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The momentum distributions of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ by intense laser pulses

  • Zhi-Xian Lei ,
  • Shu-Juan Yan ,
  • Xin-Yu Hao ,
  • Pan Ma ,
  • Sheng-Peng Zhou ,
  • Jing Guo
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  • Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China

Received date: 2022-12-22

  Revised date: 2023-02-18

  Accepted date: 2023-02-21

  Online published: 2023-05-24

Copyright

© 2023 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

We investigate theoretically the photoionization of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ by numerically solving the two-dimensional time-dependent Schrödinger equations under Bohn-Oppenheimer approximation. The results show that the photoelectron momentum distributions (PMDs) of ${{\rm{H}}}_{3}^{2+}$ with different initial states are strongly dependent on the laser ellipticities and molecular orbital symmetry, and the PMDs of degenerate electronic states E± are mirror images. Also, for degenerate electronic states E±, vortex structures appear in the PMDs by the counter-rotating circularly polarized laser pulses as the time delay between the two pulses increases, which can be explained by multi-center ionization and ultrafast photoionization model.

Cite this article

Zhi-Xian Lei , Shu-Juan Yan , Xin-Yu Hao , Pan Ma , Sheng-Peng Zhou , Jing Guo . The momentum distributions of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ by intense laser pulses[J]. Communications in Theoretical Physics, 2023 , 75(6) : 065501 . DOI: 10.1088/1572-9494/acbd95

1. Introduction

With the rapid development of attosecond (1 as = 10−18 s) laser pulses, it has been possible for investigating the electronic dynamics on its nature time scale and sub-nanometer dimension in atoms, molecules, and solids [14]. The shortest 43 pulses have been obtained, which are available for new ultrafast optical imaging [5].
The interaction of such short-pulse laser with atoms or molecules can lead to numerous novel high-order nonlinear optical phenomena, such as multiphoto ionization [6], above-threshold ionization [7, 8], nonsequential double ionization [9, 10] and high-order harmonic generation [11, 12], and so on. The ionization of electrons is one of the basic processes in the strong-field physics. The coherent electron wavepackets created by photoionization can be a powerful tool for probing the structure of atoms and molecules, imaging and controlling the molecular dynamics [1315]. The photoelectron momentum distribution (PMDs) and photoelectron angular distributions (PADs) of atoms and molecules have been very successful in probing photoionization and electron dynamics. Many efforts have been devoted to studying the PMDs and PADs of atoms or molecules [1618]. While in the past few decades, strong-field physics has mainly focused on atoms and small diatomic molecules. Consequently, the simplest molecules H2 and ${{\rm{H}}}_{2}^{+}$ have been studied in theoretical and experimental research [1921]. And the ${{\rm{H}}}_{3}^{2+}$ and H2 can be regarded as the simplest polyatomic molecules, which have also attracted considerable theoretical interest [2224].
Circularly polarized laser pulses have also been used to investigate the electron dynamics and probe atomic and molecular structures by PMDs and PADs [25, 26]. Recently, a pair of time-delayed circularly polarized attosecond UV laser pulses is adopted to produce novel phenomenon-spiral electron vortices in atomic PMDs of helium [27] and then the electron vortices of the potassium atom were measured experimentally by a sequence of counter-rotating circularly polarized laser pulses [28], which can be explained by the interference of coherent electronic wave packets created by two-pathway ionization processes. Yuan et al [18] investigated theoretically the influence of pulse helicities and angular frequencies of a pair of co- or counter-rotating circularly polarized attosecond UV laser pulses on vortex structures for ${{\rm{H}}}_{2}^{+}$. The photoionization of triangular and linear triatomic molecular ions ${{\rm{H}}}_{3}^{2+}$ has been studied by bichromatic circularly polarized attosecond UV laser pulses [29]. Ultrafast photoelectron diffraction in triatomic molecules ${{\rm{H}}}_{3}^{2+}$ by ultrafast circular soft x-ray attosecond pulses shows that the complex diffraction patterns are caused by molecular multiple center interference and charge migration process [30]. Recently, the degenerate excited electronic states with clockwise or anticlockwise orbital electron current around the z-axis have been studied. Ma et al [31] investigated the number of spiral arms in vortex patterns and controlled the relative ratio of ionization probabilities between 2p+ and 2p of Ne stom by changing the laser parameters in counter-rotating circularly polarized laser fields.
In the paper, we investigate theoretically the photoionization of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ by an elliptically polarized laser pulse and bichromatic counter-rotating circularly polarized laser pulses. We present the PMDs from different initial electronic states of triatomic molecular ${{\rm{H}}}_{3}^{2+}$ in the elliptically polarized laser field. The photoelectron distributions are obtained by solving the time-dependent Schrödinger equation (TDSE) and analyzed by multi-center ionization and the ultrafast photoionization model [30, 32]. The paper is arranged as follows: The theoretical model and computational method are described in Part 2. Part 3 presents the PMDs from the degenerate electronic states of ${{\rm{H}}}_{3}^{2+}$ by ellipcitical polarized laser pulses and bichromatic counter-rotating circularly polarized laser fields. Finally, we summarize our results in Part 4. Throughout this paper, atomic units (a.u.) e = = me = 1 are used unless otherwise stated.

