Effects of initial electronic state on vortex patterns in counter-rotating circularly polarized attosecond pulses?

Qi Zhen(甄琪), Jia-He Chen(陳佳賀), Si-Qi Zhang(張思琪),Zhi-Jie Yang(楊志杰), and Xue-Shen Liu(劉學(xué)深)

Institute of Atomic and Molecular Physics,Jilin University,Changchun 130012,China

Keywords: initial electronic state, counter-rotating circularly polarized attosecond pulses, vortex patterns,photoelectron momentum distributions

1. Introduction

The rapid development of ultrashort laser pulses provides new tools for exploring electronic dynamics on the attosecond(1 as=10?18s) time scale.[1–3]To date, the shortest single pulse with duration 43 as has been produced from superposing some orders of the high-order harmonics generation,[4]which can be used to monitor pure electronic quantum effects through attosecond imaging.[5]The ultrashort attosecond pulses can also induce electron charge migration,thus offering the possibility for probing the molecular structure and imaging the molecular reactions.[6–8]

In the research, a variety of pulses and their combinations have been used to explore electron dynamics, including the high-order harmonic generation (HHG),[9–14]nonsequential double ionization (NSDI),[15–18]and the photoelectron momentum distribution (PMD).[19–23]Recently it was found that single ionization of atom by two oppositely circularly polarized, time-delayed attosecond pulses produce vortex patterns in PMDs,[24–27]which are sensitive to the time delay between the pulses, their handedness, and their relative phase. Yuan et al.[28,29]theoretically investigated the dependence of vortex patterns in PMDs on molecular geometry by bichromatic circularly polarized attosecond laser pulses. Li et al.[30]explored the symmetry distortion of the vortex patterns by considering the dynamic Stark effect and demonstrate that the vortex patterns can also be generated by a pair of elliptically polarized laser pulses.[31]Experimentally, the electron vortices of the potassium atom in counter-rotating circularly polarized laser fields were produced and manipulated,and the experimental results are in identical with the results of numerical simulation.[32,33]In 2019, Xiao et al.[34]demonstrated a scheme to accurately measure the electron displacement using a ruler formed by vortex patterns in PMDs generated by two oppositely circularly polarized pulses.

In this paper,we investigate the effects of different initial electronic states on the vortex patterns in counter-rotating circularly polarized laser pulses. The numerical results of PMDs were presented with varying the wavelengths of attosecond laser pulses.The results show that the vortex patterns in PMDs are dependent on the initial electron density distribution. Besides, we compare the PMDs of different initial electronic states with the same wavelengths and discuss the corresponding physical mechanisms of the discrepancy in distribution.

We divide the paper into the following parts. Our theoretical model and computational method are given in Section 2.The numerical results of PMDs with the different electronic state as the initial state are presented and discussed in Section 3. Finally, we make a summary in Section 4. Atomic units(a.u.) e=ˉh=me=1 are used unless otherwise stated.

2. Theoretical methods

where r and p are the coordinate and the momentum operator of the electron,respectively;The soft-core Coulomb potential is

The soft-core parameters a=0.161 and b=2.0, which corresponds to the ionization potential IP1=2 a.u. of the ground state and IP2=0.5 a.u. of the first excited state of He+. E(t)is the electric field of a pair of counter-rotating circularly polarized pulses delayed in time by Td,

with

The research is based on solving the 2D TDSE by the fast Fourier transform technique combined with the splitoperator method.[35]We utilize the imaginary-time evolution method to obtain the initial wave function. The grid size is 409.6 a.u. containing 2048 grid points in both x and y directions. The wave packet at each step is multiplied by a cos1/8“mask function”, which is used to prevent unphysical effects from the boundary. Thus, the absorber domain ranges from|x,y|=150 a.u. to |x,y|=204.8 a.u. At final propagation time,the wave packet is multiplied by a mask function M(r),which divides the wave packet into the bounded part and ionized part.[20,36]The PMDs is obtained by Fourier transforming the wave packet of the ionized part.

3. Results and discussion

To investigate the effects of different initial electronic states on the vortex patterns, we present the PMDs of He+with varying the wavelengths of the time-delayed attosecond pulses. Figures 1(a)–1(c)display the PMDs of the vortex patterns at the different wavelengths for the time delay Td=3 o.c.for the ground state as the initial electronic state.

Figure 1(a) displays that the vortex pattern in PMDs exhibit two spiral arms at wavelengths λ1=λ2=20 nm(ω1=ω2=2.28 a.u.). We can see that there are four interference spiral arms at wavelengths λ1=λ2=40 nm(ω1=ω2=1.14 a.u.) as shown in Fig.1(b),whereas eight spiral arms are induced at wavelengths λ1=λ2=90 nm(ω1=ω2=0.5 a.u.)as shown in Fig.1(c). We also found that each spiral arm is evenly separated by the same angle as shown in Figs. 1(a)–1(c). The similar distribution patterns of H2+have been discussed systematically in Ref.[29].

The interference of the states |p,1〉 and |p,?1〉 generates a vortex with c2rotational symmetry.[25]

For the photoionization,at wavelengths λ1=λ2=40 nm(ω1=ω2= 1.14 a.u.), the corresponding angular frequencies ω1and ω2are below the ionization potential of the ground state of He+. Thus, the ionization with a pair of leftright circularly polarized pulses proceeds via the two-photon route, |s,0〉→|p,1〉→|d,2〉 and |s,0〉→|p,?1〉→|d,?2〉, respectively. The generated electron wave function is

which implies the vortex pattern having c4rotational symmetry as shown in Fig.1(b).Analogously,the left-right circularly polarized laser pulses λ1=λ2=90 nm(ω1=ω2=0.51 a.u.)ionize the He+via the four-photon routes

|s,0〉→|p,1〉→|d,2〉→|f,3〉→|g,4〉,

|s,0〉→|p,?1〉→|d,?2〉→|f,?3〉→|g,?4〉,

and yield the final superposition state electron wave function

The vortex pattern with c8rotational symmetry is shown in Fig.1(c)which generated by the interference of the states|g,4〉and|g,?4〉.

