1. Introduction

With the development of laser technology, it has become a reality to image and manipulate the nuclear and electronic dynamics at ultra-short time and ultra-micro spatial scales in photophysics and photochemsitry processes. This has prompted the researches of laser-matter interactions in atomic and molecular systems, among which one of the basic issues is the study of electronic ultrafast dynamics in molecular reactions. Attosecond (10$^{-18}$ s) pulses [1,2] have provided important extreme physical conditions for these ultrafast processes, which has become a new tool for studying and ultimately controlling the electrons dynamics at the natural time scale [3–6]. Recently, the shortest attosecond pulses with a pulse duration of 43 as are available for new ultrafast photoelectron imaging technology [7]. However, the synthesis of ultrafast pulses on the sub-second time scale for imaging and controlling electronic movement is mostly limited to the linear polarization process [1,8]. Until recently, this new capability to generate circularly polarized coherent short wavelength harmonics using intense femtosecond bichromatic circularly polarized (CP) pulses has proved the potential value in experiments, which regard the X-ray system as a new imaging tool for new application of ultrafast X-ray magnetic circular dichroism [9–11]. Currently, CP has been widely used in many aspects, such as the generation of high-order harmonic and attosecond pulses in experiments and theories [12–16].

Ultrafast charge migration (CM) induced by the coherent population excitation in multiple electronic states has attracted widespread attention in the fields of photophysics and photochemistry, or even the application of attosecond technology [17–30], which provides convenience for controlling chemical reactions and detecting molecular orbitals. Recent work has shown that attosecond CM can be detected by using iodoacetylene molecular ions to generate high harmonics in heavy collisions [31]. On this basis, the current-carrying state in a semiconductor quantum ring is completely controlled using the quantum optimal control theory for terahertz laser pulses, which proves the importance of ultrafast pulses for magneto-optics [32]. So far, many efforts have been devoted to the generation of strong magnetic fields. Researchers demonstrated that the generation of electronic ring current in aromatic molecules can produce static magnetic fields through linear [33] and CP $\pi$ UV pulses resonance with the degenerate $\pi$ orbitals [34–36]. Especially, this generated magnetic fields can be much larger than those obtained by traditional static fields [37]. Morever, the electron wave packets in H$_{2}^{+}$ generated by means of intense few-cycles attosecond CP UV pulses can also generate strong attosecond magnetic fields of several tens of Teslas [38–43]. In addition, when the current-carrying state $2p_{+}$ of Ne atom is irradiated by an intense linearly polarized laser field, an spatial-localized oscillating magnetic field with an intensity up to 47 Tesla at the center can be induced [44]. However, the influence of molecular symmetry and asymmetry on the generation of magnetic fields remains unclear, which requires further exploration.

In this work, we report the generation of electron currents and ultrafast magnetic-field on the selected different orbitals of excited states for oriented symmetric H$_{2}^{+}$ and asymmetric HeH$^{2+}$ molecular ions under the CP attosecond XUV laser pulses by numerically solving the corresponding time-dependent Schrödinger equation (TDSE). The two scenarios of (i) resonant excitation, (ii) direct ionization are considered. For different molecular ions, we compare the electron currents and the corresponding ultra-fast strong magnetic field generated under these two scenarios. The results show that, in comparison to the simplest symmetric molecular ion H$_{2}^{+}$ from ground $1s\sigma _{g}$ electronic state, in the asymmetric molecular ion HeH$^{2+}$ from first excited $2p\sigma$ electronic state, the results obtained have similar features, but overall, the electron current and corresponding ultrafast magnetic field of symmetric molecular ion are stronger [17–19].

The paper is organized as follows: We briefly describe computational methods for numerically solving 2D TDSEs of an aligned molecular ion HeH$^{2+}$ and H$_{2}^{+}$ by intense CP attosecond UV laser pulses in Sec. II. The results for aligned symmetric (H$_{2}^{+}$) and asymmetric (HeH$^{2+}$) molecular ions are discussed in Sec. III. Finally, we summarize our findings in Sec. IV. Throughout this paper, atomic units (a. u.) are used unless otherwise noted.

