III. EFFECTIVE HAMILTONIAN AND MASTER EQUATION FOR THE COUPLED MR-TQD SYSTEM IV. GROUND-STATE COOLING OF THE MR A. Master equation for the reduced density matrix of the MR 5 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) 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 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: where we have neglected a small level shift of η $^{2}$ω$_{m}$ to the states | 1 〉 and | 3 〉 and we have also defined Moreover, the equation of motion for the average phonon number, 〈 n 〉 = ∑ $_{n}$np$_{n}$ , in the MR can be obtained from Eq. (29) as 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 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 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- 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. n$_{st}$ = γn ( ω$_{m}$ ) + A$_{+}$ γ + W , (31) L $_{D}$ρ = γ [ n ( ω$_{m}$ ) + 1] D [ b ] ρ + γn ( ω$_{m}$ ) D [ b $^{†}$] ρ. (27) L $_{T}$ρ = Γ D [ a † $_{1}$] ρ + Γ D [ a † $_{2}$] ρ + Γ D [ a$_{3}$ ] ρ + η $^{2}$Γ$_{1}$ ( D [ a † $_{1}$b $^{†}$] ρ + D [ a † $_{1}$b ] ρ ) + η $^{2}$Γ$_{3}$ [ a$_{3}$b $^{†}$] ρ + [ a$_{3}$b ] ρ , (26) V = η [ 2Ω$_{1}$( a † $_{1}$a$_{3}$ − a † $_{3}$a$_{1}$ ) + Ω$_{2}$( a † $_{2}$a$_{3}$ − a † $_{3}$a$_{2}$ ) ] , (25) dρ dt = − iω$_{m}$ [ b $^{†}$b, ρ ] − i [ H$_{TQD}$, ρ ] − i [ V ( b † − b ) , ρ ] + L $_{T}$ρ + L $_{D}$ρ, (24) d 〈 n 〉 dt = − ( γ + W ) 〈 n 〉 + γn ( ω$_{m}$ ) + A$_{+}$, (30) dp$_{n}$ dt = { γ [ n ( ω$_{m}$ ) + 1] + A$_{−}$ } [( n + 1) p$_{n}$$_{+1}$ − np$_{n}$ ] +[ γn ( ω$_{m}$ ) + A$_{+}$ ][ np$_{n}$$_{−}$$_{1}$ − ( n + 1) p$_{n}$ ] , (29) B = exp[ − η ( b − b $^{†}$)] . (23) H = UH$_{total}$U † = H$_{leads}$ + H$_{ph}$ + H$_{ep}$ + ω$_{m}$b $^{†}$b − Δ$_{1}$ a † $_{1}$a$_{1}$ − Δ$_{2}$ a † $_{2}$a$_{2}$ +[Ω$_{1}$ a † $_{1}$a$_{3}$ B 2 + Ω$_{2}$ a † $_{2}$a$_{3}$ B + H . c . ] , + ∑ k [ Ω$_{1}$$_{k}$ a † $_{1}$c$_{1}$$_{k}$ B + Ω$_{2}$$_{k}$ a † $_{2}$c$_{2}$$_{k}$ +Ω$_{3}$$_{k}$ a † $_{3}$c$_{3}$$_{k}$ B † + H . c . ] , (22) ˙ µ = − i ( ω$_{m}$ + δ$_{m}$ )[ b $^{†}$b, ρ ] + 1 2 { γ [ n ( ω$_{m}$ ) + 1] + A$_{−}$ ( ω$_{m}$ ) } × [2 bµb † − ( b $^{†}$bµ + µb $^{†}$b )] + 1 2 [ γn ( ω$_{m}$ ) + A$_{+}$ ( ω$_{m}$ )][2 b $^{†}$µb − ( bb $^{†}$µ + µbb $^{†}$)] , (28) S = η ( a † $_{3}$a$_{3}$ − a † $_{1}$a$_{1}$ )( b − b $^{†}$) . (21) B. Steady-state solution