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Quantum states interrogation using a pre-shaped free electron wavefunction

We present a comprehensive theory for interrogation of the quantum state of a two-level system (TLS) based on a free-electron - bound-electron resonant interaction scheme. The scheme is based on free electrons, whose quantum electron wavefunction is pre-shaped or optically modulated by lasers in an electron microscope setup and then inelastically scattered by a quantum TLS target (e.g., atom, quantum dot, crystal defect center, etc.) upon traversing in proximity to the target. Measurement of the post-interaction energy spectrum of the electrons, probes and quantifies the full Bloch sphere parameters of a pre-excited TLS and enables coherent control of the qubit states. The exceptional advantage of this scheme over laser-based ones is the atomic-scale spatial resolution of addressing individual TLS targets. Thus, this scheme opens new horizons for electron microscopy in material interrogation and quantum information technology.


I. INTRODUCTION
Coherent control of a single quantum two-level system (TLS) using lasers is a subject of highest interest in quantum optics. The subject is central to understanding fundamental phenomena such as Rabi oscillation [1], Landau-Zener transition [2,3] and Ramsey interference [4]. It also relates to coherent laser control of atoms [5]. Additionally, it is used extensively in quantum information technologies [6,7] in connection with control and reading of quantum bits (qubits). A variety of TLS embodiments are considered in this context, including trapped ions [8], quantum dots [9] and defects in solids [10].
In this paper we propose an alternative scheme for interrogation and control of the quantum state of a TLS, with pre-shaped quantum electron wavepackets (QEW) rather than with a coherent laser beam. This scheme, based on the recently proposed concept of free-electron bound-electron resonant interaction (FEBERI) [11], differs from more conventional approaches by being able to address individual TLS targets with atomic-scale (nanometric) spatial resolution [12] in contrast to the micrometric resolution of a laser beam. This advantage stems from the fact that typical electron de-Broglie wavelength is six orders of magnitude shorter than that of the optical wavelength of a laser beam. Recent technological developments in electronquantum-optics pave the way to implementing this alternative approach for coherent control and interrogation of TLS targets using pre-shaped and optically modulated QEWs. The diagnosis of the quantum state in this scheme relies on extension of well-established techniques in electron microscopy -Electron Energy Loss Spectroscopy (EELS) and cathodoluminescence (CL).
In considering the free electron wavefunction as an alternative to a laser beam for probing and controlling the TLS, a key question in the suggested new concept is the physical interpretation of the free electron wavefunction shape and the reality of the QEW shape in its interaction with matter. The reality of the QEW and the measurability of its shape and spatial dimensions, as well as the transition from the quantum wavefunction presentation to the classical point-particle theory (the wave-particle duality), have been considered previously in the context of electron interaction with light [34][35][36][37][38][39][40][41][42]. It was shown that the transition of the QEW radiative interaction from the wave-like regime, exhibiting characteristic multi-sidebands PINEM energy spectrum [19-31, 43, 44], to the point-particle-like acceleration/deceleration regime [45,46], takes place when: Namely, the transition takes place when the wavepacket standard deviation duration σ " (or its longitudinal breadth σ ! = " # ) is short relative to the optical radiation period 2 / (or wavelength ) [35][36][37][38]46]. The reality interpretation of the QEW modulation features in the context of radiative emission, has been also extended to the all-important case of multiple modulation-correlated electron wavepackets, where coherent superradiant emission [47], proportional to the number of electrons squared # , is expected [38,[48][49][50], analogously the classical case of a pre-bunched point-particle beam [51].
In analogy with the interaction of a QEW with radiation, the reality of the QEW shaping and its modulation features were claimed to be manifested also in interaction with matter in a newly proposed effect of FEBERI [11]. Based on a simple semiclassical model, it was proposed in [11] that a QEW, passing in the vicinity of a TLS target (e.g., an atom, quantum dot, crystal color center, trapped ion), would induce transitions in the TLS which depend on the QEW profile. Specifically, it was suggested that an ensemble of optical frequency modulated QEWs would excite resonantly TLS transitions, if their modulation frequency (produced by a laser of frequency $ in a PINEM setup [19]) is a sub-harmonic of the TLS transition frequency: where $,& = $ − & is the energy separation of the TLS quantum levels. In addition, it was argued, that if all QEWs are modulation-correlated (i.e. modulated by the same coherent laser), the transition rate should be enhanced in proportion to ! , in analogy with the superradiance effect of an ensemble of atoms [47] or a bunched point-particles electron beam [31,51]. Quadratic dependence of coherently modulated electron beam scattering efficiency by bound electron quantum states has been recognized earlier also as "pulsed beam scattering" [50,52,53]. In the case of a beam of multiple near-point-particle QEWs it was recently termed "Quantum Klystron" [54] in connection to microwave frequency quantum transition. When such near-point-particle QEWs are injected as a periodic train of pulses they would produce FEBERI transition rate enhanced in proportion to ! .
The semiclassical analysis of FEBERI, based on Born's probability interpretation of the wavefunction envelope modulation [11,55], and its proposed dependence on the QEW dimensions, were questioned [56,57]. This led to a furry of recent publications on the subject [23,24,[54][55][56][57][58][59][60][61][62]. Nevertheless, recent, fully quantum-mechanical analyses (of both the free and bound electrons), have substantiated the proposition of shape-and modulation-dependence of multiple correlated QEWs interactions with a TLS and with radiation [48,55]. Moreover, technological and conceptual developments of schemes for optical control of the temporal shape of the electron wavefunction, show that attosecond-scale pre-shaping and modulation of QEWs are experimentally realizable [16,17], and can be used to realize the proposed theoretical concept.
Based on these conceptual and technological developments, we propose here that the shape-dependence of the QEW interaction with a TLS can be utilized for interrogation of the quantum excitation state of a TLS [48,55,58,61,63]. We present a comprehensive theory of quantum state interrogation by FEBERI, showing that the initial quantum state (qubit) coordinates on the Bloch sphere, which uniquely define superposition TLS quantum states (TLS-QS), can be fully extracted from the post-interaction electron energy spectrum of the interacting QEWs, and coherently controlled by their pre-shaping. This can be achieved by controlling the size of the QEWs before interaction, or pre-modulating them at optical frequency by a PINEM process. Interrogation of the TLS state has been analyzed in [58] considering an energy modulated plane-wave quantum electron wavefunction. In [61], a particular energy distribution (coherent superposition state) of the free electron wavefunction was considered and shown to be useful for evaluating the decoherence and relaxation parameters of an excited TLS. The present work depicts a more comprehensive quantum model in which the free electron wavefunction is described in terms of a QEW of general distribution in the momentum or spatial domains. The use of both presentations reveals that the entanglement of the free and bound electrons results in an energy spectrum of the post-interaction QEWs that corresponds to (and thus interrogates) the initial TLS-QS and depends on the spatiotemporal shape and size of the QEWs (the wavepacket envelope size). We show in this paper that a broad size QEW (nearly plane-wave quantum wavefunction) carries the fingerprint of the polar angle of the qubit state of a target TLS on the Bloch sphere, with little energy transfer, and therefore can be useful for its interrogation. On the other hand, interaction with narrow finite size QEWs (nearly point-particle limit) or optical-frequency density modulated QEWs, involves energy transfer between the free electron and the TLS, and enable also its coherent control.
The analysis here refers to the probability of quantum electron transitions of a single TLS and the corresponding modification of the energy density spectrum of an interacting single QEW. Obviously, measurement of the free electron energy probability distribution requires multiple QEWs and (or) multiple TLS targets. Fundamentally, inelastic scattering of an electron that results in quantum transition of a TLS-QS to the upper or lower quantum level, destroys the state, and therefore interrogation of a single TLS-QS by measuring the post-interaction EELS of multiple electrons, is possible only if the TLS-QS is re-instated after each interaction. We conjecture that coherent control and interrogation of the TLS state with the joint quantum wavefunction of multiple modulation-correlated electrons can be used for control and interrogation of the TLS state [63]. This conjecture is an extension of the recent conceptual proposals of superradiance [38,48,49] and Rabi-oscillation FEBERI [11,55] processes with multiple optical-frequency modulation-correlated electrons. However, the analysis here is confined to interaction of single pre-shaped and pre-modulated QEWs with a single TLS target. The extension to multiple particles [11,48,63] and targets [61,64] is commented upon in the discussion part.
Based on this new theory and formulation, we expect that further development of the proposed concept would lead to implementation of the FEBERI effect in numerous important applications of electron microscopy in atomic-scale probing of quantum excitations in matter, diagnostics and coherent control of qubits [65] and efficient pumping of single quantum dot quantum emitters [66].

