarily the porphyricore!, it is conceivable that the addition of the substituegroups could induce the presence of even lower frequemodes in FePPIX. For example, calculations by Findset al.40 suggest that coherences between 20–35 cm21 can beassigned to torsions of the heme vinyl groups. Thus,possible ~but speculative! explanation of the stronglycoupled, heme localized, mode near 20 cm21 is that it cor-responds to vinyl isomerization following excitation in thSoret band. Such isomerization can be viewed as analogto the behavior of conjugated polyenes and would helpexplain the absence of the;20 cm21 modes in samples thalack the vinyl groups~see Table I!.

A broad peak near 25 cm21, associated with the collective oscillations of the polypeptide chain, has also beenserved in recent high-resolution synchrotron basexperiments41 that are sensitive only to the vibrational spetrum of the 57Fe at the active site of Mb. Similar feature~;20 cm21) were observed in inelastic neutron scatterifrom hydrated protein films42,43 and in site-selected fluorescence spectra of Zn-substituted Mb.44 These experimentsalong with protein density of state calculations,45 indicate

is


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FIG. 9. Sequence of experiments uing frequency selective pump modulation on MbNO. The left panels showthe generated FCS signals~in circles!and their LPSVD fit ~solid line!,whereas the right panels show the coresponding power spectra. The insedepict the spectral regions of the pumpulse modulated by the two fork choppers.

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that the feature near 20 cm21 in the neutron scattering experiments involves a more global protein motion. SinceFCS experiments on the FePPIX model compound also donstrate that a local heme mode exists near 20 cm21, wesuggest that the active site~heme! and the protein are potentially coupled energetically through the near degeneracythese modes. Recent protein photoacoustic experimentsonstrate that the dominant displacement or strain alongreaction coordinate develops on a 2 pstime scale.13 Both theoscillatory period~1.6 ps! and damping constant~1–1.5 ps!of the observed;20 cm21 mode are in excellent agreemewith such time scales. The observed46–51 time scales ofmotion over all available length scales~from changes in thevicinity of the heme site to the global, protein-solvemotions! are approximately the same, suggestingpresence of highly correlated dynamics in myoglobin~seealso Ref. 41!.

It is also possible that the rapid transfer of the helocalized excitation energy~after the pump pulse interaction!


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to the protein material explains the much shorter lifetimethe;20 cm21 mode in the ligated samples~MbNO, HbNO!.Upon dissociation of the diatomic ligand, the initial hemlocalized forces rapidly redistribute the vibrational energyacting upon the surrounding protein. The nonconservawork resulting in the MbNo→Mb protein structural rear-rangements, done by the reaction induced forces, wouldexpected to increase the damping constant of the;20 cm21

mode in the ligated samples. Structural relaxation of the ptein can also be inferred from the changes of the Fe–frequency following the photodissociation reaction.12,52

Since the pump pulse interaction with the photostasamples~i.e., equilibrium deoxy Mb! does not lead to largenonequilibrium displacements or major protein structuralarrangements, the energy of the;20 cm21 oscillations inthese samples is dissipated at a slower rate, yieldinglonger lifetimes. Finally, it should be noted that a resoncoupling mechanism between the heme and the proteinterial ~involving the ;20 cm21 mode! is conceivably in-


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volved in the dynamics of hemoglobin cooperativity, whilinks the oxygen affinity of one heme active site to the bouor unbound state of other active sites within the protein.

In addition to the 20 cm21 mode, one also observesFig. 8 the existence of a strong;40 cm–1 mode. The pres-ence of this mode in deoxy myoglobin~and in most of theheme proteins studied to date! implies that it is unlikely thatthis mode arises from coherent population transfer betwthe ligand bound and unbound states as recently suggefor cytochrome oxidase.36 This is because the deoxy samplare not involved in ligand exchange reactions. It is possib53

that the appearance of the;40 cm21 mode in deoxy Mb is acoincidence and arises from a completely different mecnism than the;40 cm21 mode associated with ligand dissciation in Mb and cytochrome oxidase. However, the fthat the;40 cm21 mode is observed in the wavelength slective probe beam modulation experiments, which discrinates against population induced signals, speaks stroagainst the proposed coherent population transfer model36 asa source of the;40 cm21 oscillation, at least in MbNO.

