8 where to the Hamiltonian H , one obtains [up to O ( η $^{2}$)] are the respective rates of electron tunneling through the QPC when dot 3 is respectively occupied or empty [43]. Therefore, one can define the current operator as From Eq. (43), after taking the trace over the degrees of freedom of the QPC, one obtains the following master equation for the reduced density matrix elements of the coupled MR-DQD system [42]: with I$_{0}$$_{,}$$_{1}$ = e ( D ± D $^{′}$) / 2. According to the Wiener- Khintchine theorem, the power spectrum of the current through the QPC is [34] Substituting Eqs. (45) and (48) into Eq. (49), we get where δ$_{n}$ = 2Ω$_{1}$ + g $^{2}$(2 n + 1) /δ and γ$_{1}$ = 2 πg$_{s}$g$_{d}$χ $^{2}$eV$_{d}$ with g$_{s}$ ( g$_{d}$ ) being the density of states for electrons in the source (drain) of the QPC and V$_{d}$ the bias voltage across the QPC. Here γ$_{±}$ = γ$_{1}$ (1 ∓ λ$_{n}$ ), with λ$_{n}$ = δ$_{n}$/eV$_{d}$ , are the QPC-induced excitation and relaxation rates be- tween the ground state and the excited state of the DQD. Also, γ$_{d}$ is the relaxation rate resulting from the coupling of the DQD to the thermal bath. Since the dissipation rate of the MR is much smaller than that of the DQD, dissipation of the MR is neglected. In Eq. (44), the re- duced density matrix element ρ$_{in,in}$ ( i = g, e ) gives the occupation probability of the state | i, n 〉 of the coupled MR-DQD system ,while ρ$_{in,jn}$ ( i = j ) describes the co- herence between the states | i, n 〉 and | j, n 〉 . The equa- tions of motion for other elements, e.g., ρ$_{in,jn}$ ′ ( n = n $^{′}$), which are decoupled from those considered here, are not shown. Using Eq. (44) and the normalization condition p$_{n}$ = ρ$_{gn,gn}$ + ρ$_{en,en}$ , one finds where γ$_{2}$ = 2 πg$_{s}$g$_{d}$T $^{2}$V$_{d}$ , κ = ( γ$_{+}$ − γ$_{−}$ − γ$_{d}$ ) / 2 γ$_{0}$ , and S$_{0}$ = 2 eI$_{0}$ is the current-noise background. From Eq. (50), one sees that the current spectrum of the QPC consists of peaks at resonance points ω = ± δ$_{n}$ . These peaks have width γ$_{0}$ and heights increasing with the prob- ability p$_{n}$ . For instance, the peaks at the resonance point is shifted by g $^{2}$(2 n + 1) /δ from 2Ω$_{1}$. Thus, from this peak shift in the current spectrum, one can readout the phonon-number state of the MR. Figure 4 plots the current spectrum of the QPC with two different coupling strengths between the MR and the DQD. Results for three cases in which the MR is respectively in its ground state ( n = 0 . 01 ≪ 1), the first- excited state ( n = 1 . 0) or thermalized with an average phonon number n$_{st}$ = 1 . 0 are plotted. Each resonance peak in the current spectrum corresponds to a phonon- number state of the MR. For the thermally distributed case, the current spectrum shows several peaks, where the relative area under each peak gives the probability of the corresponding phonon-number state. As shown in Fig. 4(a), the distance between two adjacent peaks is smaller than the intrinsic peak width in the weak dis- persive regime, i.e., 2 g $^{2}$/δ < γ$_{0}$ , and hence the measured spectrum shows an ensemble. In this case, the phonon- number state of the MR cannot be measured. In the strong limit (2 g $^{2}$/δ > γ$_{0}$ ), however, the ensemble can be individually resolved [Fig. 4(b)], which allows us to de- tect the phonon number and also to verify the cooling where γ$_{0}$ = γ$_{1}$ + γ$_{d}$/ 2 and p$_{n}$ is the probability that the MR is at state | n 〉 . B. Current spectrum of the charge detector D = 2 πg$_{s}$g$_{d}$ ( T − χ ) $^{2}$V$_{d}$ , D ′ = 2 πg$_{s}$g$_{d}$ ( T + χ ) $^{2}$V$_{d}$ , (47) I ( t ) = I$_{0}$ + I$_{1}$σ$_{z}$ ( t ) = I$_{0}$ + I$_{1}$ρ$_{x}$ ( t ) , (48) S ( ω )=Re$^{∞}$ ∫ 0 e $^{iωτ}$dτ [ 〈 I ( t ) I ( t + τ ) 〉 − 〈 I ( t + τ 〉〈 I ( t ) 〉 ] . (49) S ( ω ) /S$_{0}$ = 1 + 2 γ$_{1}$γ$_{2}$ γ$_{1}$ + γ$_{2}$ ∑ n p$_{n}$ (1 − κp$_{n}$ ) γ$_{0}$ γ 2 0 + ( δ$_{n}$ − ω ) 2 − 2 γ$_{1}$γ$_{2}$ γ$_{1}$ + γ$_{2}$ n p$_{n}$ (1 + κp$_{n}$ ) γ$_{0}$ γ 2 0 + ( δ$_{n}$ + ω ) $_{2}$, (50) ˙ ρ$_{gn,gn}$ = − γ$_{+}$ρ$_{gn,gn}$ + ( γ$_{−}$ + γ$_{d}$ ) ρ$_{en,en}$, ˙ ρ$_{en,en}$ = γ$_{+}$ρ$_{gn,gn}$ − ( γ$_{−}$ + γ$_{d}$ ) ρ$_{en,en}$, ˙ ρ$_{gn,en}$ = − iδ$_{n}$ρ$_{gn,en}$ − ( γ$_{1}$ + γ$_{d}$ 2 ) ρ$_{gn,en}$, (44) H ≈ ω$_{m}$b $^{†}$b + 2Ω$_{1}$ + g $^{2}$(2 b $^{†}$b + 1) /δ 2 ρ$_{z}$ + g 2 2 δ ( | g 〉〈 g | + | e 〉〈 e | ) + H$_{QPC}$ + ∑ kq ( T − χρ$_{x}$ )( c † $_{Sk}$c$_{Dq}$ + H . c . ) . (43) δ$_{n}$ = 2Ω$_{1}$ + g $^{2}$(2 n + 1) δ , (51) ρ$_{gn,gn}$ ( t ) = ( γ$_{−}$ + γ$_{d}$ ) p$_{n}$ 2 γ$_{0}$ − [ ( γ$_{−}$ + γ$_{d}$ ) p$_{n}$ 2 γ$_{0}$ − ρ$_{gn,gn}$ (0) ] e − 2 γ$_{0}$$^{t}$, ρ$_{en,en}$ ( t ) = γ$_{+}$ 2 γ$_{0}$ p$_{n}$ − [ γ$_{+}$ 2 γ$_{0}$ p$_{n}$ − ρ$_{en,en}$ (0) e − 2 γ$_{0}$$^{t}$, ρ$_{gn,en}$ ( t ) = ρ$_{gn,en}$ (0) e − i ( δ n − γ$_{0}$ ) $^{t}$, (45) I ( t )= eDρ$_{11}$ + eD $^{′}$ρ$_{33}$ = e 2 ( D + D $^{′}$) + e 2 ( D ′ − D ) 〈 σ$_{z}$ 〉 , (46) The dc current through the QPC is related to the elec- tron occupation probability in the DQD and is given by [43]