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We investigate the momentum and energy distributions of the two electrons in multiphoton double ionization of He by intense attosecond xuv pulses, based on a two-dimensional model. Two different patterns of the momentum distributions are identified, corresponding to the uncorrelated and correlated channels, respectively. Our analysis of the electron correlations focuses on two-photon and three-photon double ionization processes for different pulse durations and for different time delays after the pulses. For both two-photon and three-photon cases, a clear correlation valley in energy distributions is found when both electrons are ejected in opposite directions. This is mostly attributed to the electron correlations during the ionization of the first electron. We also find that when two electrons are ejected in the same direction, their Coulomb repulsion has an significant influence on the electron energy distributions during the postionization stage. Finally, in the case of three photon double ionization, we observe that the effects of the Coulomb repulsion become much more complicated, and a new catch-up collision phenomena is observed in the energy distributions.
©2010 Optical Society of America
1. Introduction
It is well-known that many strong field phenomena including multiple ionization in atoms and molecules cannot be treated using the independent electron approximation. To understand and characterize the correlations between two active electrons is the basis for understanding the dynamics of a multielectron system in strong laser fields. In recent years, with the fast development of free-electron laser (FEL) technologies, some important FEL facilities, such as the SLAC’s linac coherent light source (LCLS), FLASH in Germany, and Spring-8 compact SASE source in Japan, can provide ultrashort xuv pulses at very high intensities (> 1014 W/cm2) [1–3]. These new laser sources make it possible to experimentally explore the role of the electron correlations in atomic double ionization by absorbing a few xuv photons, which is radically different from double electron ejections in intense infrared (IR) laser field by absorbing many photons. In the latter case, the electron correlations play a very important role in the nonsequential double ionization where two electrons are ejected simultaneously, and it is well understood based on the rescattering model [4–10]. With the availability of FELs, double ionization involving a few photons as well as a single photon has become experimentally accessible [11–14], and related theoretical researches on the roles of the electron correlations have been discussed [15–22].
Obviously, the two-photon double ionization (TPDI) is the most fundamental multiphoton process and provides a new opportunity for exploring the electron correlation effects. In the long pulse duration limit, TPDI processes are classified into two types, namely, ‘sequential’ and ‘nonsequential’ processes. For the sequential two-photon double ionization, which dominates when the photon energy is larger than the second ionization potential, the two photons ionize the two electrons one by one independently. In this case, one photon ionize the first electron from the ground state of a neutral atom, and the other photon has enough energy to kick out the second electron from the ground state of the single charged ion. On the contrary, the nonsequential double ionization dominates, when the photon energy is smaller than the second ionization potential but still larger than half of the sum of the first and the second ionization potentials. The second electron could not be liberated independently by absorbing one photon, and thus requires the reassignment of excess energy through the electron correlations. For shorter pulse durations comparable with the characteristic time scale of the electron correlations, the relative importance of the correlated effects is greatly enhanced, even in the deep sequential regime. In the short duration pulse limit, differentiating the ‘nonsequential’ and ‘sequential’ double ionizations seems to be difficult. Thus in the current work, we will use the terminologies of the processes through the ‘correlated’ and ‘uncorrelated’ channels instead of the ‘nonsequential’ and the ‘sequential’ processes.
Indeed, Ishikawa and Midorikawa [23] studied electron energy spectra of TPDI integrated over ejection angles for He by intense attosecond x-ray pulses, and found the “anomalous” component which cannot be explained in the context of sequential and nonsequential double ionizations. They explained the anomalous component in terms of the second ionization during core relaxation (SICR) [15–17], where the energy exchange between the ionized and bounded electrons is important during the first ionization. They also pointed out that the postionization (the stage after two electrons being ionized) energy exchange (PIEE) due to the Coulomb interaction, depending on the distance between the two ionized electrons, could not be completely excluded and some further investigations are needed. Barna et al. [24] inspected the angular distribution of the sequential TPDI and found that the ionization peak retains an emission pattern of a Hertz dipole during the whole pulse, while the inner region between these two peaks has a mixed emission pattern of dipole and quadrupole, which may be due to a correlated double ejection. Moreover, Ivanov et al. [25] have also found a strong deviation from dipole emission pattern outside the two energy peaks. All the above discussions indicate that two active electrons always interact with each other during the whole process of double ionization. Even in the sequential double ionization regime, where the uncorrelated channels dominate, the correlation effects can not be completely neglected.
