Imaging electronic motions in atoms by energy-resolved ultrafast electron diffraction

We propose energy-resolved ultrafast electron diffraction as a means of directly imaging target electronic motions whose space, time, and energy information can be simultaneously retrieved from time-resolved diffraction measurements. The energy-resolved diffraction images are simulated for breathing, wiggling, and hybrid modes of electronic motion in the H atom. The simulations demonstrate the capabilities of ultrafast electron diffraction to image and distinguish different kinds of electronic motion. The theoretical analysis of the scattering process identifies the requirements for time- and state-resolved imaging of electronic motion and provides interpretations of the results.

Figure 1

Schematic illustration of the experimental setup for energy-resolved ultrafast electron diffraction. The target electronic motions are initiated by the laser pulse, whereupon these motions are imaged by ultrafast (attosecond) electron pulses with controlled pump-probe delay time. The velocity analyzer determines the kinetic energies of the scattered electrons, and the detector records the time-resolved spectra. By rotating the velocity analyzer, angle-resolved spectra can be measured. By varying the pump-probe delay time, the resulting energy- and time-resolved diffraction images exhibit the target electronic motions. The coordinate system and the scattering angles $\theta$ and $\phi$ are defined here.

Figure 2

Electron densities of the breathing (left column) and the wiggling (right column) modes of electronic motion in the H atom as a function of time. The breathing mode superposes equal-parity $3p$ and $4p$ states with equal amplitude, while the wiggling mode consists of opposite-parity $3d$ and $4f$ states. Both kinds of motion share the same beat period, $\mathsf{\text{T}}$ = 6.25 fs. Owing to the periodicity, only the densities in the first half period are shown.

Figure 3

Electron pulse profiles as functions of kinetic energy $E_0$ and beam angular divergence ${\theta }_0$ for the pulse durations [full width at half maximum (FWHM)] (a) 100 as, (b) 500 as, and (c) 3.13 fs. The profiles peak at the central energy of 9.99 keV in the center-of-mass frame of the colliding electron and the H atom, while in the laboratory frame the central pulse energy is 10 keV. The momentum amplitude $a_0\left({\mathit{k}}_0\right)$ of each pulse is assumed to have a Gaussian shape whose longitudinal width is determined by the pulse duration. The angular divergence is $\pm {10}^{-4}$ rad. The contours in the bottom plane show the full width at half maximum of each pulse profile.

Figure 4

Doubly differential probabilities $d^2\mathcal{P}/d{E}_{a}d{\widehat{\mathit{k}}}_a$ for scattering of ultrafast electrons from the $3p+4p$ coherent superposition state of the H atom at pump-probe delay times $t_d=0\mathsf{T},\mathsf{\text{T}}$/4, and $\mathsf{\text{T}}$/2 are shown for three different pulse durations. The beat period of the electronic motion is $\mathsf{T}=6.25$ fs. The pulse durations (FWHM) are 100 as (left column), 500 as (middle column), and 3.13 fs (right column). The beam angular divergence is $\pm {10}^{-4}$ rad. (FWHM). See Fig. 3 for the pulse profiles. The electronic state equally superposes equal-parity $3p$ and $4p$ states, showing a breathing mode of motion (cf. Fig. 2). The azimuthal scattering angle is set at $\phi =0^{\circ }$ (cf. Fig. 1).

Figure 5

Relative variation of the differential scattering probability $d^2\mathcal{P}/d{E}_{a}d{\widehat{\mathit{k}}}_a$ at $t_d=\mathsf{T}/2$ with respect to $t_d=0$ for the three pulse durations as functions of $E_a$ and $\theta$. See Eq. (32) for the definition of the relative variation.

Figure 6

Differential probabilities $d\mathcal{P}/d{\widehat{\mathit{k}}}_a$ of a 500-as (FWHM) electron pulse for scattering from the $3p+4p$ coherent state of the H atom at the pump-probe delay times $t_d=0\mathsf{T},\mathsf{\text{T}}$/4, and $\mathsf{\text{T}}$/2. Differential probabilities are obtained without differentiating the energy of the scattered electron. Left column shows the scattering images of the de-excitation transition to the $1s$ state, while the right column corresponds to the scattering images excluding the $1s$ transition. Owing to symmetry, only the upper scattering images are shown.

