Electron vortices generation of photoelectron of H+2 by counter-rotating circularly polarized attosecond pulses

Haojing Yang(杨浩婧), Xiaoyu Liu(刘晓煜), Fengzheng Zhu(朱风筝),Liguang Jiao(焦利光), and Aihua Liu(刘爱华),5,‡

1Institute of Atomic and Molecular Physics,Jilin University,Changchun 130012,China

2School of Mathematics and Physics,Hubei Polytechnic University,Huangshi 435003,China

3College of Physics,Jilin University,Changchun 130012,China

4Helmholtz-Institut Jena,D-07743 Jena,Germany

5State Key Laboratory of Transient Optics and Photonics,Chinese Academy of Sciences,Xi’an 710119,China

Keywords: photoelectron momentum distribution,attosecond pulse,vortex

1.Introduction

The rapid development of ultrashort laser pulse technology[1–3]has provided essential tools for investigating electron dynamics in atoms,molecules,and solids,which bear the promise of electron control in matter.Many ultrafast measurement techniques, such as phasemeters,[4]streaking cameras,[5,6]and attoclocks,[7]are based upon photoionization and the subsequent induced processes.In recent years,elliptically or circularly polarized attosecond pulses and their combinations have been used to probe atomic and molecular structures using photoelectron momentum distributions(PMDs).[8–15]The PMD originating from the ionization of atoms and molecules by intense laser pulses convey valuable and fruitful information about the target electrons.Different parts of PMD not only reflect certain aspects of the target structure information but also encode different ionization dynamics in the laser-matter interaction.Both the atomic and molecular PMDs under extreme ultraviolet(XUV)laser have been investigated by many theoretical groups.For example,Murakami and Chu[16]exemplified the mechanism of symmetry breaking in the H atom PMD.Wu and He[11]showed that atomic PMD under two-color XUV lasers can be affected by both the relative phase and time delay between the two lasers.Yuanet al.[15,17]proved the dependence of molecular-frame photoelectron momentum distribution (MF-PMD) on molecular orbital symmetry.Furthermore,the interference between directly ionized electrons and rescattered electrons can also affect MF-PMD.The photoionization of H+2molecular ion by elliptically polarized UV laser pulses has been studied by Zhanget al.,[8]and it has been shown that the MF-PMD depends further on the laser ellipticity and internuclear distance.

The experimental investigations on PMDs for understanding the structure information and ionization dynamics of atoms and molecules are also continuing.[18,19]Odenwelleret al.[19]studied the electron emission from the H+2molecule ion and found that the complex laser-driven electron dynamics can be encoded in the unexpected PMD.

In recent years, creating and manipulating vortex states of light beams[20,21]and electron vortices[10,21–25]have attracted considerable attention, since vortices can be characterized by intrinsic orbital angular momentum in,e.g.,twisted light beams and electron beams.Taking the hydrogen atom as an example,the vortex quasiparticles in the atomic wavefunctions were studied under the short rectangular half-period electric field pulse.[26]When two circularly polarized laser pulses are used for ionization of atoms or molecules, the vortexshaped momentum distributions[24,27]has been discovered as a new feature of laser-induced electron wavepackets interference.The time-delayed counter-rotating circularly polarized attosecond pulses were proposed to generate vortex-shaped photoelectron wave packets in the photoionization of helium atom[22,28]and hydrogen molecular ion.[29]

Electron vortex is generated in the counter-rotating circularly polarized laser field.The electron vortex structure[22]caused by the interference effect of two pulses has a time delay that changes within one cycle.[30]The vortices in molecular photoionization[26,28,31]was studied by applying either the photoionization of monochromatic field[24,25]or dichromatic field.[10,24,26,28]In addition to the time delay, the control of other laser parameters is also an interesting topic discussed by many researchers.It is shown that the photoelectron momentum, electron ejection angle, relative carrier envelope phase,and time delay between two circularly polarized pulses are related to spiral vortices.Furthermore, vortices are found not only in photoionization, but also photodetachment[31,32]and photodissociation.[29]

The simplest molecule H+2has served as an ideal prototype to explore the ultrafast dynamics in diatomic molecules,such as high harmonic generation,[33–35]the double-slit interference effect,[36,37]the dissociation of molecule,[29,38,39]and so on.

