  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. A unique opportunity
offering new degrees of freedom for single-electron control is provided when
considering non-uniform magnetic fields. 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 in [Nedjalkov et al., Phys. Rev. B.,
2019, 99, 014423]. However, the complexity of this equation requires to derive
a more convenient formulation for linear electromagnetic fields [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
explore a new type of transport inside electronic waveguides based on snake
trajectories and also to explore the possibility to split wavepackets to
realize edge states.
