Effect of Nuclear Motion on Molecular High Order Harmonic Pump

Article pubs.acs.org/JPCA

Effect of Nuclear Motion on Molecular High Order Harmonic Pump Probe Spectroscopy Timm Bredtmann,*,†,‡ Szczepan Chelkowski,‡ and André D. Bandrauk‡ †

Physikalische und Theoretische Chemie, Institut für Chemie und Biochemie, Freie Universität Berlin, Takustrasse 3, 14195 Berlin, Germany ‡ Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1 ABSTRACT: We study pump−probe schemes for the real time observation of electronic motion on attosecond time scale in the molecular ion H2+ and its heavier isotope T2+ while these molecules dissociate on femtosecond time scale by solving numerically the non-Born−Oppenheimer time-dependent Schrödinger equation. The UV pump laser pulse prepares a coherent superposition of the three lowest lying quantum states and the time-delayed mid-infrared, intense few-femtosecond probe pulse subsequently generates molecular high-order harmonics (MHOHG) from this coherent electron−nuclear wavepacket (CENWP). Varying the pump−probe time delay by a few hundreds of attoseconds, the MHOHG signal intensity is shown to vary by orders of magnitude. Due to nuclear movement, the coherence of these two upper states and the ground state is lost after a few femtoseconds and the MHOHG intensity variations as function of pump−probe delay time are shown to be equal to the period of electron oscillation in the coherent superposition of the two upper dissociative quantum states. Although this electron oscillation period and hence the periodicity of the harmonic spectra are quite constant over a wide range of internuclear distances, a strong signature of nuclear motion is seen in the actual shapes and ways in which these spectra change as a function of pump−probe delay time, which is illustrated by comparison of the MHOHG spectra generated by the two isotopes H2+ and T2+. Two different regimes corresponding roughly to internuclear distances R < 4a0 and R > 4a0 are identified: For R < 4a0, the intensity of a whole range of frequencies in the plateau region is decreased by orders of magnitude when the delay time is changed by a few hundred attoseconds whereas in the cutoff region the peaks in the MHOHG spectra are red-shifted with increasing pump−probe time delay. For R > 4a0, on the other hand, the peaks both in the cutoff and plateau region are red-shifted with increasing delay times with only slight variations in the peak intensities. A time−frequency analysis shows that in the case of a two-cycle probe pulse the sole contribution of one long and associated short trajectory correlates with the attenuation of a whole range of frequencies in the plateau region for R < 4a0 whereas the observed red shift for R > 4a0, even in the plateau region, correlates with a single electron return within one-half laser cycle.

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INTRODUCTION Molecular processes such as the making and breaking of chemical bonds, molecular dissociation, and vibrations involve the coupled movement of both electrons and nuclei. These time dependent processes are driven by the superposition of the respective quantum states. Hence nuclear motion occurs on the femtosecond (1 fs =10−15 s) time scale due to the subelectronvolt-scale spacing of stationary electronic states. Both the real-time observation and the control of such processes is readily accessible by femtosecond pump−probe spectroscopy1 and specially tailored femtosecond laser pulses,2−4 respectively, and constitutes the field of Femtochemistry.5 In recent years, considerable progress has been achieved in Attosecond Science, involving the real-time observation of electronic motion on its natural time scale, which is 1 attosecond (asec) = 10−18 s; see, e.g., refs 6−8 for a review and further references. These changes might roughly be divided into two regimes: On the one hand, singlestate electron dynamics, i.e., the collective rearrangement of both electrons and nuclei involving nondegenerate electronic © XXXX American Chemical Society

quantum states, proceeds on the femtosecond time scale of nuclear motion; see, e.g., refs 9−12 and references therein. On the other hand, in “multistate” electron dynamics involving coherent electron−nuclear wavepackets (CENWP), the coherent superposition of electronic quantum states leads to attosecond electron motion due to the (ten to thirty) electronvolt-scale spacing of electronic states.13−20 Furthermore, it was shown that depending on the preparation of the molecular system, nuclear motion may readily lead to decoherence and “suppression” of the asec electronic motion.21 A conceptually simple way for monitoring this attosecond electron motion in molecules might be photoionization spectroscopy using two time-delayed ultrashort laser pulses, as was previously shown numerically:6,22,23 The pump pulse Special Issue: Jörn Manz Festschrift Received: June 28, 2012 Revised: September 8, 2012

