Strong-field control and spectroscopy of attosecondelectron-hole dynamics in moleculesOlga Smirnovaa,b, Serguei Patchkovskiia, Yann Mairessea,c, Nirit Dudovicha,d, and Misha Yu Ivanova,e,1

aNational Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6; bMax Born Institute, Max Born Strasse 2a, D-12489 Berlin, Germany;cCentre Lasers Intenses et Applications, Universite Bordeaux I, Unite Mixte de Recherche 5107 (Centre National de la Recherche Scientifique, Bordeaux 1,CEA), 351 Cours de la Liberation, 33405 Talence Cedex, France; dDepartment of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100,Israel; and eDepartment of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

Communicated by Stephen E. Harris, Stanford University, Stanford, CA, July 19, 2009 (received for review August 8, 2008)

Molecular structures, dynamics and chemical properties are deter-mined by shared electrons in valence shells. We show how one canselectively remove a valence electron from either � vs. � orbonding vs. nonbonding orbital by applying an intense infraredlaser field to an ensemble of aligned molecules. In molecules, suchionization often induces multielectron dynamics on the attosecondtime scale. Ionizing laser field also allows one to record andreconstruct these dynamics with attosecond temporal and sub-Ångstrom spatial resolution. Reconstruction relies on monitoringand controlling high-frequency emission produced when the lib-erated electron recombines with the valence shell hole created byionization.

high harmonic generation � multielectron dynamics �strong-field ionization

E lectrons in valence molecular shells hold keys to molecularstructures and properties. This article focuses on finding

ways to control and record their dynamics with attosecond (1asec � 10�18 sec) temporal resolution. Imaging structures anddynamics at different temporal and spatial scales is a majordirection of modern science that encompasses physics, chemistryand biology (1). Electrons in atoms, molecules and solids moveon attosecond timescale; resolving and controlling their dynam-ics are goals of attosecond and strong-field science (2–6).

A natural route in manipulating valence electrons is to applylaser pulses with electric fields comparable with the electrostaticforces binding these electrons in molecules. Combination ofpulse-shaping techniques and adaptive (learning) algorithms(7–16) turn intense ultrashort laser pulses into ‘‘photonic re-agents’’ (8), allowing one to steer molecular dynamics toward adesired outcome by tailoring oscillations of the laser electricfield. Strong-field techniques find applications in controllingunimolecular reactions (9, 14), nonadiabatic coupling of elec-tronic and nuclear motion (15), suppression of decoherence (16,17), etc. From the fundamental standpoint, one of intriguingchallenges lies in understanding and taming the complexity ofstrong-field dynamics, where multiple routes to various finalstates are open simultaneously and multiple control mechanismsoperate at the same time.

Although nuclei in molecules move on the femtosecond timescale, attosecond component of the electronic response oftenplays crucial role (see, e.g., ref. 18). For example, in polyatomicmolecules electronic excitations in intense infrared laser fieldsare created via laser-induced nonadiabatic multielectron (NME)transitions (14, 19, 20). These transitions occur on the sublasercycle time scale and determine femtosecond dynamics thatfollows. Although laser-induced NME transitions open multipleexcitation channels (14) and lead to molecular breakup into amultitude of fragments (15, 19, 20), in practice, they can be hardto control. We show that tunnel ionization in low-frequency laserfields is an alternative way of creating a set of electronicexcitations in the ion, which can be controlled by changingmolecular alignment relative to laser polarization. The controlmechanism uses symmetries of molecular orbitals to remove an

electron from either � vs. � or bonding vs. nonbonding orbital.Increasing laser wavelength � suppresses runaway nonadiabaticexcitations (19) while keeping ionization channels intact. Con-trol over the initial electronic excitation is a gateway to control-ling molecular fragmentation that follows.

Because tunnel ionization can create electronic excitations,the hole left in a molecule will move on attosecond time-scaledetermined by inverse energy spacing between excited electronicstates, � � �/�E. How can one image such motion? Usingattosecond probe pulses is problematic: necessarily high carrierfrequency of such pulses interacts with core rather than valenceelectrons. The electron current also does not record hole dy-namics, see below. However, when ionization occurs in a laserfield, the liberated electron does not immediately leave thevicinity of the parent ion: Oscillations in the laser electric fieldcan bring the electron back (21). We show how the hole dynamicsis recorded in the spectrum of the radiation emitted when theliberated electron recombines with the hole left in the molecule.

