Ученые разработали протокол для создания запутанности между атомами и свободными электронами

Abstract We propose a heralded protocol for transferring atomic single-excitation entanglement to free electrons in a sideband-resolved interaction geometry. Each electron couples locally to one two-level system, and projection of the atomic subsystem onto the all-ground state maps the atomic excitation manifold onto the electronic upper-sideband manifold. For uniform couplings and common detuning, the rotating-wave dynamics is solved analytically and yields an exact phase-matched electronic state. At resonance, the optimal intrinsic heralding probability scales as . The multipartite structure is characterized for arbitrary and illustrated for through correlation redistribution and weighted single-excitation transfer. A local sideband-measurement decomposition expresses the fidelity-based witness in terms of local observables. Robustness analyses include detuning, arm asymmetry, counter-rotating corrections, decoherence, heralding latency, and smooth coupling envelopes. The protocol provides a concrete framework for multipartite free-electron entanglement generation from localized atomic resources within quantum electron optics. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Quantum entanglement is a central resource for quantum teleportation [1], quantum cryptography [2], quantum communication [3], quantum computing [4], and tests of quantum mechanics [5, 6]. Over the past decades, entangled states have been generated in a broad range of physical platforms, including photons [7, 8], neutral atoms [9, 10], superconducting circuits [11, 12], and cavity quantum electrodynamics (cavity QED) systems [13, 14]. Experimental and theoretical progress has enabled the preparation of Bell states [15–17], Greenberger–Horne–Zeilinger states [18, 19], and states [20, 21] through a variety of mechanisms, including quantum Zeno dynamics [22–24], adiabatic passage [25–27], measurement-feedback control [28, 29], and dissipation engineering [30–32]. Among available routes to entangled-state preparation, entanglement transfer is especially useful because controlled interactions can map quantum correlations from a prepared resource to a different physical carrier [33–37]. A central obstacle to selective atom–field entanglement transfer at individual targets is the limited spatial resolution of optical addressing, which remains challenging for implementations requiring single two-level system (TLS) control. More recently, quantum electron optics (QEO) has emerged as a platform for coherent preparation and manipulation of free-electron wave packets through photon-induced near-field electron microscopy (PINEM), electron energy-loss spectroscopy (EELS), and cathodoluminescence [38–43]. QEO enables ultrafast nanoscale spectroscopy and quantum-state engineering by exploiting shaped electron–matter interactions to imprint, probe, and herald quantum correlations in emitted or scattered bosonic fields [44–50]. Direct experimental verification of free-electron–photon entanglement has been achieved through coincidence-based measurements and quantum imaging protocols in transmission electron microscopy platforms [51, 52]. These advances establish QEO as a realistic platform for generating, manipulating, and probing quantum correlations involving free-electron wave packets. In particular, free-electron–bound-electron resonant interaction (FEBERI) provides a well-defined mechanism for coupling a quantum electron wave packet (QEW) to an atomic TLS, with interaction strength governed by the longitudinal density modulation of the QEW [53]. Building on such a mechanism, engineered QEWs enable efficient excitation and quantum-state interrogation of TLSs [54–56], while recent studies have further extended free-electron quantum interactions to a broader range of bosonic and material excitations [57–62]. While recent studies have mainly focused on quantum correlations and entanglement between free electrons and photonic degrees of freedom, the controlled generation of multipartite entanglement directly between free electrons remains largely unexplored. In parallel, entanglement-swapping and quantum-eraser protocols have also been proposed as promising approaches for exploiting electron-mediated quantum correlations in nanoscale probing and quantum-enhanced measurements [63]. Free-electron entanglement is of particular interest because entangled electron pairs can support superradiant and subradiant light emission [64], while symmetric multi-electron entangled states can enable enhanced deterministic photon generation [65]. Building on the two-electron protocol in [66], we consider an -arm geometry in which each free electron couples locally to one TLS prepared as part of a single-excitation atomic resource. Within the RWA, projection onto the all-ground TLS outcome maps the atomic excitation coherently onto the electronic upper-sideband single-excitation manifold. The multipartite analysis yields the optimal scaling , the genuine-entanglement threshold , and a variance relation for fidelity loss under arm-dependent perturbations. The case illustrates time-resolved correlation transfer and the mapping of weighted single-excitation states. A local sideband-rotation scheme is developed for measuring the required populations and coherences. The discussion further quantifies detuning and symmetry-breaking errors, counter-rotating corrections, TLS decoherence and heralding latency, pulse-shape effects, representative coupling strengths, and detected event rates. The remainder of the paper is organized as follows. In section 2, we introduce the system model and derive the single-arm detuned rotating-wave dynamics. Section 3 develops the general framework of multipartite heralded entanglement transfer and includes the success probability, the entanglement structure of the heralded state, and the explicit tripartite example. The physical implications of the protocol, together with its robustness and limitations, are discussed in section 4. Section 5 concludes the paper. 