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Thus, Eq. (12) reduces to 

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where Δ$_{1}$ = ω$_{3}$ − ω$_{1}$ − ω$_{a}$ and Δ$_{2}$ = ω$_{3}$ − ω$_{2}$ − ω$_{b}$ are the frequency detunings while Ω$_{1}$ = Ω$_{a}$ / 2 and Ω$_{2}$ = Ω$_{b}$ / 2 are the effective driving strengths of the two fields. It is now clear that the Hamiltonian in Eq. (15) is formally equivalent to that of the TQD in Eq. (4). This shows that the TQD we propose here is an electronic analog of a Λ-type three-level atom driven by two electromagnetic fields. 

From the study of quantum optics, the existence of a dark state in a Λ-type three-level atom when the lower- energy states become degenerate, i.e., Δ$_{1}$ = Δ$_{2}$, is able to suppress absorption or emission. Below we demonstrate that a similar dark state also exists in the TQD [38]. 

After tracing over the degrees of freedom of the elec- trodes, the quantum dynamics of the TQD in the absence of the MR is described by 

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where ρ$_{d}$ is the reduced density matrix of the TQD and Γ$_{i}$ ( i = 1 , 2 or 3) is the rate for electrons tunneling into or out of the i th dot. The notation D for any operator A is given by 

Considering equal energy detunings of the lower energy states | 1 〉 and | 2 〉 with respect to the excited state | 3 〉 , i.e., Δ$_{1}$ = Δ$_{2}$ = Δ, the eigenstates | g 〉 , |−〉 and | + 〉 of the TQD become 

where α = cos( θ/ 2), β = sin( θ/ 2), tan θ = 2Ω / Δ, and Ω = √ Ω 2 1 + Ω 2 $_{2}$. The corresponding eigenenergies are 

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with φ = √$_{Δ}$ 2 + 4Ω $^{2}$. For simplicity, we consider equal couplings of the three quantum dots to the correspond- ing electrodes, i.e., Γ$_{1}$ = Γ$_{2}$ = Γ$_{3}$ ≡ Γ. Based on the 

## C. “Dark state” in the TQD 

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eigenstate basis of the TQD in Eq. (18), the equations of motion for the reduced density matrix elements of the TQD are obtained from Eq. (16) as 

Figure 2 schematically show effective electron tunneling processes through the TQD as described by Eq. (20). Starting from an initially empty TQD, an electron can tunnel from the left electrode into any of the three eigen- states, with tunneling rates Γ β $^{2}$, Γ and Γ α 2 for eigen- states | + 〉 , |−〉 and | g 〉 , respectively. An electron in the eigenstate | + 〉 ( | g 〉 ) then tunnels out of the TQD to the right electrode with a rate Γ α 2 (Γ β $^{2}$). However, if the electron occupies the state |−〉 , no further tunneling oc- curs because, being orthogonal to | 3 〉 , it is decoupled from the right electrode. Therefore, an electron in the TQD will be trapped in the state |−〉 , which is called the dark state in quantum optics [38]. This dark state results from the destructive quantum interference between the tran- sition | 1 〉 → | 3 〉 (i.e., the electron tunneling from state | 1 〉 to state | 3 〉 in the TQD system) and the transition 2 → | 3 〉 (i.e., the electron tunneling from state | 2 〉 to state | 3 〉 ). 

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