  The steady-state cooling of a nanomechanical resonator interacting with three
coupled quantum dots is studied. General conditions for the cooling to the
ground state with single and two-electron dark states are obtained. The results
show that in the case of the interaction of the resonator with a
single-electron dark state, no cooling of the resonator occurs unless the
quantum dots are not identical. The steady-state cooling is possible only if
the energy state of the quantum dot coupled to the drain electrode is detuned
from the energy states of the dots coupled to the electron source electrode.
The detuning has the effect of unequal shifting of the effective dressed states
of the system that the cooling and heating processes occur at different
frequencies. For the case of two electrons injected to the quantum dot system,
the creation of a two-particle dark state is established to be possible with
spin-antiparallel electrons. The results predict that with the two-particle
dark state, an effective cooling can be achieved even with identical quantum
dots subject of an asymmetry only in the charging potential energies coupling
the injected electrons. It is found that similar to the case of the
single-electron dark state, the asymmetries result in the cooling and heating
processes to occur at different frequencies. However, an important difference
between the single and two-particle dark state cases is that the cooling
process occurs at significantly different frequencies. This indicates that the
frequency at which the resonator could be cooled to its ground state can be
changed by switching from the one-electron to the two-electron Coulomb blockade
process.