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dx.doi.org/10.1021/jp3063977 | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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highest energy upon return, giving rise to the highest energy photon emission (the harmonic cutoff); see, e.g., ref 36 and references therein. Hence according to this semiclassical model, in (M)HOHG, high energy radiation is emitted at sharp wellknown return times, occurring every half laser cycle in the case of the highest energy photons in the cutoff region and twice within one-half laser cycle for lower frequency radiation in the plateau region. Thus, although the probe total duration (=5.3 fs for the two-cycle 800 nm pulses used in this study) is much longer than the attosecond time scale of the electron motion, (M)HOHG occurs on the subcycle time scale, which is confirmed by our quantum results in terms of time−frequency analysis of MHOHG spectra; see below and refs 21, 24, 33, and 34. This subcycle time scale is comparable to the electron oscillation period of about 450 asec in the case of H2+, thus allowing us to monitor attosecond electron motion with a few-femtosecond pulses.

prepares a coherent superposition of two or more electronic states whereas the time-delayed XUV probe pulse photoionizes the molecule and the resulting temporal asymmetry of the photoionization spectrum monitors the attosecond electronic motion. Note, however, that extremely short XUV probe pulses (fwhm duration of about 150 asec6,22,23) are necessary for this scheme to resolve the attosecond electron motion in the molecular ion H2+. These attosecond XUV pulses are most commonly created by high-order harmonic generation (HOHG), which occurs when atoms or molecules are exposed to strong (intensity >1014 W/cm2) low frequency (e.g., 800 nm) laser fields resulting in the emission of high energy radiation that can easily exceed 100 eV; for details see below and, e.g., ref 6. More recently we investigated the direct usage of HOHG in molecules (molecular high order harmonic generation, MHOHG) as a direct probe of this attosecond electron motion;21,24 see also ref 25. In this scheme, the attosecond XUV probe pulse (fwhm duration of 150 asec) is replaced by the much longer (fwhm duration of 5.3 fs in the case of two-cycle 800 nm probe pulses) and easier accessible intense infrared few-cycle pulse and instead of the photoelectron asymmetries the MHOHG spectrum is measured at different pump−probe delay times. Varying this time delay by a few hundred attoseconds leads to periodic MHOHG signal intensity variations by orders of magnitude that monitor the electron oscillation period in the CENWP. Our previous studies focused on the molecular ion T2+ as well as the H-atom, and different combinations of pump and probe laser pulses were investigated.21,24 Moreover, this high sensitivity of the harmonic intensity on the pump−probe time delay was explained in ref 24 in terms of the strong field approximation26−28 and the semiclassical three-step model29,30 extended to MHOHG from multiple electronic states; see also ref 31. In the present study, these investigations are extended to the dissociating H2+ molecular ion. It is shown that the attosecond electronic motion persists up to large internuclear distances of at least R = 14a0, where a0 denotes the Bohr radius, and that MHOHG pump− probe spectroscopy constitutes a very sensitive tool to measure these electronic attosecond oscillations. We show that the way in which the MHOHG spectra change as a function of pump− probe delay time depends strongly on the internuclear distance R, which is illustrated by a comparison of the isotopes H2+ and T2+. The results are analyzed in terms of time−frequency spectra32−34 that relate our quantum results to the semiclassical three-step model proposed in 1993.29,30 In this model, due to the interaction with the intense laser pulse that serves as a probe in our investigation, an electron is f irst detached from the atom or molecule via tunnel ionization, with zero initial velocity29,30 or with nonzero velocity by preionization,35 next (second step) it is accelerated in the intense laser electric field as a classical particle, and finally (third step) upon field reversal it is directed back to the parent ion, where the returning electron combines with the parent ion and emits high frequency radiation, which corresponds to the energy gained by the electron in the electric field plus the ionization potential. Electrons that are, in the first step due to tunnel ionization, born in the continuum between the peak of the electric field and about 1/20 of an optical cycle (in this study 800 nm pulses are used) follow the so-called “long” electron trajectory whereas an electron launched into the continuum between 1/20 of an optical cycle and the zerocrossings of the electric field follows “short” trajectories. For electrons born at about 1/20 of an optical cycle after the peak intensity, the short and long trajectories converge and have the

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METHODS For each fixed time-delay tdel between the pump and probe laser pulses we solve numerically the complete, three-body, onedimensional, non-Born−Oppenheimer time-dependent Schrödinger equation (TDSE) for the molecular ion H2+ and T2+ (atomic units, au, e = ℏ = mel = 1 are used): i

