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## II. A MECHANICAL RESONATOR COUPLED TO A TRIPLE QUANTUM DOT 

The device layout of a MR coupled to a TQD is shown in Fig. 1(a). The TQD is connected to three electrodes via tunneling barriers. In the TQD, dots 1 and 2 are only capacitively coupled to each other and electrons cannot tunnel directly between them. Such capacitively coupled dots have already been achieved in experiments (see, e.g., [36]). In contrast, electrons can tunnel between dots 1 and 3 as well as between dots 2 and 3. Here we focus on the strong Coulomb-blockade regime, so that at most a single electron is allowed in the TQD. Thus, only four electronic states need to be considered in the TQD, i.e., the vacuum state | 0 〉 , and states | 1 〉 , | 2 〉 and | 3 〉 corre- sponding to a single electron in the respective dot. The MR is capacitively coupled to dots 1 and 3 and this is schematically shown in Fig. 1(a). 

The total Hamiltonian of the whole system reads 

The unperturbed Hamiltonian H$_{0}$ is defined by 

where terms on the R.H.S. of Eq. (2) denote Hamiltoni- ans of the electrodes, the TQD, the MR and the thermal bath given by 

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We have put ¯ h = 1 and the energy of the state | 3 〉 is chosen as the zero-energy point. c † ik ( c$_{ik}$ ) is the creation (annihilation) operator of an electron with momentum k in the i th electrode ( i = 1 , 2 or 3) and a † i creates an electron in the i th dot. The phonon operators b † and b respectively create and annihilate an excitation of fre- quency ω$_{m}$ in the MR. In Eq. (6), the thermal bath is modeled as a bosonic bath with b † q ( b$_{q}$ ) being the creation (annihilation) operator at freqency ω$_{q}$ . 

The electromechanical coupling between the MR and dots 1 and 3 of the TQD is given by 

with a coupling strength g = ηω$_{m}$ . For a typical elec- tromechanical coupling, η ∼ 0 . 1 (see, e.g., Ref. [37]). The tunneling coupling between the TQD and the electrodes 

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## A. Model 

is 

where Ω$_{ik}$ characterizes the coupling strength between the i th dot and the associated electrode via tunneling barrier. Moreover, the coupling of the MR to the outside thermal bath is characterized by 

with a coupling strength Ω$_{q}$. 

## B. Analogy between TQD and Λ -type three-level atom in two driving electromagnetical fields 

We now show that in the absence of the MR, our TQD system is analogous to a typical Λ-type three-level atom in the presence of two classical electromagnetical fields. This field-driven three-level system is often used in quan- tum optics for producing a dark state (see, e.g., [34]). The Hamiltonian of the field-driven Λ-type three-level system can be written as 

where ω$_{i}$ ( i = 1 , 2 or 3) is the energy of the i th state in the three-level system. Also, ω$_{a}$ and ω$_{b}$ are the fre- quencies of the two driving fields and Ω$_{a}$ and Ω$_{b}$ are the corresponding driving strengths. In order to eliminate the time-dependence of the Hamiltonian in Eq. (10), we transform the system into a rotating frame defined by U$_{R}$ = e iRt with 

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The transformed Hamiltonian is 

where the first term is evaluated as (within the rotating- wave approximation) 

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and the second term gives 

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