ultrafast interferometric pump–probe correlation measurements in systems with broadened bands or...

Weida et al. Vol. 17, No. 8 /August 2000 /J. Opt. Soc. Am. B 1443

Ultrafast interferometric pump–probe correlationmeasurements in systems

with broadened bands or continua

Miles J. Weida, Susumu Ogawa, Hisashi Nagano, and Hrvoje Petek

Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama 350-0395, Japan

Received November 23, 1999; revised manuscript received April 12, 2000

Ultrafast (femtosecond) interferometric pump–probe techniques can be used to measure rates of populationand quantum phase decay in complicated media such as liquids and solids. However, the levels probed insuch systems are often inhomogeneously broadened or are part of a continuum of states. The use of broad-band ultrafast lasers thus results in multiple levels being excited and detected. The inherent averaging thatis due to this effect can alter the measured coherent response, thus affecting the information that can be re-trieved on the phase decay. The importance of these effects is considered for the representative case of two-photon photoemission from metals. The effects of (i) continuum excitation; (ii) excitation from the Fermi level,i.e., a spectral step function; (iii) excitation from broadened levels with a finite width; and (iv) photoelectronenergy analyzer resolution are determined. © 2000 Optical Society of America [S0740-3224(00)00408-2]

OCIS codes: 320.7100, 320.7150.

1. INTRODUCTIONAs a method of probing dynamics in complicated media,ultrafast (,100 fs) laser spectroscopy is becoming a stan-dard tool. Ultrafast lasers are often used in a pump–probe configuration, during which one pulse excites thesystem to a level of interest and a second, time-delayedpulse probes the ensuing dynamics. A particularly pow-erful extension of the pump–probe method is to scan thedelay between two identical or phase-related pulses inter-ferometrically. It is then possible to follow not only inco-herent processes such as population decay but also the de-cay of phase coherences between the initial and theexcited levels. With this goal in mind, interferometricmethods have been applied to studies of the quantumphase decoherence of single-particle and collective excita-tions in metals by two-photon photoemission1,2 (2PP) andsurface second-harmonic generation (SSHG) techniques.3

Ultrafast methods are often chosen for studying com-plicated media because the couplings in such systems canlead to bands of levels or inhomogeneously broadenedstates that mask fundamental properties such as excited-state lifetimes when probed by traditional frequency-resolved spectroscopies. However, the intrinsic broad-band nature of ultrafast excitation also makes it possibleto excite a broad range of states in such systems. On onehand, the coherent excitation of sparse multiple statescan lead to resolvable quantum beats that shed new lighton systems such as image-potential states on metals4 orsplittings in atomic energy levels.5 On the other hand,excitation from a continuum or multiple initial states candistort the measured pump–probe correlation, necessitat-ing careful analysis to extract the underlyingdynamics.3,6–8 It is the goal of the present study to elu-cidate the effects of continuum excitation on ultrafast in-terferometric correlation measurements and to determine

0740-3224/2000/081443-09$15.00 ©

what type of information may ultimately be extractedfrom such measurements.

There are already several examples in the literature ofcomplex systems in which ultrafast correlation measure-ments are affected by inhomogeneously broadened levels.For example, ultrafast probing7,8 of jet-cooled Cs2 resultsin simultaneous excitation of multiple dimer rotationallevels populated in the jet. The phase shifts in the inter-ferometric correlation measurements that are due to con-tributions from the different rotational energies leads toan attenuation of the interference structure that is usedto retrieve phase-decay times. In another example,SSHG studies of Au nanoparticles sample an inhomoge-neous distribution of particle sizes that leads to a signalwith contributions from different plasmon resonances.3

A careful consideration of the resulting phase shifts dem-onstrates that this too can modify the observed interfero-metric correlation. Finally, electron bands in metalspresent an extreme example of inhomogeneous broaden-ing, with a continuum of states below and above theFermi level that can have different excited-state or holelifetimes, depending on the energy. In studies of image-potential states on metals, it is found that the contribu-tions from a continuum of ground-state electron energiesmust be considered to analyze the data accurately.6

In spite of these examples, there has been no thoroughtreatment of interferometric correlation measurementsfor systems with broadened bands or continua. To ad-dress this issue, we will consider interferometric two-pulse correlation (I2PC) measurements from metals as amodel system. The basics of 2PP from metals are shownin Fig. 1(a). The pump-laser pulse excites an electronfrom below the Fermi level to an unoccupied hot-electronstate below the work function. The probe pulse then ex-cites this electron above the work function, creating a

2000 Optical Society of America

1444 J. Opt. Soc. Am. B/Vol. 17, No. 8 /August 2000 Weida et al.

photoelectron that can be detected in an energy- andmomentum-resolved manner. Photoelectrons can also becreated by direct excitation from the ground state to thefinal state by means of a two-photon transition. This sys-tem embodies nearly all the issues that arise from con-tinuum effects in interferometric measurements, sincethe initial, intermediate, and final states can each arisefrom a range of state densities. For example, the groundstate can either lie in a band of metal electron states or bea nearly discrete occupied surface state. The intermedi-ate excited state can also lie in a continuum of metal bandstates or can be a nearly discrete adsorbate or intrinsicimage-potential state. Lastly, the final photoionizedstates are part of a continuum, but only a subset of themare detected in the electron energy analyzer. With sucha wide range of possibilities, the I2PC example is thus rel-evant to the effects of broadened bands or continua on in-terferometric measurements for a variety of systems.