II. THEORETICAL FRAMEWORK
We present analysis of a general experimental scheme for interrogating bound-electron quantum states by free electrons. The free electron is modeled as a quantum electron wavepacket (QEW) propagating in proximity to a Hydrogen-like atom that models a general quantum two-level system (TLS). We assume an experimental setup, as shown in Fig. 1, in which the TLS quantum state is pre-set by a coherent laser beam that is phase locked to a laser beam that pre-shapes or pre-modulates the QEW with controlled delay. We assert that the quantum state of the TLS can be deduced from the post-interaction electron energy spectrum of the free electrons which is accumulated by repeated reinjection of identical QEWs and re-excitation of the TLS.
We start by describing the TLS as an electric dipole [55], and assuming that the interaction between the free electron and the TLS is dominated by Coulomb potential. The joint wavefunction of the free and bound electrons is governed by the relativistically modified Schrodinger equation: where 0 " = 0 "( + 0 ") , with 5 '( , 5 ') being the kinetic Hamiltonian of the free electron and the Hamiltonian of the bound electron respectively, and 5 * is the interaction Hamiltonian. For the analysis to be valid also for relativistic electrons, we use the relativistic 1D kinetic Hamiltonian for a free electron of energy ' = + # and momentum ' = + ' , where + = 9.1 × 10 ,-& is the electron mass, ' is the velocity of the free electrons ( = ' ⁄ = 0.7 in our work, is the velocity of light): This Hamiltonian was derived in Refs. [35,36] by a second order iterative approximation of Klein-Gordon equation, neglecting the spin effects. The Hamiltonian satisfies 0 "( | 〉 = * |p〉, where | ⟩ is the eigenstate of momentum operator ̂ which satisfies ⟨ | . ⟩ = ( − . ) and ∫ | ⟩⟨ | = , and the energy dispersion where / = ℏ / and | ⟩ is the eigenstate of the TLS which satisfies ⟨ | ⟩ = /0 .
The neglect of exchange energy and the spin-orbit interaction effects is valid in our simplified model under the assumption that the free and bound electrons do not overlap spatially. There is no interaction between the pre-shaped QEW and the laser field used to excite the TLS assuming that it is vanished by the time the QEW meets the TLS. Thus, the dominant interaction in the system is the Coulomb interaction of the dipole and free electron, yielding an interaction Hamiltonian 5 * = − V . ⋅ (, 1' ), where V′ is the position vector of the bound electron relative to the model Hydrogen nucleus, and V = (, 1' ) is the position vector of the free electron relative to the nucleus (See Fig. 1). In a 1D model, the free electron is presented as a QEW of negligible narrow width ( = ( , 0)). Such a relativistic electron generates an electric field ( , 1' ) at the TLS position is [67]: where 9 -and 9 . represent the directions along and perpendicular to the free electron propagation, respectively.
FIG. 1. Scheme of TLS interrogation with FEBERI. The free electron QEW is emitted from e-the gun and after acceleration gets pre-shaped or energy-modulated through interaction with the laser-induced optical nearfield of a tip/grating/membrane/nano-resonator (a PINEM process). After traversing a free-drift length 3 along the z axis, the shaped or density-modulated QEW interacts with a bound electron at coordinate (z=0, 1' + ′). The TLS is pre-excited into a quantum superposition state by a phase-locked harmonic of the same laser source with a controlled delay. The FEBERI interaction can be measured through the energy spectrum of the free electron.
We take the initial state of the free electron to be a general quantum state: is the quantum electron wavefunction in momentum space. The general timedependent pre-interaction state of the bound electron is represented as | ) ( )⟩ = sin f : # g ,/; " ! |1⟩ + /< cos f : # g ,/; # ! |2⟩, where ∈ [0, ] is the polar angle and ∈ [0,2 ] is the azimuth angle of the Bloch sphere [68]. The combined wavefunction of the free and bound electrons | 〉, is the solution of Eq. 1 in the presence of the electric dipole interaction which entangles the free and bound electron states.
It is convenient to transform to the interaction picture by using the unitary evolution operator 5 ( ) = exp[− 5 ' /ℏ] such that Eq. 3 becomes where t v ( , )〉 = 5 = ( )| ( , )〉 , and the Hamiltonian ℋ 5 * ( ) = 5 = ( ) 5 * 5 ( ) = #,& ⋅ (z V + |1⟩ being the dipole moment of the TLS, and σ { > = |2〉〈1| ( V , = |1〉〈2|) being the TLS raising (lowering) operator, see Appendix A for the detailed derivation. Thus, the wavefuction after interaction is obtained by multiplying the initial wavefunction with the scattering where is the time-ordering operator, / and ? are the initial time and final times of the interaction (see Appendix B): Since the coupling strength between free electron and bound electron is relatively weak, it is reasonable to consider only the first order expansion of the scattering matrix, which is equivalent to first order perturbation theory. Then (derived in Appendix B): is the coupling coefficient between the free electron and the TLS in For any input state  Complementary to the derivation of the QEW momentum distribution, the corresponding final occupation probabilities of the upper and lower levels of the bound electron are (see Appendix C): is an auto-correlation function [55,58].

