Coherent scattering of an optically-modulated electron beam by atoms
1 Coherent scattering of an optically-modulated electron beam by atoms
Yuya Morimoto *, Peter Hommelhoff and Lars Bojer Madsen Laser Physics, Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Staudtstraße 1, 91058, Erlangen, Germany Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark *[email protected] † [email protected] Dated: January 10, 2021.
Recent technological advances allowed the coherent optical manipulation of high-energy electron wavepackets with attosecond precision. Here we theoretically investigate the collision of optically-modulated pulsed electron beams with atomic targets and reveal a quantum interference associated with different momentum components of the incident broadband electron pulse, which coherently modulates both the elastic and inelastic scattering cross sections. We show that the quantum interference has a high spatial sensitivity at the level of Angstroms, offering potential applications in high-resolution ultrafast electron microscopy. Our findings are rationalized by a simple model.
When a free electron interacts with an optical field in the presence of a third body, the electron exchanges energy and momentum with the field [1-12]. The photon-electron coupling in the vicinity of nanomaterials forms the basis of photon-induced near-field electron microscopy [5] and chip-scale dielectric laser accelerators [13–15]. Because electrons of different kinetic energies travel with different velocities, the temporal density of the optically-modulated beam is reshaped during the propagation in vacuum, leading to the generation of attosecond electron pulses [16–23]. It has been predicted that when an optically-modulated electron beam is employed for the excitation of a two-level system located outside the beam, the excitation probability may be enhanced when the transition energy matches an integer times the photon energy of the modulating laser beam [24–27]. Recent studies showed that this process can be described by the classical electric and magnetic fields associated with the temporal density of the modulated beam [25,26]. In this work, we investigate the scattering of an optically-modulated high-energy electron wavepacket by an atomic target. By using a fully quantum mechanical theory modified from [28–31], which is beyond the standard plane wave approximation, and taking atomic hydrogen as an example, we show that a quantum interference occurs through the coherent contributions of different momentum components of the incident beam. This interference modulates both the elastic and inelastic cross sections by more than 30%. We also show that the strongest modulation of cross sections is induced when the target atom is located at the center of the electron beam, and the sign of the modulation is reversed when the target is several Angstroms away from the beam center, which can be used for sub-nanometer imaging. We hence expect potential applications in attosecond imaging, atomic collisions or radiology based on the insights into the fundamental quantum mechanical interaction between the optically-shaped electrons and atoms, as laid out in this work The physical system of this work is illustrated in Fig. 1(a). A non-relativistic electron wavepacket propagating along z -axis with group velocity 𝒗 𝑒 and central longitudinal momentum ℏ𝒌 𝑒 is coherently accelerated or decelerated by an optical field of wavelength λ and angular frequency ω , where ℏ is the reduced Planck constant. The strength of the optical modulation is characterized by a dimensionless coupling parameter 𝑔 ∝ 𝑒𝐹 𝑘 𝑒 /(𝑚 𝑒 𝜔 ) , where 𝑒 is the unit charge, 𝐹 the optical electric field amplitude and 𝑚 𝑒 the electron rest mass [8,9,32,33]. Larger | 𝑔 | yields broader energy and momentum spectra. The modulated electron wavepacket collides with a target atom located at 𝑧 = 0 where the electron beam is transversally focused. We express the target position as 𝒃 = (𝑏 𝑥 , 𝑏 𝑦 , 0) . We control the distance between the modulation stage and the target, 𝐿 𝑝 = 𝑣 𝑒 𝑡 𝑝 , where 𝑡 𝑝 is the corresponding propagation duration. Because different energy and momentum components have different phase velocities, the free-space propagation shifts the relative phase between them, which reshapes the temporal density of the electron wavepacket [34]. The momenta of the incident and the scattered electrons, ℏ𝒌 𝑖 and ℏ𝒌 𝑓 , are described by their lengths ℏ𝑘 𝑖 and ℏ𝑘 𝑓 , and polar and azimuthal angles ( 𝜃 𝑖 , 𝜑 𝑖 ) and (𝜃 𝑓 , 𝜑 𝑓 ). To consider the scattering of the broadband electron beam including its spatial structure, we adopt a time-dependent S-matrix formalism [29,31] with electron wavepackets [28,30], which is beyond the standard theory using a plane wave for the asymptotic incoming state. Within the first Born approximation, the 𝒃 - and 𝑔 -dependent total scattering probability from the initial target state characterized by the quantum number 𝑛 to the final target state characterized by 𝑚 is given by 𝑃 𝑚𝑛 (𝒃, 𝑔) = ∫|𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃, 𝑔)| 𝑑𝒌̂ 𝑓 , (1) with |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃, 𝑔)| = 𝑃 ∫ 𝑘 𝑖3 𝑑𝑘 𝑖 ∬ 𝑑𝒌̂ 𝑖 ∬ 𝑑𝒌̂ 𝑖′ × 𝑎 𝑒∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ , 𝑔)𝑎 𝑒 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝑔) 𝑇 𝑚𝑛∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ , 𝒌̂ 𝑓 )𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 )𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 , (2) where 𝒌̂ 𝑗 (𝑗 = 𝑖, 𝑓) is the unit vector along 𝒌 𝑗 , 𝑑𝒌̂ 𝑗 = sin 𝜃 𝑗 𝑑𝜃 𝑗 𝑑𝜑 𝑗 , 𝑃 is a constant, 𝑎 𝑒 is the three-dimensional momentum distribution of the incident electron beam (see below) and 𝑇 𝑚𝑛 is the atomic form factor for electron scattering given by the Fourier transform of the electron-target interaction potential [35]. In deriving Eqs. (1) and (2), we neglected exchange scattering, which is accurate for the high-energy (> 1 keV [36]) electron beams considered in this work. Furthermore, we assumed that the target atom is well localized in real space, see [34] for the full derivation and explicit expressions for 𝑇 𝑚𝑛 . Equation (2) contains the coherent term 𝑎 𝑒∗ 𝑎 𝑒 , which gives rise to quantum interference. Importantly this term is absent in conventional electron scattering but key in the present case of optically-modulated electron beams. For the optically-modulated electron beam, we consider an axially-symmetric beam whose momentum distribution is expressed by 𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 , 𝑔) = 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 , 𝑔)𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 )e 𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑖 ,𝑡 𝑝 ) . (3) The transversal momentum distribution 𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) is given by 𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) = 1(2𝜋𝜎 ⊥2 ) exp (− 𝑘 ⊥2 ⊥2 ), (4) where 𝑘 ⊥ = 𝑘 𝑖 sin 𝜃 𝑖 and ℏ𝜎 ⊥ is the root-mean-square (rms) transversal momentum width. For simplicity, we introduce the angular width 𝜎 𝜃 = 𝜎 ⊥ /𝑘 𝑒 . The optically-modulated longitudinal momentum distribution 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 , 𝑔) is expressed as a superposition of Gaussians of slightly different central momentum ℏ𝑘 𝑒 +𝑁ℏ𝛿𝑘 associated with the absorbed photon number N [9,27,32] 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 , 𝑔) = 1(2𝜋𝜎 ∥2 ) ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ exp (− (𝑘 ∥ − 𝑘 𝑒 − 𝑁𝛿𝑘) ∥2 ), (5) where 𝑘 ∥ = 𝑘 𝑖 cos 𝜃 𝑖 , ℏ 𝜎 ∥ is the rms width, 𝐽 𝑁 is the Bessel function of the first kind, ℏ𝛿𝑘 is the momentum shift corresponding to the one-photon energy gain, 𝛿𝑘 ≅ 𝑚 𝑒 𝜔/(ℏ𝑘 𝑒 ) and negative 𝑁 corresponds to the emission of photons. The phase 𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 ) in Eq. (3) represents the momentum-dependent phase shift due to the free-space propagation of duration of 𝑡 𝑝 from the optical modulation to the target, and is given by 𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 ) = 𝑡 𝑝 (𝑣 𝑒 𝑘 ∥ − ℏ𝑘 ∥2 /(2𝑚 𝑒 )) [34]. The free-space propagation reshapes the real-space density of the electron beam. An interesting case is the density bunching into attosecond pulses occurring at 𝑡 bunch = ℏ𝑘 𝑒2 /(2|𝑔|𝑚 𝑒 𝜔 ) = 𝑚 𝑒 /(2|𝑔|ℏ 𝛿𝑘 ) [19,37]. Below, for convenience, 𝑡 𝑝 is expressed in units of 𝑡 bunch . We assume a 10-keV electron beam with 𝜎 ∥ corresponding to a duration of 100 fs (full width at half maximum, FWHM) similar to an experiment [9], λ = 2 μm [21,22], and atomic hydrogen in 1s state as target, unless otherwise specified. Below, we discuss the amount of modulation of the scattering probability defined as 𝑀 𝑚𝑛 ( 𝒃, 𝑔) = 𝑃 𝑚𝑛 ( 𝒃, 𝑔 ) − 𝑃 𝑚𝑛 ( 𝒃, 𝑔 = 0 ) 𝑃 𝑚𝑛 ( 𝒃, 𝑔 = 0 ) . (6) Hence, we compare the scattering probability of the optically-shaped electron wavepackets to that without shaping. Figure 1(b) shows the total cross sections without optical modulation 𝑃 𝑚𝑛 (𝒃, 𝑔 = 0) of the elastic and inelastic (2s and 2p final states) scatterings as a function of the impact parameter 𝑏 ⊥ = √𝑏 𝑥2 + 𝑏 𝑦2 calculated at 𝜎 𝜃 = 𝑏 ⊥ , by a factor of 10 at 𝑏 ⊥ = 10 nm. Therefore, we first focus on the case of 𝑏 ⊥ = 0 , where the target is located at the center of the electron beam. Figure 2(a) compares the modulation, 𝑀 𝑚𝑛 ( 𝒃 = 0, 𝑔) , for the three processes calculated with = 5 and 𝜎 𝜃 =
5 mrad, as a function of the propagation time 𝑡 𝑝 . We observe a clear modulation; ~20% suppression at 𝑡 𝑝 ~1.7𝑡 bunch and ~10% enhancement at 𝑡 𝑝 ~3.8𝑡 bunch . All the three curves are qualitatively identical, suggesting that the modulation and interference appearing in Fig. 2(a) originate from the incident electron beam rather than the scattering processes. Equation (2) shows that different 𝒌̂ 𝑖 and 𝒌̂ 𝑖′ contribute to the same final momentum. Hence quantum interference occurs when the momentum components with different incident angles ( 𝜃 𝑖 , 𝜑 𝑖 ) and ( 𝜃 𝑖′ , 𝜑 𝑖′ ) contribute to the cross section at the same scattering angle ( 𝜃 𝑓 , 𝜑 𝑓 ). A wider angular distribution therefore leads to stronger interference. As shown in Fig. 2(b), the dipole-allowed transition of 1s→2p is dominated by forward scattering and shows a narrower angular distribution, which reduces the modulation contrast in Fig. 2(a). We now investigate the modulation dependence on the focusing angle of the electron beam ( 𝜎 𝜃 ). Figure 2(c) compares the modulation of the total elastic cross section (1s→1s) calculated with four different angular widths, 𝜎 𝜃 =