ResultsWe first consider control of optical tunneling. Consider anexample of a CO2 molecule interacting with intense infraredlaser field. The critical factors in strong-field tunnel ionizationare the ionization potential Ip and the geometry of the ionizingorbital (22, 23). Removal of an electron, which leaves the ion inan electronic state j, is visualized using the Dyson orbital �D,j �N�

j

(N�1)�NT(N)�. Here, �NT

(N) is the N-electron wave function ofthe neutral molecule and �j

(N�1) describes the ion in the state j.Dyson orbitals for the ground state X 2�g (j � X) and the firsttwo excited states A2�u (j � A) and B2�u

� (j � B) of the CO2molecule are shown in Fig. 1 A and C. Because tunnel ionizationis exponentially sensitive to Ip, one would expect that onlyground state of the ion is excited after ionization, i.e., X is theonly participating ionization channel.

However, the nodal structure of �D,X (Fig. 1 A) suppressesionization for molecules aligned parallel or perpendicular to thelaser polarization. Indeed, opposite phases of the adjacent lobesin �D,X lead to the destructive interference of the ionizationcurrents for molecular alignment angles around � 0° and �90°, as observed in the experiment (24). Due to the interferencesuppression of ionization in the X-channel, ionization channelswith higher Ip, (e.g., A, B) become important. For the ion left inthe first excited state A2�u (Fig. 1B), the same argument showsthat ionization is suppressed near � 0° but enhanced near �90°, perpendicular to the molecular axis. In the latter case, thereis no destructive interference between the adjacent lobes withthe opposite phases, because only the lobe near the ‘‘down-

Author contributions: O.S. and M.Y.I. designed research; O.S. and S.P. performed research;Y.M. and N.D. analyzed data; and O.S. and M.Y.I. wrote the paper.

The authors declare no conflict of interest.

1To whom correspondence should be addressed. E-mail: m.ivanov@imperial.ac.uk.

This article contains supporting information online at www.pnas.org/cgi/content/full/0907434106/DCSupplemental.

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bending’’ part of the ion potential will contribute into ionization.For the ion left in the second excited state B2�u

�, the Dysonorbital (Fig. 1C) shows that ionization is favored along themolecular axis. Our calculations (see Methods) confirm theseexpectations. Fig. 1D shows the degree of control for the removalof bonding (channel A) vs. nonbonding (channel X) electron orfor the ionization from � vs. � states (A or B channels). Thecontrol parameter is the molecular alignment angle . AlthoughX channel dominates ionization for all angles, the control is verysubstantial, with the ratio of amplitudes (square roots of ion-ization probabilities) for various ionization channels changing byover an order of magnitude.

Ionization leaves a hole in the valence shell of the molecule.If several electronic states of the cation are excited, will the holemove, and how? Consider alignment angles near � 90°, whenthe two ionization channels (A and X) are significant, see Fig. 1D.Note that probability of ionization for X-channel is stronglysuppressed for � 90°, but not equal to zero. Destructiveinterference occurs only for the electrons ejected along the laserfield, exactly perpendicular to the molecular axis. Although suchelectrons dominate the ionization current from each lobe, thereis a certain amount of electrons ejected in all other directions,i.e., at angles different from 90° with respect to the molecularaxis. Such electrons are responsible for the ionization probabilitywhen the laser field is orthogonal to the molecular axis. Thepresence of these electrons is a consequence of the Heisenberg’suncertainty principle and the confinement of the continuumwave packet in the direction perpendicular to the laser field.

Ionization creates an entangled electron-ion wave packetA�eI

(N)(t), where A antisymmetrizes the electrons and �eI(N)(t) �

aX(t)�X(N�1)�C,X(t) � aA(t) �A

(N�1)�C,A(t). Here aj(t) are complexamplitudes of the ionic states �j

(N�1)populated by ionization,evolving between ionization and recombination. If we were toneglect laser-induced excitations in the ion between ionizationand recombination, than aj(t) � �jexp[�iEj(t � ti)], where �j isthe complex ionization amplitude, ti is the moment of ionization

(see Methods). Continuum wave packets �C,X(t) and �C,A(t) arenormalized to unity. Due to the entanglement between the ionand the electron (the hole and the electron), the dynamics of theionic subsystem is undetermined until the measurement of thisdynamics is specified (25, 26).

