1. Introduction

Charge migration is one of the most basic processes in intense laser-matter interactions, and it plays a fundamental role in chemical reactions and biological processes [1–4]. Monitoring this process on its natural attosecond timescale is essential for understanding and controlling the electronic behavior in chemical reactions and biological processes. Usually, charge migration is initiated by the strong-field excitation and ionization [5–7]. When an electron is suddenly stripped from a neutral molecule, the residual will be in a superposition of several electronic states. The time evolution of the superposition state is called charge migration [8]. The natural time scale of the charge migration depends on the composition of the superposition state. It typically ranges from hundreds of attoseconds to a few femtoseconds [9]. Therefore, probing the charge migration requires the attosecond temporal resolutions. Recently, rapidly developing technologies, such as transient absorption spectroscopy, attosecond streaking and attosecond photoelectron interferometry have facilitated the measurement with the unprecedented temporal resolution [10–15], which provides the opportunity to monitor the charge migration.

As an alternative, the measurement based on the laser-induced electron re-collision in strong-field ionization can also achieve the attosecond resolution. It has been widely used to investigate the ultrafast dynamics in molecules [16–20]. For example, using advanced high-order harmonic spectroscopy, previous studies have measured the charge migration experimentally [21–23]. Recently, laser-induced electron diffraction has been reported to investigate the nuclear dynamics with subatomic and femtosecond resolutions [24,25]. Particularly, by tracing the phase-dependent photoelectron diffraction with a circularly polarized laser pulse [26,27], its application in probing the charge migration has been explored. Very recently, another re-collision-based measurement, strong-field photoelectron holography (SFPH) was proposed to survey the valance-shell electronic dynamics in molecules [28]. The transient location of the electron is imprinted on the phase of the transverse momentum amplitude through the strong-field tunneling ionization. By analyzing the phase distribution, it has been demonstrated the time-dependent location of the electron can be monitored. This observation is in real-time, and the method can be generalized to the complex molecules [29]. Note that the application of this method requires the phase distribution of the hologram. Strong-field photoelectron holographic structure is derived from the coherent superposition of the electron wave packets (EWPs) reaching to the detector directly after tunneling and that undergoing a forward rescattering with the target [30]. In the photoelectron momentum distributions (PEMDs), the holographic pattern is observable for the momentum region near the polarization axis of the laser pulse. While for the region away from the field-polarization axis, the forward-scattering electrons drop. The carpet-like interference between the direct electrons dominates the spectrum [31]. In experiments, the fork-like hologram is not even clear due to the laser focal volume effect [32]. This will hinder the extraction of the phase distribution from the hologram. Therefore, achieving the observation of the electron motion in molecules remains a challenging task.

SFPH is believed as a powerful tool for probing atomic/molecular structure and electronic dynamics information, and it encodes the information in the interference fringes through the transmission of the EWP. In the past years, SFPH has been extensively investigated [33–38]. Recently, it has been reported that the atomic/molecular structure information, i.e. the phase of the elastic scattering amplitude, can be extracted by examining the holographic structure [39]. The time and initial velocity information of the tunneling ionization process can be retrieved accurately with SFPH [40–43]. Most importantly, SFPH owns the attosecond temporal resolution. Thus, it can be utilized to probe the electronic dynamics in molecules. Here, we explore its application in probing the charge migration in molecules. In our study, the PEMDs from tunneling ionization of the molecules are obtained by numerically solving the time-dependent Schrödinger equation, from which the holographic interference fringes are extracted. For a non-stationary superposition state, it is shown that the holographic interference fringes in PEMDs shift significantly along the lateral direction, when the molecule is aligned perpendicular to the linearly polarized laser pulse. With the quantum-orbit analysis, we show that this shift is caused by the time evolution of the electron probability density of the superposition state. By analyzing the dependence of the shift on the parallel momentum of the electrons, the time-dependent phase and expansion coefficient ratio of the two electronic states involved in the superposition state can be retrieved accurately. The electron motion in molecules is thus monitored. The robustness of the scheme is confirmed by its application to the molecules of $\rm {H_2^{+}}$ and $\rm {HeH^{2+}}$. The obtained result shows that the observation of the electron motion does not require the phase distribution from the hologram, instead, just by tracing the shift of the central maximum of the holographic interference in PEMDs, which can be easy to achieve in experiments. Thus, our work provides an efficient approach to probe the charge migration in molecules with SFPH, which will encourage the application in experiments.

