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With the classical ensemble method, the correlated-electron dynamics of Mg atom from a doubly excited, transition Coulomb complex in few-cycle circularly polarized (CP) laser field at low laser intensity is theoretically investigated. The low energy transfer during the recollision process indicates that the two electrons cannot release directly, but it can pass through a doubly excited state, and then escape with the ionization time difference. The numerical results show that the feature of the sequential double ionization (SDI) can be observed in the nonsequential double ionization (NSDI) process. The SDI-like results demonstrate that the intermediate state has lost any memory of its formation dynamics. The distribution of the angle between the two release directions of the two electrons also depends on the ionization time difference. Finally, the influence of e-e Coulomb repulsion is discussed.
© 2017 Optical Society of America
1. Introduction
Nonsequential ionization (NSI) of atoms driven by strong laser field contains significant information about electron correlation, which has caught much attention in recent years [1–7]. So far, researchers widely hold the idea that NSI is described by a three-step model [8]. In this model, one electron tunnel ionizes when the laser field is strong enough and then is driven back to the parent ion as the oscillating electric field reverses its direction. It recollides with the parent ion and transfers a part of energy to it, which enables the second NSI event. The recollision process passing through different channels always plays a key role in NSI phenomena. For example, in nonsequential double ionization (NSDI) process, the returning electron can obtain enough energy to directly kick out the other for the higher laser intensity, which is called recollision-impact ionization (RII). For the lower laser intensity, the lower energy transfer cannot make the second electron release directly, but induces it to an excited state and then ionizing later, which is called recollision-induced excitation with subsequent ionization (RESI) [9–13]. However, at low laser intensity, researchers find that the two electrons could not be free directly but pass through a doubly excited Coulomb transition state [14–17]. This doubly excited complex (DEC) can lose memory of its formation dynamics, which emphasizes the dynamics only on the emission time from the complex in subsequent laser field.
Sequential double ionization (SDI) without recollision process driven by circularly polarized (CP) or elliptical polarized (EP) laser pulse is used to explore the fundamental physical problems, such as attoclock technique [18–20], SDI subcycle control [21] and electron angular correlation [22, 23]. In [18–20, 23], the relationship between the two emission angle and the ionization time difference is shown.
As we know, compared with linearly polarized (LP) laser pulse, small increasing of laser ellipticity can suppress the NSDI. However, NSDI in CP laser field have been observed experimentally and theoretically in Mg and other molecules [24–27], where the returning electron recollides with the other through a circular trajectory due to a larger initial transverse velocity.
In this paper, NSDI with an intermediate complex decay process Mg**→ Mg2+ + 2e− in few-cycle CP laser pulse is theoretically investigated. We use CP laser pulse to explore the NSDI from DEC and obtain SDI-like angular correlation phenomena. The SDI-like results demonstrate the effect of “lose memory”.
2. Methods
We carry out the simulated calculations with the classical ensemble approach which is proved to be useful in strong field ionization and especially in e-e correlation [28–32]. The general idea of this method is to mimic the prediction of a quantum wavefunction by using distributions of position, momenta, etc.. The initial classical ensemble is generated randomly. We use a 3D model of Mg with initial ensemble of 8 × 108 scale generated by the following equation in the absence of external fields
where Etot is the total energy of two electrons whose negative value is equal to the sum of first and second ionization potential. The subscript i = 1, 2 labels the two electrons. p⃗i is the electron momentum, and r⃗i is the electron coordinate. In Eq. (1), two terms in bracket represent kinetic energy and soft-core Coulomb potential between the ion core and the ith electron, final term is the potential between the two electrons. For Mg, Etot = −0.83 (Using atomic units throughout the paper if not stated otherwise). Softening parameters a and b are set to be 3.0 and 0.05, respectively, which could avoid autoionization. The Hamiltonian system (1) is solved numerically by symplectic method, which can preserve the symplectic structure of the system and be suitable for long-time calculations. To obtain the stable initial ensemble, the system is allowed to evolve a sufficient long time in the absence of the external laser field by classical equations of motion. With initial value available, laser field is turned on The electric field of the laser pulse can be written as here f(t) = sin2 (πt/T0), ω = 0.0599 a.u. that corresponds to a wavelength of 760 nm, the full duration of laser pulse is 5 cycles. φ is the carrier-envelope-phase with the arbitrary value. Laser intensity is 2.4 × 1013 W/cm2.3. Results and discussion
Figure 1 shows the “knee” structure of NSDI (The red dot line). The red dash line shows the SDI circumstance and the pink shadow region marks out the enhancement of double ionization due to electron correlation. The laser intensity applied in this work is around the cutoff position of the “knee”. The intensity is low and the return energy is too small to ionize both of the two electrons after recollision. Accordingly, the NSDI tends to occur through the doubly excited state.
