Non-uniform magnetic fields for single-electron control.

Controlling single-electron states becomes increasingly important due to the wide-ranging advances in electron quantum optics. Single-electron control enables coherent manipulation of individual electrons and the ability to exploit the wave nature of electrons, which offers various opportunities for quantum information processing, sensing, and metrology. Here we explore non-uniform magnetic fields, which offer unique mechanisms for single-electron control. Considering the modeling perspective, conventional electron quantum transport theories are commonly based on gauge-dependent electromagnetic potentials. A direct formulation in terms of intuitive electromagnetic fields is thus not possible. In an effort to rectify this, a gauge-invariant formulation of the Wigner equation for general electromagnetic fields has been proposed [M. Nedjalkov et al., Phys. Rev. B, 2019, 99, 014423]. However, the complexity of this equation requires the derivation of a more convenient formulation for linear electromagnetic fields [M. Nedjalkov et al., Phys. Rev. A, 2022, 106, 052213]. This formulation directly includes the classical formulation of the Lorentz force and higher-order terms, depending on the magnetic field gradient, that are negligible for small variations of the magnetic field. In this work, we generalize this equation in order to include a general, non-uniform electric field and a linear, non-uniform magnetic field. The thus obtained formulation has been applied to investigate the capabilities of a linear, non-uniform magnetic field to control single-electron states in terms of trajectory, interference patterns, and dispersion. This has led to the exploration of a new type of transport inside electronic waveguides based on snake trajectories and the possibility of splitting wavepackets to realize edge states.