III. INTERROGATION OF A TLS WITH SHAPE-CONTROLLED QEWS
In this section, we analyze the interaction between a TLS and a finite-size QEW, and present the dependence of the post-interaction incremental spectrum of the free electron on the QEW size. We demonstrate the waveparticle duality of the QEW in interaction with matter and its transition from quantum plane-wave to nearpoint-particle characteristics. This transition is analogous to the wave-particle duality and the transition from multi-photon emission/absorption (PINEM) process to acceleration/deceleration process in interaction of QEWs with light that we have analyzed earlier [36,37,38].
We model the free electron as a general Gaussian wavepacket including broadening and momentum chirp due to energy-dispersive drift from a controlled waist point located a distance : = " : before the interaction point = 0 [36] (see Appendix D). In momentum representation: where * is the wavepacket momentum spread. We lumped the quadratic term of Eq.
is the QEW longitudinal waist size. The Gaussian wavepacket in real space and time coordinates ! ( 3 ) = | § 2 ( 3 )|/ ' broadens upon propagation from the location of its waist to the interaction point = 0 after a drift time 3 = 3 / ' . This history-dependent size of the QEW can be written as [36]: where !' = ! ( 3 = 0) and we define a "longitudinal Rayleigh length" parameter P∥ = 4 -2' # R ⁄ , analogous to transverse expansion Rayleigh length of a laser beam, corresponding to the distance where the QEW envelope broadens by a factor of √2, R = ℎ/ + is the Compton wavelength.
Using Eq. (13) as the initial state 4 (') in Eq. (10), we get the post-interaction momentum density distribution of the Gaussian QEW (details shown in Appendix E): The corresponding increments of the occupation probabilities of the TLS quantum levels that are probed by this incremental energy spectrum are (from Eq. 11): Note that the decay constant Γ is the same as the decay constant (Eq. 1) of radiative interaction of a QEW in transition from wave to point-particle limit [36,69] with the substitution = #,& . So, to avoid diminishing of the first order FEBERI interaction, one should keep Γ = ω #,& ! ( 3 ) < 1 and 3 < Y (see Appendix D).
The phase is the phase of the superposition quantum state of the TLS (azimuthal angle of the qubit) at the time of the TLS excitation (see Fig. 1), and = #,& ∆ − is the phase at the arrival time of the QEW centroid to the TLS location = 0. ∆ is the time elapsed from the TLS excitation to the arrival of the QEW. It is the difference between the electron drift time 3 and the optical path delay time between the TLS excitation and the QEW formation by the phase-locked laser beams (see Fig. 1).
In this limit, the spectrum displays a PINEM-like spectrum of well distinguished two sidebands peaked at Z − ' = ± #,& . The asymmetry between the negative and positive recoil sidebands reflects the different dependencies on the polar angle of the two sideband terms in the second order increment term 4 (#) (Eq. 19), which is dominant in this limit. In general, the first order term #,& / 7 > 1, and the first order term decays then exponentially below the second order term. However, this is not necessarily the case when ! is not very large. Since the first and second order terms are proportional respectively to | | and | | # , the rigorous condition for neglecting the contribution of the first order term It is instructive to note that unlike the spectrum of PINEM [19], the post-interaction spectrum in Fig. (2d) displays only two first-order sidebands, which implies that only one quantum of energy is exchanged between the free electron and the TLS in the FEBREI process for a single electron interaction.
In this limit, the curves display an acceleration/deceleration spectrum that depends according to Eq.