1, 3, 5 and 7 mrad. We observe less than 3% modulation at 𝜎 𝜃 ≤ |𝑀 𝑚𝑛 ( 𝒃 = 0, 𝑔)| at 𝑡 𝑝 ~1.7𝑡 bunch . The amplitudes increase exponentially up to 𝜎 𝜃 ≈ 𝜃 𝑖 and 𝜃 𝑖′ , leading to the stronger modulation. We also investigate the dependence on the optical coupling strength |𝑔| . We plot and compare 𝑀 𝑚𝑛 ( 𝒃 = 0, 𝑔) for the three cases of , 2, and 5 as red circles in Figs. 3(a)-(c) at 𝜎 𝜃 = [Fig. 3(a)] shows a sinusoidal oscillation while the other two [Figs. 3(b),(c)] show non-sinusoidal shapes, suggesting that the modulation of can be described by a single sinusoidal function, however those of and 5 contain multiple contributions. At and 2, we observe only negative modulations ( 𝑀 𝑚𝑛 ( 𝒃 = 0, 𝑔) ≤ 0) while at , we observe a positive modulation as well. We note that |𝑔| -dependent modulations are also found in the inelastic scattering channels (not shown). In order to understand the above results, we invoke a simple target-independent model. By considering the limit of very small 𝜎 ∥ ( 𝜎 ∥ ≪ 𝛿𝑘 ), by assuming 𝑇 𝑚𝑛 = 1 (uniform scatterer) and by taking sets of (𝑘 𝑖 , 𝜃 𝑖 , 𝜑 𝑖 , 𝜃 𝑖′ , 𝜑 𝑖′ ) giving dominant contributions, we approximately obtain the following simplification of Eq. (2) |𝑇 𝑓𝑖model (𝒃, 𝑔)| = 𝑃 ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ ∑ 𝐽 𝑁 ′ (2|𝑔|) +∞𝑁 ′ =−∞ exp (− |𝑁 − 𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) × 𝑒 −𝑖(𝑁 −𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 𝑏 ⊥ ) , (7) where 𝑃 is a constant and 𝜔 𝛿𝑘 = ℏ𝛿𝑘 /(2𝑚 𝑒 ) = 1/(4|𝑔|𝑡 bunch ) [34]. The numerical results of Eq. (7) are plotted in Figs. 2(d) and 3(a)-(c) as black curves. Even though there is no free parameter in Eq. (7), all the curves reproduce the results of the full simulations surprisingly well. Equation (7) shows that the quantum interference and the modulation of the scattering probabilities can be described by the combinations of two different photon-exchange channels with amplitude weights 𝐽 𝑁 and 𝐽 𝑁 ′ . Because of the symmetry of the Bessel function, 𝐽 −𝑁 (𝑥) = (−1) 𝑁 𝐽 𝑁 (𝑥) , most combinations of 𝐽 𝑁 and 𝐽 𝑁′ vanish after the sum over 𝑁 and 𝑁′ except for |𝑁 − 𝑁 ′ | = 0,2,4 … Because the terms of |𝑁 − 𝑁 ′ | = 0 are independent of 𝑡 𝑝 , the modulations seen in Figs. 2 and 3 are given by the terms satisfying |𝑁 − 𝑁 ′ | = 2,4, … . The strength of the coupling is determined by the exponential term exp (−|𝑁 − 𝑁 ′ |𝛿𝑘/(2𝑘 𝑒 𝜎 𝜃2 )) , showing that a larger difference between 𝑁 and 𝑁′ gives a smaller contribution. The black dotted curve in Fig. 2(d) shows this exponential term with |𝑁 − 𝑁 ′ | = 2 and is in good agreement with the full simulations (red circles). The deviation at large 𝜎 𝜃 is due to contributions from |𝑁 − 𝑁 ′ | ≥ 4 . In order to explain the observed oscillations of 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) in Fig. 3, we consider the combinations of ( 𝑁, 𝑁 ′ ) yielding the dominant effects. At [Fig. 3(a)], we find them to be ( 𝑁, 𝑁 ′ ) = (2,0), (−2,0), (0,2) and (0, −2). The other combinations vanish or give negligibly small contributions [34]. Equation (7) shows that the phase associated with the free-space propagation is proportional to 𝑁 − 𝑁 ′2 . The above combinations give 𝑁 − 𝑁 ′2 = 4 or −4 . Using 𝑒 𝑖𝑥 + 𝑒 −𝑖𝑥 = 2 cos 𝑥 , where 𝑥 is a real number, 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) within the model of Eq. (7) is reduced to the form of −𝐴 + 𝐴 cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) with 𝐴 > 0 , which explains the sinusoidal oscillation observed in Fig. 3(a) and 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) ≤ 0 at any 𝑡 𝑝 . When we apply the same discussion to the cases of and 5 [Figs. 3(b) and (c), respectively], we find that 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) is simplified to the forms of −𝐵 − 𝐵 + 𝐵 cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐵 cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) and 𝐶 −𝐶 − 𝐶 − 𝐶 cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐶 cos(16𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐶 cos(20𝜔 𝛿𝑘 𝑡 𝑝 ), respectively, with real positive numbers 𝐵 𝑙 , 𝐶 𝑙 > 0 ( 𝑙 = 1,2, … ) , expressed in terms of Bessel functions [34]. The largest frequencies 4 𝜔 𝛿𝑘 , 𝜔 𝛿𝑘 and 𝛿𝑘 with 𝜔 𝛿𝑘 = 1/(4|𝑔|𝑡 bunch ), for , 2 and 5, respectively, suggest the appearance of the first negative peaks at 𝑡 𝑝 /𝑡 bunch = 𝜋/2 = 1.6 for all three cases. In the case of [Fig. 3(b)], the signs of the two cosine functions are both positive, which give 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) ≤ 0 . On the other hand, in the case of [Fig. 3(c)], the term cos (8𝜔 𝛿𝑘 𝑡 𝑝 ) has a negative coefficient, which leads to 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) > 0 . The frequency 𝛿𝑘 suggests the positive peak appearing at 𝑡 𝑝 /𝑡 bunch = 5𝜋/4 =3.9 [34] which agrees with the full simulation. Finally, we consider the modulation of the scattering probabilities at non-zero 𝑏 ⊥ , i.e., for a target displaced from the focus of the electron beam. According to Eq. (2), the parameter 𝒃 induces a phase term of 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 . Its physical interpretation is illustrated in Fig. 4(a) for the case of 𝒃 = (𝑏 𝑥 , 0, 0) . For an electron wave with an angle 𝜃 𝑖 , one can find the difference in the geometrical path length from the source as compared to the case of 𝒃 = 0 (green circle without filling), which is given by 𝑏 𝑥 sin 𝜃 𝑖 = 𝒌 ̂ 𝑖 ∙ 𝒃 , shown by the red arrow. This path difference corresponds to a phase shift of 𝒌 𝑖 ∙ 𝒃 . We note that the same discussion is applied to electron diffraction by molecules in which case the path length difference occurs in the scattered waves. The sign and the magnitude of the coherent term 𝑎 𝑒∗ 𝑎 𝑒 is now modified by the relative phase for 𝒌 𝑖 and 𝒌 𝑖′ , that is 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 in Eq. (2). The simulation results 𝑀 𝑚𝑛 ( 𝒃, 𝑔) for the elastic scattering using Eqs. (1)-(5) are shown in Fig. 4(d) with red filled circles. As in Figs. 2(a) and 3(c), we choose , 𝜎 𝜃 =
5 mrad and 𝑡 𝑝 = 1.5𝑡 bunch , which gives a negative modulation at 𝑏 ⊥ = 0. We observe an oscillation with 𝑏 ⊥ ; the strongest negative peak at 𝑏 ⊥ = 0 , the highest positive peak at around 𝑏 ⊥ = 0.7 nm and the second negative peak at around 𝑏 ⊥ =1.5 nm. Nearly the same oscillations are observed for 1s→2s (black open squares) and 1s→ 2p (blue diamonds). We therefore set 𝑇 𝑚𝑛 = 1 (uniform scatterer) and simulate the dependence both on 𝑏 𝑥 and 𝑏 𝑦 . The result shown in Fig. 4(b) shows a circular pattern. The vertical slice at 𝑏 𝑦 = 0 is shown in Fig. 4(d) as the green curve, well reproducing the results of 1s→1s (red circles) and 1s→2s (black open squares). On the other hand, at 𝑡 𝑝 = 4𝑡 bunch , which gives a positive modulation at 𝑏 ⊥ = 0 (see Fig. 3(c)), we obtain the result shown in Fig. 4(c). The radii of the circular patterns are nearly identical to those in Fig. 4(b), but the sign of the modulation is opposite. In both cases, the incoherent averaging over 𝑏 𝑥 and 𝑏 𝑦 gives net zero modulation [34]. We now investigate how the impact parameter dependence is scaled with the optical coupling strength |𝑔| and wavelength λ . To this end, we define the parameter 𝑏 which is the minimum impact parameter giving 𝑀 𝑚𝑛 ( 𝒃, 𝑔) = 0 , see Fig. 4(d). The simulated dependences for |𝑔| and λ are plotted as green circles in Figs. 4(e) and (f), respectively. While there is almost no dependence of 𝑏 on |𝑔| [Fig. 4(e)], we observe a monotonic increase with λ [Fig. 4(f)]. To understand these results, we return to the simple model and Eq (7). In Eq. (7), the 𝒃 dependence is given solely by the term 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 𝑏 ⊥ ) , which is independent of |𝑔| . By using 𝐽 (𝑥) = 0 at 𝑥 = 2.4 and by recalling that the dominant contribution stems from |𝑁 − 𝑁 ′ | = 2 , the model predicts 𝑏 = 2.4/√4𝛿𝑘𝑘 𝑒 = 0.42 nm at 𝜆 = 𝛿𝑘 = 𝑚 𝑒 𝜔/(ℏ𝑘 𝑒 ) ∝ 1/𝜆 gives 𝑏 ∝ √𝜆 . In Fig. 4(f), the √𝜆 dependence (black curve) reproduces well the simulated results (green circles) at 𝜆 ≤ 5 µm. The deviation between the exact results and the model at large 𝜆 , i.e., small 𝛿𝑘 , is caused by the consideration of just the dominant contributions in the simple model [34]. The wavelength dependence can also be understood from Eq. (5) which suggests that for small wavelength, i.e., larger 𝛿𝑘 , a wider range of 𝜃 𝑖 is required to cover different N components. A larger 𝜃 gives a longer path length difference [Fig. 4(a)], giving stronger impact parameter dependence and smaller 𝑏 . The dependence of 𝛿𝑘 ∝ 1/𝑘 e suggests a weak dependence on the central velocity of the electron beam [34]. In this work, we have investigated the scattering of an optically-modulated electron wavepacket by an atomic target. By virtue of the spatial focusing, the discrete longitudinal momentum components couple with each other via the scattering process and the coherent interference results in a modulation of the scattering probability. Stronger modulations were observed in elastic scattering and dipole-forbidden inelastic scatterings than in dipole-allowed inelastic scattering, which might be applied to electronic state-selective excitation. The scattering probability modulation has a strong impact parameter dependence. The largest enhancement (suppression) is predicted for a target at the center of the electron beam while the suppression (enhancement) occurs for a target only a few Angstroms away. Combined with the ability to control the spatial dependence with the optical wavelength, the quantum interference reported here might facilitate spatially selective excitation or probe of, for example, optically trapped atoms or two-dimensional solids, or even provide an opportunity towards damage-reduced microscopy and high-resolution imaging with attosecond electron pulses.
