  Two important features of mesoscopic Aharonov-Bohm (A-B) electronic
interferometers are analyzed: decoherence due to coupling with other degrees of
freedom and the coupled transport of charge and heat. We first review the
principles of decoherence of electronic interference. We then analyze the
thermoelectric transport in a ring threaded by such a flux, with a molecular
bridge on one of its arms. The charge carriers may also interact inelastically
with the molecular vibrations. This nano-system is connected to three termi-
nals; two of them are electric and thermal, held at slightly different chemical
potentials and temperatures, and the third is purely thermal. For example, a
phonon bath thermalizing the molecular vibrations. When this third terminal is
held at a temperature different from those of the electronic reservoirs, both
an electrical and a heat current are, in general, gen- erated between the
latter. Likewise, a voltage and/or temperature difference between the
electronic terminals leads to thermal current between the thermal and
electronic terminals. The transport coefficients governing these conversions
(due to energy exchange between the electrons and the vibrations) and their
dependences on the A-B flux are analyzed. Finally, the decoherence due to these
inelastic events is discussed.