Quantum wave-particle duality and anomalous FEBERI
The interpretation of the quantum electron wavefunction Ψ( , ) has been a matter of debate since the inception of quantum theory [71,72]. The accepted Born interpretation is that the expectation value of the electron wavefunction modulus in space-time -|Ψ( , )| # , represents the probability density of finding the electron at point at time t. While we use both momentum and space-time representations of the electron wavefunction, the latter is most conducive to physical understanding of the electron particle-like and wavelike regimes of interaction with light and matter in view of Born's interpretation of the wavefunction. As shown in Eq. (22), a narrow QEW in momentum (or energy) presentation interacts with a TLS as a quantum wave, exhibiting positive and negative recoils. On the other hand, the QEW interacts as a particle in the limit of a short wavepacket in space-time presentation (Eq. 23). In this case, consistently with Born's picture, the QEW is a near-point particle, experiencing acceleration or deceleration depending on its arrival time relative to the dipole moment oscillation phase of the TLS.
From the analysis of the previous section, we conclude that the interrogation of the quantum state of a pre-set TLS superposition state is possible by measuring the post-interaction energy spectrum of a passing-by QEW, as shown in Fig. 1. For probing the qubit polar angle of the TLS-QS it is necessary to employ a narrow energy spread QEW that satisfies the large recoil condition (Eq. 22). In this case the QEW is broad in real space, and its post-interaction spectrum exhibits a wave-like kind PINEM sidebands spectrum corresponding to positive and negative electron quantum recoils (Fig. 2d). The asymmetry of the sidebands pattern measures the polar angle θ of the TLS state. To probe the azimuth angle of the TLS quantum state, it is necessary to employ a short wavepacket that satisfies the near point-particle condition (Eq. 23), exhibiting an acceleration/deceleration post-interaction energy spectrum (Fig. 2a). However, the two conditions of large recoil (Eq. 22) and short wavepacket (Eq. 23) are complementary only in the case of a QEW at its waist ( 3 = 0 in Eq. 14). In the general case 3 ≠ 0, one must consider that the general QEW (Eq. 14) describes a chirped Gaussian with a longitudinal waist size ! ( 3 ) broadened relative to the waist size !' . The waist is located at an arbitrary waist location that can be controlled in principle at the QEW pre-shaping section of the experimental setup ( Fig. 1) by optical shaping and chirping techniques [16,17].
FIG. 3. Energy-time phase-space presentation and post-interaction incremental energy spectrum of the QEW. The first column displays the Wigner distribution of the QEW before interaction, and the second column displays the incremental Wigner distribution after interaction. The area of the pre-interaction Wigner distributions in the first column is the Planck constant h, the same as the square that represents the phase space area of the TLS #,& × #,& . Consequently, the phase-space topography of the initial phase-space distribution of the QEW (left column) delineates only three possible cases, corresponding to: (a) Long QEW with large recoil, (b) Short QEW and (c) Chirped QEW. The second column depicts the incremental Wigner distribution of the QEW after interaction for TLS-QS parameters = /2, = /2. The third column presents the incremental energy density distribution before and after interaction, i.e., the difference of the horizontal projections of the Wigner distributions in the second and first columns.
In the general case the QEW may be chirped. The best way to describe this case is in phase space. Fig. 3 depicts the different regimes of QEW interaction with a TLS in energy-time (or momentum-space) phase-space. The Wigner distribution of a general chirped QEW (Eq. 13 or Eq. 14) after drift, can be calculated numerically from the post-interaction density matrix of the free electron (Eq. C5 in the Appendix). The pre-interaction distribution of the QEW upon entrance to the FEBERI interaction point, is shown in color code in the left column of Fig. 3 for three cases. Cases A and B correspond to injection of the QEW at its waist (no chirp) at the large recoil (Eq. 22) and narrow wavepacket (Eq. 23) conditions respectively. Similar to the representation of anomalous PINEM in Ref. [37], we overlay the Wigner distribution of the pre-interaction QEW on a phasespace square of area #,& × #,& representing the TLS, both have the area ℎ -the Planck constant. The second column in Fig. 3 depicts the incremental Wigner distribution after interaction, and the third column presents the incremental energy distribution density ∆ 4 , which is the difference of the horizontal projections of the Wigner distributions before and after interaction. We observe that the topology of the phase-space diagrams reveals a QEW parameters regime where neither the large recoil (Eq. 22) nor the narrow wavepacket (Eq. 23) conditions are satisfied. This regime is depicted in row C of Fig. 3. This regime is analogous to "anomalous PINEM" in the general case of PINEM interaction [37]. Evidently, one should avoid this chirp-dominated topological "no-man land" in phase-space, which does not reflect the TLS-QS qubit coordinates. This can be done by optical control of the wavepacket size and chirp as shown in [16,17].
The control of the size and chirp of the QEW requires advanced optical techniques. In particular, the azimuthal phase interrogation scheme requires sub-optical-cycle shape control. Substantial progress has been achieved in QEW shape control by THz and IR chirp techniques [16,17] and other schemes [18]. Assuming such control is available, the QEW breadth, chirp and waist-location parameters can be controlled at the QEW preshaping box (Fig. 1). This control is necessary if one wants to operate at the short wavepacket regime, in order to make sure that the history-dependent QEW duration (Eq. 15) still satisfies the short wavepacket condition (Eq. 23) at the interaction point = 0 after drift, and avoid slipping into the undesirable regime of anomalous FEBERI depicted in row C of Fig. 3. Then the natural dispersive broadening of the QEW (Eq. 15) sets a practical limit on the drift length 3 before interaction. The minimum of the broadened QEW size . Therefore, to satisfy the condition of a narrow Gaussian QEW Γ = ω #,& ! ( 3 ) < √2, one needs to limit the drift length before interaction to

IV. INTERROGATION OF TLS WITH OPTICAL-FREQUENCY DENSITY MODULATED QEWS
In this section, we focus on the interaction between a TLS and an optical-frequency modulated QEW (both momentum and density modulations). In sub-section A we derive the development of density modulation in a finite size QEW after being momentum (energy) modulated in a PINEM process followed by free-space drift. In sub-section B we derive the FEBERI post-interaction spectrum of such a modulated QEW. We find good agreement between our numerical and analytical results and demonstrate the possibility of coherent control of the TLS-QS through density-modulated QEWs.

A. Density modulation of a QEW
Shaping of QEWs to sub-optical-cycle size can be done by a wavefunction chirping and drift process using THz or Infra-Red (IR) beams [16,17]. An alternative way of shaping the QEW is by applying the PINEM process (Photon-Induced Near-Field Electron Modulation/Microscopy). In this process a broad QEW interacts with the near field of a laser beam illuminated structure, and gets energy modulated at the laser frequency $ by a process of multiple emission and absorption of photon quanta ℏ $ [19][20][21][22][23]. Such optical frequency modulation of QEWs has been demonstrated experimentally also by other laser-electron interaction schemes using dielectric structures, foils and the ponderomotive potential of laser beams beat [22,25,26,44,73]. It has been demonstrated experimentally that such an energy-modulated QEW becomes density modulated after drift at the laser modulation frequency and its harmonics, and produces an array of Attosecond scale current bunches [29,33,38]. Such short sub-bunches can play the same role as a near-point-particle single QEW [11]. Thus, in the case of FEBERI interaction with pre-modulated QEWs, the pre-shaping box in Fig. 1 represents a PINEM interaction setup, and 3 is a drift distance at which the QEW develops density modulation of short (attosecond scale) sub-bunches.
The PINEM and the attosecond density modulation process have been analyzed in detail in several previous publications [19,21,[74][75][76]. We iterate the exact derivation in Appendix D, and present here a simple useful approximation for the bunching coefficient in the practical case of a long QEW (relative to the optical modulation period). Though the physics of the FEBERI process is best comprehended intuitively in space-time coordinates, it is most straightforward to present the PINEM process in momentum space. Therefore, we start with the expression for the spectral sidebands structure of a PINEM-modulated Gaussian QEW in momentum presentation [19,43]: where M = ℏ $ / ' is the electron recoil momentum, n is the order of an absorption or emission sideband, ' is the phase of the centroid of the QEW envelope relative to the near field generated by the laser at the PINEM modulation point [63] (see appendix F), M is the coupling parameter of the interaction between the laser field and the electron ( M = + #ℏ; * ∫ ( ) ,/]^2 ) [19], and it is assumed that the initial momentum (energy) spread of the QEW is small -4 ≪ M .
After drifting freely a distance 3 , the QEW in momentum representation is found by applying the evolution operator ,/_ (5 ! ) /ℏ with '( being the second order expansion of the free space Hamiltonian (Eq. 4), and where 3 = 3 / ' is the free-drift traversal time of the electron: Thus, we can derive the wavefunction in the spatial domain by Fourier transformation 2 ( , ) = ∫ 4 ( ) /42/ℏ , and the density probability distribution of the QEW is: 2 = | 2 ( , )| # . This space-time density modulation of the QEW was computed in Appendix F as a function of the drift length and the moving coordinate − ' of the wavepacket envelope centroid, for the parameters presented in Fig. 4. The density distribution is composed of a train of attosecond scale short subbunches with periodicity (nearly) equal to the laser bunching period $ = 2 / $ , with a small frequency chirp due to the quadratic electron momentum (energy) dispersion. The modulation amplitude varies with 3 in macroscopic scale (centimeters). It attains maximum modulation at 3 = 1.5 (Fig. 4b). The pattern repeats itself periodically (the temporal Talbot effect [77][78][79]).
An important figure of merit of the QEW density distribution in its interaction with radiation or other harmonic fields is the spectral bunching parameter:

