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

September 29, 2009
106 (39) 16556-16561

Abstract

Molecular structures, dynamics and chemical properties are determined by shared electrons in valence shells. We show how one can selectively remove a valence electron from either Π vs. Σ or bonding vs. nonbonding orbital by applying an intense infrared laser field to an ensemble of aligned molecules. In molecules, such ionization often induces multielectron dynamics on the attosecond time scale. Ionizing laser field also allows one to record and reconstruct these dynamics with attosecond temporal and sub-Ångstrom spatial resolution. Reconstruction relies on monitoring and controlling high-frequency emission produced when the liberated electron recombines with the valence shell hole created by ionization.
Electrons in valence molecular shells hold keys to molecular structures and properties. This article focuses on finding ways to control and record their dynamics with attosecond (1 asec = 10−18 sec) temporal resolution. Imaging structures and dynamics at different temporal and spatial scales is a major direction of modern science that encompasses physics, chemistry and biology (1). Electrons in atoms, molecules and solids move on attosecond timescale; resolving and controlling their dynamics are goals of attosecond and strong-field science (26).
A natural route in manipulating valence electrons is to apply laser pulses with electric fields comparable with the electrostatic forces binding these electrons in molecules. Combination of pulse-shaping techniques and adaptive (learning) algorithms (716) turn intense ultrashort laser pulses into “photonic reagents” (8), allowing one to steer molecular dynamics toward a desired outcome by tailoring oscillations of the laser electric field. Strong-field techniques find applications in controlling unimolecular reactions (9, 14), nonadiabatic coupling of electronic and nuclear motion (15), suppression of decoherence (16, 17), etc. From the fundamental standpoint, one of intriguing challenges lies in understanding and taming the complexity of strong-field dynamics, where multiple routes to various final states are open simultaneously and multiple control mechanisms operate at the same time.
Although nuclei in molecules move on the femtosecond time scale, attosecond component of the electronic response often plays crucial role (see, e.g., ref. 18). For example, in polyatomic molecules electronic excitations in intense infrared laser fields are created via laser-induced nonadiabatic multielectron (NME) transitions (14, 19, 20). These transitions occur on the sublaser cycle time scale and determine femtosecond dynamics that 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 to control. We show that tunnel ionization in low-frequency laser fields is an alternative way of creating a set of electronic excitations in the ion, which can be controlled by changing molecular alignment relative to laser polarization. The control mechanism uses symmetries of molecular orbitals to remove an electron from either Π vs. Σ or bonding vs. nonbonding orbital. Increasing laser wavelength λ suppresses runaway nonadiabatic excitations (19) while keeping ionization channels intact. Control over the initial electronic excitation is a gateway to controlling molecular fragmentation that follows.
Because tunnel ionization can create electronic excitations, the hole left in a molecule will move on attosecond time-scale determined by inverse energy spacing between excited electronic states, τ ≈ ℏ/ΔE. How can one image such motion? Using attosecond probe pulses is problematic: necessarily high carrier frequency of such pulses interacts with core rather than valence electrons. The electron current also does not record hole dynamics, see below. However, when ionization occurs in a laser field, the liberated electron does not immediately leave the vicinity of the parent ion: Oscillations in the laser electric field can bring the electron back (21). We show how the hole dynamics is recorded in the spectrum of the radiation emitted when the liberated electron recombines with the hole left in the molecule.