2. Theoretical model and computational methods

In this work, we perform numerical simulations on oriented single-electron molecular ions H$_{2}^{+}$ and HeH$^{2+}$ with a static nuclear frame. The corresponding TDSE is appropriately written with respect to the centre of mass of the two nuclei as

Here, a 2D (planar) model of the laser polarization in the molecular plane is adopted, which is adequate to describe electron motions in laser fields due to the 2D confinement of electron via CP laser pulses. The molecular axis is strictly oriented with the x-polarization axis. The corresponding soft-core Coulomb potential can be expressed as:

The field-molecule interaction $V_{L}\left ( \textbf {r},t\right )=\textbf {r}\cdot E\left ( t\right )$ is treated under the length gauge and dipole approximation, since by exact solutions of the TDSE, numerical results are gauge invariant. The laser fields $E\left ( t\right )$ always propagates along the z axis, which is perpendicular to the molecular (x, y) plane. The forms for the CP pulses are defined as

The 2D TDSE in Eq. (1) is numerically solved by using the second order split-operator method algorithm combined with fast Fourier transform (FFT) technique [45]:

In our calculation, the time step is taken to be $\Delta$t = 0.05 a. u. = 1.2 as. The spatial grid has 512 $\times$512 = 262144 grid points with the same step sizes for the x- and y-directions: $\Delta$x = $\Delta$y = 0.25 a.u. The grid ranges from −64 a.u. to 64 a.u. in each direction. The wave function is multiplied by a $cos^{\frac {1}{8}}$ mask function at each time step to prevent unphysical effects originating from the reflection of the wave packet from the boundary. And the domain of absorption ranges from $\left | x,y\right |$ = 50 a.u. to $\left | x,y\right |$ = 64 a.u..

The time-dependent electronic current density is defined by the quantum expression also in the length gauge,

3. Results and discussions

We study the electron currents and corresponding time-dependent magnetic fields induced by coherent electronic wave packets for symmetric H$_{2}^{+}$ and asymmetric HeH$^{2+}$ molecular ions at equilibrium aligned along the x-axis. Molecular ions are excited by using a CP XUV laser pulse with its electric field vector polarized in the (x, y) plane, propagating along the z axis, as illustrated in Fig. 1. The pulse intensity is $\ I_0 = 1\times {}{10}^{13} \ \mbox {W}/{\mbox {cm}^2}$ ($E_{0} = 1.688 \times 10^{-2}$ a.u.). The research content will follow two designed scenarios. (i) We use XUV laser pulses with photon energy $\hbar \omega$ = $\Delta E$ = $E_{e}-E_{g}$ ( $E_{e}$ and $E_{g}$ represent the eigenenergies of the excited state and ground state) to create coherent excitation and CM of the $1s\sigma _{g}-2p\sigma _{u}$ and $2p\sigma -2p\pi$ electronic state combinations. Where, $1s\sigma _{g}-2p\sigma _{u}$ resonant excitation occurs from the electronic ground state $1s\sigma _{g}$ to the electronic first excited state $2p\sigma _{u}$ for H$_{2}^{+}$. Here, the $2p\sigma -2p\pi$ resonant excitation is considered for HeH$^{2+}$. Since the ground state $1s\sigma$ of HeH$^{2+}$ is mainly localized on the He$^{2+}$, its potential energy curve is repulsive, while the first excited $2p\sigma$ state has a minimum at R = 3.89 a.u., the mean lifetime of $2p\sigma$ state is about 4 ns [47]. Of course, we also compare the results of the HeH$^{2+}$ when the initial states are $2p\sigma$ and $1s\sigma$ respectively (Supplement 1, part 1). (ii) We use the XUV ultra-short laser pulse with photon energy ($\hbar \omega >I_{p}$) to directly ionize molecular ions and explore the CM process. Figure 2(a) shows the diagram of these two scenarios. The corresponding ionization potentials $I_{p}$ for $(1s\sigma _{g})$ and $(2p\sigma _{u})$ are 1.08 a.u. and 0.65 a.u. in the H$_{2}^{+}$ (R = 2 a.u.) [48,49], and that for $1s\sigma$, $2p\sigma$ , $2p\pi$ are 2.25 a.u., 1.03 a.u., 0.74 a.u. in the HeH$^{2+}$ (R = 3.89 a.u.) [50–52]. Meanwhile, the densities of initial wavefunction of the selected electronic states for two molecular ions are presented in Fig. 2(b).

