1. Introduction

Nonsequential double ionization (NSDI) has been a hot topic in strong-field atomic and molecular physics because it reveals a highly correlated electron behavior [1, 2, 3, 4]. The correlated electron momentum distributions have provided profound insights into the physical mechanism of NSDI [5, 6, 7, 8]. Nowadays, the widely accepted picture for NSDI is the quasiclassical recollision model [9]. According to this model, the first electron tunnels through the combined barrier of the laser electric field and the Coulomb potential. Then it is driven by the oscillating electric field and returns to the parent ion as the electric field reverses its direction, recolliding with the ion inelastically and leading to the second electron freed directly in an (e, 2e) process or excited with subsequent field ionization (RESI) [10].

With the rapid advance of optical technology, the control of electronic processes in the strong field is achievable, which opens an entirely new domain of attosecond science [11]. It has been reported that the emission of the electron packet from an atom and its subsequent motion can be precisely controlled with the stabilized carrier-envelop phase (CEP) few-cycle pulses [12]. A recent paper has demonstrated that the motion of the bound electrons can also be controlled by the subcycle evolution of few-cycle pulses [13]. For NSDI, the controlling of electronic process is more challenging because of the complex dynamics of NSDI in which two electrons are involved. Recently, experimental and theoretical studies have reported that NSDI dynamics of molecules can be manipulated by changing the molecular alignment relative to the ionizing laser field [14, 15]. The momentum distributions of the doubly ionized ions from NSDI have been showed to strongly depend on the CEP of few-cycle pulses [16], which implies that NSDI can be controlled by the few-cycle pulse with stabilized phase.

As an efficient pathway for controlling the electron dynamics, the two-color scheme has been widely applied in high harmonic generation [17]. By mixing a controlling field to the driving field, the synthesized field is shaped and thus the tunnel ionization as well as the subsequent motion of the electron can be controlled by this shaped two-color field. In this paper, with the classical three-dimensional (3D) ensemble model [18], we demonstrate that the two-color few-cycle fields can be employed to steer the recollision dynamics of NSDI. For the two-color field, the correlated electron momentum distributions exhibit a surprising narrow arc-like structure, which is absent for the single-color field. With the relative phase of the two-color pulses changing, the arc-like structure moves gradually along the main diagonal of the electron momentum distributions. Back analysis reveals that the electron pairs from the arc-like region originate from the trajectories where recollisions occur in a very short time interval just before the field extremum and the recollision energies are often lower than the second ionization potential of He. The two conditions are responsible for the narrow arc-like structure in the correlated electron momentum distribution.

2. Results and discussions

The classical 3D ensemble model has been successful in understanding of NSDI [18, 19, 20] and described in detail in Ref. [18]. The ensemble size employed in our calculations is 3 million and the initial energy of each electron pair is set to be the ground-state energy of He (-2.9035 a.u.). In our calculations, the electric field of the two-color pulse is E(t) = E ₀(t) + E ₁(t), where E ₀(t) = E ₀ sin ²[π(t + NT ₀/2)/(NT ₀)]cos[ω ₀(t + NT ₀/2) + ϕ ₀]x̂ is the driving field and E ₁(t) = E ₁ sin ²[π(t + NT ₁/2)/(NT ₁)]cos[ω ₁(t + NT ₁/2) + ϕ ₁]x̂ is the controlling field. x̂ is the polarization vector. E ₀, ω ₀, T ₀ and ϕ ₀ are the amplitude, frequency, period and CEP of the driving field, respectively. E ₁, ω ₁, T ₁ and ϕ ₁ represent the amplitude, frequency, period and CEP of the controlling field, respectively. The wavelengths of the driving and controlling fields are 800 and 1600 nm respectively. Both pulses include four optical cycles, i.e., N=4. The laser intensities of the driving and the controlling fields are 6.0 × 10¹⁴ W/cm ² and 0.8 × 10¹⁴ W/cm ² respectively. The CEP ϕ ₀ is always set to be -0.5π while ϕ ₁ changes from -π to 0. The electric field of the laser pulses is shown in Fig. 1, where the CEP of the controlling field is -π and 0.

 

Fig. 1. Electric fields of the 800-nm pulse E ₀ (t) (solid black curve) and the two-color pulses with ϕ ₁ = -π (dash-dot red curve) and ϕ ₁ = 0 (dashed blue curve). P1, P2 and P3 denote three peaks of the electric fields.