∂Ψ(z ,R ,t ) = H(z ,R ,t ) Ψ(z ,R ,t ) ∂t

(1)

including both electronic and nuclear degrees of freedom, where H(z,R,t) is the exact, one-dimensional three-body Hamiltonian obtained after separation of the center-of-mass motion, z is the electron coordinate (with respect to the nuclear center of mass), and R is the internuclear distance; see refs 37 and 38 for details. Thus the Hamiltonian H(z,R,t) describes the exact dynamics (i.e., non-Born−Oppenheimer) of two protons and one electron in 1-D in interaction with the intense laser field. The electric field E(t) = Epump(t) + Eprobe(t) we use is the sum of the field of the pump UV laser pulse, which prepares the dissociating wavepackets, and a few-femtosecond intense probe pulse, which generates high-order harmonics. The shape of the laser electric field E(t) for each pulse is defined via the vector potential A(t) as E(t) = −(∂A(t))/(∂t). Sine-squared envelopes are used for the envelopes of the vector potential A(t) of each pulse; see eqs 2 and 3 in ref 39. The electric field defined in this way satisfies automatically the requirement that the area under the electric field is zero.39 The electric field of both pulses are displayed in Figure 1a, in which we define the time-delay between these pulses as the time difference between the maxima of the respective pulse envelopes. The carrier envelope phase φCEP of both pulses (as defined in refs 39 and 40) is equal to π/2; i.e., sine-like pulses are used. The total durations of the pump and probe pulse are 2.7 fs (1 fs fwhm) and 5.3 fs (1.94 fs fwhm), respectively, and the corresponding intensities and central laser frequencies are Ipump = 1013 W/cm2, λpump = 138 nm and Iprobe = 2 × 1014 W/cm2, λprobe = 800 nm. We calculate numerically the time evolution of the electron−nuclear wave function Ψ(z,R,t) using the second-order accurate split-operator method.38,41 Next, we calculate the electron acceleration a(t) = −E(t) − ⟨Ψ|(∂VC)/(∂z)|Ψ⟩, where VC(z,R) is the potential describing the Coulomb attraction from the two nuclei,37 and its Fourier transform F(ω,tdel). Note that we integrate over time until te = tend + 7.3 fs, where tend is the end of the probe pulse; i.e., we continue free evolution (no pulses) about 7 fs after the turn-off of B



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quantum states a and b, and the third interference term describing the attosecond electronic motion with oscillation period Δτab(R) = 2π/[Ea(R) − Eb(R)] depending both on the electronic energies Ea(R) and Eb(R) and, hence, on the nuclear coordinates, and on the overlap of the nuclear wavepackets χ*a (R,t) χb(R,t). In the present study, the electron dynamics is governed by the interference term (see below), and we investigate the potential of MHOHG as a tool to measure the resulting attosecond electron dynamics. For experimental and theoretical investigations concerning single-state electron dynamics; see ref 12 and refs 9−11 respectively. We suppose that initially, at t = 0, the molecular ions H2+ and T2+ are in their ground vibrational state in the electronic ground state σg1s; see Figure 1a. The pump laser pulse prepares two dissociative nuclear wavepackets on electronic states σu1s and σg2p by a one- and two-photon resonance, respectively, with final populations of ⟨χσu1s|χσu1s⟩ = 0.16 and ⟨χσg2p|χσg2p⟩ = 0.03. The corresponding nuclear wavepacket dynamics is sketched in Figure 1a and shown quantitatively for both isotopes in Figure 2

Figure 1. (a) Three lowest electronic states of H2+ and T2+ including a sketch of the nuclear wavepacket dynamics following excitation by the pump pulse. (b) Electric fields of the pump and probe laser pulses at delay time tdel = 5 fs.

the probe pulse. Finally, the power spectrum of the generated harmonics is equal to |F(ω,tdel)|221,40,42,43 and is plotted on a logarithmic scale in Figures 4 and 5. Note that before calculating the Fourier transforms of the acceleration a(t), we multiply it by a Hanning filter,32 which is zero outside the probe duration. This allows us to diminish contributions from the backgrounds and also to suppress some structures related to the fluorescence originating from the nonvanishing acceleration before and after the pulse turn-on and turn-off. We checked, however, that the periodic dependence of all spectral components as function of the pump−probe delay time (see below) is practically identical with and without the Hanning filter.

