Quantum Wave-Particle Duality in Free-Electron–Free-Electron Entanglement

{ "core_finding": "The longitudinal spatial extent of a free-electron quantum wave packet acts as a direct control parameter for entanglement generated by relativistic electromagnetic coupling between two spatially separated electrons.", "caveats": "The analysis assumes pure longitudinal Gaussian wave packets, neglects transverse deflections, spin, and radiative effects, and presumes negligible overlap-exchange interactions in separate paths.", "markdown": "## Research Question and Motivation\n\nTraditional point-particle descriptions of electron-electron interactions neglect the coherent longitudinal spatial extent of free electrons. While recent advances in ultrafast electron microscopy and quantum electron optics enable precise shaping and manipulation of free-electron wave packets, the direct link between wave-particle duality and bipartite entanglement has remained largely unexplored. This paper investigates how the spatial extent and wave-packet structure of freely propagating electrons influence mutual electromagnetic coupling and control bipartite quantum entanglement.\n\n## Theoretical Framework and Entanglement Scaling\n\nThe authors develop a relativistic two-electron wave-packet theory for particles propagating along spatially separated paths. By treating each electron as a longitudinal quantum wave packet (QEW) and expanding the accumulated interaction phase, the framework yields a mixed quadratic coefficient that defines a dimensionless entangling parameter. The resulting entanglement, quantified by the logarithmic negativity, scales quadratically with the rms longitudinal width of the wave packets in the local spatial regime. Full time-dependent Schrödinger equation calculations confirm this quadratic scaling for narrow wave packets while revealing higher-order Coulomb corrections at larger spatial extents.\n\n## Wave-Packet Control and Drift Protocols\n\nDirectly altering the initial spatial width of a Gaussian wave packet typically changes its momentum bandwidth. To overcome this limitation, the study examines free longitudinal drift, which increases the spatial width at the interaction point while preserving the momentum probability distribution. This free-drift protocol demonstrates that spatial wave-packet expansion alone can systematically enhance the subsequently generated entanglement, establishing a clear signature of wave-particle duality in quantum correlations.\n\n## Higher-Order Coulomb Entanglement and Nodes\n\nIntroducing a longitudinal centroid mismatch between the two electrons provides access to higher-order spatial contributions of the interaction kernel. Under specific mismatch conditions, the quadratic entanglement contribution vanishes, creating a node where the quadratic prediction drops to zero. However, full numerical simulations reveal a residual entanglement signal originating from cubic and higher-order spatial Coulomb terms. In the narrow-QEW regime, this residual entanglement scales cubically with the wave-packet width, confirming the presence of higher-order entangling channels.\n\n## Key Terms and Definitions\n\n- Quantum Electron Wave Packet (QEW) — A free-electron state characterized by a coherent finite longitudinal spatial and spectral structure beyond a classical point-particle description.\n- Entangling Parameter ($\Gamma_{ee}$) — A dimensionless parameter measuring the strength of the nonlocal quadratic phase coupling generated by mutual electromagnetic interactions between two wave packets.\n- Logarithmic Negativity — An entanglement monotone derived from the partial transpose of the two-electron density operator, quantifying bipartite quantum entanglement.\n- Longitudinal Effective Mass — An enhanced mass parameter, scaling with the Lorentz factor cubed, that governs the longitudinal dispersion and dynamics of relativistic electrons.\n- Centroid Mismatch — A controlled longitudinal offset between the position centers of the two interacting electron wave packets used to probe higher-order spatial interaction terms." }