1. Introduction

The electronic motion inside the molecule is of fundamental importance in determining the formation and fracture of chemical bonds. For more than two decades, many efforts have been done to study the electronic dynamics in laser-matter interactions [1–4], aiming at the control over ultrafast reactions. With the recent development of laser techniques and attosecond science [5–7], it has become feasible to steer the electron localization in dissociating molecules with the carrier-envelope phase (CEP) stabilized few-cycle laser pulses [8–11] or the sequential ultraviolet and near-infrared pulses [12–14]. The asymmetric electron localization in molecules can be understood as the quantum interference of the populations that are resonantly transferred among at least two electronics states of different parity [15–18].

More recently, the control of electron-directed reactivity in midinfrared laser pulses has been explored to enhance the electron localization probability via the match between the duration of the few-cycle pulse and the dissociation time of the molecule [19–22]. To hold the control efficiency for further heavier nuclei, one may have to use the pulse with longer wavelength [20]. While this constitutes an important step towards the control of chemical reactions in larger molecules, we are wondering whether the physical mechanism responsible for the electron localization in midinfrared pulses remains the same as that in the near-infrared regime [9]. Meanwhile, for extending the control scheme to the larger molecules, the influence of nuclear mass on the reactions must be taken into account. However, it remains unclear how the nuclear mass affects the control efficiency when the midinfrared pulses are applied, though it has been shown that the control of electron localization is weakened by the growing mass in the few-cycle near-infrared field [9, 23].

To understand the influence of mass in reactions, the use of isotopes is one of the practice means. This is because different isotopes of a given element have a similar electronic structure and chemical properties, but different masses. In fact, the isotopic effect has been an important issue in the research of laser-molecule interactions, such as high-harmonic generation [24], nuclear dynamics detection [25] and high-order above-threshold dissociation (ATD) [26]. On the other hand, thanks to the recent advance in the generation of midinfrared source, few-cycle CEP-stable pulses at 3 μm have become available in laboratory [27, 28], providing a potential for controlling as well as obtaining deeper insight into the electron localization process in heavier molecules. In this paper, we theoretically study the asymmetric dissociation of molecular hydrogen by using a few-cycle 3-μm pulse and report an anomalous isotopic effect on the electron-directed reactivity. The dissociation process is analyzed with the semi-classical model [29]. For the 3-μm field, the underlying mechanism for the electron localization is found to be based on the interferences between the one- and higher-order photon coupling channels. Due to the enhanced ATD for heavier isotopes in the intense laser field [26], the interference maxima of the spectra gradually become in phase with growing nuclear mass, resulting in the anomalous isotopic effect where the electron localization degree is larger for heavier isotopes.

2. Theoretical model

In our simulations, we have used a numerical model that has been well established for studies of electron localization in the laser-driven molecular dissociation [8, 9]. This model solves the time-dependent Schrödinger equation (TDSE) for the nuclear wavepacket in the Born-Oppenheimer (BO) representation. In the present model, we have considered the molecular ions oriented along the polarization axis. Since the time scale of the nuclear rotation is several hundred femtoseconds, two reasonable assumptions underlie our calculation [30]: (i) the nuclei do not have time to rotate during the time of interaction and (ii) the nuclear rotation after the pulse is not significant. The molecular dynamics of H₂⁺ can be well described in terms of the two lowest-lying electronic states, the 1sσ_g and 2pσ_u states. The TDSE can be written as (atomic units are used throughout unless otherwise indicated) [8]:

The time-dependent nuclear wave functions corresponding to the ‘left’ (l) and ‘right’ (r) atoms within the molecule are defined as ${\mathrm{\Psi }}_{l,r}=\left({\chi }_g\pm {\chi }_u\right)/\sqrt{2}$. The kinetic energy release (KER) spectrum S_l,r(E_k) is obtained by Fourier analysis of the real-space wave function with the bound states projected out [31]. The final probabilities P_l,r of the electron being localized on the left or right nucleus are calculated by integrating S_l,r(E_k) over the kinetic energy E_k. In respect that the preparation mechanism of H₂⁺ generally indicates an incoherent Franck-Condon (FC) distribution of vibrational states in experiments [30], we have averaged the observables over the initial vibrational states weighted by FC factors in the present calculation.

