Reproducibility of Observables and Coherent Control in Molecular

ARTICLE pubs.acs.org/JPCA

Reproducibility of Observables and Coherent Control in Molecular Photoionization: From Continuous Wave to Ultrashort Pulsed Radiation A. Gonz
alez-Castrillo,† A. Palacios,*,† F. Catoire,‡ H. Bachau,‡ and F. Martín†,§ †

Departamento de Química, M
odulo 13, Universidad Aut
onoma de Madrid, 28049 Madrid, Spain Centre des Lasers Intenses et Applications, CNRS-CEA-Universit
e de Bordeaux 1, F-33405 Talence, France § Instituto Madrile~ no de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain ‡

ABSTRACT: One-photon single ionization of molecules has been at the focus of several discussions concerning the reconstruction of observables obtained with ultrashort pulses from those obtained from continuous wave radiation (and vice versa). A related controversy on the conditions and observables that allow for coherent control in one-photon processes has been recently revisited (Science 2006, 313, 1257; J. Chem. Phys. 2010, 133, 151101). Our benchmark to investigate these issues is photoionization of the hydrogen molecule, where the autoionization events are the time-dependent processes in field-free evolution that could serve as a target for coherent control. We show that the variation of one-photon ionization probabilities with pulse duration are solely due to spectral effects and thus cannot be coherently controlled. We then discuss for which observables and under which conditions phase control of autoionization dynamics is possible.

’ INTRODUCTION Advances in controlling quantum phenomena have opened the way to new strategies to optimize yields and branching ratios of desired products in chemical reactions.1
8 In particular, electromagnetic radiation can be used to manipulate and guide different reaction pathways in molecules. The basic idea in coherent control is the ability of modifying the complex amplitudes of several interfering transition paths that reach the same final quantum state. In this respect, the remarkable progress in femtosecond laser technology in the last twenty years has been crucial. Ultrashort pulses generated by such lasers create coherent superpositions of quantum states, thus inducing the required interferences for coherent control in complex systems. Although coherent control can also be achieved by using continuous wave lasers,2,3 most applications make use of ultrashort laser pulses.4
7 In the latter works, coherent control is achieved by tuning the pulse parameters and the relative timing of multiple pulses in pump
dump and pump
pump schemes. These schemes rely on the absorption of two (or more) photons to create constructive and destructive interferences between different matter waves. In the past few years, there has been some controversy regarding the possibility of observing one-photon coherent control in isolated systems9
12 and/or in a dissipative environment16,17,13
15 by shaping ultrashort pulses. More than two decades ago, Brumer and Shapiro11 claimed that coherent control of excitation/ionization cross section ratios is not possible in one-photon absorption processes involving isolated molecules. However, in 2006, experiments by Prokhorenko et al.16 showed evidence of one-photon control in long-time isomerization yields by varying the phase properties of low-intensity light.16 The ensuing discussions13,14 showed that r 2011 American Chemical Society

there was no contradiction with the earlier predictions because the experiments were environmentally assisted.13,15,17 This spurred Spanner et al.17 to investigate the conditions under which one-photon coherent control can be achieved in open dissipative evironments. But, more interestingly, these authors also found that, under certain circumstances, one-photon coherent control may be, in fact, also possible in isolated targets. In the present work, we investigate this matter in the case of H2 photoionization. In addition to using ultrashort pulses for control purposes, the analysis of the induced dynamics can provide information on the nuclear and electronic structure of the molecule, as well as on the interatomic forces. In contrast to atoms, in molecules the excess energy from photoabsorption can be shared between electrons and nuclei in different ways, many of them with similar probabilities. Thus, when finite pulses with non-negligible spectral width are used, a full description of the break-up process requires detection of all photofragments, namely electrons and ions. In this context, an appropriate definition of the observables that contain the relevant information on the molecular dynamics launched through one-photon absorption is essential. Here we analyze differential and integrated observables, as well as the pulse length dependence, of the response obtained using continuous wave (cw) and ultrashort pulsed radiation in different photoionization channels of the hydrogen molecule. We focus on Special Issue: Femto10: The Madrid Conference on Femtochemistry Received: August 14, 2011 Revised: October 10, 2011 Published: November 21, 2011 2704

