1. Introduction

Ultrashort attosecond pulse offers a powerful tool for exploring ultrafast electronic dynamics [1–3], imaging and controlling the molecular reaction dynamics [4–6]. Many ultrafast measurement techniques, such as the phase meter [7], streaking camera [8,9] and attoclock [10], which are based on the ionization and subsequently induced processes. In recent years, the elliptically or circularly polarized attosecond pulses and their combinations have been adopted to probe atomic and molecular structures by photoelectron momentum distributions (PMDs) [11–18]. For example, Qin et al. [19] extracted successfully the phase distribution of the ionized electron wave packet from the vortex-shaped interference patterns in PMDs of hydrogen atom in the elliptically polarized laser field. Pengel et al. [20–22] observed experimentally the electron vortices of potassium atom by a sequence of two counterrotating circularly polarized femtosecond laser pulses. It has been confirmed theoretically that single ionization of atom by two oppositely circularly polarized attosecond pulses can produce vortex patterns in PMDs [23–26]. Very recently, Qi et al. [27] found that the number of spiral arms is two more than the twice number of the absorbed photons with 2p state as the initial state by counter-rotating circularly polarized attosecond pulses.

According to the Bohr’s theory, the electron moves around the nucleus on circular orbits possessing quantized orbital angular momentum associated with the ring current [28], in quantum mechanics, this motion of electron is described by magnetic quantum number ${m}$ [29]. Li et al. [30] reported experimentally the dependence of strong-field ionization rate of Ar atom by a circularly polarized light on the sign of magnetic quantum number. Xu et al. [31] studied experimentally the magnetic quantum number resolved above-threshold ionization (ATI) of Xe atom by a right circularly polarized laser. Theoretically, strong-field ionization of atoms carrying different orbital angular momenta has been studied by several different theoretical methods, such as the time-dependent Schrödinger equation (TDSE) [32–38], Perelomov-Popov-Terentev (PPT) model [29,39,40], classical trajectory Monte Carlo (CTMC) simulation [32,41], adiabatic approximation (AA) theory [33], strong-field approximation (SFA) model [42] and so on. More importantly, one can study many ultrafast processes based on the magnetic quantum number m resolved ring currents, such as the adiabatic attosecond charge migration and the generation of attosecond magnetic field pulses [43,44]. However, investigations of the magnetic quantum number resolved strong-field ionization are still quite few despite its importance.

In this paper, we numerically solve the two-dimensional time-dependent Schrödinger equations (2D-TDSE) for neon atom in a pair of counter-rotating circularly polarized attosecond pulses and study theoretically how to control the atomic-orbital-resolved vortex-shaped photoelectron momentum distributions (PMDs) and ionization probabilities. We found that the amount of spiral arms in vortex patterns is twice the number of absorbed photons when the initial state is the $\psi _{m=\pm 1}$ state, which is different from the initial $2p$ state, namely the amount of spiral arms is always two more than the twice number of absorbed photons [27]. It satisfies clearly a change from $c_{2n+2}$ to $c_{2n}$ rotational symmetry of the vortices when the initial $2p$ state is replaced by $2{p}_{+}$ (or $2{p}_{-}$) state. In addition, two- and three-photon ionization depend weakly on the magnetic quantum number of initial states, while single-photon ionization prefers when the electron and laser field corotate and the relative ratio of ionization probabilities from the initial $2{p}_{-}$ and $2{p}_{+}$ states can be controlled efficiently by varying the laser wavelength, time-delay, relative phase and amplitude ratio of two attosecond pulses.

This paper is organized as follows. In Sec. 2, we briefly introduce how to solve numerically 2D-TDSE of neon in a pair of counter-rotating circularly polarized attosecond pulses. In Sec. 3, we present the magnetic quantum number resolved PMDs and ionization probabilities at different laser intensities, wavelengths, relative phases and time delays, and also propose an efficient method to control single-photon ionization ratios between $2{p}_{-}$ and $2{p}_{+}$ by changing the amplitude ratio of two delayed attosecond pulses. Finally, our conclusions are given in Sec. 4. Atomic units are used throughout unless otherwise stated.

2. Theoretical methods

To illustrate the effects of magnetic quantum number of initial electronic states on the vortex patterns in PMDs and total ionization probabilities, we numerically solve the 2D-TDSE of the neon atom with a pair of counter-rotating circularly polarized pulses. Based on the dipole approximation and length gauge, the 2D-TDSE reads as

 

Fig. 1. The three-dimensional electric field is plotted by the red curve; the two-dimensional projections of electric field are also shown in the Time-Ex and Time-Ey planes with green and blue curves in counter-rotating circularly polarized attosecond pulses, respectively.

