Distinguishing two mechanisms for enhanced ionization of ${\rm H}_2{^+}$ using orthogonal two-color laser fields

We theoretically study the ionization enhancement of the diatomic molecular ion H2+ at two critical internuclear distances R, using orthogonal two-color laser fields. The polarization of the fundamental infrared laser field and a weak second-harmonic field is parallel and perpendicular to the molecular axis, respectively. It is observed that adding the second-harmonic field raises slightly the first ionization peak at the smaller critical R, whereas it enhances the second one at the larger critical R significantly. We further analyze the observable evidence which distinguishes two underlying mechanisms responsible for the enhanced ionization of H2+: (i) the resonant excitation along with the coherent interference of the ionizing wave packets from the 1sσg and 2pσu states and (ii) the easier ionization from the up-field site of the molecule.


I. INTRODUCTION
As the simplest molecular system in nature, the hydrogen molecular ion H 2 + has been widely applied as a prototype for studying strong-field ionization [1] and the consequential ultrafast phenomena [2] beyond atoms. While the ionization and dissociation mechanisms of H 2 + have been extensively studied in the past decades [3,4], interest remains incessant as there are still challenges and questions raised lately by new insights [5]. In this paper, we focus on one of the intricate strong-field phenomena regarding H 2 + -its enhanced ionization (EI) at critical internuclear distances .
In the 1990s, it was reported [6][7][8][9][10][11][12] that, when the molecule is stretched to a critical internuclear distance R, the ionization rate increases significantly. In particular, the earliest theoretical study [6] predicted significant ionization enhancement at two critical internuclear distances of H 2 + , based on a model with fixed R. In 2015, two ionization peaks were observed in experiment using a few-cycle-pulse pumpprobe technique [25]. However, the remaining question is whether there are indeed two ionization peaks at two critical internuclear distances or there is suppression of the enhancement that leads to the observation of two peaks. To answer this question, the underlying dynamics needs to be identified.
In Ref. [6], it was proposed that the ionization peaks at two internuclear distances are due to different mechanisms.
On one hand, the electron has a much higher probability of escaping directly from the upper-field-induced potential well at a critical R (typically in the range from 6 to 10 a.u.), due to the smaller ionization barrier compared to that for a hydrogen atom under the same electric field. This ionization process has been demonstrated theoretically [5,17] and in experiment [23]. On the other hand, it was shown [6] that the enhanced ionization at a relatively smaller R is related to the asymmetric electron localization on the nuclei. Strong-field-induced electron localization in dissociating H 2 + has been studied since 2004 [29][30][31][32][33][34][35][36][37][38]. Yet, to the best of our knowledge, little direct observable evidence has been shown to associate the EI with the asymmetric electron localization except in Ref. [38]. The underlying detail of the association between EI and electron localization in H 2 + is also unclear. The present study aims to present and analyze the observables that distinguish two mechanisms responsible for the EI of H 2 + . To this end, we apply orthogonal two-color (OTC) laser fields to interact with H 2 + . The fundamental laser field is polarized parallel to the molecular axis and triggers the ionization. The weak second-harmonic field is polarized perpendicularly to the molecular axis and plays the role of altering the interaction. It is found that, as the pulse intensity of the second-harmonic field raises, the ionization yield at the smaller critical R (the first ionization enhancement) is hardly changed, whereas that at the larger critical R (the second ionization enhancement) is significantly increased. Further analysis suggests that such observation can be explained by two qualitatively different mechanisms responsible for the EI of H 2 + .

