  We investigate the nonequilibrium population of a vibrational mode in the
steady state of a biased molecular junction, using a rate equation approach. We
focus on the limit of weak electronic-vibrational coupling and show that, in
the resonant transport regime and for sufficiently high bias voltages, the
level of vibrational excitation increases with decreasing coupling strength,
assuming a finite and non-zero value. An analytic behavior with respect to the
electronic-vibrational coupling strength is only observed if the influence of
environmental degrees of freedom is explicitly taken into account. We consider
the influence of three different types of broadening: hybridization with the
electrodes, thermal fluctuations and the coupling to a thermal heat bath. Our
results apply to vibrationally coupled electron transport through molecular
junctions but also to quantum dots coupled to a microwave cavity, where the
photon number can be expected to exhibit a similar behavior.