In this work, we investigate TPDI processes by ultrashort attosecond xuv pulses in the sequential regime, where the photon energy is larger than the second ionization potential, based on the aligned-electron model He atom [26,27]. By analyzing the main patterns in the momentum distributions, we find that this simplified model can reveal many typical characteristics of TPDI. Previous works have pointed out that the electron correlation effects can be reflected in both energy and angular distributions [23–25]. In the present paper, by doing a joint analysis of the emission direction and energy distribution, we find that the appearance of energy distributions strongly depends on the emission directions of the two electrons. Our results show that, when two electrons are ejected in different directions, the SICR mechanism dominates, and a correlation valley can be identified in energy distributions; but when two electrons are ejected in the same direction, the PIEE mechanism becomes much more important. Moreover, we investigate the three photon double ionization processes to find whether the electron correlation effects will be different when more photons are absorbed. Indeed, we find that if two ionized electrons have opposite directions, SICR mechanism still dominates, as in the case of TPDI. However, when two ionized electrons have the same direction, the PIEE mechanism becomes much more complicated. A new catch-up collision phenomena is identified in the energy distributions, which happens when the second electron has a much larger kinetic energy than the first one.
The rest of the paper is organized as follows. In Section 2, we briefly introduce our numerical method for solving the time-dependent Schrödinger equation (TDSE). In Section 3, we present and analyze our results. Two types of multiphoton processes are identified in the electron momentum distribution. The electron correlation effects at different stages are carefully examined and explained in the multiphoton double ionizations. Finally, we give a short conclusion in Section 4. Atomic units are used unless otherwise specified.
2. Numerical method
The aligned-electron model He atom restricts the motion of two electrons in one spatial coordinate along the laser polarization. It has been widely adopted [26,27] and is able to give a good description for longitudinal behaviors of two ionized electrons. Compared with the integration of full-dimensional Schrödinger equation, such a low-dimensional model is much less demanding for computation resources. Although the reduction of degrees of freedom may overestimate the influence of the Coulomb interaction, this model is still physically motivated.
The hamiltonian of the model He atom is given by
where we use soft-Coulomb potentials in order to smooth the singularities, and E(t) is the electric field strength of the linearly polarized xuv pulse. The first and the second ionization potentials of the model atom are I p1 = 0.755 a.u. and I p2 = 1.483 a.u. respectively, and they are slightly smaller than those for a real He atom of 0.903 a.u. and 2 a.u., respectively. Please note that our physical discussions below will not sensitively depend on the choice of soft parameter. In the present studies, we consider the situation where the model He atom is irradiated by a strong xuv pulse of a sin2 envelope,
where the photon energy of ω = 2.5 a.u. and the peak value of the electric field of E 0 = 0.377 a.u. (which corresponds to the peak intensity of 5 × 1015 W/cm2), are fixed throughout this study. The number of optical cycles, N = ωτ/2π, varies from 8 to 32 (from about 180 to 720 as in FWHM). The chosen photon energy ω is larger than the double ionization potential of the model He, Ip = I p1 + I p2 = 2.238 a.u., which means that an electron can be directly ejected by absorbing a single photon without any excited states involved. To numerically solve the TDSE,
we employ the finite-element discrete variable representation (FE-DVR) method [28–30] to discretize the two spatial variables of x 1 and x 2. The FE-DVR method has advantages of providing a sparse matrix representation of the kinetic energy operator and a diagonal representation of the potentials. According to our experiences, using the FE-DVR method, very little data communication is needed between adjacent CPUs when the program is coded with the message passing interface (MPI). This transparent parallelization makes our code very efficient. Temporal propagation is carried out through Arnoldi propagator [31,32], whose accuracy and stability have been verified through our convergence tests. Note that we use the zero boundary condition with a sufficiently large box size so that the reflection of the wave packets from the boundaries can be neglected.