2. System model We consider free electrons interacting with nearby TLSs, with labeling each electron–TLS pair. Each free electron couples only to its corresponding TLS, forming a local interaction geometry with independent arms. The electron labels denote spatial arm modes rather than intrinsically distinguishable particles. The wave packets remain spatially separated with negligible mutual overlap, so fermionic exchange corrections are negligible and the arm-resolved tensor-product description reproduces the properly antisymmetrized dynamics for the observables considered here. Direct inter-arm couplings are also neglected. Such a spatially local interaction picture is natural in free-electron quantum-optical settings, where an electron couples to a localized quantum system through a near field or an effective dipolar interaction [53, 55]. The resulting dynamics can therefore be built from elementary local electron–TLS exchange processes. In the sideband-resolved description, the basis state denotes a longitudinal energy component of the th propagating electron wave packet, centered at . The corresponding longitudinal momentum is determined by the free-electron dispersion relation. The index counts the net number of exchanged energy quanta relative to the reference component and does not label a phonon mode. A propagating electron wave packet may therefore be represented as a coherent superposition of different energy-sideband components. In a PINEM implementation, the electron sideband superposition can be prepared through coherent interaction with an optical premodulation field of frequency [67, 68]. For the adjacent-sideband channel considered here, , and neighboring electron components are separated by . The corresponding sideband-number operator is Such a representation is standard in free-electron quantum optics whenever the interaction induces coherent single-quantum transitions between adjacent electron energy sidebands. We therefore define the sideband translation operators and which satisfy , , and . Thus, and translate the electron along the energy-sideband ladder and are not bosonic creation and annihilation operators. For the th TLS, we denote the ground and excited states by and , and define the Pauli ladder operators , , and . The free Hamiltonian of the considered system is where is the TLS transition frequency. The first term describes the free evolution of the electron sideband ladder, while the second term gives the bare TLS splitting. The bilinear interaction Hamiltonian reads where is the effective time-dependent coupling in arm , set by the electron trajectory, the TLS transition dipole, and the finite interaction window. Equation (2) is the sideband-resolved effective form of the electric-dipole near-field interaction, assuming an isolated TLS transition and dominant single-quantum coupling between adjacent electron sidebands. The corresponding coupling scale and its dependence on the transition dipole, electron energy, and impact parameter are quantified in section 4.5. In the interaction picture with respect to , the interaction Hamiltonian becomes where and . In a PINEM-based implementation, the adjacent-sideband spacing is determined by the optical premodulation frequency, such that . The detuning can therefore be controlled by tuning or by shifting the TLS transition frequency through a Stark or Zeeman field. The first two terms in equation (3) describe excitation exchange between the electron sideband ladder and the TLS. The last two terms are counter-rotating contributions oscillating at the fast frequency . In the near-resonant regime , the rotating-wave approximation (RWA) gives Within the RWA, each local arm supports only the near-resonant single-quantum transfer channel. We assume that all incoming electrons are prepared in the same reference sideband , namely . For the atomic TLS sector, we consider the general single-excitation -type resource where are relative phases, and the symmetric state corresponds to for all . The preparation of -type resources is supported by substantial progress in atomic and atomic-like platforms. Multipartite -type entanglement has been experimentally generated and fully characterized for four to eight trapped ions [69], while deterministic generation of large-scale atomic states in cavity QED [21] and dissipative preparation of tripartite atomic states in Rydberg-atom-cavity systems [70] have also been proposed. The total initial state of the system is . In the following, we first consider the symmetric special case (), for which all local interaction arms have identical coupling envelopes. Because the RWA Hamiltonian is a sum of mutually commuting local-arm generators acting on distinct tensor factors, the full propagator factorizes as , where is the RWA propagator associated with the th local interaction arm. The dynamics of each arm is therefore determined independently by the corresponding single-arm propagator. We model the effective local coupling envelope by a ‘square pulse’, The accumulated pulse area is inside the interaction window, with . In a more realistic description, the local coupling envelope is generally smooth and determined by the electron trajectory and the spatial profile of the electron–TLS interaction region [53, 57]. For the detuned single-arm dynamics, we introduce the effective detuning parameter and the generalized pulse area as well as the associated local survival and transfer amplitudes The closed-form amplitudes apply specifically to the square pulse, with constant and throughout the interaction window. For a time-dependent coupling envelope , the amplitudes must instead be evaluated from the corresponding time-ordered evolution. Under the action of the single-arm propagator generated by the th local RWA Hamiltonian, the local basis states in the invariant subspace evolve as whereas, in the equivalent invariant subspace , the evolution is At exact resonance , the detuning-dependent amplitudes introduced in equation (8) simplify to For initial conditions in the reference sideband , the local RWA propagator yields To reconstruct the full multipartite evolution, it is convenient to introduce the -electron basis states for which only the th electron is shifted upward () or downward () by one sideband. The set spans the upper-sideband single-excitation manifold generated by the local RWA transfer process, whereas the set describes the corresponding lower-sideband sector associated with excitation of a TLS by the local electron. For the initial TLS component , the th arm is the only arm that initially carries the atomic excitation. Consequently, the th arm is the only arm that can transfer that excitation into the upper electron sideband, whereas all remaining arms contribute through local survival amplitudes and lower-sideband excitation channels according to equation (12). The resulting evolution of the basis component takes the form Summing these branch amplitudes over with the phases from the initial atomic resource gives The state contains the all-ground TLS branch, residual-TLS-excitation branches, and the associated lower-sideband electron sectors. 