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the autoionization dynamics and explore its time-dependent signature under laser-free evolution in the different observables. We show that one-photon coherent control of autoionization can be achieved by either detecting the ejected electrons or probing the system before the autoionizing decay has been completed. A straightforward relation of our numerical results with the formal arguments proposed in ref 17 on one-photon coherent control can be achieved by means of the Feshbach-like formalism employed in our calculations. The manuscript is organized as follows. In the following section, we describe our theoretical approach and its implementation for the H2 molecule. Then we briefly review first-order time-dependent perturbation theory, which will allow us to derive simple expressions that relate amplitudes and probabilities obtained by directly solving the time-dependent Schr€odinger equation with different pulses. In the subsequent section, results for total and energy-differential single photoionization probabilities of H2 are compared for different pulse durations, central frequencies, and different ionization channels. This is followed by a section in which selection of the optimum conditions and observables to achieve coherent control is discussed. The paper ends with a brief summary of the relevant conclusions.

’ THEORETICAL APPROACH AND IMPLEMENTATION The present theoretical approach has been described in detail in previous works.18,19 Here we only review the most significant methodological aspects. First we describe the spectral method used to solve the time-dependent Schr€odinger equation (TDSE). Next, we briefly describe the method used to evaluate single ionization channels, which is based on a Feshbach-like formalism where the total Hamiltonian is separated in two subspaces: one subspace contains the resonant contributions to the total wave function and the other one includes the continuum nonresonant part. The advantage of such treatment is that we can easily follow the time evolution of the doubly excited states (DES) of the molecule decaying into the continuum, as will be shown in the following sections. Finally, expressions for total and energy-differential ionization probabilities are given. Atomic units are used throughout unless otherwise stated. € dinger Equation. We solve the Time-Dependent Schro time-dependent Schr€odinger equation ∂ i Φðr, R, tÞ ¼ Hðr, R, tÞ Φðr, R, tÞ ∂t

ð1Þ

where r stands for the electronic coordinates (both r1 and r2) and R is the internuclear distance. The total Hamiltonian is given by H(r,R,t) = H0(r,R) + V(r,R,t), where H0 is the field-free Hamiltonian of the target and V(r,R,t) is the laser-molecule interaction potential. The Hamiltonian of the isolated molecule is written H 0 ðr, RÞ ¼ TðRÞ þ Hel ðr, RÞ

ð2Þ

where T(R) =
3R2/2μ is the nuclear kinetic energy, with μ the reduced mass, and Hel is the electronic Hamiltonian including the nucleus
nucleus repulsion potential term. Mass polarization terms, relativistic corrections, and nonadiabatic couplings are neglected. In the dipole approximation and the velocity gauge, the laser
molecule interaction is written in terms of the momentum operator of the electron p and the vector potential A(t). For a single pulse of photon energy ω and total pulse duration T, A(t) is written as ( A0 FðtÞ sinðωtÞε^ t ∈ ½0, T ð3Þ AðtÞ ¼ 0 elsewhere

where ^ε is the polarization vector. We use a sine squared temporal envelope for the finite pulse F(t) = sin2(πt/T). The time-dependent wave function in eq 1 is expanded in a basis set of fully correlated adiabatic vibronic stationary states (see details in ref 19):

where Ψnvn(r,R) is the vibronic state associated to the nth bound electronic state of H2 and its vn (bound or dissociative) vibrational state, Ψrvr (r,R) is the vibronic state associated to a resonant electronic state lying above the ionization threshold and its vr vibrational state, and Ψlαεα α vα ðr, RÞ is the vibronic state associated with an electronic continuum state and its vα (bound or dissociative) vibrational state. In the latter, εα is the energy of the ejected electron, l α its angular momentum and α the ionization channel. The symbol indicates a summation over vibrational bound states plus an integral over the dissociative ones, and Wx is the total energy of each vibronic state. Stationary States. The vibronic wave functions in eq 4 are described within the Born
Oppenheimer approximation, so they are written as products of an electronic and a vibrational wave function. Bound electronic states are obtained by solving the eigenvalue problem Helϕn = E n ϕn , whereas continuum and autoionizing electronic states are described using the Feshbach formalism.20 Two orthogonal complementary subspaces are defined (Q þ P = 1), respectively containing the resonant (Q ) and nonresonant (P ) contribution to the continuum electronic wave function at a given energy. We then separately solve ½Q Hel Q ϕr ðr, RÞ ¼ E r ðRÞϕr ðr, RÞ

