Over the last twenty years, the high-order harmonic generation (HHG) from atoms and molecules in the intense laser fields has been a hot subject.^[1–5] Great progress has been made in understanding and application of HHG.^[6,7] Not only does it offer the potential for creating new coherent light sources, single attosecond pulses,^[8–10] or attosecond pulse trains (APT),^[11,12] but also it is an important tool for ultra-fast imaging of a dynamic molecular system.^[13–17] So far, the physical process of HHG can be well comprehended by the semi-classical three-step model.^[18] Firstly, the electron tunnels out of the potential barrier formed by the Coulomb potential and the laser field; then the ionized electron moves in the laser field and gains additional kinetic energy; finally, it recombines with the core and a photon is emitted (high-order harmonic). Besides the atomic system, HHG has been investigated in many other systems.^[19–35]

As is well known, the harmonic spectrum from a single atom in a strong laser field presents some common and significant features.^[19–21] Typically, only odd harmonics are observed in theoretical simulations and experiments.^[21,22] One non-perturbative proof which invokes the concept of inversion symmetry has already been given by Ben-Tal et al.^[23] This theory can work well for any system of the dynamic symmetry.^[24]

In recent years, the molecular high-order harmonic generation (MHOHG) has been broadly studied, since molecules have more degrees of freedom which can be used to further control HHG.^[26–30] The odd–even-order harmonic generation from molecular systems has been also investigated.^[31–34] It has been shown that the molecules are more complicated than the atom,^[35,36] but the harmonic generation spectra (HGS) are still composed only of odd harmonics for symmetric molecules in Born–Oppenheimer approximation.^[31,32] Moreover, both odd and even harmonics were observed from asymmetric molecules with the fixed nuclei.^[31] However, for HD molecule, even harmonics were first observed in non-Born–Oppenheimer approximation.^[32] Recently, we found that the HD molecule still generated only odd harmonics in non-Born–Oppenheimer approximation though the generation of even harmonics is possible in principle.^[33] Nevertheless, so far, it is still an open question whether the HD can generate even harmonics in non-Born–Oppenheimer approximation, in which the nuclei are easier to dissociate than those in the HD molecule.

In this work, we give an affirmative answer. The HD also generated only odd harmonics. Besides, we investigate the HHG of HeH and find that both odd and even harmonics appear in the HGS. To our knowledge, the selection rules^[23] cannot explain the different odd–even property occurring in the HGS of HD and HeH. Our theoretical analysis indicates that for HD and HeH, the nuclear dipole acceleration makes a contribution to producing even harmonics, but it is about three orders of magnitude less than that of the electron. This leads to the fact that the impact of the nuclear dipole acceleration to the HGS can be ignored. Therefore, the HHG primarily depends on the contribution of the electronic dipole acceleration. For HD, the electronic dipole acceleration only results in the generation of odd harmonics. However, for HeH, it can produce both odd and even harmonics. In addition, we also invoke the concept of the broken degree of system-symmetry to interpret the different odd–even property between the harmonic spectra of HD and HeH. Generally speaking, both odd and even harmonics can be produced when the system-symmetry is destroyed. Whereas, the visible even harmonics could be generated only if the breaking of system-symmetry gets to a certain degree. For HD, the system-symmetry is broken in a very small degree. Hence, only odd harmonics appear in the HGS. For HeH, because the system-symmetry is drastically broken, there are both odd and even harmonics in the HGS.

For the numerical simulations, we have used the three-body model with two nuclei and one electron, where the motion of nuclei and electron is restricted to the polarization direction of the laser field in the one-dimensional (1D) model. Within this model, the coupling to the laser field is treated within the dipole approximation. As a result, the center-of-mass motion does not couple to the laser field. Then, the time-dependent Schrödinger equation (TDSE) can be given as (atomic units are used throughout)^[32,37,38]

Fig. 1.