Acknowledgement
This work is supported by the Gordon and Betty Moore Foundation (GBMF) through Grant No. GBMF4744 “Accelerator on a Chip International Program-ACHIP”, the FAU Emerging Talents Initiative and the Danish Council for Independent Research (GrantNo.9040-00001B). [1] N. M. Kroll and K. M. Watson,
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Yuya Morimoto , Peter Hommelhoff and Lars Bojer Madsen Laser Physics, Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Staudtstraße 1, 91058, Erlangen, Germany Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
In this supplemental material, we provide a derivation of Eqs. (1), (2) and (7) of the main text. In Sec. A1, we outline the theoretical formulation. In Sec. A2, we discuss the incident electron wavepacket. We explore several characteristics of the formulation including a discussion of the propagation phase, aspects of the modulation at zero propagation time, and the limit of plane wave scattering. We introduce the high-energy approximation, which significantly reduces computational time. In Sec. A3, we discuss a simple model, which assumes a uniform scatterer and hence highlights the physics related to the optical modulation of the electron beam. We discuss in detail the interference associated with different photon-exchange channels of the optically modulated beams. A1. BASIC THEORY
Here we consider the scattering of an electron wavepacket by an atom (A): 𝑒(𝒌 𝑖 ) + A(𝒌 𝐴,𝑖 , 𝑛) → 𝑒(𝒌 𝑓 ) + A(𝒌 𝐴,𝑓 , 𝑚), (S1) where 𝒌 𝑖 , 𝒌 𝑓 are wavevectors of the electron before and after the scattering, respectively, and 𝒌 𝐴,𝑖 and 𝒌 𝐴,𝑓 are wavevectors of the target before and after the scattering, respectively. The incident electron is assumed to be non-relativistic. The quantum numbers n and m represent the initial and final electronic states of the target atom. Because the contribution from exchange scattering is negligibly small for the high-energy electrons considered in this work, it suffices to account for the direct scattering. Below, we derive a scattering probability [Eqs. (1)-(2) of the main text] based on the full quantum mechanical framework of time-dependent S-matrix formalism. The fundamental part of this theory is based on the work by Shao and Starace [29,31]. Entrance channel (initial state)
We first describe the wavefunction of the system before the scattering, which is given by Ψ 𝑖 (𝒙 𝑒 , 𝒙 𝐴 , 𝒓, 𝑡) = ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝜒 𝑖 (𝒙 𝑒 , 𝑡)𝜒 𝐴,𝑖 (𝒙 𝐴 , 𝑡)𝜓 𝑛 (𝒓, 𝑡), (S1) where 𝑎 𝑒 (𝒌 𝑖 ) and 𝑎 𝐴 (𝒌 𝐴,𝑖 ) are (complex) amplitudes describing the distributions over the momenta of the projectile electron and the target atom, respectively. 𝜒 𝑖 (𝒙 𝑒 , 𝑡) is the plane wave part of the electron wavefunction, 𝜒 𝑖 (𝒙 𝑒 , 𝑡) = 1(2𝜋) exp (𝑖𝒌 𝑖 ∙ 𝒙 𝑒 − 𝑖𝐸 𝑖 𝑡ℏ ) , (S2) with 𝐸 𝑖 = ℏ 𝑘 𝑖2 /(2𝑚 𝑒 ) where ℏ is the reduced Planck constant and 𝑚 𝑒 is the electron rest mass. 𝒙 𝑒 is the spatial coordinate of the incident electron beam. 𝜒 𝐴,𝑖 (𝒙 𝐴 , 𝑡) is the external state of the target and given by a plane wave, 𝜒 𝐴,𝑖 (𝒙 𝐴 , 𝑡) = 1(2𝜋) exp (𝑖𝒌 𝐴,𝑖 ∙ (𝒙 𝐴 − 𝒃) − 𝑖𝐸 𝐴,𝑖 𝑡ℏ ) , (S3) with 𝐸 𝐴,𝑖 = ℏ 𝑘 𝐴,𝑖2 /(2𝑀 𝐴 ) where 𝑀 𝐴 is mass of the target. 𝒙 𝐴 is the center-of-mass coordinate of the atom. 𝒃 is the position vector of the target. 𝜓 𝑛 (𝒓, 𝑡) is the wavefunction of the initial electronic bound state of the target characterized by the quantum number n . We specify this target state as 𝜓 𝑛 (𝒓, 𝑡) = 𝜙 𝑛 (𝒓)𝑒 −𝑖𝜔 𝑛 𝑡 , (S5) where 𝒓 is the internal spatial coordinate that denotes the set of all target electrons, 𝜙 𝑛 (𝒓) is the spatial part of the eigenfunction and ℏ𝜔 𝑛 is the eigenenergy of the state n . Exit channel (Final state)
We then consider the wavefunction of the system after the scattering. The exit-channel wavefunction is given by Ψ 𝑓 (𝒙 𝑒 , 𝒙 𝐴 , 𝒓, 𝑡) = 𝜒 𝑓 (𝒙 𝑒 , 𝑡)𝜒 𝐴,𝑓 (𝒙 𝐴 , 𝑡)𝜓 𝑚 (𝒓, 𝑡), (S6) where 𝜒 𝑓 (𝒙 𝑒 , 𝑡) = 1(2𝜋) exp (𝑖𝒌 𝑓 ∙ 𝒙 𝑒 − 𝑖𝐸 𝑓 𝑡ℏ ) , (S7) with 𝐸 𝑓 = ℏ 𝑘 𝑓2 /(2𝑚 𝑒 ) is the plane wave for the scattered electron, and 𝜒 𝐴,𝑓 (𝒙 𝐴 , 𝑡) = 1(2𝜋) exp (𝑖𝒌 𝐴,𝑓 ∙ (𝒙 𝐴 − 𝒃) − 𝑖𝐸 𝐴,𝑓 𝑡ℏ ) , (S8) is the external state of the target with 𝐸 𝐴,𝑓 = ℏ 𝑘 𝐴,𝑓2 /(2𝑀 𝐴 ) . The internal final state of the target is expressed by 𝜓 𝑚 (𝒓, 𝑡) = 𝜙 𝑚 (𝒓)𝑒 −𝑖𝜔 𝑚 𝑡 , (S9) where ℏ𝜔 𝑚 is the eigenenergy of the internal final target eigenstate 𝜙 𝑚 (𝒓) . Transition amplitude
Within the first Born approximation, the transition amplitude is given by 𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃) = ∭ 𝑑𝒙 𝑒 𝑑𝒙 𝐴 𝑑𝒓 ∫ 𝑑𝑡 Ψ ∗𝑓 (𝒙 𝑒 , 𝒙 𝐴 , 𝒓, 𝑡) 𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓)Ψ 𝑖 (𝒙 𝑒 , 𝒙 𝐴 , 𝒓, 𝑡) , (S10) where 𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓) is the electron-target interaction potential, 𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓) = − 𝑒 𝑍4𝜋𝜀 𝑒 − 𝒙 𝐴 | + 𝑒 ∑ 1|𝒙 𝑒 − 𝒙 𝐴 − 𝒓 𝒊 | , 𝑍𝑖=1 (S11) where 𝑍 is the total number of electrons in the target, 𝜀 is the vacuum permittivity and 𝑒 is elementary electron charge. We here assume that the target is neutral and in Eq. (S11), 𝒓 𝒊 denotes the coordinates of the individual target electrons. By inserting the explicit expressions of the wavefunctions given above, we obtain 𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃) = ∭ 𝑑𝒙 𝑒 𝑑𝒙 𝐴 𝑑𝒓 ∫ 𝑑𝑡 ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝑒 −𝑖𝒌
𝐴,𝑖 ∙𝒃 × 𝜒 𝑓∗ (𝒙 𝑒 , 𝑡)𝜒 𝐴,𝑓∗ (𝒙 𝐴 , 𝑡)𝜓 𝑚∗ (𝒓, 𝑡)𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓) 𝜒 𝑖 (𝒙 𝑒 , 𝑡)𝜒 𝐴,𝑖 (𝒙 𝐴 , 𝑡)𝜓 𝑛 (𝒓, 𝑡), = 1(2𝜋) ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝑒 −𝑖𝒌
𝐴,𝑖 ∙𝒃 × ∫ 𝑑𝒓 𝜙 𝑚∗ (𝒓) (∫ 𝑑𝒙 𝐴 ∫ 𝑑𝒙 𝑒 𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓) exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ 𝒙 𝑒 ) exp(𝑖(𝒌 𝐴,𝑖 − 𝒌
𝐴,𝑓 ) ∙ 𝒙 𝐴 )) 𝜙 𝑛 (𝒓) × ∫ 𝑑𝑡 +∞−∞ exp (𝑖𝐸 𝑓 𝑡ℏ + 𝑖𝐸 𝐴,𝑓 𝑡ℏ − 𝑖𝐸 𝑖 𝑡ℏ − 𝑖𝐸 𝐴,𝑖 𝑡ℏ ) exp (𝑖ℏ𝜔 𝑚 𝑡ℏ − 𝑖ℏ𝜔 𝑛 𝑡ℏ ). (S12) We first consider the integrals over 𝒙 𝐴 and 𝒙 𝑒 , ∫ 𝑑𝒙 𝐴 ∫ 𝑑𝒙 𝑒 𝑉(𝒙 𝑒 − 𝒙 𝐴 , 𝒓) exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ 𝒙 𝑒 ) exp(𝑖(𝒌 𝐴,𝑖 − 𝒌
𝐴,𝑓 ) ∙ 𝒙 𝐴 ) = ∫ 𝑑𝒙 𝐴 ∫ 𝑑𝒙 𝑒 𝑉( 𝒙 𝑒 − 𝒙 𝐴 , 𝒓) exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ (𝒙 𝑒 − 𝒙 𝐴 )) exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ 𝒙 𝐴 ) × exp(𝑖(𝒌 𝐴,𝑖 − 𝒌
𝐴,𝑓 ) ∙ 𝒙 𝐴 ) = ∫ 𝑑𝒙 𝐴 exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 + 𝒌 𝐴,𝑖 − 𝒌
𝐴,𝑓 ) ∙ 𝒙 𝐴 ) ∫ 𝑑𝒙 ′ 𝑉(𝒙 ′ , 𝒓) exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ 𝒙′), (S13) where 𝒙 ′ = 𝒙 𝑒 − 𝒙 𝐴 . The integral over 𝒙 𝐴 gives a delta function, ∫ 𝑑𝒙 𝐴 exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 + 𝒌 𝐴,𝑖 − 𝒌
𝐴,𝑓 ) ∙ 𝒙 𝐴 ) = (2𝜋) 𝛿(𝑲 𝑓 − 𝑲 𝑖 ), (S14) where 𝑲 𝑓 = 𝒌 𝑓 + 𝒌 𝐴,𝑓 and 𝑲 𝑖 = 𝒌 𝑖 + 𝒌 𝐴,𝑖 . This delta function represents the momentum conservation in the scattering process. We then perform the integral over t in Eq. (S12) and obtain ∫ 𝑑𝑡 +∞−∞ exp (𝑖𝐸 𝑓 𝑡ℏ + 𝑖𝐸 𝐴,𝑓 𝑡ℏ − 𝑖𝐸 𝑖 𝑡ℏ − 𝑖𝐸 𝐴,𝑖 𝑡ℏ ) exp (𝑖ℏ𝜔 𝑚 𝑡ℏ − 𝑖ℏ𝜔 𝑛 𝑡ℏ ) = 2𝜋ℏ 𝛿(𝜀 𝑓 − 𝜀 𝑖 ), (S15) where 𝜀 𝑓 = 𝐸 𝑓 + 𝐸 𝐴,𝑓 + ℏ𝜔 𝑚 and 𝜀 𝑖 = 𝐸 𝑖 + 𝐸 𝐴,𝑖 + ℏ𝜔 𝑛 . This delta function represents the energy conservation. By inserting the results of Eqs. (S13)-(S15) into Eq. (S12), the transition amplitude becomes 𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃) = 2𝜋ℏ ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝑒 −𝑖𝒌
𝐴,𝑖 ∙𝒃 𝛿(𝑲 𝑓 − 𝑲 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ), (S16) where 𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ) = 1(2𝜋) ∫ 𝑑𝒙 ′ [∫ 𝑑𝒓 𝜙 𝑚∗ (𝒓)𝑉(𝒙 ′ , 𝒓)𝜙 𝑛 (𝒓)] exp(𝑖(𝒌 𝑖 − 𝒌 𝑓 ) ∙ 𝒙 ′ ) , (S17) is the first Born scattering amplitude for the plane-wave incident electron, i.e., the elastic or inelastic atomic form factor for electron scattering. For the first application and illustration of the physics, we consider the simplest one-electron system, the hydrogen atom, as the target. For atomic hydrogen, the spatial part of the eigenfunctions of the 1s, 2s, 2p states are known analytically and given by 𝜙 (𝒓) = √1𝜋 ( 1𝑎 ) exp (− 𝑟𝑎 ) , (S18) 𝜙 (𝒓) = √1𝜋 ( 12𝑎 ) (1 − 𝑟2𝑎 ) exp (− 𝑟2𝑎 ) , (S19) 𝜙 (𝒓) = √1𝜋 ( 12𝑎 ) 𝑟 cos𝜃 exp (− 𝑟2𝑎 ) , (S20) respectively, where 𝑟 and 𝜃 are the length of the polar angle of 𝒓 . The transition amplitudes 𝑇 𝑚𝑛 of the elastic and inelastic scatterings are given by [35] 𝑇 (𝑞) = −𝑒 𝜀 𝑎 (𝑎 𝑞 + 8)(𝑎 𝑞 + 4) , (S21) 𝑇 (𝑞) = 𝑒 𝜀 (𝑎 𝑞 + 94) , (S22) 𝑇 (𝑞) = 𝑒 𝜀 𝑞 (𝑎 𝑞 + 94) , (S23) where ℏ𝒒 = ℏ(𝒌 𝑖 − 𝒌 𝑓 ) is the momentum transfer and 𝑎 is the Bohr radius. Scattering probability