3. Heralded entanglement transfer 3.1. Heralded -electron branch The conditional state characterizes the electronic output after selecting the all-ground TLS outcome, whereas the heralding probability gives the weight of that branch in the full evolution. The distinction separates state quality from protocol efficiency. Projection of the total state in equation (15) onto the TLS outcome selects the branch in which all TLSs end in the ground state after the interaction. According to equation (15), each component initially excited in arm contributes the transfer amplitude from that arm and the survival amplitude from every remaining arm. All lower-sideband contributions are excluded because the corresponding branches retain at least one TLS excitation. The all-ground TLS sector therefore isolates the coherent upper-sideband image of the initial atomic single-excitation manifold. The resulting unnormalized heralded electron state is The associated heralding probability is The term is the transfer probability in the initially excited arm, while is the joint survival probability of the remaining spectator arms. After normalization, the conditional electron state in equation (17) becomes where denotes the phase pattern inherited from the initial atomic resource. Whenever , the normalized conditional state is independent of time and common detuning, whereas the branch probability is set by and through equation (18). For the symmetric atomic resource with for every , equation (19) reduces to The heralded protocol therefore maps the phase-resolved atomic single-excitation resource onto the corresponding electronic upper-sideband state. The conditional fidelity with respect to the phase-matched target state is where Within the symmetric detuned RWA treatment, Consequently, The unit conditional fidelity confirms coherent transfer of both the single-excitation structure and the phase pattern of the initial atomic resource. 3.2. Success probability, optimum, and large- scaling For the square pulse in equation (6) evaluated at , the dimensionless pulse area, detuning parameter, and generalized pulse area are Equation (8) yields Substitution of equation (25) into equation (18) gives the heralding probability at the end of the interaction window, At exact resonance, namely , one has and , so equation (26) reduces to The factor suppresses the heralding probability as increases. For , equation (27) gives , which coincides with the -heralding probability obtained for the two-electron protocol in [66]. For optimization in the resonant regime, the heralding probability may be written as . Introducing the variable , one obtains . The stationarity condition yields . The optimal pulse area therefore satisfies and the maximal heralding probability becomes For large , equation (29) approaches The result establishes an algebraic suppression of the optimal heralding probability rather than an exponential decrease with system size. For an internal consistency check, the same symmetric resonant RWA problem is evaluated numerically. The many-body state is reconstructed from products of the exact local propagators and equivalently satisfies with the initial condition given by the symmetric atomic resource and the product electron input state introduced above. The numerical heralding probability is then defined by projection onto the all-ground TLS outcome, where is the projector onto the TLS state with all atoms in the ground state. Based on equation (32), the numerical optimal pulse area is defined as and the corresponding numerical maximal heralding probability is Figure 1 shows agreement between the numerical construction and the analytical pulse-area dependence, optimum, and large- scaling. 3.3. Entanglement structure of the heralded state Define the upper-sideband target manifold as and its projector as . The manifold contains one upper-sideband electron and electrons in the reference sideband. For the unconditional reduced state , its population is . The branch structure in equation (15) gives because this manifold is correlated exclusively with the all-ground TLS outcome. The corresponding unconditional fidelity with respect to the ideal multipartite electron target is Because the target manifold is populated only through the heralded branch, one obtains All nonheralded branches are orthogonal to the phase-matched upper-sideband target. The operational distinction is therefore clear: the conditional state is ideal, but access to that state remains probabilistic. For the symmetric heralded state , tracing out all but an arbitrary electron pair gives where . The first term corresponds to excitation outside the selected pair, while the Bell component represents coherent sharing between electrons and . For the reduced state , partial transposition with respect to the second electron gives In the ordered basis , the matrix representation is The corresponding eigenvalues are The negative eigenvalue gives the pairwise negativity For large , equation (41) reduces to , which indicates that pairwise entanglement decreases algebraically with increasing electron number, even though the global heralded state remains genuinely multipartite entangled. Such algebraic decay is a characteristic property of the class. A single excitation distributed across parties supports a finite global correlation structure, whereas any fixed pair retains only a progressively smaller portion of that structure as increases. Certification of genuine multipartite entanglement requires a witness that excludes all biseparable states [71, 72]. For a bipartition containing electron arms and the complementary arms, the squared Schmidt coefficients of are and . The largest squared Schmidt coefficient, maximized over all bipartitions, is therefore . The maximum fidelity between the target state and any biseparable state is consequently . The corresponding witness operator is For every biseparable state , the inequality holds. For the ideal heralded target state, A sufficient condition for genuine multipartite entanglement is therefore , or equivalently . A practical evaluation of the witness requires both the populations and the coherences within the upper-sideband single-excitation manifold. For each electron arm, define the two relevant local sideband states as and . The corresponding projectors are and . The local analysis operators are defined as and . For a target state with known phases , the phase pattern can first be removed by the local operation , where . The phase-compensated conditional state is . The conditional fidelity can then be reconstructed from locally measurable quantities as where is the population of the pathway . The first term is obtained directly from sideband-resolved energy detection. The second term measures the real coherences between different excitation-transfer pathways. The required and measurements can be implemented by phase-controlled PINEM-type analysis interactions applied before energy readout. A calibrated analysis rotation maps the selected or eigenbasis onto the energy basis, after which the same sideband-resolved detector provides the projective measurement. The optical phases of the analysis interactions must remain stable across the electron arms. Phase-controlled sequential free-electron–light interactions and phase-sensitive free-electron homodyne detection provide experimental precedents for the required analysis operations [73, 74]. If independent energy measurements verify confinement of the heralded state to the upper-sideband single-excitation manifold, the spectator projectors can be omitted. The populations and coherences can then be obtained from three global measurement settings corresponding to the energy, , and bases. When leakage outside the target manifold is appreciable, pair-resolved analysis settings with energy-basis measurements on all spectator arms are required. 