2. Theoretical method

2.1 Numerically solving time-dependent Schrödinger equation

Strong-field tunneling ionization of the molecules is investigated by numerically solving the two-dimensional time-dependent Schrödinger equation (TDSE) of $i\partial \Psi (\mathbf {r},t)/\partial t=\mathrm {H}(\mathbf {r},t)\Psi (\mathbf {r},t)$, where $\Psi (\mathbf {r},t)$ is composed by the electronic states of $\Psi (\mathbf {r},t)=c_g\Psi _g(\mathbf {r})e^{-iE_gt/\hbar }+c_e\Psi _e(\mathbf {r})e^{-iE_et/\hbar +\phi _0}$. $\Psi _g(\mathbf {r})$ and $\Psi _e(\mathbf {r})$ indicate the ground and the first excited states of the molecules with the eigen-energies of $E_g$ and $E_e$, respectively. $c_g$ and $c_e$ denote the expansion coefficients. $\phi =|\Delta E|t/\hbar +\phi _0$ is the relative phase between these two states. The variation period of the superposition state is determined by $\tau =2\pi /|\Delta E|$ with $\Delta E=E_g-E_e$. Experimentally, $\Psi (\mathbf {r},t)$ can be prepared by the excitation of the ground state with an ultrashort pump pulse. $\phi _0$ is adjustable by changing the pump-probe delay [26,27].

In the TDSE, the Hamiltonian $\mathrm {H}(\mathbf {r},t)$ in length gauge is written as $\mathrm {H}(\mathbf {r},t)=-\frac {1}{2}\nabla ^2+V(\mathbf {r})+\mathbf {r}\cdot \mathbf {E}(t)$ with the soft-core potential of $V(\mathbf {r})$. Here, $V(x,y)=-1/\sqrt {x^2+(y-R_1/2)^2+0.69}-1/\sqrt {x^2+(y+R_1/2)^2+0.69}$ is adopted to produce the ground ($E_g=-0.80$ a.u.) and first excited states ($E_e=-0.66$ a.u.) with the same angular momentum as $\mathrm {H_2^+}$. $V(x,y)=-2/\sqrt {x^2+(y-R_2/2)^2+3} -1/\sqrt {x^2+(y+R_2/2)^2+3}$ is utilized to generate the ground ($E_g=-0.88$ a.u.) and first excited states ($E_e=-0.66$ a.u.) of $\mathrm {HeH^{2+}}$. The internuclear distances of the mimic $\mathrm {H_2^+}$ and $\mathrm {HeH^{2+}}$ are set to $R_1=4$ a.u. and $R_2=6$ a.u., respectively. The fixed-nuclei approximation is considered here, since that the electronics dynamics that we focus on occur within short intervals of several hundred attoseconds. The few-cycle laser pulse with the electric field of $\mathbf {E}(t)=-f(t)E_0\sin (\omega t)\hat {x}$ is employed to initiate the ionization. $f(t)=sin^2(\pi t/T_f)$ is the envelope function of the laser pulse with the duration of $T_f=3T$. $E_0$ and $\omega =2\pi /T$ indicate the amplitude and angular frequency of the laser field, respectively.

To solve the TDSE, we use the imaginary time propagation of the field-free system to prepare the initial wave function [44]. It is then propagated on a Cartesian grid, ranging from $-500$ a.u. to $500$ a.u., with the split-operator spectral method [45]. The spatial discretization is $\Delta x=\Delta y=0.2$ a.u. and the time step is fixed at $\Delta t=0.06$ a.u. After the end of the laser pulse, the wave function is further propagated for four optical periods of the laser field to ensure that all the ionized wave function is collected. The PEMD is eventually calculated by Fourier transform of the ionized part, which is obtained by filtering out the bounded part of the total wave function with the splitting function [46,47].