The remainder of this paper is organized as follows.Section 2 contains the standard description for photoexci-tation in a multilevel system of discrete levels, i.e., the op-tical Bloch equations. Section 3 is a discussion of the ef-fects of finite analyzer resolution and a continuum of finalstates. In Section 4 excitation from a full continuum inthe ground and intermediate states is considered. Sec-tion 5 concerns changing state densities and broadenedlevels and their effects on I2PC measurements. Finally,conclusions are presented in Section 6.

2. OPTICAL BLOCH EQUATIONSBefore determining the effects of continuum excitation onthe interferometric correlation, a formalism for modelingthe I2PC measurements is necessary. This formalismshould describe how the interaction of the electrons withthe radiation, lattice, and other electrons manifests itselfin the time-dependent population of the measured state.Clearly, such a description could be made arbitrarily com-plex, but a more fruitful approach is to begin with simpleassumptions and add complexity as needed. A cursory

Fig. 1. a) Schematic of two-photon photoemission. On left side,sequential two-photon excitation first lifts an electron to the in-termediate level, leaving behind an excited hole. The secondphoton lifts the excited electron above the vacuum level at whichit can be detected. It is also possible to have a direct two-photonexcitation from the ground state to the final state. b) Schematicof three-level system and the parameters representing the timescales for decay of coherences between levels, i.e., the T2 terms,and population decay, i.e., T1

1.

discussion of the optical Bloch equations, described in de-tail elsewhere,9,10 serves this purpose well.

We begin by assuming that the electrons are noninter-acting and occupy discrete energy levels, much the sameas an isolated atomic system. The coupling of these lev-els by means of optical radiation is described with first-order time-dependent perturbation theory. However,when ultrafast studies are considered, it is also importantto include the broadband nature of the excitation laser.If the laser frequency is nominally described by the car-rier frequency v l , then the spread of frequencies in thepulse makes it possible to excite transitions at frequen-cies in an envelope around v l . In a typical perturbativeapproach, the wave function of the system is expanded ina set of zero-order basis states, i.e., C (t) 5 (kak(t)fk .The frequency spread of the pulse makes it more intuitiveto expand the time-dependent coefficients as ak(t)5 exp(ikvlt)ck(t).

Multiplication by the exponential term results inslowly varying coefficients for levels at or near a multipleof v l , while levels outside the pulse energy envelope willhave rapidly oscillating coefficients that average to zeroand can thus be ignored. In this way so-called nonreso-nant transitions can be treated. The resulting differen-tial equations for the system have the form

]ck

]t5 2iDkck 1 Ck11,kck11 1 Dk21,kck21 2

ck

Tspont.

(1)

The first term of Eq. (1) describes the nonresonant energyof the level \vk with respect to a multiple of the laser-pulse energy, i.e., Dk 5 vk 2 kv l . The second and thirdterms concern one-photon coupling of nearby levels bymeans of optical radiation. The C and D coefficients con-tain terms related to the electric field of the pulse and thetransition moment between levels. The final componentof the equation is a decay term. For an isolated system,this represents the decay that is due to spontaneous ra-diative emission, which has a probability per unit time of1/Tspont . With Eq. (1), it is possible to describe the dy-namics of an isolated system interacting with optical ra-diation.

Excited electrons in a metal are not isolated, so it isnecessary to add another layer of complexity to the de-scription. The simplest way to do this is to consider anisolated system now in contact with a larger system. Thedetails of the interactions with the larger system will beignored and their effects treated phenomenologically bymeans of a density-matrix description. The density-matrix elements are calculated from the time-dependentcoefficients of the zero-order basis states with rmn5 cmcn* exp@2i(m 2 n)vlt#. Diagonal elements representpopulations, while off-diagonal elements represent opti-cally induced coherences between levels. The time de-pendence of the density matrix is written as a set of dif-ferential equations, known as the optical Bloch equations.The complete optical Bloch equations for a three-level sys-tem are given in Appendix A. As an example of the formof these equations, the time dependence of the populationof the intermediate level [shown schematically in Fig.1(b)] is described by


]r11

]t5 ~ ... ! 2 S 2

Tspont1

1

T11D r11 . (2)

The unlisted terms in the first set of parentheses repre-sent the coupling with other matrix elements by means ofthe electric field and the transition moments between lev-els and follow directly from Eq. (1). The last part of theequation, however, contains a new decay term 1/T1

1 that isdue to inelastic collisions with the larger system. Sincethe time scale for inelastic collisions of hot electrons inmetals following 1.5-eV excitation is of the order of fem-toseconds, the much slower nanosecond-time-scale spon-taneous decay will be omitted from this point on.

The time dependence of the off-diagonal density-matrixelements can be seen in the representative differentialequation for the coherence between the intermediate andthe final states of the three-level system shown in Fig.1(b):

]~c2c1* !