The transition amplitude in Eq. (S16) is the fundamental quantity from which all observables can be constructed. If we assume that we do not resolve the final momentum of the target ( ℏ𝒌 𝐴,𝑓 ) , the scattering angle of electrons ( 𝒌̂ 𝑓 ) and the kinetic energy of the scattered electrons ( 𝐸 𝑓 ) , then the total scattering probability 𝑃 𝑚𝑛 (𝒃) from the target state n to m which is located at 𝒃 is given by 𝑃 𝑚𝑛 (𝒃) = ∫ 𝑑𝑃 𝑚𝑛 (𝒃)𝑑𝒌̂ 𝑓 𝑑𝒌̂ 𝑓 , (S24) where the differential probability 𝑑𝑃 𝑚𝑛 (𝒃)/𝑑𝒌̂ 𝑓 in the time-dependent S-matrix theory is given by [12] 𝑑𝑃 𝑚𝑛 (𝒃)𝑑𝒌̂ 𝑓 = ∫ 𝑑𝒌 𝐴,𝑓 ∫ 𝑑𝐸 𝑓 (2𝜋) 𝑚 𝑒2 ℏ |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃)| . (S25) Here we used that we work in a regime where the kinetic energy of the electron beam is much higher than the electronic transition energy of the target giving |𝒌 𝑓 |/|𝒌 𝑖 | ≈ 1 . The solid angle describing the propagation direction of the final wavevector of the scattered electron is given by 𝑑𝒌̂ 𝑓 = sin 𝜃 𝑓 𝑑𝜃 𝑓 𝑑𝜑 𝑓 . The magnitude square of the transition amplitude |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃)| is given by |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒌 𝐴,𝑓 , 𝒃)| = 4𝜋 ℏ ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 ∫ 𝑑𝒌 𝑖′ ∫ 𝑑𝒌 𝐴,𝑖′ 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝑒∗ (𝒌 𝑖′ ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝑎 𝐴∗ (𝒌 𝐴,𝑖′ ) 𝑒 −𝑖(𝒌
𝐴,𝑖 −𝒌 𝐴,𝑖′ )∙𝒃 × 𝛿(𝑲 𝑓 − 𝑲 𝑖 )𝛿(𝑲 𝑓 − 𝑲 𝑖′ )𝛿(𝜀 𝑓 − 𝜀 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖′ )𝑇 𝑚𝑛∗ (𝒌 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ), (S26) where 𝒌 𝐴,𝑓 is included in the delta functions 𝛿(𝑲 𝑓 − 𝑲 𝑖 ) and 𝛿(𝑲 𝑓 − 𝑲 𝑖′ ) . Using the formula 𝛿(𝑥 − 𝑦)𝛿(𝑥 − 𝑧) = 𝛿(𝑦 − 𝑧)𝛿(𝑥 − 𝑧), (S27) the delta functions in Eq. (S26) become 𝛿(𝑲 𝑓 − 𝑲 𝑖 )𝛿(𝑲 𝑓 − 𝑲 𝑖′ ) = 𝛿(𝑲 𝑖′ − 𝑲 𝑖 )𝛿(𝑲 𝑓 − 𝑲 𝑖 ) = 𝛿 ((𝒌 ′𝑖 + 𝒌 ′𝐴,𝑖 ) − (𝒌 𝑖 + 𝒌 𝐴,𝑖 )) 𝛿 ((𝒌 𝑓 + 𝒌 𝐴,𝑓 ) − (𝒌 𝑖 + 𝒌 𝐴,𝑖 )) , (S28) and 𝛿(𝜀 𝑓 − 𝜀 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖′ ) = 𝛿(𝜀 𝑖′ − 𝜀 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖 ). (S29) We first perform an integral over 𝒌 𝐴,𝑓 in Eq. (S25) with 𝛿((𝒌 𝑓 + 𝒌 𝐴,𝑓 ) − (𝒌 𝑖 + 𝒌 𝐴,𝑖 )) . The differential probability then becomes 𝑑𝑃 𝑚𝑛 (𝒃)𝑑𝒌̂ 𝑓 = ∫ 𝑑𝐸 𝑓 (2𝜋) 𝑚 𝑒2 ℏ |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| , (S30) with |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| = 4𝜋 ℏ ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 ∫ 𝑑𝒌 𝑖′ ∫ 𝑑𝒌 𝐴,𝑖′ 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝑒∗ (𝒌 𝑖′ ) 𝑎 𝐴 (𝒌 𝐴,𝑖 ) 𝑎 𝐴∗ (𝒌 𝐴,𝑖′ ) 𝑒 −𝑖(𝒌
𝐴,𝑖 −𝒌 𝐴,𝑖′ )∙𝒃 × 𝛿 ((𝒌 𝑖′ + 𝒌 ′𝐴,𝑖 ) − (𝒌 𝑖 + 𝒌 𝐴,𝑖 )) 𝛿(𝜀 𝑖′ − 𝜀 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖 )𝑇 𝑚𝑛∗ (𝒌 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ), (S31) where 𝜀 𝑓 = 𝐸 𝑓 + ℏ |𝒌 𝑖 + 𝒌 𝐴,𝑖 − 𝒌 𝑓 | 𝐴 + ℏ𝜔 𝑚 . (S32) Second, we perform an integral over 𝒌 𝐴,𝑖′ with 𝛿((𝒌′ 𝑖 + 𝒌′ 𝐴,𝑖 ) − (𝒌 𝑖 + 𝒌 𝐴,𝑖 )) and obtain |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| = 4𝜋 ℏ ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝐴,𝑖 ∫ 𝑑𝒌 𝑖′ 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝑒∗ (𝒌 𝑖′ ) × 𝑎 𝐴 (𝒌 𝐴,𝑖 )𝑎 𝐴∗ (𝒌 𝑖 + 𝒌 𝐴,𝑖 − 𝒌 𝑖′ ) 𝑒 𝑖(𝒌 𝑖 −𝒌 𝑖′ )∙𝒃 𝛿(𝜀 𝑖′ − 𝜀 𝑖 )𝛿(𝜀 𝑓 − 𝜀 𝑖 )𝑇 𝑚𝑛∗ (𝒌 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ), (S33) where 𝜀 𝑖′ = 𝐸 𝑖′ + ℏ |𝒌 𝑖 + 𝒌 𝐴,𝑖 − 𝒌 𝑖′ | 𝐴 + ℏ𝜔 𝑛 . (S34) Third, we perform an integral over 𝒌 𝐴,𝑖 . To this end, by using 𝑀 𝐴 ≫ 𝑚 𝑒 and by assuming that the momentum distribution of the electron is narrow enough to satisfy |𝒌 𝑖′ | + |𝒌 𝑖 | ≅ 2|𝒌 𝑖 | , we obtain an approximated form of 𝛿(𝜀 𝑖′ − 𝜀 𝑖 ) as 𝛿(𝜀 𝑖′ − 𝜀 𝑖 ) = 𝛿(𝐸 𝑖′ + ℏ |𝒌 𝑖 + 𝒌 𝐴,𝑖 − 𝒌 𝑖′ | 𝐴 + ℏ𝜔 𝑛 − 𝐸 𝑖 − ℏ |𝒌 𝐴,𝑖 | 𝐴 − ℏ𝜔 𝑛 ) ≅ 𝛿(𝐸 𝑖′ − 𝐸 𝑖 ) = 𝛿(ℏ |𝒌 𝑖′ | 𝑒 − ℏ |𝒌 𝑖 | 𝑒 ) ≅ 2𝑚 𝑒 ℏ 𝛿(2|𝒌 𝑖 |(|𝒌 𝑖′ | − |𝒌 𝑖 |)) = 𝑚 𝑒 ℏ |𝒌 𝑖 | 𝛿( |𝒌 𝑖′ | − |𝒌 𝑖 |), (S35) Similarly, 𝛿(𝜀 𝑓 − 𝜀 𝑖 ) ≅ 𝛿(𝐸 𝑓 − 𝐸 𝑖 + ℏ𝜔 𝑚 − ℏ𝜔 𝑛 ). (S36) In addition, we introduce one more approximation. Because the target atom is well localized in space, its momentum distribution 𝑎 𝐴 is much wider than that of projectile electrons [31]. Then we can use the following approximation, 𝑎 𝐴∗ (𝒌 𝑖 + 𝒌 𝐴,𝑖 − 𝒌 𝑖′ ) ≅ 𝑎 𝐴∗ (𝒌 𝐴,𝑖 ). (S37)
We can now perform the integral over 𝒌 𝐴,𝑖 in Eq. (S33) |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| = 4𝜋 ℏ 𝐼 𝐴 ∫ 𝑑𝒌 𝑖 ∫ 𝑑𝒌 𝑖′ 𝑎 𝑒 (𝒌 𝑖 ) 𝑎 𝑒∗ (𝒌 𝑖′ ) 𝑒 𝑖(𝒌 𝑖 −𝒌 𝑖′ )∙𝒃 × 𝑚 𝑒 ℏ |𝒌 𝑖 | 𝛿( |𝒌 𝑖′ | − |𝒌 𝑖 |)𝛿(𝐸 𝑓 − 𝐸 𝑖 + ℏ𝜔 𝑚 − ℏ𝜔 𝑛 )𝑇 𝑚𝑛∗ (𝒌 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ), (S38) where we introduced the short-hand notation 𝐼 𝐴 for the integral 𝐼 𝐴 = ∫ 𝑑𝒌 𝐴,𝑖 |𝑎 𝐴 (𝒌 𝐴,𝑖 )| . (S39) For convenience, we use spherical coordinates in the evaluation of the integrals, that is, 𝒌 𝑖 is expressed by its length 𝑘 𝑖 and the polar and azimuthal angles (𝜃 𝑖 , 𝜑 𝑖 ) , |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| = 4𝜋 ℏ 𝐼 𝐴 ∫ 𝑘 𝑖2 𝑑𝑘 𝑖 ∫ sin 𝜃 𝑖 𝑑𝜃 𝑖 ∫ 𝑑𝜑 𝑖 ∫ 𝑘′ 𝑖2 𝑑𝑘 𝑖′ ∫ sin 𝜃 𝑖′ 𝑑𝜃 𝑖′ ∫ 𝑑𝜑 𝑖′ 𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 , 𝜑 𝑖 ) 𝑎 𝑒∗ (𝑘 𝑖′ , 𝜃 𝑖′ , 𝜑 𝑖′ ) × 𝑒 𝑖(𝒌 𝑖 −𝒌 𝑖′ )∙𝒃 𝑚 𝑒 ℏ 𝑘 𝑖 𝛿(𝑘 𝑖′ − 𝑘 𝑖 )𝛿(𝐸 𝑓 − 𝐸 𝑖 + ℏ𝜔 𝑚 − ℏ𝜔 𝑛 )𝑇 𝑚𝑛∗ (𝒌 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝒌 𝑖 , 𝒌 𝑓 ). (S40) We perform the integral over 𝑘 𝑖′ by using the presence of 𝛿(𝑘 𝑖′ − 𝑘 𝑖 ) and obtain |𝑇 𝑓𝑖 (𝐸 𝑓 , 𝒌̂ 𝑓 , 𝒃)| = 4𝜋 𝑚 𝑒 𝐼 𝐴 ∫ 𝑘 𝑖3 𝑑𝑘 𝑖 ∫ sin 𝜃 𝑖 𝑑𝜃 𝑖 ∫ 𝑑𝜑 𝑖 ∫ sin 𝜃 𝑖′ 𝑑𝜃 𝑖′ ∫ 𝑑𝜑 𝑖′ × 𝑎 𝑒 (𝑘 𝑖 , 𝒌̂ 𝑖 ) 𝑎 𝑒∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ ) 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 𝛿(𝐸 𝑓 − 𝐸 𝑖 + ℏ𝜔 𝑚 − ℏ𝜔 𝑛 )𝑇 𝑚𝑛∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ , 𝒌 𝑓 )𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌 𝑓 ), (S41) where 𝒌̂ 𝑖 and 𝒌̂ 𝑖′ are unit vectors along 𝒌 𝑖 and 𝒌 𝑖′ , respectively. Finally, we perform the integral over 𝐸 𝑓 in Eq. (S30) with 𝛿(𝐸 𝑓 − 𝐸 𝑖 + ℏ𝜔 𝑚 − ℏ𝜔 𝑛 ) and obtain 𝑃 𝑚𝑛 (𝒃) = (2𝜋) 𝑚 𝑒2 ℏ ∫ ∫|𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| sin 𝜃 𝑓 𝑑𝜃 𝑓 𝑑𝜑 𝑓 , (S42) |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| = 4𝜋 𝑚 𝑒 𝐼 𝐴 ∫ 𝑘 𝑖3 𝑑𝑘 𝑖 ∬ 𝑑𝒌̂ 𝑖 ∬ 𝑑𝒌̂ 𝑖′ 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 × 𝑎 𝑒 (𝑘 𝑖 , 𝒌̂ 𝑖 ) 𝑎 𝑒∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ ) 𝑇 𝑚𝑛∗ (𝑘 𝑖 , 𝒌̂ 𝑖′ , 𝒌̂ 𝑓 )𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ), (S43) where 𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ) = 1(2𝜋) ∫ 𝑑𝒙 ′ [∫ 𝑑𝒓 𝜙 𝑚∗ (𝒓)𝑉(𝒙 ′ , 𝒓)𝜙 𝑛 (𝒓)] exp(𝑖(𝑘 𝑖 𝒌̂ 𝑖 − 𝑘 𝑓 𝒌̂ 𝑓 ) ∙ 𝒙 ′ ) , (S44) with 𝑘 𝑓 satisfiying ℏ 𝑘 𝑓2 𝑒 = ℏ 𝑘 𝑖2 𝑒 − ℏ𝜔 𝑚 + ℏ𝜔 𝑛 . (S45) This latter equation reflects that a possible decrease in the final kinetic energy of the outgoing electron is accompanied by an excitation in the target atom. A2. INCIDENT ELECTRON WAVEPACKET
In this work, we assume an axially symmetric electron beam and decompose its momentum distribution into the longitudinal and transversal components, 𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) = 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 )𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 )𝑒 𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑖 ,𝑡 𝑝 ) , (S46) where 𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 ) is the phase associated with the free-space propagation of the electron wavepacket for the duration 𝑡 𝑝 from the optical modulation stage to the target [see Fig. 1(a)]. Transversal momentum distribution