In the present work,we investigate the interaction of H+2molecule ion with a pair of counter-rotating circularly polarized laser pulses.The ultrafast attosecond timescale allows us to ignore the rotational and vibrational effects of atomic nuclei,[40–42]which are in picosecond and femtosecond timescales,respectively.

This paper is structured as follows.In Section 2 we briefly introduce the numerical method and model system for solving the time-dependent Schr¨odinger equation(TDSE)and extracting physical observable.The simulation results and corresponding discussion are presented in Section 3.The summary and conclusion are made in Section 4.Unless specifically stated otherwise,atomic units(a.u.) ¯h=me=e=1 are used throughout this paper.

2.Theory and models

For an aligned homonuclear diatomic molecule in the presence of an ultrashort laser pulse, we can ignore the vibrational and rotational degrees of molecules and only consider electronic movement under the combination of electric field from external laser field and the electrostatic field from molecular nuclei.[37,40]In the circularly polarized laser fields,a reduced two-dimensional (2D) model can be employed to capture the essential physical significance.In this work, we adopt the reduced 2D TDSE under the frozen-nuclei approximation as follow:

where the system HamiltonianH=T+V0(x,y)+r·E(t)includes the kinetic energy term of electronT=(p2x+p2y)/2 and the field-free potentialV0(x,y),

whereRc=2 is the internuclear distance of H+2molecular ion at equilibrium, anda=0.5 is the soft-core parameter.ForRc=4,we further employ the soft-core parametera=0.73 to correctly reproduce the ground state energy of H+2molecular ion.

The interaction between external laser field and the electron is given byVI=r·E(t).The laser electric field is described by

whereE1(t) andE2(t) are two circularly polarized laser pulses in sequence with a time delaytd

whereφ1andφ2are the carrier envelope phases (CEPs) and the sign±denotes the helicity of(counter-clockwise or clockwise)circularly polarized laser pulses.A slowly varying temporal envelope function off(t)=sin2(πt/T) with durationT=nT0is employed here, whereT0=2π/ωis the optical cycle (o.c.).In this work, the pulse lasts for 10 optical cycles (n= 10).In our calculation, we focus on the singlephoton ionization process, which means that the electron can be ionized by absorbing only one photon.In the present work,we choose the two laser fields with the wavelength of 30 nm(ω= 1.52＞Ip= 1.1, whereIpis ionization energy of the ground state of H+2molecular ion atRc=2)and the peak intensity ofI0=1014W/cm2.The molecular alignment is set along thexaxis.

This study utilizes the split-operator fast-Fourier transform algorithm to solve the 2D TDSE.[43]The real time propagation of wave function fromttot+Δtcan be expressed as

The initial state is set as the ground state of H+2molecular ion,which can be obtained by imaginary time propagation with an arbitrary non-trivial state.

可溶性糖和可溶性蛋白是植物体内重要的渗透调节物质，一定程度上反映了植物的抗逆性[22]。与对照组相比，70 ～ 280 μmol/L硝基苯酚处理5 d 后，水稻幼苗根系可溶性蛋白和可溶性糖含量显著增加，且在280 μmol/L硝基苯酚处理后达到最大值，分别是对照的1.4倍和3.1倍。而560 μmol/L 硝基苯酚处理5 d后，可溶性蛋白含量显著下降且低于对照，可溶性糖含量也下降至对照水平(表1)。

In our simulation, the spatial dimension ranges from−150 to 150, which contains 1024 grid points in both thexandydimensions.A cos1/8boundary absorber placed atx,y=±130 is used to avoid the nonphysical reflection of wave functions.The step size of time propagation is set to be Δt=0.01.