Fig.1. The PMDs for the ground state as the initial electronic state of He+ by a pair of left-right circularly polarized attosecond pulses at different wavelengths. (a)λ1=λ2=20 nm,(b)λ1=λ2=40 nm,(c)λ1=λ2=90 nm. The time delay is Td=3 o.c.

Fig.2. The PMDs for the excited state as the initial electronic state of He+ by a pair of left-right circularly polarized attosecond pulses at different wavelengths. (a)λ1=λ2=20 nm,(b)λ1=λ2=40 nm,(c)λ1=λ2=100 nm. The time delay is Td=3 o.c.

We next show the signature of different spiral arms of the vortex pattern to discuss the dependence of the excited state as the initial electronic state on the PMDs,which is demonstrated in Fig.2. It is found that the spiral arms of vortex patterns are four,four,and six as shown in Figs.2(a)–2(c)at wavelengths λ1=λ2=20 nm (ω1=ω2= 2.28 a.u.), λ1=λ2=40 nm(ω1=ω2= 1.14 a.u.) and λ1=λ2=100 nm (ω1=ω2=0.46 a.u.) for the excited state as the initial electronic state.The time delay is Td=3 o.c.

To further understand the sensitivity of the initial electronic state on the vortex patterns in PMDs, the initial wave function of the different electronic states are illustrated in Fig.3. The initial wave function of the ground state depicted in Fig.3(a)shows all the electron density distribution locating in the central region. And the initial electron density distributions of the excited states as shown in Fig.3(b)are mainly distributed in two regions of the first and third quadrants.With considering the interaction of He+in the excited states with a pair of attosecond pulses, the initial wave packets distributed in two regions create interference effects in the photoelectron spectra,which leads to two more spiral arms than in the ground state as the initial electronic state by absorbing the same number of photons. The characteristics of the vortex patterns [in Figs. 1 and 2] agree well with the above analysis, indicating that the difference of initial electronic state causes the change in the vortex patterns in PMDs.

Fig.3. Electron density distribution of(a)ground state and(b)excited state of He+.

To compare the effect of the initial electronic state on PMDs, we select Figs. 1(b) and 2(b) to show the discrepancy. And the corresponding schematic energy-level diagram of Figs.1(b)and 2(b)are illustrated in Figs.4(a)and 4(b),respectively. As illustrated in Fig.4(a), the vortex pattern with four spiral arms(Fig.1(b))can be generated by absorbing twophoton from the first laser pulse through path Γ1, absorbing two-photon from the second laser pulse through path Γ2, and absorbing a photon from different pulses through path Γ12(or Γ21).[24]The vortex pattern showed in Fig.2(b)is generated by absorbing a photon from each of the two pulses through paths Γ1and Γ2,which is shown in Fig.4(b).

Fig.4. Schematic energy-level diagrams for the ionization of He+ in counter-rotating circularly polarized laser pulses delayed in time by Td, in the case of (a) ground state and (b) excited state as the initial electronic state. The wavelengths of both pulses are λ1=λ2=40 nm. The ionization pathways Γ1 and Γ2 are shown,respectively.

It can be found that the two vortex patterns are similar,and both exist the difference in the brightness(intensity of the PMDs). In Fig.1(b), one sees that the two spiral arms along the horizontal axis have the same brightness,those two along its perpendicular direction are less bright;In Fig.2(b),the intensities of the two spiral arms in the first and third quadrants are maximum. However, the intensities of the other two spiral arms in second and fourth quadrants are minimum. We note that the physical mechanisms which give rise to differences in distribution intensities are different. It is found that the difference in the brightness of the vortex patterns showed in Fig.1(b)caused by the contribution of the ionization cross channel.[24]The authors in Ref. [24] first proposed the cross channel contributions and pointed out that the brightness of the vortex patterns depends on the time delay. However,the physical mechanism underlying the creation of the discrepancy in distribution shown in Fig.2(b)is based on the initial electron density distribution. From Fig.3(b) we get that the electron density distributions at the first and third quadrants are maximum, leading to larger momentum distributions in these regions than in others.

4. Conclusions

In summary, we have investigated the effects of the initial electronic state on vortex patterns in PMDs by a pair of counter-rotating circularly polarized attosecond pulses. Simulations are performed on He+by numerically solving the 2D TDSE. The initial electronic state plays an essential role in the PMDs. We demonstrate the effects of the initial electronic state on vortex patterns with varying wavelengths. It concludes that the number of spiral arms in vortex patterns is equal to the number of the absorbed photons when the initial state is the ground state. And the number of spiral arms in vortex patterns is always two more than the number of the absorbed photons when the initial state is the excited state.The initial electron density distributions are presented to interpret the difference of PMDs on the initial electronic state.Besides, we compare the PMDs of different initial electronic states with the same wavelengths. The results show that the vortex patterns in PMDs are similar. However, the physical mechanisms underlying the creation of the discrepancy in distribution are different. The study of different physical mechanisms can be employed to effectively control the distribution of electron vortices.