26)
This parameter, also termed the coherence coefficient [58,80], is consistent with the corresponding definition of point-particle beam spectral bunching parameter in accelerator and FEL physics terminology [51,81]. In the practical case of a long QEW envelope, the frequency chirp is small, and the density distribution may be written as a product of the envelope and density modulation functions [38] analogously to presentation in the classical bunched point-particle beam regime [82]: where the envelope function is the absolute value square of the history-dependent broadened Gaussian QEW (Eq. 14) , $ M is the carrier envelope phase (CEP), and the modulation function is periodic and composed of harmonics of the bunching frequency: corresponds to an over-bunching position. The beam energy is ' = 205 , the (laser) bunching frequency is $ = 3 × 10 &\ / , the envelope size of the QEW at = 0 is !' = 2 $ = 4.2 , and the phase ' = 0. (b) The density distribution at different drift lengths. Continuous dependence of the density distribution on the drift length is shown in supplementary Movie S4 [70].
Thus, the spectral bunching parameter is where for the Gaussian modulated QEW (Eq. 25), the bunching parameter O of harmonic frequency m can be expressed in absolute value by the simple expression (Appendix F) [58,80]: This parameter attains its maximal value for the -th harmonic is the argument that makes the -th order Bessel function maximal. The maximum bunching point is at a drift length location: where f = 4 --+ -/ℏ $ # is the temporal Talbot distance [58,80]. Interestingly enough, f = 2 # Y for the resonant case #,& = $ , where Y was defined in Eq. (21) in connection to broadening of a finite Gaussian QEW in free space, both phenomena originate from the quadratic term in the energy dispersion HG|U * | for = 1, which is the classical limit expression of maximal bunching [58,80], and consistent with corresponding expressions for maximum point-particle beam bunching in accelerator and FEL theory [51,81].

B. FEBERI with a modulated QEW
The analysis of the FEBERI interaction with a modulated QEW (Eq. 25) is similar to the analysis for a single finite size QEW, except that instead of (Eq. 13), one should substitute the modulated wavefunction expression (Eq. 25) in the generic expressions for the post-interaction free electron energy density distribution (Eq. 10).
The initial momentum distribution in this case is 4  Equations (32)(33)(34) and Figs. (5a) and (5b) show that after interaction, the multiple sidebands structure of the PINEM-modulated QEW remains peaked at = ' + M , but the momentum density distribution is redistributed due to FEBERI induced recoils between the sideband orders. Similar to the case of unmodulated QEW, the zero-order modification of the density distribution is negative and independent of the TLS-QS qubit parameters, the second order term depends on only, and the first order term depends on both and (through ). Evidently, with | ≪ 1|, the zero-order and second-order terms (proportional to | | # ) are much smaller than the first-order term (proportional to | |), so Δ (&) is the dominant term for the incremental spectrum, except near the poles of the Bloch sphere.
We first consider the case without drift, 3 = 0 or short drift, such that the recoil phase shift is The It is instructive to note that substitution of 3 = 0 (or | M | = 0) diminishes all the harmonics except = 0 and reduces Eq. (35) to the expression for an unmodulated Gaussian QEW (Eq. 20). When 3 = 0 , there is significant first order incremental change in all sidebands of the energy spectrum (Eq. 33, Figs. 5a, 5b), however, there are no first order incremental transitions of the TLS (the second term in Eq. 35 vanishes). Finite first order transition probability of the TLS and corresponding acceleratin/deceleration of the QEW are possile only when 3 ≠ 0. Their dependence on ( , ) is depicted in Figs. (5e) and (5f). The physical explanation for that is that only then density modulation takes place, and in time-space domain the QEW sub-bunches acquire phase, and can interact with the TLS like near-point-particles in correspondence to the Born interpretation of the probability density of the QEW. Note that in the case of a modulated QEW (Figs. 5a-5d), the spectral peaks of the energy spectrum are three orders of magnitude larger than with a long unmodulated QEW (Fig. 2d). This is true for both cases of 3 = 0 and 3 ≠ 0, However in the first case no first order quantum transitions take place, and the TLS-QS stays nearly intact, and in the second case there is first order change of the ocupation probabilities of the TLS and the polar angle of the qubit in the Bloch sphere. For this reason it is preferable to use the short drift case for non-destructive probing of the TLS-QS, and the maximum bunching finite drift length case -for cohernt quantum control.