The hamiltonian is invariant under the particle exchange (x 1 ↔ x 2), so that there is no transition between the states with different symmetries. In all the calculations below, we consider the symmetric states which belong to the spin singlet states. To accelerate our calculations, we only calculate the upper triangle of hamiltonian matrix, namely, the part satisfying x 1 ≥ x 2, while the lower triangle part is obtained by exchanging the spatial coordinates. We solve the TDSE with the ground state as the initial state. The bound states of the model He atom can be obtained by repeatedly relaxing an arbitrary test wavefunction in the imaginary time propagation. For our momentum-space analysis, we split the two-dimensional coordinate space into three types of regions as shown in Fig. 1, namely, the bound state excitation region B, the single ionization region S, and the double ionization region D. After the end of the pulse, the total wavefunction is freely propagated for a sufficiently long time so that even the low energy electrons enter into regions S and D. At a certain time delay Td after the laser pulse is over (tf = τ + Td), we apply a Fourier transformation to the doubly ionized wavepackets in region D, which is extracted by multiplying a window function to the total wavefunction so that the influences of the bound excited states and the single ionizing states are removed,
Fig. 1. The two dimensional space is split by the thick solid lines into three type of regions: B, S and D. Type B, where |x 1| ≥ a and |x 2| = a, contains all the bound states; type D, where |x 1| > a and |x 2| > a, represents the double ionization region; type S, the rest of the space, stands for the single ionization region. Here a denotes the width of the window function.
where k 1 and k 2 are the momenta of the two electrons, and the explicit form of the window function is
Here a and b denotes the width and the dumping rate of the window function, respectively. Typically, we set a = 35 a.u. and b = 15 a.u. Note that all our results in this work are not sensitive to the parameters of the window function. The momentum distribution of the doubly ionized electrons is obtained by
For doubly ionized electrons, their total kinetic energy is given by E = ½(k 1 2 + k 2 2), and the total kinetic energy distribution is
where k and θ are satisfied with tan θ = k 2/k 1 and k = √k 1 2 + k 2 2 = √2E.
Fig. 2. Energy levels of the model He atom of interest in the present work. The first and the second ionization potentials of the model atom are I p1 = 0.755 a.u. and I p2 = 1.483 a.u. respectively. Also shown are four double ionization processes: (a) one photon double ionization; (b) two photon double ionization; (c) three photon double ionization with the first electron absorbing only one photon; (d) three photon double ionization with the first electron absorbing two photons.
3. Results and discussions
In this section, we will analyze our numerical results. Figure 2 shows four basic double ionization processes of our interest in the current work. According to the chosen photon energy ω > Ip, there are mainly four double ionization processes. The first is one photon double ionization, as shown in Fig. 2(a), but it is not the emphasis in this paper. The second is the uncorrelated TPDI, as shown in Fig. 2(b), where the second electron is ionized from the He+. This induces a double peak structure in the energy distribution [33]. The other two processes belong to the uncorrelated three photon double ionizations, as shown in Fig. 2(c) and Fig. 2(d), where their difference depends on which electron absorbs two photons to be ionized. If we consider two electrons are completely uncorrelated, there are four peaks in energy distributions for three photon processes, namely, ω - I p1, 2ω - I p1, ω - I p2, and 2ω - I p2.
In the following, we will first look at the total energy and momentum distributions to identify the multiphoton double ionizations. Then we will focus on the two-photon double ionization in Fig. 2(b), identifying the electron correlations in different ionization stages. Finally, we will present the results on three-photon double ionization in Figs. 2(c) and Fig. 2(d), in which a catch-up collision is observed.