3.4. Explicit tripartite example: 3.4.1. Heralded tripartite free-electron state The case is the minimal genuinely multipartite realization and makes the correlation transfer explicit. The initial atomic resource is taken as The relevant upper-sideband electron basis states are These states specify the three possible arms in which the transferred single excitation can appear in the electron upper-sideband manifold. Projection of the evolved state onto the all-ground TLS outcome gives The corresponding heralding probability is The normalized heralded three-electron state is The transferred excitation is coherently delocalized over the three electrons. At the end of a detuned square pulse, the heralding probability follows from equation (26) as At resonance, the expression simplifies to Optimization of equation (50) gives , , and . For , the multipartite witness in equation (42) takes the explicit form For the ideal heralded state, the witness expectation value in equation (43) is The negative value certifies genuine tripartite entanglement and places the heralded state outside the biseparable set. For the tripartite state, experimental evaluation of the witness requires the three pathway populations associated with , , and , together with the real parts of the three pairwise coherences. The quantities can be obtained using the local sideband rotations and energy-resolved measurements described above. Tracing out one electron, for example electron , gives the reduced state of electrons and : The same reduced-state structure holds for and by symmetry. The corresponding negativity is Every two-electron reduction therefore retains nonzero bipartite entanglement, as expected for the class. 3.4.2. Time-resolved entanglement redistribution A time-resolved analysis distinguishes the unconditional electron entanglement, the conditional entanglement of the heralded branch, and the success-weighted postselected yield. The numerical evolution follows equation (31). For the evolved pure state , the atomic reduced state is For any chosen atomic pair , the corresponding two-TLS reduced state is and the pairwise atomic entanglement is quantified by the negativity where denotes partial transposition with respect to subsystem , and is the trace norm. The atomic average pairwise negativity is then evaluated numerically as The unconditional electron reduced state is where the trace is taken over the three TLS degrees of freedom. For any chosen electron pair , the corresponding two-electron reduced state is and its pairwise entanglement is quantified analogously by The time-resolved average pairwise entanglement of the unconditional electron subsystem is In the fully symmetric tripartite protocol, the reduced electron state remains permutation symmetric under exchange of the three electron arms. Consequently, , so that equation (62) reduces to . The quantity is determined by the full electron reduced state and therefore contains contributions from both the heralded branch and all nonheralded branches. The numerical heralding probability associated with the TLS outcome is and the corresponding postselected electron state is For any chosen electron pair , the corresponding conditional two-electron reduced state is and the conditional pairwise electron entanglement is quantified by The numerical conditional electron average pairwise negativity is therefore and the corresponding numerical success-weighted conditional electron entanglement yield is defined as By contrast, the conditional electron pairwise entanglement admits a closed-form analytical benchmark. Postselection of the TLS outcome projects the electron subsystem onto the ideal three-electron -type state within the symmetric RWA treatment. The conditional electron average pairwise negativity is time independent, and provides the analytical benchmark. Strictly speaking, at times for which the conditional state is not operationally accessible, but the constant value in equation (69) remains the correct entanglement content of the ideal heralded branch whenever that branch is successfully selected. Another useful quantity is the success-weighted conditional electron pairwise entanglement, , which characterizes the practically accessible electron entanglement yield after postselection. In the symmetric detuned RWA case, , so that At exact resonance, equation (70) reduces to Figure 2 shows the progressive loss of atomic pairwise entanglement and the transient growth of unconditional electron entanglement. The latter subsequently decreases because the reduced electron state contains both heralded and nonheralded branches. The conditional negativity remains at the ideal value whenever the branch is accessible, and its agreement with the analytical result provides an internal numerical check. The success-weighted yield peaks near the optimal heralding point and earlier than the unconditional electron negativity, demonstrating that maximal branch accessibility and maximal entanglement of the full reduced electron state occur at different times. Download figure: Standard image High-resolution image 3.4.3. Entanglement transfer of weighted single-excitation states To illustrate the mapping beyond equal weights, we consider the real-amplitude single-excitation family where . The amplitudes are parametrized as with and . Under the symmetric resonant RWA protocol, postselection of preserves the amplitudes and yields The protocol therefore realizes an isometric mapping from the atomic single-excitation manifold to the heralded electronic upper-sideband manifold. For pairwise correlations, define the initial atomic average pairwise negativity as and the heralded electron average pairwise negativity as For the one-versus-rest bipartite structure of the initial three-TLS resource, we use the von Neumann entropy where is the reduced density matrix of TLS obtained by tracing over TLS and TLS . For the one-versus-rest bipartite structure of the heralded three-electron state, we use the von Neumann entropy Here denotes the reduced density matrix of electron in the conditional output state after tracing over electrons and . Exact preservation of the amplitudes in equation (74) leads to the analytical relations and The heralding probability, by contrast, is determined only by the dynamical coefficients and is independent of the weight distribution . For the symmetric tripartite protocol, , which reduces to equation (50) at