For the transversal momentum distribution, we assume a Gaussian distribution given in the main text by Eq. (3) and repeated here for ease of reference, 𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) = 1(2𝜋𝜎 ⊥2 ) exp (− (𝑘 𝑖 sin 𝜃 𝑖 ) ⊥2 ) , (S47) where ℏ𝜎 ⊥ is the standard deviation of the transversal momentum distribution. We assume that 𝜎 ⊥ = 𝑘 𝑒 sin 𝜎 𝜃 ≈ 𝑘 𝑒 𝜎 𝜃 , (S48) where 𝜎 𝜃 is the standard deviation of the divergence (convergence) angle and 𝑘 𝑒 is the central wavenumber of the longitudinal momentum distribution (see below). In electron microscopes, the typical value of 𝜎 𝜃 is in the range of 1-10 mrad. The rms spatial size at the focus in the transverse direction is given by ⊥ ) . For 10 keV electrons, as considered in this work, 𝑘 𝑒 =
51 Å -1 , the angular width is 𝜎 𝜃 = 1 mrad in Fig. 1(b), and the rms beam size in the transverse direction at the focus is 0.97 nm. Longitudinal momentum distribution
The longitudinal momentum distribution of the optically modulated electron beam is given by [9,27,32], 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 ) = 1(2𝜋𝜎 ∥2 ) ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ 𝑒 𝑖𝜙 𝑁 exp (− (𝑘 𝑖 cos 𝜃 𝑖 − 𝑘 𝑒 − 𝑁𝛿𝑘) ∥2 ) , (S49) where |𝑔| represents the coupling strength of the optical modulation (see main text for the explicit expression), 𝐽 𝑁 denotes a Bessel function of the first kind and ℏ𝜎 ∥ is the standard deviation of the longitudinal momentum distribution. The photon-exchange number ( N ) dependent phase 𝜙 𝑁 =𝑁 arg (−𝑔) [9] is set to be zero, since this choice makes the temporal density after the optical modulation match experimental observations [17-23]. The rms spatial width is ~1/(2𝜎 ∥ ) , which corresponds to the rms temporal duration of ~1/(2𝑣 𝑒 𝜎 ∥ ) . ℏ𝛿𝑘 is the momentum shift corresponding to the one-photon energy gain, ℏ (𝑘 𝑒 + 𝛿𝑘) 𝑒 − ℏ 𝑘 𝑒2 𝑒 = ℏ𝜔, (S50) where 𝜔 is the angular frequency of the modulating optical field. The wavenumber shift 𝛿𝑘 is approximately given by 𝛿𝑘 ≅ 𝑚 𝑒 𝜔ℏ𝑘 𝑒 . (S51) According to classical mechanics, the maximal number of photons absorbed or emitted by the electron is 𝑁 = 2|𝑔| . Therefore, the maximal velocity shift is given by Δ𝑣 𝑚𝑎𝑥 = 2|𝑔| × ℏ𝛿𝑘/𝑚 𝑒 = 2|𝑔|𝜔/𝑘 𝑒 . Propagation phase and temporal density modulation
The propagation phase 𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 ) is obtained by the following procedure. The evolution of the real-space amplitude is given by the Fourier transform of the momentum-space amplitude [32], 𝑎 𝑒 (𝒙, 𝑡) = 1(2𝜋ℏ) ∫ 𝑑𝒌 𝑎 𝑒 (𝒌) exp (𝑖𝒌 ∙ 𝒙 − 𝑖𝐸 𝑘 𝑡ℏ ) , (S52) where the norm squared of 𝑎 𝑒 (𝒙, 𝑡) gives the probability distribution of the electron in real-space, and the energy is given by 𝐸 𝑘 = ℏ 𝑘 /(2𝑚 𝑒 ) . We now consider the propagation of the electron wavepacket over a distance of 𝐿 𝑝 = 𝑣 𝑒 𝑡 𝑝 = ℏ𝑘 𝑒 𝑡 𝑝 /𝑚 𝑒 , where 𝑣 𝑒 is the group velocity of the electron wavepacket taken to be along the z -axis, see Fig. 1(a). The phase term associated with the propagation can be expressed by exp (𝑖𝑘 ∥ 𝑣 𝑒 𝑡 𝑝 − 𝑖𝐸 𝑘 𝑡 𝑝 ℏ ) , (S53) where 𝑘 ∥ = 𝑘 𝑖 cos 𝜃 𝑖 . This expression, however, needs to be modified as we will now discuss. The kinetic energy term 𝐸 𝑘 contains not only the longitudinal component but also the transversal component. The transversal component changes the electron beam diameter at the position of the target depending on the value of 𝑡 𝑝 . In other words, depending on the value of 𝑡 𝑝 , the location of the transversal beam focus moves along the z -axis with respect to the target. In order to compensate this contribution and to obtain the beam focus at the target plane 𝑏 𝑧 = 0 , we define the propagation phase as exp (𝑖𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 )) = exp (𝑖𝑘 ∥ 𝑣 𝑒 𝑡 𝑝 − 𝑖𝐸 𝑘 𝑡 𝑝 ℏ ) exp (− 𝑖𝐸 𝑘,⊥ (−𝑡 𝑝 )ℏ ) = exp (𝑖𝑘 ∥ 𝑣 𝑒 𝑡 𝑝 − 𝑖𝐸 𝑘,∥ 𝑡 𝑝 ℏ ) , (S54) where 𝐸 𝑘,⊥ = ℏ 𝑘 ⊥2 /(2𝑚 𝑒 ), 𝑘 ⊥ = 𝑘 𝑖 sin 𝜃 𝑖 and 𝐸 𝑘,∥ = ℏ 𝑘 ∥2 /(2𝑚 𝑒 ). The temporal evolution of the wavepacket and the corresponding change of the temporal density calculated with 𝜙 prop (𝑘 𝑖 , 𝜃 𝑖 , 𝑡 𝑝 ) are consistent with experiment results [17-23], see below. The optical energy and momentum modulations induce a temporal density modulation during propagation in the free-space vacuum. An interesting case is the compression or bunching into attosecond electron peaks. The propagation time after the modulation stage for the attosecond electron bunching to occur is given by [19,37], 𝑡 bunch = 𝑣 𝑒 𝜔 Δ𝑣 𝑚𝑎𝑥 = ℏ𝑚 𝑒 𝑘 𝑒2 = 𝑚 𝑒 . (S55) For a 10-keV electron beam ( 𝑣 𝑒 = 5.9 × 10 m/s, 𝑘 𝑒 = 51 Å −1 ), a laser wavelength of 2 µm ( ℏ𝜔 = 0.62 eV) and a laser-electron coupling strength of , the attosecond peaks show up at the instant of time 𝑡 bunch = 6.9 ps. An example of the propagating electron wavepacket is shown in Fig. S1(a), which is for 𝑡 𝑝 =𝑡 bunch , i.e., the case where the attosecond bunching occurs at the focal position of the electron beam. The simulation is performed with 𝜎 𝜃 = 1 mrad and . At 𝑡 = 0 (left panel), the energy and momentum of the electron wavepacket are modulated by an optical field. The other plots show the instantaneous density of the propagating wavepacket. After a propagation duration of 𝑡 𝑝 ( 𝑡 = 𝑡 𝑝 , middle panel), the transversal size reaches its minimum. At 𝑡 > 𝑡 𝑝 (two panels on the right side), the beam is diverging. Simultaneously with the transversal focusing dynamics, the longitudinal (or temporal) density modulation can also be seen. Fig. S1. Free-space propagation dynamics of the electron wavepacket after the optical modulation. (a) Evolution of the optically-modulated 10-keV electron beam, calculated using Eq. (S46). At 𝑡 = 0 (left panel), the energy and longitudinal momentum distributions are modulated by a laser field of 2-µm wavelength but the real-space density has a Gaussian profile. After some free-space propagation ( 𝑡 > 0 ), the longitudinal (temporal) density modulation can be seen. Simultaneously with the longitudinal density modulation, the transversal focusing/divergence occurs with the propagation. The transversal focus is at the target ( 𝑧 = 0) . (b) Temporal density at the target position ( 𝑧 = 0) at the modulation strengths of (left panel), 2 (middle panel) and 5 (right panel), the cases for the results in Fig. 3. Vertical profile at each 𝑡 𝑝 shows the longitudinal density profile. The propagation time t p from the optical modulation to the target, or equivalently, the location of the optical modulation, controls the temporal density and the scattering probabilities, see main text. Right after the optical energy modulation ( 𝑡 = 0 , left panel), the longitudinal density is still a Gaussian. However, after the free-space propagation ( 𝑡 > 0 ), the longitudinal density gets modulated. At 𝑡 = 𝑡 𝑝 =𝑡 bunch (middle panel), sharp peaks appear in the density. Figure S1(b) compares the temporal density at the transversal focus, i.e., at the position of the target, as a function of the propagation time ( 𝑡 𝑝 ) for the three cases of and 5, which are the same as in Fig. 3 of the main text. In contrast to Fig. S1(a), the attosecond bunching occurs not only at the transversal focal position of the electron beam ( 𝑡 bunch = 𝑡 𝑝 ) but also before ( 𝑡 bunch < 𝑡 𝑝 ) and after (𝑡 bunch > 𝑡 𝑝 ) the focus. Following the change of the propagation time 𝑡 𝑝 from the optical modulation to the target, the temporal density of the electron beam at the target changes. The sharp density peaks separated by an optical cycle (6.7 fs here) can be seen at 𝑡 𝑝 = 𝑡 bunch in each panel. At larger 𝑡 𝑝 , the bunched peaks are temporally dispersed and overlap with the neighboring peaks. These overlaps induce interference among the peaks and produce the complex temporal density profile. The calculated temporal density evolutions in Fig. S1(b), especially the one in the right panel ( ), are consistent with previous experimental [17-23] and theoretical [8,25,32] reports, showing the validity of the wavefunction and the propagation phase used in this study.
