Generation, Characterization and Manipulation of Quantum Correlations in Electron Beams

Entanglement engineering plays a central role in quantum-enhanced technologies, with potential physical platforms that outperform their classical counterparts. However, free electrons remain largely unexplored despite their great capacity to encode and manipulate quantum information, due in part the lack of a suitable theoretical framework. Here we link theoretical concepts from quantum information to available free-electron sources. Specifically, we consider the interactions among electrons propagating near the surface of a polariton-supporting medium, and study the entanglement induced by pair-wise coupling. These correlations depend on controlled interaction interval and the initial electron bandwidth. We show that long interaction times of broadband electrons extend their temporal coherence. This in turn is revealed through a widened Hong-Ou-Mandel peak, and associated with an increased entanglement entropy. We then introduce a discrete basis of electronic temporal-modes, and discriminate between them via coincidence detection with a shaped probe. This paves the way for ultrafast quantum information transfer by means of free electrons, rendering the large alphabet that they span in the time domain accessible.


INTRODUCTION
Quantum degrees of freedom occupy a large parameter space compared with their classical counterparts. This property renders them challenging for simulation on classical computers. Nonetheless, it also endows them with a vast information capacity, useful for novel computational and metrologic paradigms [1][2][3]. Entangled photon pairs have long been the work-horse of quantum enhancement demonstrations in the optical arena, with applications in metrology [4,5], imaging [6][7][8][9][10][11] and spectroscopy [11][12][13][14]. A key concept in generation of such useful states, is initiation of well-monitored interactions between continuous variables. The latter exhibit rich entanglement spectra and large state-space on which information can be recorded and accessed [15][16][17][18][19]. These concepts have not yet been addressed in the well-established field of free-electron based techniques, such as spectroscopy and microscopy [20]. Designing controlled entanglement of freeelectron sources constitutes the main challenge, and this is precisely what we address here.
Extraordinary electron-beam-shaping capabilities have been recently demonstrated in electron microscopes combining light ultrafast optics * Shahaf.Asban@icfo.eu † Javier.GarciaDeAbajo@icfo.eu elements [21][22][23]. Revolutionary concepts such as free-electron qubits [24] and cavity-induced quantum control [25][26][27] are becoming available, pointing towards the emergence of nextgeneration quantum light-electron technologies. While photons maintain coherence over large distances, electrons decohere rapidly due to their strong environmental coupling. Combined with the control schemes above, this suggests electrons are valuable quantum probes. Lately, we have shown that electrons passing by polaritonsupporting media, can experience geometrically controlled attractive interaction [28]. This effect is closely related to Amperean pairing of electrons discussed in [29][30][31], which in turn induces an entangled EPR state in the long interaction time limit [28].
Here, we study the quantum correlations generated by abrupt interactions of electron pairs with a neighboring medium, as depicted in Fig. 1, for a controlled time interval T I . We explore the transient state generated by abrupt interactions, as well as the steady state limit in the pertubative regime. By varying two control parameters -interaction time T I and initial electron bandwidth σ e -we effectively scan the degree of entanglement. The entanglement in the longitudinal dimension is characterized by the Schmidt decomposition of the wave-function. We then calculate the coincidence probability and display it versus the degree of entanglement. We denote the resulting eigenstates electronic temporal modes (ETMs) in analogy to their photonic counterparts [32,33]. Finally, we propose a technique that is useful for real-time discrimination between ETMs, essential for state tomography and related quantum information processing applications. x,y . After an interaction time TI , a correlated pair |Ψ (TI ) is obtained.

The pair-amplitude
The electron-pair amplitude is obtained from the underlying electron-polariton coupling. We consider free electrons traveling with mean momentum k 0 , as depicted in Fig. 1. The full Hamiltonian is given by three contributions: H = H e +H φ +H e−φ . The electrons kinetic term is described by H e , the electromagnetic field degrees of freedom combined with the surface polaritons are contained in H φ [34,35] and the electron-field coupling is H e−φ (see Methods and Sec. S1 of SI). The electrons illustrated in Fig. 1, are assumed to be prepared in an initial state with statisti-cal independence, reflected by the product state Here |k i , s i represents an electron state of momentum k i and spin s i . The single-electron amplitude α (i) si (k i ) is determined by the preparation process, and we assume it to be a Gaussian centered around k 0 along the propagation axis in our calculations. As the electrons pass in vicinity to the film, they exchange energy with the medium. The interaction mediated by the polaritons decays exponentially with the distance from the medium, validating the use of pertubative approach [28]. Expanding the evolution in the interaction picture to second order, we obtain the electron-pair wave-function in its generic form where λ labels a set of control parameters. In the present configuration, λ parametrizes the dimensionless interaction time T I and the initial electron bandwidth σ e . We are interested in the dynamics of the longitudinal component of the electron pair. By tracing the transverse momenta, we obtain an expression for Φ λ s1,s2 (k 1 , k 2 ), which we denote as the pair amplitude (see Eq. 5 in the Methods section). The pair amplitude exhibits continuous variable entanglement throughout most of the explored parameter space.

