Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation?

TL;DR

This paper investigates whether quantum collision models based on Wigner functions can simultaneously satisfy complete positivity and energy conservation, highlighting the practical challenges and proposing explicit microscopic models as a solution.

Contribution

The study demonstrates that while no fundamental restriction prevents satisfying both conditions, practical collision models require explicit microscopic states for consistency.

Findings

Collision models can violate energy conservation.

No fundamental restriction prevents simultaneous positivity and energy conservation.

Explicit microscopic models are recommended for practical implementations.

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 modelling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner-Weyl transformation, the modelling 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, since they allow to model collisions of each electron individually in a controlled way satisfying both conditions.