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This work concerns the theoretical description of the quantum dynamics ofmolecular junctions with thermal fluctuations and probability losses To thisend, we propose a theory for describing non-Hermitian quantum systems embeddedin constant-temperature environments. Along the lines discussed in [A. Sergi etal, Symmetry 10 518 (2018)], we adopt the operator-valued Wigner formulation ofquantum mechanics (wherein the density matrix depends on the points of theWigner phase space associated to the system) and derive a non-linear equationof motion. Moreover, we introduce a model for a non-Hermitian quantumsingle-molecule junction (nHQSMJ). In this model the leads are mapped to atunneling 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 describesphenomenologically probability losses. Finally, the temperature of the moleculeis controlled by means of a Nos\'e-Hoover chain thermostat. A numerical studyof the quantum dynamics of this toy model at different temperatures isreported. We find that the combined action of probability losses and thermalfluctuations assists quantum transport through the molecular junction. Thepossibility that the formalism here presented can be extended to treat bothmore quantum states (about 10) and many more classical modes or atomicparticles (about 10^3 - 10^5) is highlighted.

Evolution of a Non-Hermitian Quantum Single-Molecule Junction at Constant Temperature