Download Full Size | PDF

The time-dependent wave function $\psi (x,y,t)$ is propagated using the split-operator method on a Cartesian grid. The spatial range is from −200 a.u. to 200 a.u. and the spatial step is 0.1 a.u. in both x- and y- directions. A $\textrm{cos}^{1/8}$ absorber placed at $x,y=\pm 180$ a.u. is used to avoid the reflection of wave functions from the boundary.

To build the 2p orbital having $m=\pm 1$, we first obtain the ground state and then calculate the excited states $\psi _{2p_{x}}$ and $\psi _{2p_{y}}$ by filtering out the ground state in each propagation step by using imaginary time propagation method. The $m=\pm 1$ states can be defined as $|\psi _{m=\pm 1}\rangle =(|\psi _{2p_{x}}\rangle \pm i|\psi _{2p_{y}}\rangle )/\sqrt {2}$. In Fig. 2, we plot electron density distributions of the fictitious ground state (1s), valence orbital (2p), the current-carrying orbitals $2p_{+}$ and $2p_{-}$ of the neon atom, respectively.

 

Fig. 2. Electron density distributions of Ne for the (a) fictitious ground state (1s); (b) valence orbital (2p); (c) the current-carrying orbital $2p_{+}$; (d) the current-carrying orbital $2p_{-}$.

Download Full Size | PDF

After the laser field is finished, wave functions are further propagated for an additional eight optical cycles to ensure that all the ionized components away from the core, we then filter out the wave function in the range $\sqrt {x^{2}+y^{2}}<50$ a.u., which was regarded as the unionized wave packet. Therefore, the ionized wave packet can be obtained by $\psi _{ion}(x,y)=[(1-M(r_{b})]\psi _{final}(x,y)$. Here, $\psi _{final}(x,y)$ is the wave packet at final time and $M(r_{b})$ is a mask function

3. Results and discussion

To investigate the effects of magnetic quantum number of initial states on the vortex pattern in PMDs. Figure 3 shows the PMDs of neon atom in a pair of time delayed attosecond pulses for the $2p_{+}$ or $2p_{-}$ electronic state as the initial state, respectively. In the present calculations, we use $E_{0}=0.11$ a.u., $T_{d}=3$ o.c. and three different wavelengths of 20 nm, 100 nm and 130 nm. In Figs. 3(a) and 3(d), the vortex pattern in PMDs exhibit two spiral arms at wavelength $\lambda =20$ nm ($\omega =2.28$ a.u.) and spiral arms come from the interference of wave packets ionized by absorbing a left or right circularly polarized photon, respectively. Similarly, the spiral arms are 4 (1st ATI peak) and 6 (2nd ATI peak) for two-photon ionization ($\lambda =100$ nm) as shown in Figs. 3(b) and 3(e), 6 (1st ATI peak) and 8 (2nd ATI peak) for three-photon ionization ($\lambda =130$ nm) (see Figs. 3(c) and 3(f)). That means the number of spiral arms is twice the number of absorbed photons, which satisfies the $c_{2n}$ rotational symmetry. In addition, the maxima of PMDs corresponding to the $2p_{+}$ state and $2p_{-}$ state have significant angular offsets for single-photon ionization (see Figs. 3(a) and 3(d)).

 

Fig. 3. Comparison of the PMDs for the current-carrying orbital $2p_{+}$ (top row) and $2p_{-}$ (bottom row) as the initial state of neon atom by a pair of left-right circularly polarized attosecond pulses at different wavelengths: (a),(d) $\lambda =20$ nm; (b),(e) $\lambda =100$ nm; (c),(f) $\lambda =130$ nm. Note that different normalization factors are used respectively for two ATI peaks in each of Figs. 3(b), (c), (e) and (f). The amplitude ratio $\beta =1$, the electric field amplitude $E_{0}=0.11$ a.u., the carrier envelope phase $\phi _{1}=\phi _{2}=0$ and time delay $T_{d}=3$ o.c. are used in the present calculations.