Results

We first consider control of optical tunneling. Consider an example of a CO2 molecule interacting with intense infrared laser field. The critical factors in strong-field tunnel ionization are the ionization potential Ip and the geometry of the ionizing orbital (22, 23). Removal of an electron, which leaves the ion in an electronic state j, is visualized using the Dyson orbital ΨD,j = N〈Ψj(N−1)∣ΨNT(N)〉. Here, ΨNT(N) is the N-electron wave function of the neutral molecule and Ψj(N−1) describes the ion in the state j. Dyson orbitals for the ground state 2Πg (j = ) and the first two excited states Ã2Πu (j = Ã) and 2Σu+ (j = ) of the CO2 molecule are shown in Fig. 1 A and C. Because tunnel ionization is exponentially sensitive to Ip, one would expect that only ground state of the ion is excited after ionization, i.e., is the only participating ionization channel.
Fig. 1.
Dyson orbitals corresponding to different ionization channels of CO2. (A) . (B) Ã. (C) . 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 Θ: green and blue curves show control over electron detachment from Π vs. Σ orbital, the red curve shows control over electron detachment from bonding (channel Ã) vs. nonbonding (channel-) 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 λ = 1,140 nm, and the intensity I = 2.2 × 1014 W/cm2.
However, the nodal structure of ΨD,X̃ (Fig. 1A) suppresses ionization for molecules aligned parallel or perpendicular to the laser polarization. Indeed, opposite phases of the adjacent lobes in ΨD,X̃ lead to the destructive interference of the ionization currents for molecular alignment angles around Θ = 0° and Θ = 90°, as observed in the experiment (24). Due to the interference suppression of ionization in the -channel, ionization channels with higher Ip, (e.g., Ã, ) become important. For the ion left in the first excited state Ã2Πu (Fig. 1B), the same argument shows that ionization is suppressed near Θ = 0° but enhanced near Θ = 90°, perpendicular to the molecular axis. In the latter case, there is no destructive interference between the adjacent lobes with the opposite phases, because only the lobe near the “down-bending” part of the ion potential will contribute into ionization. For the ion left in the second excited state 2Σu+, the Dyson orbital (Fig. 1C) shows that ionization is favored along the molecular axis. Our calculations (see Methods) confirm these expectations. Fig. 1D shows the degree of control for the removal of bonding (channel Ã) vs. nonbonding (channel ) electron or for the ionization from Π vs. Σ states (Ã or channels). The control parameter is the molecular alignment angle Θ. Although channel dominates ionization for all angles, the control is very substantial, with the ratio of amplitudes (square roots of ionization probabilities) for various ionization channels changing by over 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 hole move, and how? Consider alignment angles near Θ = 90°, when the two ionization channels (Ã and ) are significant, see Fig. 1D. Note that probability of ionization for -channel is strongly suppressed for Θ = 90°, but not equal to zero. Destructive interference occurs only for the electrons ejected along the laser field, exactly perpendicular to the molecular axis. Although such electrons dominate the ionization current from each lobe, there is a certain amount of electrons ejected in all other directions, i.e., at angles different from 90° with respect to the molecular axis. Such electrons are responsible for the ionization probability when the laser field is orthogonal to the molecular axis. The presence of these electrons is a consequence of the Heisenberg's uncertainty principle and the confinement of the continuum wave packet in the direction perpendicular to the laser field.
Ionization creates an entangled electron-ion wave packet ÂΨeI(N)(t), where  antisymmetrizes the electrons and ΨeI(N)(t) = a(t(N−1)ΦC,X̃(t) + aÃ(t) ΨÃ(N−1)ΦC(t). Here aj(t) are complex amplitudes of the ionic states Ψj(N−1)populated by ionization, evolving between ionization and recombination. If we were to neglect laser-induced excitations in the ion between ionization and recombination, than aj(t) = αjexp[−iEj(tti)], where αj is the complex ionization amplitude, ti is the moment of ionization (see Methods). Continuum wave packets ΦC,X̃(t) and ΦC(t) are normalized to unity. Due to the entanglement between the ion and the electron (the hole and the electron), the dynamics of the ionic subsystem is undetermined until the measurement of this dynamics is specified (25, 26).
