Strong-Field Control and Spectroscopy of Attosecond Electron-Hole Dynamics in Molecules

Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules

Olga 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 Libe´ration, 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- electron from either ⌸ vs. ⌺ or bonding vs. nonbonding orbital. mined by shared electrons in valence shells. We show how one can Increasing laser wavelength ␭ suppresses runaway nonadiabatic selectively remove a valence electron from either ⌸ vs. ⌺ or excitations (19) while keeping ionization channels intact. Con- bonding vs. nonbonding orbital by applying an intense infrared trol over the initial electronic excitation is a gateway to control- laser field to an ensemble of aligned molecules. In molecules, such ling molecular fragmentation that follows. ionization often induces multielectron dynamics on the attosecond Because tunnel ionization can create electronic excitations, time scale. Ionizing laser field also allows one to record and the hole left in a molecule will move on attosecond time-scale reconstruct these dynamics with attosecond temporal and sub- determined by inverse energy spacing between excited electronic Ångstrom spatial resolution. Reconstruction relies on monitoring states, ␶ Ϸប/⌬E. How can one image such motion? Using and controlling high-frequency emission produced when the lib- attosecond probe pulses is problematic: necessarily high carrier erated electron recombines with the valence shell hole created by frequency of such pulses interacts with core rather than valence ionization. electrons. The electron current also does not record hole dynamics, see below. However, when ionization occurs in a laser high harmonic generation ͉ multielectron dynamics ͉ field, the liberated electron does not immediately leave the strong-field ionization vicinity of the parent ion: Oscillations in the laser electric field can bring the electron back (21). We show how the hole dynamics lectrons in valence molecular shells hold keys to molecular is recorded in the spectrum of the radiation emitted when the Estructures and properties. This article focuses on finding liberated electron recombines with the hole left in the molecule. ways to control and record their dynamics with attosecond (1 asec ϭ 10Ϫ18 sec) temporal resolution. Imaging structures and Results dynamics at different temporal and spatial scales is a major We first consider control of optical tunneling. Consider an direction of modern science that encompasses physics, chemistry example of a CO2 molecule interacting with intense infrared and biology (1). Electrons in atoms, molecules and solids move laser field. The critical factors in strong-field tunnel ionization on attosecond timescale; resolving and controlling their dynam- are the ionization potential Ip and the geometry of the ionizing ics are goals of attosecond and strong-field science (2–6). orbital (22, 23). Removal of an electron, which leaves the ion in A natural route in manipulating valence electrons is to apply an electronic state j, is visualized using the Dyson orbital ⌿D,j ϭ ͌ ͗⌿(NϪ1)⌿(N)͘ ⌿(N) laser pulses with electric fields comparable with the electrostatic N NT . Here, NT is the N-electron wave function of j ⌿(NϪ1) forces binding these electrons in molecules. Combination of the neutral molecule and j describes the ion in the state j. 2 pulse-shaping techniques and adaptive (learning) algorithms Dyson orbitals for the ground state X˜ ⌸g (j ϭ X˜) and the first ˜ 2⌸ ϭ ˜ ˜2⌺ϩ ϭ ˜ (7–16) turn intense ultrashort laser pulses into ‘‘photonic re- two excited states A u (j A) and B u (j B) of the CO2 agents’’ (8), allowing one to steer molecular dynamics toward a molecule are shown in Fig. 1 A and C. Because tunnel ionization desired outcome by tailoring oscillations of the laser electric is exponentially sensitive to Ip, one would expect that only field. Strong-field techniques find applications in controlling ground state of the ion is excited after ionization, i.e., X˜ is the unimolecular reactions (9, 14), nonadiabatic coupling of elec- only participating ionization channel. tronic and nuclear motion (15), suppression of decoherence (16, However, the nodal structure of ⌿D,X˜ (Fig. 1A) suppresses 17), etc. From the fundamental standpoint, one of intriguing ionization for molecules aligned parallel or perpendicular to the challenges lies in understanding and taming the complexity of laser polarization. Indeed, opposite phases of the adjacent lobes strong-field dynamics, where multiple routes to various final in ⌿D,X˜ lead to the destructive interference of the ionization states are open simultaneously and multiple control mechanisms