Interference between Quantum Paths in Coherent Kapitza-Dirac Effect

In the Kapitza-Dirac effect, atoms, molecules, or swift electrons are diffracted off a standing wave grating of the light intensity created by two counter-propagating laser fields. In ultrafast electron optics, such a coherent beam splitter offers interesting perspectives for ultrafast beam shaping. Here, we study, both analytically and numerically, the effect of the inclination angle between two laser fields on the diffraction of pulsed, low-energy electron beams. For sufficiently high light intensities, we observe a rich variety of complex diffraction patterns. These do not only reflect interferences between electrons scattered off intensity gratings that are formed by different vector components of the laser field. They may also result, for certain light intensities and electron velocities, from interferences between these ponderomotive scattering and direct light absorption and stimulated emission processes, usually forbidden for far-field light. Our findings may open up perspectives for the coherent manipulation and control of ultrafast electron beams by free-space light.


Introduction
Coherent control of the shape of quantum wave functions has set the way towards bond-selective chemistry [1], quantum computing [2,3], and ultrafast control of plasmons [4]. Quantum coherent control originates from the ability to manipulate the interference between quantum paths towards realizing a desired shape of a target wave function by means of shaped, coherent, and/or strong laser excitation. Theoretically, such quantum paths may intuitively be studied using Feynman's path integral approach [5]. This has been instrumental in interpreting observations of above-threshold ionization [6,7] and high-harmonic generation [8], but also for selective control of the quantum paths using circular and elliptical polarizations [9]. The success of path integrals in understanding coherent control lies in the offered ease of selecting few, physically significant paths from a wealth of mathematically available options. Moreover, important concepts such as the action are easily related to classical quantities.
So far, coherent control has mostly been used to shape the wave function of bound-electron states [10], while applications in controlling free electron waves have emerged only recently [11]. Specifically, the inelastic interaction of electrons with optical near-fields can cause attosecond longitudinal electron bunching. Population amplitudes of certain electron states can be controlled precisely by the laser phase in a Ramsey-type experiment [12]. It has been learned that the inelastic processes involved in electronnear-field scattering require moderate laser intensities and are well understood in a minimal coupling Hamiltonian, neglecting ponderomotive forces [13].
For future progress in ultrafast electron microscopy, it appears desirable to avoid the need for matterbased near-field interactions and to simply use light waves in free space to control and shape pulsed electron beams. As a step in this direction, we show here how to use the elastic interaction of freeelectron waves with focused, freely-propagating laser-fields to coherently control the transversal distribution of electron wave packets. This is achieved by a generalization of the Kapitza-Dirac (KD) effect [14,15] to the concomitant utilization of standing-wave and travelling-wave light patterns. In the normal KD effect, electron waves, travelling through a standing-wave pattern of light, are diffracted to transversely populated momentum states at multiples of twice the momentum of free space light. We show that the KD effect can be generalized to the realization of arbitrary momentum states of the electron wavepacket by controlling the interference between quantum pathways originating from distinctly different parts, absorptive and ponderomotive, of the interaction Hamiltonian. This offers fundamentally new degrees of freedom for designing light-controlled phase masks for free-space electron pulses.

Results
In a normal KD effect, electrons are propagating through a standing-wave pattern of the light intensity and in a direction perpendicular to the momentum of the light. We assume here that the standing wave pattern of light is formed by two counter-propagating light waves with the wave vectors 1 phk k y  and  [16,17]. Here, e and 0 m are the elementary charge and mass respectively, the Planck constant, and  the angular frequency of the light wave. Essentially, the spatially varying light intensity introduces a periodic ponderomotive potential along the y-axis which results in a bunching of electron density in regions of low light intensity. After leaving the interaction zone, this electron wave packet transforms into a coherent superposition of plane waves propagating in direction Theoretically, the dynamics of such free-space electron wave packets in KD gratings are commonly described in terms of the Wolkow representation of the electron wavefunction propagating through a vector potential   , A r t . This representation is a special case of the Eikonal approximation where the wavefunction amplitude is taken as constant and spatial gradients of the vector potential are neglected.
Then, the wavefunction can be represented as [18] For this, we have 16  along the transverse direction. Using the Jacobi-Anger transform [21], this can be expressed as a series of Bessel functions corresponding to the different diffraction orders of the electron beam. This is the last term in Eq.
(2). The probability for populating the -th diffraction order then is given as Moreover, the argument of the exponential inside the summation in Eq. (2) can be written as   ph 2 1 2 i nk y in k k r    . This specifies that the final electron wave function can be regarded as a superposition of plane wave electrons occupying the momentum states , nn  in the complete basis specified by the momentum states of the two incident light beams, leading to the final momentum of Importantly, the phase modulation introduced by the second exponential in Eq. (1) is usually discarded in the description of the Kapitza-Dirac effect. As shown in more detail in the Supporting Information, this term causes an ultrafast oscillation of the phase, which for plane wave electrons averages to zero. Hence, the second exponential in Eq. (1) is simply replaced by a multiplicative factor of unity in Eq. (2).