 

Fig. 1. Illustration of the attosecond magnetic field pulses $B\left ( \textbf {r},t\right )$. It is generated in the molecular ion by a right-handed circular attosecond pulses E(t) in the $\left ( x,y\right )$ plane. The corresponding electron currents $j\left ( \textbf {r},t\right )$ are induced in the (x, y) plane and the magnetic field is perpendicular to $j\left ( \textbf {r},t\right )$, along the z axis.

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Fig. 2. Illustration of the possible photoexcitation pathways to produce electron currents and magnetic fields by CP laser pulses. (a) The left and right column represent the two scenarios designed for HeH$^{2+}$ and H$_{2}^{+}$. (b) The initial wavefunction densities of selected electronic states. The upper row represents the initial $2p\sigma$ excited state and the second excited state $2p\pi$ for HeH$^{2+}$, and the lower row represents the ground state $1s\sigma _{g}$ and the first excited state $2p\sigma _{u}$ for H$_{2}^{+}$.

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We first consider the scenario (i). For HeH$^{2+}$, the $2p\sigma - 2p\pi$ resonant excitation from the electronic first excited state $2p\sigma$ ($|\psi _{g0}\left ( \textbf {r}\right )\rangle$ with the eigenenergy $E_{g0}$) to the electronic excited $2p\pi$ state ($|\psi _{e0} \left ( \textbf {r}\right )\rangle$ with the eigenenergy $E_{e0}$) is studied. To induce molecular resonant excitation, the wavelength $\lambda$=150 nm ( $\omega$ = $E_{e0}$ - $E_{g0}$ = 0.3 a.u.) of the CP XUV pulses is required. For H$_{2}^{+}$, the $1s\sigma _{g}-2p\sigma _{u}$ resonant excitation from the electronic ground state $1s\sigma _{g}$ ($|\psi _{g1} \left ( \textbf {r}\right )\rangle$ with the eigenenergy $E_{g1}$) to the electronic excited $2p\sigma _{u}$ state ($|\psi _{e1} \left ( \textbf {r}\right )\rangle$ with the eigenenergy $E_{e1}$) is studied. The wavelength with $\lambda$=100 nm ($\omega$= $E_{e1}$ - $E_{g1}$ = 0.455 a.u.) of the CP pulses is used to produce such resonant excitation. Then, a coherent superposition of the two electronic states is created due to a strong charge-resonance excitation for both molecular ions, respectively.

The coherent electron dynamic is composed of two electronic state densities, $\mathcal {A}^{(gn)}$ = $|c_{gn}\left ( t\right )\psi _{(gn)}\left ( \textbf {r}\right )|^{2}$ and $\mathcal {A}^{(en)}$ = $|c_{(en)}\left ( t\right )\psi _{(en)}\left ( \textbf {r}\right )|^{2}$, which are insensitive to the time. Their interfering superposition can be expressed: $\mathcal {A}^{(g,e)}$ = $2\left | c_{(gn)}\left ( t\right )c_{(en)}\left ( t\right )\right |\psi _{(gn)}\left ( \textbf {r}\right )\psi _{(en)}\left ( \textbf {r}\right )cos\left ( \Delta Et\right )$. Whereas the coherent superposition term $2\left | c_{(gn)}\left ( t\right )c_{(en)}\left ( t\right )\right |\psi _{(gn)}\left ( \textbf {r}\right )\psi _{(en)}\left ( \textbf {r}\right )cos\left ( \Delta Et\right )$ is determined by the time t.