Download Full Size | PDF

The physical mechanism underlying the generation of electron vortices can be well understood as the superposition of two time delayed photoelectron wave packets with different magnetic quantum number m. The valence electron in the $2p_{+}$ or $2p_{-}$ state ($I_{2p_{+}}=I_{2p_{-}}=0.793$ a.u.) can be ionized by absorbing respectively a single photon from the left- and right- circularly polarized laser pulses ($\omega =2.28$ a.u.) and generate the vortex pattern with two spiral arms in Figs. 3(a) and 3(d), which are different from four spiral arms for 2p state as the initial state [27]. When the $2p_{+}$ ($2p_{-}$) state as the initial state, the electron can be ionized by the first left-handed circularly polarized pulse via the following route, $|p,+1\rangle \rightarrow |d,0\rangle$ ($|p,-1\rangle \rightarrow |d,-2\rangle$) or by the second right-handed circularly polarized pulse via another route, $|p,+1\rangle \rightarrow |d,+2\rangle$ ($|p,-1\rangle \rightarrow |d,0\rangle$) as shown in Figs. 4(a) and 4(b), respectively. In addition, due to the time delay between two attosecond pulses, the final electronic wave packet will generate a phase of $e^{-iE'T_{d}}$ by the first pulse, in which $E'$ represents the photoelectron energy [21]:

 

Fig. 4. Ionization channel scheme for photoionization of neon atoms with the initial electronic state of $2p_{+}$ and $2p_{-}$ by left- and right-handed circularly polarized light pulses ($E_{L}$ and $E_{R}$). (a),(b) single-photon ionization; (c),(d) two-photon ionization; (e),(f) three-photon ionization.

Download Full Size | PDF

We next study the effect of magnetic quantum number of the initial state on total ionization probabilities by using a pair of time-delayed circularly polarized pulses. Figure 5(a) presents the ratios of ionization probabilities of $2p_{-}$ and $2p_{+}$ (i.e., $2p_{-}/2p_{+}$) as a function of wavelength. One can see that the $2p_{-}$ have a larger ionization probabilities than those of $2p_{+}$ at wavelengths covering from 10 nm to 70 nm for three different field amplitudes, while the ionization probabilities of $2p_{-}$ is almost the same as that of $2p_{+}$ when the wavelengths are larger than 100 nm. In Figs. 5(b) and 5(c), we also compare the photoelectron energy spectra calculated from the $2p_{+}$ (or $2p_{-}$) state at wavelengths of 20 nm and 130 nm, respectively. As can be seen, the first ATI peak obtained from $2p_{-}$ is much stronger than that from $2p_{+}$ for single-photon ionization ($\lambda =20$ nm) as shown in Fig. 5(b). However, there is very weak magnetic quantum number dependence on the ATI spectra for three-photon ionization ($\lambda =130$ nm) (see Fig. 5(c)). For the $2p_{+}$ and $2p_{-}$ orbital, the magnetic quantum number of the electron will decrease by 1 (i.e., $\Delta m=-1$) by absorbing a left-rotating circularly polarized photon and increase by 1 (i.e., $\Delta m=+1$) by absorbing a right-rotating circularly polarized photon, which meets the Fano propensity rules in photoionization [45]. In addition, it is well known that the single-photon ionization is preferential when the circularly polarized corotates with the electron [46,47]. In Figs. 5(a) and 5(b), one can see that single-photon ionization probability of the $2p_{-}$ orbital is much larger than that of the $2p_{+}$ orbital even though both orbitals experience the left- and right- rotating attosecond pulses. We also found that ionization probability depend on the sequence of corotating and counter-rotating when two attosecond pulses are overlapped. However, for the multi-photon ionization, the dependence of the ionization probability on the sequence of corotating and counter-rotating is weak (see Figs. 5(a) and 5(c)).

 

Fig. 5. (a) Ratios of ionization probabilities of $2p_{-}$ and $2p_{+}$ as a function of wavelength for three different field amplitudes of $E_{0}=0.09$ a.u. (black squares), $E_{0}=0.11$ a.u. (red circles) and $E_{0}=0.14$ a.u. (blue triangles), respectively. The ATI spectra at (b) 20 nm and (c) 130 nm. We take the time delay $T_{d}=3$ o.c., the carrier envelope phase $\phi _{1}=\phi _{2}=0$ and the amplitude ratio $\beta =1$.