2. Theoretical model and computational methods

In the work, we investigate the photoionization of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ by solving the two-dimensional time-dependent Schrödinger equations (2D-TDSE), which can be expressed as
$\begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial \psi ({\boldsymbol{r}},t)}{\partial t}=\left[\displaystyle \frac{{{\boldsymbol{p}}}^{2}}{2}+V({\boldsymbol{r}})+{V}_{L}({\boldsymbol{r}},t)\right]\psi ({\boldsymbol{r}},t),\end{eqnarray}$
where ψ(r, t) is the time-dependent electron wave function, p is the momentum of electron in the (x, y) plane. The soft-core Coulomb potential in (x, y) plane can be expressed as
$\begin{eqnarray}\begin{array}{rcl}V(x,y) & = & -\displaystyle \frac{1}{\sqrt{{\left(x-0\right)}^{2}+{\left(y-\tfrac{2}{3}R\cos \tfrac{{\rm{\Theta }}}{2}\right)}^{2}}}\\ & & -\displaystyle \frac{1}{\sqrt{{\left(x+R\sin \tfrac{{\rm{\Theta }}}{2}\right)}^{2}+{\left(y+\tfrac{1}{3}R\cos \tfrac{{\rm{\Theta }}}{2}\right)}^{2}+c}}\\ & & -\displaystyle \frac{1}{\sqrt{{\left(x-R\sin \tfrac{{\rm{\Theta }}}{2}\right)}^{2}+{\left(y+\tfrac{1}{3}R\cos \tfrac{{\rm{\Theta }}}{2}\right)}^{2}+c}},\end{array}\end{eqnarray}$
the soft parameter c = 0.5 is used to remove the singularity of Coulomb potential and obtain the precise energies of certain electronic state for ${{\rm{H}}}_{3}^{2+}$ molecule. R is the distance between two protons and the bond angle is Θ = 60°. The molecular geometry with equilateral triangle structure is shown in figure 1(a), and for triangular molecule ${{\rm{H}}}_{3}^{2+}$ with equivalent internuclear distance R = 2 a.u. [30], the three lowest electronic states ${A}^{{\prime} }$, E+ and E are used as initial states. The excited electronic states E± corresponding angular momentum m = ± 1 around the z axis are degenerate with E-symmetry which can be obtained by two orthogonal real states E(1) and E(2) from solving numerically TDSE with imaginary-time propagation method. The wave functions can be expressed as $| {E}^{\pm }\rangle =(| {E}^{(1)}\rangle \pm {\rm{i}}| {\text{}}{E}^{(2)}\rangle )/\sqrt{2}$. Where, E(1) and E(2) are two orthogonal degenerate real states, corresponding to E(x) and E(y). They are the first excited state and the third excited state obtained by imaginary-time evolution method. In figures 1(b), (c), (d) and (e), we plot the initial electron density distribution of the first excited state E(1), the third excited state E(2), the three lowest electronic states ${A}^{{\prime} }$, E+ and E. The corresponding ionization potentials are respectively ${I}_{p}({A}^{{\prime} })$ = 1.69 a.u. and Ip(E+) = Ip(E) = 1.08 a.u. [30]. And the term VL(r, t) is the interaction between the electron and laser pulses treated in the length gauge and dipole approximation, which can be written as ${V}_{L}({\boldsymbol{r}},t)={{xE}}_{x}{\hat{e}}_{x}+{{yE}}_{y}{\hat{e}}_{y}$ in the (x, y) plane.
Figure 1. (a) The ${{\rm{H}}}_{3}^{2+}$ molecular ion is oriented in the (x, y) plane. R = 2 a.u. is the distance between two protons and the bond angle is Θ = 60. The electron density distributions of molecular ion ${{\rm{H}}}_{3}^{2+}$ for (b) the first excited state E(1), (c) the third excited state E(2), (d) the ground state ${A}^{{\prime} }$ and (e) the degenerate excited electronic states E±.
The 2D-TDSE can be solved by the fast Fourier transform (FFT) technique combined with a split-operator method [33, 34]:
$\begin{eqnarray}\begin{array}{l}\psi ({\boldsymbol{r}},t+{\rm{\Delta }}t)\\ \quad ={{\rm{e}}}^{-{\rm{i}}(\displaystyle \frac{{{\boldsymbol{p}}}^{2}}{4}){\rm{\Delta }}t}{{\rm{e}}}^{-{\rm{i}}[V({\boldsymbol{r}})+{V}_{L}({\boldsymbol{r}},t)]{\rm{\Delta }}t}{{\rm{e}}}^{-{\rm{i}}\left(\displaystyle \frac{{{\boldsymbol{p}}}^{2}}{2}\right){\rm{\Delta }}t}\psi ({\boldsymbol{r}},t)+{\rm{O}}({\rm{\Delta }}{t}^{3}),\end{array}\end{eqnarray}$
with time step Δt = 0.05 a.u., combined with FFT technique in spatial steps Δx = Δy = 0.04 a.u.. The spatial range is 409.6 a.u., which is enough to prevent unphysical effects from the reflection of the wave packet from the boundary. And a mask function of the form ${\cos }^{(1/8)}$ is used, and the domain of absorption ranges from ∣x, y∣ = 150.0 a.u. to ∣x, y∣ = 204.8 a.u.. Therefore, the ionized wave packets can be obtained by [1 − M(r, t)]ψ(r, t) and $r=\sqrt{{x}^{2}+{y}^{2}}$
$\begin{eqnarray}M({\boldsymbol{r}})=\left\{\begin{array}{ll}1, & {\boldsymbol{r}}\leqslant {r}_{b}\\ \exp \left[-\beta ({\boldsymbol{r}}-{r}_{b})\right], & {\boldsymbol{r}}\gt {r}_{b},\end{array}\right.\end{eqnarray}$
where β = 1 a.u., and the boundary for ionized electron wave functions and bounded wave functions is set to be rb = 30 a.u.. Finally, the wave functions in momentum space can be obtained by Fourier transforming the ionized wave functions in the coordinate space, and then squaring it to obtain the PMDs, the photoelectron angular distributions are obtained from the PMDs.

