Even harmonic generation in isotropic media of dissociating homonuclear molecules

Isotropic gases irradiated by long pulses of intense IR light can generate very high harmonics of the incident field. It is generally accepted that, due to the symmetry of the generating medium, be it an atomic or an isotropic molecular gas, only odd harmonics of the driving field can be produced. Here we show how the interplay of electronic and nuclear dynamics can lead to a marked breakdown of this standard picture: a substantial part of the harmonic spectrum can consist of even rather than odd harmonics. We demonstrate the effect using ab-initio solutions of the time-dependent Schrödinger equation for and its isotopes in full dimensionality. By means of a simple analytical model, we identify its physical origin, which is the appearance of a permanent dipole moment in dissociating homonuclear molecules, caused by light-induced localization of the electric charge during dissociation. The effect arises for sufficiently long laser pulses and the region of the spectrum where even harmonics are produced is controlled by pulse duration. Our results (i) show how the interplay of femtosecond nuclear and attosecond electronic dynamics, which affects the charge flow inside the dissociating molecule, is reflected in the nonlinear response, and (ii) force one to augment standard selection rules found in nonlinear optics textbooks by considering light-induced modifications of the medium during the generation process.

Common wisdom suggests that even harmonics should disappear for long driving laser pulses, restoring the expected symmetry of the overall process and balancing contributions from the adjacent laser half-cycles. However, results of our study of high harmonic generation from + H 2 , + D 2 and + T 2 , using a full dimensionality model for the electronic and vibrational degrees of freedom, contrast with this expectation even for rather long laser pulses. We find that the nuclear motion has an even more dramatic effect than anticipated in the previous work. For sufficiently long pulses, the HHG spectrum of the lighter molecules can exclusively consist of even harmonics in the plateau region. Furthermore, by changing the pulse duration, one can control the region of the plateau where the even harmonics appear. We unambiguously link the appearance of even harmonics to electron localization on one side of the molecule. This induces a permanent dipole during dissociation, even for pulses containing tens of laser cycles. The linear Stark effect associated with the permanent dipole introduces a relative phase between the two consecutive electron round trips initiated in the adjacent laser half-cycles. As this phase approaches π, the odd harmonics are replaced by the even ones.
Several works on even harmonic generation in diatomic molecules have been presented before [23][24][25][26] . However, the origin of even harmonic generation reported there is either the nuclear mass asymmetry 25,26 or contributions from the nuclear quadrupole moment 23,24 . In contrast, here we deal with a homonuclear molecule and a purely dipole effect. When electron localization is present, it gives the strongest contribution to the even harmonics signal, on par with the strength of the odd harmonics. It is also important to emphasize that, in the present work, electron localization leading to even harmonic emission is purely induced by the IR field, at variance with earlier work 27 , where it results from the coherent superposition of gerade and ungerade molecular states arising from the interaction with a UV pump pulse that precedes IR irradiation. Figure 1 shows the calculated HHG spectra for the three different isotopes of the + H 2 molecule and different pulse durations. For the shortest pulse, 5 optical cycles, our results are very close to our earlier results for low dimensionality models 20,21 . The spectra from different isotopes are generally very similar, with several broad even harmonic lines between orders 36-40. The effect is independent of the isotope and hence of the nuclear motion; it is associated with the frequency chirp induced by the changing laser intensity between successive half-cycles in an ultra-short pulse. As expected, the effect disappears for the longer, 10-cycle pulse, and for the heaviest isotope, where the nuclear motion is negligible. However, for the lightest isotope even harmonics remain very prominent for 10, 14, and 20-cycle pulses. As one moves to higher orders, the harmonic peaks in + H 2 experience a red shift, up to a point that only even harmonics are observed for high enough orders. The spectral region where even harmonics dominate shifts to higher orders with increasing pulse duration. For the 20-cycle pulse even harmonics also appear for the + D 2 isotope. Additionally, for the longer pulses, the harmonics from + D 2 and + T 2 are strongly suppressed in the plateau region compared to + H 2 , with rather dramatic modifications of the shape of the harmonic lines. What could be the origin of even harmonics in + H 2 , their shift with the pulse duration, and the emergence of two spectral regions where they are seen for 20-cycle pulse (the plateau and the cutoff)? What could be the origin of the combination of harmonic suppression and the dramatic harmonic line shape modifications for + T 2 ?