After the laser field is finished, wave functions are further propagated for an additional five optical cycles to ensure that all the ionized components of electron wave function are far away from the nuclei.Then we seperate the photoelectron wavepacket by applying the mask function.The ionized wave packet is obtained byψion(x,y)=[1−M(rb)]ψfinal(x,y).Here,ψfinal(x,y)is the wave packet at final time andM(rb)is a mask function

withα=1,andrb=10 which corresponds to the boundary of the bound-electron wave function.Finally,wave functions in momentum space can be obtained by applying Fourier transform to the ionized wave function as

The 2D MF-PMD is given by

3.Results and discussion

In this work, the initial state of the hydrogen molecular ion H+2is the ground state,which is denoted as 1sσg.As the internuclear distance varies,the initial electron density distribution of the H+2molecular ion also changes.This is illustrated in Figs.1(a)and 1(b)for internuclear distances ofRc=2 and 4,respectively.The wave function of the H+2molecular ion can be approximately described as the symmetric superposition of the wave functions of two hydrogen atoms positioned along the molecular axis at±Rc/2.In the case of equilibrium atRc=2,the electron densities of the two hydrogen atoms overlap significantly,resulting in an ionization potential energy ofIp=1.1.However,whenRc=4,the overlapping between the two hydrogen atoms is reduced, leading to a decrease in the ionization potential energy toIp=0.79.

Fig.1.The initial electron density distributions of H+2 in the ground state(1sσg)at two different internuclear distances: (a)Rc=2,(b)Rc=4.

We first investigate the MF-PMDs of H+2molecule ions under the influence of a pair of counter-rotating circularly polarized attosecond pulses at various time delaystd.In the left panels of Fig.2, we demonstrate the MF-PMDs of the H+2molecule ion by left and right circularly polarized laser pulses with time delays of 0, 1, and 1.5 o.c., respectively.The right panels depict the corresponding Lissajous figures of the electric fields.The comparison among these three situations indicates that different time delay results in very different distribution patterns in the MF-PMDs.In the top two panels,when there is no time delay between two counter-rotating pulses,the total electric field is reduced to a purely linearly polarized pulse along thexaxis, and parallel to the molecular alignment axis.In this case, the MF-PMD is a typical six-petal structure.[41,44]The two panels in the middle row display the MF-PMD (left) and corresponding Lissajous curve (right) in the case of 1 o.c.time delay between two counter-rotating pulses in sequence.From Fig.2(b), it can be seen that whentd=1 o.c.,the electric field mainly lies along thexaxis.The corresponding MF-PMD show a significant vortex structure.When the time delay is continuously increased totd=1.5 o.c.,the total field has bothxandycomponents,with peak intensity inyaxis stronger than that inxaxis.The MF-PMD demonstrates dominant distribution in thepydirection,which is similar to the MF-PMD of H+2in the perpendicular geometry, as discussed in Ref.[44].

Fig.2.The MF-PMDs(left panels)and corresponding Lissajous curves of electric field(right panels).The H+2 sits at its equilibrium internuclear distance Rc=2.From top to bottom panels,the time delay are 0,1,and 1.5 o.c.,respectively.The laser parameters are: central carrier frequency ω =1.52,peak intensity I0=1014 W/cm2,the carrier envelope phase φ1=φ2=0,and pulse duration τ =1.0 fs.

Our numerical results can be understood by the attosecond perturbation ionization theory.[45,46]For H+2molecule ions with two identical centers, the corresponding MF-PMD can be approximated by the superposition of the ionization probabilities of two identical hydrogen atoms.For circularly polarized pulses,the photo-ionization in thexandydirections can be obtained by calculating the modular squares of the transition amplitude in thexandydirections,[15]

whereσ(x)andσ(y)arex-andy-geometry transition cross sections, andθis the angle between the electron momentumpand molecular axis (that isxaxis).The total photoelectron momentum distributionPcan be decomposed into two components

wherePxandPyare the ionization probabilities due to thexandycomponents of the electric field,respectively.The photoelectron momentum distribution is linearly related to the electric field amplitude and transition cross section.