3. Results and discussion

In the present work, we respectively present the photoelectron distributions for different initial states of tri-atom molecular ions ${{\rm{H}}}_{3}^{2+}$ by elliptically polarized laser pulse and bichromatic counter-rotating circularly polarized laser pulses. As shown in figure 1(a), the molecule is pre-oriented before photoionization which can be readily achieved with current orientational laser technology [35]. Simulations of PMDs are obtained from numerical solutions of the 2D TDSEs.

3.1. The PMDs of different initial states for ${{\rm{H}}}_{3}^{2+}$ in elliptically polarized laser fields.

Firstly, we consider the PMDs for the triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ of different initial states by elliptically polarized laser pulses. The three lowest electronic states of ${{\rm{H}}}_{3}^{2+}$, ${A}^{{\prime} }$, E+ and E, are used respectively as the initial states. The elliptically polarized laser pulses can be expressed as
$\begin{eqnarray}E(t)={E}_{0}f(t)[\displaystyle \frac{1}{\sqrt{1+{\varepsilon }^{2}}}\cos (\omega t){\hat{e}}_{x}+\displaystyle \frac{\varepsilon }{\sqrt{1+{\varepsilon }^{2}}}\sin (\omega t){\hat{e}}_{y}],\end{eqnarray}$
where &ohgr; = 2.28 a.u. (λ = 20 nm) is the angular frequency of laser pulses, E0 = 0.11 a.u. (I0 = 4 × 1014 W cm−2) is the maximum field strength, ϵ is the ellipticity of the laser field. The pulse duration is T = 12T0 (T0 = 2π/&ohgr;) and the pulse envelope $f(t)=\exp [-2(\mathrm{ln}2){t}^{2}/{\tau }^{2}]$ is used with τ = 3T0. In the calculation, the single-photon process is considered which means that the electron can be ionized by absorbing only one photon for both ground and degenerate excited electronic states.
In figures 2 and 3, we show that the dependence of the PMDs and PADs with different initial states of the molecular ion ${{\rm{H}}}_{3}^{2+}$ on the laser ellipticities. As shown in figure 2, for the ground state ${A}^{{\prime} }$, as the ellipticity increases, the distributions of photoelectrons in the y-direction become stronger. And we can see that the PMDs have a relative small tilted angle which comes from the effect of Coulomb potential and the helicity of the laser field [36]. When ϵ = 1.0 (circularly polarized laser field) the PMDs present a six-lobe symmetric structure in figure 2(d1). In figure 3 we compare the PMDs and PADs of degenerate excited electronic states for triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ together. From figure 3 one sees that the PMDs and PADs present a three-lobe structure. Of note is that the PMDs and PADs of the two degenerate excited states, E+ and E are mirror images as shown in figure 3, the reason is that the corresponding angular momenta are around the z-axis, which cause the currents with mirror symmetry in (x, y) plane, and the PMDs are the reflection images of each current. We describe the dependence of PMDs on laser ellipticities by an ultrafast ionization model [30, 32].
Figure 2. The 2D PMDs and PADs of ${{\rm{H}}}_{3}^{2+}$ for the ground state ${A}^{{\prime} }$ at equilibrium R = 2 a.u. by elliptically polarized laser pulses (λ = 20 nm, &ohgr; = 2.28 a.u.). The pulse intensities and durations are I0 = 4 × 1014 W cm−2 (E0 = 0.11 a.u.) and T = 12T0 (T0 = 2π/&ohgr;). The ellipticity in (a1) and (a2), (b1) and (b2), (c1) and (c2), and (d1) and (d2) is ϵ = 0.3, 0.5, 0.75, and 1, respectively. The upper row is PMDs and the lower row is PADs.
Figure 3. The 2D PMDs of ${{\rm{H}}}_{3}^{2+}$ for degenerate excited electronic states E± at equilibrium R = 2 a.u. by elliptically polarized laser pulse. The laser parameters are the same as those used in figure 2. The inserts(white lines) are the corresponding PADs (photoelectron angular distributions). The ellipticity in (a1) and (a2), (b1) and (b2), (c1) and (c2), and (d1) and (d2) is ϵ = 0.3, 0.5, 0.75, and 1, respectively. The upper row is PMDs and PADs of E+ electronic state and the lower row is PMDs and PADs of E electronic state.
For single-photon ionization process, the transition amplitude can be written as