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	Fig. 1. Harmonic spectra generated from (a) H, (b) HD, and (c) HeH by a laser with the peak intensity of 4 × 10 W/cm and the wavelength of 770 nm. The corresponding 3D simulation results are shown in the right column. The insets are the enlarged area of the harmonic spectra from 40th to 50th order.

Before the time evolution of the wave function, an initial state is requisite. For H and HD, the initial states can be obtained by propagation in imaginary time. We choose the soft-core parameters and such that the energy of the ground state and equilibrium distance are −0.7763 a.u. and 2.745 a.u., respectively. However, a bound state cannot be obtained by this method for HeH, because the lowest state 1s is dissociated.^[40] In order to solve this problem, we adopt the spectral method.^[39] Then, the first excited state 2p which is one bound state of HeH is chosen to be the initial state. The soft-core parameters α = 0.3 and β = 0.01 are adopted to reproduce the ionization energy of 1.03 a.u. and equilibrium distance of 3.89 a.u.^[40]

After the initial states have been obtained, we solve Eq. (1) using the Crank–Nicholson method^[41]

The harmonic spectra generated by H, HD, and HeH are shown in Fig. 1. The left column is the results of 1D simulations. In our calculation, we use a grid from −400 a.u. to 400 a.u. containing 8000 grid points in the z-axis, and a grid from 0 to 100 a.u. containing 1000 grid points in the R-axis. The time step of wave function propagation is set to be 0.05 a.u. In order to avoid spurious reflections of the wave function from the boundary = ± 400 and R = 100, the wave function is multiplied by a cos1/8 mask function over a range of –400 and –100 per evolving step. From Figs. 1(a)–1(c), one can see that the HGS of H and HD are composed only of odd harmonics, while the HGS of HeH are composed of both odd and even harmonics. In fact, for HeH, a two-plateau structure and a continuum spectrum have been observed for long wavelength.^[42,43] But in this paper, the wavelength of laser is just 770 nm, so the spectra of HeH just exhibit one cut-off, which is in agreement with the results of Refs. [42] and [43] in short wavelength. In addition, we also display the results of 3D simulations in the right column. It is clear that in two cases (1D and 3D), HD produces only odd harmonics, while HeH produces both odd and even harmonics.

The result of H can be well understood with the selection rules in Ref. [23], which can be described in the following way. The probability to produce the nth harmonic from a system is^[23]

As shown in Ref. [33], the nuclear dipole acceleration may play a role in the HHG beyond Born–Oppenheimer approximation. It can be seen from Eq. (6) that for H, due to the asymmetry parameter . However, for HD and HeH, the asymmetry parameter , and thus the nuclear dipole acceleration can contribute to the HHG. In order to examine the role of nuclear dipole acceleration, figure 2 presents the harmonic spectra calculated with for HD and HeH. As shown in Fig. 2, the cutoff of the harmonic spectrum is smaller than that in Fig. 1 for both cases (harmonic order lower than 15). This is because the nuclei are easy to dissociate when the only electron of HD and HeH is ionized by the intense laser pulse. Besides, it is found that there are even harmonics in both of the harmonic spectra. Consequently, in principle, the generation of even harmonics is possible for HD and HeH. But the intensity of harmonics is lower than that in Fig. 1.

Fig. 2.

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	Fig. 2. The harmonic spectra calculated with for (a) HD and (b) HeH.

For further capturing the signatures of nuclear dipole acceleration, figure 3 shows the electronic dipole acceleration and the nuclear dipole acceleration for HD and HeH. For comparison, is amplified by times. Clearly, for both HD and HeH, the electronic dipole acceleration is about three orders of magnitude larger than the nuclear dipole acceleration. Then we can conclude that the contribution of the nuclear dipole acceleration to HHG can be ignored. In order to verify it, we also present the harmonic spectra of HD and HeH calculated with in Figs. 3(c) and 3(d). It is found the harmonic spectra calculated with are the same as that calculated with . This implies that the influence of the nuclear dipole acceleration to HHG can be indeed ignored for both HD and HeH, though it can contribute to the even-order harmonic generation. Apparently, it is not the reason why the odd–even property of the HGS of HD and HeH is different.