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RESULTS AND DISCUSSION Consider a molecular system prepared in a coherent superposition of two electronic states a and b, which may be expressed as the superposition of two Born−Oppenheimer electronic states for the present purpose:

Figure 2. Illustration of the nuclear wavepacket dynamics on electronic states σg1s (black dashed line), σu1s (blue), and σg2p (red dashed) for H2+ (upper panel) and for T2+ (lower panel).

by solid blue lines for the dynamics on electronic state σu1s and by dashed red lines on state σg2p. Due to comparable gradients on these excited electronic states, the overlap of the dissociative nuclear wavepackets is preserved, leading to coherent electron− nuclear wavepackets (CENWP) up to large internuclear distances of at least R = 14a0. The speed of the electronic motion in this coherent superposition is determined by the energy spacing of the coherently coupled electronic states, resulting in electronic motion on the subfemtosecond time scale in the present case, which is confirmed in Figure 3. In Figure 3a we plot the electron oscillation period Δτab of a CENWP consisting of the electronic ground and first excited state, Δτ1,2(R) = 2π/[Eσg1s(R) − Eσu1s(R)] (blue lines) and of the first two excited states, Δτ2,3(R) = 2π/[Eσu1s(R) − Eσg2p(R)] (green lines) as a function of the internuclear distance R. In the present case, the overlap of the

Ψ(r,R,t ) = χa (R,t ) ϕa(r; R)e−iEa(R)t + χb (R,t ) ϕb(r;R)e−iE b(R)t

(2)

where χi(R,t) and ϕi(r;R) denote the nuclear and electronic wave functions, respectively, depending on the nuclear and electronic coordinates, R and r with corresponding probability density |Ψ(r,R,t )|2 = |χa (R,t )|2 |ϕa(r;R)|2 + |χb (R,t )|2 |ϕb(r;R)|2 + 2Re(χa* (R,t ) χb (R,t ) ϕa(r;R) ϕb(r;R)e−i(Ea(R) − E b(R)t ))

(3)

Hence, the coupled electron−nuclear dynamics is governed by three terms, the first two describing the collective rearrangement of both electrons and nuclei involving the single electronic C



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In the following, we show that MHOHG pump−probe spectroscopy is a very useful tool for the time-resolved measurement of this attosecond electron motion, which is very sensitive to nuclear motion, illustrated by a comparison between the isotopes H2+ and T2+: Figure 4a shows the MHOHG spectra

Figure 3. (a) Electron oscillation period Δτ(R) as function of internuclear distance R for an electronic wavepacket consisting of the two lower electronic states, Δτ1,2(R) = 2π/[Eσg1s(R) − Eσu1s(R)] (blue line) and of the two upper electronic states Δτ2,3(R) = 2π/[Eσu1s(R) − Eσg2P(R)] (green line). (b) Induced dipole moment by the pump laser pulse in T2+. The lasting oscillations after the end of the pump pulse at t = 2.75 fs are due to the coherence of electronic states σu1s and σg2p. (c) Electron oscillation period Δτ of the induced dipole moment as function of time, defined as the midpoint between two successive maxima. Figure 4. (a) Harmonic spectra for H2+ (upper panel) and T2+ (lower panel) for two selected delay times, tdel = 4.35 fs (red) and tdel = 4.55 fs (black). For H2+ the peaks are red-shifted when the delay time is increased whereas for T2+ different mechanisms in the plateau and cutoff region are observed: In the cutoff region, the same red shift as in H2+ is seen whereas in the plateau region, the intensity of a whole range of frequencies changes by orders of magnitude when the pump−probe delay time is changed. Details of the red shift for H2+ (upper panel) are given in the inset, which shows the shift of the peak centered around harmonic 21 at tdel = 4.35 fs when the pump−probe delay time tdel is increased. (b) shows horizontal cuts through both panels of (a) for harmonic 31 as a black line for H2+ and as a red line for T2+.

dissociating wavepackets and the quasi-static ground state is practically lost for internuclear distances R > 4a0 and the electronic motion is dominated by contributions from the two upper surfaces σu1s and σg2p. This is confirmed by Figure 3b,c, which shows the induced dipole, d(t) = ⟨Ψ(t)|z|Ψ(t)⟩, and the corresponding electron oscillation period, respectively, for the molecular ion T2+. Note that the maximal oscillation period of Δτ ≈ 460 asec reached after about 12 fs coincides nicely with the associated oscillation period calculated from the energy gap at R = 6.5a0, Figure 3a, and the corresponding position of the nuclear wavepacket at t = 12 fs, Figure 2b. D