Fig. 1 The “knee” structure of NSDI (The red dot line) is clearly shown in classical simulation. The red dash line is inserted by hand to indicate the SDI predicted by ADK theory. The pink shadow region marks out the enhancement of double ionization due to electron correlation.
We define an electron to be ionized if its energy ( , i = 1, 2) turns positive, and define the recollision time as the instant when the repulsion energy ( ) reaches its maximum after the first electron ionized. Figure 2(a) shows the energy trajectories (electron energy versus time), which indicates that there is one time recollision in the few-cycle laser pulse. The energy spectrum forks into two branches at around 1 o.c. (optical cycle). The upper and the lower branches represent the ionized electron and the bounded electron, respectively. The two branches converge after 2.5 o.c.. It means that the first electron is captured while the second electron is excited to a highly excited state, which forms a clear platform. The platform represents the DEC formation, which is roughly marked by gray area as shown in Fig. 2(a). The energy of the platform is about −0.2 ∼ −0.1 a.u., which is higher than the bound state (time before 1.0 o.c.) and below zero. The two energy branches all converge in this platform, so both of them have the same aspect and symmetric around the platform. Finally, all of the energy trajectories exceed zero after around 4 o.c., which means that the double ionization occurs from the DEC. In order to observe it clearly, we zoom in the gray area and picture the typical energy trajectories of the two electrons as shown in Fig. 2(b). Notice that they are not the average of all. The green and the blue lines represent the two energy trajectories. There is a delay time between recollision and final ionization which are written as Δt1 and Δt2 (Here, we define Δt1 as the first ionizing time after recollision), and Δt is the ionization time difference of the two electrons.
Fig. 2 (a) Energy of electrons versus time. Because of its low recollision energy the first ionizing electron becomes bound during the process and the DEC is created. The gray area roughly marks out the DEC formation. (b) Two typical electron energy trajectories (not the average of all). Δt1 is the delay time between recollision and ionizing of the one electron and Δt2 is the other. Δt is the ionization time difference of the two electrons. The distribution is on a logarithmic.
Figure 3 shows the distribution of the Δt1, Δt2 and Δt. The similar process has been investigated and intuitively showed the different time duration for their excited state [17]. From Fig. 3(a), we find that the first electron ionized mainly with 0.2 ∼ 0.4 o.c. time delay after recolliding, and 0.4 ∼ 1.0 o.c. for the second. The distribution of Δt1 and Δt2 locates above the diagonal due to distinguishing them by means of final ionization order. Red line in Fig. 3(b) represents the distribution of Δt, which indicates that the ionization time difference is almost less than 0.8 o.c.. We mark out three typical range with roughly equivalent counts (the areas of the three shadows are about equal) and label them as I, II and III. With the three range, we could qualitatively analyze electron correlation for the shorter, middle and longer ionization time difference. The choice of the three intervals is arbitrary in a way if it can show the comparison.
Fig. 3 (a) The distribution of Δt1 and Δt2 (Attach each projection as shown by red line). (b) The distribution of ionization time difference Δt (red line). Three typical range mainly explored in this paper are marked by I, II and III (See details in the text).
With “simpleman theory”, Eckel et al in the attoclock experiments [18–20] and Wang et al. in works [23] give an angular correlation in SDI process of Ar atom
where Δθ is the angle between the two release directions of the two electrons (See inset in Fig. 4) and it is proportional to the ionization time difference Δt. The scale factor is just the laser frequency ω. Notice that, in general, this conclusion is correct in the SDI process driven by CP laser pulse with the zero initial momentum. The stronger recollision in NSDI may twist the electron trajectories and enable electrons to leave the core with the non-zero initial momentum, which causes the shift from Eq. (4). However, as mentioned above, DEC is an intermediate transition state that like a “buffer” and loses memory of its formation dynamics. Final electron correlation behavior merely depends on the subsequent ionization process after decaying of the DEC (Mg** → Mg2+ + 2e−). As a consequence, the NSDI of Mg atom in lower CP laser intensity could show SDI-like results as shown in Fig. 4, which is similar to the discussions in [23]. The solid lines and the dashed lines show the distribution of the angle Δθ and the distribution of ωΔt, respectively. The red, blue and green lines correspond to the three ionization time difference areas I, II and III, respectively.Fig. 4 The distribution of the angle between the two releases direction of the two electrons, where the solid lines and the dashed lines show the distribution of θ and the distribution of ωΔt. The red, blue and green lines correspond to the three ionization time difference areas I, II and III, respectively. The inset illustrates the effect of enlarging the angle by the e-e Coulomb repulsion with a shorter ionization time difference (circumstance in area I).