V. OFF RESONANCE FEBERI AND TEMPORAL EVOLUTION OF TLS TRANSITIONS
Except for the small second-order term, the dominant first order term in Eq. (35) indicates that the FEBERI process of a modulated QEW requires satisfaction of a resonance condition (Eq. 2) in order to attain controlled TLS quantum transitions. However, this resonance requirement is not sharp when the size of the envelope of the QEW 2' , is finite. This is also depicted by the numerically computed curves in Fig. 6 that are in full agreement with the analytical expression. The shorter the size of the QEW envelope, the wider is the tolerance of transition probability to off-resonance interaction, however the envelope size must be kept short enough to keep Γ 3 < 1, to avoid the exponential decay in Eq. (35).
It is intriguing to get deeper insight into the off-resonance decay of the transition probability by examining the temporal evolution of the transition probability during the interaction period of the modulated QEW with the TLS. The numerically computed incremental occupation probability of the upper TLS level is shown in Fig.  7 as a function of interaction time comparatively for the case of resonant and off-resonant interactions. Comparing the curves of temporal dependence of TLS quantum transition probability (Fig. 7b) to the density probability curves of the modulated QEW (Fig. 7a), we see that the "quantum jumps" of the TLS take place at times commensurate with the period of the quantum transition frequency #,& = 2 / #,& whether the modulated QEW is at resonance or not. When the bunching is out of resonance, the increments of quantum jump probability diminish towards the end of the interaction time, and consequently the measurable postinteraction incremental transition probability is smaller than in resonance setting. Evidently, this happens because the QEW bunching gets out of phase with the natural dipole oscillation of the TLS.  It is instructive to examine the quantum transition process from the point of view of Born's interpretation of the electron wavefunction [72]. This is possible in the present theory since the quantum wavepacket formulation encompasses both limits of the particle-wave duality in spatiotemporal presentation. First consider a finite size (unmodulated) QEW (Eq. 20). In the wavelike limit of a long QEW and small energy spread (Eq. 22), the first order transition probability decays exponentially, and what is left in the transition probability (Eq. 20) is only the second order term, which is independent of the azimuth angle of the qubit. On the other hand, in the near-point-particle limit of the QEW (Eq. 23), the dominant term in the transition probability (Eq. 22) is the azimuth-phase-dependent first order term. This is Consistent with the probabilistic Born interpretation, since in the latter case there is high probability for arrival of the QEW at the FEBERI interaction point with well-defined phase relative to the oscillating dipole moment of the TLS.
The case of a density-modulated QEW is more ambiguous. On one hand, the envelope of the modulated QEW is long relative to the modulation period, but on the other hand, the duration of the density modulation sub-bunches (attosecond scale -see Fig. 4) satisfies the short wavepacket condition (Eq. 23). Since we deal here with a single electron, the sub-bunches do not represent individual electrons as in the case of the semiclassical "optical klystron" [54]. Instead, all the sub-bunches in the time-space distribution shown in Fig.  (4b) represent a single electron. According to the Born interpretation, the electron has highest probability to arrive to the FEBERI interaction point at a time corresponding to one of the peaks of the sub-bunches and appear as a near-point-particle QEW. The analytical expression (Eq. 35) and Fig. 7 affirm the intuitive interpretation of the sub-bunches as near-point-particle wavepackets. Evidently, the "quantum jumps" temporal evolution curves of Fig. 7 do not represent the evolution of the TLS due to passage of a single electron, but only the expectation value of # over multiple modulation-phase correlated QEWs, whereas in each individual FEBERI event, only one of the sub-bunches of the QEW is responsible for the "quantum jump". With multiple correlated electrons, these increment jumps of the transition probability would take place in synchronization with the TLS dipole oscillation frequency #,& . This happens around the random times, of the QEWs centroid arrival within the QEW envelope duration. This observation is crucial for the generalization of the FEBERI effect to the case of multiple modulation-correlated QEWs, where coherent build-up of TLS transitions proportional to # is expected [11,48,55,56]. It gives a lead to a Born interpretation inspired concept, in which the joint wavefunction of a pulse of modulation-correlated QEWs (e.g., as proposed in [57]) can interrogate and coherently control the TLS-QS collectively.