resonance. At the optimal pulse area , the success probability becomes independent of the initial weighted resource. The conditional fidelity with respect to the target heralded electron state is defined as where and . Within the ideal symmetric RWA treatment, Figures 3(a) and (b) follow the diagonal identities in equations (79) and (80), confirming preservation of the relevant bipartite spectra. Different parameter pairs can produce the same initial average pairwise negativity, which accounts for repeated horizontal coordinates in figure 3(c). The small vertical spread is numerical; analytically, the optimal heralding probability is independent of the weight distribution. Figure 3(d) likewise remains at unit conditional fidelity, confirming the isometric amplitude mapping. Download figure: Standard image High-resolution image Numerical methods and convergence. The general- resonant results above are evaluated directly in the symmetry-reduced RWA exchange sector, and the optimal pulse areas are obtained by bounded one-dimensional minimization. The explicit tripartite state-vector calculations use the truncated electron basis and are propagated with the adaptive MATLAB solver ode, using relative and absolute tolerances of and . The RWA and complete-Hamiltonian calculations use and , respectively, while the Lindblad calculations use with matrix-exponential propagation. All adopted cutoffs were verified by increasing by one, and the state-vector calculations were repeated with tighter integration tolerances. The resulting changes remained below the numerical resolution of the reported results. 4. Discussion 4.1. Robustness to detuning and symmetry breaking Common detuning preserves arm symmetry, whereas coupling or detuning mismatch produces arm-dependent amplitudes and phases. For , figure 4 agrees with equation (26): common detuning lowers the heralding probability away from resonance, while the conditional fidelity and witness remain unchanged. Thus, symmetric detuning affects branch accessibility but not the normalized postselected state. Download figure: Standard image High-resolution image We next turn to weak symmetry breaking between different local arms. In realistic implementations, such imperfections may arise from small variations in coupling strength, interaction duration, or local detuning, which represent typical challenges in distributed quantum entanglement protocols. The postselected state then remains confined to the upper-sideband single-excitation manifold, but the ideal symmetric state is replaced by a weighted superposition. For the th arm, we define with , and . After postselecting the TLS outcome , the unnormalized heralded electron state becomes Introducing the normalized heralded state can be written as Weak symmetry breaking preserves the upper-sideband single-excitation manifold but distorts the relative pathway amplitudes . Define the phase-corrected amplitudes as . The conditional fidelity with respect to the phase-matched ideal target is For weak arm-dependent deviations, write Expansion to second order gives The leading fidelity loss is therefore set by the relative variance of the phase-corrected pathway amplitudes, while a common rescaling leaves the conditional state unchanged. For the tripartite protocol, two representative perturbation scans are particularly instructive. A coupling-mismatch scan may be introduced as while a detuning-mismatch scan may be introduced as The first probes unequal local transfer strengths, whereas the second introduces relative phase mismatch between different transfer pathways. In both cases, the most informative observables are the heralding probability, the conditional fidelity, the pathway-weight distribution and the witness expectation value. Figures 5 and 6 confirm the variance-based picture. Coupling mismatch mainly unbalances pathway weights, whereas detuning mismatch also accumulates relative phases and produces a stronger fidelity reduction over the displayed ranges. Common detuning primarily lowers the success probability, while arm-dependent perturbations convert the symmetric target into a weighted state. The numerical scans therefore show that conditional state quality is more sensitive to broken arm symmetry than the heralding probability. Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image 4.2. Beyond-RWA corrections and asymptotic validity The RWA provides an analytically transparent description of the multipartite transfer mechanism. A realistic treatment must nevertheless include the counter-rotating contributions present in the complete interaction Hamiltonian. The leading effect of the rapidly oscillating terms is a Bloch–Siegert-type renormalization of the detuning [75, 76]. The effective detuning is given by A second-order elimination of the terms oscillating at gives for the square-pulse ladder-coupled model, hence . The coefficient is derived for the present interaction rather than imported from the standard harmonically driven TLS. Under the fixed-area condition , the resonance displacement scales as . The corresponding dimensionless detuning parameter becomes Substitution of into equation (49) gives the Bloch–Siegert-corrected approximation for the tripartite heralding probability. The correction accounts for the leading resonance displacement but does not describe population leakage into additional sideband channels. For a square pulse, rapidly oscillating counter-rotating channels give Both the resonance-induced probability correction and the off-resonant leakage are therefore algebraically suppressed for long pulses. The complete interaction Hamiltonian remains necessary in the short-pulse regime, while the RWA is recovered in a controlled manner as increases. Figure 7(a) shows the resonance displacement extracted from the complete interaction Hamiltonian. The short-pulse results deviate from the second-order linear relation, while the fit over gives the finite-interval value . Figure 7(b) shows the corresponding pointwise coefficient. Despite short-pulse oscillations, the long-pulse values approach the perturbative prediction . The difference between the global fit and unity mainly originates from finite-duration and higher-order corrections at the shorter interaction times, which carry greater weight because the resonance displacement decreases as . Download figure: Standard image High-resolution image Figure 7(c) compares the tripartite heralding probability obtained from the complete Hamiltonian, direct RWA integration, the bare RWA expression, and the Bloch–Siegert-corrected expression. The RWA numerical and analytical results are nearly indistinguishable, providing an internal consistency check rather than independent validation. The complete-Hamiltonian result deviates from the RWA predictions for short pulses and approaches the RWA limit as increases. The bare and corrected analytical probabilities remain close because the probability is locally even in the effective detuning around resonance. Figure 7(d) presents the residual discrepancies on a log–log scale. The slopes evaluated over are close to , in agreement with the asymptotic estimate in equation (96). The proximity of the two residual curves confirms that the Bloch–Siegert correction captures the leading resonance displacement but does not remove leakage, finite-pulse contributions, or higher-order corrections. 