Beam parameter dependence
The dependence of the modulation 𝑀 𝑚𝑛 (𝒃, 𝑔) on the transversal momentum width ℏ𝜎 ⊥ (or 𝜎 𝜃 ) is discussed in the main text with Fig. 2. Here we consider the dependence on the longitudinal momentum width ℏ𝜎 ∥ . Figure S2(a) compares the modulation of elastic scattering probabilities at 𝒃 = 0 and at five different values of ℏ𝜎 ∥ . The corresponding FWHM initial wavepaket durations are shown next to the curves. For the durations longer than 50 fs (red, black and green curves), the modulations are nearly identical. However, at 25 fs (blue curve), some deviation is observed especially at large propagation time 𝑡 𝑝 ≥ 4𝑡 bunch . This deviation is due to the dispersion occurring in each 𝑁 -photon component, which becomes stronger at shorter duration, i.e., larger ℏ𝜎 ∥ . At the very short duration of 10 fs (purple curve), which is comparable to the laser cycle (6.7 fs at 2 µm wavelength), significant deviation occurs already at 𝑡 𝑝 = 2𝑡 bunch . We then consider the dependence on the kinetic energy of the incident electron beam ℏ 𝑘 𝑒2 /(2𝑚 𝑒 ) . Green circles in Fig. S2(b) shows the zero-impact parameter 𝑏 (see main text and Fig. 4) as a function of the electron-beam kinetic energy, calculated with 𝑇 𝑚𝑛 = 1. At 𝜎 𝜃 = 5 mrad, we observe noticeable modulations at energies only above 3 keV. Below that, the coupling of different 𝑁 -photon components are negligibly small at the angular divergence. At the energies below 20 keV, 𝑏 is almost constant at around 0.4 nm, in good agreement with the simple model prediction (black curve). However, at energies higher than 50 keV, significant deviation from 0.4 nm can be seen. This is because the approximation used in the simple model Eqs. (S81)-(S82) (below) is not perfectly accurate at high kinetic energy, i.e., large 𝑘 𝑒 . The approximation is also not fully accurate for long wavelength, i.e., small 𝛿𝑘 , which can be seen in Fig. 4(f). Both cases yield 𝑏 larger than 0.4 nm. Fig. S2. Beam parameter dependence. (a) Modulations 𝑀 𝑚𝑛 (𝒃 = 0, 2|𝑔| = 5) of total elastic scattering probability at 10 keV for five different longitudinal momentum widths ℏ𝜎 ∥ . The corresponding FWHM durations √2 ln 2 /(𝑣 𝑒 𝜎 ∥ ) are shown. At durations longer than 50 fs, the modulation is nearly identical. At shorter durations, the dispersion of each N -photon component becomes significant and the deviations from the results with longer durations become noticeable at large 𝑡 𝑝 . (b) Zero-modulation impact parameter 𝑏 (see Fig. 4) as a function of the kinetic energy of the incident electron beam. Black curve shows the prediction of the simple model. Below 20 keV, 𝑏 is almost constant at 𝑏 ~ Limit of infinitely large electron beam diameter
We here consider the limit of an infinitely large electron beam in the transversal direction, 𝜎 ⊥ → 0 . This limit corresponds to that of scattering by a plane wave with a fixed wavenumber and longitudinal propagation direction. The amplitude that describes the transversal momentum components becomes 𝑎 𝑒,⊥ → 1√2 𝛿((𝑘 𝑖 sin 𝜃 𝑖 ) ) = 1√2𝑘 𝑖2 𝛿(sin 𝜃 𝑖 ). (S56) Thus only 𝜃 𝑖 = 𝜃 𝑖′ = 0 is allowed in Eq. (S43). Therefore, at this limit, as expected, the incident beam is a superposition of plane waves propagating along the z -axis and the 𝜑 𝑖 and 𝜑 𝑖′ dependences are lost. The scattering probability becomes |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| → constant × ∫ 1𝑘 𝑖 𝑑𝑘 𝑖 |𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 = 0)| |𝑇 𝑚𝑛 (𝑘 𝑖 , 𝜃 𝑖 = 0, 𝒌̂ 𝑓 )| . (S57) Therefore, the scattering probability is given by the incoherent sum over each 𝑘 𝑖 component. Accordingly, the coherence and associated interference effect discussed in the main text are lost in this limit. Spatial distribution of the target atoms: incoherent averaging
Here we consider an ensemble of target atoms placed at different positions 𝒃 and consider the incoherent average over contributions from different values of 𝒃 to obtain the total scattering probability with simultaneous excitation from the internal target state 𝑛 to target state 𝑚. When the spatial distribution of the target atoms is given by 𝜌(𝑏 𝑥 , 𝑏 𝑦 , 𝑏 𝑧 ), the total scattering probability is given by 𝑃 𝑚𝑛incoh = ∭ 𝜌(𝒃) 𝑃 𝑚𝑛 (𝒃)𝑑𝒃. (S58) We consider two extreme cases. First, we consider target atoms uniformly distributed in the x-y plane, 𝜌(𝑏 𝑥 , 𝑏 𝑦 , 𝑏 𝑧 ) = 𝜌 𝑧 (𝑏 𝑧 ) . We note that ∬ 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 𝑑𝑏 𝑥 𝑑𝑏 𝑦 +∞−∞ = (2𝜋) 𝛿 (𝑘 𝑖 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑥 ) 𝛿 (𝑘 𝑖 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑦 ) , (S59) where (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑥 and (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑦 are the x and y components of 𝒌̂ 𝑖 − 𝒌̂ 𝑖′ . The product of the two delta functions is equivalent to 𝛿(𝜃 𝑖 − 𝜃 𝑖′ )𝛿(𝜑 𝑖 − 𝜑 𝑖′ ). Under this condition, the coherent term becomes 𝑎 𝑒∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) = |𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 )| . Therefore, the scattering probability is independent on the phase of 𝑎 𝑒 and no modulation occurs, i.e., 𝑀 𝑚𝑛 , of Eq. (6) of the main text attains the value 𝑀 𝑚𝑛 = 0 . Second, we consider target atoms uniformly distributed along the z -axis, i.e., 𝜌(𝑏 𝑥 , 𝑏 𝑦 , 𝑏 𝑧 ) =𝜌 𝑥,𝑦 (𝑏 𝑥 , 𝑏 𝑦 ) . By using ∫ 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 𝑑𝑏 𝑧 = +∞−∞ 𝑖 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑧 ) , (S60) where (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑧 is the z- component of 𝒌̂ 𝑖 − 𝒌̂ 𝑖′ . The delta function 𝛿 ((𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) 𝑧 ) is equivalent to 𝛿(𝜃 𝑖 − 𝜃 𝑖′ ) . Therefore, 𝑀 𝑚𝑛 = 0 and no modulation is observed as expected. High-energy approximation
The results shown in Figs. 1-4 are obtained with Eqs. (S42)-(S45) [see also Eqs.(1)-(2) of the main text], which contain integrals over seven parameters in total. We introduce here a high-energy approximation which allows to perform the integral over 𝑘 𝑖 analytically and to reduce computational time significantly. When the central energy of the electron beam is much higher than the energy bandwidth after the optical modulation, i.e., 𝑘 𝑒 ≫ 𝛿𝑘 , and 𝑘 𝑒 ≫ 𝜎 ∥ , which is the case for all the reported experiments [16-23], the transversal momentum distribution 𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) and the scattering form factor 𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ) are nearly constant over the variation of 𝑘 𝑖 within the momentum distribution of the electron beam. Therefore, they can be represented by their values at 𝑘 𝑖 = 𝑘 𝑒 , 𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) ≈ 𝑎 𝑒,⊥ (𝑘 𝑖 = 𝑘 𝑒 , 𝜃 𝑖 ), (S61) and 𝑇 𝑚𝑛 (𝑘 𝑖 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ) ≈ 𝑇 𝑚𝑛 (𝑘 𝑖 = 𝑘 𝑒 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ). (S62) Under this approximation, the differential scattering probability |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| in Eq. (S43) can be expressed as |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| = 4𝜋 𝑚 𝑒 𝐼 𝐴 ∬ 𝑑𝒌̂ 𝑖 ∬ 𝑑𝒌̂ 𝑖′ 𝑎 𝑒,⊥∗ (𝑘 𝑖 = 𝑘 𝑒 , 𝜃 𝑖′ )𝑎 𝑒,⊥ (𝑘 𝑖 = 𝑘 𝑒 , 𝜃 𝑖 ) × 𝑇 𝑚𝑛∗ (𝑘 𝑖 = 𝑘 𝑒 , 𝒌̂ 𝑖′ , 𝒌̂ 𝑓 ) 𝑇 𝑚𝑛 (𝑘 𝑖 = 𝑘 𝑒 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 ) ∑ ∑ 𝐽 𝑁 ′ (2|𝑔|) 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞+∞𝑁 ′ =−∞ 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 ) (S63) where 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 ) = ∫ 𝑘 𝑖3 𝑑𝑘 𝑖 exp (𝑖𝑘 𝑖 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) ∙ 𝒃) × exp (𝑖𝑘 𝑖 𝑣 𝑒 𝑡 𝑝 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ ) − 𝑖ℏ𝑘 𝑖2 𝑡 𝑝 𝑒 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ )) × exp (− (𝑘 𝑖 cos 𝜃 𝑖 − 𝑘 𝑒 − 𝑁𝛿𝑘) ∥2 ) exp (− (𝑘 𝑖 cos 𝜃 𝑖′ − 𝑘 𝑒 − 𝑁 ′ 𝛿𝑘) ∥2 ) . (S64) This integral over 𝑘 𝑖 can be performed analytically. Because we consider 𝜃 𝑖 and 𝜃 𝑖′ of the order of mrad, we can consider the Taylor expansions of cos 𝜃 𝑖 and cos 𝜃 𝑖′ and take the leading orders, cos 𝜃 𝑖 = 1 − 𝜃 𝑖2 /2 and cos 𝜃 𝑖′ = 1 − 𝜃 𝑖′2 /2 . By using 𝑘 𝑒 ≫ 𝛿𝑘, 𝜎 ∥ , we obtain approximately 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 ) ≈ √2𝜋 𝜎 ∥ 𝑘 𝑒2 {𝑘 𝑒 + 2𝑖𝜎 ∥2 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) ∙ 𝒃 + 𝑖𝜎 ∥2 𝑣 𝑒 𝑡 𝑝 (𝜃 𝑖2 − 𝜃 𝑖′2 )}(1 + 𝑖𝜎 ∥2 ℏ 𝑡 𝑝 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 )) exp ( − {𝑘 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 ) − 2𝛿𝑘(𝑁 ′ − 𝑁)} ∥2 (1 + 𝑖𝜎 ∥2 ℏ 𝑡 𝑝 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 ))) × exp (−𝑖 𝑘 𝑒 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) ∙ 𝒃1 + 𝑖𝜎 ∥2 ℏ 𝑡 𝑝 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 )) exp ( −𝑖 (𝑁 + 𝑁 ′ )𝑣 𝑒 𝑡 𝑝 𝛿𝑘(𝜃 𝑖2 − 𝜃 𝑖′2 )4 (1 + 𝑖𝜎 ∥2 ℏ 𝑡 𝑝 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 ))) . (S65) Fig. S3. Validity of the high-energy approximation. (a) Simulation result for 10-keV electrons