Abstract
To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpretation). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interacting with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation.
Highlights
Electron devices are quantum systems outside of thermodynamic equilibrium with many interacting particles
In this subsection we show the evolution of the Wigner function defined from a single electron evolving in free space and undergoing scattering through the collision models explained in the Sects. 3.2 and 3.3
We conclude that the evolution of the Wigner function of an electrons scattered in free space behaves equivalently with both models, the one of Sect. 3.2 with energy conservation and the one of Sect. 3.3 with momentum conservation
Summary
Electron devices are quantum systems outside of thermodynamic equilibrium with many interacting particles (electrons, atoms, photons, etc). The second difficulty appears because the collisions are, at best, a reasonable approximation of the real interaction between the simulated and non-simulated degrees of freedom (but never an exact result) Both difficulties make the evaluation of the physical soundness of a collision model for quantum transport a difficult task. We clarify that we are not referring here to the fact that the Wigner function is a quasi-probability in the phase space [24], but to the creation of negative probability of finding electrons at some positions in the physical space It is well-known that the typical use of a Boltzmann-like superoperator in the description of the collision term in the Wigner formalism can produce such regions of negative probability presence [6, 7, 25]. 2, we will define the two conditions, complete positivity and energy conservation, for modeling collisions in a general density matrix formulation of a quantum system.
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