3.1. Momentum and energy distributions of doubly ionized electrons
First, we show the total kinetic energy distribution, P DI, for the 32 cycle pulse and time delay Td = 80 a.u. in Fig. 3(a). Clearly there exist a series of peaks due to multiphoton double ionization, which is similar to the usual above threshold ionization peaks for a single electron atom. Namely, p1 corresponds to the total kinetic energy of 2.76(4) a.u., indicating that it is due to TPDI from the ground state (2ω - Ip = 2.762 a.u.). Meanwhile, p2 with a total kinetic energy of 5.26(8) a.u. corresponds to a higher order process, three photon double ionization (3ω - Ip = 5.262 a.u.). Similarly, we also attribute p3 with a total kinetic energy of 7.76(1) a.u. to the four photon double ionization (4ω - Ip = 7.762 a.u.). There are actually some small peaks at about 4.2 a.u., 6.8 a.u. and 8.3 a.u., which are possibly due to the traces of grid pattern in Fig. 3(b). And we do not show energy spectrum below E = 2, the region not concerned about here.
Fig. 3. (a) Total kinetic energy distribution of the doubly ionized electrons by an attosecond xuv pulse with ω = 2.5 a.u., N = 32 cycles, and the peak intensity of 5 × 1015 W/cm2. (b) Joint momentum distributions of the doubly ionized electrons with the same laser parameters. The ring r1, r2, and r3 corresponds to the peak p1, p2, and p3 in (a) respectively.
To investigate the multiphoton double ionization further, we plot the momentum distribution as shown in Fig. 3(b). Obviously, there exist two major patterns: grids and rings. The grid pattern can be described by the direct product of the transition probabilities of two individual electrons, and thus it represents multiphoton double ionization which can be decomposed into two completely independent above threshold ionization processes involving only one electron. We call this double ionization process as the one through the uncorrelated channels. For example, if the first electron is ionized by absorbing n 1 photons (n 1 ≥ 1), and the second is independently ionized from the He+ ion ground state by absorbing n 2 photons (n 2 ≥ 1), these two independent above threshold ionization processes make up an uncorrelated (n 1 + n 2) photon double ionization. In contrast, the ring pattern represents the double ionization process having the energy conservation , where n is the number of photons absorbed by two electrons. In this process, two ionized electrons exchange their energy to ensure k 1 2 + k 2 2 = constant through the strong electron correlations. We thus call this process as the one through the correlated channels. Note that rings, r1, r2, and r3 in Fig. 3(b) correspond to peaks, p1, p2, and p3, respectively in Fig. 3(a).
Indeed the present results for model He are very similar to those for the full-dimensional He in Ref. [34]. This means that the present low-dimensional model is physically motivated and is able to reflect many important characteristics of the multiphoton double ionization. Panfili [35] investigated 13-photon double ionization at the laser wavelength of 248 nm, based on the same model He atom. The author mainly focused on the double ionization process through the uncorrelated channels and investigated the time evolution of the grid pattern during a laser pulse. In this paper, we investigate the double ionization processes of such a model He atom in the xuv regime.
3.2. Two photon double ionization
Let us focus on the process of TPDI from the ground state, namely, peak p1 in Fig. 3(a) or ring r1 in Fig. 3(b). This is the most fundamental multiphoton double ionization. The traditional relationship of kinetic energy and momentum (i.e., ) loses the information about the electron emission direction. For the two dimensional model, this disadvantage can be avoided by redefining the electron energy as following [35],
Fig. 4. (a) Energy distributions for one of the TPDI electrons from the ground state He by a pulse with ω = 2.5 a.u. and the peak intensity of 5 × 1015 W/cm2 for different pulse duration as indicated. All the curves have been normalized to unity at their maximal value for better comparison of the relative importance of the correlation valley. (b) The total energy distribution for the case 32 cycles (red dashed line) by adding up the negative part (green dashed-dotted line) and the positive part (blue solid line) in (a).
where E 1 is the kinetic energy for one of the two electrons. It is negative when both electrons are ejected in opposite directions, and positive when they are ejected in the same direction. We will adopt this energy definition below unless otherwise stated.