Download Full Size | PDF

Let us turn to investigate how the interference patterns in PMDs depend on the time delay and carrier envelope phase of two pulses. In our simulations, we choose a pair of left-right circularly polarized attosecond pulses with different time delays. According to the polarization gating [48], the time-dependent ellipticity of the combined attosecond pulse is 0 for $T_{d}=0$ o.c.. For the smaller time delay, the polarization state of the combined attosecond pulse changes between elliptical and linear in the overlap region. When $T_{d}$ is large enough, the combined field is made of a pair of circularly polarized attosecond pulses. To reveal how interference patterns depend on the time delay, Fig. 6 compares the PMDs at different wavelengths and time delays for $2{p}_{+}$ and $2{p}_{-}$ as the initial state, respectively. One can see that the interference structures in PMDs for $2{p}_{+}$ and $2{p}_{-}$ depend weakly on the time delay in single-photon ionization. However, interference structures depend strongly on the time delay and clear vortices are produced at the time delay of 3 o.c. for the multiphoton ionization. In addition, for these two ionization regimes, each spiral arm in PMDs becomes longer and thinner if the time delay of two pulses further increases, which consists with the theoretical and experimental results in Ref. [22,25,49]. Figure 7 shows the carrier envelope phase effect on the vortex patterns in PMDs for the $2{p}_{+}$ and $2{p}_{-}$ as initial state, respectively. Both for the single-photon ionization and three-photon ionization, we can see that the interference structures in PMDs for $2{p}_{+}$ and $2{p}_{-}$ depend strongly on the CEP. The vortex structure rotates a cycle (i.e., 2$\pi$) when the relative phase ($\phi _{2}-\phi _{1}$) changes from 0 increase to 2$\pi$.

 

Fig. 6. Comparison of the PMDs for the $2p_{+}$ and $2p_{-}$ as the initial state of neon atom by a pair of left-right circularly polarized attosecond pulses at different wavelengths and time delays: (a) - (h) $\lambda =20$ nm; (i) - (p) $\lambda =130$ nm; and (a), (e) $T_{d}=0$ o.c.; (b), (f) $T_{d}=2$ o.c.; (c), (g) $T_{d}=4$ o.c.; (d), (h) $T_{d}=6$ o.c.; (i), (m) $T_{d}=1$ o.c.; (j), (n) $T_{d}=2$ o.c.; (k), (o) $T_{d}=3$ o.c.; (l), (p) $T_{d}=4$ o.c.; Note that different normalization factors are used respectively for two ATI peaks in each of Figs. 6(i) - (p). The amplitude ratio $\beta =1$, the carrier envelope phase $\phi _{1}=\phi _{2}=0$ and the electric field amplitude $E_{0}=0.11$ a.u. are used in our calculations.

Download Full Size | PDF

 

Fig. 7. Comparison of the PMDs for the $2p_{+}$ and $2p_{-}$ as the initial state of neon atom by a pair of left-right circularly polarized attosecond pulses at different carrier envelope phases: (a) - (j) $\lambda =20$ nm; (k) - (t) $\lambda =130$ nm; and (a), (f), (k), (p)$\phi _{1}=\phi _{2}=0$; (b), (g), (l), (q) $\phi _{1}=0$, $\phi _{2}=\pi /2$; (c), (h), (m), (r) $\phi _{1}=0$, $\phi _{2}=\pi$; (d), (i), (n), (s) $\phi _{1}=0$, $\phi _{2}=3\pi /2$; (e), (j), (o), (t) $\phi _{1}=0$, $\phi _{2}=2\pi$; The time delay $T_{d}=3$ o.c., the amplitude ratio $\beta =1$ and the electric field amplitude $E_{0}=0.11$ a.u. are used in our calculations.

Download Full Size | PDF

Figure 8(a) demonstrates the ratios of ionization probabilities as the function of time delay. For single-photon ionization (i.e. $\lambda =20$ nm and 50 nm), the ratios of ionization probabilities increase from 1.0 to the maximum at $T_{d}=4$ o.c. and then decrease back to 1.0 as the time delay increases. However, the ratios change slightly around 1.0 for two-photon (i.e. $\lambda =70$ nm and 100 nm) and three-photon (i.e. $\lambda =130$ nm) ionization processes, which indicates ionization depends weakly on the magnetic number of the initial state. In Figs. 8(b) and 8(c), we compare the ATI spectra at different time delays for 20 nm and 130 nm, respectively. As can be seen, probability of the ATI peak decreases significantly and the ATI peak shifts toward higher energy (i.e., blue-shift) as the time delay decreases (i.e., intensity of the combined field increases) for single-photon ionization in Fig. 8(b). The situation is very different for the multiphoton ionization case, namely probabilities of all the ATI peaks increase slightly as the time delay decreases and the spectral red-shift can be clearly observed in Fig. 8(c). The spectral red-shift as intensity of the combined field increases for multiphoton ionization has been explained successfully by the AC-Stark shifts of the atomic bound states induced by laser with the longer wavelength [50]. The spectral blue-shift for single-photon ionization can be well understood as the upward shift of the ground-state energy in a high-frequency laser [51,52].

 

Fig. 8. Ratios of ionization probabilities of $2p_{-}$ and $2p_{+}$ as a function of time delay, for wavelength $\lambda =20$ nm (black squares), $\lambda =50$ nm (red circles), $\lambda =70$ nm (blue triangles), $\lambda =100$ nm (green triangles) and $\lambda =130$ nm (violet squares). ATI spectra with different time delays of (b) 20 nm and (c) 130 nm. The electric field amplitudes $E_{0}=0.11$ a.u., the carrier envelope phase $\phi _{1}=\phi _{2}=0$ and the amplitude ratio $\beta =1$, respectively.