3. Results and discussion

We study the isotopic effect on the asymmetric dissociation of H₂⁺ with a three-cycle (FWHM), 3000-nm laser pulse with an intensity of 1×10¹⁴ W/cm². The asymmetry parameter of the final electron localization probability is defined as A = (P_l − P_r)/(P_l + P_r). Figures 1(a)–1(c) reveal the asymmetries after FC averaging as a function of CEP in the 3000-nm pulse for H₂⁺ and its isotopes, respectively. For comparison, a similar simulation is performed for a three-cycle, 800-nm pulse with the same pulse intensity [20, 32]. The results are shown in Figs. 1(d)–1(f). First of all, it is obvious that the asymmetry is enlarged in the midinfrared field for each isotope, as predicted in [19,20]. Then, more interestingly, we can obtain two contrastive observations from Fig. 1. (i) The asymmetry for the 800-nm pulse displays a CEP dependence that decreases with growing mass. This is coincident with the results of the previous studies [23]. But curiously, the asymmetry for the 3000-nm pulse shows an inverse isotopic behavior in which the degree of the asymmetry is even higher for heavier isotopes. (ii) The curves of the asymmetry for the 800-nm pulse show a regular sine form, but they become irregular for the 3000-nm pulse. In addition, there is a phase shift of the CEP-dependent electron localization asymmetry between different isotopes as well as different wavelengths. This can be explained as follows. After the pulse is applied, partial nuclear wavepacket is driven away to the larger internuclear distance and passes through the one- (or multi-) photon coupling region. Then, populations are transferred between the gerade and ungerade states by the external field. Here, the coupling strength depends on the instantaneous electric field that is sensitive to the CEP of the pulse. Therefore, for the molecular ions with different masses, the nuclear wavepackets reach the coupling region at different times and, consequently, the asymmetries respond differently to the CEP. The mechanism for the phase shift of the asymmetry had also been discussed in detail in our previous work [19].

 

Fig. 1 The CEP dependence of the electron localization asymmetry of H₂⁺ and its isotopes, D₂⁺ and T₂⁺, for the pulse wavelengths of 3000 (first row) and 800 nm (second row). The third row shows the results (for the 3000-nm pulse) calculated by the molecular model of one-dimensional nuclear motion and one-dimensional electronic motion. The results have been averaged over all the initial vibrational states weighted by FC factors.

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In order to check the influence of the higher excited states as well as the ionization on the isotopic phenomenon revealed by the two-state model, we have repeated the simulations in the 3000-nm field with the same pulse parameters by using the molecular model of one-dimensional nuclear motion and one-dimensional electronic motion [19, 33]. The results are present in Figs. 1(g)–1(i). One can see that the asymmetries for three isotopes have revealed the isotopic behavior that are predicted by the two-state model. Although the CEP-dependent asymmetries are not exactly the same for the two sets of results, it is still acceptable because different approximations are made in the two models and, more importantly, their energy curves of 1sσ_g and 2pσ_u states that play important roles in determining the electron localization are quite different [33]. In addition, the probabilities for transitions to higher excited states and for ionization may also affect the results but they do not change the essence of the isotopic behavior after all. Therefore, the two-state model is considered to be sufficient and reliable to reveal the isotopic effect on the electron localization in the present study.

To gain insight into the interaction dynamics under the 3-μm field, we present the absolute asymmetry as a function of the CEP and the kinetic energy in Fig. 2, with the same laser parameters as in Figs. 1(a)–1(c). The absolute asymmetry is given by A(E_k, ϕ) = S_l(E_k, ϕ) − S_r(E_k,ϕ). Generally, the asymmetry shows a strong dependence on both the CEP and the KER for all isotopes. In detail, we see, on one hand, two major “undulate tilted stripes” corresponding to the left-right asymmetry. As the nuclear mass increases, the stripes gradually become more and more precipitous and they are almost vertical for T₂⁺. Additionally, as indicated by the horizontal dashed line, the asymmetry is almost stop at about 1 eV for H₂⁺, but the strong asymmetry extends to higher energy with growing mass. Consequently, after the integral over the KER, the degree of the asymmetry turns out to be larger for the heavier isotope. On the other hand, we can see that the stripes of the spectra contain several peaks. In the case of H₂⁺ in Fig. 2(a), the peaks at different KER respond inconsistently to the CEP, leading to the irregular asymmetry curve after the KER integral, as shown in Fig. 1(a). But, for T₂⁺ in Fig. 2(c), the alignment of the peaks are almost in step. Therefore, the asymmetry for T₂⁺ in Fig. 1(c) becomes closer to a sine-like curve.