Consider first an example of the measurement, which ‘‘pre-pares’’ the electron wave packet in the ion. One way of obtainingsuch wave packet is to project the continuum wave packets�C,X(t) and �C,A(t) on the same continuum state, e.g., the statehaving a momentum p at the detector: �I

(N�1)(t) � pA�eI(N)(t)� �

aX(t)p�C,X(t)��X(N�1) � aA(t) p�C,A(t)��A

(N�1). Here, we havepostulated that there is no exchange between the continuumelectron at the detector and the electrons remaining in theion.�I

(N�1)(t) represents a coherent superposition of the ionicstates �X

(N�1), �A(N�1), provided that both p�C,X(t)� � 0 and

p�C,A(t)� � 0. Note, that for the channel X no electrons escapewith p exactly perpendicular to the molecular axis, and hence, forthis choice of p�, the projection p�C,X� � 0, and the hole is‘‘static,’’ with only �A

(N�1) present in the superposition.The hole wave packet can be visualized via the overlap of the

ionic wave packet with the neutral ground state: �hole(t) ��NT

(N)�I(N�1)(t)�. Phases of aj(t) will evolve, reflecting the hole

dynamics. The Dyson orbitals for the ‘‘plus’’ and ‘‘minus’’superpositions of the two states, �D

(�) � �NT(N)�X

(N�1) � �A(N�1)�

are shown in Fig. 2. The hole moves along the molecular axis witha period of 1.18 fsec, determined by the energy splitting betweenthe ionic states X and A, �EX,A � 3.5 eV.

But where does this motion begin? Will the hole start on theleft side of the molecule and move to the right, or vice versa? Fora molecule aligned exactly at � 90°, there is no preferencebetween the two directions. What breaks the symmetry? Again,the answer requires the analysis of the entanglement between theparent ion and the liberated electron (see ref. 27 for a similarexample in the core-hole localization in N2). The symmetry isbroken by the direction of the electron escape, characterized bythe final momentum p. Its angle relative to the molecular axis

A B

DC

Fig. 1. Dyson orbitals corresponding to different ionization channels of CO2. (A) X. (B) A. (C) B. Red and orange correspond to positive and negative lobes. Theorbitals are shown at the level of 90% of the electron density. (D) Shown is the degree of control over strong-field ionization amplitudes, as a function ofmolecular alignment angle : green and blue curves show control over electron detachment from � vs. � orbital, the red curve shows control over electrondetachment from bonding (channel A) vs. nonbonding (channel-X) orbital. The plotted value is the ionization amplitude (square root of probability) at the peakof the electric field, which is the relevant quantity for the process of harmonic generation discussed later. The laser wavelength is � � 1,140 nm, and the intensityI � 2.2 � 1014 W/cm2.

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determines the relative phase of the projections p�C,X� andp�C,A� and, hence, the direction of the hole motion.

Let the ionizing electric field point vertically downward in Fig.1 A and B, so that the electron tunnels upwards. Since thetunneling wave packets �C,X, �C,A are confined transversely, theHeisenberg uncertainty relationship dictates that, for both wavepackets, the outgoing electron can have a range of momenta withnonzero projections on the molecular axis. For the X channel,the symmetry of �D,X� also dictates that (i) the tunneling wavepacket �C,X will have a nodal plane along the vertical direction,and (ii) the projection p�C,X� will change its sign depending onthe direction of the electron escape. In the tunneling limit, fordirection of electron escape pointing into the upper left quadrantof Fig. 1 A and B, both p�C,X�and p�C,A� are positive at themoment of ionization (‘‘plus’’ superposition �D

(�), Fig. 2 light anddark blue) and the hole starts on the left side. If direction ofelectron escape points into the upper right quadrant, then at themoment of ionization p�C,X� � 0 is negative, but p�C,A� �0 is positive (superposition �D

(�), Fig. 2, red and orange) and thehole starts on the right side. Both directions of the electronescape are equally probable. Thus, equal number of holeswill move right to left and vice versa, with the electron-holeentanglement and the electron momentum p selected by themeasurement defining the direction.