Consider first an example of the measurement, which “prepares” the electron wave packet in the ion. One way of obtaining such wave packet is to project the continuum wave packets ΦC,X̃(t) and ΦC(t) on the same continuum state, e.g., the state having a momentum p at the detector: ΨI(N−1)(t) = 〈p∣ÂΨeI(N)(t)〉 = a(t)〈p∣ΦC,X̃(t)〉Ψ(N−1) + aÃ(t) 〈p∣ΦC(t)〉ΨÃ(N−1). Here, we have postulated that there is no exchange between the continuum electron at the detector and the electrons remaining in the ion.ΨI(N−1)(t) represents a coherent superposition of the ionic states Ψ(N−1), ΨÃ(N−1), provided that both 〈p∣ΦC,X̃(t)〉 ≠ 0 and 〈p∣ΦC(t)〉 ≠ 0. Note, that for the channel no electrons escape with p exactly perpendicular to the molecular axis, and hence, for this choice of ∣p〉, the projection 〈p∣ΦC,X̃〉 = 0, and the hole is “static,” with only ΨÃ(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: Ψhole(t) = 〈ΨNT(N)∣ΨI(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, ΨD(±) = 〈ΨNT(N)∣Ψ(N−1) ± ΨÃ(N−1)〉 are shown in Fig. 2. The hole moves along the molecular axis with a period of 1.18 fsec, determined by the energy splitting between the ionic states and Ã, ΔE = 3.5 eV.
Fig. 2.
Qualitative picture of the hole dynamics: evolution of the coherent superposition of the Dyson orbitals corresponding to and à 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 motion. The snapshots are shown at the 40% level of the electron density. The initial conditions in the figure correspond to the “minus” superposition, ΨD(−) = 〈ΨNT(N)∣Ψ(N−1) − ΨÃ(N−1)〉.
But where does this motion begin? Will the hole start on the left side of the molecule and move to the right, or vice versa? For a molecule aligned exactly at Θ = 90°, there is no preference between the two directions. What breaks the symmetry? Again, the answer requires the analysis of the entanglement between the parent ion and the liberated electron (see ref. 27 for a similar example in the core-hole localization in N2). The symmetry is broken by the direction of the electron escape, characterized by the final momentum p. Its angle relative to the molecular axis determines the relative phase of the projections 〈p∣ΦC,X̃〉 and 〈p∣ΦC〉 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 the tunneling wave packets ΦC,X̃, ΦC are confined transversely, the Heisenberg uncertainty relationship dictates that, for both wave packets, the outgoing electron can have a range of momenta with nonzero projections on the molecular axis. For the channel, the symmetry of ∣ΨD,X̃〉 also dictates that (i) the tunneling wave packet ΦC,X̃ will have a nodal plane along the vertical direction, and (ii) the projection 〈p∣ΦC,X̃〉 will change its sign depending on the direction of the electron escape. In the tunneling limit, for direction of electron escape pointing into the upper left quadrant of Fig. 1 A and B, both 〈p∣ΦC,X̃〉and 〈p∣ΦC〉 are positive at the moment of ionization (“plus” superposition ΨD(+), Fig. 2 light and dark blue) and the hole starts on the left side. If direction of electron escape points into the upper right quadrant, then at the moment of ionization 〈p∣ΦC,X̃〉 < 0 is negative, but 〈p∣ΦC〉 > 0 is positive (superposition ΨD(−), Fig. 2, red and orange) and the hole starts on the right side. Both directions of the electron escape are equally probable. Thus, equal number of holes will move right to left and vice versa, with the electron-hole entanglement and the electron momentum p selected by the measurement defining the direction.
The situation changes gradually as we rotate the electric field away from Θ = 90°, so that tunneling along the electric field is no longer orthogonal to the molecular axis. For the channel, this changes the relative number of electrons escaping toward the upper left vs. the upper right quadrant of Figs. 1 (a). For example, for Θ < 90°, more electrons will escape toward the upper right quadrant, and hence more holes will start on the right side.
We now turn to attosecond spectroscopy of valence shell holes as a tool for visualizing control over strong-field ionization that can be achieved. To record hole dynamics, which reflect the population of several ionic states, one has to record the interference between different ionization channels with variable time delay after ionization. To record the interference, a measurement must bring the system to the same final state. Only then different quantum pathways (here, ionization channels) will interfere. 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 to different ionization channels will not interfere.
