Two-particle dark-state cooling of a nanomechanical resonator

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 in which 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 to 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 occurring 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.

Figure 1

Schematic diagram of the system composed of three quantum dots connected to the source and drain electrodes through tunneling barriers. Dots 1 and 2 are capacitively coupled to each other without electrons tunneling directly between them while dots 1 and 2 are coupled coherently to dot 3 with the tunneling amplitudes $T_1$ and $T_2$, respectively. A NMR is capacitively coupled to dots 2 and 3 only.

Figure 2

The steady-state average number of phonons ${\langle n\rangle }_s$ plotted as a function of the mechanical frequency ${\omega }_m$ for $\alpha =2\Gamma$, initial temperature $T_p=100$ mK (corresponding to ${\overline{n}}_p=21$), ${\gamma }_p=2\times {10}^{-4}\Gamma$, ${\Gamma }_1={\Gamma }_2=\Gamma$, $T_1=T_2=10\Gamma$, with ${\Gamma }_3=0.5\Gamma$ and different ${\Delta }_2$: ${\Delta }_2=0.5\Gamma$ (solid line), ${\Delta }_2=\Gamma$ (dashed line), ${\Delta }_2=2\Gamma$ (dot-dashed line), and ${\Delta }_2=4\Gamma$ (dotted line).

Figure 3

The mean phonon number ${\langle n\rangle }_s$ as a function the mechanical frequency ${\omega }_m$ for $\alpha =2\Gamma$, initial temperature $T_p=100$ mK (corresponding to ${\overline{n}}_p=21$), ${\Gamma }_1={\Gamma }_2=\Gamma$, ${\gamma }_p=2\times {10}^{-4}\Gamma$, ${\Gamma }_3=0.5\Gamma$, ${\Delta }_2=\Gamma$, average tunneling amplitude $T=(T_1+T_2)/2=10\Gamma$, and different ratios $\beta =T_1/T_2$: $\beta =1$ (solid line), $\beta =4$ (dotted line), $\beta =9$ (dashed line), and $\beta =19$ (dot-dashed line).

Figure 4

Schematic diagram of possible cooling and heating transitions from the dark state $|{\Phi }_9\rangle$ to other dressed states of the system in the case of the symmetric coupling between the dots. The absorption (cooling) processes correspond to the transitions indicated by solid arrows while the emission (heating) transitions are indicated by dashed arrows.

Figure 5

Two-electron dressed states of the quantum-dot system in the case of the asymmetric coupling between the dots. The solid arrows indicate dominant cooling transitions from the dark state $|{\Phi }_7\rangle$ to the upper states $|{\Phi }_2\rangle$ and $|{\Phi }_3\rangle$, and dashed arrows indicate emission (heating) transitions to the lower energy states $|{\Phi }_8\rangle$ and $|{\Phi }_9\rangle$.

Figure 6

Dependence of the steady-state mean phonon number ${\langle n\rangle }_s$ on frequency ${\omega }_m$ for the asymmetric case of ${\delta }_w=T$ with initial temperature $T_p=100$ mK (corresponding to ${\overline{n}}_p=21$), ${\Gamma }_1={\Gamma }_2=\Gamma$, $T=10\Gamma$, $\alpha =2\Gamma$, ${\delta }_u=0$, ${\Delta }_i=0(i=1,2,3)$, ${\gamma }_p=2\times {10}^{-4}\Gamma$, ${\delta }_{33}=2T$, and different tunneling rates ${\Gamma }_3$: ${\Gamma }_3=0.5\Gamma$ (solid line), ${\Gamma }_3=\Gamma$ (dashed line), ${\Gamma }_3=2\Gamma$ (dot-dashed line).

Figure 7

The dependence of minimum value of ${\langle n\rangle }_s$ on the tunneling rate ${\Gamma }_3$ for the asymmetric case of ${\delta }_w=T$ with initial temperature $T_p=100$ mK (corresponding to ${\overline{n}}_p=21$), ${\Gamma }_1={\Gamma }_2=\Gamma$, ${\omega }_m=2T$, $\alpha =2\Gamma$, ${\delta }_u=0$, ${\Delta }_i=0(i=1,2,3)$, ${\gamma }_p=2\times {10}^{-4}\Gamma$, ${\delta }_{33}=2T$, and different $T$: $T=5\Gamma$ (solid line), $T=10\Gamma$ (dashed-line), $T=15\Gamma$ (dash-dotted line).