4.3. TLS decoherence and heralding latency TLS relaxation and pure dephasing introduce incoherent errors in addition to the coherent imperfections considered above. During the interaction window, the joint electron–TLS density operator obeys where and are the local TLS relaxation and pure-dephasing rates, respectively. The Lindblad dissipator is defined as The corresponding relaxation and coherence times are and . Relaxation during the interaction can transfer a TLS excitation to the environment without producing the intended electron-sideband transition. The resulting trajectory may nevertheless contribute to the all-ground TLS outcome and therefore produces a false heralding event. Pure dephasing leaves the TLS populations unchanged but suppresses the coherence between the indistinguishable transfer pathways. Relaxation can therefore affect both the heralding probability and conditional fidelity, whereas pure dephasing mainly degrades the coherence of the conditional electron state. A finite interval between the end of the interaction and the TLS measurement introduces an additional heralding error. Let denote the heralding latency. During , the electron–TLS interaction is switched off, while the residual TLS state continues to evolve under relaxation and dephasing. Residual excited-state population may decay before the measurement and be incorrectly accepted as an all-ground outcome. The all-ground TLS projector is where is the identity operator on the electron Hilbert space and is defined in equation (16). The delayed heralding probability is The corresponding normalized conditional electron state is For the symmetric protocol, the delayed conditional fidelity is where the ideal electronic target state is defined in equation (20). We define the manifold-leakage false-herald probability as the weight of all-ground TLS outcomes whose electron state lies outside the target upper-sideband single-excitation manifold, where is the target-manifold projector introduced above. Pure dephasing can reduce the conditional fidelity without directly producing a false-herald event, whereas relaxation and heralding latency can increase both the false-herald fraction and the absolute false-herald probability. The leading dissipative corrections are governed by the dimensionless quantities , , and . Reliable heralding requires the interaction duration and readout latency to remain short compared with the TLS relaxation and coherence times. The coherent limit is recovered when , , and . Figure 8(a) shows that both TLS relaxation and pure dephasing reduce the conditional fidelity. The dependence on is more pronounced over the scanned range because relaxation can generate all-ground TLS outcomes without the intended electron-sideband transfer. Pure dephasing mainly suppresses the coherence between indistinguishable transfer pathways and therefore lowers the fidelity without directly changing the TLS populations. Figure 8(b) shows that the total heralding probability increases with , whereas the dependence on remains comparatively weak. The increase does not represent improved protocol performance. Relaxation converts residual TLS excitation into additional all-ground outcomes, including false heralding events, which explains the simultaneous increase in and decrease in . Download figure: Standard image High-resolution image Figure 8(c) demonstrates that a finite heralding latency further reduces the conditional fidelity. Residual TLS excitation can decay after the interaction and before the projective measurement, producing all-ground outcomes that are not accompanied by the target electronic transfer. The corresponding false-herald probability in figure 8(d) increases with both and . Near the coherent limit, the conditional fidelity approaches unity and the false-herald probability vanishes. The numerical results therefore emphasize that a large all-ground detection probability alone is insufficient for assessing protocol performance, and that short interaction and readout times are required to preserve reliable heralding. 4.4. Gaussian envelopes and pulse-shape effects on final electron entanglement A realistic electron–TLS coupling follows the spatially varying near field sampled along the electron trajectory and is generally smooth [43, 60]. Spatiotemporal shaping of electron wave packets provides additional control over such interaction profiles [38, 42]. We model the envelope as with peak coupling and temporal width . The accumulated area is At exact resonance within the RWA, Hamiltonians at different times commute and the evolution depends only on this area. Equal-area square and Gaussian pulses therefore produce the same resonant RWA evolution, whereas the fixed-peak comparison used below changes as varies. We quantify the final unconditional three-electron entanglement by Figure 9(a) compares square and Gaussian envelopes with the same peak coupling. Both retain the same resonant structure, while the Gaussian result is lower near resonance because the finite-width profile accumulates a smaller area over the fixed interaction window. Figure 9(b) shows that increasing enlarges the accumulated area and raises the final electron entanglement toward the square-pulse limit. The pulse-shape dependence under fixed-peak normalization is therefore governed mainly by the effective interaction area, while the resonant single-excitation transfer mechanism remains unchanged. Download figure: Standard image High-resolution image 4.5. Physical implications, resource requirements, and limitations The genuinely multipartite constraint arises from the global all-ground heralding condition rather than from the factorized local RWA dynamics. Unlike the two-electron protocol [66], the selected branch considered here occupies the upper-sideband single-excitation manifold and preserves the amplitudes and phases of the atomic resource. An implementation requires a spatially separated array of TLSs prepared in a single-excitation resource, synchronized electron trajectories coupled predominantly to designated targets, and all-ground TLS readout. No direct electron–electron interaction is required. Certification additionally needs phase-sensitive sideband measurements in every electron arm, for which recent free-electron correlation and imaging experiments provide relevant precedents [51, 52]. To assess the required interaction strength, the