The time-dependent superposition term $\mathcal {A}^{\left ( g,e\right )}\left (\textbf {r},t \right )$ describes the attosecond electronic coherence with the oscillation period $\Delta \tau = 2 \pi /\Delta E = 506.8$ as, where $\Delta E$ = $E_{e0}$ - $E_{g0}$ = 0.3 a.u. for HeH$^{2+}$. The coherent electron density probability $\mathcal {A}\left ( \textbf {r},t\right )$ oscillates between the two protons with time t (Fig. 3(a)). At t = 5 $\tau$, the coherent electron density is mainly localized at H$^{+}$, while, as time increases, the electron is gradually localized at ion He$^{2+}$. That is, at 5.0 $\tau$ $\leq$ t $\leq$ 5.25 $\tau$, the electron migration from H$^{+}$ to He$^{2+}$. As the time t increases further, the electron density come back to the H$^{+}$, finally, the electron migrates from H$^{+}$ back to He$^{2+}$ at t = 6 $\tau$. Therefore, the CM presents a periodic discipline i.e., the electron oscillates between the H$^{+}$ and He$^{2+}$, and the time is exactly one optical cycle.

 

Fig. 3. Electronic dynamics in CM for HeH$^{2+}$ at equilibrium R = 3.89 a.u. from the initial $2p\sigma$ excited state. (a) and (c) show the electronic density probabilities $\mathcal {A}\left (\textbf {r},t \right )$ at five different times (top row). (b) and (d) are the electronic current densities $j\left ( \textbf {r},t\right )$ at five different times. The white arrows in (b) and (d) label the directions of the electronic currents. All these are obtained by using right-handed CP pulses ( $\xi$= −1) with $\lambda$ = 150 nm in (a-b) and 30 nm in (c-d). Arbitrary units of distributions are used.

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The electronic density probabilities intuitively reflect the distribution of charges in space, while the electronic current can show the distribution of electron velocity in space. To visualize the coherent electronic density, we present the electronic current distribution at different times (Fig. 3(b)). At t = 5.0 $\tau$, the electronic current is localized at H$^{+}$, while at time t = 5.25 $\tau$, the current density is completely localized at He$^{2+}$. The electron current directions for two moments above are both to the left, then, the direction starts to point to the right at t = 5.5 $\tau$, and finally at t = 6 $\tau$, the direction migrate to the left again. The currents alternate between He$^{2+}$ and H$^{+}$ along the molecular x axis with the change of time $\tau$. Actually, under the resonance excitation, the coherent electronic density cannot be ignored, and it is accompanied by the periodic oscillation at time $\tau$, since $\mathcal {A}^{\left ( g,e\right )}\left (\textbf {r},t \right )\sim cos(\Delta Et)$. Meanwhile, the superposition state is the non-stationary state. Therefore, the electronic density distributions oscillates back and forth between the two protons. We can see that the process of CM between the two protons is significant by resonance excitation between $2p\sigma$ and $2p\pi$ electronic states, which can better explain the enhancement of the CM efficiency.

Next, we discuss the scenario (ii), which uses ultrashort pulses with $\lambda$ = 30 nm ($\omega$ = 1.5 a.u., $\omega >I_{p}$) to directly ionize initial $2p\sigma$ state of HeH$^{2+}$. The electronic density probability and the electron current density distribution over time are calculated, as shown in Fig. 3(c-d). It can be seen that the electronic density probability does not change with time, and the CM between the He$^{2+}$ and H$^{+}$ is not significant, which result in the low CM efficiency. And from the perspective of the direction of electron currents, the electron no longer alternately migrate between two protons over time, while exhibits a periodic and clockwise rotation with an optical cycle of 506 as. That is, at time t = 5.0 $\tau$, 5.25 $\tau$, 5.5 $\tau$, 5.75 $\tau$ and 6.0 $\tau$, the electron current directions are downward, left, upward, right and downward, respectively. Since at higher laser frequency $\omega >I_{p}$, the molecular ion is ionized after absorbing one photon, the ionized electrons with relatively small kinetic energies move away directly with the velocity perpendicular to the molecular R axis, that is, the y direction, due to a nonzero drift velocity in the y direction [53]. In fact, the induced electron currents (Fig. 3(d)) result from the electron wave packets in the continuum state generated by the driving CP pulses, and the coherence of electronic wave packet is very weak and almost negligible. Therefore, the electron wave packets with near zero initial velocities move in the (x, y) plane as a free electron, following the driving circularly polarized pulses. It explains the electron currents rotating clockwise in an optical cycle, and the corresponding CM efficiency decreases.