The cross sectionσ(x)related to the parallel geometry transition toσu(m=0), andσ(y)is connected to the vertical transitionπu(m=±1),therefore they can also be rewritten asσ‖andσ⊥, respectively.AtRc=2, the transition cross sections from the ground state toσuandπuareσ⊥=528.5 kb andσ‖=38.95 kb, respectively.[41]The contribution of theπu(vertical transition geometry) channel is more significant than that of theσu(parallel transition geometry)channel.The significant difference between parallel and perpendicular geometries are demonstrated obviously in Figs.2(a) and 2(c).In panel (a), theycomponent of the electric field is canceled due to the 0 time delay between two counter-rotating pulses,therefore the MF-PMD is characterized by the typical structure in the parallel geometry.[40]In Fig.2(c),although the electric field inxaxis does not vanish,its magnitude is much smaller than that inyaxis.Considering thatσ⊥≫σ‖,the contribution of parallel geometry is negligible,and perpendicular geometry is dominant,no obvious vortex structure can be observed.

However,we can adapt the relative ratio between the parallel and perpendicular geometries,which makes them comparable.In Fig.2(b),when the time delay is 1 o.c.,the total combined electric field has much smallerycomponent than thexcomponent.Considering the fact thatσ⊥≫σ‖,the ionization yield in parallel and perpendicular geometries are expected to be comparable, strong interference effect and electron vortex structure can be observed.Although this is single photon single ionization,we should expect two spiral arms only,but there are“six spiral arms”.Such“six-spiral-arm”structure is due to the contribution of parallel geometry which has six-petal structure from the confinement effect.[41]The difference phase in parallel and perpendicular geometries produces strong interference between them, which gives birth to the vortex structure.

In the scenario of atomic spirals,[22,24,30,47], the interference between±m(m/=0)partial waves plays a crucial role in the formation of vortex structures.To understand the vortex structure of H+2, it is essential to examine themvalue of the photoelectron in the final state.Clearly, in case of Figs.2(a)and 2(d), only them=0 partial wave contributes to the final state of the electron.On the other hand, for the other cases,bothm=0 andm=±1 partial waves can be present in the final state.The interference of all three partial waves will eventually results in more intricate vortex structures than the twospiral-arm vortex,assuming that the yield of each partial wave is comparable.

From the above discussion,we know that the vortex structure appears if the yields of parallel and perpendicular geometries are comparable.To achieve such a condition, we can manipulate the relative phase and time delay simultaneously.If the time delay is small enough to ensure a large overlapping between two pulses,we can always adjust the laser parameters to keep

Fig.3.The MF-PMDs (left panels) and corresponding Lissajous curves of electric field (right panels).From top to bottom panels: (a) and (d)td=0.5 o.c.,φ1=π,φ2=0;(b)and(e)td=0.75 o.c.,φ1=0.5π,φ2=0;(c)and(f)td=1 o.c.,φ1=φ2=0.The rest laser parameters are:central carrier frequency ω =1.52, peak intensity I0 =1014 W/cm2, and pulse duration τ =1.0 fs.The H+2 sits at its equilibrium internuclear distance Rc=2.

In Fig.3, we demonstrate several examples that produce visible“six-spiral-arm”vortex structure.In the top panels,td= 0.5 o.c.,φ1=π,φ2= 0; in the middle panels,td=0.75 o.c.,φ1=0.5π,φ2=0;and the bottom panels havetd=1 o.c.,φ1=φ2=0.In top panels, when the time delay is 0.5 o.c., the relative phase isφ2−φ1=−π.When the time delay is increased to 0.75 o.c., the relative phase should be adjusted asφ2−φ1=−π/2.Then in the bottom panels,if the time delay is 1 o.c., the relative phase difference need to be set asφ2−φ1=0.All graphs in left panels give very similar“six-spiral-arm”vortex structure.To maintain the similar MF-PMDs,2πrelative phase compensation is required for per optical cycle time delay increasing.As the time delay expands, the spiral arms in MF-PMDs become more elongated and slender.[48]In addition, as the time delay increases, the yield of they-component laser field will continue to grow,and the yield of perpendicular geometry will become more pronounced.

Fig.4.The MF-PMDs(left panels)and corresponding Lissajous curves of electric field(right panels).The H+2 sits at its equilibrium internuclear distance Rc =2.From top to bottom panels, the time delay are 0.5, 1.5, 2.5,3.5,and 4.5 o.c.,respectively.The rest laser parameters are: central carrier frequency ω =1.52, peak intensity I0 =1014 W/cm2, the carrier envelope phase φ1=π,φ2=0,and pulse duration τ = 1.0 fs.