]t5 ~ ... ! 2 F i~D2 2 D1! 1

1

2T12

11

2T11 1

1

T212Gc2c1* . (3)

Once again, the unlisted terms in the first set of paren-theses on the right-hand side describe the coupling be-tween levels by the electric field and transition moments.The remaining terms describe the time dependence of thecoherence as influenced by the energy shifts of the levelsand the lifetime of each level. The additional term, i.e.,T2

12 , is included to represent the time scale for phase-changing elastic collisions of the isolated system with thelarger system. To contrast the difference between popu-lation and phase decay, the T1 terms are often referred toas the time scale for incoherent population decay, whilethe T2 terms are referred to as the time scale for coherentphase decay. The coherent phase decay is sometimes de-scribed in terms of an effective phase decay that consistsof contributions from both the T1 and the T2 terms in Eq.(3). We will not use this notation, and the T2 terms hereare the true intrinsic phase-decay lifetimes. Although itis possible to move to more complicated descriptions of2PP than embodied in the optical Bloch equations, suchas semiconductor formalisms11 that include interactionsbetween electron–hole pair quasi particles, we have foundthese equations to be sufficient for describing most of theobserved features in the 2PP data and will thereforechoose them as the stopping point in adding complexity.

Consider now a sample interferogram for a three-levelsystem coupled by the laser-pulse sequence E(t) 1 E(t2 t), where E(t) is the electric field of the pulse and t isthe delay between the pump and the probe pulses. It isassumed that (i) the initial population is in the groundstate, (ii) transitions are in the weak-field limit where thepopulation of the ground state is not changed signifi-cantly, and (iii) the final photoelectron has an infinite life-time. For a given t and energy of the intermediate andfinal state, i.e., D1 and D2 , the optical Bloch equations arenumerically integrated over the complete pulse sequenceto determine the population in the final state after laserexcitation; the resulting interferogram as a function of de-

lay and level energy is written as I(t, D1 , D2). A repre-sentative interferogram with an inset comparing inter-ferograms for resonant and near-resonant (D25 40 meV) excitation is shown in Fig. 2. The interfero-gram is contrasted with the envelope of an interferomet-ric second-order autocorrelation for the laser pulse.There is a net offset in the wings that is due to the finitedecay time of the intermediate level, set to T1

1 5 30 fs forthis calculation. The broadening of the oscillatory enve-lope comes from the increased time scale for coherent de-cay, T2

01 5 T212 5 5 fs and T2

02 5 20 fs. Finally, the pres-ence of 2v oscillations in the wings, i.e., twice thefrequency of the laser fundamental, is due to two-photontransitions that couple the ground and final states di-rectly.

The most relevant aspect of the example in Fig. 2 to thepresent study, however, is the comparison between theI2PC’s calculated with D2 5 0 and 40 meV, a pair of final-state energies that are both accessible, given a 20-fs laserpulse with a full width half-maximum (FWHM) frequencyenvelope of the order of 100 meV. A shift in the final-state energy results in a different frequency response ofthe system. As the phase coherences induced by thepump and probe pulses interfere, this change in fre-quency will lead to a phase shift of the interference pat-tern, clearly evident in the comparison of the two I2PC’sin Fig. 2. This phase shift is also plotted explicitly in Fig.2 for the difference between the 2v oscillations for the twotraces. If these two interferograms were averaged to-gether, as happens when the analyzer resolution is finite,

Fig. 2. Simulated I2PC (solid curve) to show effects of differentoptical Bloch equation parameters. Net displacement in thewings from the laser second-order autocorrelation (19-fs electricfield FWHM hyperbolic-secant-squared pulse, v l 5 1.55 eV:heavy dashed curve) is due to the incoherent lifetime of the in-termediate state, T1

1 5 30 fs. The oscillatory envelope outsidethe laser autocorrelation is due to the finite decay time for coher-ences between the levels, in this case T2

01 5 T212 5 5 fs and T2

02

5 20 fs. The longer T202 lifetime results in oscillations in the

wings at twice the frequency, i.e., 2v, of the laser fundamental.The inset shows the I2PC magnified by 2 compared with an I2PCwith the same parameters but with a shift in the final-state en-ergy of D2 5 40 meV (offset vertically to highlight differences).The slightly different driving frequency results in a phase shift ofthe oscillations. The relative phase shift between the D25 40-meV and 0-meV traces for the 2v oscillations (thin dashedcurve) demonstrates that they are completely out of phase for adelay of 60 fs.


then the resulting interferogram would be modified, ow-ing to the phase shifts. The remainder of this paper con-cerns the application of the optical Bloch equations tovarious situations in which such effects are significant,such as when the electron energy levels either are not dis-crete or are numerous enough to be considered a con-tinuum.

3. FINAL-STATE ENERGY RESOLUTIONMany pump–probe correlation measurements detect a fi-nal photoionized or photoelectron state belonging to acontinuum of energies. Owing to the broadband natureof ultrafast excitation, a range of final-state energies willbe present in such experiments. In some experiments,such as interferometric three-photon photoion correla-tions of alkali atoms and dimers,5,7,8,12 there is no energyresolution in the final state; all ions produced by photo-ionization are detected. In other experiments such as2PP, the final-state photoelectrons are detected in anenergy-resolved fashion in an electron energy analyzer.As discussed in Section 2, shifts in the final-state energycan lead to phase shifts in the interferogram; averagingover the final-state energies thus results in a loss of infor-mation in the observed interferogram. The degree towhich the final-state energy resolution affects the oscilla-tory envelope is first considered for the case of energy-resolved final states, then final states detected with noenergy resolution.