ð5aÞ

½P Hel P ϕα, l α , ε ðr, RÞ ¼ E α, ε ðRÞϕα, l α , ε ðr, RÞ

ð5bÞ

The eigenvalue equations for the bound states and the resonant component of the continuum (eq 5a) are solved using a configuration interaction method in a basis of H2+ orbitals.21 These orbitals are previously obtained in single center expansion using spherical harmonics with angular momenta up to l = 16 for the angular part and 180 B-splines functions of order k = 8 in a box of size 60 au for the radial part. Nonresonant continuum electronic states, eq 5b, are calculated using a multichannel L2 close-coupling procedure.22,23 The nuclear wave functions χvx (R) are calculated by solving the one-dimensional Schr€odinger equation: ½TðRÞ þ E x ðRÞχvx ðRÞ ¼ Wx, vx χvx ðRÞ

ð6Þ

where x is a bound, resonant or continuum electronic state, and E x is the corresponding potential energy curve previously obtained. Nuclear wave functions are written in a basis set of 240 B-splines defined in a box of size 12 au. Ionization Probability. Ionization amplitudes are extracted by projecting the time-propagated wave packet into the final continuum states. Because the wave packet is written in terms of the stationary states of the system, the expansion coefficients 2705

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directly give the corresponding amplitudes, which allows for a straightforward analysis of the spectral components (both total and energy- and angle-differential) in the wave packet. Therefore, for a given final energy Wεαvα and a given energy sharing, Wεαvα = Evα + εα, between electrons (εα) and nuclei (Evα), the ionization probability is given by d2 Pl α ðEvα , εα , TÞ ¼ jClαεα α vα ðt ¼ tmax Þj2 dEvα dεα

ð7Þ

where l α indicates the angular momentum of the ejected electron in the α ionization channel and T is the pulse duration. The vibrationally resolved ionization probability is simply given by Z dPðEvα , TÞ dεα jClαεα α vα ðt ¼ tmax Þj2 ¼ ð8Þ dEvα lα

∑

where Evα= Wεαvα
εα. For nondissociative ionization, Evα is the vibrational energy of the residual H2+ ion. In eqs 7 and 8, tmax is the integration time, i.e., the maximum time for which the integration of the TDSE is carried out. The choice of tmax is a key point in the energy region where doubly excited states appear, because it should be chosen large enough (tmax > T) to ensure that the stationary regime has been reached, i.e., that autoionization has been completed. This aspect will be discussed in detail in the forthcoming sections.

’ TIME-DEPENDENT PERTURBATION THEORY In the particular case of weak fields, an accurate description can be achieved using time-dependent perturbation theory (TDPT), and therefore, amplitudes can be extracted over the range of energies within the bandwidth of the pulse.24,25 As is well-known, the ionization amplitude for a transition from an initial state ψi with energy Ei = pωi to a final state ψf with a total energy Wf = Evα + εα by absorption of one photon can be written using first-order TDPT as i ~ ðωif , ω, TÞ Clαεα α vα ðtf∞Þ ¼
ÆΨlαεα α vα jμjΨi æA ð9Þ p where ÆΨlαεα α vα jμjΨi æ is the dipole matrix element between the initial and the final state; the energy difference between the initial and the final state is pωif = Wf
E0 = Evα + εα
E0 (notice that f stands for a given final state with total energy Wf and an angular momenta l for the ejected electron in the α ionization channel); ~ (ωif, ω, T) ω corresponds to the central frequency of the pulse; and A is the Fourier transform of the vector potential defined in eq 3 which, using a sine-squared temporal envelope and the rotating wave approximation,25 has the following analytical expression: ~ ðωif , ω, TÞ ¼ A

Z T

¼ A0

0

AðtÞe
iωif t dt

 
2iπ2 ðωif
ωÞT sin 2 2 ðωif
ωÞf4π2
½ðωif
ωÞT g

ð10Þ with T being the pulse duration. The ionization probability is d2 Pl α ðEvα , εα , TÞ ~ ðωif , ω, TÞj2 ¼ jÆΨlαεα α vα jμjΨi æj2 3 jA dEvα dεα ð11Þ Therefore, differential photoionization probabilities in both electron and nuclear energies can be easily reconstructed for

Figure 1. Schematic representation of the relevant potential energy curves: ground state of H2, first (1sσg) and second (2pσu) ionization potentials, and doubly excited states from Q1 (red) and Q2 (blue) series. Thick dashed lines indicate the energy reached using photons of 20, 28, and 33 eV. Vertical boxed shadow correspond with the Franck
Condon region from the v = 0 vibrational state in the ground state of the neutral.

