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Abstract
We theoretically investigate the photodissociation dynamics of H$_2^+$ using the methodology of ultrafast X-ray photoelectron diffraction (UXPD). We use a femtosecond infrared pulse to prompt a coherent excitation from the molecular vibrational state (v = 9) of the electronic ground state (1sσg) and then adopt another time-delayed attosecond X-ray pulse to probe the dynamical properties. We have calculated photoionization momentum distributions by solving the non-Born-Oppenheimer time-dependent Schrödinger equation (TDSE). We unambiguously identify the phenomena associated with the g − u symmetry breakdown in the time-resolved photoelectron diffraction spectra. Using the two-center interference model, we can determine the variation in nuclear spacing with high accuracy. In addition, we use a strong field approximation (SFA) model to interpret the UXPD profile, and the SFA simulations can reproduce the TDSE results in a quantitative way.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Rapid developments of advanced technologies for generating the specific tailored intense laser pulses have afforded a miraculous doorway that lead to profound insights into the nuclear motions in real time during transitions [1–3]. The ultrafast absorption and emission spectroscopic techniques have widely adopted in monitoring the chemical reactions [4,5], but are limited by their reliance on local chromophores and their associated ladders of quantum states rather than global structural characterization. Conventional X-ray scatting method already can reach few-femtosecond pulse duration [6,7] and has enhanced the temporal resolving power for probing the ultrafast molecular dynamics [8–10]. However, for gas-phase investigation, this approach has the coarse spatial resolution and the low scattering cross sections. Electron diffraction is complementary to X-ray scattering, but features much larger cross sections and smaller de Broglie wavelength than X-rays that allow the study of surface phenomena and molecular structures of gas phase samples [11,12]. Unfortunately, the electron scattering suffers from space charge broadening, which decreases the temporal resolution [13].
Other imaging techniques such as the light-induced electron diffraction (LIED) [14–25] and the ultrafast X-ray photoelectron diffraction (UXPD) [26–32] are also in progress. Different from the conventional electron beam method in which the scattered electrons come from an electron source off a molecular target, LIED and UXPD are the self-imaging method based on coherent electron scatting, where the geometric and orbital structure information of the molecule can be encoded in the photoionization momentum distributions (PMDs). However, the LIED process usually involves several recollision and diffraction events, which complicate the diffraction patterns. In the UXPD process, a photoelectron wave is emitted from the parent molecules directly, and then it interferes with the waves elastically scattered by neighboring atoms. Because of the high energy of the X-ray photons, the ionization process is rather fast and the rescattering events are greatly reduced. In addition, the UXPD cross sections are much larger than those for X-ray scattering. For these reasons, the UXPD is very suitable for detecting the molecular inner structure and monitoring the ultrafast dynamics.
Based on UXPD one can measure molecular geometry structure [28], can monitor the nuclear motion [33] and even can track dissociation and elimination process in chemical reactions [32]. Recent Born-Oppenheimer quantum simulations showed that UXPD can be used to measure phase and amplitudes of coherent electron wave packets in molecular charge migration [34,35]. However, in the electron transition process, the nuclear motion also gives rise to the dephasing of electron coherence and charge resonance-enhanced ionization [36,37]. In order to obtain deep physical insights into electron dynamics and molecular dynamics the non-Born-Oppenheimer quantum simulations are needed.
In this paper, we propose a pump-probe scheme based on UXPD to image the spatiotemporal coherent electron dynamics and the photochemical reaction. The simplest molecular ion H$_2^+$ as a benchmark model [38–40] is used to illustrate the dynamics. Initially, a coherent wave packet is generated by a circularly polarized (CP) 800 nm pulse from the molecular vibrational state ($v_i = 9$) of the electronic ground state ($1s\sigma _g$). And then, another CP soft X-ray pulse (5 nm) is introduced at different time delays to act as the probe pulse. The merit of using the CP laser field is that the electric field is uniformly distributed in radial direction and the initial molecular rotation effect can be ignored. The pump-probe process is simulated by numerically solving the non-Born-Oppenheimer time-dependent Schrödinger equation (TDSE). In addition, we introduced a strong field approximation (SFA) model to interpret the TDSE results [41]. Both the TDSE simulations and SFA results show that, the phase of the electron wave packet and the changes of internuclear distance can be measured simultaneously by using the proposed imaging technique. The rest of this paper is organized as follows: In Section 2, we briefly introduce the theoretical model and computational method. In Section 3, we present the numerical results and corresponding discussions. A short summary is given in Section 4.