Figure 7

(a) Doubly differential probability of a 500-as (FWHM) electron pulse scattered from the wiggling mode of electronic motion in the H atom at delay times $t_d=0\mathsf{T}$. (b),(c) The differences of the doubly differential probabilities at (b) $t_d=\mathsf{T}/4$ and (c) $3\mathsf{\text{T}}$/4 with respect to the one at zero delay time in (a) [cf. Eq. (33)]. The wiggling electronic motion is produced by synthesizing the opposite-parity $3d$ and $4f$ states with equal amplitude (cf. Fig. 2). Right and left columns correspond to the spectra at azimuthal scattering angle $\phi =0^{\circ }$ (positive $z$ axis) and ${180}^{\circ }$ (negative $z$ axis) (cf. Fig. 1); their comparison shows the asymmetry. The spectra of (b) and (c) above 10 keV are multiplied by a factor $5\times {10}^5$ to accentuate the variation.

Figure 8

Left column: Differential probability $d{\mathcal{P}}_{1s}/d{\widehat{\mathit{k}}}_a$ of transition to the $1s$ state for scattering of a 500-as electron pulse from the $3d+4f$ coherent state of the H atom at pump-probe delay times $t_d=0\mathsf{T},\mathsf{\text{T}}$/4, $\mathsf{\text{T}}$/2, and $3\mathsf{\text{T}}$/4. Due to symmetry, only the upper diffraction patterns are shown. Right column: Differential probability for four scattering angles $\theta$ at three delay times as a function of azimuthal scattering angle $\phi$.

Figure 9

(a)–(d) Differential scattering probabilities $d\mathcal{P}/d{\widehat{\mathit{k}}}_a$ of the 100-as (left column) and the 500-as (right column) electron pulses from a hybrid mode of electronic motion at scattering angles $\theta =0.5^{\circ }$ and $1.0^{\circ }$ at $t_d=0\mathsf{T},\mathsf{\text{T}}$/4, and $\mathsf{\text{T}}$/2 as a function of azimuthal scattering angle $\phi$. (e),(f) The asymmetry of $d\mathcal{P}/d{\widehat{\mathit{k}}}_a$ at delay time $t_d=\mathsf{T}/8\approx 782$ as. The hybrid mode consists of slow breathing and rapid wiggling motions, comprised of 90% population in the $1s$ state, 5% populations in the odd-parity $3p$ and $4p$ states, and zero relative phase in their amplitudes $c_n$ at time zero. See Eq. (34) for the definition of the asymmetry.

Figure 10

(a),(b) Real and imaginary parts of the modified wave function ${\varphi }_{\mathrm{coh}}^{\prime }(\mathit{k},t)$ of the target coherent state along the $z$ axis as a function of time. (c) Time-dependent momentum distribution $|{\varphi }_{\mathrm{coh}}{(\mathit{k},t)|}^2$ along the $z$ axis. The opposite-parity $3d$ and $4f$ states are equally superposed in the coherent state ${\varphi }_{\mathrm{coh}}(\mathit{k},t)$. The modified wave function ${\varphi }_{\mathrm{coh}}^{\prime }(\mathit{k},t)$ retains only the relative phase between the $3d$ and $4f$ states. The real parts of ${\varphi }_{\mathrm{coh}}^{\prime }(\mathit{k},t)$ at $t=0\mathsf{T}$ (red) and $\mathsf{\text{T}}$/2 (blue) are identical, as are the imaginary parts of ${\varphi }_{\mathrm{coh}}^{\prime }(\mathit{k},t)$ at $t=\mathsf{T}/4$ (green) and $3\mathsf{\text{T}}$/4 (black).

Figure 11

Imaginary parts of the modified coherent momentum wave function ${\varphi }_{\mathrm{coh}}^{\prime }(\mathit{k},t)$ as a function of time and the momentum space wave function of the $1s$ state of the H atom, with both plotted vs $k_z$. The coherent state consists of $3p$ and $4p$ states of the H atom with equal amplitude; only the relative phase between the $3p$ and $4p$ states is kept in the modified wave function. For display purposes, the $1s$ wave function is multiplied by a factor of 10.