Entanglement spectrum and ETMs
It is useful to explore the parameter space of the pair amplitude by performing a Schmidt decomposition. The Schmidt-Mercer theorem allows us to express an inseparable state as a superposition of separable ones, where spin labels are omitted for brevity. The longitudinal eigenstates {ψ n , φ n } appear in pairs of ETMs. If the state ψ n is detected, its counterpart occupies the state φ n with absolute certainty. The eigenvalues p n reflect the probability of detecting the n th mode. The joint momentum-representation of the pair amplitude is displayed in Fig. 2a for selected values of the control parameters (i.e., a dimensionless interaction time T I and the electron bandwidth σ e λ p ). The dimensionless interaction time is given by c is the electron velocity relative to of the speed of light, L is the length of the medium along the main propagation direction and λ p is the polariton wavelength in the film (see Sec. S1 of SI). We employ the collision (Rényi) entropy H 2 [λ] = − log n p 2 n and Schmidt number κ ≡ 2 H2[λ] as measures for entanglement [16,17,36,37]. The entropy quantifies the degree of uncertainty with respect to the instantaneous ETM, while κ is the effective number of participating ETMs. Sweeping the control parameters reveals two opposite highly correlated regimes, as depicted in Figs. 2bc. For large σ e λ p and T I we observe increasing entanglement and anticorrelated momenta due to energy conservation combined with long exchange times, in agreement with the results obtained in [28]. For short interaction times, narrow-band electrons present correlated momenta. In this regime we observe a rapidly growing degree of entanglement. The entanglement spectrum plotted in Fig. 2d is obtained from the Schmidt decomposition of the amplitude displayed in the extreme left of panel (e), for which κ ≈ 6. The first (lowest-order) three ETMs are visualized along with their corresponding cross-sections.

Coincidence detection
A common approach to probe quantum correlations is by measuring the coincidence probability , assuming an experimental setup as sketched in Fig. 3a. We consider balanced beam splitters (BSs) and obtain where n, m label ETM and I nm (δl) = dk φ * n (k) φ m (k) e − i E k δl /v (see Sec. S3 of the SI). Figure 3b displays P 12 (δl) as a function of the BS displacement, arranged in growing degree of entanglement. The probability ranges from 1 /2 (completely random) to unity (utterly antibunched) as expected from fermionic Hong-Ou-Mandel interference, usually revealed by a Pauli dip in matter systems [38][39][40]. In the left panel of Fig. 3b we scan T I while fixing σ e = 4π /λp in the broadband range. Interestingly, we find that the for higher degree of entanglement the probability peak extends over a wider range of δl. be attributed to temporal expansion of the electron wave-function due to long interaction times.
In the inset we see that κ grows with T I in a piecewise linear manner. On the right panel σ e is varied while T I is fixed in the long-interaction range and a similar behavior is found. We find that κ ∝ σ 2 e and consider it a direct result of the initial Gaussian wave-packet, together with the emergent linear relations of κ and the interaction time.
In Fig. 4a, the coincidence detection of the instantaneous incoming ETM with a known probe mode labeled φ p is presented. The first three ETMs are extracted from the Schmidt decomposition of the amplitude displayed in Fig. 2e. These modes are the eigenstates of the reduced single-electron density matrix, therefore in each realization one such mode is detected with probability p n . When the incoming mode matches the shaped probe mode, the coincidence signal exhibits a peak for a vanishing path difference, as depicted in Fig. 4b. By counting the appearance rate of each mode separately we can deduce the probability vector p n , and thus characterize the quantum state. Beyond state tomography this could also be used in coincidence with parallel operations on its ETM-twin, realizing more sophisticated information processing protocols. Figure 4. ETM discrimination. (a) An incoming electron pair prepared in a superposition of ETMs is separated by a first BS, then combined with a (shaped) probe mode φp and finally measured in coincidence. (b) Coincidence outcomes of the probe with three possible incoming modes n, p ∈ {1, 2, 3}, as a function of path difference δl. The interference pattern displays increased response for identical probe and incoming ETM.