Interference of different ionization channels is recorded in the spectrum of radiation emitted when the liberated electron recombines with the hole left in the molecule, Fig. 3A. This process, known as high-harmonic generation, returns the molecule to the same ground state it has started from, ensuring interference of different electron-hole states evolving between ionization and recombination. In the language of pump-probe spectroscopy, strong-field ionization acts as a “pump,” and recombination acts as a “probe” (28, 29). Recombination occurs within a fraction of the laser cycle after ionization, Fig. 3B. The time 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 a different “pump-probe” delay, providing a “frame” for the attosecond “movie.”
Fig. 3.
The origin of high temporal resolution in HHG spectroscopy. (A) Physical mechanism: Intense IR laser field liberates an electron. Within a fraction of the laser cycle, the electron is returned to the parent ion by the oscillating field. Recombination is accompanied by the emission of a harmonic 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 delay between ionization and recombination Δt = trti is mapped onto harmonic number: Nω = Eet) + Ip. Here, Ee is the energy of the returning electron.
This idea has already been used to track the motion of protons on attosecond time scale (28), but extending the method to electrons proved problematic. Indeed, in the pump-probe spectroscopy an important ingredient is the ability to calibrate the pump-probe signal against that obtained without pump-induced excitations. Unfortunately, the HHG process does not allow one to turn the pump off while keeping the probe on. Thus, one needs a different calibration scheme. The key ingredient of the approach (28, 29) was the ability to slow protons down by replacing them with heavier deuterons, thus calibrating intensity modulation in the harmonic spectra and reconstructing the dynamics. But how to slow down the electrons? We solve this problem by using molecular alignment to control the hole. The comparison of harmonic signals for different alignment angles simplify interpretation of our results.
Formally, the harmonic emission results from recombination, which is described by the matrix element D(t) = 〈ΨNT(N)∣ÂΨeI(N)(t)〉. Emitted light is given by the Fourier transform of (t). Relative phases, accumulated by aj(t) in the wave packet ΨeI(N)(t) between ionization and recombination, lead to constructive or destructive interference of different ionization channels in emission, suppressing or enhancing harmonics. This is the amplitude modulation we are looking for. Changing molecular alignment, we control the relative amplitudes of the channels and, hence, the modulation depth.
In addition to the phase accumulated in the ion, the relative phase between the channels includes contributions from the oscillatory continuum motion in the laser field and the scattering phases encoded in the recombination matrix elements. Analysis is simple for high-energy harmonic photons Nhω ≫ EÃE (see supporting information (SI)). Then the relative phase accumulated in the ion is φ ≈ (EÃE)τ, where τ is the channel-average time delay between ionization and recombination. The phases of the oscillatory continuum motion differ much less, 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 laser intensity I = 1.7 × 1014W/cm2, for pulse duration 90 fsec and λ = 1,140 nm (see Methods and SI). The results are averaged over alignment distribution typical for modern experiments with aligned molecular ensembles (30), and the so-called long trajectories are filtered out, see SI. In a typical experimental setup, alignment is achieved by applying a linearly polarized aligning pulse, which excites molecular rotations. Polarization of the pulse defines the axis of the aligned molecular ensemble. Harmonics are generated by the second linearly polarized pulse, with variable angle Θ relative to the polarization of the aligning pulse. This is the angle Θ in Fig. 4A. Currently, most of the experiments on HHG in molecules are done with 800 nm driving laser filed. Using longer wavelength of the driving field allows one to record longer movies (number of frames (harmonics) ∝λ3) and increase time resolution (number of frames per femtosecond ∝λ2). On the other hand, the efficiency of harmonic generation strongly decreases for longer wavelength typically as ∝λ−5 − λ−6 (31). Here, we use the wavelength λ = 1,140 nm, which is just long enough to observe the whole period of the hole motion by looking only at short trajectories. Note that observing both short and long trajectories simultaneously almost doubles the length of the movie for a given wavelength.
Fig. 4.
Recording attosecond hole dynamics in harmonic spectra. Laser wavelength is λ = 1,140 nm, intensity I = 1.7 × 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 (red squares) and à (green triangles), for Θ = 90°. 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 Θ = 90° normalized by the spectrum at Θ = 70° 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.

