4 FIG. 2: (color online) Effective tunneling processes of elec- trons through a TQD represented in the eigenstate basis. An electron can tunnel from the left electrode into the three eigenstates | + 〉 , |−〉 and | g 〉 , with rates Γ β $^{2}$, Γ and Γ α $^{2}$, re- spectively. Note that the total tunneling rate is 2Γ because an electron tunnels from the left electrode to the TQD via two tunnel barriers (each having a tunneling rate Γ). In the eigenstate | + 〉 ( | g 〉 ), it will tunnel out to the right electrode with a rate Γ α 2 (Γ β $^{2}$). However, if the electron occupies the dark state |−〉 , no further tunneling occurs and the electron is trapped. Thus, Eq. (12) reduces to 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 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 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 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 〉 ). 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 C. “Dark state” in the TQD ˙ ρ$_{d}$ = L $_{TQD}$ρ$_{d}$ = − i [ H$_{TQD}$, ρ$_{d}$ ] + Γ$_{1}$ D [ a † $_{1}$] ρ + Γ$_{2}$ D [ a † $_{2}$] ρ$_{d}$ + Γ$_{3}$ D [ a$_{3}$ ] ρ$_{d}$, (16) ˜ H$_{Λ}$ = − Δ$_{1}$ a † $_{1}$a$_{1}$ − Δ$_{2}$ a † $_{2}$a$_{2}$ +Ω$_{1}$( a † $_{1}$a$_{3}$ + a † $_{3}$a$_{1}$ ) + Ω$_{2}$( a † $_{2}$a$_{3}$ + a † $_{3}$a$_{2}$ ) , (15) D [ A ] ρ = AρA † − 1 2 [ A $^{†}$Aρ + ρA $^{†}$A ] . (17) | g 〉 = β | 3 〉 − α Ω (Ω$_{1}$ | 1 〉 + Ω$_{2}$ | 2 〉 ) , |−〉 = 1 Ω (Ω$_{2}$ | 1 〉 − Ω$_{1}$ | 2 〉 ) , | + 〉 = α | 3 〉 + β Ω (Ω$_{1}$ | 1 〉 + Ω$_{2}$ | 2 〉 ) , (18) E$_{g}$ = − Δ + φ 2 , E$_{−}$ = − Δ , E$_{+}$ = − Δ − φ 2 , (19) ˙ ρ$_{00}$ = − 2Γ ρ$_{00}$ + Γ β $^{2}$ρ$_{gg}$ + Γ α $^{2}$ρ$_{++}$ + Γ αβ ( ρ$_{+}$$_{g}$ + ρ$_{g}$$_{+}$ ) , ˙ ρ$_{gg}$ = Γ α $^{2}$ρ$_{00}$ − Γ β $^{2}$ρ$_{gg}$ − Γ 2 αβ ( ρ$_{+}$$_{g}$ + ρ$_{g}$$_{+}$ ) , ˙ ρ$_{−−}$ = Γ ρ$_{00}$, ˙ ρ$_{++}$ = Γ β $^{2}$ρ$_{00}$ − Γ α $^{2}$ρ$_{++}$ − Γ 2 αβ ( ρ$_{+}$$_{g}$ + ρ$_{g}$$_{+}$ ) , ˙ ρ$_{+}$$_{g}$ = − i ( E$_{+}$ − E$_{g}$ ) ρ$_{+}$$_{g}$ − Γ 2 ρ$_{+}$$_{g}$ − Γ 2 αβ ( ρ$_{++}$ + ρ$_{gg}$ ) − Γ αβρ$_{00}$. (20)