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for H2+ (upper panel) and T2+ (lower panel) at the delay times tdel = 4.35 fs (red lines) and tdel = 4.55 fs (black lines). We observe that in both cases, changing the pump−probe delay time by 200 asec changes the intensity of all spectral components in general by orders of magnitude. Additionally, we observe distinct differences in the characteristics of the spectra of the two isotopes due to nuclear movement: In the case of H2+ (upper panel), changing the pump−probe delay time results in general in a red shift of the peaks with only slight variations in the peak intensities whereas for T2+ (lower panel) the situation in the plateau and cutoff region is different: In the cutoff region (frequencies > harmonic 33) we observe the same red shift as for H2+ whereas in the broad plateau region, the intensity of a broad range of frequencies is lowered by orders of magnitude when the delay time is changed by 200 asec. This feature (which manifests via appearance of vertical valleys) does not appear for H2+ because at tdelay > 3 fs already a significant part of the nuclear wave function is nonzero at R > 4a0 and at this internuclear distance the contribution of two upper states predominates due to the loss of the overlap between the nuclear wave packets on the two lower surfaces. Consequently, the ground state surface no longer contributes to the interference pattern. Further details of the red shift in the case of H2+ are shown in the inset in the upper panel of Figure 4a, which shows the migration of one specific peak centered around harmonic 21 at delay time tdel = 4.35 fs and ending up being centered around harmonic 19 for delay time tdel = 4.55 fs by the respective peak positions at delay times tdel = 4.4, 4.45, and 4.5 fs. Besides, we notice that the peaks are roughly separated by 2ω0 in the case of H2+ and by ω0 in the case of T2+, where ω0 is the central laser frequency. In Figure 4b we fix arbitrarily harmonic 31 and plot its intensity as function of pump−probe delay time. One sees that indeed its intensity changes by orders of magnitude with an oscillation period around 440 asec monitoring the attosecond electron movement in the coherent superposition of electronic states σu1s and σg2p (cf. Figure 3c). We observe two considerable dips in the intensities that occur for H2+ in the range 4.5 fs < tdel < 7 fs and for T2+ at later delays in the range 7 fs < tdel < 9.5 fs, manifesting again the effect of nuclear motion on the harmonic spectra. Both major effects of nuclear motion on the harmonic spectra described so far are nicely illustrated for the whole range of delay times considered in this study in Figure 5, which shows contour plots of the harmonic spectra as function of tdel for both isotopes. Figure 5a (H2+) shows that the previously described dip in the intensity of harmonic 31 in the range 4.5 fs < tdel < 7 fs extends to all higher lying frequencies, thereby diminishing considerably the cutoff in the indicated ranges of delay times. Figure 5b shows that an analogous suppression of a whole range of harmonics occurs likewise for T2+, which starts, however, at delays tdel ≈ 7 fs due to the slower nuclear movement. Furthermore, the above-described red shift of the peaks for H2+ (Figure 4a) is manifested in the contour plot by the occurrence of diagonal ridges of strong intensity (Figure 5a). In the case of T2+ (Figure 5b) these diagonal ridges appear for the broad plateau region considerably later at around tdel = 5 fs, which roughly corresponds to internuclear distances R = 4a0. Before, one observes rather synchronized alternating strong and low intensities of the harmonic spectrum in the plateau region (cf. Figure 4b). Note that, because the laser probe pulse is extremely short, any frequency can be generated.40 Moreover, even for longer pulses nonharmonic peaks ω = 2nω0 + (Eb − Ea) (where Eb, Ea are energies in au of two electronic states) are expected to appear31

Figure 5. Dependence of harmonic spectra on pump−probe delay time tdel. (a) The diagonal ridges for H2+ indicate the red shift of the respective peaks with increasing pump−probe delay time tdel. (b) For T2+, the same red shift appears at delay times tdel > 5 fs. For tdel < 5 fs, the mechanism is split into two parts: In the cutoff region (frequencies > harmonic 33) the peaks are red-shifted as in H2+ whereas in the broad plateau region the intensity of a broad range of frequencies is lowered by orders of magnitude when the delay time is changed.

when harmonics are coherently generated from two electronic states having opposite parity. Thus, if (Eb − Ea)/ω0 is an even number, we expect even harmonics to appear. Thus, when harmonics are generated with a short pulse whose spectrum is broad, in principle, any high order frequency can be generated. This is very clearly seen in the inset of Figure 4a where we see that, by varying the time delay continuously from trmdel = 4.35 to 4.55 fs, the peak shifts continuously from harmonic 21 down to harmonic 20. Thus we see that any frequency from this interval can be generated. For analysis of the differences in the harmonic spectra of H2+ and T2+, we calculate the time profile of the spectra shown in Figure 4a using the Morlet-wavelet transform W(t,ω) of the acceleration a(t) (as defined in refs 32−34) W (t ,ω) =

E

∞ ω a(t ′) exp{[−i(t ′−t )ω]} 1/2 −∞ σπ ⎧⎡ ω 2(t ′−t )2 ⎤⎫ × exp⎨⎢ − ⎥⎬ d t ′ 2σ 2 ⎦⎭ ⎩⎣

∫

⎪

⎪

⎪

⎪

(4)



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Figure 6. Time frequency spectra corresponding to harmonic spectra shown in Figure 4a for H2+ (a, upper panel) and T2+ (b, lower panel) for pump− probe time delays tdel = 4.35 fs (left) and tdel = 4.55 fs (right). For H2+, we observe one strong electron return within one-half laser cycle corresponding to short electron trajectories (upper right panel) whereas for T2+ the intensity of the first short trajectory is considerably decreased and we observe two strong electron returns within the second half laser cycle corresponding to short and long electron trajectories (lower right panel). These strong returns vanish for tdel = 4.35 fs (lower left panel) (cf. Figure 4a, lower panel).