currents for molecular alignment angles around ⌰ϭ0° and ⌰ϭ operate at the same time. 90°, as observed in the experiment (24). Due to the interference Although nuclei in molecules move on the femtosecond time suppression of ionization in the X˜-channel, ionization channels scale, attosecond component of the electronic response often with higher Ip, (e.g., A˜, B˜) become important. For the ion left in plays crucial role (see, e.g., ref. 18). For example, in polyatomic 2 the first excited state A˜ ⌸u (Fig. 1B), the same argument shows molecules electronic excitations in intense infrared laser fields that ionization is suppressed near ⌰ϭ0° but enhanced near ⌰ϭ are created via laser-induced nonadiabatic multielectron (NME) 90°, perpendicular to the molecular axis. In the latter case, there transitions (14, 19, 20). These transitions occur on the sublaser is no destructive interference between the adjacent lobes with cycle time scale and determine femtosecond dynamics that the opposite phases, because only the lobe near the ‘‘down- follows. Although laser-induced NME transitions open multiple excitation channels (14) and lead to molecular breakup into a multitude of fragments (15, 19, 20), in practice, they can be hard Author contributions: O.S. and M.Y.I. designed research; O.S. and S.P. performed research; to control. We show that tunnel ionization in low-frequency laser Y.M. and N.D. analyzed data; and O.S. and M.Y.I. wrote the paper. fields is an alternative way of creating a set of electronic The authors declare no conflict of interest. excitations in the ion, which can be controlled by changing 1To whom correspondence should be addressed. E-mail: [email protected]. molecular alignment relative to laser polarization. The control This article contains supporting information online at www.pnas.org/cgi/content/full/ mechanism uses symmetries of molecular orbitals to remove an 0907434106/DCSupplemental.

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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. The orbitals 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 of molecular alignment angle ⌰: green and blue curves show control over electron detachment from ⌸ vs. ⌺ orbital, the red curve shows control over electron detachment from bonding (channel A˜ ) vs. nonbonding (channel-X˜ ) orbital. The plotted value is the ionization amplitude (square root of probability) at the peak of the electric ﬁeld, which is the relevant quantity for the process of harmonic generation discussed later. The laser wavelength is ␭ ϭ 1,140 nm, and the intensity I ϭ 2.2 ϫ 1014 W/cm2.

bending’’ part of the ion potential will contribute into ionization. (see Methods). Continuum wave packets ⌽C,X˜(t) and ⌽C,A˜(t) are ˜2⌺ϩ For the ion left in the second excited state B u , the Dyson normalized to unity. Due to the entanglement between the ion orbital (Fig. 1C) shows that ionization is favored along the and the electron (the hole and the electron), the dynamics of the PHYSICS molecular axis. Our calculations (see Methods) confirm these ionic subsystem is undetermined until the measurement of this expectations. Fig. 1D shows the degree of control for the removal dynamics is specified (25, 26). of bonding (channel A˜) vs. nonbonding (channel X˜) electron or Consider first an example of the measurement, which ‘‘pre- for the ionization from ⌸ vs. ⌺ states (A˜ or B˜ channels). The pares’’ the electron wave packet in the ion. One way of obtaining control parameter is the molecular alignment angle ⌰. Although such wave packet is to project the continuum wave packets X˜ channel dominates ionization for all angles, the control is very ⌽C,X˜(t) and ⌽C,A˜(t) on the same continuum state, e.g., the state ⌿(NϪ1) ϭ͗ ˆ⌿(N) ͘ϭ substantial, with the ratio of amplitudes (square roots of ion- having a momentum p at the detector: I (t) p A eI (t) ͗ ⌽ ͘⌿(NϪ1) ϩ ͗ ⌽ ͘⌿(NϪ1) ization probabilities) for various ionization channels changing by aX˜(t) p C,X˜(t) X˜ aA˜(t) p C,A˜(t) A˜ . Here, we have over an order of magnitude. postulated that there is no exchange between the continuum Ionization leaves a hole in the valence shell of the molecule. electron at the detector and the electrons remaining in the ⌿(NϪ1) If several electronic states of the cation are excited, will the hole ion. I (t) represents a coherent superposition of the ionic ⌰ϭ ⌿(NϪ1) ⌿(NϪ1) ͗ ⌽ ͘ move, and how? Consider alignment angles near 90°, when states X˜ , A˜ , provided that both p C,X˜(t) 0 and the two ionization channels (A˜ and X˜) are significant, see Fig. 1D. ͗p⌽C,A˜(t)͘0. Note, that for the channel X˜ no electrons escape Note that probability of ionization for X˜-channel is strongly with p