We plot the TPDI energy distributions along ring r1 as a function of E 1 for different pulse durations in Fig. 4(a). First, we analyze the distributions in the negative energy regime where two electrons are ejected in opposite directions. For the 32 cycle pulse, we find two well-defined peaks exist at -1.747 a.u. and -1.026 a.u., as expected for the uncorrelated channels, where the first electron is ionized from the model He atom with the kinetic energy of E 1 = -(ω - I p1) = -1.745 a.u., and the second is ionized from the He+ with the kinetic energy of E 1 = -(ω - I p2) = -1.017 a.u. There is also an apparent valley between two peaks, which is very flat E 1 ~ -1.4 a.u. For shorter pulses of 16 cycles, the valley becomes more prominent. For the 8-cycle pulse, the two energy peaks merge together and the valley thoroughly converts into a single peak. Note that for the laser pulse parameters adopted here, when pulse duration is 16 cycles and 8 cycles, the corresponding energy uncertainty is about 0.07 a.u. and 0.14 a.u., if full width at half maximum (FWHM) is taken into account. However, in Fig. 4, the two peaks originally separate by 0.73 a.u. can not be identified for 8 cycle laser pulse. So there must be some other reasons responsible for the disappearance of the two peaks structure . Although two energy peaks are broadened due to the limited duration of laser pulse, this valley could hardly be explained as the overlap of the spectrum broadening of two pulses [20,21]. We call it as the correlation valley in the following.
Fig. 5. Electron energy distributions of TPDI from the ground state of He for different observation time Td as indicated. The parameters of the attosecond xuv pulse are taken to be ω = 2.5 a.u., N = 12 cycles, and the peak intensity of 5 × 1015 W/cm2.
Next, we look at the TPDI distributions in the positive energy regime, where two ionized electrons have the same direction. We find that two well-defined peaks stably exist similar to those in the negative energy regime, but the correlation valley never appears even for the shortest pulse duration of 8 cycles. Further, for the 32 cycle pulse, a careful inspection of the energy peaks shows that the faster electron ionized from He ground state has the kinetic energy of 1.776 a.u., slightly larger than the expected value 1.745 a.u. in accordance with the uncorrelated channels, while the slower electron ionized from He+ has the kinetic energy of 0.971 a.u., smaller than the expected value 1.017 a.u. For the shorter pulses, the faster electron becomes much faster while the slower one becomes slower.
In Fig. 4(b), we further plot the energy distributions by adding the negative and the positive part together in the traditional definition of energy for the 32 cycle pulse. Generally speaking, no matter two electrons are ejected in the same or opposite directions, they will contribute to the energy peaks corresponding to the uncorrelated double ionization. However, the inner region around the valley between the two peaks, is completely attributed to the double electron ejection in opposite directions, while the outer region of the two peaks is mostly due to the double ionization in the same direction.
All above phenomena indicate that in the positive and negative energy regimes, different underlying mechanisms govern the dynamics of the two electrons. Let us look into the details of the electron correlation effects for each case. For the case where two energetic electrons are ejected in opposite directions due to the back-to-back motion, the distance between them increases very rapidly with the time evolution, so that the effects of Coulomb repulsion between the two ionized electrons, or the PIEE effects, could be neglected. Thus, the electron correlations during the first ionization, or the SICR effects, are considered to be the most important factor in this case. If the time interval between two ionization events is so large that the He+ has relaxed to its ground state during the second ionization, the first electron may take away all the dielectric interaction energy in the initial state [36]. If the time interval is so short that the first electron still stays around the core during the second ionization, then both electrons will get a part of the dielectric interaction energy. For an extreme case in which two electrons are ionized simultaneously and ejected back to back, both of them would feel the same ionization threshold, and have similar kinetic energies due to the symmetry of the system. This is the origin of the correlation valley between the two peaks due to the SICR effects in the traditional energy distribution in Ref. [23].