We set the width of the Gaussian time window to σ = 2π. These time−frequency spectra are shown in Figure 6a for H2+ and Figure 6b for T2+ for the two time delays tdel = 4.35 fs (left) and tdel = 4.55 fs (right) (cf. Figure 4a). For cutoff harmonics we observe two electron returns around 0.9 and 1.4 optical cycles where we define t = 0 as the start of the respective probe pulses. For plateau harmonics these electron returns usually split into contributions from short and long electron trajectories; see the Introduction and refs 6 and 36 for details. For H2+, however, the intensities of the short trajectories significantly exceed those from the long trajectories (Figure 6a). This mechanism persists over a long range of internuclear distances (see Figure 7a, which shows the time profile of the plateau harmonic 29 as function of pump− probe delay time). For delays tdel < 5 fs, strong intensities of short trajectories returning around 0.75 and 1.25 optical cycles are seen whereas the intensities of the long trajectories are considerably lower. Starting at tdel > 6 fs, the second short trajectory is shifted to larger return times, which corresponds to the expected return times in the cutoff region. In the regime 5 fs < tdel < 7 fs very weak electron returns for both short and long trajectories are observed, which corresponds to the dip and lowered cutoff in the harmonic spectra described above (cf. Figures 4b and 5a). For T2+, on the other hand, a completely different pattern of electron returns is observed. Even for early delay times, e.g., for tdel = 4.55 fs, right panel in Figure 6b, the second long trajectory returning shortly

after 1.4 optical cycles constitutes the most intense electron return followed by the associated short trajectory whereas the contributions of both short and long trajectory returning around 0.9 fs are both practically negligible. Note that for T2+ the time− frequency spectra depend strongly on the pump−probe time delay illustrated by the significantly lowered intensity of all electron returns at tdel = 4.35 fs, left panel of Figure 6b. This is due to the fact that in this range of internuclear distances, a whole range of harmonics is suppressed when the delay is changed by few hundred attoseconds (cf. the lower panel in Figure 4a). For H2+, on the other hand, this dependence is less pronounced because the averaging over several harmonics does not allow us to resolve the above-described red shift of harmonic peaks (Figure 6a). For delays tdel > 5 fs, we observe for T2+, in analogy to H2+, major contributions from the short trajectories returning shortly before 0.9 and 1.4 optical cycles whereas the intensity of the hitherto strong long trajectory decays practically to zero. The observed red shift of the peaks in the MHOHG spectra as a function of the delay time, therefore, correlates with the contributions of a single electron return within one-half laser cycle, even in the plateau region. For H2+, these returns are associated with electron trajectories at early delay times, whereas for large delay times (tdel > 6 fs) and associated larger internuclear distance R, one of these two short electron trajectories is shifted F



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of dissociating H2+, due to the continuing overlap of the dissociative nuclear wavepackets on the two upper electronic states σu1s and σg2p (Figure 2) prepared by the UV pump laser pulse, the coherent attosecond electronic motion persists over large internuclear distances and is mapped on the MHOHG spectra. Although the electron oscillation period and hence the periodicity of the harmonic spectra are quite constant over a wide range of internuclear distances, we observe a strong dependence on the internuclear distance of both the shapes of the MHOHG spectra and the way in which these spectra change as function of the pump−probe delay time. This is illustrated by interchanging the H2+ molecular ion by its heavier isotope T2+, hence reducing the speed of nuclear movement (Figure 2). Whereas for H2+ the variations of the MHOHG spectra involve a red shift of the associated peaks in the spectrum with increasing pump−probe time delay, we observe that for T2+ up to delay times tdel ≈ 5 fs, which corresponds roughly to internuclear distances R = 4a0, the variations involve a collective decrease of the intensity of a whole range of harmonics in the plateau region (Figures 4a and 5) while the peaks in the cutoff region are also red-shifted. Furthermore, at specific ranges of internuclear distances, the MHOHG cutoff is considerably decreased, which occurs of course earlier in the case of H2+ compared to T2+ (Figures 4b and 5). The analysis of the corresponding time−frequency spectra (Figure 6 and Figure 7) shows again strong signatures of nuclear movement. For H2+, up to pump−probe time delays tdel ≈ 6 fs two short electron trajectories contribute mainly to the MHOHG spectra in the plateau region over large ranges of internuclear distances, whereas for tdel > 6 fs the second short trajectory is shifted to larger return times, which coincide with those return times expected for cutoff harmonics. For T2+, on the other hand, up to delays tdel ≈ 5 fs, which corresponds to internuclear distances of roughly R = 4a0, the strongest intensity is due to one long electron trajectory with minor contributions from the associated short one. Therefore, the observed red shift of the peaks in the MHOHG spectra as a function of pump−probe delay time correlates with the sole contribution of one electron return within one-half laser cycle whereas the sole contribution of one long and associated short trajectory correlates with the attenuation of a whole range of harmonics when the delay time is changed by a few hundred attoseconds. The tentative explanation why the peak is red-shifted was given in Figure 11 in ref 24, which gives the plot of the harmonic order as a function of the electron return time together with the plot of the electron velocity in the two-state system. One assumes that maxima in HHG occur when the returning electron and electron oscillating in the two-state system propagate in the same direction. This figure shows that for longer time delays the maxima in the electron velocities shift to the left, coinciding with the positive slope in the left part (short trajectories) of the plot of the harmonic order as a function the electron return time.