 

Fig. 2 The asymmetry parameter A(E_k, ϕ) as a function of the CEP and the kinetic energy for (a) H₂⁺, (b) D₂⁺, and (c) T₂⁺ in the 3000-nm pulse. The other laser parameters are the same as in Figs. 1(a)–1(c).

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While the KER distributions give us a well understanding of the features of the CEP-dependent asymmetry in Fig. 1, the physical mechanism for the anomalous isotopic behavior of the electron localization has remained unclear: Why are there asymmetry peaks at different kinetic energy and why do those peaks gradually become in step with growing nuclear mass? In order to uncover the underlying dynamical process, we analyze the molecular dynamics with the semi-classical model in terms of the quasi-static states [29], which are related to the left and right localized wave functions Ψ_l,r via ${\psi }_{1,2}=\left[\left(\text{cos}\theta \pm \text{sin}\theta \right){\mathrm{\Psi }}_l\pm \left(\text{cos}\theta \mp \text{sin}\theta \right){\mathrm{\Psi }}_r\right]/\sqrt{2}$, with 2θ = tan⁻¹[−2V_gu/(V_u −V_g)]. The corresponding quasi-static eigenvalues can be obtained by

In Fig. 3, from up to down the left and right columns depict the time evolution of the electric field E, the quasi-state eigenvalues V_1,2, the internuclear distance R, the transition probability P_ts, and the localization asymmetry for the 800- and 3000-nm pulses, respectively. According to the semi-classical model, the electron localization is established though the region where 0 < P_ts < 1, as shown by the shadows in the figure. We find that the evolutions of P_ts in the 3000-nm pulse [Fig. 3(h)] are quite different from those in the 800-nm pulse [Fig. 3(g)]. This indicates a distinct physical mechanism responsible for the electron localization in the 3000-nm field.

 

Fig. 3 The time evolution of the physical quantities of the electron localization process in the 800- and 3000-nm fields for H₂⁺ (thick curve) and D₂⁺ (thin curve). (a)–(b) The electric field. (c)–(d) The quasi-state eigenvalues. (e)–(f) The internuclear distance. (g)–(h) The transition probability. (i)–(j) The localization asymmetry. The shadows indicate the regions where the electron localization is established.

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In the case of the near-infrared pulse (the left column of Fig. 3), the nuclei separate slowly [Fig. 3(e)]. At the early time of the interaction, the energy gap between V_1,2 is large and the dynamics is adiabatic. As the interaction continues, the energy gap becomes smaller. For H₂⁺ (thick solid curve), the dynamics becomes non-adiabatic when the internuclear distance sequentially passes through the three- and one-photon crossings, as indicated by the dashed lines in Fig. 3(c). This can be understood as that the electron localization is established by interference between the dissociative populations that are generated through the three- and one-photon coupling channels. After that, the electric field is no longer strong enough to overcome the energy gap and the dynamics becomes adiabatic again. For D₂⁺ (thin solid curve), however, due to the slower nuclear motion, the pulse becomes much weaker when the internuclear distance passes the coupling region. Meanwhile, the energy gap is still large for D₂⁺ [Fig 3(c)], so the transition probability is near upon zero, resulting in the small asymmetry. Therefore, under the mechanism of the near-infrared field, the electron localization is more difficult to be achieved for heavier molecules.