A. Energy-Resolved CaseIn 2PP experiments a subset of the final-state energies isdetected in an energy-resolved fashion by use of an elec-tron energy analyzer with a finite resolution DE that var-ies typically between 1 and 100 meV. Since the analyzerresolution can constitute a significant portion of the pulsewidth, it is possible to detect electrons that have beendriven by different frequency components of the pulse.To determine the effect of analyzer resolution, the initialand intermediate states are taken to be discrete with aseparation \v l , as shown in Fig. 3. The energy analyzeris centered \v l above the intermediate state in a con-tinuum of final states. A range of final state energies canbe excited either coherently or incoherently. The inco-herent case can be treated by summing together inter-ferograms for different final-state energies. In the caseof coherent excitation, however, an incoherent summationwill not reproduce oscillations in the interferogram due tocoherences between the final-state energies. This is notan issue for a continuum, since the net quantum-beatstructure for coherent continuum excitation averages tozero.13 However, in the case of simultaneous observationof multiple discrete states, it would be necessary to treatthe averaging effects of analyzer resolution by includingthe additional states directly in the optical Bloch equationformalism. Since we are concerned only with a con-tinuum here, the incoherent summation approach is ap-propriate.

Each interferogram in the incoherent sum is calculatedwith the optical Bloch equations and is described byI(t, D1 , D2), where D1 and D2 are the energy shifts withrespect to a resonant one- or two-photon transition to the

intermediate or final level, respectively. The effect ofanalyzer resolution for the situation in Fig. 3 is calculatedwith

Itot~t! 5 E f~D2!g~D1 5 0, D2!I~t, D1 5 0, D2!dD2 , (4)

where f is the line shape of the analyzer and g(D1 , D2) isthe weighting factor of each interferogram determined inthe optical Bloch equation analysis. It is assumed thatthe analyzer has a Gaussian resolution function. Equa-tion (4) is approximated by a sum with a discrete step sizein D2 . The convergence of the sum is tested by increas-ing the integration limits and decreasing the step size un-til there is no significant change in Itot(t). For a 20-meVanalyzer resolution, integration limits of 640 meV with astep size of 5 meV lead to convergence for a 19-fs electricfield FWHM hyperbolic-secant-squared pulse.

This formalism is used to consider the effects of a 20-meV electron energy analyzer resolution on the measuredI2PC, similar to our actual experimental resolution.1 Forthis calculation, the coherent phase-decay times involvingthe final state are made quite long to highlight the effectthat the analyzer resolution has on the oscillatory enve-lope of the I2PC. The results of a calculation for T1

1

5 T201 5 5 fs and T2

02 5 T212 5 100 fs are shown in Fig. 3.

As can be seen, the analyzer resolution does affect the os-cillatory envelope. Similar to the calculated period of abeat note, all coherent oscillatory information involvingthe final state is averaged out for time scales * h/DE.Moreover, the analyzer resolution affects the oscillatoryenvelope at even shorter time scales. From Fig. 3, a ruleof thumb is that the oscillatory envelope will be attenu-ated for delays * 0.25h/DE.

However, analyzer resolution does not affect the mea-sured incoherent decay times, i.e., the net offset of the

Fig. 3. Effects of analyzer resolution on I2PC oscillatory enve-lope. Inset shows level scheme for discrete initial and interme-diate states and a continuum of photoelectron states. The en-ergy analyzer samples a finite range, DE, of the continuum.The calculated I2PC envelopes are for (A) a second-harmonic au-tocorrelation of the 19-fs electric field FWHM hyperbolic-secant-squared pulse, (B) I2PC averaged over 20-meV analyzer resolu-tion for T1

1 5 T201 5 5 fs and T2

02 5 T212 5 100 fs, and (C) I2PC

for infinitely sharp analyzer resolution and same parameters asat (B). Note that analyzer resolution can completely wash outcoherent oscillatory structure for time scales * h/DE.


I2PC from the laser autocorrelation shown in Fig. 2. Nordoes it affect the oscillatory structure that is due to coher-ences between levels other than the final detected state.This last fact can be seen immediately by considering Eq.(4). Accounting for final-state resolution requires aver-aging over D2 but not D1 . When the interferograms areadded together, the phase shifts in the oscillatory struc-ture involving coherences with the final state will washout the oscillations. But coherences between the groundand the intermediate states will result in oscillations thatare not phase shifted for each interferogram, since D1 isconstant. When averaged together, these oscillationswill add constructively, and no modification of the oscilla-tory envelope will be observed. For example, if the samecalculation demonstrated in Fig. 3 is performed for a sys-tem with a long phase-coherence time of T2

01 5 100 fs, theoscillatory envelope that is due to this term will not be af-fected by the final-state resolution.