Figure 7. Time frequency spectrum at fixed frequency corresponding to harmonic 29 as function of pump−probe delay time tdel for H2+ (upper panel, a) and T2+ (lower panel, b). Short and long electron trajectories are marked for this plateau harmonic. Note that for delay times tdel > 6 fs, the second short trajectory in the case of H2+ (upper panel) is shifted to longer electron return times, which coincide with those return times expected for cutoff harmonics.

to larger return times, which corresponds to the expected return times in the cutoff region. The sole appearance of short and long trajectories shortly before and after 1.4 optical cycles, on the other hand, correlates with the attenuation of a whole range of harmonics. We think that it is possible that the proton (or triton) motion during the electron sojourn in the continuum affects the ratio between short and long trajectories because the longer is this sojourn the continuum, the larger is the loss of the overlap between the corresponding nuclear wave packets; see refs 36 and 44 and references therein for more details about this effect.

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SUMMARY AND CONCLUSIONS Our non-Born−Oppenheimer TDSE simulations show that molecular high order harmonic pump−probe spectroscopy is a very promising tool for the real time observation of attosecond electron motion, which is at the same time sensitive to nuclear motion. In the present study, we use 800 nm probe pulses of total duration of 5.3 fs to monitor attosecond electronic motion with an oscillation period of about 450 asec (cf. Figure 3). This is possible because (M)HOHG occurs on the subcycle time scale, as suggested by the semiclassical three-step model and confirmed by time−frequency analysis of the MHOHG spectra (cf. Figures 6 and 7, which show that the time-width of emission of a given harmonic is on the order of (1/10) of the laser cycle). In the case

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS T.B. thanks Deutsche Forschungsgemeinschaft (DFG, project Ma 515/25-1) for financial support. G



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(41) Bandrauk, A. D.; Shen, H. J. Chem. Phys. 1993, 99, 1185. (42) Diestler, D. J. Phys. Rev. A 2008, 78, 033814. (43) Bandrauk, A.; Chelkowski, S.; Diestler, D.; Manz, J.; Yuan, K. Phys. Rev. A 2009, 79, 023403. (44) Patchkovskii, S. Phys. Rev. Lett. 2009, 102, 253602.