ultrashort pulses from those of long pulses, and vice versa, as long as the Fourier transform of the pulses are well-defined:11,24,25 1 d2 Pl α ðEvα , εα , T 0 Þ ~ ðωif , ω0 , T 0 Þj2 dEvα dεα jA ¼

1 d2 Pl α ðEvα , εα , TÞ 2 ~ ðωif , ω, TÞj dEvα dεα jA

ð12Þ

Notice that the expression in eq 12 is simply equal to the dipole matrix element between the initial i and the final f state, which is a consequence of the exact factorability of the time dependence in the expression for the transition probability resulting from first-order TDPT (eq 9). Thus any dipole matrix element can be directly extracted from the calculated ionization amplitudes in the range of energies where the amplitude is significantly different from nonzero, i.e., within the range covered by the spectral bandwidth of the pulse. In the long pulse limit, ~ (ωif,ω,T0 )|2 takes the constant value (8/3T)(1/|A0|2)2πδ1/|A (ωif
ω), which is consistent with the cross section definitions in previous works where the long-time limit is initially assumed.26 Experimentally, it is not always possible to perform coincident measurements for all ejected particles; i.e., fully energy differential distributions may not be accessible. In this case, eq 12 is useless unless monochromatic light (cw radiation) is used. In the case of finite pulses, when only ionization probabilities integrated over electron (or proton) energies are available, some considerations must be taken into account to appropriately extract information on the molecular structure and dynamics. This will be discussed in the following section.

’ DIFFERENTIAL PROBABILITY DISTRIBUTIONS Here, single photoionization of the hydrogen molecule is investigated with finite pulses in the weak field regime. Throughout the manuscript, the peak intensity is fixed at 1012 W cm
2, which keeps us within the perturbative regime.26 Linearly polarized light parallel to the molecular axis is used; thus onephoton absorption can only lead to final states of 1Σ+u symmetry. 2706

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Figure 2. Dissociative ionization probabilities differential in the nuclear and electron energies for pulses of central frequencies of 20, 28, and 33 eV (respectively indicated in each graph). The x-axis corresponds to nuclear kinetic energy (NKE) and the y-axis to electron kinetic energy (EKE). Panels on the left: pulse duration of 1 fs. Panels on the right: pulse duration of 10 fs. Diagonal magenta dashed lines indicate the position of the central frequency of the pulse.

We perform calculations for three central frequencies (Figure 1) where different regions are explored:27 (i) 20 eV, for which only direct photoionization through the first ionization threshold (1sσg) is possible, (ii) 28 eV, for which contributions from both direct photoionization and autoionization from the Q1 series of doubly excited states are possible, and (iii) 33 eV, for which, in addition to the former, direct photoionization through the second threshold 2pσu and autoionization from the Q2 series of DES is open. Q1 states decay into the 1sσg continuum and Q2 states decay to both the 1sσg and 2pσu continua. For each central frequency we have carried out calculations for pulse durations of 1 and 10 fs. In Figure 2, dissociative ionization probabilities differential in both electron and nuclear energies (eq 7) are plotted as they would appear in a proton
electron coincidence measurement. Notice that nuclear kinetic energy (NKE) is referred to the center of mass of the molecule, so that it is defined as twice the proton kinetic energy (PKE) when the laboratory frame is considered. For a 10 fs pulse duration (right column), we obtain narrow distributions (the energy bandwidth of a 10 fs pulse is =0.6 eV) centered at the excess total energy. Signatures from different open ionization channels vary with the photon energy absorbed. For 20 eV, we observe a structureless

distribution with a rapid decrease with proton energies, corresponding to the direct photoionization process to the first ionization threshold. For 28 eV, we also find structures around 2
8 eV of NKE arising from autoionization from the Q1 DES. For 33 eV, a second peak appears at the highest values of NKE corresponding to direct photoionization through the 2pσu ionization threshold. DES in the Q2 series are hardly populated and lead to a negligible contribution to the ionization probability. A key point to consider, nonexistent in atoms, is the fact that molecular autoionization mainly leaves its signature for electron (or proton) kinetic energies that are not those associated with direct photoionization from the ground state. It implies that, instead of the well-known Fano profiles found in atoms, we may observe Lorentzian-type shapes for fully differential (in energy) probabilities as the photon energy varies. Moreover, their widths depend on the kinetic energies of electrons and nuclei because autoionization widths depend on the internuclear distance.28 For a 1 fs pulse duration (left column in Figure 2), we obtain energy distributions widened by the energy bandwidth (for a 1 fs pulse is =6 eV). The diagonal magenta dashed line indicates the position for the expected energy excess at the central frequency of the pulse. Distributions do not appear exactly centered at this 2707