2.2 Quantum-orbit model

In the PEMDs, the forward-scattering holographic structure comes from the coherent superposition of the direct and forward rescattering EWPs, and it is described by $M^2(\mathbf {p})=|M_d(\mathbf {p})|^2+|M_r(\mathbf {p})|^2+2|M_d(\mathbf {p})||M_r(\mathbf {p})|\cos [\Delta \Phi (\mathbf {p})]$. Here, $|M_d(\mathbf {p})|$ and $|M_r(\mathbf {p})|$ are the transition amplitudes of the direct and rescattering EWPs, respectively. $\Delta \Phi (\mathbf {p})$ denotes their phase difference, which is given by

With the quantum-orbit (QO) analysis [48–50], the phase difference of $\Delta \Phi _F$ can be calculated by

3. Result and discussion

A 1000 nm few-cycle laser pulse with the intensity of $2.5\times 10^{14}$ W/cm$^2$ is used to ionize $\mathrm {H_2^+}$. The laser field is linearly polarized along the $x$ axis and the molecule is aligned to the laser polarization direction with the angle of $\pi /2$. The obtained PEMDs for strong-field tunneling ionization of the ground and superposition states are presented in Figs. 1(a) and 1(b), where the expansion coefficient ratio of the two electronic states involved in the superposition state is $c_e/c_g=0.2$. The time-dependent phase is $\phi =0.14t+\phi _0$ with $\phi _0=0$. The parameters serve as an example to illustrate our scheme, which is not restricted to a specific value. It is shown that several kinds of interference patterns appear in these PEMDs. The annular structure visible in the low momentum region is the intercycle interference, known as the above-threshold ionization peaks. It arises from the interference of the direct EWPs tunneling ionized from different laser cycles. The nearly vertical fringes are visible on the $p_x$ axis, which is referred as the intracycle interference. It results from the interference of the direct EWPs ionized during the adjacent quarter cycles of the laser pulse [51–53]. The fork-like fringes in the PEMDs are the forward-scattering holographic interference. It originates from the interference of the tunneled EWPs reaching to the detector directly and that undergoing a nearly forward re-collision with the parent ion [30]. The holographic interference encodes the information of the electron dynamics within molecules and it is the most pronounced interference pattern in the PEMDs for the near-infrared and mid-infrared laser pulses [54,55]. In real experiments, the holographic interference fringes still exist, although the other interference structures are often wiped out due to the laser focal volume effect. Therefore, we will focus on this interference pattern below.

 

Fig. 1. (a) PEMD for strong-field tunneling ionization of the ground state ($1s\sigma _g$) $\mathrm {H_2^+}$ in a few-cycle pulse. (b) Same as (a) but for the superposition state with the initial phase of $\phi _0=0$ and expansion coefficient ratio of $c_e/c_g=0.2$. The color bars are logarithm scaled. (c) Several cuts of (a) at: (A) $p_x=-1.2$ a.u., (B) $p_x=-0.9$ a.u., (C) $p_x=-0.6$ a.u. and (D) $p_x=-0.3$ a.u. (d) Same as (c) but extracted from (b). For a better view, these cuts are normalized such that the maximum is unity. The dashed line indicates the location of the central maximum of the holographic interference fringes.

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A closer inspection of Figs. 1(a) and 1(b) shows that there is an interesting difference between the holographic structures. For Fig. 1(a), it is observed that the holographic interference is exactly symmetric about $p_y=0$. While for Fig. 1(b), the interference fringes are asymmetric. More obviously, the difference can be seen from Figs. 1(c) and 1(d), where we wash out the vertical interference fringes by averaging the PEMDs with a window function [39], and present several cuts of the obtained PEMDs at different $p_x$. It is shown that for the ground state, the holographic interference fringes always maximize at $p_y=0$, while for the non-stationary superposition state, the interference minimum/maximum is shifted towards $-p_y$ direction. This shift gradually varies with $p_x$, and the phenomenon agrees with the recent study [28], where the asymmetry holographic interference has been pointed out. Here, as we will demonstrate, this shift of the holographic interference is closely related to the time-dependent superposition state. It can enable one to monitor the ultrafast electron motion in the molecule, by examining the locations of the holographic interference fringes. In our work, the PEMDs are symmetric about $p_x=0$. Thus, we take the hologram in $p_x<0$ as an example for analysis in the following.

