9 of the MR. In this regime, the coupling between the MR and the DQD induces a MR-phonon-number dependent shift of the transition frequency of the DQD. Thus the phonon-number state which characterizes the cooling re- sult of the MR can be detected by measuring the shifts of the resonance peaks in the current spectrum of a nearby charge detector. This work is supported by the National Basic Re- search Program of China Grant Nos. 2009CB929300 and 2006CB921205, the National Natural Science Foundation of China Grant Nos. 10534060 and 10625416, and the Research Grant Council of Hong Kong SAR project No. 500908. In this appendix, we derive the master equation [Eq. (28)] for the reduced density matrix of the MR from the master equation [Eq. (24)] of the coupled MR-TQD sys- tem by eliminating the degrees of freedom of the TQD. In general, the dissipation rate of the MR is much smaller than the decay rate of the TQD, i.e., [ n ( ω$_{m}$ ) + 1] γ ≪ Γ. The TQD hence attains its steady-state quickly and its perturbation to the MR can be regarded as part of the en- vironment [27, 39]. Up to the second order in η , Eq. (24) can be rewritten as result of the MR. Indeed, a relatively strong coupling be- tween a MR and a quantum dot has been recently demon- strated [12]. The strong dispersive regime is thus achiev- able and one can apply the proposed coupled MR-DQD system to verify the cooling of the MR via measuring the current spectrum of a nearby charge detector (e.g., QPC). Our proposal on ground state cooling of the MR re- quires that the TQD is able to evolve into the dark state. However, the dephasing of the TQD due to coupling to other degrees of freedom in the environment can project the TQD into one of the three localized states | 1 〉 , | 2 〉 , and | 3 〉 and drive the system away from the dark state [38]. However, the dephasing between the two localized states | 1 〉 and | 2 〉 depends on the coupling strength and the en- ergy detunning between dots 1 and 2 [44]. Here in our system no direct coupling exists between the two local- ized states | 1 〉 and | 2 〉 , and thus the dephasing almost has negligible effects on the cooling efficiency of the MR. In summary, we have studied the cooling of a MR by capacitive coupling to a TQD. We show that when the two lower-energy localized states become degenerate, the TQD will be trapped in a dark state which is decou- pled from the excited state in the absence of the MR. With the MR in resonance with the transition between the dark state and the excited eigenstate in the TQD, we have shown that the MR can be cooled to its ground state in the non-resolved sideband cooling regime . More- over, we have proposed a coupled MR-DQD system in the strong dispersive regime for verifying the cooling result The MR and the TQD are decoupled. Since the TQD is at its steady state most of the time, one has ρ ( t ) = ρ st d ⊗ Tr$_{d}$ { ρ } with ρ st d denoting the reduced density ma- trix of the TQD at steady-state and Tr$_{d}$ {· · · } the trace over the TQD’s degrees of freedom. Equation (A5) has an infinite number of steady-state solutions. These so- lutions can be expanded in the basis of the eigenvectors ρ st d ⊗ | n 〉〈 n $^{′}$| of the Liouville operator L 0 with eigenval- ues λ$_{nn}$ ′ = − i ( n − n $^{′}$) ω$_{m}$ , i.e., L $_{0}$ρ$_{nn}$ ′ = λ$_{nn}$ ′ ρ$_{nn}$ ′ [33]. are respectively the Liouvillians to zeroth, first, and sec- ond orders in η . At zeroth order in η , the quantum dy- namics of the whole system is described by where dρ dt = L ρ = [ L 0 + L 1 + L $_{2}$] ρ, (A1) L $_{0}$ρ = − iω$_{m}$ [ b $^{†}$b, ρ ] − i [ H$_{TQD}$, ρ ] +Γ$_{1}$ D [ a † $_{1}$] ρ + Γ$_{2}$ D [ a † $_{2}$] ρ + Γ$_{3}$ D [ a$_{3}$ ] ρ, (A2) L $_{1}$ρ = − i [ V ( b † − b ) , ρ ] , (A3) $_{2}$ρ = η $^{2}$Γ$_{1}$ ( D [ a † $_{1}$b $^{†}$] ρ + D [ a † $_{1}$b ] ρ ) + η $^{2}$Γ$_{3}$ ( [ a$_{3}$b $^{†}$] ρ + D [ a$_{3}$b ] ρ ) + L $_{D}$ρ, (A4) ˙ ρ ( t ) = L $_{0}$ρ ( t ) . (A5) VI. DISCUSSION AND CONCLUSION Appendix A: Master equation for the reduced density matrix of the MR Acknowledgments FIG. 5: (color online) Power spectrum of the current through the QPC when the phonon number in the MR are respec- tively n = 0 . 01 (black solid line), n = 1 (red dashed line), or given by the thermal distribution (blue dotted line), i.e., p n = n $_{st}$/ (1 + n st ) $^{n}$+1 with n st = 1. The coupling strength between the MR and the DQD is g = 0 . 1 ω m (a) and g = 0 . 3 ω m (b). The other parameters are ω m = 100 MHz, Ω 1 = 2 ω m , γ 2 = 0 . 01 ω m , γ 1 = 0 . 2 γ 2 , and γ d = 2 γ 2 .