1. Introduction

Photoionization of atoms and molecules by the strong laser field [1], as the doorway step of many strong field processes [2–9], continued attracting much attention over the past thirty years. The simple-man model [10, 11], that the electron firstly leaves the parent ion with zero longitudinal momentum by tunneling at maximum of the electric field and then is determined by the laser field, has obtained great success in understanding the dynamics of the strong field ionization [12, 13]. Because of the increasingly deep study in the strong field ionization, more and more novel phenomena [14–18] beyond the scope of the simple-man model are found. The peculiar low-energy structure [14, 15] of ATI electron spectra is generally investigated and the role of the Coulomb potential of the parent ion has been identified [19,20]. The counterintuitive angular shift in the photoelectron momentum distribution for atoms in strong few-cycle circularly polarized laser pulses is also attributed to an intimate interplay between the external field and the binding potential [16].

Furthermore, the situation for the molecule is more complex in comparison with the atom. By solving the time-dependent Schrödinger equation, Takemoto etal. found multiple bursts of ionization of ${\text{H}}_2^{+}$ by a linearly polarized laser pulse within a half-cycle of the laser field [17], which deviates from the widely accepted tunnel ionization picture [21]. The very recent experiment by Odenweller etal. shows the unexpected electron angular distribution with a tilt angle relative to the expected direction (perpendicular to the molecular axis) for ${\text{H}}_2^{+}$ aligned parallel to the polarization plane [18]. The unexpected electron angular shift is related to the attosecond time lag of electron emissions with respect to the instant when the electric field is parallel to the molecular axis in [18]. However, they do not definitely confirm that most of electrons are emitted with a time lag. Here we expect by the classical ensemble model to obtain the direct evidence of the delayed emission of the electron.

In a strong laser field with intensity above 10¹⁴ W/cm², the discrete character of the energy spectrum of a molecule is washed out, since the energy levels can be shifted by the electric field to an amount comparable to the field-free energy level differences. Thus classical models including an infinite number of excited states often behave very well and reveal many detailed dynamics processes in the strong field ionization [22–27]. Moreover, previous studies have demonstrated that back analysis of classical trajectories is a powerful tool to explore the origins of the strong field phenomena [22,28,29]. It allows us, at the end of a simulation, to tag individual key trajectories, and then carefully back examine their histories and analyze their dynamics in detail. The laser intensity used in Odenweller’s experiment [18] is 6×10¹⁴ W/cm², which is well in the over-the-barrier ionization regime for ${\text{H}}_2^{+}$. Therefore the classical model is expected to well describe the electron emission of ${\text{H}}_2^{+}$ in this laser intensity. In this paper, we employ the classical fermion molecular dynamics model (FMD) [30–32] to investigate electron emissions from ${\text{H}}_2^{+}$ by circularly polarized laser pulses. Our classical calculations well reproduce the observed electron angular shift in Odenweller’s experiment [18]. Moreover, by back analysis of classical trajectories we find that most of electrons are emitted not at the instant when the electric field is parallel to the molecular axis, but hundreds of attoseconds later, providing the direct evidence for the delayed emission of the electron. Furthermore, the strong wavelength dependence of the angular shift is investigated.

2. The fermion molecular dynamics model

The classical fermion molecular dynamics model (FMD) was originally introduced by Kirschbaum and Wilets etal. [30, 31]. In order to prevent autoionization and collapse of the molecule, they add two momentum-dependent pseudopotentials to the usual Hamiltonian. The two pseudopotentials (V_H, V_P) constrain the motion to satisfy the Heisenberg uncertainty and the Pauli exclusion principles. Then Cohen introduces two additional terms V_m1, V_m2 to improve the treatment of the ${\text{H}}_2^{+}$ and H₂ molecules [32]. V_m1 is a one-electron operator, and V_m2 is a two-electron operator. This extension avoids the overbinding of ${\text{H}}_2^{+}$ and H₂ and successfully reproduces the Born-Oppenheimer ground state energies of ${\text{H}}_2^{+}$ and H₂. Very recently, the FMD model was further developed to study successfully the dynamics of laser-driven ${\text{D}}_3^{+}$ by Lötstedt etal. [33].