exactly perpendicular to the molecular axis, and hence, for suppressed for ⌰ϭ90°, but not equal to zero. Destructive this choice of p͘, the projection ͗p⌽C,X˜͘ϭ0, and the hole is ⌿(NϪ1) interference occurs only for the electrons ejected along the laser ‘‘static,’’ with only A˜ present in the superposition. field, exactly perpendicular to the molecular axis. Although such The hole wave packet can be visualized via the overlap of the electrons dominate the ionization current from each lobe, there ionic wave packet with the neutral ground state: ⌿hole(t) ϭ ͗⌿(N)⌿(NϪ1) ͘ is a certain amount of electrons ejected in all other directions, NT I (t) . Phases of aj(t) will evolve, reflecting the hole i.e., at angles different from 90° with respect to the molecular dynamics. The Dyson orbitals for the ‘‘plus’’ and ‘‘minus’’ ⌿(Ϯ) ϭ͗⌿(N)⌿(NϪ1) Ϯ⌿(NϪ1)͘ axis. Such electrons are responsible for the ionization probability superpositions of the two states, D NT X˜ A˜ when the laser field is orthogonal to the molecular axis. The are shown in Fig. 2. The hole moves along the molecular axis with presence of these electrons is a consequence of the Heisenberg’s a period of 1.18 fsec, determined by the energy splitting between uncertainty principle and the confinement of the continuum the ionic states X˜ and A˜, ⌬EX˜,A˜ ϭ 3.5 eV. wave packet in the direction perpendicular to the laser field. But where does this motion begin? Will the hole start on the Ionization creates an entangled electron-ion wave packet left side of the molecule and move to the right, or vice versa? For ˆ⌿(N) ˆ ⌿(N) ϭ ⌰ϭ A eI (t), where A antisymmetrizes the electrons and eI (t) a molecule aligned exactly at 90°, there is no preference ⌿(NϪ1)⌽ ϩ ⌿(NϪ1)⌽ aX˜(t) X˜ C,X˜(t) aA˜(t) A˜ C,A˜(t). Here aj(t) are complex between the two directions. What breaks the symmetry? Again, ⌿(NϪ1) amplitudes of the ionic states j populated by ionization, the answer requires the analysis of the entanglement between the evolving between ionization and recombination. If we were to parent ion and the liberated electron (see ref. 27 for a similar neglect laser-induced excitations in the ion between ionization example in the core-hole localization in N2). The symmetry is and recombination, than aj(t) ϭ ␣jexp[ϪiEj(t Ϫ ti)], where ␣j is broken by the direction of the electron escape, characterized by the complex ionization amplitude, ti is the moment of ionization the final momentum p. Its angle relative to the molecular axis

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Fig. 2. Qualitative picture of the hole dynamics: evolution of the coherent superposition 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. Two snapshots, shown in (i) red and orange and (ii) light and dark blue are separated by half-period and correspond to the two turning points of the hole Fig. 3. The origin of high temporal resolution in HHG spectroscopy. (A) motion. The snapshots are shown at the 40% level of the electron density. The Physical mechanism: Intense IR laser field liberates an electron. Within a ⌿(Ϫ) fraction of the laser cycle, the electron is returned to the parent ion by the initial conditions in the figure correspond to the ‘‘minus’’ superposition, D ϭ͗⌿(N)⌿(NϪ1) Ϫ⌿(NϪ1)͘ oscillating field. Recombination is accompanied by the emission of a harmonic NT X˜ A˜ . photon. (B) The moment of ionization ti is linked to the moment of recombination tr. Solid blue curve shows one oscillation of the laser field Elaser, straight lines connect moments of ionization ti and recombination tr.(C) The time determines the relative phase of the projections ͗p⌽C,X˜͘ and delay between ionization and recombination ⌬t ϭ tr Ϫ ti is mapped onto ͗p⌽C,A˜͘ and, hence, the direction of the hole motion. harmonic number: N␻ ϭ Ee(⌬t) ϩ Ip. Here, Ee is the energy of the returning Let the ionizing electric field point vertically downward in Fig. electron. 1 A and B, so that the electron tunnels upwards. Since the tunneling wave packets ⌽C,X˜, ⌽C,A˜ are confined transversely, the Heisenberg uncertainty relationship dictates that, for both wave states and, hence, the electronic wave packets correlated to packets, the outgoing electron can have a range of momenta with different ionization channels will not interfere. nonzero projections on the molecular axis. For the X˜ channel, Interference of different ionization channels is recorded in the the symmetry of ⌿D,X˜͘ also dictates that (i) the tunneling wave spectrum of radiation emitted when the liberated electron packet ⌽C,X˜ will have a nodal plane along the vertical direction, recombines with the hole left in the molecule, Fig. 3A. This and (ii) the projection ͗p⌽C,X˜͘ will change its sign