with the high-energy approximation with Eq. (S63) (purple curve) compared to that of full simulation with Eq. (S43) (red circles). The two results match well. (b) Impact parameter dependence at 𝑡 𝑝 = 1.5𝑡 bunch for the three collisional processes of the hydrogen atom. Curves show the results of the high-energy approximation with Eq. (S63). Circles, squares and diamonds show the simulation results with Eq. (S43), the same as in Fig. 4(d). The results of the full simulations are well reproduced. In order to confirm the validity of this high-energy approximation, we compare in Fig. S3(a) the modulations 𝑀 𝑚𝑛 (𝒃 = 0, 2|𝑔| = 5) of elastic scattering calculated with Eq. (S43) and Eq. (S63). The two curves are almost identical. In Fig. S3(b), we show the modulation at non-zero impact parameter, 𝑀 𝑚𝑛 (𝒃 ≠ 0, 2|𝑔| = 5) . The results given by the full simulation with Eq. (S43) depicted by circles, squares and diamonds are well reproduced by the results with the high-energy approximation of Eq. (S63) shown in lines. These results demonstrate the validity of the high-energy approximation. The high-energy approximation speeds up the computations by more than an order of magnitude compared to the evaluation of the full integrals. The good agreement facilitates its application in future works. Modulation at zero propagation duration
We consider the differential scattering probability |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| at 𝑡 𝑝 = 0 . The numerical results in Figs. 2(a), (c) and 3 show that 𝑀 𝑚𝑛 (𝒃 = 0, 𝑔) ≅ 0 . This suggests that even though the energy and momentum spectra of the electron beam are already broad at 𝑡 𝑝 = 0 , some free-space propagation is required to give a non-zero modulation. With the high-energy approximation, whose accuracy is shown above, the integral 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 ) becomes 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) = √2𝜋 𝜎 ∥ 𝑘 𝑒2 (𝑘 𝑒 + 2𝑖𝜎 ∥2 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) ∙ 𝒃) exp (− {𝑘 𝑒 (𝜃 𝑖2 − 𝜃 𝑖′2 ) − 2𝛿𝑘(𝑁 ′ − 𝑁)} ∥2 ) exp(−𝑖𝑘 𝑒 (𝒌̂ 𝑖 − 𝒌̂ 𝑖′ ) ∙ 𝒃) (S66) Equation (S66) shows that the integral 𝐼 𝑘,𝑁,𝑁 ′ (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) depends only on the difference of 𝑁 and 𝑁 ′ , i.e., 𝑁 ′ − 𝑁 , not the absolute numbers. We therefore express the above integral as 𝐼 𝑘,𝑁 ′ −𝑁 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) and consider the sum over 𝑁 and 𝑁′ in Eq. (S63), ∑ ∑ 𝐽 𝑁 ′ (2|𝑔|) 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞+∞𝑁 ′ =−∞ 𝐼 𝑘,𝑁 ′ −𝑁 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) = 𝐼 𝑘,0 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) ∑ 𝐽 𝑁2 (2|𝑔|) +∞𝑁=−∞ + ∑ 𝐼 𝑘,𝑚 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) ∑ 𝐽 𝑁−𝑚 (2|𝑔|) 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞+∞𝑚=−∞𝑚≠0 . (S67) By using ∑ 𝐽 𝑁2 (𝑥) +∞𝑁=−∞ = 1 and ∑ 𝐽 𝑁 (𝑥)𝐽 𝑁+𝑚 (𝑥) +∞𝑁=−∞ = 0, (S68) for 𝑚 ≠ 0 , Eq. (S67) becomes ∑ ∑ 𝐽 𝑁 ′ (2|𝑔|) 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞+∞𝑁 ′ =−∞ 𝐼 𝑘,𝑁 ′ −𝑁 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0) = 𝐼 𝑘,0 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0). (S69) We therefore obtain the approximate form of the differential scattering probably at 𝑡 𝑝 = 0 as |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| = 4𝜋 𝑚 𝑒 𝐼 𝐴 ∬ 𝑑𝒌̂ 𝑖 ∬ 𝑑𝒌̂ 𝑖′ 𝑎 𝑒,⊥∗ (𝑘 𝑖 = 𝑘 𝑒 , 𝜃 𝑖′ )𝑎 𝑒,⊥ (𝑘 𝑖 = 𝑘 𝑒 , 𝜃 𝑖 ) × 𝑇 𝑚𝑛∗ (𝑘 𝑖 = 𝑘 𝑒 , 𝒌̂ 𝑖′ , 𝒌̂ 𝑓 ) 𝑇 𝑚𝑛 (𝑘 𝑖 = 𝑘 𝑒 , 𝒌̂ 𝑖 , 𝒌̂ 𝑓 )𝐼 𝑘,0 (𝜃 𝑖 , 𝜃 𝑖′ , 𝒃, 𝑡 𝑝 = 0). (S70) Notably, |𝑇 𝑓𝑖 (𝒌̂ 𝑓 , 𝒃)| is now independent on the optical modulation strength |𝑔|. Therefore, the modulation of the scattering probability for any 𝒃 is approximately zero, 𝑀 𝑚𝑛 (𝒃, 𝑔) ≅ 0 , at 𝑡 𝑝 = 0 . A3. TARGET INDEPENDENT MODEL
Here we derive a simple model for the qualitative discussion of the simulation results. For simplicity, we consider the case 𝑇 𝑚𝑛 = 1 . We refer to this case as the case of a uniform scatterer. Since we take 𝑇 𝑚𝑛 = 1 , scattering effects related to the detailed nature of the target are neglected. In this sense this model highlights physical effects related directly to the optically modulated electron beam. The validity of this model is illustrated in Fig. 4(d). We consider the integrals over 𝜑 𝑖 and 𝜑 𝑖′ in Eq. (S43), 𝐼 𝑏 = ∫ 𝑑𝜑 𝑖 ∫ 𝑑𝜑 𝑖′ 𝑒 𝑖𝑘 𝑖 (𝒌̂ 𝑖 −𝒌̂ 𝑖′ )∙𝒃 . (S71) For 𝒃 = (𝑏 𝑥 , 𝑏 𝑦 , 𝑏 𝑧 ) and using spherical coordinates, we obtain 𝐼 𝑏 = exp (𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ )) × ∬ 𝑑𝜑 𝑖2𝜋0 𝑑𝜑 𝑖′ exp(−𝑖𝑘 𝑖 𝑏 𝑥 (sin𝜃 𝑖 cos 𝜑 𝑖 − sin 𝜃 𝑖′ cos 𝜑 𝑖′ )) exp (−𝑖𝑘 𝑖 𝑏 𝑦 (sin𝜃 𝑖 sin 𝜑 𝑖 − sin 𝜃 𝑖′ sin 𝜑 𝑖′ )) . (S72) By using ∫ 𝑒 −𝑖𝐴 sin 𝑥−𝑖𝐵 cos 𝑥 𝑑𝑥 = 2𝜋𝐼 (√−𝐴 − 𝐵 ) , (S73) where 𝐼 𝑛 is the modified Bessel function, we obtain 𝐼 𝑏 = 4𝜋 exp (𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ ))𝐼 (𝑘 𝑖 sin𝜃 𝑖 √−(𝑏 𝑥2 + 𝑏 𝑦2 )) 𝐼 (𝑘 𝑖 sin𝜃 𝑖′ √−(𝑏 𝑥2 + 𝑏 𝑦2 )) = 4𝜋 exp(𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ )) 𝐽 (𝑘 𝑖 sin𝜃 𝑖 √𝑏 𝑥2 + 𝑏 𝑦2 ) 𝐽 (𝑘 𝑖 sin𝜃 𝑖′ √𝑏 𝑥2 + 𝑏 𝑦2 ) . (S74) Equation (S43) now becomes |𝑇 𝑓𝑖model (𝒃, 𝑔)| = 16𝜋 𝑚 𝑒 𝐼 𝐴 ∫ 𝑘 𝑖3 𝑑𝑘 𝑖 ∫ sin 𝜃 𝑖 𝑑𝜃 𝑖 ∫ sin 𝜃 𝑖′ 𝑑𝜃 𝑖′ 𝑎 𝑒∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) × exp(𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ )) 𝐽 (𝑘 𝑖 sin𝜃 𝑖 √𝑏 𝑥2 + 𝑏 𝑦2 ) 𝐽 (𝑘 𝑖 sin𝜃 𝑖′ √𝑏 𝑥2 + 𝑏 𝑦2 ) . (S75) In order to further simplify the above equation, we focus on the sets of ( 𝑘 𝑖 , 𝜃 𝑖 , 𝜃 𝑖′ ) that maximize the values of 𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) and 𝑎 𝑒∗ (𝑘 𝑖 , 𝜃 𝑖 ) of Eq. (S46). Specifically, we first consider only values of 𝜃 𝑖 and 𝜃 𝑖′ which give zero arguments in the exponential functions in 𝑎 𝑒,∥ (𝑘 𝑖 , 𝜃 𝑖 ) and 𝑎 𝑒,∥∗ (𝑘 𝑖 , 𝜃 𝑖 ), that is [see Eq. (S49)] 𝑘 𝑖 cos 𝜃 𝑖 − 𝑘 𝑒 − 𝑁𝛿𝑘 = 0 , (S76) and 𝑘 𝑖 cos 𝜃 𝑖′ − 𝑘 𝑒 − 𝑁 ′ 𝛿𝑘 = 0 . (S77) The coherent term in 𝑎 𝑒∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) in Eq. (S75) becomes 𝑎 𝑒∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒 (𝑘 𝑖 , 𝜃 𝑖 ) = 𝑎 𝑒,⊥∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) 1(2𝜋𝜎 ∥2 ) ∑ ∑ 𝐽 𝑁 ′ (2|𝑔|) 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞+∞𝑁 ′ =−∞ 𝑒 𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑖 , 𝑡 𝑝 )−𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑖′ , 𝑡 𝑝 ) . (S78) By denoting the angles 𝜃 𝑖 and 𝜃 𝑖′ that satisfy these equations as 𝜃 𝑁 and 𝜃 𝑁 ′ ′ , which are functions of 𝑘 𝑖 , the term for the propagation phase in Eq. (S78) becomes 𝑒 𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑁 , 𝑡 𝑝 )−𝑖𝜙 prop (𝑘 𝑖 ,𝜃 𝑁′′ , 𝑡 𝑝 ) = exp (𝑖𝑘 𝑖 cos 𝜃 𝑁 𝑣 𝑒 𝑡 𝑝 − 𝑖ℏ(𝑘 𝑖 cos 𝜃 𝑁 ) 𝑒 𝑡 𝑝 ) exp (−𝑖𝑘 𝑖 cos 𝜃 𝑁 ′ ′ 𝑣 𝑒 𝑡 𝑝 + 𝑖ℏ(𝑘 𝑖 cos 𝜃 𝑁′′ ) 𝑒 𝑡 𝑝 ) = exp (𝑖(𝑁 − 𝑁 ′ )𝛿𝑘 𝑣 𝑒 𝑡 𝑝 − 𝑖ℏ2𝑚 𝑒 𝑡 𝑝 {(𝑘 𝑒 + 𝑁𝛿𝑘) − (𝑘 𝑒 + 𝑁′𝛿𝑘) }) = exp(−𝑖(𝑁 − 𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 ), (S79) where 𝜔 𝛿𝑘 = ℏ𝛿𝑘 𝑒 = 14|𝑔|𝑡 bunch . (S80) As a second approximation, we only consider values of 𝑘 𝑖 which give zero arguments in the exponential function in 𝑎 𝑒,⊥ or 𝑎 𝑒,⊥∗ , that is, sin 𝜃 𝑁 = 0 (𝑁′ ≤ 𝑁) or sin 𝜃′ 𝑁′ = 0 ( 𝑁′ ≥ 𝑁) . In the case of sin 𝜃 𝑁 = 0 , we obtain from Eqs. (S76) and (S77), 𝑘 𝑖 = 𝑘 𝑒 + 𝑁𝛿𝑘, (S81) 𝑘 𝑖 cos𝜃 𝑁 ′ ′ = 𝑘 𝑒 + 𝑁 ′ 𝛿𝑘, (S82) and 𝑎 𝑒,⊥∗ (𝑘 𝑖 , 𝜃 𝑖′ )𝑎 𝑒,⊥ (𝑘 𝑖 , 𝜃 𝑖 ) in Eq. (S78) becomes 𝑎 𝑒,⊥∗ (𝑘 𝑖 )𝑎 𝑒,⊥ (𝑘 𝑖 ) = exp (− (𝑘 𝑖 sin 𝜃 𝑁 ′ ′ (𝑘 𝑖 )) ⊥2 ) = exp ( − 𝑘 𝑖2 (1 − (𝑘 𝑒 + 𝑁 ′ 𝛿𝑘) 𝑘 𝑖2 )4(𝑘 𝑒 𝜎 𝜃 ) ) = exp (− 𝑘 𝑖2 − (𝑘 𝑒 + 𝑁 ′ 𝛿𝑘) 𝑒2 𝜎 𝜃2 ) = exp (− (𝑘 𝑒 + 𝑁𝛿𝑘) − (𝑘 𝑒 + 𝑁 ′ 𝛿𝑘) 𝑒2 𝜎 𝜃2 ) ≈ exp (− (𝑁 − 𝑁 ′ )𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) , (S83) where 𝛿𝑘 is neglected. The term for the impact parameter dependence 𝐼 𝑏 of Eq.(S74) becomes 𝐼 𝑏 = 4𝜋 exp (𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑁 − cos 𝜃 𝑁′′ ))𝐽 (0) 𝐽 (𝑘 𝑖 sin 𝜃 𝑁 ′ ′ √𝑏 𝑥2 + 𝑏 𝑦2 ) = 4𝜋 exp (𝑖𝑏 𝑧 (𝑁 − 𝑁′)𝛿𝑘)𝐽 (𝑘 𝑖 √1 − (𝑘 𝑒 + 𝑁 ′ 𝛿𝑘) 𝑘 𝑖2 √𝑏 𝑥2 + 𝑏 𝑦2 ) ≈ 4𝜋 exp(𝑖𝑏 𝑧 (𝑁 − 𝑁 ′ )𝛿𝑘) 𝐽 (√2(𝑁 − 𝑁 ′ )𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ). (S84) On the other hand, in the case of sin 𝜃′ 𝑁′ = 0 ( 𝑁′ ≥ 𝑁) , we obtain from Eqs. (S76) and (S77) and following the procedure in Eqs. (S81)-(S84), 𝑎 𝑒,⊥∗ (𝑘 𝑖 )𝑎 𝑒,⊥ (𝑘 𝑖 ) ≈ exp (− (𝑁 ′ − 𝑁)𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) , (S85) and 𝐼 𝑏 = 4𝜋 exp(𝑖𝑘 𝑖 𝑏 𝑧 (cos 𝜃 𝑖 − cos 𝜃 𝑖′ )) 𝐽 (𝑘 𝑖 sin𝜃 𝑖 √𝑏 𝑥2 + 𝑏 𝑦2 ) 𝐽 (0) ≈ 4𝜋 exp(𝑖𝑏 𝑧 (𝑁 − 𝑁 ′ )𝛿𝑘) 𝐽 (√2(𝑁 ′ − 𝑁)𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ). (S86) In total, we obtain |𝑇 𝑓𝑖model (𝒃, 𝑔)| = 𝑃 × ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ ∑ 𝐽 𝑁 ′ (2|𝑔|) +∞𝑁 ′ =−∞ exp(−𝑖(𝑁 − 𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 ) × exp (− |𝑁 − 𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) exp(𝑖𝑏 𝑧 (𝑁 − 𝑁 ′ )𝛿𝑘) 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ) . (S87) where 𝑃 is a constant. This equation shows that the scattering probability can be estimated by considering sums of products of 𝐽 𝑁 and 𝐽 𝑁′ , reflecting different photon-exchange channels. The strength of the coupling between different channels is given by the term of exp (− |𝑁−𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) . The propagation effect, i.e., the dependence on 𝑡 𝑝 , is given by exp(−𝑖(𝑁 − 𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 ). The impact parameter dependence is given by 𝐼 𝑏 = 4𝜋 exp (𝑖𝑏 𝑧 (𝑁 − 𝑁′)𝛿𝑘) 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ) . In the case of no optical modulation, the laser-electron coupling vanishes, , and |𝑇 𝑓𝑖model (𝒃, 𝑔 = 0)| = 𝑃 × ∑ |𝐽 𝑁 (0)| = 𝑃 . (S88) The modulation of the scattering probability is given by 𝑀 𝑚𝑛model (𝒃, 𝑔) = |𝑇 𝑓𝑖model (𝒃, 𝑔)| − |𝑇 𝑓𝑖model (𝒃, 𝑔 = 0)| |𝑇 𝑓𝑖model (𝒃, 𝑔 = 0)| = −1 + ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ ∑ 𝐽 𝑁 ′ (2|𝑔|) +∞𝑁 ′ =−∞ exp(−𝑖(𝑁 − 𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 ) × exp (− |𝑁 − 𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) exp(𝑖𝑏 𝑧 (𝑁 − 𝑁 ′ )𝛿𝑘) 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ) . (S89) Zero propagation