REFERENCES

(1) Zewail, A. H. J. Phys. Chem. A 2000, 104, 5660. (2) Daniel, C.; Full, J.; González, L.; Lupulescu, C.; Manz, J.; Merli, A.; Vajda, S.; Wöste, L. Science 2003, 299, 536. (3) Peirce, A. P.; Dahleh, M. A.; Rabitz, H. Phys. Rev. A 1988, 37, 4950. (4) Kosloff, R.; Rice, S.; Gaspard, P.; Tersigni, S.; Tannor, D. Chem. Phys. 1989, 139, 201. (5) Femtosecond Chemistry; Manz, J., Wöste, L., Eds.; VCH: New York, 1995. (6) Krausz, F.; Ivanov, M. Rev. Mod. Phys. 2009, 81, 163. (7) Attosecond Molecular Dynamics; Bandrauk, A., Manz, J., Vrakking, M., Eds.; Chem. Phys. (8) Quantum Dynamics Imaging; Bandrauk, A., Ivanov, M., Eds.; CRM Series in Mathematical Physics; Springer, 2011. (9) Barth, I.; Hege, H.; Ikeda, H.; Kenfack, A.; Koppitz, M.; Manz, J.; Marquardt, F.; Paramonov, G. Chem. Phys. Lett. 2009, 481, 118. (10) Bredtmann, T.; Manz, J. Angew. Chem., Int. Ed. 2011, 50, 12652. (11) Andrae, D.; Barth, I.; Bredtmann, T.; Hege, H.; Marquardt, F.; Paulus, B. J. Phys. Chem. B 2011, 115, 5476. (12) Wörner, H. J.; Bertrand, J. B.; Kartashov, D. V.; Corkum, P. B.; Villeneuve, D. M. Nature 2010, 466, 604. (13) Dutoi, A.; Cederbaum, L. S. J. Phys. Chem. Lett. 2011, 2, 2300. (14) Łabuda, M.; González-Vázquez, J.; Gonzlez, L. Phys. Chem. Chem. Phys. 2010, 12, 5439. (15) Klinkusch, S.; Klamroth, T.; Saalfrank, P. Phys. Chem. Chem. Phys. 2009, 11, 3875. (16) Okuyama, M.; Takatsuka, K. Chem. Phys. Lett. 2009, 476, 109. (17) Yonehara, T.; Takatsuka, K. Chem. Phys. 2009, 366, 115. (18) Nagashima, K.; Takatsuka, K. J. Phys. Chem. A 2009, 113, 15240. (19) Sansone, G.; et al. Nature 2010, 465, 763. (20) Goulielmakis, E.; Loh, Z.; Wirth, A.; Santra, R.; Rohringer, N.; Yakovlev, V. S.; Zherebtsov, S.; Pfeifer, T.; Azzeer, A. M.; Kling, M. F.; Leone, S. R.; Krausz, F. Nature 2010, 466, 739. (21) Bredtmann, T.; Chelkowski, S.; Bandrauk, A. D. Phys. Rev. A 2011, 84, 021401. (22) Bandrauk, A. D.; Chelkowski, S.; Nguyen, H. S. Int. J. Quantum Chem. 2004, 100, 834. (23) Bandrauk, A. D.; Chelkowski, S.; Corkum, P. B.; Manz, J.; Yudin, G. L. J. Phys. B: At. Mol. Opt. Phys. 2009, 42, 134001. (24) Chelkowski, S.; Bredtmann, T.; Bandrauk, A. D. Phys. Rev. A 2012, 85, 033404. (25) Niikura, H.; Villeneuve, D.; Corkum, P. Phys. Rev. Lett. 2005, 94, 083003. (26) Lewenstein, M.; Balcou, P.; Ivanov, M. Y.; L’Huillier, A.; Corkum, P. B. Phys. Rev. A 1994, 49, 2117. (27) Lewenstein, M.; Salères, P.; L’Huillier, A. Phys. Rev. A 1995, 52, 4747. (28) Milošević, D. B.; Becker, W. Phys. Rev. A 2002, 66, 063417. (29) Corkum, P. B. Phys. Rev. Lett. 1993, 71, 1994. (30) Krause, J. L.; Schafer, K. J.; Kulander, K. C. Phys. Rev. Lett. 1992, 68, 3535. (31) Milošević, D. B. J. Opt. Soc. Am. B 2006, 23, 308. (32) Chandre, C.; Wiggins, S.; Uzer, T. Physica D 2003, 181, 171. (33) Bandrauk, A. D.; Chelkowski, S.; Kawai, S.; Lu, H. Phys. Rev. Lett. 2008, 101, 153901. (34) Bandrauk, A. D.; Chelkowski, S.; Lu, H. In Quantum Dynamics Imaging; CRM Series in Mathematical Physics; Springer: New York, 2011; p 37. (35) Bandrauk, A. D.; Chelkowski, S.; Goudreau, S. J. Mod. Opt. 2005, 52, 411. (36) Marangos, J. P.; Baker, S.; Kajumba, N.; Robinson, J. S.; Tisch, J. W. G.; Torres, R. Phys. Chem. Chem. Phys. 2007, 10, 35. (37) Bandrauk, A. D.; Chelkowski, S.; Corkum, P. B.; Manz, J.; Yudin, G. L. J. Phys. B: At. Mol. Opt. Phys. 2009, 42, 134001. (38) Chelkowski, S.; Foisy, C.; Bandrauk, A. D. Phys. Rev. A 1998, 57, 1176. (39) Chelkowski, S.; Bandrauk, A. D. Phys. Rev. A 2002, 65, 061802. (40) de Bohan, A.; Antoine, P.; Milošević, D. B.; Piraux, B. Phys. Rev. Lett. 1998, 81, 1837. H


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