3 II. A MECHANICAL RESONATOR COUPLED TO A TRIPLE QUANTUM DOT A. Model B. Analogy between TQD and Λ -type three-level atom in two driving electromagnetical fields 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 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$_{q}$. 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 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 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 The transformed Hamiltonian is 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 where the first term is evaluated as (within the rotating- wave approximation) 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 and the second term gives i ˙ U − 1 R U$_{R}$ = ω$_{a}$a † $_{1}$a$_{1}$ + ω$_{b}$a † $_{2}$a$_{2}$ − ω$_{3}$ ( a † $_{1}$a$_{1}$ + a † $_{2}$a$_{2}$ + a † $_{3}$a$_{3}$ ) . (14) U − 1 R H$_{Λ}$U$_{R}$ = ω$_{1}$a † $_{1}$a$_{1}$ + ω$_{2}$a † $_{2}$a$_{2}$ + ω$_{3}$a † $_{3}$a$_{3}$ + Ω$_{a}$ 2 ( a † $_{1}$a$_{3}$ + a † $_{3}$a$_{1}$ ) + Ω$_{b}$ 2 ( a † $_{2}$a$_{3}$ + a † $_{3}$a$_{2}$ ) , (13) H$_{int}$ = − g ( a † $_{3}$a$_{3}$ − a † $_{1}$a$_{1}$ )( b † + b ) , (7) ˜ H$_{Λ}$ = U − 1 R H$_{Λ}$U$_{R}$ + i ˙ U − 1 R U$_{R}$, (12) R = ω$_{a}$a † $_{1}$a$_{1}$ + ω$_{b}$a † $_{2}$a$_{2}$ − ω$_{3}$ ( a † $_{1}$a$_{1}$ + a † $_{2}$a$_{2}$ + a † $_{3}$a$_{3}$ ) . (11) H$_{ph}$ = ∑ q ω$_{q}$b † $_{q}$b$_{q}$ . (6) H$_{R}$ = ω$_{m}$b $^{†}$b, (5) H$_{TQD}$ = − Δ$_{1}$ a † $_{1}$a$_{1}$ − Δ$_{2}$ a † $_{2}$a$_{2}$ +(Ω$_{1}$ a † $_{1}$a$_{3}$ + Ω$_{2}$ a † $_{2}$a$_{3}$ + H . c . ) , (4) H$_{leads}$ = ∑ ik E$_{ik}$c † $_{ik}$c$_{ik}$ , (3) H$_{Λ}$ = ω$_{1}$a † $_{1}$a$_{1}$ + ω$_{2}$a † $_{2}$a$_{2}$ + ω$_{3}$a † $_{3}$a$_{3}$ +Ω$_{a}$ cos( ω$_{a}$t )( a † $_{1}$a$_{3}$ + a † $_{3}$a$_{1}$ ) +Ω$_{b}$ cos( ω$_{b}$t )( a † $_{2}$a$_{3}$ + a † $_{3}$a$_{2}$ ) , (10) H$_{0}$ = H$_{leads}$ + H$_{TQD}$ + H$_{R}$ + H$_{ph}$, (2) H$_{total}$ = H$_{0}$ + H$_{int}$ + H$_{T}$ + H$_{ep}$. (1) H$_{ep}$ = ∑ q Ω$_{q}$( b † $_{q}$b + H . c . ) , (9) H$_{T}$ = ∑ ik (Ω$_{ik}$ a † i c$_{ik}$ + H . c . ) , (8) is