2. Theoretical model and numerical methods
The three-body non-Born-Oppenheimer model of H$_2^+$ is shown in the inset of Fig. 1. The electron motion is restrained to the $x$-$z$ plane. Because the rotation of the molecule is neglected another dimension $R$ is used to depict the internuclear motion. For simplicity $R$ is considered along the $x$ axis. The propagation direction of the laser pulses is set to be perpendicular to the $x$-$z$ plane. Thus, the effect of the CP laser fields on H$_2^+$ is always confined on the $x$-$z$ plane over the whole time. In this coordinate system, the time dependent Schrödinger equation (TDSE) can be written as (atomic units are used unless stated otherwise):
The laser fields used in our simulation can be expressed as:
Fig. 1. The pump-probe process and potential energy curves for $1s\sigma _g$ and $2p\sigma _u$ electronic states within the Born-Oppenheimer approximation. The inset stands for the three-body 3D model ($R,x,z$) of H$_2^+$.
The 3D TDSE is solved by means of the splitting-operator fast-Fourier transform technique. At each propagation step, a strategy of imparity in the time scales of the nuclear and electronic motion is adopted to speed up the computer time [44]. The time evolution operator can be expressed as:
The sizes of the simulation box are 0.1 to 25 a.u. in $R$, -128 to 128 a.u. in the $x$ and $z$ directions. The grid points for these three coordinates are 256, 1024 and 1024, respectively. The time step of dynamical evolution is 0.01 fs. The electronic coordinates is updated $N$ times more often than the nuclear coordinates, and $N$ is of 20 in the simulation. The wavefunction is multiplied with a Gaussian shaped damping function at each time step to enforce the proper outgoing wave boundary condition. The absorption domains are $20$-$25$ a.u. and $120$-$128$ a.u. for $R$ and $|x, z|$, respectively. The total simulation time is 80 fs, where the wave packet on the excited state arrives at the asymptotic region and has not yet reached the boundary. After the probe field is expired, the ionization part is recorded as $[1-M(r_b)]\Psi$, where $M(r_b)$ is expressed in the $(x, z)$ space as [45,46]:
where $r=\sqrt {x^2+z^2}$ and the boundary for the wave function is set to be $r_b=20$ a.u. A Fourier operation is used to transform the ionized wavefunction from position space to momentum space. Finally, the PMD is obtained via $|\Psi (p)|^2$.In order to fully account for the ionization of H$_2^+$ ions in strong field, one has to solve the TDSE. However, to obtain more basic physical insight, the simple solvable models, like the SFA is vital [47]. In this treatment, the Coulomb attraction of the parent ion to the ionized electron is neglected, and the transition probability amplitude between a molecular bound state and a Volkov state can be written as: [48–51]
The relation between the phased Bessel function $X_n(Z)$ and the ordinary Bessel function $J_n(Z)$ is $X_n(Z)=J_n(|Z|)e^{in\phi }$, with $Z=|Z|e^{i\phi }$.
For circular polarization, the arguments in the generalized Bessel function $\chi _n(Z,\eta )$ are defined as:
In Eq. (8), $\varphi (p)$ is the electronic wave function in momentum space, and it is extracted from the first step of the TDSE simulation where only the pump field acts on the molecule. The molecule axis is set along the $x$ direction and the electron motion is restricted to the polarization plane. According to Eq. (8), the transition probability amplitude is calculated by setting $\theta =\pi /2$ and varying $\phi$ from 0 to 2$\pi$.