dimensionless single-pass coupling for a transition dipole oriented along the transverse impact-parameter direction can be estimated within the FEBERI dipole model as where is the elementary charge, is the vacuum permittivity, is the transverse transition dipole moment, is the adjacent-sideband frequency, is the electron impact parameter, is the electron velocity, and is the corresponding Lorentz factor. The function denotes the modified Bessel function of the second kind of order one [53, 56]. The quantity is the time-integrated single-pass coupling corresponding to , whereas in equation (2) is the instantaneous coupling rate. For a electron, and , where is the speed of light. A representative optical transition with , , and gives , consistent with previous FEBERI estimates [56]. The intrinsic heralding probability accounts only for the unitary electron–TLS dynamics. A detected multipartite event additionally depends on atomic-resource preparation, synchronized electron delivery, TLS readout, and electron detection. Let denote the experimental cycle rate, the preparation probability of the atomic resource, the probability of synchronized delivery of all electrons, the probability of correctly identifying the ground state of one TLS, and the sideband-resolved detection efficiency for one electron. The detected coincidence rate for correctly heralded -electron events is where is the intrinsic probability of the all-ground TLS branch and is the dimensionless interaction pulse area. For the square-pulse model, , while the single-pass estimates below identify with . For resonant symmetric square pulses, the intrinsic heralding probability is The optimal intrinsic probability scales as . Independent TLS-readout and electron-detection losses introduce the additional factors and . Defining the single-TLS readout error as , the probability of correctly identifying the all-ground result is equivalently . The detected event rate therefore contains both the intrinsic algebraic scaling and exponential efficiency losses associated with multipartite readout and detection. The intrinsic algebraic suppression is less severe than an exponential decrease, but the detected coincidence rate can become small for large because preparation, synchronization, readout, and electron-detection efficiencies accumulate across all arms. The principal scalability limitation is therefore the experimental event yield rather than the conditional quality of a successfully heralded state, which remains an ideal state under symmetric conditions. Improved per-arm efficiencies and higher experimental repetition rates are essential for extending the protocol to larger electron numbers. An optimistic component-level benchmark is obtained using , , and . is based on a trapped-ion fluorescence-readout fidelity of obtained within a detection interval [77]. is based on the quantum efficiency reported for a K2 direct electron detector operated in the electron-counting regime [78]. Deterministic preparation of four- to eight-ion states supports the use of deterministic atomic-resource preparation as an upper component-level benchmark [69]. Accordingly, is used in table 1. Realistic preparation and synchronization probabilities below unity reduce the listed rates by the factor . Table 1. Representative TLS parameters, estimated single-pass couplings, and component-level detected event rates. All coupling estimates use a electron and equation (107). The rates are calculated from equations (108) and (109) using , , , and . The cited references provide the transition parameters, whereas , , and are evaluated in the present work. The two Rydberg lifetimes correspond to the and states, respectively. Representative atomic, solid-state, and Rydberg TLS parameters are summarized in table 1. The Rb and Cs estimates use the impact parameter adopted in the FEBERI benchmark. The values for the perovskite nanocrystal and for the Rydberg transition are geometric assumptions introduced for the present estimates. The Cs dipole moment and coupling represent upper estimates because the reduced dipole matrix element is used before the relevant angular-momentum coefficient is applied. The perovskite transition energy, dipole moment, and lifetime are representative values reported for related lead-halide perovskite nanocrystals and do not correspond to one identical specimen. The resonant pulse area that maximizes is . The corresponding values are , , and for , , and , respectively. Using the component-level efficiencies specified above gives the optimal detected rates before preparation and synchronization losses are included. The candidate single-pass couplings in table 1 remain below the corresponding optimal pulse areas. Enhancement factors ranging from approximately to are required to reach the optimum. The estimates therefore do not demonstrate that the optimal pulse area is attainable in a current single-pass FEBERI experiment. Rather, the comparison quantifies the enhancement required for operation near the theoretical optimum. Collective dipole enhancement, resonant nanophotonic structures, repeated coherent interactions, or reduced electron–target separation could reduce the remaining gap. The Rydberg transition gives the largest single-pass coupling in table 1 because of the large electric-dipole matrix element [84]. The associated transition frequency of corresponds to an electron-sideband separation of only . Conventional optical-frequency EELS cannot directly resolve such a small separation. A Rydberg implementation would therefore require a phase-sensitive electron measurement or a higher-frequency transition. The Rydberg entry should consequently be regarded primarily as an upper coupling benchmark rather than a directly compatible realization of the proposed sideband-resolved measurement. The fast-electron interaction time near a nanometer-scale target is much shorter than the lifetimes in table 1. As a representative example, the CsPbBr nanocrystal has an excited-state lifetime of approximately and an assumed impact parameter . For a electron with , a characteristic interaction length of approximately gives and . Relaxation during the electron–TLS interaction is therefore negligible on this timescale. For comparison, heralding latencies of , , and correspond to , , and , respectively. Relaxation during the heralding latency therefore exceeds relaxation during the ultrashort electron interaction by several orders of magnitude for a short-lived optical TLS and constitutes the dominant relaxation-related contribution to false heralding. Reliable heralding consequently requires the full preparation-to-readout sequence to remain short compared with the excited-state lifetime. A complete false-herald analysis also requires the TLS readout confusion matrix, while rapid detection, shelving, or an equivalent platform-specific protocol reduces readout-induced