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The Journal of Physical Chemistry A energy due to the variation of the dipole couplings with the final energy. As expected from first-order TDPT, for a given total final energy, the ionization probabilities for different pulse durations only differ by a constant factor (see eq 12). The 1 fs pulse is simply uncovering a wider range of energies; i.e., the light source is exposing a broader (1 fs) or narrower (10 fs) region on the dipole matrix elements between the ground state of the neutral and the final scattering states. In other words, the analytical expression in eq 12 can be used to reconstruct the ionization distributions differential in both electronic and nuclear energies from those obtained with (arbitrary) longer pulses by scanning the same range of final energies. The origin of the observed structures and contributions from each ionization threshold and autoionizing states belonging to the Q1 and Q2 series of DES have been discussed in detail in previous papers within the frame of time-independent perturbation theory (TIPT).28,29 TIPT is able to describe one-photon molecular ionization in weak fields once the stationary regime has been reached, because only spectral effects are responsible for the variation of ionization probabilities with the pulse duration. Note that, although for the present work we use linearly polarized light, the same analysis and conclusions apply to circular or elliptical polarizations as long as the polarization is the same for all pulses involved in the comparison. Experimentally, it is not always possible to achieve detection of all fragments in coincidence, and thus integrated (in electron or proton) energy quantities are the only available data. Nevertheless, despite the loss of information in one of the variables, a large amount of structural and dynamical information can be retrieved, as long as some basic considerations on the pulse length are accounted for. In Figure 3, we show the ionization probabilities for both 1 and 10 fs pulses, now integrated over the electron kinetic energy. We also include the nondissociative ionization probabilities (as a function of the quantum number of the bound vibrational state in the remaining molecular ion). In the dissociative channel, we plot the density of probability distributions that we would find in an experiment where only the protons were detected. The PKE distributions for both pulse durations present a reasonable agreement at 20 eV, where only direct photoionization is present. The peak appearing around 9 eV is the contribution from the two-photon absorption (barely visible in the 2D plot in Figure 2 due to the larger scale). However, for 28 and 33 eV, as already discussed, the features coming from the Q1 DES are smoothed out for a 1 fs pulse. After integrating over electron energies, we are effectively averaging the probability distributions over all photon energies contained within the pulse. On the other hand, vibrational distributions for the bound states, leaving behind H2+ (left panels in Figure 3) exhibit a slow variation with the pulse duration, which is expected because, in this energy region, direct photoionization is the dominant process. Note that for a 10 fs pulse, PKE distributions imply that we have integrated over a very narrow region of electron energies (=0.6 eV), whereas for 1 fs, integration is carried out in a range of =6 eV of electron energies, therefore washing out the existing distinct features. The larger the pulse, the finer the energy resolution to observe any structure in the integrated quantity.18 We now compare the ionization probabilities that are directly obtained from the full calculation for a pulse of 10 fs with those obtained by using various reconstruction procedures. PKE and bound vibrational states distributions integrated over electron energies are plotted in Figure 4 for three different simulations:

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Figure 3. Bound (left panels) and dissociative (right panels) vibrational distributions for H2+ for pulse durations of 1 fs (red thin bars and red dashed line) and 10 fs (black thick and full black line); for central frequencies of 20, 28, and 33 eV. For the dissociative channel, plotted as a function of proton kinetic energy (PKE), results correspond to probability density. Notice that probabilities are normalized with the pulse duration (probability/T (fs)) to make the magnitudes comparable.