Discussion

Consider first the structures around Θ = 90°, marked with arrows. Fig. 4B shows amplitudes (square roots of intensities) of the total signal and of the channels à and , for Θ = 90°. The channels give comparable contributions for a broad range of harmonics, which record channel interference. Constructive interference occurs around H39, whereas destructive interference occurs around H71. Another destructive minimum becomes visible when the signal at Θ = 90° is normalized by the signal at smaller angles (here, Θ = 70°), Fig. 4C. As Θ changes from Θ = 90° to Θ = 70 − 60°, the channel begins to dominate the spectrum. Thus, such normalization allows one to calibrate the “pump-probe” signal against the “time-independent” background. As Θ changes from Θ = 90° to Θ = 60°, the minima do not move to different harmonic order, but gradually disappear, reflecting the decreasing contribution of the channel Ã. Time delays between ionization and recombination for different harmonics are given in Fig. 4C, bottom axis. The minima appear at H21–23 and H71. Time-delay between the frames H21–23 and H71 is the period of the wave-packet motion in the ion, which is here equal to 1.05 ± 0.12 fsec, close to the field-free period of 1.18 fsec. The error in the time-energy mapping is larger for low harmonic numbers such as H21–23. The time delay between the frames H71 and H39 gives half a period of the hole motion 0.63 ± 0.08 fsec, yielding the period 1.26 ± 0.11 fs (see SI for the discussion of the harmonic order-time delay mapping and error bars).
Now consider the amplitude structures near Θ = 0°, see Fig. 4A and Fig. 5A. Here, ionization creates a wave packet of and states, which corresponds to a breathing motion perpendicular to the molecular axis, see Fig. 5C. The example at Θ = 0° illustrates additional difficulties in the reconstruction. The contribution of the channel is modulated by the deep structural minimum around H55–H59, associated with the geometry of ΨD,X̃. As a result, the relative contributions of the two channels into the total signal vary significantly, obscuring their interference. Furthermore, the way the period of the hole motion is recorded in the harmonic spectrum depends on the relative phase of recombination between the two channels. Accurate recording of this period in the harmonic spectrum requires that this phase does not strongly depend on energy. This requirement is violated in the vicinity of the structural minimum in the channel , where the phase of recombination changes by about π. Thus, the presence of the structural minimum will affect not only our ability to accurately reconstruct the period of the hole motion, but also our ability to accurately record it in the harmonic spectra.
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 Θ = 0° (black diamonds) and amplitudes of channels (red squares) and (blue circles). (B) Calibrating the harmonic signal at Θ = 0° against that for nearly “static” hole. Calibration is done by normalizing the total signal at Θ = 0° by the signals at Θ = 10,20°; cuts are taken from Fig. 4A. (C) Shown is cosine of relative phase between the channels and . (D) Shown is the evolution of the coherent superposition of the Dyson orbitals corresponding to and 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 background, we again calibrate the results against Θ = 10°,20°, where geometry-induced modulation of the channel is still present, but the contribution of the channel is less and the hole is almost static. Therefore, by normalizing to Θ = 10,20° we accentuate the effect of the hole motion while decreasing the effect of structural modulations. The interference minimum appears at H43 (see Fig. 5B). Searching for the second minimum, one has to take into account the sign flip of the recombination matrix element for channel , turning second position of the destructive interference around H77 into constructive (see Fig. 5B). Note, that it also turns constructive interference around H61 into destructive. The delay between H77 and H43 should correspond to the period of hole motion and is 0.76 ± 0.1 fsec. However, the relative phase between the channels and (Fig. 5C) indicates that the first destructive minimum should appear at H39. The delay between H77 and H39 is 0.84 ± 0.1 fsec, close to the field-free period of 0.96 fsec. The appearance of the amplitude minimum at H43–H45 instead of H39 is due to the fast change in both phase and amplitude of the recombination matrix element for channel as discussed above. For the two “turning points,” delayed by half-period of the hole motion, the corresponding Dyson orbitals are shown in Fig. 5D.
At the first glance, one might expect that for different wavelengths λ of the driving laser field the dynamical minimum—the position of the destructive interference between the channels—will occur for the same time delay τ* between ionization and recombination. Generally, this expectation is wrong. The reason is the energy dependence of the phase of recombination for each channel. Indeed, for the same τ*, the energy of the returning electron will change with λ, and hence the relative phase of recombination between the two channels will also change with λ, for the same τ*, affecting the position of the dynamical minimum. This effect is particularly strong near the structural minimum of one of the participating channels, because there, the phase of recombination for the channel is changing rapidly. For example, we find that at 800 nm the destructive interference between the channels and occurs not at τ* = 1.65 fsec but at τ* = 1.2 fsec.
Reconstructed periods are close to those of the field-free motion for several reasons. First, in both cases we monitor hole dynamics with the laser field orthogonal to the motion. Second, laser-induced dynamics between the , Ã, and states of the CO2+ ion between ionization and recombination, which is included in our calculation, is weak in the IR range. Finally, the laser-induced polarization, e.g., relative Stark shifts, which is also included in our calculation, is also small. The deviation from the field-free periods for the hole motion perpendicular to molecular axis (Fig. 5D) is due to the structural minimum in the channel as discussed above.
We have demonstrated the possibility to control the removal of bonding (channel Ã) vs non-bonding (channel ) electron or for the ionization from Π vs Σ states (Ã or channels) on CO2 molecule. The control parameter is the molecular alignment angle Θ. We used HHG spectroscopy to visualize the control that can be achieved. The dynamics of hole recorded in harmonic spectra indicates and characterizes population of several ionic states after ionization.
The potential of HHG spectroscopy is not limited to the example considered here. Analogous technique can be applied to study nonadiabatic multielectron dynamics (19) excited by the laser field and determining fragmentation pathways for polyatomic molecules subjected to strong laser field. Attosecond dynamics in the ion can also be induced by spin-orbit coupling (32). Strong-field ionization of a state with well-defined orbital angular momentum creates coherent superposition of ionic ground states with different total momenta. Mapping the evolution of this superposition into harmonic spectrum resolves spin-orbit coupling in time.
Another example is hole dynamics induced by one-photon ionization with a single attosecond XUV pulse, phase-locked to the 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 is less than quarter-cycle of the IR, time resolution is determined by the principle shown in Fig. 3. Controlling polarization of the ionizing XUV pulse relative to the molecular axis and to the polarization of the IR field offers the possibility of controlling hole excitation. Delaying ionizing pulse relative to the phase of the oscillations of the IR field provides additional control knob in decoding attosecond-resolved signal. Such arrangement should provide a versatile setup for attosecond spectroscopy of multielectron dynamics.