II. NUMERICAL METHODS
Following Ref. [6], we employ for H 2 + the model with fixed internuclear distance R to study the dependence of the ionization dynamics on the internuclear distance. Such a theoretical model is justified for experiments where a fewcycle pulse is applied to probe the ionization of dissociating H 2 + at certain internuclear distances [25]. We solve numerically the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) in the Cartesian coordinate system for H 2 + . The molecule is aligned along the x axis. The fundamental infrared laser field is polarized in the x direction and the second-harmonic field is polarized in the y direction. The TDSE is given by (in atomic units) . The vector potential of the few-cycle OTC pulse is given by (within the dipole approximation) for 0 t 3T , with ω, T = 2π/ω, E i , and φ i (i = 1 and 2) being the fundamental laser frequency, the optical cycle of the fundamental field, the field amplitudes, and the carrier-envelope phases (CEPs), respectively. The corresponding electric field is calculated by E(t ) = −∂A(t )/∂t. The TDSE is numerically solved using the split-operator spectral method [39]. The 3D grid ranges both for x and y from −107.925 to 107.925 a.u. and for z from −47.875 to 47.875 a.u. with the spacing steps of x = y = 0.15 a.u. and z = 0.25 a.u. The grid contains 1440 × 1440 × 384 grid points. In our simulations, the nuclei are placed in the middle of two grid points in the x axis, in order to avoid the singularity of the Coulomb potential [40]. The time step for the evolution of the wave function is chosen as t = 0.01 a.u. to guarantee the numerical convergency. The initial stationary wave functions are obtained by the imaginary-time propagation method. The imaginary-time propagation is continued until the calculated ground energy is converged down to 10 −12 in each time interval of 0.2 a.u. In real-time evolution of the wave function, the in-box and outgoing wave packets are split by an ellipsoid-shaped absorbing function, which is defined as with r e = [x 2 + (x max /y max ) 2 y 2 + (x max /z max ) 2 z 2 ] −1/2 , where x max , y max , and z max indicate the maxima of the grid in the x, y, and z dimensions, respectively. The absorber parameters are r 0 = 100 a.u. and r = 3.5 a.u., which are the ideal choice for our box size according to our tests. To avoid the unphysical reflections at boundaries of the calculation box, the absorber is applied every 40 time steps (0.4 a.u. in time). The in-box wave function is then replaced by new (t ) = V abs (t ). After the laser pulse is off, the field-free propagation continues for an additional 15 fs so that the majority ionizing wave packet is absorbed. At the end of the propagation (t = t end ), the ionization yield is calculated by Y (R, ε) = 1 − (r, t end )| (r, t end ) for the given R and the field amplitude ratio ε = E 2 /E 1 of the OTC field. We adopt four significant digits for the plot of the ionization yield. Figure 1 depicts the electric field of a three-cycle, 800-nm fundamental laser pulse with E 1 = 0.06 a.u. and φ 1 = 0 (black thick curve) and those of the orthogonal second-harmonic pulses with E 2 = εE 1 (ε = 1/8, 2/8, 3/8, and 4/8) and φ 2 = 0. In this scheme, we increase the intensity of the secondharmonic field after each interaction but keep it relatively weak, in order to see how the ionization yield is altered. We have calculated the ionization yield as a function of R and ε. The results are shown in Fig. 2(a). In general, two ionization peaks are observed, in agreement with the previous study [6]. But what attracts our attention is that as the pulse intensity of the second-harmonic field increases, the first ionization peak (at R 1 = 4.5 a.u.) barely raises while the second ionization peak (at R 2 = 8.1 a.u.) is further enhanced significantly.