VI. DISCUSSIONS AND CONCLUSIONS
In this article we present a scheme for diagnostics and coherent control of the superposition quantum state of a TLS in a variety of possible targets of interest, including, but not limited to atoms, quantum dots and crystal electron defect centers. Such targets can be addressed by an electron beam with atomic-scale spatial resolution. Hence, this scheme may be useful for such applications as full characterization of the TLS state as a multi-qubit on a single Bloch sphere in quantum information technology [6,7], measurement of decoherence time [61] and selective coherent excitation of quantum emitters [66].
The interrogation scheme is based on the recently proposed FEBERI process [11]. The electron quantum wavefunction is pre-shaped after its emission from the cathode by coherent processes of optical chirping and streaking at THz or IR frequencies, followed by drift [16,17]. This causes it to arrive at the TLS location in a controlled shape -longer or shorter than the TLS transition cycle #,& = 2 / #,& (of the order of femtosecond). Alternatively, the QEW may be energy modulated before the FEBERI by a laser beam in a PINEM interaction process at a sub-harmonic frequency of the TLS transition frequency #,& . After free drift, the energy modulated QEW becomes periodically density-modulated (bunched) at the PINEM modulation frequency with tight (attosecond) sub-bunches, and contains high frequency spatio-temporal harmonics of the modulation frequency. In either case, the FEBERI interaction with an excited TLS-QS results in a postinteraction spectrum of multiple repeated measurements that reflects the quantum state of the TLS. This can be used for diagnostics of the TLS-QS qubit parameters on the Bloch sphere, and for coherent control of the TLS-QS.
Determination of the polar angle of the TLS-QS can be carried out by using narrow energy spread QEWs that satisfy the large recoil condition #,& > 2 7 (Eq. 22, Fig. 3a). This requires sub-eV electron energy spread, which is readily available in electron microscopy. The post-interaction incremental energy density distribution of such an electron beam depicts a PINEM-kind spectrum of two asymmetric sidebands (Fig. 2d, Movie S2). The degree of asymmetry indicates the polar angle of the pre-set interrogated TLS-QS. The determination of the TLS-QS azimuthal angle is less straightforward. It necessitates forming narrow QEWs that satisfy the near-point-particle limit condition 2 ! ( 3 ) < #,& (Eq. 23, Fig. 3b) at the FEBERI interaction location. The resulting post-interaction incremental energy density distribution depicts in this limit an acceleration/deceleration spectrum that depends on the phase increment of the TLS-QS, accumulated from the time of TLS pre-excitation up to the arrival time of the QEW centroid: = #,& 3 − − M . For this measurement, the TLS excitation and the timing of the QEW beam must be phase locked, and the TLS-QS azimuth angle can be then determined by judicial adjustment of the optical path difference between the two laser beams that determines the reference phase M (see Fig. 1).
The alternative proposed method for interrogating the TLS-QS using optical frequency modulated QEWs is expected to be more accessible in the lab. It requires energy modulation of the QEWs before the FEBERI point by means of the well-established PINEM process. We have demonstrated that such modulated QEWs may be used for interrogating the TLS-QS qubit parameters ( , ) in two different modes: without drift after PINEM ( 3 = 0) and with finite drift ( 3 ≠ 0). In the first case the QEW is only energy modulated (PINEM sidebands) and there is no density modulation, the post-interaction incremental spectrum is either symmetric (for = ) or antisymmetric = , and is dependent on both ( , ) (see Figs. 5a, 5b). There is no net energy transfer between the TLS and the QEW in this case. In the second case, density bunching of the modulated QEW is formed due to the pre-interaction free drift, and the post-interaction incremental spectrum is asymmetric and dependent on ( , ) (Figs. 5c, 5d). There is then net energy transfer between the QEW (that is accelerated or decelerated -see Figs. 5e, 5f) and the TLS. Consequently, the TLS changes its quantum levels occupations ratio (measured by the qubit parameter ).
In both cases of modulated QEW, with and without drift, the incremental spectrum signal is significantly larger than in the case of an unmodulated long QEW (for the parameters used -three orders of magnitude more in comparing Figs. (5a-5d) to Figs. (2d), (2e). It is however of the same order of magnitude as in the case of a short QEW (Fig. 2a). The reason is that the azimuthal phase-dependent first order term ( ∆ (&) ∝ | |) dominates in both cases over the second order term (∆ (&) ∝ | | # ) for | | ≪ 1. Further, we draw attention to the fact that there is energy transfer only in the case of short sub-cycle QEW (Fig. 2b, 2c) and the case of modulated QEW after drift 3 ≠ 0 (Figs. 5e, 5f). This should be related to the development of bunching with attosecond-scale sub-bunches after drift, as shown in Fig. 4. In the spatio-temporal presentation, we are led by Born's interpretation of the quantum electron wavefunction to conclude that these sub-bunches can be considered as multiple near-point-particles at the FEBERI interaction point. Therefore, similarly to sub-cycle QEWs in the near-point-particle limit, they have well defined phase relative to the phase of the TLS dipole oscillation, and in both cases the phase dependent first order term (∆ (&) ∝ | |) in (Eq. 20, Eq. 35) facilitates the transfer of energy between the QEW and the TLS. This observation and the analysis of the temporal development of TLS transitions in Fig. 7 of the previous section, have important ramification to the extension of the single modulated QEW FEBERI concept to multiple modulation-correlated QEWs [11,48].
The laboratory realization of the proposed concept of TLS-QS interrogation and coherent control using free electrons, faces substantial theoretical and conceptual challenges as well as technological implementation challenges that we address below. In the first place, any projective measurement of a TLS quantum state by EELS technique is destructive upon every EELS data measurement event. Therefore, as we indicated in section II, the measurement of our predicted post-interaction energy density distribution by EELS requires reinstating the TLS-QS after each EELS measurement data collection event in a way similar to experimental practice in super-elastic scattering research [83]. This limits the application of the proposed scheme to cases where the TLS-QS targets are reconstructable, and the multiple QEWs are identical. For example, we conceive an application of monitoring the Bloch sphere time-evolution trajectory of laser beam coherent control of a TLS [84] with an electron beam. Realization of such an application requires that the laser coherent control process is reinstated identically after every destructive EELS data collection by the interrogating QEWs. We exclude consideration of nondestructive measurements. These have been demonstrated and proposed with photons [85,86] and electrons [87].
On top of this fundamental limitation on EELS measurement, there are also technical impediments on realizing a true energy spectral distribution by means of multiple single electron EELS measurements. In the first place, the cross section for an inelastic scattering event is very small (in our examples, ~10 ,l per electron for a broad unmodulated QEW, and ~10 ,per electron for a modulated QEW). Another limitation is that any single event measurement of the qubit parameters ( , ) must take place within the decoherence time # and the relaxation time & . One can conceive a scheme of accumulating EELS data with a pulse of multiple QEWs interacting with multiple TLS centers, excited all with a laser pulse to the same state (for example, employing multiple crystal defect centers implanted on a crystal surface). However, the single atom resolution advantage of the FEBERI scheme is then lost, and for the same reason, there is no possibility then of interrogating the azimuthal phase , only the average polar angle of the TLS ensemble can be interrogated.
Further investigations are needed for identifying TLS targets of bigger dipole moment and coupling coefficient , e.g., perovskite [88]. We point out here also to another potential scheme for enhancing the FEBERI interaction process and amplifying the spectral energy distribution signal in the interrogation of a TLS-QS. In this scheme one would place multiple ( ) ) TLS targets within a volume smaller than ' / #,& , such that they can all be excited in phase to the same quantum state by the exciting laser. In this case, a pulse of uncorrelated QEWs interacting each with a single TLS in the lump of TLS can produce multiple EELS data points without destruction of the qubit state of other TLS targets in the lump. Furthermore, if the TLSs are lumped within the range of the QEWs field, they may act as a "super-qubit" with dipole moment and interaction coupling parameter that are ) times bigger, and then the EELS probing signal, as well as the transition probability, would be enhanced correspondingly. This situation is similar to the concept of superradiant emission, induced by free electrons [64]. In fact, besides enhancing the EELS measurement, such an excited multiple-TLS "super-qubit" is expected to emit superradiant luminescence proportional to ) # after the radiative relaxation lifetime of the TLS [61].
With a "super-qubit" of many TLS targets, it is possible to keep the high spatial resolution addressing advantage of a focused electron beam. However, if the electrons of the beam are uncorrelated, it is still necessary to reinstate the TLS-QS after each measurement event, in order to collect enough data for constructing the EELS. The "Holly Grail" of TLS-QS interrogation by FEBERI would be achieved if a pulse of optical frequency modulation-correlated QEWs can be produced within the duration of the decoherence or relaxation time. In this case one may conceive that the joint wavefunction of a pulse of modulation-correlated QEWs (e.g., as proposed in [48]) can interrogate and coherently control the TLS-QS collectively, similarly to a laser beam, but with high spatial resolution. This concept requires further intended theoretical considerations beyond the scope of the present publication (for example of the entanglement between the interacting electrons), and certainly further technological development.
Finally, we draw attention to the rapid development of the field of quantum electron-optics and the marriage of integrated photonics with electron microscope technology, such as enhanced coupling of light through micro-resonators [89,90,91] and embedding single ions in a nanophotonic cavity [7]. These have the potential of bringing about new development directions of the FEBERI concept, providing more degrees of freedom in control and measurement of electron-light-TLS interaction. For example, we may envision schemes for extension of TLS decoherence time by suppression of spontaneous emission in a microcavity and electronlight interaction schemes with multi-cavity, multi-TLS integrated photonics circuits.
Author contributions: A.G., A.Y., J.S., and D.R. conceived the concept. B.Z. and D.R. performed the theoretical derivations and prepared the figures and videos. B.Z., D.R. and A.G. wrote the paper with contributions from R.I. and A.F. All authors reviewed and discussed the manuscript and made significant contributions to it. The interaction Hamiltonian in the interaction picture is defined as: where the operator e ps ( K t"/ℏ is the translation operator in coordinate domain, and σ { > = |2〉〈1| ( V , = |1〉〈2|) is the TLS raising (lowering) operator.

