content_image/404140.jpg content_image/404136.jpg content_image/404135.jpg content_image/404134.jpg content_image/404133.jpg content_image/404142.jpg content_image/404139.jpg content_image/404141.jpg content_image/404138.jpg content_image/404137.jpg content_image/404132.jpg
## III. EFFECTIVE HAMILTONIAN AND MASTER EQUATION FOR THE COUPLED MR-TQD SYSTEM 

We now study the coupled MR-TQD system. Rather than analyzing directly the energy exchange between the MR and the TQD which involves tedious algebra, we apply a canonical transform U = e S on the whole system, where 

The transformed Hamiltonian is given by 

<img src='content_image/404140.jpg'>

where we have neglected a small level shift of η $^{2}$ω$_{m}$ to the states | 1 〉 and | 3 〉 and we have also defined 

To describe the quantum dynamics of the coupled MR- TQD system, we derive a master equation (under the Born-Markov approximation) by tracing over the degrees of freedom of both the electrodes and the thermal bath. Up to second order in η , the master equation can be written as 

<img src='content_image/404136.jpg'>

where 

<img src='content_image/404135.jpg'>

<img src='content_image/404134.jpg'>

<img src='content_image/404133.jpg'>

Here, the Liouvillian operator L $_{T}$ρ accounts for the dissi- pation due to the electrodes and L $_{D}$ρ represents the dis- sipation at the MR induced by the thermal bath. Also, γ denotes the decay rate of excitations in the MR induced by the thermal bath and n ( ω$_{m}$ ) is the average boson num- ber at frequency ω$_{m}$ in the thermal bath. 

<img src='content_image/404142.jpg'>

<img src='content_image/404139.jpg'>

## IV. GROUND-STATE COOLING OF THE MR 

## A. Master equation for the reduced density matrix of the MR 

In the limit γ ≪ g ≪ ω$_{m}$ , the TQD is weakly coupled to the MR and can be regarded as part of the environ- ment experienced by the MR. The degrees of freedom of the TQD can then be adiabatically eliminated [27, 39] and the master equation for the reduced density matrix µ of the MR is given by (see Appendix) 

<img src='content_image/404141.jpg'>

where δ$_{m}$ is the driving-induced shift of the MR fre- quency. In Eq. (28), the additional terms A$_{+}$ and A$_{−}$ are induced by the coupling with the TQD. With this mas- ter equation, one obtains the equation of motion for the phonon-number-probability distribution, p$_{n}$ = 〈 n | µ | n 〉 , of the MR: 

<img src='content_image/404138.jpg'>

Moreover, the equation of motion for the average phonon number, 〈 n 〉 = ∑ $_{n}$np$_{n}$ , in the MR can be obtained from Eq. (29) as 

<img src='content_image/404137.jpg'>

where W = A$_{−}$ − A$_{+}$ . In order to cool the MR, one needs W > 0 (i.e., A$_{−}$ > A$_{+}$ ). 

From Eq. (30), the steady-state average phonon num- ber in the MR is 

where the term γn ( ω$_{m}$ ) in the numerator is due to the thermal bath while A$_{+}$ results from the scattering pro- cesses by the TQD. We assume that the MR is initially at equilibrium with the thermal bath, so that the initial phonon number in the MR is n ( ω$_{m}$ ). In order to cool down the MR significantly, one needs a large cooling rate W ≫ γ to overcompensate for the heating effect of the thermal bath. At the end of Sec. IV, we show that this can be achieved using typical experimental parameters. 

Here we consider Δ$_{1}$ = Δ$_{2}$ ≡ Δ so that the dark state exists. The transition rates A$_{±}$ are found to be (see Ap- 

## B. Steady-state solution 

<img src='content_image/404132.jpg'>