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Figure 2 shows the correlated electron momentum distributions in the direction parallel to the laser field. It is clearly seen that for the single-color (800 nm) field the electron momentum distribution clusters in two circular regions, which are located in the first and the third quadrants, respectively. The DI yield from the first quadrant is higher than that from the third quadrant, which is consistent with previous studies [21, 22, 23]. For the two-color fields the electron momentum distributions exhibit arc-like structures. For ϕ ₁ = -π, the arc-like structure is located in the first quadrant. When ϕ ₁ changes from -π to -0.75π, the arc-like structure moves from the first quadrant toward the origin along the main diagonal k ₁ = k ₂ (k ₁, k ₂ are the parallel momenta of the two electrons) and from the third quadrant to the origin when ϕ ₁ changes from - 0.5π to 0. For ϕ ₁ = -0.75π and 0, the arc-like structures overlap circular structures, as shown in Figs. 2(c) and 2(f). These results reveal that the correlated electron momentum distributions have a strong dependence on the relative phases of the two-color fields. This implies that the correlated electron momentum distribution can be controlled wonderfully by the two-color field.

 

Fig. 2. Correlated electron momentum distribution in the direction parallel to the laser polarization for DI of helium by (a) the 800-nm field E ₀(t) and by the two-color fields with ϕ ₁ = -π (b), -0.75π (c), -0.5π (d), -0.25π (e) and 0 (f). k ₁, k ₂ denote the parallel momenta of the two electrons. The intensity of the 800-nm field is 6.0 × 10¹⁴ W/cm ² and that of the controlling field is 0.8 × 10¹⁴ W/cm ².

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In order to understand the detailed dynamics of NSDI by the two-color fields, we trace back all the DI trajectories. Figure 3 shows the counts of the DI trajectories from Fig. 2 versus the recollision time. The recollision time is defined to be the instant of the closest approach of the two electrons after the first departure of one electron from the core [20]. The dashed red curves denote the electric field and the dash-dot magenta curves denote -A(t), where A(t) = - ∫^t-_∞ E(t)dt is the vector potential. Forthe single-color field, two groups of trajectories significantly contribute to NSDI (indicated by GI and GII in Fig. 3(a)), while for the two-color field and ϕ ₁ = -π, only one group of trajectories significantly contribute to NSDI, where the recollision time ranges from -0.5T ₀ to 0. The absence of another group of trajectories is due to the fact that the electric field at P3 is weakened when the controlling pulse with ϕ ₁ = -π is added to the 800-nm pulse (as shown in Fig. 1), thus the returning energy of the recolliding electron ionized around P2 is insufficient to free the bound electron.

Similarly, for the relative phases ϕ ₁ = -0.5π and -0.25π, only one group of trajectories contribute to NSDI, for which the recollision time ranges from 0 to 0.5T ₀. When the phases ϕ ₁ are -0.75π and 0, two groups of trajectories contribute to NSDI. From Fig. 3 we find that in all cases recollisions occur near the extrema of the electric fields. For the single-color field recollisions occur after the field extrema, while for the two-color fields the peaks of the recollision time distributions are so nearly before the field extrema, especially for ϕ ₁ = -0.75π and 0. The recollision time distributions for the first groups (labeled as GI in Figs. 3(c) and 3(f)) of DI trajectories are entirely before the field extrema. In NSDI, the recollision time of the first electron ionized around the field extremum is determined by the subsequent accelerating field. For ϕ ₁ = 0, as shown in Fig. 1, the accelerating field (P3) is much stronger than the ionizing field (P2), thus the electrons ionized around P2 can be driven back before the next field extremum. Moreover, it is clearly showed in Fig. 3. that recollisions of the first groups of trajectories are concentrated in a very short time interval for ϕ ₁ = -0.75π and 0. The short time interval is about 1/5 laser cycle (several hundreds attosecond). Below we will reveal that this behavior is responsible for the narrow arc-like structure in the correlated electron momentum distributions.

 

Fig. 3. Counts of DI trajectories vs recollision time. Panels (a)-(f) correspond to figure 2(a)-(f), respectively. The laser field is shown by the dashed red curve. The dash-dot magenta curve indicates -A(t) with the scale given on the right-hand ordinate.