$\begin{eqnarray}{A}^{(x)}=\langle {\psi }_{c}| {\boldsymbol{r}}\cdot {{\boldsymbol{E}}}_{x}| {\psi }_{0}\rangle ={\sigma }^{x}{E}_{x}(\omega )\cos (\theta ),\end{eqnarray}$
$\begin{eqnarray}{A}^{(y)}=\langle {\psi }_{c}| {\boldsymbol{r}}\cdot {{\boldsymbol{E}}}_{y}| {\psi }_{0}\rangle ={\sigma }^{y}{E}_{y}(\omega )\sin (\theta ),\end{eqnarray}$
where σx = ⟨ψcxψ0⟩ and σy = ⟨ψcyψ0⟩ are the transition matrix elements in the x and y directions. θ is the ejection angle between the photoelectron momentum p and the x-axis. ∣ψ0⟩ and ⟨ψc∣ are the ground state and continuum state, respectively. And ${E}_{x}(k)={E}_{x0}(k)\tfrac{1}{\sqrt{1+{\varepsilon }^{2}}}$ and ${E}_{y}(k)={E}_{y0}(k)\tfrac{\varepsilon }{\sqrt{1+{\varepsilon }^{2}}}$ are the pulse frequency shape as a Fourier transform of the pulse along the x and y directions, where ${E}_{x0}(k)={\int }_{-\infty }^{\infty }{E}_{0}f(t)\cos (\omega t){{\rm{e}}}^{{\rm{i}}{\text{}}{kt}}{\rm{d}}t$ and ${E}_{y0}(k)={\int }_{-\infty }^{\infty }{E}_{0}f(t)\sin (\omega t){{\rm{e}}}^{{\rm{i}}{\text{}}{kt}}{\rm{d}}t$.
For triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$, there is a 1s orbital for each H atom. According to molecular orbital theory, the electronic states are linear combinations of the three 1s orbitals at position Ri(i = 1, 2, 3). The molecular wavefunction for the ground state ${A}^{{\prime} }$ are given by [37]
$\begin{eqnarray}{\psi }_{{A}^{{\prime} }}({\boldsymbol{r}})=[{\psi }_{s1}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{1})+{\psi }_{s2}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{2})+{\psi }_{s3}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{3})]/\sqrt{3}.\end{eqnarray}$
According to the multi-center interference based on the ultrafast ionization model [32], the transition amplitude to a continuum can be written as
$\begin{eqnarray}F({\boldsymbol{p}})\propto {\left(2\pi \right)}^{-\tfrac{2}{3}}\int {{\rm{e}}}^{-{\rm{i}}{\boldsymbol{p}}\cdot {\boldsymbol{r}}}{{\rm{e}}}^{-{\rm{i}}{\boldsymbol{F}}\cdot {\boldsymbol{r}}}{\psi }_{0}({\boldsymbol{r}}){\rm{d}}{\boldsymbol{r}}.\end{eqnarray}$
The vector potential F of the laser field can be ignored to describe high frequency photoionization process since the intensities produce small ponderomotive energies UpIp and Fp. For the ground state ${A}^{{\prime} }$, setting the molecular orbital ${\psi }_{0}({\boldsymbol{r}})={\psi }_{{A}^{{\prime} }}({\boldsymbol{r}})$, one then obtains the momentum distribution of photoelectron which can be written as
$\begin{eqnarray}\begin{array}{l}{F}^{{A}^{{\prime} }}\propto \left[{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{12}/2)+{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{13}/2)\right.\\ \quad \left.+{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{23}/2)\right]{\psi }_{1s}^{2}({\boldsymbol{p}})\\ \quad \propto \left[{\cos }^{2}({pR}\cos (\theta )/2)+{\cos }^{2}({pR}\cos (\theta +\pi /3)/2)\right.\\ \quad \left.+{\cos }^{2}({pR}\cos (\theta -\pi /3)/2)\right]{\psi }_{s}^{2}({\boldsymbol{p}}),\end{array}\end{eqnarray}$
where Rij = RjRi (i, j = 1, 2, 3) and Rij = R = 2 a.u. is the molecular internuclear distance, as illustrated in figure 1(a). From equation (10) we can obtain the maxima of distributions at angles θ = π/6 + nπ/3, n = 0, ±1, ±2, ⋯ as illustrated in figure 4(a). One sees that these angle nodes predicted from equation (10) are in good agreement with the numerical results in figure 2.
Figure 4. Results of photoelectron angular distributions for (a) the ground state ${A}^{{\prime} }$ obtained by equation (10), excited electronic state E+ (b) and E (c) predicted by equations (12) and (13).
As laser ellipticities increase, the distributions of photoelectrons in the y-direction become more intense, so we focus on the modulus square of transition amplitude along the y-direction.