depending on process, known as high-harmonic generation, returns the mol- the direction of the electron escape. In the tunneling limit, for ecule to the same ground state it has started from, ensuring direction of electron escape pointing into the upper left quadrant interference of different electron-hole states evolving between of Fig. 1 A and B, both ͗p⌽C,X˜͘and ͗p⌽C,A˜͘ are positive at the ionization and recombination. In the language of pump-probe ⌿(ϩ) moment of ionization (‘‘plus’’ superposition D , Fig. 2 light and spectroscopy, strong-field ionization acts as a ‘‘pump,’’ and dark blue) and the hole starts on the left side. If direction of recombination acts as a ‘‘probe’’ (28, 29). Recombination occurs electron escape points into the upper right quadrant, then at the within a fraction of the laser cycle after ionization, Fig. 3B. The moment of ionization ͗p⌽C,X˜͘Ͻ0 is negative, but ͗p⌽C,A˜͘Ͼ time of ionization is linked to the time of recombination (Fig. ⌿(Ϫ) 0 is positive (superposition D , Fig. 2, red and orange) and the 3B), the latter is mapped onto the harmonic number (Fig. 3C). hole starts on the right side. Both directions of the electron Thus, each harmonic makes a snapshot of the dynamics for a escape are equally probable. Thus, equal number of holes different ‘‘pump-probe’’ delay, providing a ‘‘frame’’ for the will move right to left and vice versa, with the electron-hole attosecond ‘‘movie.’’ entanglement and the electron momentum p selected by the This idea has already been used to track the motion of protons measurement defining the direction. on attosecond time scale (28), but extending the method to The situation changes gradually as we rotate the electric field electrons proved problematic. Indeed, in the pump-probe spec- away from ⌰ϭ90°, so that tunneling along the electric field is troscopy an important ingredient is the ability to calibrate the no longer orthogonal to the molecular axis. For the X˜ channel, pump-probe signal against that obtained without pump-induced this changes the relative number of electrons escaping toward the excitations. Unfortunately, the HHG process does not allow one upper left vs. the upper right quadrant of Figs. 1 (a). For to turn the pump off while keeping the probe on. Thus, one needs example, for ⌰Ͻ90°, more electrons will escape toward the a different calibration scheme. The key ingredient of the ap- upper right quadrant, and hence more holes will start on the right proach (28, 29) was the ability to slow protons down by replacing side. them with heavier deuterons, thus calibrating intensity modu- We now turn to attosecond spectroscopy of valence shell holes lation in the harmonic spectra and reconstructing the dynamics. as a tool for visualizing control over strong-field ionization that But how to slow down the electrons? We solve this problem by can be achieved. To record hole dynamics, which reflect the using molecular alignment to control the hole. The comparison population of several ionic states, one has to record the inter- of harmonic signals for different alignment angles simplify ference between different ionization channels with variable time interpretation of our results. delay after ionization. To record the interference, a measure- Formally, the harmonic emission results from recombination, ment must bring the system to the same final state. Only then which is described by the matrix element D(t) ϭ ͗⌿(N)ˆˆ⌿(N) ͘ different quantum pathways (here, ionization channels) will NT d A eI (t) . Emitted light is given by the Fourier trans- interfere. Thus, measuring the tunneled electrons will not do— form of D¨(t). Relative phases, accumulated by aj(t) in the wave ⌿(N) different ionization channels leave the ion in different final packet eI (t) between ionization and recombination, lead to

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Fig. 4. Recording attosecond hole dynamics in harmonic spectra. Laser wavelength is ␭ ϭ 1,140 nm, intensity I ϭ 1.7 ϫ 1014 W/cm2, pulse duration 60 fsec. Results are averaged over typical angular distribution for aligned molecules, see SI.(A) Harmonic spectrum as a function of the alignment angle of the molecular ensemble. (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 ⌰ϭ90°. Total spectrum records the relative phase between the channels by mapping it into amplitude modulations reﬂecting constructive (H39) and destructive (H23 and H71) interference. (C) Harmonic spectrum at ⌰ϭ90° normalized by the spectrum at ⌰ϭ70° to accentuate the positions of dynamical minima. Bottom axis shows ‘‘pump-probe’’ delay for different harmonics (top axis). The interference minima at H21–23 and H71 allow one to reconstruct the hole dynamics.