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In Figs. 4(a) and 4(b) we display the correlated electron momentum distributions of the first (GI) and the second (GII) groups of trajectories from Fig. 3(f), respectively. It is well known that in the (e,2e) process, the correlated electron momentum is distributed in a circle centered at -A(t_i) (t_i is the ionization time) with radius √2E_exc, where A(t_i) is the vector potential and E_exc is the excess energy of the recolliding electron [10]. For the second group, where recollisions occur mainly after the field extremum (see Fig. 3(f)), the population is concentrated in a circular region, which is consistent with previous studies [10, 23]. While for the first group of trajectories, for which recollisions occur in a short time interval just before the field extremum, the electron pairs are clustered in a very narrow arc-like region. Back analysis shows that for other relative phases of the two-color fields, the electron pairs clustering in the narrow arc-like region also originate from the trajectories where recollisions occur in a short time interval just before the field extremum. Why are the correlated electron momenta of these trajectories concentrated in the narrow region and why do the momentum distributions exhibit an arc-like structure?

In Fig. 4(c) we show the returning energy of the recolliding electron for ϕ ₁ = 0. It is clearly seen that the returning energies of the first group of trajectories are often lower than 2.0 a.u., i.e., the ionization potential of He ⁺. For these trajectories, after recollision the electrons are ionized with negligible velocity when the potential barrier is suppressed by the electric field and their final momenta are determined by -A(t_i) (t_i is the ionization time and where A(t_i) is nearly zero). The two conditions that the recollision time is restricted in a short interval and most recollision energies are lower than the second ionization potential of He, result in the narrow correlated electron momentum distribution.

 

Fig. 4. Correlated electron momentum distributions for (a) the first and (b) the second groups of trajectories in Fig. 3(f). (c) The returning energy of the recolliding electron for the trajectories from Fig. 3(f). The solid green curve indicates the electric field. (d) The energies of the electrons at the time 0.03 T ₀ after recollision. Only the events within region A have been selected. (e) Same as (d) but for the events in region B. (f) Accumulated fraction of double ionization vs time delay between recollision and double ionization for the events in region A (the solid red curve) and region B (the dashed blue curve).

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Further, we divide the momentum distribution of Fig. 4(a) into two regions: region A where ∣k ₁ -k ₂∣ > 1.0 a.u. and region B where ∣k ₁ -k ₂∣ < 1.0 a.u. The energies of the electrons at the time 0.03T0 after recollision for regions A and B are shown in Figs. 4(d) and 4(e), respectively. In Fig. 4(f), we show the fraction of DI versus time delay between DI and recollision. For the trajectories from region A, the two electrons have similar energies after recollision (see Fig.4(d)), and both electrons are ionized over the suppressed barrier immediately after recollision. As a result, the electron pairs achieve similar final momenta. For the trajectories from region B, because of the unequal energy sharing during recollision (see Fig. 4(e)), there is a longer time delay between recollision and DI. The electron with a higher energy is freed immediately after recollision while the other with a lower energy is freed after a longer time after recollision. As a consequence, the two electrons achieve different final momenta, resulting in the distribution in region B.

As mentioned above, the arc-like structures of the electron momentum distributions evolve gradually from the third quadrant to the origin along the diagonal k ₁ = k ₂ when ̅ ₁ changes from -0.5π to 0 and move from the first quadrant toward the origin whenf1 changes from -π to -0.75π (see Figs. 2(b)-(f)). The electrons from region A of the arc-like shape are ionized immediately after recollision and their momenta are determined by -A(t_i). Thus the moving of the momentum distribution indicates the change of vector potential at the time of recollision. In other words, it implies that the recollision time can be exactly controlled by the relative phases of the two-color fields.

By changing the relative phase of the two-color field, the asymmetry of the electric field in the adjacent half cycles is significantly modulated. As a result, the recollisions of the DI trajectories can be restricted in an attosecond time interval. This process is insensitive to the laser intensity, frequency and duration of the few-cycle pulses. Our further calculations confirmed this conclusion. However, in the limit of long pulses, the narrow arc-like structure in the electron momentum distribution is indistinct even though a great part of recollisions are restricted in a short time interval before the field extremum. This is due to the fact that in the long pulses, the contribution of the RESI pathway increases [4], and the electrons ionized through RESI obscure the arc-like distribution.

3. Conclusion

In conclusion, using the classical 3D ensemble model, we have demonstrated that the recol-lision dynamics of NSDI can be manipulated exactly by changing the relative phase of the two-color pulse. The recollision time can be restricted in an attosecond time interval and at the same time the recollision energy for most recollisions is limited to be lower than the second ion-ization potential of He. As a result, the correlated electron momentum distribution from NSDI exhibits a narrow arc-like structure. This novel structure implies a unique electronic correlation in NSDI by the two-color few-cycle pulses. The physical mechanism for this correlation will be investigated in our further work. Our results suggest that the two-color field can serve as a powerful tool to investigate the recollision dynamics of NSDI in attosecond time precision.