B. Non-Energy-Resolved CaseIn ultrafast interferometric correlation studies of alkaliatoms and dimers, a pump–probe pulse sequence is usedto excite the system between bound levels coherently.For example, the Cs atom5 can be viewed as a three-levelsystem in which the intermediate level is far enough outof resonance that it cannot be effectively excited. Tran-sitions are then predominately two-photon transitions toa bound excited electronic state that is split by spin-orbitcoupling. Cs dimer7,8 is a two-level system in which one-photon transitions excite different vibrational levels ofthe first bound electronic state. One option for detectingthe final state is to photoionize it with either the pump orprobe pulse in a one-photon transition for Cs or in a two-photon transition for Cs2. In this instance the full mea-surement becomes an interferometric three-photon photo-ion correlation with a continuum of final ionized states.If this type of experiment were performed with energyresolution in the final state, then in analogy to the two-photon case, the interferogram would contain additionaloscillatory structure due to coherences between four dif-ferent levels. For example, there would be 3v oscilla-tions due to three-photon transitions that couple theground and ionized states directly.

However, these experiments are typically performedwithout energy resolution in the final state. It is possibleto use a variant of Eq. (4) for a four-level system to calcu-late the effects of no energy resolution. In analogy to thethree-level system, it involves averaging over all possiblefinal-state energies accessed by the broadband pump–probe pulse sequence. The line shape of the analyzer inEq. (4) is replaced by a constant, and the integration isperformed as a discrete sum. We have found that aver-aging over all possible final-state energies leads to a com-plete cancellation of oscillations in the interferogram dueto coherences involving the final state. However, as dis-cussed above, this type of averaging does not affect oscil-lations involving coherences between states other thanthe final state. Therefore, observing an interferometricthree-photon photoion correlation for Cs with no final-state energy resolution is effectively the same as observ-ing an interferometric two-photon correlation betweendiscrete levels. Similarly, Cs2 observed in the same way

effectively results in an interferometric one-photon corre-lation between discrete levels. This same conclusion hasbeen reached for Cs2 on the basis of a differentargument.7 It would be interesting to perform these ex-periments with final-state energy resolution to provideadditional information on the time scale for elastic colli-sions in the excited states. Such information might beuseful for experiments that observe alkali atoms in com-plex media, such as recent experiments of K adsorbed onHe clusters.12

Finally, an interesting variation on final-state energyresolution issues is the observation of quantum beats thatare due to splittings between intermediate levels in mul-tiphoton measurements. A prime example is the obser-vation of quantum beats in two-color noninterferometric2PP correlation measurements of image-potential stateson metal surfaces.4 By the above arguments, oscillatorystructure in the I2PC between the ground state and thecoherently excited intermediate levels is not affected byaveraging, owing to limited resolution in the final state.In fact, oscillatory structure that is due to the coherencebetween the excited intermediate states alone is notwashed out by averaging, even in the limit of fast phasedecay between the ground and intermediate states. Oursimulations confirm that it is possible to observe quantumbeats between intermediate levels in both interferometricand noninterferometric measurements, even in the casein which there is no energy resolution in the final state or(as will be discussed below) in which the initial statearises from a continuum.

4. EXCITATION FROM A FULLCONTINUUM OF INITIAL ANDINTERMEDIATE STATESMetals present a challenging system to study by means of2PP, owing to the presence of continuous bands withwidths of several eV. This is not common for most sys-tems studied with ultrafast techniques, in which thereare at most inhomogeneously broadened levels or limitedbands. However, if I2PC from metals is to be treated cor-rectly, excitation from a full continuum must be consid-ered.

The simplest way to understand the effect of con-tinuum excitation is to contrast it with photoexcitationbetween discrete levels, shown schematically in Fig. 4(a).The initial level is fixed, but the energy spread present inthe pulse makes it possible to excite levels of different en-ergies, described by D1 5 \v l 2 E1 . Consider now theabsorption of an ultrafast pulse for a continuum, shownschematically in Fig. 4(b). A good example might be one-photon photoemission from a metal. In such an experi-ment electrons are excited from a near-continuum ofstates below the Fermi level EF to a continuum of statesabove the vacuum level Evac . Electrons are observed ata particular energy Eobs with the electron energy ana-lyzer, which we will now assume to have infinitely sharpresolution (analyzer resolution effects can be consideredas a final modification of the interferogram predicted forinfinitely sharp resolution). Although Eobs is fixed, theelectrons observed at this energy might result from tran-sitions from different initial levels. As shown in Fig.


4(b), excitation of electrons from an initial energy level re-sults in a distribution of excited electrons determined bythe energy spread of the pulse. The energy analyzer se-lects a particular energy from this distribution that canbe conveniently described by D1 5 \v l 2 E1 , where E15 \v l 1 E0 is the peak of the excited distribution andE0 is the energy of the initial level (E0 5 0 for D1 5 0).To describe completely the electrons observed by the ana-lyzer, we must average over all of the initial energies thatcontribute to the observed electron distribution. For afixed Eobs , D1 effectively describes the energy of the ini-tial state, so averaging over all initial states is tanta-mount to averaging over D1 .