(i) The black full line corresponds to the 10 fs results obtained by directly solving the time-dependent Schr€odinger equation (TDSE). (ii) The red dotted line corresponds to first solving the TDSE for a 1 fs pulse and then reconstructing the ionization probabilities for a 10 fs pulse using eq 12: d2 Pl α ðEvα , T 0 ¼ 10 fsÞ ¼ dEvα

Z



d2 Pl α ðEvα , εα , T ¼ 1 fsÞ εα dEvα dεα ~ ðωif , ω, T 0 ¼ 10 fsÞj2 jA dεα ~ ðωif , ω, T ¼ 1 fsÞj2 jA ð13Þ

where we have employed the identity in eq 12. In using this procedure, one assumes that one has access to a coincidence spectrum in which the energies of the ejected electron, the ejected proton and the remaining H2+ molecular ion is measured. Finally, (iii) the green dashed line in Figure 4 corresponds to the results obtained by applying the approximate formula: d2 Pl α ðEvα , T 0 ¼ 10 fsÞ d2 Pl α ðEvα , T ¼ 1 fsÞ ¼ dEvα dEvα Z ~ ðωif , ω, T 0 ¼ 10 fsÞj2 dεα jA εα Z ~ ðωif , ω, T ¼ 1 fsÞj2 dεα jA εα

ð14Þ which is the only alternative when PKE distributions are the only available measurements. The use of eq 14, instead of the formally correct eq 13, implies a partial loss of information on the electron kinetic energy dependence. This leads necessarily to larger discrepancies as the pulse shortens and, particularly, as Figure 4 shows, in the autoionization region. At this point, it is worth emphasizing that a 2708

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Figure 4. Bound and dissociative vibrational distributions for single ionization of H2 for central frequencies of 20, 28, and 33 eV using a pulse duration of 10 fs. Three different calculations are plotted. (i) Black full line: results from the ab initio calculation directly solving TDSE. (ii) Dotted red line: results for a 10 fs pulse extracted from a TDSE calculation for 1 fs and applying eq 13. (iii) Dashed green line: results for a 10 fs pulse extracted from a TDSE calculation for 1 fs and applying eq 14.

Figure 5. Dissociative ionization probability density as a function of proton kinetic energy obtained at different times during the pulse. Results are for a 1 fs (upper panel) and a 10 fs pulse (bottom panel), both with a central frequency of 33 eV.

photon energy scan performed with monochromatic light (as synchrotron radiation) can be used to fully reconstruct the ionization rates, even if only ejected protons and bound molecular ions are detected. Because, in this case, the Fourier transform of the pulse becomes a Dirac delta in energy, distributions in Figure 2 will appear as an infinitely narrow diagonal and hence the electron kinetic energy (EKE) distributions will be the specular image of the PKE distributions (in other words, this is simply the consequence of energy conservation). For long enough pulses, with bandwidths narrower than the structures coming from autoionization (T > 1/Γ, where Γ is the autoionization width averaged over the internuclear distances inside the Franck
Condon region), the same criteria applies, but the energy resolution decreases to the order of such energy bandwidth. Finally, for pulses shorter than the average resonance

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Figure 6. Dissociative ionization probability density as a function of proton kinetic energy obtained at different times after the end of the pulse. Results are for a 1 fs (upper panel) and a 10 fs pulse (bottom panel), both with a central frequency of 33 eV. Vertical dashed lines are placed at those kinetic energy releases at which angular distributions are plotted in Figures 7 and 8.

lifetime (T < 1/Γ), photoionization fragments should be measured in coincidence to properly apply the expressions from firstorder TDPT, as the comparisons shown in Figure 4 demonstrate. Another important remark that complements the above analysis is the fact that we are always assuming that autoionization has completely finished when the charged fragments are detected, i.e., once a stationary regime has been reached independently of the pulse duration. To fulfill this requirement in our calculations, once the pulse is turned off, we keep integrating in time until the DES have totally decayed into the continuum. Results converge for integrations carried out 4 fs after the end of the pulse, because the lifetimes of the most populated DES (mainly the lowest one in the Q1 series) are around 3 fs. In a real experiment, the detection of electrons and protons ejected in a one-photon ionization process takes much longer. Therefore, the measured probability distributions will satisfy all the properties discussed above. In the following section we discuss what would happen if one observes the system before autoionization has been completed. As we will see, this opens the possibility to observe and/or trace the dynamics of autoionization processes launched by one-photon absorption.

