Thermoelectric information engine driven by an autonomous Maxwell demon across quantum-to-classical transitions
We study a three-terminal thermoelectric engine, focusing on the role of quantum coherence and information flow. A double-dot connects two reservoirs at different chemical potentials, while a third dot monitors their occupation via Coulomb interaction and can be interpreted as an autonomous Maxwell demon. Within the parameter range where the device operates as an engine, we identify conditions under which this interpretation holds. The system dynamics is described within a Redfield master equation that allows us to identify two distinct dynamical regimes with steady states well captured by suitable Lindblad approximations. These two regimes define a first quantum-to-classical transition controlled by the interdot tunneling strength. We further consider the effect of a phonon bath coupled to the double-dot, which induces a second quantum-to-classical transition by generating incoherent transport and decoherence in the small interdot tunneling regime. We identify a parameter region where phonon-induced decoherence suppresses both the coherent transport contribution and the information flow toward the monitoring dot, suggesting that coherence can enhance the demon mechanism in this regime. By tracking information and transport properties across these crossovers, our model shows how coherent tunneling, decoherence, and incoherent phonon-assisted transport compete in an autonomous information engine, while clarifying which thermodynamic Lindblad description is appropriate in each regime.