  This work concerns the theoretical description of the quantum dynamics of
molecular junctions with thermal fluctuations and probability losses To this
end, we propose a theory for describing non-Hermitian quantum systems embedded
in constant-temperature environments. Along the lines discussed in [A. Sergi et
al, Symmetry 10 518 (2018)], we adopt the operator-valued Wigner formulation of
quantum mechanics (wherein the density matrix depends on the points of the
Wigner phase space associated to the system) and derive a non-linear equation
of motion. Moreover, we introduce a model for a non-Hermitian quantum
single-molecule junction (nHQSMJ). In this model the leads are mapped to a
tunneling two-level system, which is in turn coupled to a harmonic mode (i.e.,
the molecule). A decay operator acting on the two-level system describes
phenomenologically probability losses. Finally, the temperature of the molecule
is controlled by means of a Nos\'e-Hoover chain thermostat. A numerical study
of the quantum dynamics of this toy model at different temperatures is
reported. We find that the combined action of probability losses and thermal
fluctuations assists quantum transport through the molecular junction. The
possibility that the formalism here presented can be extended to treat both
more quantum states (about 10) and many more classical modes or atomic
particles (about 10^3 - 10^5) is highlighted.