This shows, that in conventional treatments of the Kapitza-Dirac effect, only the ponderomotive forces on the electrons, expressed by the Hamiltonian (1) induces only highly oscillatory phase terms, which -in a phase-cycling approach -are neglected. This is justified whenever considering sufficiently long interaction times between electron and field. This interaction can be however shortened dramatically to a few or even less than a single cycle of the light field, by letting electrons interact with spatially confined optical near fields. In this case, it is the absorptive part 2 H of the Hamiltion which becomes dominant and causes inelastic electron photon interactions, as for example in photon-induced near-field electron microscopy [13,16,19,22].
The above mentioned approximations lead to the general assumption that free-electrons and light waves cannot inelastically interact in free space. In contrast, for restricted electron-light interaction times, achieved, e.g., by ultrafast laser excitations and/or employing slow-electron pulses, at energies below 100 eV, none of the previous assumptions necessarily holds true. This has recently been evidenced, e.g., by the direct acceleration and bunching of electrons with laser pulses in free space [23,24]. In this case, 1 H may not only contribute to the inelastic but also to the elastic interaction, which will be demonstrated below. As a result, one might observe, even in free space, interferences between quantum paths arising from both parts of the Hamiltonian. We show that, for light gratings formed by optical beams with finite inclination angles, two different paths can reach the momentum recoil of 5 These quantum path interferences appear as modulations in the diffraction pattern of the electron waves, different to the diffraction orders observed in the normal KD effect, and hence can -in principlebe observed by a regular position-sensitive detector. To verify these conclusions, we present comprehensive analytical and numerical studies, directly from first principles.
For the sake of simplicity of our analytical model, we first consider the interaction of an electron wave packet with two plane waves of light propagating at angles  and   with respect to the direction of propagation of the electron (see Figure 1c). In this way, we construct a standing wave pattern along the y-axis and perpendicular to the initial propagation direction of the electron, whereas along the -axis, the superposition of the light beams imposes a traveling wave. The -and -components of the vector potential will be given by (3)    , will also occur for incoherent electron beams since their interaction with the pondermotive potential is phase insensitive.
To further investigate the formation of KD diffraction orders under the pure effect of ponderomotive forces, we simulated the interaction of a pulsed electron wave packet with two inclined continuous optical beams with . At this velocity, electron experiences many ponderomotive diffraction orders up to = 18 (see Figure 2a for lower order diffraction probabilities). This electron wave packet, which has a pulse duration of about 0.1 fs, propagates along the x-axis, and its overall interaction time with the focused light field is roughly 6 fs, corresponding to about 60 oscillation cycles of the light field ( Figure 2b). In the simulations we assume a fixed relative phase between light field and electron wavepacket, as commonly is the case in ultrafast laser-driven electron generation schemes. At the exit of the interaction zone, we notice a gradual formation of symmetric and transversal electron bunching effects with up to 12 distinct lobes ( Figure  2c). This generation of KD diffraction orders is better visualized by the momentum representation of the electron wave function (Figure 2d). Longitudinal electron bunching along the direction of the propagation of the electron is not observed, which confirms that inelastic scattering processes are not effective. The generation of up to 28 KD diffraction orders, however, is nicely seen. The number of diffraction orders in the momentum space is significantly larger than the observed number of lobes in the real space. This is related to the fact that many of the observed diffraction orders in the momentum space may result from a pure modulation of the phase of the wave function in the real space -thus they do not become apparent in the amplitude of the wave function in the real space. Considering that the detector is not energy selective, the integration of the probability amplitude along the x k -axis as gives the probability distribution of finding the electron in the transversal This can be further recast into    Interestingly, for fast electrons, the electron wave function populates higher momentum states already for much lower field amplitude since now the coupling scales linearly with the field amplitude and electron momentum. This behavior is in distinct contrast with the action of the pondermotive potential on the electron, for which larger field amplitudes will be necessary to observe higher order diffractions of the electron by the light. At a specific photon energy and electron velocity, two other parameters, i.e., As a result the probability of finding the electron at the final , lomomentum state will be given as This establishes an interesting interference pattern for each given final , lomomentum state ( Figure 4).