Methods

Harmonic generation results from time-dependent polarization D(t) induced in a molecule by the incident laser pulse. Generalizing (33), we write D(t) as:
This general expression provides a convenient framework, which allows one to go beyond such approximations as the single-active electron approximation and the strong-field approximation. N-electron wave function ΨNT(N) describes evolution of the neutral molecule during the laser pulse, including depletion by ionization. Ψj(N−1) describes multielectron dynamics of the ion starting in state j at the moment of ionization ti = ti(t), until recombination at t. Continuum electron is described by the scattering state ΦC,j, correlated to the state of the ion Ψj(N−1) and characterized by the (asymptotic) kinetic momentum k(t) acquired from the laser field (see SI). Â is antisymmetrization operator. Angle Θ characterizes molecular alignment. We also include autocorrelation function associated with nuclear motion for each channel (29, 36) (see SI). Note, that Eq.1 obtains by the evaluation of the integral over ionization times using the saddle point method, and thus includes the sum over all saddle points ti. The applicability of the method relies on the large action accumulated by the continuum electron in the strong laser field (34). In our calculations we include only short trajectories to model experimental conditions, thus a single ti corresponds to each t.
The amplitude of each channel includes ion-state-specific subcycle ionization 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 or excited by non-adiabatic transitions (35) in the ion between ionization and recombination, and we include both mechanisms. The bound states, Stark shifts, dipole couplings between different ionic states are calculated using quantum chemistry methods (see SI). Harmonics are given by the Fourier transform of Eq 1.

Acknowledgments.

