11 M =     − Γ$_{1}$ − Γ$_{1}$ − Γ$_{1}$ 0 0 − i Ω$_{1}$ i Ω$_{1}$ 0 0 Γ$_{2}$ Γ$_{2}$ Γ$_{2}$ 0 0 0 0 − i Ω$_{2}$ i Ω$_{2}$ 0 0 − Γ$_{3}$ 0 0 i Ω$_{1}$ − i Ω$_{1}$ i Ω$_{2}$ − i Ω$_{2}$ 0 0 0 − i Δ$_{d}$ 0 − i Ω$_{2}$ 0 0 i Ω$_{1}$ 0 0 0 0 i Δ$_{d}$ 0 i Ω$_{2}$ − i Ω$_{1}$ 0 − i Ω$_{1}$ 0 i Ω$_{1}$ − i Ω$_{2}$ 0 λ$_{1}$ 0 0 0 i Ω$_{1}$ 0 − i Ω$_{1}$ 0 i Ω$_{2}$ 0 λ ∗ 1 0 0 0 − i Ω$_{2}$ i Ω$_{2}$ 0 − i Ω$_{1}$ 0 0 λ$_{2}$ 0 0 i Ω$_{2}$ − i Ω$_{2}$ i Ω$_{1}$ 0 0 0 0 λ ∗ 2    $^{,}$ (A18) and B = (Γ$_{1}$ , Γ$_{2}$ , 0 , 0 , 0 , 0 , 0 , 0 , 0) $^{T}$. Here we have defined Δ$_{d}$ ≡ Δ$_{1}$ − Δ$_{2}$, λ$_{1}$ ≡ − ( i Δ$_{1}$ + 1 $_{2}$Γ$_{3}$ ), and λ$_{2}$ ≡ − ( i Δ$_{2}$ + 1 $_{2}$Γ$_{3}$ ). From Eq. (A17), the steady-state solution of the vector 〈 ˆ σ 〉 is calculated as Here the matrix L is defined as L = ( sI − M ) − 1 where I denotes the identity matrix. Assuming that the TQD has already attained its steady-state at initial time t = 0, i.e., 〈 ˆ σ (0) 〉 = 〈 ˆ σ $^{st}$〉 , using Eqs. (A15), (A16), (A19), (A21) and the quantum regression theorem [34], one can obtain the correlation function G ( s ) and the scattering rates A$_{±}$ as given in Eq. (A12). Applying the Laplace transform to Eq. (A17), one ob- tains Moreover, the i th component of the vector $^{〈}$˜ σ ( s ) 〉 is given $^{〈}$˜ σ$_{i}$ ( s ) 〉 = 9 ∑ k =1 L$_{ik}$ [ 〈 ˆ σ$_{k}$ (0) 〉 + B$_{k}$ s ] . (A21) 〈 ˆ σ $^{st}$〉 = M − $^{1}$B. (A19) s $^{〈}$˜ σ ( s ) 〉 − 〈 ˆ σ (0) 〉 = M $^{〈}$˜ σ ( s ) 〉 + B s . (A20) by [1] K. C. Schwab and M. L. Roukes, Phys. Today 58 , 36 (2005). [2] X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, Nature (London) 421 , 496 (2003). [3] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science 304 , 74 (2004). [4] V. Braginsky and S. P. Vyatchanin, Phys. Lett. A 293 , 228 (2002). [5] E. Buks and B. Yurke, Phys. Rev. E 74 , 046619 (2006). [6] S. Savel’ev, X. Hu, and F. Nori, New J. Phys. 8 , 105 (2006). [7] S. Savel’ev, A. L. Rakhmanov, X. Hu, A. Kasumov, and F. Nori, Phys. Rev. B 75 , 165417 (2007). [8] D. Vitali, S. Gigan, A. Ferreira, H. R. B¨ ohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. As- pelmeyer, Phys. Rev. Lett. 98 , 030405 (2007). [9] M. J. Hartmann and M. B. Plenio, Phys. Rev. Lett. 101 , 200503 (2008). [10] L. Tian and P. Zoller, Phys. Rev. Lett. 93 , 266403 (2004). [11] T. F. Li, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, J. S. Tsai, and H. Im, Appl. Phys. Lett. 92 , 043112 (2008). [12] A. K. H¨ uttel, G. A. Steele, B. Witkamp, M. Poot, L. P. Kouwenhoven, and H. S. J. van der Zant, Nano Lett. 9 , 2547 (2009). [13] C. H. Metzger and K. Karrai, Nature (London) 432 , 1002 (2004). [14] S. Gigan, H. R. Bohm, M. Paternostro, F. Blaster, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger, Nature (London) 444 , 67 [27] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippen- berg, Phys. Rev. Lett. 99 , 093901 (2007); I. Wilson-Rae, [26] S. Zippilli, G. Morigi, and A. Bachtold, Phys. Rev. Lett. 102 , 096804 (2009). [25] S. H. Ouyang, J. Q. You, and F. Nori, Phys. Rev. B 79 , 075304 (2009). [24] I. Wilson-Rae, P. Zoller, and A. Imamoglu, Phys. Rev. Lett. 92 , 075507 (2004). [23] J. Q. You, Y. X. Liu, and F. Nori, Phys. Rev. Lett. 100 , 047001 (2008). [22] P. Zhang, Y. D. Wang, and C. P. Sun, Phys. Rev. Lett. 95 , 097204 (2005); Y.D. Wang, Y. Li, F. Xue, C. Bruder, and K. Semba, Phys. Rev. B 80 , 144508 (2009). [21] T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Phys. Rev. Lett. 95 , 033901 (2005). [20] J. D. Teufel, C. A. Regal, and K. W. Lehnert, New J. Phys. 10 , 095002 (2008). [19] M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, Phys. Rev. Lett. 99 , 017201 (2007). [18] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97 , 243905 (2006). [17] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C. Schwab, Nature (Lon- don) 443 , 193 (2006). [16] O. Arcizet, R. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444 , 71 (2006). [15] D. Kleckner and D. Bouwmeester, Nature (London) 444 , 75 (2006). (2006).