constructive or destructive interference of different ionization channels give comparable contributions for a broad range of channels in emission, suppressing or enhancing harmonics. This harmonics, which record channel interference. Constructive is the amplitude modulation we are looking for. Changing interference occurs around H39, whereas destructive interfer- molecular alignment, we control the relative amplitudes of the ence occurs around H71. Another destructive minimum be- channels and, hence, the modulation depth. comes visible when the signal at ⌰ϭ90° is normalized by the In addition to the phase accumulated in the ion, the relative signal at smaller angles (here, ⌰ϭ70°), Fig. 4C.As⌰ changes phase between the channels includes contributions from the from ⌰ϭ90° to ⌰ϭ70 Ϫ 60°, the channel X˜ begins to dominate oscillatory continuum motion in the laser field and the scattering the spectrum. Thus, such normalization allows one to calibrate phases encoded in the recombination matrix elements. Analysis the ‘‘pump-probe’’ signal against the ‘‘time-independent’’ back- ␻ ϾϾ Ϫ is simple for high-energy harmonic photons Nh EA˜ EX˜ ground. As ⌰ changes from ⌰ϭ90° to ⌰ϭ60°, the minima do (see supporting information (SI)). Then the relative phase not move to different harmonic order, but gradually disappear, PHYSICS ␾ Ϸ Ϫ ␶ ␶ accumulated in the ion is (EA˜ EX˜) , where is the reflecting the decreasing contribution of the channel A˜. Time channel-average time delay between ionization and recombina- delays between ionization and recombination for different har- tion. The phases of the oscillatory continuum motion differ much monics are given in Fig. 4C, bottom axis. The minima appear at less, see SI. The phases of the recombination matrix elements, H21–23 and H71. Time-delay between the frames H21–23 and when different for the two channels, affect their interference. H71 is the period of the wave-packet motion in the ion, which is Fig. 4A shows calculated harmonic spectra for CO2 at laser here equal to 1.05 Ϯ 0.12 fsec, close to the field-free period of ϭ ϫ 14 2 ␭ ϭ intensity I 1.7 10 W/cm , for pulse duration 90 fsec and 1.18 fsec. The error in the time-energy mapping is larger for low 1,140 nm (see Methods and SI). The results are averaged over harmonic numbers such as H21–23. The time delay between alignment distribution typical for modern experiments with the frames H71 and H39 gives half a period of the hole motion aligned molecular ensembles (30), and the so-called long trajec- 0.63 Ϯ 0.08 fsec, yielding the period 1.26 Ϯ 0.11 fs (see SI for the tories are filtered out, see SI. In a typical experimental setup, discussion of the harmonic order-time delay mapping and error alignment is achieved by applying a linearly polarized aligning bars). pulse, which excites molecular rotations. Polarization of the Now consider the amplitude structures near ⌰ϭ0°, see Fig. pulse defines the axis of the aligned molecular ensemble. Har- ˜ monics are generated by the second linearly polarized pulse, with 4A and Fig. 5A. Here, ionization creates a wave packet of X and ⌰ B˜ states, which corresponds to a breathing motion perpendicular variable angle relative to the polarization of the aligning pulse. ⌰ϭ This is the angle ⌰ in Fig. 4A. Currently, most of the experiments to the molecular axis, see Fig. 5C. The example at 0° on HHG in molecules are done with 800 nm driving laser filed. illustrates additional difficulties in the reconstruction. The con- ˜ Using longer wavelength of the driving field allows one to record tribution of the channel X is modulated by the deep structural longer movies (number of frames (harmonics) ϰ␭3) and increase minimum around H55–H59, associated with the geometry of ⌿ time resolution (number of frames per femtosecond ϰ␭2). On the D,X˜. As a result, the relative contributions of the two channels other hand, the efficiency of harmonic generation strongly into the total signal vary significantly, obscuring their interfer- decreases for longer wavelength typically as ϰ␭Ϫ5 Ϫ ␭Ϫ6 (31). ence. Furthermore, the way the period of the hole motion is Here, we use the wavelength ␭ ϭ 1,140 nm, which is just long recorded in the harmonic spectrum depends on the relative enough to observe the whole period of the hole motion by phase of recombination between the two channels. Accurate looking only at short trajectories. Note that observing both short recording of this period in the harmonic spectrum requires that and long trajectories simultaneously almost doubles the length of this phase does not strongly depend on energy. This requirement the movie for a given wavelength. is violated in the vicinity of the structural minimum in the channel X˜, where the phase of recombination changes by about Discussion ␲. Thus, the presence of the structural minimum will affect not Consider first the structures around ⌰ϭ90°, marked with only our ability to accurately reconstruct the period of the hole arrows. Fig. 4B shows amplitudes (square roots of intensities) of motion, but also our ability to accurately record it in the the total signal and of the channels A˜ and X˜, for ⌰ϭ90°. The harmonic spectra.