$\begin{eqnarray}\begin{array}{rcl}| {A}^{(y)}{| }^{2} & \propto & {E}_{y0}^{2}(k)\displaystyle \frac{{\varepsilon }^{2}}{1+{\varepsilon }^{2}}\sin (\theta ){\cos }^{2}(p\cdot R/2)\\ & & \times [{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{12}/2)\\ & & +{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{13}/2)+{\cos }^{2}({\boldsymbol{p}}\cdot {{\boldsymbol{R}}}_{23}/2)]{\psi }_{s}^{2}({\boldsymbol{p}}).\end{array}\end{eqnarray}$
The values for $\tfrac{{\varepsilon }^{2}}{1+{\varepsilon }^{2}}$ are 0.08, 0.2, 0.36 and 0.5, when ϵ = 0.3, 0.5, 0.75 and 1, respectively, the intensity of the momentum distributions in the y-direction is directly proportional to the value of $\tfrac{{\varepsilon }^{2}}{1+{\varepsilon }^{2}}$ , which causes the enhancement of PMDs in the y-direction as the ellipticity increases as shown in figures 2(a1)–(d1). The PMDs have a relative small tilted angle which comes from the effect of Coulomb potential and the helicity of the laser field.
The molecular wavefunctions for the two degenerate excited electronic states are given by [37]
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{{E}^{+}}({\boldsymbol{r}}) & = & \left[{\psi }_{s1}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{1})+{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{2\pi }{3}}{\psi }_{s2}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{2})\right.\\ & & \left.+{{\rm{e}}}^{-{\rm{i}}\displaystyle \frac{2\pi }{3}}{\psi }_{s3}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{3})\right]/\sqrt{3},\end{array}\end{eqnarray}$
for the E+ electronic state, and
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{{E}^{-}}({\boldsymbol{r}}) & = & [{\psi }_{s1}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{1})+{{\rm{e}}}^{-{\rm{i}}\displaystyle \frac{2\pi }{3}}{\psi }_{s2}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{2})\\ & & +{{\rm{e}}}^{{\rm{i}}\displaystyle \frac{2\pi }{3}}{\psi }_{s3}({\boldsymbol{r}}-{{\boldsymbol{R}}}_{3})]/\sqrt{3},\end{array}\end{eqnarray}$
for the E electronic state. For the wave functions of the degenerate excited electronic states ${\psi }_{{E}^{+}}({\boldsymbol{r}})$ and ${\psi }_{{E}^{-}}({\boldsymbol{r}})$, we can obtain that they are reflection of each other ${\psi }_{{E}^{+}}({\boldsymbol{r}})={\psi }_{{E}^{-}}^{* }({\boldsymbol{r}})$. Then the corresponding PMDs in equations (9) and (10) are given by
$\begin{eqnarray}\begin{array}{rcl}{F}^{{E}^{+}}({\boldsymbol{p}}) & \propto & \left[{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta -{\rm{\Theta }})+\displaystyle \frac{\pi }{3}\right)\right.\\ & & +{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta )+\displaystyle \frac{2\pi }{3}\right)\\ & & \left.+{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta +{\rm{\Theta }})+\displaystyle \frac{\pi }{3}\right)\right]{\psi }_{1s}^{2}({\boldsymbol{p}}),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{F}^{{E}^{-}}({\boldsymbol{p}}) & \propto & \left[{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta -{\rm{\Theta }})-\displaystyle \frac{\pi }{3}\right)\right.\\ & & +{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta )-\displaystyle \frac{2\pi }{3}\right)\\ & & \left.+{\cos }^{2}\left(\displaystyle \frac{1}{2}{pR}\cos (\theta +{\rm{\Theta }})-\displaystyle \frac{\pi }{3}\right)\right]{\psi }_{1s}^{2}({\boldsymbol{p}}).\end{array}\end{eqnarray}$
From equations (10), (14) and (15) one notes that, in the multi-center interference model, the interference for triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ can be seen as a simply sum of the contributions from all centers. The electron wave packets ionized from different centers can interfere with each other. The terms of cosine triangle functions present the interference effects of the three outgoing electron wave packets emanating from the three nuclear centers. The results are shown in figure 4(c) (predicted by equation (14)) and figure 4(d) (predicted by equation (15)) which are in good agreement with the simulations obtained by TDSEs in figure 3. And photoelectron distributions for the degenerate excited E+ and E electronic states are the mirror images satisfied equation ${F}^{{E}^{+}}(p,{\theta }_{p})={F}^{{E}^{-}}(p,{\theta }_{p}\pm \pi )$.