We consider |𝑇 𝑓𝑖model (𝒃, 𝑔)| and 𝑀 𝑚𝑛model (𝒃, 𝑔) at zero propagation duration 𝑡 𝑝 = 0 . We can rewrite Eq. (S87) as |𝑇 𝑓𝑖model (𝒃, 𝑔)| /𝑃 = ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ ∑ 𝐽 𝑁 ′ (2|𝑔|) +∞𝑁 ′ =−∞ 𝑓(𝑁 − 𝑁 ′ , 𝒃) = ∑ 𝐽 𝑁2 (2|𝑔|) +∞𝑁=−∞ + ∑ ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞𝑁 ′ ≠𝑁 𝐽 𝑁 ′ (2|𝑔|) 𝑓(𝑁 − 𝑁 ′ , 𝒃) = ∑ 𝐽 𝑁2 (2|𝑔|) +∞𝑁=−∞ + ∑ 𝑓(𝑚, 𝒃) ∑ 𝐽 𝑁 (2|𝑔|)𝐽 𝑁−𝑚 (2|𝑔|) +∞𝑁=−∞+∞𝑚=−∞𝑚≠0 (S90) where 𝑓(𝑁 − 𝑁 ′ , 𝒃) = exp (− |𝑁−𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) exp(𝑖𝑏 𝑧 (𝑁 − 𝑁 ′ )𝛿𝑘) 𝐽 (√2|𝑁 − 𝑁 ′ |𝛿𝑘𝑘 𝑒 √𝑏 𝑥2 + 𝑏 𝑦2 ) and 𝑓(0, 𝒃) = 1. By using ∑ 𝐽 𝑁2 (𝑥) +∞𝑁=−∞ = 1 and Eq. (S68) for 𝑚 ≠ 0 , we obtain |𝑇 𝑓𝑖model (𝒃, 𝑔)| = 𝑃 , (S91) and 𝑀 𝑚𝑛model (𝒃, 𝑔) = 0, (S92) at 𝑡 𝑝 = 0 . Therefore, the simple target independent model yields zero modulation at 𝑡 𝑝 = 0 , which is consistent with the scattering theory, see above. Examples
Here we consider a few examples using the simple model above. For simplicity, we limit our considerations to the cases of 𝒃 = 0 , where the 𝒃 -dependent scattering probabilities attain their maxima (see main text and Fig. 1(b)). Equation (S87) is simplified to |𝑇 𝑓𝑖model (𝒃 = 0, 𝑔)| = ∑ 𝐽 𝑁 (2|𝑔|) +∞𝑁=−∞ ∑ 𝐽 𝑁 ′ (2|𝑔|) +∞𝑁 ′ =−∞ exp(−𝑖(𝑁 − 𝑁 ′2 )𝜔 𝛿𝑘 𝑡 𝑝 ) exp (− |𝑁 − 𝑁 ′ |𝛿𝑘2𝑘 𝑒 𝜎 𝜃2 ) , (S93) where the constant 𝑃 in Eq. (S87) is set to one, because we are interested only in the relative quantities and the modulation in Eq. (S89) is given by 𝑀 𝑚𝑛model (𝒃 = 0, 𝑔) = |𝑇 𝑚𝑛model ( 𝒃 = 0, 𝑔)| − 1. (S94) The black curves in Fig. 2(d) and 3 of the main text are calculated using Eq. (S94). First, at [Fig. 3(a)], because of the symmetry of the Bessel function 𝐽 −𝑛 (𝑥) =(−1) 𝑛 𝐽 𝑛 (𝑥) , many of the combinations of 𝑁 and 𝑁′ cancel with each other, see Table S1. Since the absolute values of the Bessel functions significantly decrease for |𝑁| > 2|𝑔| , it suffices to consider the range of −2 ≤ 𝑁, 𝑁′ ≤ 2 . In this case we obtain |𝑇 𝑓𝑖model ( 𝒃 = 0, 2|𝑔| = 1)| ≈ ∑ |𝐽 𝑁 (1)| + 4𝐽 (1)𝐽 (1) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) (exp(−𝑖4𝜔 𝛿𝑘 𝑡 𝑝 ) + exp(𝑖4 𝜔 𝛿𝑘 𝑡 𝑝 )) = ∑ |𝐽 𝑁 (1)| + 8 𝐽 (1)𝐽 (1) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) , (S95) Because we here consider the range of −2 ≤ 𝑁, 𝑁′ ≤ 2 , Eq. (S95 ) does not satisfy Eqs. (S91) and (S92). In order to be consistent with these equations, we take only the modulation term, i.e., the second term of Eq. (S95) and assume 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 1) = 0 at 𝑡 𝑝 = 0 . We then obtain 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 1) = −𝐴 + 𝐴 cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) , (S96) where 𝐴 = 8 𝐽 (1)𝐽 (1) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0. This equation suggests that the modulation 𝑀 𝑚𝑛 (𝒃 =0, 2|𝑔| = 1) oscillates sinusoidally with the propagation time 𝑡 𝑝 and with a period given by 𝛿𝑘 ) =𝜋 𝑡 bunch . Therefore, the first negative peak in the modulation 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 1) is expected to appear at 𝑡 𝑝 = 𝜋 𝑡 bunch /2 = 1.6𝑡 bunch , which is consistent with the result of the full quantum simulation in Fig. 3(a). Next, at [Fig. 3(b)], when we consider the combinations of (𝑁, 𝑁 ′ ) in the range of −3 ≤𝑁, 𝑁′ ≤ 3 , there are three combinations giving non-zero contributions (see Table S2), |𝑇 𝑓𝑖model (𝒃 = 0, 2|𝑔| = 2)| ≈ ∑ |𝐽 𝑁 (2)| + 8 𝐽 (2)𝐽 (2) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) +8 𝐽 (2)𝐽 (2) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) + 8 𝐽 −1 (2)𝐽 (2) exp (− 2𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) . (S97) However, when is 𝜎 𝜃 is not very large and exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) ≫ exp (− 𝑒 𝜎 𝜃2 ) is satisfied, which is the case at 𝜎 𝜃 = 5 mrad used in this work, we can neglect the last term and obtain |𝑇 𝑓𝑖model (𝒃 = 0, 2|𝑔| = 2)| ≈ ∑ |𝐽 𝑁 (2)| + 8 exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) { 𝐽 (2)𝐽 (2)cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐽 (2)𝐽 (2) cos(8𝜔 𝛿𝑘 𝑡 𝑝 )}, (S98) and 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 2) = −𝐵 − 𝐵 + 𝐵 cos(4𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐵 cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) , (S99) with 𝐵 = 8 𝐽 (2)𝐽 (2) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0 and 𝐵 = 8 𝐽 (2)𝐽 (2)exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0 . The faster oscillation period is 𝛿𝑘 ) = 𝜋 𝑡 bunch . Because the signs of the two cosine functions 𝐵 and 𝐵 are both positive, the modulation 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 2) is always negative. As a third example, we consider the case of [Fig. 3(c)]. In Table S3, we show all the combinations of (𝑁, 𝑁 ′ ) but now in the range of −6 ≤ 𝑁, 𝑁′ ≤ 6 and |𝑁 − 𝑁′| ≤ 4 . Using the symmetry of the Bessel function 𝐽 −𝑛 (𝑥) = (−1) 𝑛 𝐽 𝑛 (𝑥) , we find in total 9 combinations giving non-zero contributions. When we take the terms of |𝑁 − 𝑁′| = 0 and 2, and neglect the terms containing 𝐽 and 𝐽 −2 , which are relatively smaller than the other terms ( | 𝐽 (5)| = | 𝐽 −2 (5)| = 0.047 ), we obtain |𝑇 𝑓𝑖model (𝒃 = 0, 2|𝑔| = 5)| ≈ ∑ |𝐽 𝑁 (5)| + 8exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) { 𝐽 (5)𝐽 (5)cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐽 (5)𝐽 (5) cos(16𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐽 (5)𝐽 (5) cos(20𝜔 𝛿𝑘 𝑡 𝑝 )} (S100) and 𝑀 𝑚𝑛model (𝒃 = 0, 2|𝑔| = 5) = 𝐶 − 𝐶 − 𝐶 − 𝐶 cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐶 cos(16𝜔 𝛿𝑘 𝑡 𝑝 ) + 𝐶 cos(20𝜔 𝛿𝑘 𝑡 𝑝 ) (S101) with 𝐶 = −8 𝐽 (5)𝐽 (5) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0, 𝐶 = 8 𝐽 (5)𝐽 (5) exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0, and 𝐶 = (2)𝐽 (2)exp (− 𝛿𝑘𝑘 𝑒 𝜎 𝜃2 ) > 0 . The fastest oscillation period is 𝛿𝑘 ) = 𝜋 𝑡 bunch . Because the coefficient of cos(8𝜔 𝛿𝑘 𝑡 𝑝 ) , −𝐶 , is negative while the other two coefficients +𝐶 and +𝐶 are positive, the modulation 𝑀 𝑚𝑛 (𝒃 = 0, 2|𝑔| = 5) can be positive. A positive peak is expected to appear at 𝑡 𝑝 =1/2 × 2𝜋/(8𝜔 𝛿𝑘 ) = 3.9𝑡 bunch , which is consistent with the result in Fig. 3(c). Table 1. Combinations of (𝑁, 𝑁 ′ ) at with nonvanishing contribution to the scattering probability in the simple model. The numbers N (or N ’) shown in red give negative value of 𝐽 𝑁 (2|𝑔|) . The gray shaded columns are the combinations of (𝑁, 𝑁 ′ ) , giving non-zero contributions. |𝑁 − 𝑁 ′2 | |𝑁 − 𝑁′| (𝑁, 𝑁 ′ ) Net contribution 1 1 (1,0), (-1,0), (0,1), (0,-1) 0 3 1 (2,1), (-2,-1),(1,2), (-1,-2) 0 3 3 (2,-1), (-2,1), (-1,2), (1,-2) 0 4 2 (2,0), (-2,0), (0,2), (0,-2) (1)𝐽 (1) > 0 Table 2. As caption of Table 1, but for . |𝑁 − 𝑁 ′2 | |𝑁 − 𝑁′| (𝑁, 𝑁 ′ ) Net contribution 1 1 (1,0), (-1,0), (0,1), (0,-1) 0 3 1 (2,1), (-2,-1),(1,2), (-1,-2) 0 3 3 (2,-1), (-2,1), (-1,2), (1,-2) 0 4 2 (2,0), (-2,0), (0,2), (0,-2) (2)𝐽 (2) > 0 5 1 (3,2), (-3,-2),(2,3), (-2,-3) 0 5 5 (3,-2), (-3,2), (-2,3), (2,-3) 0 8 2 (3,1), (-3,-1),(1,3), (-1,-3) (2)𝐽 (2) > 0 8 4 (3,-1), (-3,1), (-1,3), (1,-3), −1 (2)𝐽 (2) < 0 9 3 (3,0), (-3,0),(0,3), (0,-3) 0 Table 3. As caption of Table 1, but for and |𝑁 − 𝑁′| ≤ 4 . |𝑁 − 𝑁 ′2 | |𝑁 − 𝑁′| (𝑁, 𝑁 ′ ) Net contribution 1 1 (1,0), (-1,0), (0,1), (0,-1) 0 3 1 (2,1), (-2,-1),(1,2), (-1,-2) 0 3 3 (2,-1), (-2,1), (-1,2), (1,-2) 0 4 2 (2,0), (-2,0), (0,2), (0,-2) (5)𝐽 (5) < 0 5 1 (3,2), (-3,-2),(2,3), (-2,-3) 0 7 1 (4,3),(-4,-3),(3,4),(-3,-4) 0 8 2 (3,1), (-3,-1),(1,3), (-1,-3) (5)𝐽 (5) < 0 8 4 (3,-1), (-3,1),(1,-3), (-1,3) −1 (5)𝐽 (5) > 0 9 3 (3,0), (-3,0),(0,3), (0,-3) 0 9 1 (5,4), (-5,-4), (4,5), (-4,-5) 0 11 1 (6,5), (-6,-5), (5,6), (-5,-6) 0 12 2 (4,2), (-4,-2), (2,4), (-2,-4) (5)𝐽 (5) > 0 15 3 (4,1), (-4,-1), (1,4), (-1,-4) 0 16 2 (5,3), (-5,-3),(3,5), (-3,-5) (5)𝐽 (5) > 0 16 4 (4,0), (-4,0), (0,4), (0,-4), (5)𝐽 (5) < 0 20 2 (6,4), (-6,-4), (4,6), (-4,-6) (5)𝐽 (5) > 0 21 3 (5,2), (-5,-2),(2,5),(-2,-5) 0 24 4 (5,1), (-5,-1), (1,5),(-1,-5) (5)𝐽 (5) < 0 27 3 (6,3),(-6,-3),(3,6),(-3,-6) 0 32 4 (6,2),(-6,-2),(2,6),(-2,-6) (5)𝐽 (5) > 0> 0