  Molecular electronics is a rapidly developing field focused on using
molecules as the structural basis for electronic components. It is common in
such devices for the system of interest to couple simultaneously to multiple
environments. Here we consider a model comprised of a double quantum dot (or
molecule) coupled strongly to vibrations and weakly to two electronic leads
held at arbitrary bias voltage. The strong vibrational coupling invalidates
treating the bosonic and electronic environments simply as acting additively,
as would be the case in the weak coupling regime or for flat leads at infinite
bias. Instead, making use of the reaction coordinate framework we incorporate
the dominant vibrational coupling effects within an enlarged system
Hamiltonian. This allows us to derive a non-additive form for the lead
couplings that accounts properly for the influence of strong and non-Markovian
coupling between the double dot system and the vibrations. Applying counting
statistics techniques we track electron flow between the double dot and the
electronic leads, revealing both strong-coupling and non-additive effects in
the electron current, noise and Fano factor.
