Ground-state cooling of a nanomechanical resonator with a triple quantum dot via quantum interference

We employ double quantum interference to propose a ground-state cooling scheme for a nanomechanical resonator coupled to a triple quantum dot (TQD), of which the third dot is driven by a microwave field. In our scheme, the carrier transition is diminished by the destructive interference (the occurrence of a dark state) arising from the coherent tunnelings of electrons among the three dots. Furthermore, the additional transition induced by the microwave field enables the system to possess double quantum interference effect. Under certain conditions, the processes which might increase one phonon through the blue sideband transitions are prohibited due to the double destructive interference analog to double electromagnetically induced transparency mechanism. Thus, both the carrier and blue sideband transitions are canceled. The cooling process occurs when the electron trapped in the dark state absorbs one phonon of the resonator through the transition from the dark state to the bright state and then to the excited state and finally tunnels to the drain. As a result, the phonon occupation can be reduced to zero in the ideal case and the resonator can be cooled down to its ground state with high efficiency even when the dephasing of the TQD is present.

Figure 1

Schematic of TQD connected to source and drain electrodes. Dots 1 and 2 are capacitively coupled to each other without electrons tunneling directly between them. Both levels $|1\rangle$ and $|2\rangle$ are designed to align with the excited level of the third dot such that they can be coupled with tunneling amplitudes $T_1$ and $T_2$, respectively. A microwave field of the Rabi frequency $\Omega$ with frequency ${\omega }_L$ is applied to drive the third dot and ${\Delta }_g$ is the detuning between the driven field frequency and the transition $|g\rangle \leftrightarrow |e\rangle$. A NMR is capacitively coupled only to dot 1.

Figure 2

Contour plot of ${\langle n\rangle }_{\text{st}}$ and ${\gamma }_{\text{tot}}$ as a function of $\omega$ and ${\Delta }_g$. To optimize the parameters, we set ${\Delta }_1={\Delta }_2=$ 0 such that the electron is trapped in the dark state of the TQD in absence of the NMR. All the parameters are scaled with $\Gamma =\pi \times 7.5$ MHz and other parameters are ${\Gamma }_j=3\Gamma (j=1,2,g,e)$, $T_1=T_2=8\Gamma$, $\Omega =20\Gamma$, ${\kappa }_{1,2,3}=0$, $\eta =0.05$, ${\overline{n}}_p=21$ ($T_p=100$ mK), and ${\gamma }_p=0$.

Figure 3

Same as in Fig. 2 with ${\gamma }_p=2\times {10}^{-5}\Gamma$. A NMR with frequency $\omega =2\pi \times 100$ MHz ($\Gamma \simeq 2\pi \times 7.5$ MHz) can be cooled from $T_p=100$ mK $({\overline{n}}_p=21)$ to $T_m=0.9$ mK $({\langle n\rangle }_{\text{st}}=5\times {10}^{-3})$.

Figure 4

Schematic of transitions in coupled NMR-TQD system with $T^{\prime }=T_1{T}_2/T^2$. Under the condition of Eq. (20), the heating processes depicted by the dashed arrow are prohibited due to the destructive interference between the two-step transition $|D,n\rangle \to |B,n+1\rangle \to |e,n+1\rangle$ and the auxiliary transition $|g,n+1\rangle \to |e,n+1\rangle$. Meanwhile, the cooling processes depicted by the solid arrow are allowed. Thus the electron may transfer to the state $|e,n-1\rangle$, taking away one phonon of the NMR mode and subsequently escaping into the drain.

Figure 5

${\langle n\rangle }_{\text{st}}$ as a function of $\omega$ for $\Omega =20\Gamma$ and definite $T=8\Gamma$ in difference ratio $\beta =T_1/T_2$: $\beta =1$ (solid line), $\beta =\sqrt{23}/3$ ($T_2=6\Gamma$, dashed line), $\beta =\sqrt{103}/5$ ($T_2=5\Gamma$, dot-dashed line), and $\beta =\sqrt{7}$ ($T_2=4\Gamma$, dotted line). Other parameters are ${\Gamma }_i=3\Gamma$, ${\kappa }_{1,2,3}=0$, $\eta =0.05$, ${\overline{n}}_p=21$, and ${\gamma }_p=2\times {10}^{-5}\Gamma$.

Figure 6

Plot of average phonon number ${\langle n\rangle }_{\text{st}}$ as a function of frequency $\omega$ for different dephasing rates, ${\kappa }_1={\kappa }_2={\kappa }_3=0$, (solid line), $5\times {10}^{-4}\Gamma$ (dashed line) and $1\times {10}^{-3}\Gamma$ (dot-dashed line). Other parameters are $\Omega =20\Gamma$, $T_1=T_2=8\Gamma$, ${\Gamma }_i=3\Gamma$, $\eta =0.05$, ${\overline{n}}_p=21$, and ${\gamma }_p=2\times {10}^{-5}\Gamma$.

Figure 7

Average phonon number ${\langle n\rangle }_{\text{st}}$ of NMR mode as a function of frequency $\omega$ of NMR for $\Omega =20\Gamma$ and $T_1=T_2=8\Gamma$ for different values of tunneling rate ${\Gamma }_e$, for which the electron escapes into the drain, ${\Gamma }_e=2\Gamma$ (solid line), ${\Gamma }_e=3\Gamma$ (dashed line), and ${\Gamma }_e=5\Gamma$ (dot-dashed line). Other parameters are ${\kappa }_{1,2,3}={10}^{-4}\Gamma$, $\eta =0.05$, ${\overline{n}}_p=21$, and ${\gamma }_p=2\times {10}^{-5}\Gamma$.