The observation of oscillatory signals in the pump bewavelength selective modulation experiments performedNO bound myoglobin yields further information regardinthe photodissociation mechanism. Since the technique issitive only to processes triggered by the intervention of tpump fields of different color, the signals arising from trasient population dynamics are almost completely eliminatIn the nonadiabatic dissociation model,39 the vibrational co-herence is triggered by photon absorption~two fields of thesame color! and subsequent rapid electronic surface crossassociated with the ligand dissociation induced iron spstate change~from S50 to S52). In this model, the coherent signal is expected to diminish when using the wavelenselective pump modulation, since the population termsdiscriminated against. This expectation is contradicted byexperimental results presented in Fig. 9, which demonstthat the vibrational coherence observed in the reacsamples is generated directly from the optical frequencavailable in the femtosecond pump pulse. This is a remaable observation, particularly for the 220 cm21 Fe–Hismode, since it is not observed in the resonance Raman strum of the reactant~MbNO! and is not expected to bcoupled to the resonant optical transition. The dissociasurface accessed by the pump pulse fields evidently suppthe 220 cm21 oscillation, providing strong evidence thaligand dissociation proceeds directly upon photoexcitatio

A related observation is the previously reported phmeasurement of the 220 cm21 Fe–His stretching mode inMbNO.11 Its phase is measured to be either 0 orp as thecarrier frequency of the laser pulse is tuned across the puct state~deoxy Mb! absorption band. These values of tabsolute phase imply that the coherence of the Fe–His mis generated nearly instantaneously, without a significantlay associated with the reaction time, which would shift tphase of the 220 cm21 mode away from 0 orp. For ex-ample, if we assume that the reaction time is;30 fs~half theperiod of the Fe–NO mode!, a comparison to the period othe Fe-His mode~150 fs! yields a nonzero initial phase (F>72°). Nevertheless, since the Fe–His mode is Raman


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active in the NO bound sample, the observed signal musassociated with the deoxy myoglobin product state.

One possible explanation for these observations isan adiabatic~strong coupling! model provides a better description of the dissociation process. In such a model the ispin states of the reactant~MbNO! and the product~deoxyMb! are not pure (S50 andS52, respectively!, but ratherhave a certain degree of admixture associated with thallowing for direct electric dipole coupling between the ractant and product states. In the nonadiabatic model,generally assumed39 that the electric dipole coupling between the reactant~MbNO, S50) and product~Mb, S52)states is forbidden based on spin selection rules. This methat only a chemical reaction~i.e., rapid ligand dissociationafter the fields have ceased to evolve! can access the finaproduct state and lead to coherence in the coupled degrefreedom.

On the other hand, there are experimental results sgesting that iron spin admixtures do exist in deoxy Mb adeoxy Hb. Measurements of the magnetic momentsphosphate-free and phosphate-bound deoxy Hb generdifferent results,54 and it has been suggested that molecuvibrations can affect the spin state of the iron atom.55 Thecalculated56–58energy gap between the ground spin state aother higher lying electronic spin states is at most a fhundred cm21, which is well within the range of thermallyexcited nuclear vibrations. The existence of direct elecdipole coupling between the reactant and product stateslows the pump pulse to directly generate product state coences, as the photodissociation reaction takes place.presence of such coupling is consistent with the observathat multiple optical frequencies within the pump pulse aneeded for the generation of coherences in MbNO. It is aconsistent with the fact that the phase of the~Raman inac-tive! Fe–His mode is found to be 0 orp in the MbNOsample. Such observations provide strong support foradiabatic strong coupling model in MbNO.

The presence of spin state admixtures could impactcharacteristics of NO recombination, following the photodsociation reaction. The large amplitude (I g;95%) and fastrate (tg;10–100 ps)59,60of the NO geminate recombinatioto deoxy Mb differ considerably from the correspondincharacteristics of CO (I g;5% with tg;100 ns)61 and O2

(I g;58% with tg;25 ns).62 The geminate rate is likely tobe affected by the iron spin changes associated withligand recombination. Thus, a strong iron spin admixtureMbNO could help to explain the increased NO geminarates and the dramatic differences between the recombinacharacteristics of NO, CO, and O2.

ACKNOWLEDGMENTS

This work was supported by NSF MCB9904516 aNIH DK35090.

APPENDIX: ANALYSIS OF PUMP WAVELENGTHSELECTIVE MODULATION EXPERIMENTS

We consider a linearly coupled two electronic level sytem, initially in the ground electronic state, with a vibration


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density matrixrT . The pulse-induced nuclear density matrin the excited state is given by~to second order in the lasefields!

re5m2

\2E2`

`

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`

dt9E~ t8!E~ t9!

3$eiV00(t82t9)eiH et8/\e2 iH gt8/\rTeiH gt9/\e2 iH et9/\%,

~A1!

where Hg and He are, respectively, the adiabatic BornOppenheimer Hamiltonians for the ground and excited etronic states,m is the electronic dipole moment, andV00 isthe 0–0 transition energy. In order to calculate the denmatrix relevant for wavelength selective modulation of tpump pulse, we split the electric field into components onred and blue sides of the carrier frequency

E~ t !5cos~VRt !ER~ t !1cos~VBt !EB~ t !, ~A2!

where

ER~ t !51

pE0

vc2dv/2

dvG~v2vc!cos~vt !, ~A3a!