Figure 4 demonstrates MF-PMDs for fixed phase difference atφ2−φ1=π, but vary the time delay from 0.5 o.c.to 4.5 o.c.by 1-o.c.increment.On the top two rows, where the time delaytd＜2 o.c., the “six-spiral-arm” vortex structure are clearly visible.In the middle panels, whentd=2.5 o.c.,the vortex structure still exists, but the two arms on thexaxis diminish, and connect to the other arms.Since now, it does not guarantee the comparable yields in both parallel and perpendicular geometries.If we further enlarge the time delay, as displayed in bottom two rows,,the yield in parallel geometry is smeared out,and the MF-PMD reduces to “two-spiral-arm” vortex, which is typical vortex of single photon single ionization by a pair of counter-rotating circularly polarized attosecond pulses.[24]

Other than the equilibrium internuclear distance, the hydrogen molecular ion H+2can vary its internuclear distance.As mentioned above,e.g., H+2withRc= 4 has a different ground state energies.Actually, it also has different transition cross sections.Some early studies[41]showed that whenRc= 4,σ‖=413.9 kb,σ⊥=284.8 kb, the transition cross section in thexdirection is slightly larger than that in theydirection.Therefore, we can expect different behavior when H+2is radiated by the same counter-rotating circularly polarized pulses.In Fig.5,we display the MF-PMDs(left panels)and the Lissajous curves of electric field(right panels)for H+2withRc=4.All the laser parameters are the same as Fig.2.In the top two rows, whereandσ‖ ＞σ⊥, the yield from parallel geometry dominates the MF-PMDs in both cases, because the yield of perpendicular geometry transition vanishes.Only in the case of bottom panels,when time delaytd=1.5 o.c.,,the yield from perpendicular geometry can be compared to that from parallel geometry transition,a clear“six-spiral-arm”vortex structure occurs.If we ignore the nodes that caused by the confinement effect,[41,49]this vortex shows typical“two-spiral-arm”vortex in MF-PMD.

Fig.5.The same as Fig.2 but with different internuclear distance Rc=4.

4.Summary and conclusions

In summary,by numerically solving the two-dimensional TDSE in the frozen-nuclei approximation, we have investigated the MF-PMDs of H+2molecule ion by a pair of counterrotating circularly polarized laser pulses with time delay.When the time delay is small enough that one component of the electric laser field is partially canceled, the regular “twospiral-arm” vortex in single photon single ionization turns into“six-spiral-arm”vortex or smear out the vortex structure.These effects are due to the combination effect of different transition cross sections in parallel and perpendicular geometries,and the confinement effect of diatomic molecules.By simultaneously varying the phase and time delay of laser pulses,we can control the appearance and disappearance of such“sixspiral-arm”.

For H+2at non-equilibrium internuclear distance, the same behavior can be observed.However, due to the variation in transition cross sections and their relative ratio overRc,the vortex generation parameters are different.For instance,the cases ofRc=2 and 4 have opposite MF-PMDs when they have the same 0 relative phase and sametd=1 o.c.or 1.5 o.c.time delay.It implies the possibility of using molecular electron vortex as strong spectroscopic instruments to investigate molecular structure and dynamics.The clear distinction in MF-PMDs provides us a novel tool to identify the molecular structure information,e.g., the chemical bond length.Additionally, the molecular vortex structure produced by counterrotating circularly polarized pulses may be used to probe the electron dynamics in chiral molecules by photoelectron circular dichroism.[50]

In this work, we have only investigated the simplest molecular prototype system.Future research may be extended to more complex molecule, such as carbon dioxide, carbonyl sulfide or benzene which have many electrons.

Acknowledgements

Project supported by the Natural Science Foundation of Jilin Province, China (Grant No.20220101016JC), the National Key Research and Development Program of China(Grant No.2022YFE0134200), the National Natural Science Foundation of China (Grant Nos.12174147, 91850114, and 11774131),the Open Research Fund of State Key Laboratory of Transient Optics and Photonics.Part of the numerical simulation was done on the high-performance computing cluster Tiger@IAMP in Jilin University.