Consider now two-photon photoemission from a con-tinuum of initial and intermediate levels, shown sche-matically in Fig. 4(c). Photoexcitation from a particularenergy below the Fermi level results in a distribution ofelectrons in the intermediate level. The next photoexci-tation step involves transitions from this distribution ofstates to a distribution of photoionized final states. Theelectron energy analyzer is tuned to observe photoelec-trons at a particular energy Eobs . As with the one-photon case, however, the observed electrons can origi-nate from many different initial states and now fromintermediate states as well. In summary, there are nowmany different paths that an electron can take and stillhave a final energy Eobs . Clearly, one interferogram forone set of three possible levels will be insufficient to de-scribe the final observed electron distribution. Instead,we will need to calculate an interferogram for each pos-sible pathway and then average over all the interfero-grams to obtain the observed interferometric correlationtrace. As discussed above, an incoherent sum of the in-

Fig. 4. Examples of one- and two-photon photoexcitation from acontinuum. Here the solid curves represent the resonant tran-sition, while the dashed and dotted–dashed curves represent lev-els and distributions that are sampled to the red (D1R) and blue(D1B) of the resonant frequency, respectively. a) In the discreteone-photon case, D1 represents the position of the final level withrespect to the laser frequency envelope. b) For the continuumone-photon case, D1 represents the shift of Eobs with respectto the peak of the final photoelectron distribution from a spe-cific ground-state energy. Averaging over D1 is the same asaveraging over all ground state energies. c) The two-photoncontinuum case is similar, except that averaging over theground-state energies is the same as averaging over D2 . Foreach D2 , it is also necessary to average over all intermediatestate energies, i.e., D1 .

terferograms is appropriate for the case of either incoher-ent or coherent excitation of a continuum of states in theintermediate level but is inappropriate for the case of co-herent excitation of multiple discrete intermediate states.Once again, we are considering only the continuum case.

The procedure for calculating the averaged interfero-gram is relatively straightforward. In analogy with theone-photon continuum case, D2 defines the separation be-tween Eobs and the initial energy level E0 , so all possibleD2 are averaged to take into account the contributionsfrom different initial states. However, for each D2 , wemust now also average over all of the possible D1 for theintermediate state. The only thing tricky in the use ofthe optical Bloch equations in this fashion is that D2nominally corresponds to the final-state level. If we av-erage over D2 , then the relative separation of theintermediate- and final-state energies is changing, whenwe want to change only the ground- to intermediate-stateseparation. To correct for this, an effective shift of theintermediate state D18 is defined, and the total shift of theintermediate level becomes D1 5 D18 1 D2 . In this way,averaging over the ground state does not affect theintermediate- to final-state separation. To average overthe intermediate state, it suffices to average over D18 . Itis assumed that the density of states, transition moments,and decay lifetimes are constant for the initial-,intermediate-, and final-state continua sampled by the la-ser pulse; this is justified for Drude absorption from thenearly free electron bands of metals. The total averagedinterferogram becomes

Itot~t! 5 EEg~D1 5 D18 1 D2 , D2!

3 I~t, D1 5 D18 1 D2 , D2!dD18dD2 , (5)

where g(D1 , D2) is the weighting factor of each interfero-gram determined in the optical Bloch equation analysis.Because each I(t, D1 , D2) must be calculated numeri-cally, the integral in Eq. (5) is approximated by a sum.Convergence is tested by increasing the integration limitsand decreasing the step size in D18 and D2 until changes inthe resulting interferogram are insignificant. For a con-tinuum case, convergence occurs for integration limits of6200 meV and a 10-meV step.

On the basis of these assumptions, Itot(t) is calculatedin Fig. 5 for a Gaussian laser pulse with an electric fieldFWHM of 19 fs and energy \v l 5 1.55 eV. The incoher-ent population decay is set at T1

1 5 25 fs, while the coher-ent terms are set to T2

01 5 T212 5 20 fs and T2

02 5 50 fs.For comparison, the I2PC for a discrete system with noaveraging, as well as the second-order autocorrelation forthe laser pulse, are also shown in Fig. 5. The discretesystem shows an oscillatory envelope that is significantlybroader than the laser autocorrelation. The averagingthat is due to the continuum, however, almost completelywashes out this extended oscillatory structure, leavingonly the net offset from the laser autocorrelation that isdue to the incoherent time scale for population decay T1

1.The results are identical over the tested range (1 to 100fs) of incoherent and coherent lifetimes: Continuum ex-citation erases any information about the coherent life-times involving the continuum levels. However, it does


not affect the incoherent decay times. For example, theaveraged interferogram can be reproduced almost exactlyby calculating an interferogram with the above param-eters and a fast phase-decay time, i.e., T2

01 5 T212 5 T2

02

5 0.5 fs.Given the limitations that these results place on what

can be determined from interferometric data for a con-tinuum case like this, it is worth trying to understandwhy the coherent information is lost. Consider the sche-matic in Fig. 4(c). At Eobs there are equal contributionsfor electrons being driven at higher and lower frequenciesthan the carrier frequency v l . Discrete level interfero-grams for these lower/higher energy distributions havefaster/slower oscillations, indicative of the higher/lowerfrequencies to which these electrons are responding. Av-eraging these different envelopes together results in a netcancellation of the coherent oscillations outside the auto-correlation of the laser. If some of our approximationsbreak down (for example, if the density of states, transi-tion moments, etc., are not constant over the range of con-tributing states), the cancellation is not complete, and theresulting interferograms are broader.