III. RESULTS AND DISCUSSION
Before looking into the underlying mechanisms responsible for such an observation, we have checked the influence of the second-harmonic field on the ionization yield at R 1 and R 2 by varying E 2 and φ 2 while the fundamental pulse is unchanged. The results in Fig. 2(b) show that for both φ 2 = 0 and φ 2 = π/2 the ionization yield at R 2 is enhanced by more The dashed arrow shows the further enhanced ionization with increasing ε. (b) Relative increasement of the ionization yields at R 1 and R 2 after adding the second-harmonic field with φ 2 = 0 and φ 2 = π/2, respectively. (c) Ionization yield at R 1 and R 2 by the single orthogonal second-harmonic field with φ 2 = 0. than 70% for E 2 = 0.03 a.u., while that at R 1 is increased less than 20%. It indicates that the CEP of the second-harmonic pulse does not change essentially the distinctive tendencies of the ionization yields for R 1 and R 2 in the OTC fields, as long as the fundamental pulse remains the same. The relative enhancement of the ionization yield for φ 2 = π/2 with respect to that for φ 2 = 0 is due to the relatively larger amplitude of the OTC field at t = 1.5T for φ 2 = π/2 (|E| max = √ E 2 1 + E 2 2 ). Note that without the fundamental pulse, the single second-harmonic field would hardly cause considerable ionization of H 2 + at either R 1 or R 2 [see Fig. 2(c)], since it is relatively weak and is polarized perpendicularly to the molecular axis. In particular, the ionization probability is less than 0.0006 even for the strongest case as shown in Fig. 2(c), whereas the absolute ionization yield at R 2 is increased by about 0.043 (more than 70 times to 0.0006) after adding the perpendicular pulse to the fundamental pulse. It is unlikely that the weak perpendicular component of the OTC field could independently lead to the considerably increased ionization yield at R 2 in the perpendicular direction to the molecule. This is consistent with the previous experimental observation [28] which showed that the laser pulse perpendicularly polarized with respect to the molecular axis can hardly lead to the enhanced ionization. Meanwhile, we did not find any considerable resonant excitation in the orthogonal direction by the 400-nm photons. Note that the present observation under the OTC pulses differs essentially from that in the previous study [41] where a very strong laser pulse polarized perpendicularly to the molecular axis was used. In the following, we analyze the underlying dynamics for the first ionization peak and then discuss the underlying dynamics for the second one.
First, let us revisit the explanation for two EI peaks given in the previous study [6]. As shown in Figs. 3(a) and 3(b), the energy level on the up-field site is lifted by approximately E 1 R/2 with respect to the ground state level of H 2 + and, as R increases, it becomes higher than the inner and outer potential barriers, eventually resulting in the enhanced ionization from the up-field site peaked at a critical R. Meanwhile, they found the significant asymmetric electron localization at a smaller critical R and believed that the enhanced population on the up-field site increases further the ionization, leading to another ionization peak. Such an explanation is certainly reasonable, and there have been studies [5,17,23] demonstrating the ionization from the up-field site of the molecules at certain internuclear distances. According to the present results shown in Fig. 2, however, the underlying dynamics for the first ionization peak could be more complicated. The reason is as follows. As shown in Fig. 1(b), the OTC field is expanded near t = 1.5T with respect to the fundamental linearly polarized (LP) one. Accordingly, we depict in Fig. 1(c) the field amplitude of the OTC field as a function of its direction with respect to the −e x direction. It shows that near the peak, the OTC field is still in the vicinity ( 10 • ) of the direction parallel to the molecule. Then, because the ionization rate depends exponentially on the field amplitude and the total ionization yield is the integration of the ionization rate over time, we expect, in general, more ionization yield near the peak of the OTC field with respect to the case of the LP field. In particular, near the peak of the fundamental pulse, the up-field levels at R 1 and R 2 are lifted by the fundamental field and the gaps between the lifted states and the barriers are comparable to each other for R 1 and R 2 [see Figs. 3(a) and 3(b)]. Then, if the molecule proceeds with the same mechanism (i.e., the easier ionization directly from the up-field site) at both R 1 and R 2 , one would expect close relative enhancement of two ionization peaks after adding the perpendicular second-harmonic field. Especially when more population is asymmetrically localized on the up-field nucleus at R 1 , the increased ionization yield at R 1 in OTC fields could have been even more pronounced if the ionization mechanism were the same for both R 1 and R 2 . However, we find in contradiction that the ionization yield at R 1 is hardly increased in the OTC fields while that at R 2 is significantly increased (see Fig. 2). Therefore, the mechanism responsible for the first ionization peak should qualitatively differ from the one responsible for the second ionization peak. field hardly modifies the evolution of the bound density in the x dimension, while it slightly modifies the interference pattern of ionizing wave packets. Second, it is noticeable that for R 1 the electron density is localized on the up-field nucleus (x = −R 1 /2) around the field maximum, which is in agreement with the previous study [6]. It indicates that the first ionization peak is somehow associated with the asymmetric electron localization, which will be explained later.