To survey the shift of the holographic interference for tunneling ionization of $\mathrm {H_2^+}$, we trace the time evolution of the electron density distribution for the superposition state. By taking a series of snapshots on the electron probability density in the coordinate space, the results for the instants of $t=1.25T,1.35T,1.45T$ and $1.55T$ ($1T=137.9$ a.u. $=3.3$ $fs$) are presented in Figs. 2(a1)–2(a4), respectively. It is shown that for the superposition state composed by the ground $1s\sigma _g$ ($|g>$) and the first excited states $2p\sigma _u$ ($|e>$), the electron probability density changes on an attosecond time scale. More clearly, the time-dependent localized electron populations of $P_U(t)=<U|\psi (t)>^2$ and $P_D(t)=<D|\psi (t)>^2$ are presented in Fig. 2(b), where $|D>=(|g>-|e>)/\sqrt {2}$ and $|U>=(|g>+|e>)/\sqrt {2}$ represent the electron localized at each molecular center. It is shown that the populations periodically oscillate with the time $t$, implying that the EWP moves back and forth between the two molecular centers. The oscillation period of the superposition state reads $\tau =0.32T=44.8$ a.u.. It is determined by the eigen-energies of the ground and the first excited states through $\tau =2\pi /|\Delta E|$. For $t$ ranging from $1.38T$ to $1.54T$, $P_D(t)>P_U(t)$ indicates that the EWP is closer to the molecular center in $p_y<0$. For $t\in [1.22T,1.38T]$, $P_D(t)<P_U(t)$ shows that the EWP approaches the other molecular center.

 

Fig. 2. (a) Several snapshots of the electron probability density for the superposition state at (a1) $t=1.25T$, (a2) $t=1.35T$, (a3) $t=1.45T$ and (a4) $t=1.55T$. The color coding is on a logarithmic scale. (b) Time-dependent populations of the electron localized at each molecular center.

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Figure 2 illustrates the electron dynamic process inside the molecule. In strong-field tunneling ionization, the few-cycle laser pulse polarized along $x$ axis is employed to ionize the molecule, as shown in Fig. 3(a). Subsequently, the electrons are released from the parent ion at the falling edge of the laser field. Accelerated by the oscillating electric field of the laser pulse, some of the electrons re-collide with the molecular geometric center, interfering with those without re-collision. It results in the holographic pattern in PEMDs. For the atom, the electrons are usually emitted with zero initial displacement in the $y$ direction, since the tunneling ionization is limited along the direction of the instantaneous electric field. While for the molecule, the initial lateral displacement of the electron is affected by the electronic structure and the molecular alignment [56]. For our laser parameters, the longitudinal tunneling exit of the EWP in tunneling ionization estimated by -$I_p/E(t)$ is about $10-15$ a.u. Therefore, in Figs. 3(b) and 3(c) we monitor the electron probability density at $x = 15$ a.u. for $t=1.35T$ and $1.45T$, respectively. The dashed and solid curves indicate the amplitudes of the electron wave function for the ground and superposition states, respectively. It is observed that for the ground state, the electron wave function is exactly symmetric about the $x$ axis, and its maximum is located at $y = 0$. While for the superposition state, the wave function is asymmetric. The maximum of the electron wave function shifts towards $y$ direction for $t=1.35T$ and shifts towards $-y$ direction for $t=1.45T$. The results in Figs. 3(b) and 3(c) show the transverse electron probability density of the ground state and the superposition state immediately after tunneling ionization, where the location of the maximum determines the initial lateral displacement of the tunneling EWP [57]. For the ground state, it is obtained that the electron is released from the aligned molecule with a zero lateral displacement, while for the superposition state, there is a nonzero initial lateral displacement. Due to the time evolution of the superposition state, the lateral displacement periodically varies with the time $t$, as shown in Figs. 3(d) and 3(e). In strong-field tunneling ionization, the PEMD could be approximately considered as the Fourier transform of the tunneled EWP in position space [58,59]. According to the delay theorem of Fourier transformation, a shift of the position for the EWP corresponds to a phase in the momentum distribution. Therefore, considering the tunneling ionization in the position space, a nonzero initial lateral displacement of the EWP can induce an additional phase in the transverse momentum distribution. This phase encoded in the hologram of the PEMD leads to a shift of the holographic interference.