Before leaving this example, it is natural to test theseresults against experiment. In recent interferometricmeasurements of two-photon photoemission from Cu(111)and Cu(100) with 1.55-eV excitation, we have carefullycharacterized the electric field of the laser pulse, using in-terferometric SSHG measurements. Using this informa-tion in optical Bloch equation fits for a three-level system,we find, for Drude absorption for electrons originatingfrom energies up to 1 eV below the Fermi level, that thefitted T2

01 , T212 , and T2

02 lifetimes are all extremely fast,,0.5 fs. However, analysis of interferograms for elec-trons originating within 100 meV of the Fermi level, as

Fig. 5. Simulated interferograms for discrete and continuumtwo-photon excitation with a constant density of states. Bothinterferograms are for a system with the same decay times (T1

1

5 25 fs, T201 5 T2

12 5 20 fs, T202 5 50 fs), but one is averaged

over a continuum of possible initial and intermediate states witha discrete final state, while the other is for three discrete levels.The averaging that is due to the continuum washes out the in-terference fringes outside the laser autocorrelation trace. How-ever, the incoherent population decay time results in a net offsetin the wings, compared with the laser autocorrelation. Simula-tions are for a pulse with \v l 5 1.55 eV and a 19-fs FWHM elec-tric field Gaussian envelope.

described below, suggest that the T2 decay times are con-siderably longer than this. The averaging that is due tocontinuum absorption appears to have completely erasedthis information, just as predicted.

5. EXCITATION FROM CHANGING STATEDENSITIES AND INHOMOGENEOUSLYBROADENED LEVELSOne of the most puzzling aspects about interferometricdata recorded from metals is the sudden broadening ofthe oscillatory envelope at the Fermi level, along with anapparent driving frequency higher than v l ,1 shown inFig. 6(a). The magnitude of this effect seems to be incon-sistent with the abrupt loss of coherent broadening ob-served for electrons at slightly lower energies. However,continuum effects can easily explain what is occurring.Consider the schematic of the Fermi edge in Fig. 7. Thiscase is qualitatively different than the continuum ex-ample above, for now the population of the initial stateschanges rapidly with energy. Figure 7 suggests that theobserved electron distribution should therefore beskewed, with a larger contribution from electrons re-

Fig. 6. a) Observed and b) simulated I2PC from the Fermi edgeof 50 K cesiated Cu(100) with 1.55-eV excitation. The observedtrace is much broader than the laser second-harmonic autocorre-lation. In addition, the phase shift between the laser autocorre-lation and the I2PC demonstrates that the interferogram is duepredominately to electrons driven by the higher-frequency com-ponents of the laser pulse. The observed I2PC can be com-pletely reproduced by the continuum model. In a least-squaresfit of the data, this model reveals extremely long phase-coherencetimes of .100 fs between the ground and final state.


sponding at frequencies higher than v l . In addition,there will be an incomplete cancellation of the coherentoscillations because of the skewed distribution. There-fore the observed oscillatory envelope should be broaderand indicative of a driving frequency greater than v l .

Because the Fermi distribution is known exactly, it ispossible to model this effect explicitly. To do so, we makethe above assumptions that the incoherent and coherentlifetimes, as well as the transition moments, are constantas a function of energy over the relevant range of energiessampled by the pulse. An additional weighting term isthen included in Eq. (5),

Itot~t! 5 EEf~Eobs 2 2\v l 2 D2!g~D1 5 D18 1 D2 , D2!

3 I~t, D1 5 D18 1 D2 , D2!dD18dD2 , (6)

where f(E) is the Fermi distribution with EF 5 0. Onceagain, the integral is approximated by a sum, with con-vergence occurring for integration widths of 6200 meVand a step size of 10 meV. This simple Fermi weightingreproduces the observed trends almost exactly. Con-sider, for example, the simulation based on a fit of datataken at the Fermi level for 1.55-eV excitation from 50 Kcesiated Cu(100) in Fig. 6(b). Both the apparent fre-quency shift of the oscillatory envelope, shown as a phaseshift between the laser autocorrelation and the interfero-gram in Fig. 6(a), and the broadening that is due to both1v and 2v oscillations are completely reproduced. How-ever, the most important aspect of this data from theFermi level is that the averaging effects from continuumexcitation no longer wash out the coherent oscillations.The true T2

01 , T212 , and T2

02 lifetimes can be retrieved in aleast-squares fit of the data. For the example in Fig. 6,T2

02 . 100 fs, as expected from hole lifetimes at the Fermilevel,1 while the T2

01 and T212 terms are significantly

longer than the measured population decay times of T11

' 3 fs. That these extremely long phase-decay lifetimesare not observed for electrons originating from energiesslightly lower than the Fermi level is a testament to theeffects of averaging on phase-coherence information.However, the Fermi level, with its known change in statedensity, offers an opportunity to retrieve these param-eters in I2PC measurements from metals. This same ap-proach can also be applied to other discrete bands in 2PPspectra, provided that the distribution function is known.

