Klein's paradox is beautifully visualized in the phase space through the relativistic Wigner function.
In introductory quantum mechanics we learn about quantum tunneling and how it allows the transmission of particles through potentials that are otherwise insurmountable in mechanics.
In relativity we have another twist, the transmission is not only present but it can be the dominant effect for strong potentials. In the following animation we illustrate this effect where we clearly see that the transmitted wave-packet is composed of antiparticles (negative momentum but moving to the positive direction).
This type of effect can be seen in effective Dirac materials in terms of an enhanced tunneling. The animation of such process is
These effects are well know in the literature. My contribution in [1] was to show that the presence of quantum decoherence does not affect them in a significant measure.
[1] Renan Cabrera, Andre G. Campos, Denys I. Bondar, and Herschel A. Rabitz, Dirac open-quantum-system dynamics: Formulations and simulations , Phys. Rev. A 94, 052111 (2016) https://doi.org/10.1103/PhysRevA.94.052111
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