Figure 4 shows that the distributions of the angle Δθ are in good agreement with the distributions of the angle ωΔt, which expressed as Eq. (4), in the areas II and III. However, the distribution of the angle Δθ dose not agree with the distribution of the angle ωΔt in the area I, which doesn’t match Eq. (4). We infer that this shift is caused by e-e Coulomb repulsion. This influence is different from the circumstance in LP laser pulse. In [17], Hang et al. demonstrate that the final-state e-e Coulomb repulsion plays a vital role in the formation of the line-shaped structural momentum distribution. In LP laser field, movement of the electrons is restricted in the polarized direction. Thus, the e-e Coulomb repulsion is also mainly along the polarized direction which leads to one electron being accelerated and the other being decelerated, and then an unequal energy sharing appears. However, in CP laser field, the e-e Coulomb repulsion is perpendicular to longitudinal direction, which could enlarge the angle between the two releases directions of the two electrons as shown in Fig. 4 inset. This is the reason that the distributions of the angle Δθ dose not agree with that of the angle ωΔt in the area I. However, the two electrons are not synchronously driven by the external laser field for the lager ionization time difference, the distributions of the angle Δθ in II and III can not be influenced by the e-e Coulomb repulsion and are in agreement well with Eq. (4).
The ion momentum distribution is widely used to illustrate the physics mechanism in strong field ionizing in CP or EP laser field [18–20, 23, 27, 33, 34]. Experimentally, this can be measured using the so-called Cold-Target-Recoil-Ion-Momentum-Spectroscopy (COLTRIMS) technique. An ion-electron system with zero net momentum enables us to write the ion momentum distribution as P⃗Mg2+ = −(p⃗1 + p⃗2). Figure 5(a)–5(d) show the ion momentum distribution for all Δt and in the three typical areas. Small angle Δθ leads to a large radius ring structure (area I), and vice versa (area III). A visualized illustration can be seen in the insets in Figs. 5(b)–5(d). The recollision process just contributes to the formation of the doubly excited state and enlarge the double ionization probability, but can not influence the SDI-like release of the two electrons after DEC decays. The information about DEC formation dynamics finally disappears. Accordingly, this result proves the effect of “lose memory” in NSDI from DEC in CP laser field.
Fig. 5 The ion momentum distribution. (a) All of the counts. (b)–(c) Correspond to area I, II and III. The insets illustrate the relationship that small angle Δθ leads to a large radius ring structure, and vice versa due to ion-electron system with zero net momentum. The distributions are normalized.
4. Conclusion
In conclusion, we have investigated the NSDI of Mg atom in few-cycle CP laser field at low laser intensity. The low energy transfer during recollision process leads to the DEC formulation. In the process of ionizing from decaying of the DEC, angular correlation is dependent on the ionization time difference from the transition state. This relationship proved the effect of “lose memory”. The distribution of the angle Δθ and the ring structure of the ion momentum exhibit the SDI-like feature. Furthermore, the result in shorter ionization time difference is affected by the e-e Coulomb repulsion. The repulsion will enlarge the angle between the two release directions of the two electrons. It has the guiding significance in further experimental works. And it also helps one understand or compare the different circumstance for the e-e correlation such as the two-center Coulomb potentials NSDI with delocalized electrons in [31] and the anticorrelation through simultaneous electron emission (SEE) regime in [35]. In SEE process, there is no time for excitation. The strong e-e repulsion pushes them apart and leads to the anticorrelation. However, the doubly excited state with the delay time could relief the repulsion and just shows the feature of the DEC dynamics.
In general, the electron dynamics driven by two dimensional fields is more complex comparing with that in LP fields. There are more tunable factor to explore the electron dynamics with DEC. For example, At other intensity the DEC may also exist with different yields. We infer that the proportion of DEC in all double ionization events could decrease with increase of intensity. The ellipticity-dependent circumstances could also be explored in further works. The small ellipticities (NSDI can also be observed in rare gases) will limit the motions of the electrons to the major polarized direction. A transition phenomenon (circumstance from EP laser to LP laser) in major direction of EP fields may be investigated. The SDI-like phenomena in NSDI helps one further understand the correlated electron dynamics in attoclock technique as well as the Coulomb effect for angle-to-time relationship.
Funding
National Natural Science Foundation of China (Grants No. 61575077 and No. 11271158).
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