HHG Spectra
Two effects may be responsible. The first is electron localization, prominent in molecular dissociation 7,28 . Electron localization breaks the symmetry of the system and hence can lead to even harmonics 20,21,29 . The second is the asymmetry introduced by molecular dissociation between the raising and descending parts of the IR pulse 22 , in particular due to the shift in the characteristic ionization potential I p . As pointed out in ref. 22, this asymmetry leads to the red shift. The analysis of the harmonic spectra for different pulse durations and isotopes allows us to distinguish the contributions of these two effects. We argue that both are important but have quite different impact in multi-cycle pulses. While electron localization controls the interference of the emission bursts from successive laser half-cycles, the front-back asymmetry pertains to the longer time-scale and hence leads to finer-scale modifications in the harmonic spectrum, for long pulses.
We first analyze the impact of dissociation-induced asymmetry between the raising and the falling edges of the pulse. As discussed above, the harmonic spectrum is formed by the interference of emission bursts produced during successive half-cycles. If the system response during several successive half-cycles is nearly identical, odd harmonics will form during the raising part of the pulse. The same would apply at the falling edge of the pulse, even if the system has changed due to dissociation. The interference of odd harmonics generated at the front and at the back of the pulse will then lead to modification of harmonic lineshapes, within their width. However, it will not turn odd harmonics into even. If the contributions from the raising and the falling parts of the pulse are phase-shifted by π in some spectral window, strong reshaping and suppression in this spectral window may result. This is precisely what we find for the heaviest isotope, + T 2 , for a 20-cycle pulse, between harmonics 33-45: overall suppression, strong modification of line-shapes, but prominent odd harmonics nevertheless. To shift the harmonic lines on the scale of ħω IR , where ω IR is the laser frequency, one has to modify the interference between successive half-cycles. We now show that the generation of a permanent dipole moment due to electron localization is the origin of such modification and the appearance of even harmonics.

Electron Localization
The dissociative dynamics leading to eventual electron localization is well understood 28 . A sketch of this dynamics is shown in Fig. 2. During the first few cycles, the IR field, which is not yet intense enough to significantly ionize the molecule, induces Rabi-type oscillations between the 1sσ g and 2pσ u states, which lie much closer in energy than the 1sσ g state and the ionization continuum (process 1). As a result, excited vibrational states associated with the 1sσ g electronic state can be efficiently populated 30 , thus creating a vibrational wave packet. A similar vibrational wave packet is formed in the 2pσ u state. These wave packets then move towards larger internuclear distances (process 2), until they reach a region of internuclear distances where the 1sσ g and 2pσ u electronic states are very close in energy and are strongly coupled by the IR field. Therefore they mix, leading to localized states 1sσ g ± 2pσ u . The characteristic value of the internuclear distance R = R C where the onset of localization occurs can be estimated by using the criterion from ref. 31, , where E 0 is the field amplitude and ω gu the energy difference between the 1sσ g and 2pσ u states at R = R C . This criterion yields . Fig. 2 with a pink vertical line. Localization opens the barrier for dissociation, through the process called bond softening 32 . By then, the IR field has reached (or nearly reached) its peak intensity and tunnel ionization becomes prominent (process 3). This enhancement of tunnel ionization in the long R region is caused by both the reduced ionization potential (compared to the equilibrium geometry), which decreases with R, and the enhanced ionization caused by electron localization 33,34 . It is important to stress that localization of the electron on one or the other side of the molecule depends on the phase of the laser field 35 and, therefore, can be controlled by the carrier-envelope phase 28 . Thus, the ulterior recombination process, i.e., harmonic generation, is acutely sensitive to the window of R where the medium symmetry is broken.