DISCUSSION
Near-fields evolving at the surface of polariton supporting materials provide a novel approach to generate and shape quantum correlations in charged particles, and in particular in free electrons. While such pairing mechanisms are suppressed in matter due to ambient noise (e.g., thermal), electrons structured in a beam undergo significantly less scattering events, thus enabling coherent interactions to persist over longer space-time intervals. We have shown that electron pairs near polariton-supporting material boundaries undergo nontrivial coupling that generates entanglement. Such correlations are mathematically expressed by the apparent inseparability of the pair amplitude in Eq. 1, giving rise to the results displayed in Fig. 2. The Schmidt decomposition allows us to express the pair amplitude using a set of factorized states, providing useful measures for bipartite entanglement [16,17,[41][42][43][44][45][46][47]. This framework reveals simple relations between the control parameters and the resulting evolution of quantum correlations of the above setup. Such properties are desirable for entanglement engineering.
The large Hilbert-space dimensionality occupied by the ETMs, renders them as appealing ultrafast quantum information carriers. This has potential applications in quantum-enhanced electron metrology, as proposed using optical setups [14,48]. For example, measuring the momentum of one of the electrons in the pair and the position of the other, one may obtain superesolved imaging. Such class of quantum enhancements benefits from the fact that single-particle (local) observables are not Fourier conjugates of the (extended) composite state. This work raises multiple open questions concerning vicarious temperature effects, the imprint of the medium topology on the entanglement spectrum, entanglement along the transverse plane and higher order matter-electron quantum correlations. These are just a few examples of the emerging field of quantum free-electron metrology. From the information theoretic point of view, the large alphabet spanned by ETMs, promotes their candidacy for electron-beam quantum information processing and communication tasks. This raises questions regarding information capacity of the channel in the presence of noise, providing a direction for future study.

Pair-amplitude derivation
The pair amplitude is obtained pertubatively in the interaction picture (Sec. S1 of the SI). The full Hamiltonian of the system contains three contributions: H = H e + H φ + H e−φ . The electrons kinetic term is given by H e = k,s k c † k,s c k,s , where the operator c k,s c † k,s creates (annihilates) an electronic mode with momentum k and spin s obeying the anticommutation relations c k,s , c k ,s = δ kk δ ss . The term H φ describes the electromagnetic-field degrees of freedom combined with the surface polaritons in the framework of macroscopic quantum electrodynamics [34,35]. The electron-field coupling is expressed using the Hamiltonian [28,35,49] where λ C = h /mec is the Compton wavelength of the electron, while e and m e are its charge and mass, respectively. We have employed the Weyl-gauge, setting the scalar potential to zero and introduced A (q), the vector field operator in momentum space. The vector-field in macroscopic quantum electrodynamics is expressed in terms of the Green tensor, encapsulating the geometric and spectral properties of the medium. Proceeding to calculate the first nontrivial order (second), we obtain the general form of Eq. 1. We consider Gaussian initial states of mean distance y 0 = 5 nm from the thin film and σ y = 0.5 nm (see Fig. 1). Taking the long l x limit and choosing σ x = 2π λp , we trace the transverse components and obtain Here, N is a normalization constant, T is the interaction time, and χ (q) is obtained by tracing the lateral wave vector q = (q x , q y ) in the interaction picture (Sec. S1 of SI). Additionally, we have invoked the nonrecoil approximation for small momentum exchanges relative to k 0 , resulting in a linear electron-energy exchange where q is the polariton wave vector and v is the electron velocity.

ETMs calculation
To find the set of ETMs {ψ n , φ n } and their weights p n , we solve the integral eigenvalue equations p n ψ n (k) = dk K 1 (k, k ) ψ n (k ) and p n φ n (k) = dk K 2 (k, k ) φ n (k ) (Sec. S2 of SI). The kernels, which are found from the reductions K 1 (k, k ) = dk 2 Φ λ s1s2 (k, k 2 ) Φ λ * s1s2 (k , k 2 ) and K 2 (k, k ) = dk 1 Φ λ s1s2 (k 1 , k) Φ λ * s1s2 (k 1 , k ), can be interpreted as single-electron correlation functions. To obtain the Schmidt spectrum and characterize the degree of entanglement, we discretize the kernels and numerically solve the integral eigenvalue equations. We have used a 800 × 800 discretization of the kernel and repeated the procedure for each control parameter separately. The pair amplitude used for the generation of the kernels involves integration over the polariton degrees of freedom. We have done this numerically for each set of control parameters λ using straightforward numerical integration of Eq. 5 on a uniform grid. The step size was varied to satisfy the convergence of the Schmidt number.
The convergence criterion adopted in this scheme is max{2 (κ N +1 − κ N ) / (κ N +1 + κ N )} ≤ 0.05, where N is the number of data points within a constant range in the given kernel size.

DATA AVAILABILITY
The main results of this manuscript are composed of analytical and numerical calculations. All data generated, analyzed or required to reproduce the results of this study are included in this article and its Supplementary Information file.