For a deeper insight, we define the wave functions σ ± = ( 1sσ g ± 2pσ u )/2 to represent the temporary states in which the wave packet is localized on the left (σ − ) and right (σ + ) nucleus, respectively, with 1sσ g and 2pσ u being the stationary wave functions of the ground and first excited states. We have calculated the time-dependent population of σ ± by Y σ ± (t ) = | σ ± | (t ) | 2 . The results are shown in Fig. 5, where the solid and dashed curves represent the cases using the LP and OTC pulses, respectively. In Fig. 5(a), for R 1 , it is shown that before the electric field hits the maximum (t = 1.5T ) the population is mostly on the up-field site (σ − ), creating a hole on the down-field site. Then, the population on the up-field site starts decreasing and, meanwhile, the temporary population appears on the down-field site (σ + ). Such a feature can also be seen in Figs. 4(a) and 4(c), where the dotted circles outline the temporary localization of the electron on the down-field site. Therefore, it is most likely that the ionizing electron originates from the up-field site, but instead of escaping directly from the up-field site, it has a certain probability to recombine with the down-field hole and then proceeds to a tunneling ionization from the down-field site. For R 1 , the tunneling from the downfield site has been confirmed by the Gaussian distribution of the lateral photoelectron momentum in our previous study [17].
At R 1 , the ionization is taking place while the population is being intertransferred between the σ ± states at the same time due to the resonant coupling. It is therefore difficult to exactly tell from Fig. 5(a) whether the depletion of a given state (σ + or σ − ) is due to the ionization or due to the population transfer to the other state. Here, we seek from an alternative perspective for the explanation of the striking ionization enhancement at R 1 . The physical picture is shown in Fig. 3(c). First of all, instead of considering the left and right sites of the molecule, we treat it as one whole system since the ground state level E 1s is higher than the inner barrier. According to the R-dependent energy levels of H 2 + , the one-photon (800 nm) crossing between 1sσ g and 2pσ u is at R = 4.8 a.u. So, in this diabatic picture, the first excited state is populated during the interaction via the one-photon resonant excitation near the crossing. Then, the excited system is ready to be ionized much more easily, which is the prerequisite for the enhanced ionization near R 1 . Meanwhile, the ground state of H 2 + at R 1 has also certain probability to be ionized due to the thin laser-dressed barrier [see Fig. 3(c)]. In this picture, one can consider that the fundamental field is the superposition of two pulses with the same wavelength, which play different roles during the interaction. One is to trigger the resonant excitation and the other is to drive the ionization by bending the Coulomb potential. Once the ground and first excited states are populated, there is the possibility for the ionization from either state in the external field and the ionizing wave packets from two states will interfere with each other. Eventually, from the viewpoint of energy reservation, the ionizing wave packet having absorbed N photons from the ground state and that having absorbed N − 1 photons from the first excited state share the same kinetic energy and thus coherently interfere in continuum. Depending on the phase difference, the constructive or destructive interference will enhance or reduce the ionization, respectively. Note that because of the band width of the laser pulse, the physical picture discussed here is still justified when R is around the one-photon crossing and when Stark shifts of the energy levels are taken into account. In short, we propose that the first ionization peak near the one-photon crossing is determined by the resonant excitation and the coherent interference of the ionizing wave packets from the 1sσ g and 2pσ u states.
The results further supporting our explanation of the underlying mechanism can be found in Fig. 6. On one hand, by varying the CEP of the few-cycle pulse, the interference process is modified and we indeed observe in Fig. 6(a) that the first peak is altered gradually when the CEP is changed. On the other hand, we compare in Fig. 6(b) the ionization yields for the 800-and 1200-nm pulses with otherwise the same parameters. The peak for the 1200-nm pulse is observed at a larger internuclear distance, as the resonant excitation for the 1200-nm photon is at R = 5.3 a.u. Besides, the underlying dynamics can also be modified by the intensity [31,32], the wavelength [35,36], or even the pulse duration [33] of the fundamental LP pulse, leading to the modification of the first ionization enhancement under different laser parameters. This explains that among the previous studies [7,10,[15][16][17][18], the first ionization peak is observed at different R but it is always near the one-photon crossing. On the other hand, in the present study the second-harmonic electric field perpendicular to the molecular axis does not contribute to the coupling between 1sσ g and 2pσ u and, thus, it hardly affects the phase shifts between the ionizing wave packets from these two states. Only the transverse drifts of the ionizing wave packets in the OTC fields cause small modification of their interference, as shown in Figs. 4(a) and 4(c). Therefore, it explains the observation in Fig. 2 that, with a fixed fundamental pulse, the first ionization peak is hardly changed by adding the perpendicular secondharmonic field. Furthermore, according to the results shown in Fig. 6(a), we expect that the CEP-averaged R-dependent ionization yields under the fundamental LP pulse would still exhibit the double-peak structure, though the locations of the ionization peaks differ from those shown in the previous studies [25,41] due to the laser parameters being chosen differently.