In order to examine the purely repulsive of the potential energy in the ground state $1s\sigma$ of HeH$^{2+}$, we also show the electronic density probability and electronic currents distribution at different times, as shown in Supplement 1, part 1. It was found that the generation of electronic current and corresponding magnetic field induced by resonance excitation $1s\sigma -2p\sigma$ are both weaker than those induced by resonance excitation $2p\sigma -2p\pi$. Interestingly, the electron current induced by former is sensitive to the helicity of CP laser pulses, which is similar to the scenarios (ii) from the initial $2p\sigma$ state.

Then, we turn our attention to the symmetric molecular ion H$_{2}^{+}$. Figure 4(a-b) present the electron density probabilities $\mathcal {A}\left (\textbf {r},t \right )$ and the density distribution $j\left ( \textbf {r},t\right )$ of electron currents for a resonant excitation ($1s\sigma _{g}-2p\sigma _{u}$) by using right-handed CP laser pulses ($\xi$ = −1) with $\lambda$ = 100 nm. It can be seen that both of them show the oscillating behavior similar to that in resonant excitation ($2p\sigma -2p\pi$ of HeH$^{2+}$). Meanwhile, the corresponding $\mathcal {A}\left (\textbf {r},t \right )$ and $j\left ( \textbf {r},t\right )$ under the case of direct ionization are also calculated by using same ultrashort pulses with $\lambda$ = 30 nm (Fig. 4(c-d)). We found that the results obtained were consistent with that of direct ionization for HeH$^{2+}$. Thus, it can be known that no matter in the scenario (i) or (ii), the two molecular ions have similar disciplines. That is, the electron current induced by the alternating CM between two parts, the CM efficiency is high, and it is not sensitive to the helicity of circular polarization in the scenario (i). However, in the scenario (ii), the CM efficiency is low, and it is extremely sensitive to the helicities of the laser pulses.

 

Fig. 4. Electronic dynamics in CM for H$_{2}^{+}$ at equilibrium R = 2 a.u. from the initial $1s\sigma _{g}$ ground state. (a) and (c) show the electronic density probabilities $\mathcal {A}\left (\textbf {r},t \right )$ at five different times (top row). (b) and (d) are the electronic current densities $j\left ( \textbf {r},t\right )$ at five different times. The white arrows in (b) and (d) label the directions of electronic currents. All these are obtained by using right-handed CP pules ($\xi$ = −1) with $\lambda$ = 100 nm in (a-b) and 30 nm in (c-d). Arbitrary units of distributions are used.

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In order to better observe above phenomena of the H$_{2}^{+}$ and HeH$^{2+}$, we display the axial electronic current (i.e., $j_{x}\left ( x,t\right )$) as a function of position x and time t (in units of period 5 $\tau$ $\sim$ 6 $\tau$ = 506.8 as). Here, we operate on Eq. (5) with $j_{x}\left ( x,t\right )=\int _{-\infty }^{+\infty }dyj_{x}\left ( \textbf {r},t\right )$ (i.e., by integrating over the coordinate y), where $j\left (\textbf {r},t \right )=j_{x}\left ( \textbf {r},t\right )\hat {e}_{x}+j_{y}\left ( \textbf {r},t\right )\hat {e}_{y}$, and $\hat {e}_{x}$, $\hat {e}_{y}$ are the unit vector along x and y directions. Taking scenario (i) as an example, we display the results of axial electronic currents for the $2p\sigma -2p\pi$ resonant excitation of HeH$^{2+}$ in Fig. 5(a) and for the $1s\sigma _{g}-2p\sigma _{u}$ resonant excitation of H$_{2}^{+}$ in Fig. 5(c). It can be seen that the direction of axial electron currents as a function of time remains the same between 5 $\tau$ and 5.25 $\tau$, while the direction changes when the time evolves from 5.25 $\tau$ to 5.5 $\tau$, then at time from 5.5 to 5.75 $\tau$, the direction remains unchanged. Finally, the reverse of direction occurs between t = 5.75 $\tau$ and t = 6 $\tau$. These phenomena can more clearly reflect the enhanced resonance. And it is consistent with the electron current direction shown in Fig. 3(b)-Fig. 4(b) (white arrow). In addition, the result under the scenario (ii) is also calculated (Fig. 5(b-d)). We can see that only at t= 5.5 $\tau$, the direction of axial electron current reverses, which is consitent with Fig. 3(d) and Fig. 4(d).