’ OBSERVABLES FOR PHASE CONTROL As mentioned in the Introduction, there is some controversy on one-photon coherent phase control in isolated systems and on the conditions under which it would be observable for a general target.13,15
17 In ref 17, Spanner and collaborators present a careful analysis of these conditions, which, for the present case, are summarized in the following paragraph: “coherent control is achievable in isolated systems as long as one can measure observables corresponding to operators that are time dependent under laser-free evolution, therefore, operators that do not commute with the Hamiltonian of the isolated target (in our case H0, see eq 2). Those observables that commute with H0 are not phase-controllable”. A particular case is autoionization. Our Feshbach-like implementation allows for a straightforward connection to the previous statement, because the doubly excited states belong to the subspace Q . Their decay on time is described through the P HQ coupling, 2709

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Figure 7. Fully, in angular and proton and electron energies, differential ionization probabilities at different times after the pulse is turned off. The upper row corresponds to 1 fs results (probabilities in 10
6) and the bottom row corresponds to 10 fs results (probabilities in 10
5), both with a central frequency of 33 eV. Each column corresponds to a given PKE (Ea = 1 eV, Eb = 6.3 eV, and Ec = 7.4 eV), indicated in Figure 6 with vertical dashed lines, with electron energies 12.92, 2.32, and 0.12 eV, respectively. Proton ejection and polarization vector directions for all plotted distributions are indicated in the lower left pannel.

which tends to zero as time goes to infinity. As long as this coupling is nonzero, the Hamiltonian in each subspace (Q HQ and P HP ) does not commute with the total Hamiltonian; it only does upon reaching the stationary regime (t = ∞). It is in this limit that the amplitudes of the eigenstates in each subspace coincide with those of the total Hamiltonian, and therefore it is pertinent to project into them (see details in refs 18, 28, and 29). As autoionization occurs not only during but also after the end of the pulse, the time dependence of the amplitudes in the subspace allows one to monitor the decay of the quasi-bound DES. This also opens the door for coherent phase control of this decay. In the following, we study the field-free evolution of the ionization probability and the dependence of different observables with the pulse length from the end of the pulse until the stationary regime is reached. We choose a pulse with a central frequency of 33 eV, for which both the Q1 and Q2 DES can be populated. We first analyze the evolution of the proton kinetic energy distributions in the presence of the field (see Figure 5). These probabilities are obtained for those values of time at which the vector potential is zero, so that the projection into field free eigenstates is meaningful. A similar theoretical analysis has been

made for long pulses in a previous work.30 At the latest sampling time (840 as), the 1 fs pulse is almost over but the 10 fs pulse is at the start of its rising flank. As we use the same peak intensity for both pulses, the amplitudes are thus much larger for the 1 fs pulse. Figure 5 shows that during the first femtosecond of the 10 fs pulse, the system does not yet “know” the final pulse duration, and the “instantaneous” bandwidth is equivalent to that of a 1 fs pulse. In the presence of the field, the PKE distributions obtained at different times are almost identical to those obtained with a pulse whose duration is equal to the corresponding sampling time. As shown in Figure 5, at such short times, there is almost no trace of autoionization: the observed ionization probability is almost exclusively due to direct ionization through the 1sσg (at low PKE) and 2pσu (at high PKE) channels. The ionization probability is close to zero in the PKE region in between, i.e., where autoionization is expected to be seen. We now analyze the PKE distributions after the pulse, for 1 and 10 fs pulse durations. These are shown in Figure 6. For the 1 fs pulse and in the absence of the field (t > T), the ionization probability increases in the region of intermediate PKE. This is due to decay of the doubly excited states. Autoionization 2710

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Figure 8. Electron angular distributions integrated over electron energy. Ionization probabilities (probabilities in 10
6 for 1 fs results and 10
5 for 10 fs results) are plotted for different times once the pulse is finished. Each column corresponds to a given PKE (Ea = 1 eV, Eb = 6.3 eV, and Ec = 7.4 eV), indicated in Figure 6 with vertical dashed lines. Proton ejection and polarization vector directions for all plotted distributions are indicated in the lower left pannel.

structures are completely built up 3 fs after the end of the 1 fs pulse, when the stationary regime is reached. In contrast, for the 10 fs pulse, the PKE distribution hardly changes after the pulse, because most autoionization has already occurred while the field is still active. A small fraction of autoionization, corresponding to the energy absorption in the tail of the temporal envelope, will take place once the pulse is turned off. But this is small in comparison with autoionization produced earlier. In practice, one can assume that for finite pulses whose half-duration is larger than the lifetime of the DES, most of the autoionization, launched at maximum amplitude, has taken place. Hence, for pulse durations such that T/2 < 1/Γ, the decay could be traced and the relative phases of the populated final states could be extracted. The phase dependence of photofragment ejections is better observed in the electron angular distributions for a given electron and vibrational energy (i.e., in the fully differential ionization probabilities). They are shown in Figure 7 for three different proton kinetic energies, Ea = 1 eV, Eb = 6.3 eV, and Ec = 7.4 eV. Independently of the pulse duration, an asymmetry is observed for Eb and Ec once one reaches the stationary regime. As has been shown in previous works,30,31 asymmetries in the angular distributions appear in H2 one-photon ionization when both the