3.2. Vortex structure of the degenerate electronic states for ${{\rm{H}}}_{3}^{2+}$ in the bichromatic counter-rotating circularly polarized laser fields

We next consider the PMDs in bichromatic counter-rotating circularly polarized laser fields for the two degenerate electronic states E+ and E of the triatomic molecule ion ${{\rm{H}}}_{3}^{2+}$. The excited electronic states E± corresponding to the angular momentum m = ±1 around the z axis are degenerate with E-symmetry. The counter-rotating circularly polarized laser pulses can be expressed by
$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{E}}}_{1}(t) & = & {E}_{0}f(t)[\cos ({\omega }_{1}t){\hat{e}}_{x}+\sin ({\omega }_{1}t){\hat{e}}_{y}]\\ {{\boldsymbol{E}}}_{2}(t) & = & {E}_{0}f(t-{T}_{d})[\cos ({\omega }_{2}(t-{T}_{d})){\hat{e}}_{x}\\ & & -\sin ({\omega }_{2}(t-{T}_{d})){\hat{e}}_{y}],\end{array}\end{eqnarray}$
where &ohgr;1 = 0.7 a.u. (λ1= 65 nm) and &ohgr;2 = 1.4 a.u. (λ2= 32.5 nm) are the angular frequencies of laser pulses, and Td is the time delay between the two pulses. We define one cycle duration of 65 nm pulses as the unit, i.e. 1 o.c. = T0 = 2π/&ohgr;1. Since Ip(E+) = Ip(E) = 1.08 a.u. corresponds to a laser wavelength λ = 42 nm, direct ionization occurs at λ2 = 32.5 nm whereas two-photon ionization is induced at λ1 = 65 nm. The other laser parameters are the same as those in figure 2.
From figure 5 we can see that the PMDs of the degenerate excited electronic states E+ and E are sensitive to the time delay between the two pulses in bichromatic counter-rotating circularly polarized laser fields. When Td = 0 o.c., the PMDs of E± electronic states present a six-lobed ring structure, and no vortex structures appear, shown in figures 5(a) and (c). For E+ electronic states and the E electronic states, the PMDs have three strong lobes and three weak lobes, and for E+ state the differences among the intensity of the distribution for six lobes are little. The weak lobe in E+ corresponds to the strong lobe in E, vice versa. When the delay time is added to Td = 5 o.c., vortex structures appear in PMD of the two degenerate states E± in figures 5(b) and (d). For the triatomic molecule ${{\rm{H}}}_{3}^{2+}$, the electron wave packets initiate from the three different nuclear centers. Meanwhile, the electrons can also absorb two &ohgr;1 photons or one &ohgr;2 photon to reach the continuum with the same final kinetic energies, and then interfere with each other, leading to a three-center ionization spectra. Likewise, multi-center interference based on the ultrafast ionization model which has been successfully used to describe the ultrafast ionization processes for ${{\rm{H}}}_{2}^{+}$ [30, 32] is adopted to describe these electron dynamics.
Figure 5. The PMDs of the two degenerate excited electronic states E+ (upper row: (a) and (b)) and E (lower row: (c) and (d)) of triatomic molecular ion ${{\rm{H}}}_{3}^{2+}$ in bichromatic couner-rotating circularly polarized laser fields (&ohgr;1 = 0.7 a.u., &ohgr;2 = 1.4 a.u.). The time delay Td between the two pulses are, respectively, Td = 0 o.c. (left column: (a), (c)) and Td = 5 o.c. (right column; (b), (d)).