 

Fig. 3. (a) The electric field of the few-cycle laser pulse (cyan curve), and its vector potential (orange curve). (b) The electron probability density of the ground state (dashed curve) and superposition state (solid curve) at the cut of $x=15$ a.u for the moment of $t=1.35T$. (c) Same as (b) but for $t=1.45T$. The results of (b) and (c) are normalized such that the maximum is unity. (d) The time evolution of the transverse electron probability density for the ground state of $\mathrm {H_2^+}$. (e) Same as (d) but for the superposition state.

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Quantitatively, to reveal the origin of the shift of the holographic interference we employ the QO analysis. In the QO model (as demonstrated in Sec. 2.2), the holographic structure in the PEMDs is well described by the phase difference (Eq. (1)) between the rescattering and direct electrons in strong-field tunneling ionization [28,39]. For the near-forward rescattering holographic pattern, the interference fringes exist in the momentum range of $|p_y| < 0.3$ a.u. and $p_x\in [-1.4,-0.3]$ a.u. In this region, the direct electron is ionized by the laser field with an initial lateral momentum close to $p_y$. For the near-forward rescattering electron, it tunnels with a small initial lateral momentum of $k_y$, and gets the final lateral momentum $p_y$ through a re-collision. In the laser polarization direction, the electron momentum is almost unchanged during the “soft re-collision” process, because the near-forward rescattering electron interacts weakly with the target. Thus, it allows for the assumption of $p_x\doteq k_x$. With the result, it is obtained $t_0^d\approx t_0^r$ through Eqs. (3) and (4). The approximations have been confirmed by the recent calculations in Ref. [41,60]. Thus, they are reasonable. In the following, we will use $t_0$ to represent the ionization times of both the direct and rescattering electrons.

Further, the phase difference of $\Delta \Phi _F$ in Eq. (1) can be expanded as

Equation (10) indicates that the shift of the holographic interference for the superposition state is induced by the nonzero initial lateral displacement of the tunneling EWP, and it relates to the time difference between the re-collision and ionization of the electron. The re-collision and ionization times can be obtained from Ref. [41], which agrees with the QO model. The Coulomb interaction between the parent ion and the electron influences the locations of the interference fringes, but when we consider the relative shift of the interference fringes, it can be safely eliminated with a “reference molecule”. Therefore, we can focus on the central maximum of the holographic interference, which is visible in experiments. By examining the shift of the holographic interference for the superposition state relative to that for the ground state, the nonzero initial lateral displacement of the tunneling EWP can be retrieved with Eq. (10). The lateral displacement reflects the lateral location of the maximum of the electron probability density at the longitudinal tunneling exit. In PEMDs, the horizontal axis $p_x$ can be transferred to the ionization time through the saddle-point equation of Eq. (3). When the charge migration occurs in molecules, the electron probability density changes with time, and the initial lateral displacement of the EWP will change correspondingly. By tracing the momentum-dependent shift of the holographic interference with our method and treating $p_x$ as the time axis, the time-dependent charge migration in molecules can thus be monitored. It should be mentioned that when the tunneling ionization is initiated by the multiple-cycle laser pulse, the electrons ionized from different laser periods can contribute to the same final momentum, and thus the dynamical information of the electrons within a single period will be obscured in the holographic interference. So, the few-cycle pulse with stable carrier-envelope phase is required in our scheme [61,62].