3. Results and discussions
The initial molecular wavefunction is constructed by using the Lanczos algorithm [52]. A wavelength of 800 nm CP laser pulse (18 cycle, duration of 48 fs) with a peak intensity of $I_1=3\times 10^{13}$W/cm$^2$ is used to excite the molecule. Given that the $v=9$ vibrational state of the electronic ground state $1s\sigma _g$ is resonance with the $2p\sigma _u$ electronic state at the 800 nm excitation [53–55]. The nuclear wave packet is mainly promoted to the $2p\sigma _u$ electronic state and then evolves in the dissociation channel. As shown in Fig. 2, most parts of the wave packet move to a large internuclear distance and yield molecular fragments. Such photodissociation process can be monitored with a time-delayed probe pulse. In the following, we perform an UXPD step to probe the electron and nucleus dynamics during the excitation-dissociation process.
Fig. 2. Nuclear probability density for the initial vibrational state $v_i=9$ under the influence of a 18-cycle CP laser pulse with 800 nm wavelength and peak intensity $I_0=3\times 10^{13}$W/cm$^2$.
The wavelength of the time-delayed X-ray pulse is set to be 5 nm. It is an appropriate value for observing the diffraction phenomenon in H$_2^+$ molecular ions [45,56]. The time duration lasts for 10 cycles (about 165 as), and the time propagation is continued for another 10 cycles to allow for some time relax. Here, the XPD is a single-photon ionization process, we can obtain the momentum of the ionized electrons on the basis of energy conservation law by $p_i=\sqrt {2(\omega _2-\varepsilon _b)}$. The estimated photoelectron momentum is about 4.05 a.u. When the pump and the probe fields interact with the molecule simultaneously, the PMDs are difficult to analyze. Thus, we use a filter function to multiply with the electron wavefunction in momentum space to exclude the influence of the pump field on PMDs. The step formed filter function is written as
with $p_r=\sqrt {p_x^2+p_z^2}$.Figure 3 shows the electron diffraction patterns for several delay times under the influence of the ultrashort CP X-ray pulse. As shown in the figure, clear diffraction fringes can be observed in the PMDs. The center position of these fringes in the radical direction is well compared with the value from the energy conservation law corresponding to the momentum region of the ejected electrons by single-photon ionization. The oscillatory behavior in the PMDs can be viewed as a microscopic version of Young’s double-slit experiment [57]. The emitted electrons, which are initially shared between the two indistinguishable nucleus, would be coherent when their de Broglie wavelengths are comparable to the molecular internuclear distance. Thus, when the electrons rescatter with different nuclei under the influence of the external field, the interference between the amplitudes arising from different scattering center would appear in the PMDs.
Fig. 3. Photoelectron diffraction patterns for different time delays. The optical cycle, wavelength and peak intensity of the probe pulse are 10, 5 nm and $5\times 10^{14}$W/cm$^2$, respectively.
Since the photoelectron diffraction patterns depend on the initial electronic state and encode the information of molecular orbital symmetries. The information which is associated with the $g-u$ symmetry breaking should be implied in the time-resolve PMDs. As shown in Fig. 3, the PMDs display slight asymmetry of the stripe structure about $p_y$ axis. In order to show it more clearly, Fig. 4 shows the $p_x$ dependence of PMDs by integrating the 2D PMDs over $p_y$. The SFA results are also shown in the figure. As expected, the trends of SFA curves are in well agreement with the TDSE results especially at small $p_x$ values. This further confirms the validity of the SFA model in the X-ray region. In addition, the local maximum observed in Fig. 4(a) transforms into the local minimum in Fig. 4(d) near the region of $p_x=0$. In the following, we show that such conversion can be directly connected with the $g-u$ symmetry breaking.
Fig. 4. The normalized photoelectron momentum distribution along the $p_x$ direction by integration over $p_y$ for several time delays. The black lines stand for the results of TDSE calculations and the red lines are calculated by the SFA model.