The situation changes gradually as we rotate the electric fieldaway from � 90°, so that tunneling along the electric field isno longer orthogonal to the molecular axis. For the X channel,this changes the relative number of electrons escaping toward theupper left vs. the upper right quadrant of Figs. 1 (a). Forexample, for � 90°, more electrons will escape toward theupper right quadrant, and hence more holes will start on the rightside.

We now turn to attosecond spectroscopy of valence shell holesas a tool for visualizing control over strong-field ionization thatcan be achieved. To record hole dynamics, which reflect thepopulation of several ionic states, one has to record the inter-ference between different ionization channels with variable timedelay after ionization. To record the interference, a measure-ment must bring the system to the same final state. Only thendifferent quantum pathways (here, ionization channels) willinterfere. Thus, measuring the tunneled electrons will not do—different ionization channels leave the ion in different final

states and, hence, the electronic wave packets correlated todifferent ionization channels will not interfere.

Interference of different ionization channels is recorded in thespectrum of radiation emitted when the liberated electronrecombines with the hole left in the molecule, Fig. 3A. Thisprocess, known as high-harmonic generation, returns the mol-ecule to the same ground state it has started from, ensuringinterference of different electron-hole states evolving betweenionization and recombination. In the language of pump-probespectroscopy, strong-field ionization acts as a ‘‘pump,’’ andrecombination acts as a ‘‘probe’’ (28, 29). Recombination occurswithin a fraction of the laser cycle after ionization, Fig. 3B. Thetime of ionization is linked to the time of recombination (Fig.3B), the latter is mapped onto the harmonic number (Fig. 3C).Thus, each harmonic makes a snapshot of the dynamics for adifferent ‘‘pump-probe’’ delay, providing a ‘‘frame’’ for theattosecond ‘‘movie.’’

This idea has already been used to track the motion of protonson attosecond time scale (28), but extending the method toelectrons proved problematic. Indeed, in the pump-probe spec-troscopy an important ingredient is the ability to calibrate thepump-probe signal against that obtained without pump-inducedexcitations. Unfortunately, the HHG process does not allow oneto turn the pump off while keeping the probe on. Thus, one needsa different calibration scheme. The key ingredient of the ap-proach (28, 29) was the ability to slow protons down by replacingthem with heavier deuterons, thus calibrating intensity modu-lation in the harmonic spectra and reconstructing the dynamics.But how to slow down the electrons? We solve this problem byusing molecular alignment to control the hole. The comparisonof harmonic signals for different alignment angles simplifyinterpretation of our results.

Formally, the harmonic emission results from recombination,which is described by the matrix element D(t) ��NT

(N)dA�eI(N)(t)�. Emitted light is given by the Fourier trans-

form of D(t). Relative phases, accumulated by aj(t) in the wavepacket �eI

(N)(t) between ionization and recombination, lead to

Fig. 2. Qualitative picture of the hole dynamics: evolution of the coherentsuperposition of the Dyson orbitals corresponding to X and A states of the ion.The hole moves along the molecular axis with period of 1.18 fsec. Twosnapshots, shown in (i) red and orange and (ii) light and dark blue areseparated by half-period and correspond to the two turning points of the holemotion. The snapshots are shown at the 40% level of the electron density. Theinitial conditions in the figure correspond to the ‘‘minus’’ superposition, �D

(�)

� �NT(N)�X

(N�1) � �A(N�1)�.

A

CB

Fig. 3. The origin of high temporal resolution in HHG spectroscopy. (A)Physical mechanism: Intense IR laser field liberates an electron. Within afraction of the laser cycle, the electron is returned to the parent ion by theoscillating field. Recombination is accompanied by the emission of a harmonicphoton. (B) The moment of ionization ti is linked to the moment of recombi-nation tr. Solid blue curve shows one oscillation of the laser field Elaser, straightlines connect moments of ionization ti and recombination tr. (C) The timedelay between ionization and recombination �t � tr � ti is mapped ontoharmonic number: N� � Ee(�t) � Ip. Here, Ee is the energy of the returningelectron.

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constructive or destructive interference of different ionizationchannels in emission, suppressing or enhancing harmonics. Thisis the amplitude modulation we are looking for. Changingmolecular alignment, we control the relative amplitudes of thechannels and, hence, the modulation depth.