In the FMD model, the classical mechanical Hamiltonian of the ${\text{H}}_2^{+}$ ion is written

The proton mass m_p=1836, and the electron mass m_e=1. The precise value of the constant α is unimportant. Here α is chosen to be 4.0. The parameters ξ_H =0.9428, ξ_m1=0.90 are selected to fit the ground-state energies of H and ${\text{H}}_2^{+}$.

The evolution of the molecular system is determined by the Hamilton’s classical equations of motion:

3. Results and discussions

Figure 1(a) displays the two-dimensional photoelectron momentum distribution from ${\text{H}}_2^{+}$ aligned perpendicular to the polarization plane of circularly polarized laser pulses. The donut-shaped final electron momentum distribution shown in Fig. 1(a) agrees with the simple man model [10]. Figure 1(c) shows the radial momentum distribution of the photoelectron in the polarization plane, and its maximum is located at 1.62 a.u. which is exactly equal to the peak amplitude A₀ of the vector potential. The two-dimensional momentum distribution in z-y plane [Fig. 1(b)] is also consistent with the recent observation [Fig. 1(b) in [34]] for atoms.

 

Fig. 1 Photoelectron momentum distributions from ${\text{H}}_2^{+}$ aligned along z axis for a clockwise circularly polarized 800 nm laser pulse at a peak intensity of 6×10¹⁴ W/cm². (a) and (b) show the two-dimensional photoelectron momentum distributions in x-y and z-y planes, respectively. (c) is the photoelectron radial momentum distribution. The electric field E(t) rotates in x-y plane.

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When ${\text{H}}_2^{+}$ is aligned perpendicular to the polarization plane, the electron momentum distribution in the polarization plane is an uniform donut. However, for the case of ${\text{H}}_2^{+}$ aligned parallel to the polarization plane, the ionization rate is expected to be largest when E(t) is aligned along the axis of the molecule [35, 36]. Therefore, two maximum parts are expected to appear in the direction perpendicular to the molecular axis in the final electron momentum distribution. This prediction deviates from the recent experimental observation where two maximum parts are found in the first and third quadrants. Here we obtain the two-dimensional photoelectron momentum distribution in polarization plane for ${\text{H}}_2^{+}$ aligned along x axis by the classical FMD model, as shown in Fig. 2(a). It is obvious that there are two maximum parts in the first and third quadrants not in the expected y direction. This deviation from the expected direction is in qualitative agreement with recent experimental result [see Fig. 1(e) in [18]]. The momentum in the direction perpendicular to the polarization plane is almost zero [Fig. 2(b)]. Figure 2(c) shows the radial momentum distribution with a maximum of 1.5 a.u., which is slightly smaller than the peak amplitude A₀ of the vector potential. This is also qualitatively consistent with the experimental observation.

 

Fig. 2 Photoelectron momentum distributions from ${\text{H}}_2^{+}$ aligned along x axis for a clockwise circularly polarized 800 nm laser pulse at a peak intensity of 6×10¹⁴ W/cm². (a) and (b) show the two-dimensional photoelectron momentum distributions in x-y and z-y planes, respectively. (c) is the photoelectron radial momentum distribution. The electric field E(t) rotates in x-y plane.

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In [18] the angular shift is related to the attosecond time lag of electron emissions with respect to the instant when the electric field is parallel to the molecular axis. However, they do not definitely confirm that most of electrons are emitted with a time lag. Here we attempt to find the direct evidence of the delayed emission of the electron by tracing the classical trajectory. Back analysis [22, 37, 38] is used to investigate the electron emission process. The ionization time is defined as the instant when the energy of the electron becomes positive [27, 39], where the energy of the electron contains the kinetic energy, potential energy of the electron-proton interaction and the two momentum-dependent pseudopotentials (V_H, V_m1) [38]. According to back analysis, the electrons are predominantly ionized at internuclear distances of 6–9 a.u. in our calculations which is well in the so-called enhanced ionization regime [18, 40, 41]. A sample trajectory from Fig. 2(a) is shown in Fig. 3 from the beginning of the laser field to the end. Panels (a), (b) and (c) correspond to the electron energy, the electron coordinate x and y, and the electron momentum p_x and p_y as functions of the time in unit of laser cycle, respectively. Panel (d) displays the electron trajectory in x-y plane. At first the electron is bound [Fig. 3(a)] with negative energy and at the vicinity of the parent ion [Fig. 3(b)]. At the instant of 5.6 T₀ the electron gets enough energy to ionize, and then shifts out of the parent ion [Figs. 3(b) and 3(c)]. Obviously, the electron is ionized after the instant when the electric field is parallel to the molecular axis. At the end of the laser field the electron has the positive momenta in both of x and y directions [Fig. 3(c)], i.e., the electron is emitted into the first quadrant [Fig. 3(d)].

