| Summary: | 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 Nosé-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 (<inline-formula><math display="inline"><semantics><mrow><mo>∼</mo><mn>10</mn></mrow></semantics></math></inline-formula>) and many more classical modes or atomic particles (<inline-formula><math display="inline"><semantics><mrow><mo>∼</mo><msup><mn>10</mn><mn>3</mn></msup><mo>−</mo><msup><mn>10</mn><mn>5</mn></msup></mrow></semantics></math></inline-formula>) is highlighted.
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