In addition to the phase accumulated in the ion, the relativephase between the channels includes contributions from theoscillatory continuum motion in the laser field and the scatteringphases encoded in the recombination matrix elements. Analysisis simple for high-energy harmonic photons Nh� �� EA � EX(see supporting information (SI)). Then the relative phaseaccumulated in the ion is � � (EA � EX)� , where � is thechannel-average time delay between ionization and recombina-tion. The phases of the oscillatory continuum motion differ muchless, see SI. The phases of the recombination matrix elements,when different for the two channels, affect their interference.

Fig. 4A shows calculated harmonic spectra for CO2 at laserintensity I � 1.7 � 1014W/cm2, for pulse duration 90 fsec and � �1,140 nm (see Methods and SI). The results are averaged overalignment distribution typical for modern experiments withaligned molecular ensembles (30), and the so-called long trajec-tories are filtered out, see SI. In a typical experimental setup,alignment is achieved by applying a linearly polarized aligningpulse, which excites molecular rotations. Polarization of thepulse defines the axis of the aligned molecular ensemble. Har-monics are generated by the second linearly polarized pulse, withvariable angle relative to the polarization of the aligning pulse.This is the angle in Fig. 4A. Currently, most of the experimentson HHG in molecules are done with 800 nm driving laser filed.Using longer wavelength of the driving field allows one to recordlonger movies (number of frames (harmonics) ��3) and increasetime resolution (number of frames per femtosecond ��2). On theother hand, the efficiency of harmonic generation stronglydecreases for longer wavelength typically as ���5 � ��6 (31).Here, we use the wavelength � � 1,140 nm, which is just longenough to observe the whole period of the hole motion bylooking only at short trajectories. Note that observing both shortand long trajectories simultaneously almost doubles the length ofthe movie for a given wavelength.

DiscussionConsider first the structures around � 90°, marked witharrows. Fig. 4B shows amplitudes (square roots of intensities) ofthe total signal and of the channels A and X, for � 90°. The

channels give comparable contributions for a broad range ofharmonics, which record channel interference. Constructiveinterference occurs around H39, whereas destructive interfer-ence occurs around H71. Another destructive minimum be-comes visible when the signal at � 90° is normalized by thesignal at smaller angles (here, � 70°), Fig. 4C. As changesfrom � 90° to � 70 � 60°, the channel X begins to dominatethe spectrum. Thus, such normalization allows one to calibratethe ‘‘pump-probe’’ signal against the ‘‘time-independent’’ back-ground. As changes from � 90° to � 60°, the minima donot move to different harmonic order, but gradually disappear,reflecting the decreasing contribution of the channel A. Timedelays between ionization and recombination for different har-monics are given in Fig. 4C, bottom axis. The minima appear atH21–23 and H71. Time-delay between the frames H21–23 andH71 is the period of the wave-packet motion in the ion, which ishere equal to 1.05 � 0.12 fsec, close to the field-free period of1.18 fsec. The error in the time-energy mapping is larger for lowharmonic numbers such as H21–23. The time delay betweenthe frames H71 and H39 gives half a period of the hole motion0.63 � 0.08 fsec, yielding the period 1.26 � 0.11 fs (see SI for thediscussion of the harmonic order-time delay mapping and errorbars).

Now consider the amplitude structures near � 0°, see Fig.4A and Fig. 5A. Here, ionization creates a wave packet of X andB states, which corresponds to a breathing motion perpendicularto the molecular axis, see Fig. 5C. The example at � 0°illustrates additional difficulties in the reconstruction. The con-tribution of the channel X is modulated by the deep structuralminimum around H55–H59, associated with the geometry of�D,X. As a result, the relative contributions of the two channelsinto the total signal vary significantly, obscuring their interfer-ence. Furthermore, the way the period of the hole motion isrecorded in the harmonic spectrum depends on the relativephase of recombination between the two channels. Accuraterecording of this period in the harmonic spectrum requires thatthis phase does not strongly depend on energy. This requirementis violated in the vicinity of the structural minimum in thechannel X, where the phase of recombination changes by about�. Thus, the presence of the structural minimum will affect notonly our ability to accurately reconstruct the period of the holemotion, but also our ability to accurately record it in theharmonic spectra.