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Fig. 5. Recording and reconstructing the hole motion perpendicular to the molecular axis in the harmonic spectra. (A) Cut of the total spectral amplitude (square root of intensity) shown in Fig. 4A at ⌰ϭ0° (black diamonds) and amplitudes of channels X˜ (red squares) and B˜ (blue circles). (B) Calibrating the harmonic signal at ⌰ϭ0° against that for nearly ‘‘static’’ hole. Calibration is done by normalizing the total signal at ⌰ϭ0° by the signals at ⌰ϭ10,20°; 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 orbitals corresponding 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 correspond to the two turning points of the hole motion. The snapshots are shown at the 40% level of the electron density.

To identify channel interference over a strongly modulated structural minimum of one of the participating channels, because background, we again calibrate the results against ⌰ϭ10°,20°, there, the phase of recombination for the channel is changing where geometry-induced modulation of the channel X˜ is still rapidly. For example, we find that at 800 nm the destructive present, but the contribution of the channel B˜ is less and the hole interference between the channels X˜ and B˜ occurs not at ␶* ϭ is almost static. Therefore, by normalizing to ⌰ϭ10,20° we 1.65 fsec but at ␶* ϭ 1.2 fsec. accentuate the effect of the hole motion while decreasing the Reconstructed periods are close to those of the field-free effect of structural modulations. The interference minimum motion for several reasons. First, in both cases we monitor hole appears at H43 (see Fig. 5B). Searching for the second minimum, dynamics with the laser field orthogonal to the motion. Second, laser-induced dynamics between the X˜, A˜, and B˜ states of the one has to take into account the sign flip of the recombination ϩ matrix element for channel X˜, turning second position of the CO2 ion between ionization and recombination, which is in- destructive interference around H77 into constructive (see Fig. cluded in our calculation, is weak in the IR range. Finally, the 5B). Note, that it also turns constructive interference around laser-induced polarization, e.g., relative Stark shifts, which is also H61 into destructive. The delay between H77 and H43 should included in our calculation, is also small. The deviation from the correspond to the period of hole motion and is 0.76 Ϯ 0.1 fsec. field-free periods for the hole motion perpendicular to molec- However, the relative phase between the channels X˜ and B˜ (Fig. ular axis (Fig. 5D) is due to the structural minimum in the channel X˜ as discussed above. 5C) indicates that the first destructive minimum should appear We have demonstrated the possibility to control the removal at H39. The delay between H77 and H39 is 0.84 Ϯ 0.1 fsec, close of bonding (channel A˜) vs non-bonding (channel X˜) electron or to the field-free period of 0.96 fsec. The appearance of the for the ionization from ⌸ vs ⌺ states (A˜ or B˜ channels) on CO2 amplitude minimum at H43–H45 instead of H39 is due to the fast molecule. The control parameter is the molecular alignment change in both phase and amplitude of the recombination matrix ⌰ ˜ angle . We used HHG spectroscopy to visualize the control that element for channel X as discussed above. For the two ‘‘turning can be achieved. The dynamics of hole recorded in harmonic points,’’ delayed by half-period of the hole motion, the corre- spectra indicates and characterizes population of several ionic sponding Dyson orbitals are shown in Fig. 5D. states after ionization. At the first glance, one might expect that for different The potential of HHG spectroscopy is not limited to the ␭ wavelengths of the driving laser field the dynamical mini- example considered here. Analogous technique can be applied mum—the position of the destructive interference between the to study nonadiabatic multielectron dynamics (19) excited by the channels—will occur for the same time delay ␶* between ion- laser field and determining fragmentation pathways for poly- ization and recombination. Generally, this expectation is wrong. atomic molecules subjected to strong laser field. Attosecond The reason is the energy dependence of the phase of recombi- dynamics in the ion can also be induced by spin-orbit coupling nation for each channel. Indeed, for the same ␶*, the energy of (32). Strong-field ionization of a state with well-defined orbital the returning electron will change with ␭, and hence the relative angular momentum creates coherent superposition of ionic phase of recombination between the two channels will also ground states with different total momenta. Mapping the evo- change with ␭, for the same ␶*, affecting the position of the lution of this superposition into harmonic spectrum resolves dynamical minimum. This effect is particularly strong near the spin-orbit coupling in time.