To demonstrate the retrieval of the electron dynamics information encoded in the hologram, we extract the holographic interference fringes from the PEMDs. With the procedure introduced in [39], we wash out the vertical fringes and eliminate the envelope of the PEMDs induced by the ionization amplitude of the electrons. The interference term of $\cos [\Delta \Phi (\mathbf {p})]$ for the hologram is thus obtained. Two examples of $\cos [\Delta \Phi (\mathbf {p})]$ for the superposition state with $\phi _0=0$ and $\phi _0=\pi$ are presented in Figs. 4(a) and 4(b), respectively. Clearly, the fork-like holographic interference and its shift can be observed. Then, we trace the central maximum of the holographic interference fringes for each $p_x$, and record its location. The shift of the holographic interference for the superposition state with $\phi _0=0$ and $\phi _0=\pi$ relative to that of the ground state is obtained, as shown in Figs. 4(c) and 4(d). For comparison, the results of the first maximum and the first minimum of the holographic interference are presented in Figs. 4(c) and 4(d) as well. It is shown that the relative shift of $\Delta p_y$ gradually varies with $p_x$. For different interference fringes, the variations are the same. This confirms the accuracy of Eq. (10), which enables one can take the central maximum of the holographic interference fringe as an example to analyze. The re-collision and ionization times obtained from the QO method are the injective function of the final parallel momentum of the rescattering electron, as presented in Figs. 4(e) and 4(f). Based on the correspondence, the lateral position $y_0^s$ is therefore retrieved through Eq. (10), as shown by the dotted curves in Fig. 4(g).

 

Fig. 4. (a) and (b) The extracted interference term of $\cos (\Delta \Phi )$ in the range of $p_x\in [-1.4, -0.3]$ a.u. for the superposition state with $\phi _0=0$ and $\pi$, respectively. (c) and (d) The shift of the holographic interference for the superposition state with $\phi _0=0$ and $\phi _0=\pi$ relative to that of the ground state, respectively. The diamonds, circles and solid curves indicate the results of the first maximum ($1^{max}$), the central maximum ($0^{max}$) and the first minimum ($1^{min}$) of the holographic interference fringes, respectively. (e) and (f) The ionization and rescattering times of the electron, which are scaled by the laser period of $T$. (g) The circles represent $y_0^s$ extracted from the hologram. The dashed curve indicates the time-dependence peak position of the electron probability density at $x=15$ a.u. for the superposition state.

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To check the accuracy, we refer to the electron probability density of the superposition state. Monitoring the time evolution of the electron probability density at the longitudinal tunneling exit of $x=15$ a.u., the location of the maximum is displayed in Fig. 4(g) by the dashed curve. The peak location of the transverse electron probability density shows the initial lateral displacement of the tunneling EWP in molecules [57]. The retrieved $y_0^s$ nicely agrees with the dashed curve, indicating that the lateral launching displacement of the EWP is retrieved accurately. It should be mentioned that the laser field may affect the displacement of $y_0^s$. Here, the agreement in Fig. 4(g) implies that $y_0^s$ is insensitive to the laser field. The conclusion is consistent with the other molecules reported in experiments [19,21]. Further, we repeat our extraction for the superposition state with $\phi _0=0$, $\pi /2$ and $\pi$, as presented in Fig. 4(g). It is shown that the cosine like $y_0^s$ changes from the maximum to the minimum with a time interval of $\tau$/2 = 0.16T = 22.4 a.u. (from 1.3T to 1.46T), and it maximizes at $t=1.30$T with the amplitude of $Y_0^s=1.9$ a.u. The time-dependent $y_0^s$ is caused by the periodic changes of the electron density distribution. Thus, the dotted curve in Fig. 4(g) exactly provides the time-dependent phase of $\phi =2\pi \cdot t/\tau =0.14t$ for the superposition state.