On the other hand, when the two outgoing electrons are ejected in the same direction, due to the side by side motion, they may be much more closer with each other in space for some longer time. In this case, the Coulomb repulsion between the two ionized electrons can largely influence the postionization dynamics of the two electrons. The faster electron is accelerated and the slower one is decelerated, due to the energy transfer. This is the reason why in the positive energy regime, the faster electron gets even faster and the slower gets even slower. In Fig. 4(a), we also observe that the distance between the two positive energy peaks become larger as the pulse duration decrease from 32 cycles to 8 cycles, which indicates a larger energy transfer from the slower electron to the faster one. If the two side-by-side electrons are ionized nearly at the same time, the strong Coulomb repulsion between them would induce a much larger energy transfer. Thus, it is impossible to observe a stable correlation valley in the positive energy regime.
Our interpretation can be further consolidated by investigating the energy distributions at different time delay Td after the end of pulse. We show the time delay dependence of TPDI energy distributions for the N = 12 cycle pulse in Fig. 5. Traces of the Coulomb repulsion can be clearly observed by examining the shift of the positive energy peaks. We find that the distance between the two positive peaks increases by increasing time delay Td, indicating that the faster electron is accelerated at the expense of deceleration of the slower one. We also observe that the negative energy peaks are not at all sensitive to the time delay, verifying the conclusion that the electron correlations between the two ionized electrons in opposite directions after the pulse ends, has little influence on the electron energy distributions.
Here based on the model atom, we have clarified different effects of the electron correlations at different stages. The electron correlations during the first ionization, or the SICR, are mostly responsible for the correlation valley in the inner region of two peaks. This effect is apparent only when both electrons have opposite directions, because in this case the influence of Coulomb repulsion between two ionized electrons can be neglected. One can further infer that, for the real He atom, the effect of SICR could be significantly observed when the two electrons are ejected in opposite hemispheres. Meanwhile, the Coulomb repulsion between two ionized electrons, or PIEE, is mostly responsible for the outer region of two peaks. This effect is most apparent only when two ionized electrons have the same direction and close to each other in space. We can also infer that, for the real He atom, the effect of PIEE may be more clear when two electrons are ejected in the same hemisphere.
3.3. Three photon double ionization
Up to this point, we have focused on the role of the electron correlations in TPDI. In this subsection, we turn our attention to the double ionization processes involving three photons. Does the electron correlations become less important when two electrons become much more energetic by absorbing more photons? Or does the role of electron correlation at different stages change? Our discussion will be focused on the three photon double ionization from ground state, namely peak p2 or ring r2 in Fig. 3.
In Fig. 6, we first investigate the energy distributions in the negative energy regime, where two electrons are ejected back to back. When the pulse duration is 32 cycles in Fig. 6(a), there are totally four peaks from the left to the right, corresponding to the first ionization from He by absorbing two photons (P4, - (2ω - I p1) = -4.245 a.u.), the second ionization from He+ by absorbing two photons (P3, - (2ω - I p2) = -3.517 a.u.), the first ionization from He by absorbing one photon (P2, - (ω - I p1) = -1.745 a.u.), and the second ionization from He+ by absorbing one photon (P1, - (ω -I p2) = -1017 a.u.). The outer two peaks, P1 and P4, correspond to three photon double ionization in which the first electron absorbs two photons, as shown in Fig. 2(d). The inner two peaks, P2 and P3, correspond to three photon double ionization in which the second electron absorbs two photons, as shown in Fig. 2(c). Obviously, we find these four peaks are symmetric about E 1 = -2.632 a.u. That is because the restriction of k 1 2 + k 2 2 = constant. Two correlation valleys are clearly seen at E 1 ~ -1.5 a.u., between two one photon ionization peaks, P1 and P2, and at E 1 ~ -4 a.u. between two two-photon ionization peaks, P3 and P4. If we shorten the pulse duration from 32 cycles in Fig. 6(a) to 8 cycles Fig. 6(c), two correlation valleys are gradually enhanced. The two connected peaks move closer to and finally merge with each other for the 8 cycle pulse. Comparing with Fig. 4, we find that the correlation valleys in three-photon double ionization have very similar characteristics with those in TPDI process. It is reasonable to explain the two valleys in terms of the electron correlations during the first ionization, namely, the SICR effects. The first electron is ejected by absorbing one photon, and the second is ionized during the core relaxation by absorbing two photons, or in turn the first absorbs two photons, and the second absorbs only one when the core is still relaxing. Note the two peaks connected by the correlation valley is asymmetric in this case.