The $g/u$ symmetry stems from the phase difference of the electronic wave packet between the $1s\sigma _g$ and $2p\sigma _u$ electronic states. According to the LCAO theory, the phase-related molecular wave function of the two-center model in the momentum space can be expressed as [50,58]
The trigonometric functions describe the interference of the electronic wave packet, which stems from the double-core feature of molecules. Thus, at $R=0$, PMDs from $\sigma _g$ electrons and $\sigma _u$ electrons will display the construction interference and destructive interference respectively. Figures 4(a)-4(d) vividly show a $\pi$ phase shift in the interference patterns when the electron transits from the $1s\sigma _g$ to the $2p\sigma _u$ electronic state. This enables us to capture the phase of the electron wave packet during the electron transition process.
The change of the internuclear separation can affect the density of interference fringes. As shown in Figs. 3 and 4, the number of diffraction fringes increases with the delay time between the two laser pulses. The variation of diffraction patterns as the increase of the internuclear distance is one of the key points of UXPD, which allows us to image the bond breaking process in the photodissociation of H$_2^+$. Now, we will show how the changes in internuclear spacing can be mapped accurately in the time-dependent PMDs. The measured $R$ from the PMDs can be obtained in accordance with the classical Young’s double-slit experiment via
in which $\Delta p_R$ denotes the momentum separation between the adjacent interference peaks along the $R$ direction. Given that the $R$ axis coincides with the $x$ axis, $\Delta p_R$ is approximately equal to $\Delta p_x$. Thus, the $p_x$ dependence of PMDs can be used to extract the interference information. Moreover, since only the direction of the rescattered electrons is perpendicular to the molecular axis the PMD can have clear and equal spaced interference fringes. Under the action of the CP laser field, the direction of rescattered electrons may be messy. Thereby average value of the interference fringe spacing in the range of $-2.0<p_x<2.0$ a.u. is used to estimate the internuclear distance. The estimated internuclear distance as a function of the delay time is shown in Fig. 5. The center peak position (CPP) of the wave packet density of the dissociated part is also shown in the figure. The agreement with the estimated result and with the CPP result is quite good, which affirms the accuracy of the proposed probe scheme. However, at the delay time of $t_d$=20 fs, the estimated result deviates from the CP result significantly. The deviation can be attributed to the estimation error when extracting the value of $\Delta p_R$ from the time-dependent PMDs. As shown in Fig. 4(b), the main peak is distortion and even has a width wider than the case of Fig. 4(a). This makes the measured $R$ values to be much smaller than its true value at the delay time of 20 fs.Fig. 5. Estimated internuclear distance from photoelectron diffraction patterns as a function of the delay time $t_d$ (black quadrate line). The blue dashed line stands for the center peak position of the nuclear probability density.
4. Conclusions
In conclusion, we have theoretically reported that the electron transition and the bond breaking in the photodissociation dynamics of H$_2^+$ can be snapshotted via UXPD. The proposed pump-probe scheme includes a molecular resonance excitation procedure excited by an infrared CP laser pulse and a subsequent ionization step induced by a time-delayed CP soft X-ray pulse. The first step is implemented by solving the non-Born-Oppenheimer TDSE. In the second step, the momentum distributions of the emitted photoelectrons are calculated by using both the TDSE method and a SFA model. The obtained PMD curves from the two approaches are in good agreement with each other, which further confirms the validity of the SFA theory in the X-ray region. The time-resolved photoelectron diffraction spectra clearly display the breaking of the $g-u$ symmetry in the electron transition process. The estimated internuclear distance from the diffraction pattern of the PMDs reflects the center peak position of the dissociated wave packet with a high accuracy. Recent experimental advances in photoelectron holography [59] and ultrafast extreme ultraviolet coherent spectroscopy technique [60] have obtained the temporal resolution of several attoseconds, which can be applied to image the ultrafast dynamics of electrons to some extent. We hope that the presented scheme can provide guidance for experiments in improving the temporal and spatial resolutions.
Funding
National Natural Science Foundation of China (11704173, 11764041, 11874192); Taishan Scholar Foundation of Shandong Province (ts201511055).
Disclosures
The authors declare no conflicts of interest.
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