Additionally, we can establish a map between the amplitude of $Y_0^s$ and the expansion coefficient ratio of $c_e/c_g$ for the superposition state. By monitoring the time-dependent peak location of the electron probability density at each $x$, the amplitude of the result for different $c_e/c_g$ is presented in Fig. 5(a). For the cut of $x=15$ a.u. [shown in Fig. 5(b)], one can see that $Y_0^s$ gradually increases with $c_e/c_g$. This illustrates the effect of the expansion coefficient ratio on the electron probability density of the superposition state, which allows one to retrieve $c_e/c_g$, by seeking $Y_0^s$ from the hologram and comparing it with the map of Fig. 5(b). With this point, we extract $Y_0^s$ from the interference shift at $p_x=1.2$ a.u. for tunneling ionization of the superposition state. The obtained result is displayed in Fig. 5(b), where the diamond is well located on the solid curve, confirming that the expansion coefficient ratio of $c_e/c_g=0.2$ is determined. With the retrieved time-dependent phase and the expansion coefficient ratio of the electronic states involved in the superposition state, the electron motion in molecules is monitored. The wave function of the superposition state can be reconstructed. Therefore, our work establishes a method to probe the charge migration in molecules with the SFPH.

 

Fig. 5. (a) $Y_0^s$ as a function of $c_e/c_g$ for different $x$. (b) The solid curve indicates the cut of (a) at $x=15$ a.u. The diamond indicates $Y_0^s$ extracted from the holographic interference.

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Moreover, we apply the method to survey the electronic dynamic process in the asymmetric molecule of $\mathrm {HeH^{2+}}$. For the aligned $\mathrm {H_2^+}$, the electron probability density of the ground state is symmetric about $x$ axis, while for $\mathrm {HeH^{2+}}$, it is not. Interestingly, our scheme still works well, as long as the holographic interference is observable in the PEMDs. We use the same laser field as Fig. 3(a) to ionize $\mathrm {HeH^{2+}}$. The obtained PEMDs for the superposition state with $\phi _0=-\pi /2$ and $\pi /2$ are presented in Figs. 6(a) and 6(b), respectively. Then, the ground state of $\mathrm {HeH^{2+}}$ is employed as a reference, and it is exposed to the same laser pulse as the superposition state. With the procedure, the central maximum of the holographic interference is traced, and the locations of the interference fringes are obtained, as presented in Fig. 6(c). It is shown that for the superposition state with the expansion coefficient ratio of $c_e/c_g=0.2$, the central maximum of the holographic interference indicated by the dots and diamonds is obviously shifted along the $p_y$ direction, due to the nonzero initial lateral position of the tunneling EWP. Different from $\mathrm {H_2^+}$, the interference fringes for the ground state of $\mathrm {HeH^{2+}}$ indicated by the dashed curve are also shifted towards $p_y$ direction. This results from the asymmetric electron probability density of the ground state $\mathrm {HeH^{2+}}$, which leads to the electron released from the molecule with a nonzero initial lateral displacement of $y_0^g$. To calculate the relative shift $\Delta p_y$ of the holographic interference, we subtract the data of the ground state with that of the superposition state. The obtained results are presented in Fig. 6(d), where $\Delta p_y$ gradually varies with $p_x$. Considering this relationship and converting $p_x$ to the time axis, the relative lateral position $\Delta y_0=\Delta y_0^s-\Delta y_0^g$ of the tunneling EWP is eventually obtained by Eq. (10). The result is shown by the dotted curve in Fig. 6(e), where $\Delta y_0$ indicates the initial displacement of the EWP launching from the superposition state with respect to the ground state in the direction perpendicular to the laser polarization. For comparison, we also trace the time-dependent electron probability density for the superposition state at the tunneling exit. The data is shifted along $y$ direction with $y_0^g$. The peak position of the electron probability density at $x=15$ a.u. is presented in Fig. 6(e) by the dashed curve. The dotted curve agrees excellently with the dashed curve, indicating that the initial lateral displacement of the tunneling EWP is retrieved accurately. It is shown that $\Delta y_0$ periodically oscillates over $t$ with a period of $\tau /2$ = 0.20T = 28.5 a.u. (from 1.25T to 1.35T). It indicates the time-dependent phase of $\phi =2\pi \cdot t/\tau =0.22t$ for the superposition state. The cosine-like oscillation of $\Delta y_0$ maximizes at $t=1.25$T with the amplitude of $\Delta Y_0=3.0$ a.u. It provides the expansion coefficient ratio of $c_e/c_g=0.2$ for the electronic states involved in the superposition state, by comparing the data with the map of $\Delta Y_0$ in Figs. 6(f) and 6(g). Thus, the movement of the EWP in the asymmetric molecule of $\mathrm {HeH^{2+}}$ is monitored accurately.