According to the multi-center interference, for the two degenerate excited E+ and E+ electronic states, the corresponding photoelectron distributions are predicted in equations (14) and (15) And the maxima of distributions at angles $\theta =\tfrac{\pi }{3},\pi ,\tfrac{5\pi }{3}$ as shown in figure 4(b) for E+ electronic state and, for E electronic state, the maxima of distributions at angles $\theta =0,\tfrac{2\pi }{3},\tfrac{4\pi }{3}$ as illustrated in figure 4(c). Of note is that the results of the interference for the electron wave packets emanating from the three nuclei are three-lobed structures. The six-lobed structures of the PMDs, therefore, are affected by the interference between the two-pathway ionization processes.
We next consider the two-pathway ionization interference. When λ1 = 65 nm and λ2 = 32.5 nm, the transition amplitude for two-photon ionization W(1) under left-handed circularly polarized pulse and for one-photon ionization W(2) under right-handed circularly polarized pulse can be written as
$\begin{eqnarray}{W}^{(1)}=\displaystyle \sum _{n}\langle {\psi }_{c}| {\boldsymbol{D}}\cdot {{\boldsymbol{E}}}_{1}| {\psi }_{n}\rangle \langle {\psi }_{n}| {\boldsymbol{D}}\cdot {{\boldsymbol{E}}}_{1}| {\psi }_{0}\rangle ,\end{eqnarray}$
$\begin{eqnarray}{W}^{(2)}=\langle {\psi }_{c}| {\boldsymbol{D}}\cdot {{\boldsymbol{E}}}_{2}| {\psi }_{0}\rangle {{\rm{e}}}^{-{\rm{i}}{\omega }_{1}{T}_{d}},\end{eqnarray}$
where E1(k) = e1E1(k) and E2(k) = e2E2(k) represent the Fourier transform of E1(t) and E2(t); D = nD is the electric dipole operator, n is the unit vector direction; ${{\rm{e}}}^{-{\rm{i}}{\omega }_{1}{\text{}}{T}_{{\text{}}d}}$ is the phase caused by the time delay. Therefore, the equations (17) and (18) can also be expressed as
$\begin{eqnarray}{W}^{(1)}=\displaystyle \sum _{n}\langle {\psi }_{c}| {{DE}}_{1}(k)| {\psi }_{n}\rangle \langle {\psi }_{n}| {{DE}}_{1}(k)| {\psi }_{0}\rangle {\left({\boldsymbol{n}}\cdot {{\boldsymbol{e}}}_{1}\right)}^{2},\end{eqnarray}$
$\begin{eqnarray}{W}^{(2)}=\langle {\psi }_{c}| {{DE}}_{2}(k)| {\psi }_{0}\rangle ({\boldsymbol{n}}\cdot {{\boldsymbol{e}}}_{2}){{\rm{e}}}^{-{\rm{i}}{\omega }_{1}{T}_{d}}.\end{eqnarray}$
We take σ(1) = ⟨ψcDE1(k)∣ψn⟩⟨ψnDE1(k)∣ψ0⟩ and ${f}^{(1)}\,={\left({\boldsymbol{n}}\cdot {{\boldsymbol{e}}}_{1}\right)}^{2}={{\rm{e}}}^{2{\rm{i}}\theta }$, then W(1) = σ(1)f(1). Likewise, W(2) = σ(2)f(2), where σ(2) = ⟨ψcDE2(k)∣ψ0⟩ and ${f}^{(2)}=({\boldsymbol{n}}\cdot {{\boldsymbol{e}}}_{2}){{\rm{e}}}^{-{\rm{i}}{\omega }_{1}{T}_{d}}\,={{\rm{e}}}^{-{\rm{i}}\theta }{{\rm{e}}}^{-{\rm{i}}{\omega }_{1}{T}_{d}}$. The final PMDs could be expressed as the sum of the square of the two amplitudes and an interference term of the cross products of the two one-photon and one two-photon ionization amplitudes in equations (19) and (20) which can be expressed as
$\begin{eqnarray}\begin{array}{rcl}W & = & |{W}^{\left(1\right)}+{W}^{\left(2\right)}{|}^{2}\\ & = & |{W}^{\left(1\right)}{|}^{2}+|{W}^{\left(2\right)}{|}^{2}+{W}^{\left(1,2\right)},\end{array}\end{eqnarray}$
the interference term W(1,2) can be simply written as
$\begin{eqnarray}{W}^{(\mathrm{1,2})}={W}^{(1)* }{W}^{(2)}+{W}^{(1)}{W}^{(2)* }={\sigma }^{(1)}{\sigma }^{(2)}{f}^{(\mathrm{1,2})}.\end{eqnarray}$