One might observe, at specific ranges of field amplitude and electron velocities, quantum coherent interference paths in the diffraction orders. In contrast to previously reported Rabi oscillations in photon-induced near-field electron microscopy [11,12], the quantum interferences discussed here will occur in free space, and may be utilized as a neat way of controlling the transversal distribution of the electron wave packets, without changing their energy. Interestingly, even in the presence of the  kk  unravel from the non-equilibrium population distribution at 1,0  and 0, 1  states, as well as higher order 0, 2  and 2,0  states, which only last for ultrashort times during the interaction (Figure 5b). Moreover, these states provide means to form the desired interference paths between various quantum levels (see Figure 1d), resulting in diffraction orders which are not located at , ll  states (Figure 5d and e). These diffraction orders appear as ultrathin electron bunches which are caused by the interference Interestingly, all the diffraction levels will form along the circumference of an Ewald-like sphere (marked in Figure 5d), demonstrating the striking similarity between the elastic interaction of electrons with light and crystalline matter, even though the interaction Hamiltonian is completely different (Figure 5d). The radius of the Ewald sphere is equal to the wavenumber of the electron el k , as expected. This further strengthens the fact that the inelastic light-matter interaction, though temporarily happening during the interaction of an electron wave packet with continuous-wave light in vacuum, cannot give rise to a steady-state outcome. This is due to the fact that (i) both light waves specified by wave vectors 1 k and between the above-mentioned quantum paths (Figure 5c).  the Hamiltonian become active as well, and the resulting diffraction pattern is an outcome of the interference between the quantum paths associated with each part of the Hamiltonian. By increasing the field amplitude, the overall number of diffraction orders will be also increased.
. Thus, the lower the electron velocity, the lower becomes the required field amplitude (Figure 7a). Additionally, the electric field amplitude is related to the vector potential as 00 E  . In other words, it will be possible to employ lower field amplitudes by using oscillating fields at lower frequencies (Figure 7b) (Figure 7c and d). Such conditions can certainly be created by currently available light sources, or by including near-field field-enhancement effects [25].
Dynamics of sideband modulations in the scattering of single-particle electron wavepackets with freespace light can be described as the outcomes of a quantum walk [26] in the discrete momentum states specified by the classical electromagnetic waves. This behavior is akin to the random walks of photons in classical reliable interferometers with low losses and high stability performances [27]. As a result of the coherent action of the unitary operator specified in Eq. (1) on the single electron wavepacket, at each given time the electron will be left entangled between different momentum states. As expected, features of this quantum walk are interferences and boson sampling [27] from a sea of many possible states, as for example selectively selecting few elastic or inelastic scatterings, as a result of such inferences. The ability to dynamically control the outcome of the random walk by few parameters as the polarization, wavelength, intensity, and the inclination angle of the incident Gaussian beams, and particularly by avoiding matter and hence electron-electron and electron-core interactions, make the proposed system a possible promising candidate for bosonic-fermionic random walks and new generation of boson-sampling devices [26,28,29].
As a summary, we have generalized the KD effect to the inclusion of two laser fields which propagate at an inclination angle with respect to the electron trajectory. The spatio-temporal behavior of the introduced light waves appears as a standing wave pattern transverse to the electron momentumsimilar to the normal KD effect. In addition, a travelling wave pattern forms longitudinal to the initial electron velocity -in contrast with the normal KD effect. We have shown, both theoretically and numerically, that the interaction of the electron wavepacket with the introduced light beams, results in an exotic momentum distribution in the final electron wavepacket, which can be described as the quantum path interferences between two parts of the Hamiltonian, namely ponderomotive and absorptive channels. These interference paths and their coupling strengths can be further controlled in general by the shape of the light waves and in particular by the inclination angles and intensity of the two Gaussian beams, but also by tuning the interaction time and the electron velocity. We anticipate that the aforementioned interference effects can be proposed as new boson sampling device for

Materials and Methods
For our first principle calculations, we have used a time-dependent propagator combined with a pseudospectral Fourier method [30,31], which conserves the norm of the wave function altogether, and propose a convergent numerical scheme. The accuracy and stability of the numerical method can be controlled by an appropriate choice of the time steps. All the simulations were performed in twodimensional space. This is particularly rationalized by the choice of polarizations for the fields (magnetic field normal to the simulation domain, electric field lying in the plane). In this way, the classical trajectory of the electron and its experienced quantum mechanical recoils all remain in the simulation plane. Moreover, only a single electron wavepacket is considered. For calculating the optical Gaussian beams, we have used the analytical solutions based on the paraxial approximations, which perfectly model the laser excitations, and is valid for focus regions larger than 1.5 , where  is the optical wavelength. Here the waist of the introduced optical beams are 2 . We however have benchmarked this approximation by comparing our results with those obtained using a self-consistent Maxwell-Schrödinger numerical toolbox, and the same diffraction orders and probability amplitudes have been noticed. 32