In contrast to the nearly continuous bands of metals,most systems exhibit inhomogeneously broadened levels

Fig. 7. Continuum excitation from the Fermi level. The popu-lation of the initial states changes rapidly with energy at theFermi edge, skewing the final distribution of energies to elec-trons that have been predominately excited by higher-frequencycomponents of the laser pulse. Averaging results in an incom-plete cancellation of the oscillatory envelope from coherentterms, and the skewed distribution results in an apparent shiftin the oscillation frequency from v l .

of finite width. These too can be treated by performingan incoherent sum of interferograms based on Eq. (6).Instead of the Fermi distribution, f(E) is replaced by theline shape of the broadened level. Similar to the SSHGstudy of plasmon lifetimes for inhomogeneous size distri-butions of Au nanoparticles,3 broadened levels lead to anattenuation of the oscillatory envelope but do not affectthe measured incoherent decay times. It is tempting totry to derive a rule of thumb for the effect of inhomoge-neous broadening on the oscillatory envelope, but it isbest to treat each example on a case-by-case basis. Forexample, a three-level system consisting of a discrete ini-tial state and final state, but a broadened intermediatestate, will show an attenuation of the 1v oscillations inthe interferogram but not the 2v oscillations, since theyresult from two-photon transitions that couple the initialand final states directly.

As a final note, an interesting variation to the apparentdriving frequency of the laser at the Fermi level can occurfor broadened levels. If Eobs is selected such that theresonant levels lie on the low-energy side of a broadenedlevel, then the observed interferogram will exhibit an ap-parent driving frequency lower than the laser frequency.In other words, when the low-energy edge of a peak is ob-served, the distribution of electrons will be skewed tothose responding to lower-frequency components of the la-ser pulse. Similarly, detection on the high-energy side ofa peak can lead to an apparent higher-driving frequency.This has been observed in the case of 2PP from the occu-pied surface state on Cu(111).1,14 These effects can alsobe easily calculated on a case-by-case basis with the aboveformalism.

6. CONCLUSIONSTo extract information on phase-coherence lifetimes fromultrafast interferometric correlation measurements oncomplex systems, it is imperative that averaging effectsthat are due to a continuum of states or broadened levelsbe taken into account. Although the inherent averagingthat is due to broadband excitation in such systems doesnot affect the measured incoherent population lifetimes,it can completely erase information on coherent phase-decay lifetimes for the extreme case of excitation betweencontinua. It is possible to retrieve the phase-decay life-times from systems that have broadened levels or aknown change in the density of states, such as at theFermi level, by an iterative procedure of fitting the ob-served data to averaged interferograms calculated withthe optical Bloch equation formalism. However, if this isnot done rigorously, the observed coherent phase-decaytimes can be taken only as a lower limit on the actualphase-decay times. In addition, averaging in such in-stances can lead to an apparent shift in the driving fre-quency of the system, an effect that can be modeled with acontinuum approach. Finally, even in systems with dis-crete levels, the final photoionized state is often part of acontinuum. Averaging effects that are due to energyanalyzer resolution also need to be considered whenphase coherence lifetimes are extracted.


APPENDIX A: OPTICAL BLOCHEQUATIONS FOR A THREE-LEVEL SYSTEMIt is useful to write down the complete optical Bloch equa-tions for a three-level system, both as a reference and tohighlight any assumptions implicit in their use. Thecomplex electric field that couples the levels is written as

E~t ! 512 @ e exp~iv lt ! 1 e* exp~2iv lt !#. (A1)

Here

e~t ! 5 e~t !exp@if~t !#, (A2)

where e(t) is the real electric field envelope of the pulseand f(t) is the time-dependent phase. The dipole tran-sition moment between levels is written as

dij 5 udu^ f iu ru f j&, (A3)

where udu is the intrinsic transition moment magnitude,f i is a basis function for the three-level system, and r isthe dipole transition moment operator. With the defini-tions laid out in Section 2, the differential equations ofthe density-matrix elements are written as

]r11

]t5

i

2\d01~ e* c0c1* 2 ec1c0* !

1i

2\d12~ ec2c1* 2 e* c1c2* ! 2

1

T11 r11 , (A4)

]r22

]t5

i

2\d12~ e* c1c2* 2 ec2c1* ! 2

1

T12 r22 , (A5)

]r00

]t5 2

]r11

]t2

]r22

]t, (A6)

]~c1c0* !

]t5

i

2\d01e* ~r00 2 r11! 1

i

2\d12ec2c0*

2 S iD1 11

2T11 1

1

T201D c1c0* , (A7)

]~c2c1* !

]t5

i

2\d12e* ~r11 2 r22! 2

i

2\d01ec2c0*

2 F i~D2 2 D1! 11

2T12 1

1

2T11 1

1

T212Gc2c1* ,

(A8)

]~c2c0* !

]t5

i

2\e* ~d12c1c0* 2 d01c2c1* !

2 S iD2 11

2T12 1

1

T202D c2c0* . (A9)

Equation (A6) indicates that the three-level system isclosed, meaning that the sum of the diagonal elements ofthe density matrix is one. As discussed in Section 2, the

time scale for spontaneous radiative decay of an excitedlevel is expected to be several orders of magnitude longerthan hot-electron lifetimes in metals and is thus omittedfrom these equations. In an actual calculation of theI2PC, these differential equations are split into real andimaginary parts, leading to eight unique coupled differen-tial equations.

ACKNOWLEDGMENTSM. J. Weida thanks the National Science Foundation andthe Center for Global Partnership for support (NSF grantINT-9819100).

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