The angular scalar product of the interference term, f(1,2) can be expressed as
$\begin{eqnarray}{f}^{(\mathrm{1,2})}={f}^{(1)* }{f}^{(2)}+{f}^{(1)}{f}^{(2)* }=2\cos (3\theta +{\omega }_{1}{T}_{d}).\end{eqnarray}$
We can see that the interference has a $\cos (3\theta )$ dependence for bichromatic counter-rotating circularly polarized laser pulses. The maximum distribution satisfies the condition $\cos (3\theta )$ = 1, i.e. $\theta =0,\tfrac{2\pi }{3},\tfrac{4\pi }{3}$ as shown in figure 6. The total momentum distributions of the two degenerate excited states are determined by the three centers ionization in equations (14) and (15) and the two-pathway ionization in equation (23). Consequently, there are six maximal radiation peaks located at $\theta =\tfrac{\pi }{3},\pi ,\tfrac{5\pi }{3}$ in equation (14) and $\theta =0,\tfrac{2\pi }{3},\tfrac{4\pi }{3}$ in equation (23) for E+ electronic state shown in figure 5(a), while for E electronic state the theoretical prediction shows that there are only three maximal radiation peaks located at $\theta =0,\tfrac{2\pi }{3},\tfrac{4\pi }{3}$ in equations (15) and (23) as shown in figure 6, which is conflict with TDSE result of figure 5(c). As the time delay increases, when Td = 5 o.c., there are vortex structures appearing in both the PMDs of the degenerate electron states E±, which are obviously sensitive to the time delay of the two pulses. The vortex structures of PMDs are the result of the coherent superposition of photoelectron wave packets ionized by the left-handed and right-handed circularly polarized laser pulses. By the coherent superposition, the total PMDs are distorted, and the vortex structure appears.
Figure 6. Results of the two degenerate excited electronic states predicted from the two-pathway ionization models in equation (23).

4. Conclusions

We present the PMDs of the triatomic molecule ion ${{\rm{H}}}_{3}^{2+}$ in the elliptically polarized and bichromatic counter-rotating circularly polarized laser fields, respectively. For the elliptically polarized laser field, the three lowest electronic states, ${A}^{{\prime} }$, E+ and E of ${{\rm{H}}}_{3}^{2+}$, are used respectively as the initial states, and we found that the PMDs show the dependence on laser ellipticities and molecular orbital symmetry. Meanwhile, the PMDs of the degenerate excited electronic states E± are mirror images since the corresponding angular momentum is m = ±1 a round the z axis, i.e. reflection images of each current. The different interference patterns of PMDs illustrate the various symmetries of the initial electronic state, thus it provides a powerful tool for the imaging of molecular orbitals. The ultrafast ionization model is adopted to describe these phenomena. For bichromatic counter-rotating circularly polarized laser fields, we found that, for the degenerate excited electronic states E±, the PMDs are determined by the multi-center and two-pathway interferences. And we also found that the electron vortex of PMDs is sensitive to the time delay between the two pulses.
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Outlines

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