However, the underlying dynamics is different in the midinfrared field (the left column of Fig. 3). As shown in Fig. 3(d), the evolutions of V_1,2 are almost the same for both molecular ions. According to the transition probability in Fig. 3(h), the dynamics becomes non-adiabatic when the internuclear distances pass about R = 3.9 a.u. [corresponding to the nine-photon crossing of 3000 nm, indicated by the dashed line in Fig. 3(f)]. Then the electron localizations for both molecules begin to be established at almost the same time. Due to the fast separation of the nuclei, the dynamics has been diabatic when the pulse is turned off and, consequently, the electron localization is frozen. Similarly to the situation in 800-nm field, the effect of nuclear mass in the 3000-nm field is to slow the nuclear motion. But the midinfrared pulse is strong and long enough to overcome the energy gap. Moreover, a distinct difference is that the higher-order multi-photon crossing is open in the midinfrared field. The population is therefore transferred between the gerade and ungerade states through those multi-photon coupling channels. Thus the pronounced electron localization can still be achieved for heavier isotopes.

Based on the semi-classical analysis above, next we will introduce the dissociation model illustrated in Fig. 4(a) and then discuss the reason for the anomalous isotopic effect in the 3000-nm field. The dissociation process can be understood as follows. During the interaction, the outgoing wavepacket sequentially passes through the 9ω to 1ω crossings shown by the text-arrows in Fig. 4(a). Then, the populations are resonantly transferred between the 1sσ_g and 2pσ_u states via absorption or emission of photons at the crossing. Note that for the few-cycle pulses, some of the crossings may be “switched off” and the dissociation pathways would be changed if the CEP is varied. This is because the instantaneous electric field may be zero when the wavepacket reaches the crossing. Therefore, the molecule could finally dissociate via net-absorption of one or more photons at different initial coupling channels. For example, the dashed arrow in Fig. 4(a) depicts one of the possible pathways that start at the 7ω coupling channel: The wavepacket first absorbs seven photons at the 7ω crossing, then passes “5ω” without emission of photons but emits three photon at “3ω”, and finally reabsorbs one photon at “1ω”, leading to the net five-photon absorption at the initial crossing of “7ω”.

 

Fig. 4 (a) Illustration of the dissociation model for H₂⁺ in the 3000-nm field. The vertical lines with arrows reveal the coupling channels that are noted by the text-arrows. The thin gray curves indicate the BO potentials dressed by net absorbed numbers of photons. The blue dashed arrow shows one of the possible dissociation pathways. (b) The absolute value of a cut of the asymmetry A(E_k, ϕ) normalized to the maximum asymmetry at the given CEP of ϕ = 0.5π, as marked by the vertical dashed lines in Fig. 2. (c) The KER spectra normalized to the maximum value for the H₂⁺ dissociation under the 3000-nm pulse with the CEP of 0.5π and 1.0π.

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In Fig. 4(a), the centers of the KER distribution for different dissociation pathways have been shown by the horizontal dashed or solid lines. We also show the FC-averaged KER spectra normalized to the maximum value for the H₂⁺ dissociation under the 3000-nm pulse with the CEP of 0.5π and 1.0π in Fig. 4(c). Both spectra range from 0 to 2.5 eV and involve several peaks that originate from the different dissociation pathways. The change of the peak values with the CEP demonstrates that the contributions from different pathways are indeed affected by the CEP as mentioned above.

Note that those horizontal dashed and solid lines in Fig. 4(a) suggest the dissociative populations on the 1sσ_g and the 2pσ_u states, respectively. Due to the large laser bandwidth of the pulse, the KER distributions for the pathways of different parity overlap in energy, leading to the interferences and thus the peaks of the asymmetry at different KER. By changing the CEP of the pulse, the contributions from different pathways will be modulated, so that the electron localization exhibits the CEP- as well as KER-dependent asymmetry shown in Fig. 2.

For comparison, in Fig. 4(b) we present the asymmetry parameter A_N as a function of kinetic energy for three isotopes, where A_N is the absolute value of a cut of the asymmetry A(E_k, ϕ) normalized to the maximum asymmetry at ϕ = 0.5π, as marked by the vertical dashed lines in Fig. 2. One can see that all the coupling channels, from “9ω” to “1ω”, contribute to the asymmetry at lower KER between 0–0.55 eV. But the asymmetry of higher KER is mainly contributed from the relative higher-order coupling channels. According to the semi-classical analysis, the strength of the relative lower-order coupling channels will be shrunk for heavier isotopes due to slower motion of the wavepacket which reaches the coupling region when the electric field is weaker. Relatively, the contributions of the higher-order channels will become more important to the dissociation of heavier isotopes. This process can also be understood as the enhanced high-order ATD [26]. Therefore, the asymmetry is gradually enhanced in higher energy regions with growing mass, as indicated by the horizontal dashed lines in Fig. 4(b) and Fig. 2.