 

Fig. 5. Profiles of axial electronic currents $j_{x}\left ( x,t\right )$) as a function of position x and time t (in units of period $\tau$ = 350 as (refers to the optical period of the H$_{2}^{+}$ in the case of resonant excitation)). (a-b) The $j_{x}\left ( x,t\right )$ for resonant excitation and direct ionization of HeH$^{2+}$. (c-d) The $j_{x}\left ( x,t\right )$ for resonant excitation and direct ionization of H$_{2}^{+}$. Dotted white lines indicate positions of protons. Scale of magnitude and direction are shown at right.

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For further quantifying the CM process in the above two scenarios, such as the level of migration efficiency, we define the axial electronic yield along the molecular axis. That is, a change in the probability of finding an electron from an area of one nucleus position to another nucleus position during a period of 5$\tau$ $\leq$ t $\leq$ 6$\tau$. It can be expressed as $y_{x}=\int _{5\tau }^{6\tau }dtj\left ( x,t\right )$, where $j\left ( x,t\right )=\int _{-\infty }^{+\infty }dyj\left (\textbf {r},t \right )$ . We exhibit the axial electronic yield in Fig. 6.

 

Fig. 6. Axial electronic yield $y_{x}$ for two molecular ions. (a) The $y_{x}$ for resonant excitation of $2p\sigma -2p\pi$ (blue solid line) and $1s\sigma -2p\sigma$ (green solid line), direct ionization (orange solid line) in HeH$^{2+}$. (b) The $y_{x}$ for resonant excitation of $1s\sigma _{g}-2p\sigma _{u}$ (blue solid line ) and direct ionization (orange solid line) in H$_{2}^{+}$. The black arrows denote positions of protons.

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For HeH$^{2+}$, the maximum electronic yield (0.266 a.u.) of $2p\sigma -2p\pi$ resonant excitation is much higher than that (0.008 a.u.) of $1s\sigma -2p\sigma$ resonant excitation. They are both higher than that (0.004 a.u.) of direct ionization. These indicate that the maximum electronic yield of resonant excitation is higher that of direct ionization. This feature also applies to H$_{2}^{+}$, for which the yields of $1s\sigma _{g}-2p\sigma _{u}$ resonant excitation and direct ionization are 0.436 a.u. and 0.006 a.u.. Additionally, the electronic yield of HeH$^{2+}$ is different from that of H$_{2}^{+}$. On the one hand, the $y_{x}$ of H$_{2}^{+}$ has an maximum at x = 0, but for HeH$^{2+}$, it does not appear. This is because the spatial reflection symmetry or parity of the different nuclear charges in HeH$^{2+}$ with respect to the midpoint of the two nuclei is no longer a conserved quantity. On the other hand, the yields of H$_{2}^{+}$ is almost twice higher than that of HeH$^{2+}$ in the scenario (i), which also shows that this symmetric molecular ion has a better CM efficiency and is more likely to induce stronger electronic currents and magnetic fields than asymmetric HeH$^{2+}$ molecular ion.