In addition, we can now understand the association of the asymmetric electron localization with the first ionization peak. The electron localization results from the superposition of the ground and first excited states, and the asymmetry depends on the relative phase shift between these two states. Meanwhile, the first ionization peak also relies on the phase shifts of the ionizing wave packets from these two states. As a result, when the first excited state is resonantly populated at R 1 , the enhanced ionization and the asymmetric electron localization would take place at the same time under the same phase shift between the ground and first excited states. This finding agrees well with the previous observation that the enhanced ionization is always associated with the electron localization [6].
Finally, we turn to the underlying dynamics for the second ionization peak, which has been well demonstrated in the previous studies [5,6,17,23]. At R 2 , two nuclei are well separated by the inner barrier. As shown in Fig. 3(b) and discussed previously, the energy level on the up-field site is lifted and, thus, the overbarrier ionization from the up-field site leads to the enhanced ionization at R 2 . From Figs. 4(b) and 4(d), we indeed observe roughly the transportation of the population from the up-field site to the continuum, as indicated by the dotted arrows. We can also see from the color scale that the ionizing wave packet is enhanced in the OTC field [ Fig. 4(d)] with respect to that in the LP field [ Fig. 4(b)]. In order to confirm where the additional ionizing population comes from in the case of OTC fields, we have calculated the evolutions of the population of σ ± at R 2 for the OTC and LP pulses and show them in Fig. 5(b). Because of the relatively larger internuclear distance at R 2 , the asymmetric electron localization is suppressed. Thus, from the straight depletion of the population of σ − near t = 1.5T , as indicated by the arrow in Fig. 5(b), we can generally tell that for both OTC and LP pulses the ionization from the up-field site is more pronounced than that from the down-field site. Meanwhile, as indicated by the vertical-line shadow, the population of σ − (up-field site) in the OTC field is additionally depleted with respect to that in the LP field.
For a deeper insight into the enhanced ionization at R 2 after adding the orthogonal second-harmonic field, we show in Fig. 7 the time-dependent depletion differences between the LP and OTC cases [i.e., the difference between the solid and dashed curves shown in Fig. 5(b)] for the σ + and σ − states, respectively, along with the fundamental electric field. As shown by the solid arrow, the depletion difference for σ − increases significantly near t = 1.5T , indicating that the up-field population is much more depleted after adding the orthogonal field. In contrast, as shown by the dotted arrow, the depletion difference for the down-field state (i.e., σ + ) grows slower, indicating that the down-field population is also further depleted after adding the orthogonal field, but the increased depletion is not as significant as that of the up-field site. Note that the faster depletion of the up-field population can also be observed at the other two local peaks of the electric field where σ + is on the up-field site, as shown in Fig. 7. The results suggest that the ionization yield from the up-field site at R 2 is further enhanced when the intensity of the second-harmonic field is increased, supporting the mechanism of easier ionization from the up-field site of the molecule.

IV. CONCLUSION
In conclusion, we have theoretically studied the enhanced ionization of H 2 + using the OTC fields. The results have shown the distinctive patterns of the ionization at two critical internuclear distances, which can be considered as the observable evidence demonstrating two qualitatively different underlying mechanisms for the enhanced ionization. In particular, we propose that the ionization peak near the one-photon crossing can be explained by the resonant excitation along with the coherent interference of the ionizing wave packets from the 1sσ g and 2pσ u states. Also, we confirm that the ionization enhancement at the larger critical internuclear distance is due to the easier ionization from the up-field site of the molecule.