 

Fig. 6. (a) and (b) The interference term for the superposition state of $\mathrm {HeH^{2+}}$ composed by the ground and the first excited states with $\phi _0=-\pi /2$ and $\pi /2$, respectively. (c) The location of the central maximum of the holographic interference for the ground (dashed curve) and superposition states with $\phi _0=-\pi /2$ (diamonds) and $\phi _0=\pi /2$ (circles). (d) The central maximum shift of the holographic interference for the superposition state with $\phi _0=-\pi /2$ (diamonds) and $\pi /2$ (circles) relative to that for the ground state. (e) The time-dependent peak position of the electron probability density at $x=15$ a.u. for the superposition state, which is shifted along $y$ direction with $y_0^g$ (the dashed curve ). The dotted curve denotes the $\Delta y_0$ extracted from the hologram with Eq. (10). (f) $\Delta Y_0$ at different $c_e/c_g$. It is obtained by monitoring the time-dependent $\Delta y_0$ in the electron probability density of the superposition state and recording its amplitude for each $x$. (g) The cut of (f) at $x=15$ a.u. The circle indicates $\Delta Y_0$ extracted from the holographic interference.

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Charge migration in molecules is of fundamental importance in intense laser-matter interactions and it has attracted considerable interest during the past years. Here, using SFPH we have demonstrated a scheme to probe the charge migration. In our approach, when the charge migration occurs in molecules, the electron probability density varies with time, resulting in the time-dependent initial lateral displacement of the tunneling EWP. This displacement is uniquely mapped to the phase of the transverse momentum amplitude in strong-field tunneling ionization, and it then manifests as the shift of the holographic fringes in PEMDs. By examining the momentum-dependent shift of the hologram and converting the final parallel momentum of the electron to the ionization time, we illustrate that the charge migration in molecules can be monitored. Our work provides an efficient approach for observing the ultrafast electron motion in molecules. Recently, the imprint of the coupled electronic and nuclear dynamics in the hologram has been reported [35]. For more complex molecules, the hologram in PEMDs has been observed in experiments successfully [63]. Thus, our study will encourage one to apply the SFPH to investigate the electronic dynamics in complex molecules as well as the coupled nuclear and electronic dynamics.

4. Conclusion

In summary, by numerically solving TDSE we have investigated the PEMDs in strong-field tunneling ionization of molecules, from which the holographic interference is exacted. For the non-stationary superposition state, it is shown that the holographic interference distinctly shifts along the lateral direction, when the molecule is aligned perpendicular to the laser polarization direction. With QO analysis, we reveal that this shift is caused by the time evolution of the electron density distribution of the superposition state. Considering the “reference molecule” method, the shift of holographic interference for the superposition state relative to the ground state is acquired, where the complicated effect of the Coulomb interaction can be safely counteracted. By analyzing the dependence of the relative shift on the parallel momentum of the electrons, we demonstrate that the time-dependent phase and the expansion coefficient ratio of the electronic states involved in the superposition state can be retrieved accurately. The electron motion in molecules is thus monitored. The validity of our method is confirmed by its application to the molecules of $\rm {H_2^{+}}$ and $\rm {HeH^{2+}}$. The obtained result shows that the electron motion in molecules can be traced, just by examining the central maximum shift of the holographic interference in PEMDs, which can be easily achieved in experiments. Therefore, our work provides an efficient approach to probe the charge migration with the concept of SFPH. It will encourage the investigation of the electronic dynamics in complex molecules.