Nonequilibrium phonon occupation in carbon nanotube quantum dots*

We present a formalism for electron transport through a coulomb blockaded quantum dot strongly coupled with vibrations and weakly with leads and the thermal environment. By calculating the joint electron-phonon probability distribution, we show that recently observed anomalous conductivity through single-walled carbon nanotube (SWCNT) quantum dots arises from `hot' phonons that are generated by the current at a faster rate than their extraction rate by the surrounding. We explain semi-quantitative details of the experiment and predict a nontrivial temperature dependence of the phonon population arising from a subtle interplay between phonon emission and absorption rates at specific bias voltage values.