To quantify this picture and confirm our explanation, we have evaluated an R-dependent HHG spectrum, S(R, ω), as the square of the Fourier transform of , R is the internuclear distance, and  is the dipole acceleration operator. The calculated R-dependent HHG spectrum is shown in Fig. 3 for the 14-cycles pulse. As can be seen, even harmonics appear near the localization distance  R R C a.u., far beyond the equilibrium distance R eq = 1.9 a.u., in agreement with our arguments. This also explains the lack of even harmonics for heavier isotopes, which dissociate slower and do not reach the localization region before the pulse is over. Figure 4 shows the time-windowed Fourier transforms of the calculated wave functions (Gabor profiles) and the time evolution of the nuclear wave packets in + H 2 for 10 and 20 cycle pulses. For each pulse duration we select the time, t C , at which the density around the critical internuclear distance, R C = 3.7 a.u., is largest. By looking at the Gabor profile at t > t C and at the harmonics that are emitted at t C , we can predict the location of the even harmonics in the HHG spectra. For the 10-cycles pulse, R C is reached when the intensity of the laser pulse is already decreasing. This leads to even harmonics in the lower region of the HHG spectrum. In contrast, for the 20-cycles pulse, R C is already reached when the intensity of the laser pulse is still increasing. Consequently, even harmonics appear at higher energies in the spectrum. Note that even harmonics arise for sufficiently long pulses, when the nuclear wave packet has had enough time to reach R C . By controlling the pulse duration one thus controls the region of the plateau where even harmonics appear.
It is interesting to point out that the magnitude of the nuclear wave packets shown in Figure 4 outside the Franck-Condon region, e.g., at R~R C , is much larger than that obtained from earlier low-dimensionality models 20 . In other words, nuclear dynamics is actually much faster in the true three-dimensional space. Consequently, the intensity of the even harmonic peaks predicted by low dimensionality models is much lower than that of the odd harmonic peaks, which dominate the spectrum for all harmonic orders, in contrast with the results of the present work.
The physical picture that identifies electron localization as the origin of even-harmonic generation is further substantiated by analyzing the phase-shift in the emission bursts during successive half-cycles, accumulated due to the induced dipole moment. Figure 5 shows a sketch of the typical trajectory followed by an electron starting in a delocalized and localized initial bound state. In the second case, the localized bound state experiences a linear Stark shift, leading to additional phase difference accumulated between the ionization t i and recombination t r times, For trajectories generated in consecutive half-cycles, the accumulated phase difference is twice as large, In the last equality we have used the three-step model of HHG: for the electron starting with zero velocity at t i , the integral of the laser field E(t) between t i and t r gives its instantaneous electron velocity v(t r ) at the return time t r . Thus, we can rewrite this phase difference in terms of the electron recombination energy E kin = v 2 (t r )/2, or the emitted photon energy, Nω IR , Figure 6 shows this phase difference as a function of the harmonic order and the internuclear distance. We have checked, by solving the classical equations of motion numerically, that the inclusion of the molecular potential adds an extra phase, which is however much smaller than that shown in Fig. 6, so that the global picture remains unchanged. In the vicinity of .  R 3 7 C a.u., where localization takes place, there is a large range of harmonic orders where the additional phase difference between the right and the left trajectories is ≈ π, thus leading to even harmonic generation. According to Fig. 6, this occurs approximately between harmonic orders 20 and 40. Remarkably, Fig. 6 predicts a phase difference of ≈ 3π, hence a revival of even harmonic emission, between harmonic orders 60 and 80, in excellent agreement with the appearance of even harmonics in the cut-off region predicted by the ab initio calculations for + H 2 (see Fig. 1c,d). Finally, Fig. 6 also shows that the positions of the harmonic lines experience a slow frequency shift across the spectrum as R C is varied, again in agreement with our ab-initio observations.

