Direct Visualization of Valence Electron Motion Using Strong-Field Photoelectron Holography

Watching the valence electron move in molecules on its intrinsic timescale has been one of the central goals of attosecond science and it requires measurements with subatomic spatial and attosecond temporal resolutions. The time-resolved photoelectron holography in strong-field tunneling ionization holds the promise to access this realm. However, it remains to be a challenging task hitherto. Here we reveal how the information of valence electron motion is encoded in the hologram of the photoelectron momentum distribution (PEMD) and develop a novel approach of retrieval. As a demonstration, applying it to the PEMDs obtained by solving the time-dependent Schrödinger equation for the prototypical molecule ${\mathrm{H}}_2^{+}$, the attosecond charge migration is directly visualized with picometer spatial and attosecond temporal resolutions. Our method represents a general approach for monitoring attosecond charge migration in more complex polyatomic and biological molecules, which is one of the central tasks in the newly emerging attosecond chemistry.

Figure 1

(a) Photoelectron momentum distribution for tunneling ionization of the ground state ($1s{\sigma }_g$) ${\mathrm{H}}_2^{+}$ by a single-cycle pulse with wavelength of $\lambda =2000\mathrm{nm}$ and intensity of $3.0\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (b) Same as (a) but for tunneling ionization of the superposition state of ${\mathrm{H}}_2^{+}$ with ${\theta }_0=0$ and $c_2/c_1=0.5$.

Figure 2

Cuts of averaged PEMDs for the ground state (orange curve) and the superposition state with initial relative phase ${\theta }_0=0$ (green curve) at (a) $k_z=4.0$ and (b) $k_z=6.0\mathrm{a}.\mathrm{u}.$ (c) The phase difference $\mathrm{\Delta }{\phi }_{+,-}$ for $k_{\perp }>0$ (purple curve) and $k_{\perp }<0$ (blue curve) extracted from the cuts (green curve) in (a). The inset presents the quantity $\mathrm{\Delta }{\phi }_{+}-\mathrm{\Delta }{\phi }_{-}$ as a function of $|k_{\perp }|$. (d) Same as (c) but for the PEMD in (b).

Figure 3

(a) The transverse momentum distribution amplitude of $1s{\sigma }_g$ (orange) and $2p{\sigma }_u$ (green) states of ${\mathrm{H}}_2^{+}$, respectively. The magnitudes and phases are represented by solid (left ordinate) and dashed (right ordinate) curves, respectively. (b) The phase difference $\mathrm{\Delta }{\varphi }_{+}-\mathrm{\Delta }{\varphi }_{-}$ extracted from the PEMD in Fig. 1 at four different $k_z$. Dotted lines are the fitted results using the bases in (a). (c) The retrieved relative phase $\theta$ of the superposition state as a function of time (blue squares). The black line is the linear fitting of $\theta (t)$. The inset presents the electric field used in our calculations. The shadowed area indicates the time window where the electron dynamics is retrieved from the PEMDs. The error bars indicate the 95% confidence interval in fitting. (d) The reconstructed modulus (left) and phase (right) of electron wave functions at selected instants marked with solid squares in (c). The lower row of (d) shows the field-free evolution of the superposition state wave function for comparison.

Figure 4

(a) The phase difference $\mathrm{\Delta }{\varphi }_{+}-\mathrm{\Delta }{\varphi }_{-}$ extracted from the PEMDs (as presented in the Supplemental Material [26], Fig. S2) for $k_z$ ranging from 4.3 to 6.2 a.u. The upper and lower panels show the results for ${\theta }_0=0$ and $\pi$, respectively. Here the superposition state of ${\mathrm{H}}_2^{+}$ with populations $c_1:c_2=4:1$. (b) Example cuts from the upper panel of (a) for $k_z=4.5$, 5.6, 5.8, and 6.1 a.u. (c) Back row, electron density distributions in coordinate space as a function of time. The corresponding peak position of electron density distribution at the longitudinal tunneling exit (marked as blue dotted line) is presented as blue cycles in the front row. (d) The position of the oscillating valence electron wave packet retrieved from PEMDs for both ${\theta }_0=0$ (green circles) and $\pi$ (orange circles). The peak position of electron density distribution at the longitudinal tunneling exit obtained from (c) as a function of time is indicated by the blue solid curve for comparison. The inset presents the electric field used in our calculations. The shadowed area indicates the time window where the electron dynamics is retrieved from the PEMDs. The error bars indicate the 95% confidence interval in fitting.