Fig. 3.

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Fig. 3. (color online) The electronic dipole acceleration (red dotted line), the nuclear dipole acceleration (violet dashed-dotted line), and the total dipole acceleration (black solid line) for (a) HD and (b) HeH. For clarity, is amplified by times. (c) Harmonic spectra generated from HD. Black solid line is calculated with , and red dotted line is calculated with . (d) Same as panel (c) but for HeH.

As is already known, the contribution of the nuclear dipole acceleration to the HHG can be ignored. Therefore, the electronic dipole acceleration mainly contributes to the harmonic generation. In order to confirm our inference, the total dipole acceleration is displayed in Fig. 3. As shown in Figs. 3(a) and 3(b), for both HD and HeH, the total dipole acceleration and the electronic dipole acceleration are almost equal. Moreover, it can be clearly seen from Figs. 3(c) and 3(d) that the harmonic spectra calculated with are the same as that calculated with . All of these imply that the HHG mainly relies on the contribution of the electronic dipole acceleration. The detailed theoretical derivation is discussed in Ref. [33], which is in good agreement with our conclusion.

In order to further understand the difference between the harmonic spectra of HD and HeH, figure 4 shows the Coulomb potential for H, HD, and HeH. In Fig. 4, for H, the potential well is perfectly symmetric. Certainly, H obeys the selection rules in Ref. [23]. For HD, the potential well is still in symmetry, though it has a small translation in the axis. One can see that the potential well of HD and H could coincide by a small translation in the axis. By contrast, for HeH, the potential-well symmetry is drastically destroyed. Thus, one concept of the broken degree of system-symmetry can be proposed to explicate the difference of the HGS of HD and HeH. In detail, for HD, the system-symmetry is only destroyed into a small extent, which shows that the nuclear acceleration can contribute to the HHG. However, its contribution can be ignored. Therefore, only odd harmonics are observed for HD. While for HeH, the system-symmetry is drastically destroyed. Hence, there are both odd and even harmonics in the spectrum.

Fig. 4.

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	Fig. 4. (color online) Coulomb potential for H (black solid line), HD (red dashed line), and HeH (blue dashed-dotted line). For H and HD, R = 2.745 a.u. For HeH, R = 3.89 a.u.

To examine the influence of the broken degree of system-symmetry, we used the model molecule with different (. Within this model, one can change the broken degree of system-symmetry by manipulating , where the bigger is, the more completely the broken extent is. The results are shown in Fig. 5. It can be seen that as increases, the even harmonics gradually appear and the intensity becomes higher and higher. Hence, we invoke the concept of the broken degree of system-symmetry to explain the different odd–even property between the harmonic spectra of HD and HeH. In other words, to observe obvious even harmonics, the system-symmetry should be broken to some extent.

Fig. 5.

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	Fig. 5. The harmonic spectra with , 0.01, and 0.1. The insets are the enlarged area of the harmonic spectra from order 10 to 20.

In summary, we have studied the HHG of H, HD, and HeH in non-Born–Oppenheimer approximation. It is found that HD generates only odd harmonics, while HeH generates both odd and even harmonics. Note that the selection rules^[23] cannot explain this difference. Then theoretical analysis implies that the dipole acceleration of the electron and the nuclei jointly contribute to HHG. However, for both HD and HeH, the nuclear dipole acceleration has no effects on the HHG, since it is three orders of magnitude lower than that of the electron. Thus, the electronic dipole acceleration is the key of the HHG. For HD, the electronic dipole acceleration only leads to the odd-order harmonic generation, but for HeH, it leads to the odd–even-order harmonic generation. Additionally, we proposed the concept of the broken degree of system-symmetry to explain the difference of the HGS of HD and HeH. It is found that the visible even harmonics could be generated only if the breaking of system-symmetry gets to a certain degree. For HD, due to the tiny destruction of system-symmetry, it generates only odd harmonics. However, for HeH, both odd and even harmonics are produced, which is attributed to the drastic breaking of system-symmetry.