 

Fig. 3 A sample trajectory from Fig. 2. Panels (a), (b) and (c) show the electron energy, the electron coordinate x and y, and the electron momentum p_x and p_y versus time in unit of laser cycle, respectively. Panel (d) displays the electron trajectory in x-y plane. The arrows mark the ionization time. The black dash-dot curve is the sketch of the electric field along the molecular axis.

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Figure 4 shows the ionization yield as a function of the ionization time in unit of laser cycle for ${\text{H}}_2^{+}$ aligned along z axis [Fig. 4(a)] and aligned along x axis [Fig. 4(b)]. The blue dashed curves mark those instants that the electric field is parallel to the molecular axis. For the perpendicular alignment, the ionization yield firstly increases and then decreases with the time, which is similar to the laser envelope. Compared with Fig. 4(a), it is clear that the ionization yield curve in Fig. 4(b) shows a series of peaks separated by a half-cycle of the laser field. Moreover, these ionization peaks are not located at those instants that the electric field is parallel to the molecular axis, but hundreds of attoseconds later. This provides the direct evidence for the delayed emission of the electron. Without the effect of the Coulomb potential on the free electron considered, the electron final momentum is obtained as p_f =p_i–A(t_i). A(t_i) is the vector potential at the instant of ionization t_i, and p_i is the initial electron momentum. Because the p_i is small relative to A(t_i), the final direction is significantly determined by the vector potential A(t_i) at the instant of ionization. If the electron is emitted at the instant when the electric field is parallel to the molecular axis, the electron will appear in the direction (y axis) perpendicular to the molecular axis. Because of the time lag of hundreds of attoseconds relative to those instants when the electric field is parallel to the molecular axis, the electron gains a momentum with a tilt angle relative to the direction perpendicular to the molecular axis from the subsequent electric field. Thus the photoelectron momentum distribution shows an angular shift relative to the expected direction for the case of the parallel alignment.

 

Fig. 4 Panels (a) and (b) show the ionization yield versus the ionization time in unit of laser cycle for the cases of Figs. 1 and 2, respectively. The vertical blue dashed curves mark those instants when the electric field is parallel to the molecular axis. The green dash-dot curve is the sketch of the electric field along the molecular axis.

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Our classical calculations have well reproduced the unexpected angular shift in the electron momentum distribution from ${\text{H}}_2^{+}$ aligned parallel to the polarization plane. The attosecond time lag of electron emissions, which is responsible for the angular shift, is confirmed by the back analysis of classical trajectories. This implies that classical simulations can well describe the electron emission of ${\text{H}}_2^{+}$ by circularly polarized laser pulses. By solving the two-dimensional TDSE of ${\text{H}}_2^{+}$, the angular shift feature of the experimental data are reproduced as a part [peaks II of Fig. 1(f)] of the numerical simulation in [18]. However, the additional two prominent parts I and III near the axes [Fig. 1(f)], which do not exist in the experimental result, also appear in the numerical simulation. In our classical simulations the two additional parts are not found. In order to analyze the origin of the delayed electron emission with our model, we carefully examine the evolution of the classical ensemble in the circularly polarized laser pulses. We find the numbers of the electrons near the two protons are almost same at the instant when the electric field is parallel to the molecular axis. However, just after the electric field is parallel to the molecular axis many electrons are transferred to one of the protons and thus the electrons near one proton are much more than those near the other proton. This scenario is very similar to that reported by Takemoto et al. for linear polarization [17]. We consider that the transient electron localization at one of the protons results in the dominant electron emissions just after the instant when the electric field is parallel to the molecular axis.