APPENDIX B: DERIVATION OF THE SCATTERING OPERATOR G . , 2
With the explicit form of the interaction Hamiltonian, we can calculate the wavefuction after the interaction by multiplying the initial wavefunction with the scattering operator: where is the time-ordering operator. The scattering operator can be expanded to a Dyson series where The first order term in this series is easily obtained: Replacing the integral variable t by = + ' yields The integration on the field presents the coupling coefficient between the free electron and the TLS: g = & ℏ@ ( #,& ⋅ ∫ ( , 1' ) ,/; #," u/@ ( ,`. Substituting the field (z V, 1' ) into (Eq. S8) yields: Considering laboratory available parameter values -TLS energy gap of #,& = ℏ #,& = 1.97 , free electron velocity of ' = 2.1 × 10 v / (β = 0.7) and impact parameter r 1' = 2.0 nm, we find the coupling coefficient g to be in the order of 10 ,-. Thus, it is reasonable to neglect the higher order terms in Dyson series. Hence, the scattering matrix can be written as

APPENDIX C: DENSITY DISTRIBUTION AND TRANSITION PROBABILITY AFTER INTERACTION
For any input state tψ v / 〉 = t ' ( 〉 ⊗ t ' ) 〉, the general form of the final state is obtainted by applying the scattering operator: . Since the free electron and bound electron are entangled after the interaction, it is convenient to represent the final state by the density matrix In order to obtain the density distribution of the free electron and the transition probability of the TLS, we partially trace the density matrix of the final state, setting ρ V ( ? = Tr w • V ? -, ρ V ) ? = Tr x • V ? -. Consequently, the momentum density distribution of the free electron is given by: is the autocorrelation function [42], then the final density matrix of the TLS is The additional phase term is determined by the electron energy dispersion relation 4 Combining the quadratic term of ( − ' ) yields We define the momentum spread after propagation as § 4 ( ) = î Consequently, the wavefunction in coordinates domain is given by: where the chirped wavepacket size is and σ y' = ℏ/2 4 . When considering the density distribution | 2 ( )| # , the wavepacket size (standard deviation of the density functionenvelope) is Thus, even if the initial wavepacket σ "' approaches 0, the real wavepakcet is always finite. According to the inequality of arithmetic and geometric means | | + | | ≥ 2Å| |, we can derive the lower limit of the wavepacket size for any given free drift time 3 : We find that in order to mitigate the QEW envelop broadening effect, and satisfy the near point-particle condition at the FEBERI interaction point Γ = ω #,& ! ( 3 ) < 1, the QEW free drift length must be kept within the limitation 3 < Y , where

APPENDIX E: DERIVATION OF POST_INTERACTION ELECTRON MOMENTUM DISTRIBUTION AND TLS TRANSITION PROBABILTY FOR THE CESE OF A GAUSSIAN QEW
The general form of the first-order free electron post-interaction density distribution (Eq. C5) is With the initial electron wavefunction (') , we obtain: Substituting Eqs. (E3) and (E4) into the expression for the first-order incremental momentum distribution, Eq. (E1), yields where the envelop decay factor due to energy dispersion and drift is defined as We finally get The numerically computed incremental electron energy distribution Δ 7 ! is depicted in Fig. 2 of the main text.
The continuous parameters dependence version of the cases shown in Fig. 2 are presented in movie: (S1) the dependence on in the case of a short QEW; (S2) the dependence on in the case of a long QEW; (S3) the dependence on both and in an intermediate case:

APPENDIX F: DETAILED ANALYSIS OF DENSITY BUNCHING
In this section, we derive the bunched density distribution ( , ) of the QEW after energy modulation and drift. We start with the energy modulation of the QEW modeled by a Gaussian distribution in momentum domain, as given in Appendix D, the QEW can be written as |ψ The energy modulation can be described by the scattering operator • ( , ) ⋅̂ describes the coupling between the free electron and the laser-induced optical near-field, ' represent the the arrival time of the QEW centroid to the center of the interaction region and * is the interaction time. For simplicity we consider here the case of PINEM modulation in a case of "transition radiative interaction, namely the case where the spatial extent of the near field of a laser milluminated nanostructure (e.g. a tip) is shorter than the optical wavelength of the laser [19]. In this case the QEW hardly senses the spatial variation of the time-varying near field during the interaction, and the vector potential can be generally written as • ( , ) = ' sin ƒ M f 2@ ( − g + / ", where / is the initial phase of the near-field at the interaction time ' . Since the initial momentum distribution of the QEW (Eq. D1) is narrow, 4 ≪ ℏ is valid to assume | ⟩ ≈ ' | ⟩. Thus, the explicit form of scattering operator is According to the Jacobi-Anger expansion, the scattering matrix can be presented in terms of a serries: where the new coefficient 4 of the energy-modulated QEW has the form where M = ℏ M ' ⁄ . Thus, the momentum presentation of the wavefuction after drift: is the same as Eq. (D3), which using the standard Gaussian integral results in Since in the case of a modulated QEW the envelop must be broad, necessarily 4 = ℏ/2 2 ( → 0, so the chirp effect is ignorable and After neglecting high orders of 4 , we find that the spatial wavefunction ( , 3 can be written as: where ( ) = − ' 3 . Thus, the density distribution of the modulated electron is is the Talbot length and 3 = ' 3 is the free drift propagation length. According to Graf's addition theorem, the density distribution is The evolution of the density distribution of a modulated QEW, corresponding to Fig. 4, is shown in the Movie S4.

APPENDIX G: DERIVATION OF POST_INTERACTION MOMENTUM DISTRIBUTION AND TLS TRANSITIOIN PROBABILITY FOR A MODULETED QEW
In section S. C, we derived a generic expression (Eq. C5) for the post-interaction density distribution of the free electron 4 ? . For the case of a Gaussian QEW that is momentum-modulated by a PINEM process, and then traverses a drift-time 3 , we plug in there the initial state in momentum presentation 4 where ' is the initial phase of the modulation laser and M = ℏ $ / ' , $ is the frequency of the modulating laser. With this wavefunction, Eq. (C5) results in the following expression for the post-interaction density distribution of the free electron: We first calculate the zero order and second order incremental density distribution terms in (Eq. G1). Since the quantum recoil induced by the modulating laser is much larger than the momentum spread of the QEW ℏ; * @ ( ≫ 4 , the overlap between the sidebands is negligible, and we can make the approximation: Thus, for being odd, the final incremental spectrum would be symmetric, while for being even, it is antisymmetric.
We proceed now to calculate the incremental transition probability of the TLS that corresponds to the incremental distribution of the modulated QEW. For this we need first to evaluate the autocorrelation function for the modulated Gaussian case: With discretization of the time domain, the evolution of entangled free-electron and bound electron system is expressed as • with = />& − / . Finally, we get