EB~ t !51

pEvc1dv/2

`

dvG~v2vc!cos~vt !. ~A3b!

In the above expressions,G(v2vc) is the Fourier transformof the pump pulse spectral envelope, which is taken aGaussian centered at the carrier frequencyvc . dv is thespacing between the blades of the choppers, which indedently modulate the red and blue spectral regions ofpulse at frequenciesVR andVB , respectively (VR andVB

are much smaller than the optical frequency!.When Eqs.~A2!–~A3b! are used in Eq.~A1!, four terms

result, proportional to the productsEREB , EBER , ERER ,andEBEB . Of these four terms, wavelength selective modlation selects only the first two terms, which are modulatedthe sum frequencyVR1VB . Considering only these termswe evaluate the elements of the excited state density m( re

RB)vv85^veureRBuve8&, where uve& is the vth vibrational

eigenstate of the excited electronic state. Assuming thatsystem is initially in the ground vibrational level of the eletronic stateug&, i.e., rT5u0g&^0gu, we get

~ reRB!vv85

m2Fv,v8

p2\2 E0

vc2dv/2

dvEvc1dv/2

`

dv8G~v2vc!

3G~v82vc!E2`

`

dt8E2`

`

dt9 ei (V001v8v0)s

3ei (v2v8)v0t8@cos~vt8!cos~v8t9!

1cos~vt9!cos~v8t8!#, ~A4!

whereFv,v85^veu0g&^0guve8& ands5t82t9. Note thatFv,vis simply the Franck–Condon overlap factor betweenground and excited vibrational states. Performing the ingrals above and ignoring small contributions from nonrenant terms~rotating wave approximation!, we get


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e--

~ reRB!vv85

m2Fv,v8

2p\2 E0

vc2dv/2

dvEvc1dv/2

`

dv8G~v2vc!

3G~v82vc!@Fv~v!d~v2v82kv0!

1Fv8~v!d~v2v81kv0!#, ~A5!

wherek5v2v8 and

Fv~v!5E2`

`

ds ei (v2V002vv0)s52pd~v2V002vv0!

~A6!

is defined as a spectral line shape function. To obtain~A5!, we have used the relationFv8(v2kv0)5Fv(v).Since v,v8 ~which follows from the integration limits,with dv.0), the first term in the square brackets in E~A5! contributes forv,v8 and the second term contributewhenv.v8. It is also verified that (re

RB)vv85( reRB)v8v

* .

It is clear from Eq. ~A5! that (reRB)vv850 if dv

.u(v2v8)uv0 , since the two frequency integrals in E~A5! are nonoverlapping. This accounts for the sharp fquency selection of vibrational modes asdv is varied. Inparticular, we see that the zero frequency diagonal eleme( re

RB)vv50, provideddv.0. This in turn accounts for theabsence of population terms in the signal channel detecteVR1VB .

The above expressions assumed that there were notronic dephasing mechanisms. One way to introduce etronic dephasing is to makeV00 complex. This wouldamount to introducing a factor exp(2Gcusu) in Eq. ~A1!. Gc

could be identified with dephasing of the electronic cohence between the ground and excited electronic states33,62

Since V00 appears in the above expressions only throuFv(v) in Eq. ~A6!, it is clear that the sharp frequency seletion and the suppression of the population terms would sbe feasible using a wavelength selective modulationlowed by detection atVR1VB .

In contrast to electronic dephasing processes, vibratiodephasing can affect the resolution of the wavelength setive modulation. To see this, we introduce vibrational daming through the factor exp(2gvutu/2) on the bra and ket sideinteractions in Eq.~A1!. This corresponds to the transformtion He→He6 igv/2. In this case, we immediately find

~ reRB!vv85

m2Fv,v8

4p2\2 E0

vc2dv/2

dvEvc1dv/2

`

dv8G~v2vc!

3G~v82vc!@Fv~v!Fv8~v8!

1Fv8~v!Fv~v8!#, ~A7!

where the line shape function is now a Lorentzian rather ta delta function

Fv~v!5E2`

`

ds e2gvusu/2ei (v2V002vv0)s. ~A8!

It is clear from Eq.~A7! that even when the conditiondv.u(v2v8)uv0 is satisfied, (re

RB)vv8 does not necessarilyvanish. This is due to the spectrally broad line shape futions that take the place of the delta functions in Eq.~A5!. It


a

tin

. A

er

V

hy

.

nica

B

o-

, J

ec

s.

s.

.

m

m.

.

oesnth as aA

ther

14–d a

hyl

on,

ett.

en,

m-

s.

A.

.

u,

h-


is easily verified that whengv50, the line shape reduces todelta function as in Eq.~A6!, and if we use the identityd(a2x)d(b2x)5d(a2x)d(a2b), (re

RB)vv8 reduces tothe result in Eq.~A5!.

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