Fig. 6. Electron energy distributions of three photon double ionization from the ground state He by an attosecond pulse with ω = 2.5 a.u. and the peak intensity of 5 × 1015 W/cm2 for pulse duration: (a) N = 32 cycles, (b) N = 16 cycles, and (c) N = 8 cycles. In each frame, the energy distributions at different time delay Td are shown to identify the role of the electron correlations in the positive energy regime.
When we pay attention to the energy distribution in the positive energy regime, we find there are also about 4 peaks, labeled as P′1, P′2, P′3, and P′4. These four peaks could generally exist even for the 8-cycle pulse in Fig. 6(c), which is in contrast to the negative energy regime for this case. If we plot the energy distributions at different time delay Td, we find the outer two peaks P1 and P4 are very stable, while the inner two peaks P′2 and P′3 change rapidly. A hump structure in Fig. 6 is also identified at E 1 ~ 2.632 a.u. for short time delays. The hump completely disappears at longer time delay Td. These characteristics are different from those in TPDI. By inspecting the corresponding energies of these four peaks, we find the outer two peaks, P1 and P4, represent a three photon process demonstrated as in Fig. 2(d), where the first electron absorbs two photons. Because the first electron is faster than the second one, the distance between the two electrons increases rapidly, and the effects of the Coulomb repulsion in the final state are greatly weakened. Thus, the corresponding two peaks in the energy distributions are very stable. On the other hand, for the inner two peaks, P2 and P3, the first electron absorbs only one photon, while the second absorbs two photons, which is demonstrated as in Fig. 2(c). The second electron is much more energetic, and it may catch up with the first one and induce a collision. Much energy may be transferred between the two electrons, which is different from the energy transfer in TPDI case. The hump shows that two electrons may have a similar kinetic energy at a certain time, which means that there exists a catch up process between the two electrons. After that, the hump disappears because one of the electrons get accelerated and the other gets decelerated due to the Coulomb repulsion. Note that this catch up collision process is very complicated. We even find that for a 16-cycle pulse in Fig. 6(b), there are six peaks in the positive energy distributions. The reason is still not clear. For a real He atom in the full-dimensional space, there is an angle between two electrons’ emission directions, so the energy transfer will be less than that in the present model atom. However, this catch-up mechanism may still be present for the real He case.
4. Conclusions
In this paper, we have investigated the dynamics of multiphoton double ionization in the xuv regime with a model He atom. Although this simple model neglects some angular information it is still physically motivated and is able to provide us with important insights into the underlying mechanisms. We illustrate that, for the TPDI process, a correlation valley exists between the two peaks in energy distributions. When the pulse duration gets shorter, such a correlation valley would be greatly enhanced.
We also identify the influence of electron correlations at different stages. When the two electron are ejected in opposite directions, the electron correlations during the fist ionization play an important role and induce the correlation valley. We also give an explanation that when two electrons are ejected in the same direction, their Coulomb interaction in the final state is much more important.
Finally, we study the electron correlations in the three-photon double ionization case. Our analysis about the correlations at different stages is still valid, even when more excess photons are involved. But differently, the Coulomb repulsion in the final state may be very complicated. And a new catch-up collision process is identified in the energy distributions. Currently, we are carrying out much more complete investigations on these newly found effects using a full-dimensional solution of a real He.
Acknowledgment
This work has been supported by National Natural Science Foundation of China under Grant Nos. 10704003, 10974007, and 10821062, and the National Basic Research Program of China under grant No. 2006CB806007. T.M. is also financially supported by Grant-in-Aid from the Japan Society for the Promotion Science (JSPS) and the PRESTO program of the Japan Science and Technology Agency (JST).
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