Conclusion
Using the example of one-electron homonuclear diatomic molecules, we have shown how dynamics induced in the molecule can lead to a dramatic breakdown of the standard selection rules in high harmonic generation, including the nearly complete suppression of odd and the appearance of even harmonics for multi-cycle laser pulses, in a broad window of the harmonic spectrum. Our analysis links strong shifts of the harmonic lines with the appearance of a permanent dipole moment in dissociating homonuclear molecules, caused by electron localization. This dipole moment introduces phase shifts between the emission bursts during successive laser half-cycles, which can approach and exceed π. The ultimate origin of symmetry breaking is the sensitivity of the overall process to the carrier envelope phase of the laser pulse. The fact that minute changes in the driving laser field, associated with the carrier-envelope phase of a multi-cycle (20-cycle) laser pulse, can lead to strong effects in the harmonic spectrum, reflects very strong sensitivity of the underlying dissociation -localization dynamics to the details of the driving field. In classical systems, such extreme sensitivity is characteristic of dynamical chaos. Thus, our results suggest that high harmonic generation might also be a sensitive probe for the onset of dynamical chaos in light-driven systems.
At first sight, the pulse durations required to see dominant even-harmonic emission are much longer than what one would normally expect when studying electron localization by detecting dissociating fragments. Electron localization, when observed in molecular fragmentation, is a 'zero background' signal, i.e., with no contamination from non dissociative channels. In this case, the shorter the laser pulse, the better, since asymmetry in the pulse temporal profile due to the carrier envelope phase (CEP) matters. This is why pioneering experiments on electron localization in molecular fragmentation 7,28 have been most successful for relatively short pulses. However, this is not the case for HHG, where non-dissociated molecules also contribute to the signal. For short pulses, when the fraction of dissociating molecules is still tiny, their response and the associated even harmonic signal drown in the response of the non-dissociating molecules. Only when the pulses are long enough to produce a significant number of dissociating molecules, emission of even harmonics will be observed. On the other hand, if the pulse is too long, the effect should disappear, because the CEP does not matter any more and, therefore, the probabilities to localize the electron on the left or on the right of the dissociating molecule are identical. Clearly, there must be a window of pulse durations for even harmonics in HHG. Calculations performed for 30-cycle pulses show that this is indeed the case: as can be seen in Fig. 7, for such a long pulse, one begins to see a decrease in the strength of the even harmonic signal.
One could also ask whether the effect we observe would still be present in the macroscopic response. First, we note that as far as initial conditions are concerned, we start with the vibrational ground state. This is adequate for + H 2 even at room temperature. As far as the phase matching is concerned, our analysis shows nothing special regarding the phases of even harmonics compared to those of the odd ones. Thus, we do not see any reason for the propagation to quench the single-molecule effects.

Methods
Our theoretical method has been described in detail in ref. 36. Briefly, we solve the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) in cylindrical coordinates, ρ and z for the electron, R for the internuclear distance, where z coincides with the linear polarization direction of the electric field. We assume that the molecules are aligned with the linearly polarized driving IR field, neglecting their rotations. The electron azimuthal coordinate φ is removed by the cylindrical symmetry of the problem. The TDSE reads , , ) [ ( , , ) ( ) ( , )] ( , , , ), el where H el = T el + V eN + 1/R is the electronic Hamiltonian of + H 2 or its molecular isotopes, T is the nuclear kinetic energy operator, V = zE(t) describes the interaction with the laser field in the length gauge, T el is the nuclear kinetic energy operator, and the Coulomb potential V eN describes the electron -nuclei interaction. Atomic units are used throughout unless stated otherwise. The external laser field is 2 E 0 the electric field amplitude, ω the central frequency corresponding to the wavelength λ = 800 nm, and T the total pulse duration ranging between 5 and 20 optical cycles (13.34-53.36 fs). The peak intensity in all calculations is 3 × 10 14 W/cm 2 .
We have used a non-equidistant cubic 3D grid with |z| < 55, ρ < 50 and R < 30, and grid spacings Δ z = 0.1, Δ ρ = 0.075 and Δ R = 0.05 a.u. at the center of the grid. The grid spacings increase gradually from the origin to the box boundaries 36 . The Crank-Nicolson propagator with a split-operator method was used for time-propagation, with a time step Δ t elec = 0.011 a.u. for the electrons and Δ t nuc = 0.11 a.u. for the nuclei. The convergence of these parameters was checked. The initial, ground, state of the + H 2 molecule was obtained by diag-