A B C

Fig. 4. Recording attosecond hole dynamics in harmonic spectra. Laser wavelength is � � 1,140 nm, intensity I � 1.7 � 1014 W/cm2, pulse duration 60 fsec. Resultsare averaged over typical angular distribution for aligned molecules, see SI. (A) Harmonic spectrum as a function of the alignment angle of the molecularensemble. (B) Hole motion along the molecular axis, recorded in the harmonic spectra. Total amplitude (square root of intensity) (black diamonds) and amplitudes(square roots of intensities) of channels X (red squares) and A (green triangles), for � 90°. Total spectrum records the relative phase between the channels bymapping it into amplitude modulations reflecting constructive (H39) and destructive (H23 and H71) interference. (C) Harmonic spectrum at � 90° normalizedby the spectrum at � 70° to accentuate the positions of dynamical minima. Bottom axis shows ‘‘pump-probe’’ delay for different harmonics (top axis). Theinterference minima at H21–23 and H71 allow one to reconstruct the hole dynamics.

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To identify channel interference over a strongly modulatedbackground, we again calibrate the results against � 10°,20°,where geometry-induced modulation of the channel X is stillpresent, but the contribution of the channel B is less and the holeis almost static. Therefore, by normalizing to � 10,20° weaccentuate the effect of the hole motion while decreasing theeffect of structural modulations. The interference minimumappears at H43 (see Fig. 5B). Searching for the second minimum,one has to take into account the sign flip of the recombinationmatrix element for channel X, turning second position of thedestructive interference around H77 into constructive (see Fig.5B). Note, that it also turns constructive interference aroundH61 into destructive. The delay between H77 and H43 shouldcorrespond to the period of hole motion and is 0.76 � 0.1 fsec.However, the relative phase between the channels X and B (Fig.5C) indicates that the first destructive minimum should appearat H39. The delay between H77 and H39 is 0.84 � 0.1 fsec, closeto the field-free period of 0.96 fsec. The appearance of theamplitude minimum at H43–H45 instead of H39 is due to the fastchange in both phase and amplitude of the recombination matrixelement for channel X as discussed above. For the two ‘‘turningpoints,’’ delayed by half-period of the hole motion, the corre-sponding Dyson orbitals are shown in Fig. 5D.

At the first glance, one might expect that for differentwavelengths � of the driving laser field the dynamical mini-mum—the position of the destructive interference between thechannels—will occur for the same time delay �* between ion-ization and recombination. Generally, this expectation is wrong.The reason is the energy dependence of the phase of recombi-nation for each channel. Indeed, for the same �*, the energy ofthe returning electron will change with �, and hence the relativephase of recombination between the two channels will alsochange with �, for the same �*, affecting the position of thedynamical minimum. This effect is particularly strong near the

structural minimum of one of the participating channels, becausethere, the phase of recombination for the channel is changingrapidly. For example, we find that at 800 nm the destructiveinterference between the channels X and B occurs not at �* �1.65 fsec but at �* � 1.2 fsec.

Reconstructed periods are close to those of the field-freemotion for several reasons. First, in both cases we monitor holedynamics with the laser field orthogonal to the motion. Second,laser-induced dynamics between the X, A, and B states of theCO2

� ion between ionization and recombination, which is in-cluded in our calculation, is weak in the IR range. Finally, thelaser-induced polarization, e.g., relative Stark shifts, which is alsoincluded in our calculation, is also small. The deviation from thefield-free periods for the hole motion perpendicular to molec-ular axis (Fig. 5D) is due to the structural minimum in thechannel X as discussed above.

We have demonstrated the possibility to control the removalof bonding (channel A) vs non-bonding (channel X) electron orfor the ionization from � vs � states (A or B channels) on CO2molecule. The control parameter is the molecular alignmentangle . We used HHG spectroscopy to visualize the control thatcan be achieved. The dynamics of hole recorded in harmonicspectra indicates and characterizes population of several ionicstates after ionization.