Furthermore, the phase shift of the asymmetry peaks at different KER depends on the relative phase between the overlapped populations on the gerade and ungerade states [17]. For the dissociation pathways starting at different initial crossings, the corresponding dissociative wavepackets have the different initial phases and thus respond inconsistently to the CEP. Consequently, the phase of the CEP-dependent asymmetry shifts with the KER, resulting in the tilted asymmetry stripes shown in Fig. 2. But due to the relative enhancement of the higher-order coupling channels for heavier isotopes, the CEP response of the asymmetry at higher KER gradually becomes consistent to that at lower KER. Therefore, the alignment of the asymmetry peaks is more vertical (KER-independent) for heavier isotopes, ultimately leading to the isotopic behavior where the degree of electron localization increases with growing nuclear mass.

In order to further confirm the analysis above, we would like to present additional simulations for the electron localization asymmetry in the 3000-nm pulses with two weaker pulse intensities, 7.11 × 10¹² and 5 × 10¹³ W/cm². Note that, the ponderomotive energy of the 3000-nm pulse with the former intensity is the same to that of the 1 × 10¹⁴ W/cm², 800-nm pulse. The results are shown in Fig. 5. On the one hand, the asymmetries of heavier isotopes gradually become larger than those of lighter ones when the intensity increases, as shown in the upper rows of Fig. 5 and Fig. 1. On the other hand, from the lower row of Fig. 5, combined with the KER spectra in Fig. 2, one can see that strong asymmetries for all isotopes extend to higher kinetic energy as the intensity increases, and the higher KER is further enhanced by the heavier nuclei. It indicates that the higher-order ATD is a significant contribution to the enhanced asymmetries of heavier isotopes. Therefore, the additional results in Fig. 5 have further demonstrated and confirmed that the mechanism of higher-order transitions play an important role in enhancing the asymmetry of heavier molecules. In addition, from the perspective of the laser parameters, the enhanced high-order ATD can also be attributed to the high ponderomotive energy of the 3000-nm pulse. One may expect that the high-order ATD could be enhanced by increasing the intensity of the near-infrared field as well. However, extremely high driving field will lead to the ionization saturation of the molecule and will consequently weaken the control of electron localization on the dissociating nuclei [19]. Thus, it is effective to control the electron localization by using the midinfrared pulses rather than the near-infrared ones.

 

Fig. 5 The CEP dependence of the electron localization asymmetry (upper row) and the CEP-dependent KER spectra of the asymmetry (lower row) for the dissociation of H₂⁺ and its isotopes in the 3000-nm pulse with two different intensities, 7.11 × 10¹² (left part) and 5 × 10¹³ W/cm² (right part). The other laser parameters are the same as in Figs. 1(a)–1(c).

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4. Conclusion

In conclusion, we have studied the electron localization in H₂⁺ and its isotopes by a few-cycle 3-μm pulse and have found an anomalous isotopic effect on the electron-directed reactivity. Compared to the situation in near-infrared regime, we show that the distinct underlying dynamics in the intense midinfrared fields is the opening of the higher-order (more than three) multi-photon coupling channels. Pronounced electron localization, even for heavier isotopes, can be achieved through interferences between the dissociation pathways that start at the high-order crossings. As the nuclear mass increases, the contribution of the high-order coupling channels is relatively enhanced and thus the interference maxima of the spectra gradually become in step. After the energy integral, the electron localization asymmetry turns our to be larger for heavier isotopes. This unexpected isotopic behavior has provided us deep insights into the electronic dynamics in molecular dissociation and the high-order multi-photon coupling channels appear to be an important role in the control over electron-directed reactivity of larger molecules with midinfrared pulses.