errors. Long-lived hyperfine or metastable states provide storage for the resource, although their direct coupling is weak. Coherent mapping to a dipole-allowed transition immediately before the electron interaction provides an alternative approach. Direct inter-arm effects can be estimated for a representative parallel geometry. Let denote the transverse separation between neighboring electron trajectories, the length over which the trajectories propagate in parallel, and the relativistic electron momentum, where is the electron rest mass. The relativistically reduced pairwise transverse Coulomb force and the corresponding deflection angle are estimated by For electrons, , and , the resulting deflection angle is . The corresponding pairwise Coulomb phase scale is For symmetric trajectories and arrival times, the leading Coulomb contribution is common to all single-excitation branches and does not change the conditional entanglement. Geometric or timing asymmetries can instead produce differential phases that require calibration. Although the number of interacting electron pairs grows as , only branch-dependent phase differences affect the fidelity of the heralded state. Near-field cross-talk occurs when an electron couples to a noncorresponding TLS. For identical transition dipoles and orientations, let denote the coupling in equation (107) evaluated at transverse separation . The unwanted-to-target coupling ratio is For , , and , the ratio is . The corresponding probability-level contribution per unwanted electron–TLS pair is . Even after summing over all noncorresponding pairs for , the estimated correction remains of order for the stated geometry. Free propagation from the interaction regions to the detectors produces deterministic sideband phases. Let denote the reference electron energy, the propagation distance in arm , and the electron wave number at energy . The propagation phase of sideband in arm is Defining the path-length mismatch between arms and as , the relative phase between the corresponding components of the heralded state is, to first order in , For a electron and , . Adopting a representative tolerance requires relative path-length control of approximately . Equal propagation distances produce only a common phase. Residual arm-dependent phases can be calibrated or compensated through local phase-controlled electron–light interactions [67, 68]. Several limitations remain. The protocol is probabilistic, and the detected yield decreases with both the intrinsic scaling and accumulated preparation, readout, and detection losses. An ideal symmetric output requires uniform couplings, detunings, and propagation phases, while short pulses or strong coupling require the complete Hamiltonian. Entanglement certification demands phase-stable local sideband rotations because energy detection alone does not recover pathway coherences. The model omits multiple-excitation resources, mixed inputs, electron loss, partial pathway distinguishability, correlated or non-Markovian noise, and detector-specific readout channels. The inter-arm and rate estimates depend on the assumed geometry and combine parameters from different platforms. Table 1 therefore provides a component-level upper benchmark rather than a design for an integrated apparatus. 5. Conclusion and outlook We have developed a heralded protocol that maps an atomic single-excitation resource onto the upper-sideband manifold of propagating electrons. For a symmetric -arm geometry, the RWA solution produces an exact phase-matched electronic state after all-ground TLS postselection, with optimal intrinsic probability . The multipartite structure was characterized for arbitrary and illustrated for through correlation redistribution and weighted-state transfer. A decomposition into local sideband rotations and energy measurements expresses the fidelity-based witness in terms of local observables. Common detuning mainly reduces branch accessibility, whereas arm-dependent perturbations distort the conditioned state. Counter-rotating effects vanish algebraically in the long-pulse regime, while TLS relaxation, dephasing, and heralding latency reduce the conditional fidelity and produce false all-ground heralding events. Smooth coupling envelopes preserve the transfer mechanism, with fixed-peak pulse-shape effects governed mainly by the accumulated area. Representative parameter estimates further quantify the relation between the required interaction strength, TLS lifetime, heralding latency, and multipartite event rate. Further studies will address collective enhancement, resonant nanophotonic structures, repeated coherent interactions, and reduced electron–target separation. Extension to multiple-excitation Dicke resources provides a route toward higher-dimensional electronic excitation manifolds. Additional theoretical models will incorporate electron-wave-packet dispersion, electron loss, partial distinguishability, correlated noise, and complete detector-error channels. The present results establish a quantitative framework for transferring localized multipartite matter entanglement to propagating free electrons. Acknowledgments This work is supported by the Natural Science Foundation of Chongqing (Grant No. CSTB2025NSCQ-GPX0416) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202401437). S Liu acknowledges the support of the Natural Science Foundation of Hubei Province (Grant No. 2024AFB200), the Scientific Research Foundation of the Education Department of Hubei Province (Grant No. B2025165), and the Research Start-up Fund of Hubei University of Arts and Sciences (Grant No. qdf2022033). Y-D L acknowledges support from the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-K202401403) and the scientific research project of the Science and Technology Bureau of Fuling (Grant No. FLKJ2025AAG2003). A. G and R. I acknowledge the support of the Israel Science Foundation (Grant No. 2992124). Data availability statement The data that support the findings of this study are not publicly available because the numerical results were generated directly from the theoretical model using MATLAB and are not maintained as a separate dataset. The MATLAB codes used to generate the numerical results are publicly available as described in the Code Availability statement. Author contributions D R and A G conceived the project and developed the theoretical framework. J-Y L, D J, and S L designed the methodology. D R, R I, and Z-L H visualized the results and wrote the first draft. J-Y B and Y-D L developed the computational framework and performed the simulations. All authors discussed the results, reviewed and edited the manuscript, and approved the final version. Conflict of interest The authors declare no competing interests. Code availability The codes supporting the numerical results are accessible at [https://doi.org/10.5281/zenodo.22076175].