The potential of HHG spectroscopy is not limited to theexample considered here. Analogous technique can be appliedto study nonadiabatic multielectron dynamics (19) excited by thelaser field and determining fragmentation pathways for poly-atomic molecules subjected to strong laser field. Attoseconddynamics in the ion can also be induced by spin-orbit coupling(32). Strong-field ionization of a state with well-defined orbitalangular momentum creates coherent superposition of ionicground states with different total momenta. Mapping the evo-lution of this superposition into harmonic spectrum resolvesspin-orbit coupling in time.

A B

D C

Fig. 5. Recording and reconstructing the hole motion perpendicular to the molecular axis in the harmonic spectra. (A) Cut of the total spectral amplitude (squareroot of intensity) shown in Fig. 4A at � 0° (black diamonds) and amplitudes of channels X (red squares) and B (blue circles). (B) Calibrating the harmonic signalat � 0° against that for nearly ‘‘static’’ hole. Calibration is done by normalizing the total signal at � 0° by the signals at � 10,20°; cuts are taken from Fig.4A. (C) Shown is cosine of relative phase between the channels X and B. (D) Shown is the evolution of the coherent superposition of the Dyson orbitalscorresponding to X and B states of the ion. Two snapshots, shown in (i) red and orange and (ii) light and dark blue are separated by half-period and correspondto the two turning points of the hole motion. The snapshots are shown at the 40% level of the electron density.

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Another example is hole dynamics induced by one-photonionization with a single attosecond XUV pulse, phase-locked tothe oscillation of the IR laser field. The liberated electron,oscillating in the laser field, can still recombine with the hole,emitting harmonics. As long as the duration of the UV pulse isless than quarter-cycle of the IR, time resolution is determinedby the principle shown in Fig. 3. Controlling polarization of theionizing XUV pulse relative to the molecular axis and to thepolarization of the IR field offers the possibility of controllinghole excitation. Delaying ionizing pulse relative to the phase ofthe oscillations of the IR field provides additional control knobin decoding attosecond-resolved signal. Such arrangementshould provide a versatile setup for attosecond spectroscopy ofmultielectron dynamics.

MethodsHarmonic generation results from time-dependent polarization D(t) inducedin a molecule by the incident laser pulse. Generalizing (33), we write D(t) as:

D� t ,����j,ti

a ion,j� t i,��aprop,j� t , t i,�� �

� �NT�N�� t ,��| d|A� j

�N � 1�� t ,���C, j� t ,k� t� ,��� [1]

This general expression provides a convenient framework, which allows oneto go beyond such approximations as the single-active electron approxima-tion and the strong-field approximation. N-electron wave function �NT

(N) de-

scribes evolution of the neutral molecule during the laser pulse, includingdepletion by ionization. �j

(N�1) describes multielectron dynamics of the ionstarting in state j at the moment of ionization ti � ti(t), until recombination att. Continuum electron is described by the scattering state �C,j, correlated tothe state of the ion �j

(N�1) and characterized by the (asymptotic) kineticmomentum k(t) acquired from the laser field (see SI). A is antisymmetrizationoperator. Angle characterizes molecular alignment. We also include auto-correlation function associated with nuclear motion for each channel (29, 36)(see SI). Note, that Eq.1 obtains by the evaluation of the integral over ioniza-tion times using the saddle point method, and thus includes the sum over allsaddle points ti. The applicability of the method relies on the large actionaccumulated by the continuum electron in the strong laser field (34). In ourcalculations we include only short trajectories to model experimental condi-tions, thus a single ti corresponds to each t.

The amplitude of each channel includes ion-state-specific subcycle ioniza-tion amplitude aion,j at the moment ti and the propagation amplitude aprop,j

between ti and t. Different ionic states can be populated by ionization orexcited by non-adiabatic transitions (35) in the ion between ionization andrecombination, and we include both mechanisms. The bound states, Starkshifts, dipole couplings between different ionic states are calculated usingquantum chemistry methods (see SI). Harmonics are given by the Fouriertransform of Eq 1.

ACKNOWLEDGMENTS. We appreciate stimulating discussions with Paul Cor-kum, David Villeneuve, Michael Spanner, Albert Stolow, Aephraim Steinberg,and Jon Marangos. This work was partially supported by a Natural Sciencesand Engineering Research Council Special Research Opportunity grant. O.S.acknowledges a Leibniz Senatsausschuss Wettbewerb award, M.I. acknowl-edges support from the Alexander von Humboldt Foundation.

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