In addition, our calculations obtain a final radial momentum of 1.5 a.u., which is slightly smaller than the vector potential (1.62 a.u.). In [18] the low final momentum is attributed to the non-zero initial momentum of the electron at the instant of ionization. Here by tracing the classical trajectory it is obtained that the electron is emitted with an initial momentum of 0.5 a.u., and there is a 106 degree angle between the initial momentum p_i and the inverse of the vector potential at the instant of ionization −A(t_i). If the Coulomb attraction between the parent ion and the escaping electron is neglected, the electron final momentum is 1.56 a.u., which is a little smaller than the vector potential but bigger than the final momentum obtained by our numerical calculations. This indicates that the non-zero initial momentum is one reason for the low final momentum but not the only reason. We suggest that Coulomb attraction between the parent ion and the escaping electron could also result in the decrease of the electron momentum.

We further explore the effect of the laser wavelength on the angular shift of aligned ${\text{H}}_2^{+}$ by circularly polarized laser pulses. Figures 5(a) and 5(c) show the two-dimensional photoelectron momentum distributions from ${\text{H}}_2^{+}$ aligned along x axis for circularly polarized laser pulses at I=6×10¹⁴ W/cm², λ=1200 nm and 1600 nm, respectively. The electron momentum distributions for 1200 nm and 1600 nm both show clear angular shifts. Furthermore, by comparing Figs. 2(a), 5(a) and 5(c), it is obvious that the tilt angle decreases with the laser wavelength increasing. By analyzing the ionization yield curves for different laser wavelengths [Figs. 4(b), 5(b) and 5(d)], one can easily find that the longer the laser wavelength is, the smaller the time lag in the electron emission is. This wavelength dependence of the time lag is responsible for the decrease of the angular shift with the laser wavelength increasing. Since the final emission direction of the electron depends on the laser phase of ionization, the ionization time lag is presented in unit of laser cycle in this paper. In our calculations we find that the time lags in atomic units for different wavelengths are almost same. For the same time lag in atomic units the long wavelength corresponds to a smaller value of time lag in unit of laser cycle in comparison with the short wavelength. Thus the time lag in unit of laser cycle decreases with the laser wavelength increasing. In order to more clearly display the strong wavelength dependence of the angular shift, we present the tilt angle as a function of the laser wavelength in Fig. 6.

 

Fig. 5 The two-dimensional photoelectron momentum distributions from ${\text{H}}_2^{+}$ aligned along x axis for a clockwise circularly polarized laser pulse at I=6×10¹⁴ W/cm², λ=1200 nm (a) and 1600 nm (c). (b) and (d) show the ionization yield versus the ionization time in unit of laser cycle for the cases of (a) and (c) respectively. The vertical red dashed curves mark those instants when the electric field is parallel to the molecular axis. The electric field E(t) rotates in x-y plane.

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Fig. 6 Dependence of the tilt angle on the laser wavelength.

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4. Conclusion

In conclusion, we have investigated the electron momentum distributions from ${\text{H}}_2^{+}$ by circularly polarized laser pulses with the classical fermion molecular dynamics model. For ${\text{H}}_2^{+}$ aligned perpendicular to the polarization plane, the electron momentum distribution exhibits donut-shaped structure which is similar to the case of atoms. For ${\text{H}}_2^{+}$ aligned parallel to the polarization plane the ionization events are clustered in the first and third quadrants in the electron momentum distribution, i.e., the ejection direction of the electron has a tilt angle relative to the y axis direction (the expected direction), which is in good agreement with the recent experimental result. By back analysis of classical trajectories we definitely confirm that this angular shift in the electron momentum distribution originates from the delayed emission of the electron [18], i.e., most of electrons are emitted not at the instant when the electric field is parallel to the molecular axis, but hundreds of attoseconds later. Further study indicates that the angular shift from ${\text{H}}_2^{+}$ by circularly polarized laser pulses decreases with increasing the laser wavelength.