Based on the above results, we can reasonably speculate that electronic currents induced by the excitation resonance ($1s\sigma _{g}-2p\sigma _{u}$ or $2p\sigma -2p\pi$ ) will be stronger than that induced by direct ionization. To further illustrate the physical mechanism about this phenomenon, the electronic currents are calculated, based on the principle of the continuity of the electronic current in quantum mechanics, that is, the probability of the measured electron in any cross-sections generated through the center point is equal [44]. Therefore, we can examine the dependence of the electronic currents on time by integrating $j\left ( \textbf {r},t\right )$ over the line along the y axis through the section -$\infty$ to 0. It is expressed by [44]

 

Fig. 7. The time-dependent electronic currents $J\left (t \right )$ and magnetic fields for two molecular ions. (a) and (c) show the $J\left (t \right )$ passing through the section -$\infty$ to 0 along y axis in equation (9) of HeH$^{2+}$ and H$_{2}^{+}$ respectively. The required wavelength of resonance excitation, intensity of the laser pulses are marked in the figure and at intensity $\ I_0=1\times {}{10}^{13} \ \mbox {W}/{\mbox {cm}^2}$. (b) and (d) represent magnetic fields $B\left (\textbf {r}=+R_{2},t \right )$ for HeH$^{2+}$, and $B\left (\textbf {r}=0,t \right )$ for H$_{2}^{+}$. Both magnetic fields $B\left (\textbf {r}, t\right )$ are perpendicular to the laser (x, y) polarization plane (shown in Fig. 1) by using laser pulses at $\lambda$ = 150 nm ($\omega$= 0.3 a.u.) (pink solid line) and $\lambda$ = 30 nm ($\omega$ = 1.5 a.u.) (black solid line) for HeH$^{2+}$, and at $\lambda$= 100 nm ($\omega$ = 0.455 a.u.) (pink solid line) and $\lambda$=30 nm ($\omega$=1.5 a.u.) (black solid line) for H$_{2}^{+}$. The durations are t=10 $\tau$, where $\tau$=2$\pi$/$\omega$ (i.e., 1 o.c. is 506.8 as for $\lambda$=150 nm) is fixed.

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4. Conclusion

In summary, we focused on the electronic coherence dynamics and explored the ultrafast oscillating electron current and corresponding magnetic field generated from CM for aligned symmetric molecular ion (H$_{2}^{+}$) and asymmetric molecular ion (HeH$^{2+}$) by numerically solving TDSE. The two scenarios were considered for HeH$^{2+}$ and H$_{2}^{+}$, including (i) resonance excitation and (ii) direct ionization. In scenario (i), the electronic density distribution and density distribution of electronic current exhibit the oscillating behavior as a function of time, which illustrates the strong electron coherence. Moreover, it can be known that the process of CM is very significant by resonance excitation ($1s\sigma _{g}-2p\sigma _{u}$ for H$_{2}^{+}$ or $2p\sigma -2p\pi$ for HeH$^{2+}$), which can better reflect the enhancement of the CM efficiency. And electronic current is not sensitive to the helicity of circular polarization of the laser pulses. In addition, the resonance excitation of HeH$^{2+}$ from the ground electronic state $1s\sigma$ was also studied. It was found that a weak magnetic field is generated and the electron current is sensitive to the helicity of circular polarization. However, in the scenario (ii), the CM is not significant, and the CM efficiency is low. Weaker magnetic fields are also obtained. By comparing the two scenarios, we found that in scenario (i), the electron current and magnetic field of resonance excitation for HeH$^{2+}$ from the initial first excited $2p\sigma$ electronic state and H$_{2}^{+}$ from the ground electronic state $1s\sigma _{g}$ are much stronger than that of ionization for HeH$^{2+}$ and H$_{2}^{+}$ in scenario (ii). Moreover, the magnetic field generated by H$_{2}^{+}$ is stronger than that by HeH$^{2+}$ through scenario (i). The generation of the ultrafast oscillating magnetic field provides possibility to detect and manipulate the ultrafast magnetic and chiral dynamics, and study electron dynamics in ultrafast magneto-optics. We expect the magnetic source with high oscillation frequency will be applied in other areas, such as the high-speed information processing and storage.