7 In this case, a MR can be cooled from, e.g., an initial temperature T = 100 mK corresponding to n ( ω$_{m}$ ) = 21 down to T = 0 . 8 mK with n$_{st}$ = 0 . 017. In contrast, for sideband cooling of a MR [24–27], the resolved-sideband cooling condition ω$_{m}$ ≫ Γ must be fol- lowed for ground-state cooling of a MR. For a relatively large decay rate, only MR with a very high frequency (which becomes fragile in experiments) can be cooled. On the other hand, for cooling via quantum interfer- ence in the TQD proposed here, the cooling conditions Ω 2 = ω$_{m}$ ( ω$_{m}$ − Δ) and Δ ≫ Γ do not require a high MR frequency. the DQD is zero. The DQD has the eigenstates To verify whether the MR is cooled or not, we pro- pose a scheme in which the MR is coupled to a two-level system realized by a double quantum dot (DQD). The state of the DQD is in turn measured by a nearby charge detector in the form of, e.g., a quantum point contact (QPC). This setup is schematically shown in Fig. 4. Ex- perimentally, the cooling and the verification setups are all fabricated on the same chip with shared components. Dots 1 and 3 in the TQD from the above cooling setup can make up the DQD after a gate voltage is applied to decouple dot 2 from dot 3. Also, the QPC should be decoupled from the rest of the system during cooling by applying a high gate voltage. The Hamiltonian of the system is given by with | g 〉 ( | e 〉 ) being the ground (excited) state. Rewriting the Hamiltonian in Eq. (38) on the eigenstate basis of the DQD, we have where ρ$_{z}$ = | e 〉〈 e | − | g 〉〈 g | and ρ$_{x}$ = | e 〉〈 g | + | g 〉〈 e | are the Pauli matrices. We consider the coupled MR-DQD system in the strong dispersive regime where the coupling strength is much smaller than the difference between the transition frequency 2Ω$_{1}$ of the DQD and that of the MR, i.e., g ≪ δ = 2Ω$_{1}$ − ω$_{m}$ . This regime was previously consid- ered to study whether the vibration of a MR coupled to a superconducting circuit is classical or quantum mechani- cal [41]. In this regime, the phonon in the MR is only vir- tually exchanged between the DQD and the MR. Thus, the coupling of the DQD to the MR does not change the occupation probability of the electron in the DQD, but only results in phonon-number-dependent Stark shifts on energy levels of the DQD. Moreover, the Stark shifts can be detected by measuring the full-frequency current spec- trum of the QPC. The Hamiltonian H$_{R}$ of the MR and the coupling H$_{int}$ be- tween the MR and the DQD are already given in Eqs. (5) and (7). Here H$_{DQD}$ , H$_{QPC}$ and H$_{det}$ are respectively the Hamiltonians of the DQD, the QPC and the coupling be- tween them and are given by Applying both a rotating-wave approximation and a canonical transformation U ′ = e s ′ with where σ$_{z}$ = a † $_{3}$a$_{3}$ − a † $_{1}$a$_{1}$ and σ$_{x}$ = a † $_{3}$a$_{1}$ + a † $_{1}$a$_{3}$ are the Pauli matrices. Also, c$_{ik}$ ( c † $_{ik}$) is the annihilation (cre- ation) operator for an electron with momentum k in ei- ther the source ( i = S ) or the drain ( i = D ) of the QPC. T is the transition amplitude of an isolated QPC and χ is the variation of the transition amplitude caused by the DQD. For simplicity, we assume that the detunning Δ of H$_{DQD}$ = − Δ 2 σ$_{z}$ + Ω$_{1}$ σ$_{x}$, H$_{QPC}$ = ∑ kq ω$_{Sk}$c † $_{Sk}$c$_{Sk}$ + ω$_{Dq}$c † $_{Dq}$c$_{Dq}$ , H$_{det}$ = kq ( T − χσ$_{z}$ )( c † $_{Sk}$c$_{Dq}$ + H . c . ) , (39) s ′ = η ( ρ$_{−}$b † − ρ$_{+}$b ) , η = g/δ, (42) H = ω$_{m}$b $^{†}$b + Ω$_{1}$ ρ$_{z}$ − gρ$_{x}$ ( b † + b ) + H$_{QPC}$ + ∑ kq ( T − χρ$_{x}$ )( c † $_{Sk}$c$_{Dq}$ + H . c . ) , (41) H = H$_{R}$ + H$_{DQD}$ + H$_{QPC}$ + H$_{int}$ + H$_{det}$. (38) | g 〉 = √$_{2}$ 2 ( | 1 〉 − | 3 〉 ) , | e 〉 = √$_{2}$ 2 ( | 1 〉 + | 3 〉 ) , (40) A. Quantum dynamics of coupled MR-DQD system in the presence of a charge detector V. A SCHEME FOR VERIFYING THE COOLING OF THE MR FIG. 4: (color online) Schematic diagram of a MR capaci- tively coupled to a DQD which is under measurement by a nearby QPC. The energy detuning between the two dot states in the DQD is Δ and the interdot coupling strength between them is Ω 1 .