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

Supporting Information

Supporting Information (PDF)
Supporting Information

References

1
AH Zewail, 4D ultrafast electron diffraction, crystallography, and microscopy. Annu Rev Phys Chem 57, 65–103 (2006).
2
PB Corkum, F Krausz, Attosecond science. Nat Phys 3, 381–387 (2007).
3
M Drescher, et al., Time-resolved atomic inner-shell spectroscopy. Nature 419, 803–806 (2002).
4
M Uiberacker, et al., Attosecond real-time observation of electron tunnelling in atoms. Nature 446, 627–630 (2007).
5
AL Cavalieri, et al., Attosecond spectroscopy in condensed matter. Nature 449, 1029–1032 (2007).
6
M Kling, et al., Control of electron localization in molecular dissociation. Science 312, 246–249
7
R Judson, H Rabitz, Teaching lasers to control molecules. Phys Rev Lett 68, 1500–1503 (1992).
8
H Rabitz, Shaped laser pulses as reagents. Science 299, 525–528 (2003).
9
A Assion, et al., Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses. Science 282, 919–922 (1998).
10
T Brixner, et al., Photoselective adaptive femtosecond quantum control in the liquid phase. Nature 414–417, 57–60 (2001).
11
G Vogt, et al., Femtosecond pump-shaped-dump quantum control of retinal isomerization in bacteriorhodopsin. Chem Phys Lett 433, 211–215 (2006).
12
VI Prokhorenko, et al., Coherent control of retinal isomerization in bacteriorhodopsin. Science 313, 1257–1261 (2006).
13
W Wohlleben, et al., Coherent control for spectroscopy and manipulation of biological dynamics. Chem Phys Chem 6, 850–857 (2005).
14
RJ Levis, GM Menkir, H Rabitz, Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses. Science 292, 709–713 (2001).
15
BJ Sussman, et al., Dynamic stark control of photochemical processes. Science 314, 278–281 (2006).
16
MPA Branderhorst, et al., Coherent control of decoherence. Science 320, 638–643 (2008).
17
E Shapiro, I Walmsley, M Ivanov, Suppression of decoherence in a wave packet via nonlinear resonance. Phys Rev Lett 98, 050501. (2007).
18
F Remacle, RD Levine, An electronic time scale in chemistry. Proc Natl Acad Sci USA 103, 6793–6798 (2006).
19
M Lezius, et al., Nonadiabatic multielectron dynamics in strong field molecular ionization. Phys Rev Lett 86, 51–89 (2001).
20
A Markevitch, et al., Nonadiabatic dynamics of polyatomic molecules and ions in strong laser fields. Phys Rev A 69, 013401. (2004).
21
PB Corkum, Plasma perspective on strong field multiphoton ionization. Phys Rev Lett 71, 1994–1997 (1993).
22
J Muth-Böhm, A Becker, FHM Faisal, Suppressed molecular ionization for a class of diatomics in intense femtosecond laser fields. Phys Rev Lett 85, 2280–2283 (2000).
23
XM Tong, ZX Zhao, CD Lin, Theory of molecular tunneling ionization. Phys Rev Lett 85, 2280–2283 (2000).
24
D Pavicic, Direct measurement of the angular dependence of ionization for N2, O2, and CO2 in intense laser fields. Phys Rev Lett 98, 243001 (2007).
25
M Shapiro, EPR breakup of polyatomic molecule. J Phys Chem A 110, 8580–8584 (2006).
26
K Molmer, Reply to comment on “Optical coherence: A convenient fiction”. Phys Rev A 58, 4247 (1998).
27
MS Schoffler, et al., Ultrafast Probing of Core Hole Localization in N2. Science 320, 920–923 (2008).
28
S Baker, et al., Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006).
29
M Lein, Attosecond probing of vibrational dynamics with high-harmonic generation. Phys Rev Lett 94, 053004. (2005).
30
H Stapelfeldt, T Seideman, Aligning molecules with strong laser pulses. Rev Mod Phys 75, 543–557 (2003).
31
J Tate, et al., Scaling of wave-packet dynamics in an intense midinfrared field. Phys Rev Lett 98, 013901. (2007).
32
R Santra, RW Dunford, L Young, Spin-orbit effect on strong-field ionization of krypton. Phys Rev A 74, 043403. (2006).
33
MY Ivanov, T Brabec, NH Burnett, Coulomb corrections and polarization effects in high-intensity high-harmonic emission. Phys Rev A 54, 742–745 (1996).
34
P Salieres, et al., Feynman's path integral approach for intense laser-atom interactions. Science 292, 902–905 (2001).
35
NB Delone, VP Krainov Atoms in Strong Light Fields (Springer, Berlin, 1985).
36
S Patchkovskii, Nuclear dynamics in polyatomic molecules and high-order harmonic generation. Phys Rev Lett 102, 253602 (2009).

Information & Authors

Information

Published in

Go to Proceedings of the National Academy of Sciences
Go to Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
Vol. 106 | No. 39
September 29, 2009
PubMed: 19805337

Classifications

Submission history

Received: August 8, 2008
Published online: September 29, 2009
Published in issue: September 29, 2009

Keywords

  1. high harmonic generation
  2. multielectron dynamics
  3. strong-field ionization

Acknowledgments

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

Notes

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

Authors

Affiliations

Olga Smirnova
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6;
Max Born Institute, Max Born Strasse 2a, D-12489 Berlin, Germany;
Serguei Patchkovskii
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6;
Yann Mairesse
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6;
Centre Lasers Intenses et Applications, Université Bordeaux I, Unité Mixte de Recherche 5107 (Centre National de la Recherche Scientifique, Bordeaux 1, CEA), 351 Cours de la Libération, 33405 Talence Cedex, France;
Nirit Dudovich
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6;
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel; and
Misha Yu Ivanov1 [email protected]
National Research Council of Canada, 100 Sussex Drive, Ottawa, ON, Canada K1A 0R6;
Department of Physics, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom

Notes

1
To whom correspondence should be addressed. E-mail: [email protected]
Communicated by Stephen E. Harris, Stanford University, Stanford, CA, July 19, 2009
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.

Competing Interests

The authors declare no conflict of interest.

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