1sσg and 2pσu ionization channels are populated. The coherent superposition of final states with different parity leads to a symmetry breaking that will be observed in the molecular frame electron angular distributions. In the present case, a central photon energy of 33 eV reaches the Q11 Σ+u DES that decay into the continuum populating an interval of proton kinetic energies between 1 and 8 eV in the 1sσg channel. Direct ionization through the 2pσu channel leads to PKE between 5 and 8 eV. As a consequence, there is a region of PKE (5
8 eV) where a coherent superposition of H2+ states of different parity is possible. The coherent superposition can be observed in measurements distinguishing between protons going upward and going downward. The molecular frame electron angular distributions shown in Figure 7 correspond to the case of protons going upward. Note that the asymmetry appears for Eb and Ec, where the 1sσg and 2pσu contributions overlap, but not for Ea. As expected, for the 1 fs pulse, the asymmetry appears a few femtoseconds after the end of the pulse (time needed by the DES to decay), whereas in the 10 fs case, the angular distributions do not vary with time after the pulse is over because one is already in the stationary regime. Results for a 1 fs pulse (Figure 7, upper row) and 10 fs pulse (bottom row) only differ if traced during the decay of the 2711

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The Journal of Physical Chemistry A DES; otherwise, they all look as those corresponding to the phaseindependent stationary limit for one-photon absorption. Thus, the time-evolution of the asymmetry reflects the phase changes during autoionization. These dynamics could be investigated, for instance, by using pump
probe schemes in which the pump is a single pulse with T/2 < 1/Γ32 or by using probe pulses through attosecond transient absorption.33 Finally, as for the energy-integrated quantities discussed in the previous section, special attention should be paid when angular distributions integrated over electron energy are analyzed. As can be seen in Figure 8, in the 1 fs case, the angular distributions remain symmetric due the mixing of different frequency components contained in the bandwidth. This is not the case for the narrow bandwidth of a 10 fs pulse (or equivalently stated in the time domain: T . 1/Γ).

’ SUMMARY AND CONCLUSIONS Pulse duration effects have been investigated in one-photon single ionization of H2 in the weak field regime, with special attention to energy regions where autoionization manifests. Once the system reaches a stationary regime, upon autoionization, the variation of ionization probabilities with pulse duration is solely due to spectral effects. Hence, ionization probabilities obtained for ultrashort pulses can be reconstructed from cw radiation results for weak fields11 by simply using the ionization rate expression derived from first-order time-dependent perturbation theory. In contrast, to reconstruct ionization probabilities from those obtained for a finite pulse, first-order TDPT expressions are only applicable provided that all fragments are detected in coincidence. In their corresponding integrated quantities (as PKE or EKE distributions), part of the information is washed out by the broad spectral width; thus reconstruction is no longer possible. For these singly differential probabilities, the condition to observe signatures of autoionization as those observed in the infinite long pulse limit is that T/2 (i.e., half of the total pulse duration) is larger than 1/Γ (i.e., the DES lifetime). The lifetimes of the DES that are effectively populated by onephoton absorption in H2 are of the order of a few femtoseconds. Therefore, current one-photon experiments can only detect fragments once the stationary limit has been reached. However, molecular autoionization can be experimentally traced as long as the system is probed/detected before autoionization has been completed. For DES whose lifetimes are longer than the detection/probing time, coherent phase control of this decay is possible. In particular, the symmetry breaking induced by autoionization in the molecular frame electron angular distributions could be used to trace the different phases in the created wave packet. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT Calculations performed in BSC-MareNostrum and CC-UAM. Work partially supported by the MICINN projects FIS201015127, ACI2008-0777 and CSD 2007-00010, the ERA-Chemistry project PIM2010EEC-00751, the European grants MCITN CORINF and MC-RG ATTOTREND, and the European COST Action CM0702. A.P. acknowledges Dr. J. Feist for insightful discussions and Prof. J. L. Sanz-Vicario for carefully

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reading the manuscript. A.P. also thanks her Juan de la Cierva Contract from MICINN.

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