16560 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0907434106 Smirnova et al. Downloaded by guest on September 26, 2021 Another example is hole dynamics induced by one-photon scribes evolution of the neutral molecule during the laser pulse, including ⌿(NϪ1) ionization with a single attosecond XUV pulse, phase-locked to depletion by ionization. j describes multielectron dynamics of the ion ϭ the oscillation of the IR laser field. The liberated electron, starting in state j at the moment of ionization ti ti(t), until recombination at t. Continuum electron is described by the scattering state ⌽C,j, correlated to oscillating in the laser field, can still recombine with the hole, ⌿(NϪ1) emitting harmonics. As long as the duration of the UV pulse is the state of the ion j and characterized by the (asymptotic) kinetic momentum k(t) acquired from the laser field (see SI). Aˆ is antisymmetrization less than quarter-cycle of the IR, time resolution is determined operator. Angle ⌰ characterizes molecular alignment. We also include auto- by the principle shown in Fig. 3. Controlling polarization of the correlation function associated with nuclear motion for each channel (29, 36) ionizing XUV pulse relative to the molecular axis and to the (see SI). Note, that Eq.1 obtains by the evaluation of the integral over ioniza- polarization of the IR field offers the possibility of controlling tion times using the saddle point method, and thus includes the sum over all hole excitation. Delaying ionizing pulse relative to the phase of saddle points ti. The applicability of the method relies on the large action the oscillations of the IR field provides additional control knob accumulated by the continuum electron in the strong laser field (34). In our in decoding attosecond-resolved signal. Such arrangement calculations we include only short trajectories to model experimental condi- should provide a versatile setup for attosecond spectroscopy of tions, thus a single ti corresponds to each t. multielectron dynamics. The amplitude of each channel includes ion-state-specific subcycle ionization amplitude aion,j at the moment ti and the propagation amplitude aprop,j Methods between ti and t. Different ionic states can be populated by ionization or excited by non-adiabatic transitions (35) in the ion between ionization and Harmonic generation results from time-dependent polarization D(t) induced recombination, and we include both mechanisms. The bound states, Stark in a molecule by the incident laser pulse. Generalizing (33), we write D(t) as: shifts, dipole couplings between different ionic states are calculated using ͑ ⍜͒ϰ͸ ͓ ⍜͔ ͓ ⍜͔ ϫ quantum chemistry methods (see SI). Harmonics are given by the Fourier D t, aion,j ti, aprop,j t,ti, transform of Eq 1. j,ti ϫ ͗⌽͑N͒͑ ⍜͒ ˆ ˆ ⌽͑N Ϫ 1͒͑ ⍜͒⌽ ͑ ͑ ͒ ⍜͒͘ ACKNOWLEDGMENTS. We appreciate stimulating discussions with Paul Cor- NT t, |d|A j t, C, j t,k t , [1] kum, David Villeneuve, Michael Spanner, Albert Stolow, Aephraim Steinberg, and Jon Marangos. This work was partially supported by a Natural Sciences This general expression provides a convenient framework, which allows one and Engineering Research Council Special Research Opportunity grant. O.S. to go beyond such approximations as the single-active electron approxima- acknowledges a Leibniz Senatsausschuss Wettbewerb award, M.I. acknowl- ⌿(N) tion and the strong-field approximation. N-electron wave function NT de- edges support from the Alexander von Humboldt Foundation.

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