Download Full Size | PDF

Since single-photon ionization prefers when the electron and attosecond pulse corotate [46,47], thus it is possible to control by suppressing or enhancing selectively its favourite ionization channels, namely $|p,+1\rangle \rightarrow |d,+2\rangle$ for $2p_{+}$ or $|p,-1\rangle \rightarrow |d,-2\rangle$ for $2p_{-}$, respectively. Here, we propose an efficient method to control the ratios of ionization probabilities of $2{p}_{-}$ and $2{p}_{+}$ by varying the field amplitude ratio $\beta$ ( i.e., remain unchanged amplitude of left circularly polarized pulse and decrease amplitude of right circularly polarized pulse) between the left- and right- circularly polarized attosecond pulses. In Fig. 9(a), it is clear that ionization probabilities of $2{p}_{+}$ are reduced seriously due to its favourite ionization channel (i.e., $|p,+1\rangle \rightarrow |d,+2\rangle$) is suppressed, while those of $2{p}_{-}$ almost keep the same, thus the ratios of ionization probabilities increases from 2.3 to 3.8 as $\beta$ decrease for $T_{d}=4$ o.c..In Fig. 9(b), we further show the significant enhancement of ionization ratios by decreasing $\beta$ for five different time delays.

 

Fig. 9. (a) Ionization probabilities and their ratios of $2p_{-}$ and $2p_{+}$ as a function of $\beta$: $2p_{+}$ (black squares), $2p_{-}$ (blue circles) and ratios of ionization probabilities (i.e., $2p_{-}/2p_{+}$) (red triangles) for two attosecond pulses with $T_{d}=4$ o.c.; (b) ratios of the ionization probabilities for five time delays: $T_{d}=1$ o.c. (black squares), $T_{d}=3$ o.c. (red circles), $T_{d}=4$ o.c. (blue triangles), $T_{d}=5$ o.c. (green triangles), $T_{d}=7$ o.c. (violet rhombus). The field amplitude $E_{0}=0.11$ a.u., the carrier envelope phase $\phi _{1}=\phi _{2}=0$ and wavelength $\lambda =50$ nm are used, respectively.

Download Full Size | PDF

An issue should be addressed that the present simulations are based on the approximated 2D-TDSE. So far, the electron vortices have also been studied by solving the more elaborate three-dimensional TDSE (3D-TDSE) [13,37,38,53,54] and even the six-dimensional TDSE (6D-TDSE) for the helium atom [24]. Since the polarization vector of laser fields is planar (such as in the O-xy plane) and we thus expect the present 2D-TDSE is a good approximated model. Hopefully, the present magnetic quantum number resolved vortex patterns and the relative ratios of ionization probabilities of neon atoms driven by counter-rotating circularly polarized attosecond pulses can be examined by the more accurate 3D-TDSE simulations and experiments in the near future.

4. Conclusion

In summary, we theoretically study the magnetic quantum number resolved vortex-shaped PMDs and ionization probabilities by numerically solving the 2D-TDSE of neon in a pair of counter-rotating circularly polarized attosecond pulses. We found that the number of spiral arms in vortex patterns is twice the number of the absorbed photons (i.e., vortices have $c_{2n}$ rotational symmetry) when the initial state is the $\psi _{m=\pm 1}$ state and the spiral arm becomes longer and thinner with time delay increasing. In addition, the vortex structure depends strongly on the CEP of attosecond pulses and rotates a cycle when the relative phase $\phi _{2}-\phi _{1}$ changes from 0 to $2\pi$. The vortex pattern in PMDs are well explained as the interference of wave packets come from different ionization channels. We also found the spectral red-shift and blue-shift due to the AC-Stark shifts for multiphoton and single-photon ionization, respectively. Furthermore, single-photon ionization is preferred when the electron and laser field corotate and the relative ratios of ionization probabilities (i.e., $2{p}_{-}$/$2{p}_{+}$) can be efficiently controlled by varying the laser wavelength, time delay, relative phase and amplitude ratio of two attosecond pulses. Finally, we emphasize that the magnetic quantum number resolved strong-field ionization can pave the way for retrieving the atomic-orbital-symmetry resolved structure information (e.g., differential cross section (DCS)) and studying ultrafast processes on a nanoscale, including negative charge migration in many biological and chemical reactions and chemical bond formation, as well as for the generation of intense magnetic field pulses.