10 where Eq. (A11) is simply the master equation (28) for the re- duced density matrix of the MR derived in Sec. IV. In Eq. (A9), the correlation function G ( s ) is given by Here, | n 〉 ( n = 0 , 1 , 2 , · · · ) denotes the n th state of the MR and ( n − n $^{′}$) ω$_{m}$ represents the energy difference be- tween the states | n 〉 and | n $^{′}$〉 . For η = 0, these states with different n are weakly coupled by the perturbative terms L 1 and L $_{2}$. To obtain the quantum dynamics of the MR, we project the system onto the subspace with a zero eigenvalue ( n = n $^{′}$) of L $_{0}$. The projection operator P is defined by Noting g ≪ ω$_{m}$ (i.e., η ≪ 1), a second order perturbation expansion gives the following closed equation for P ρ [25] Substituting Eqs. (A3) and (A4) into Eq. (A7) and taking the trace over the TQD degrees of freedom, the first term in Eq. (A7) becomes [39] and the second term gives where µ = Tr$_{d}$ {P ρ } is the reduced density matrix of the MR and ρ st 00 is the probability of an empty TQD at the steady state. Here, we have defined Thus, from Eqs. (A7), (A8) and (A9), one has where S$_{i}$ ( s ) = $^{〈}$˜ σ$_{i}$ ( s ) V (0) 〉 with $^{〈}$˜ σ$_{i}$ ( s ) 〉 being the Laplace transform of 〈 ˆ σ$_{i}$ ( t ) 〉 . From Eq. (16), we find that 〈 ˆ σ$_{i}$ ( t ) 〉 obeys the equation of motion: where where the average value of each component is 〈 ˆ σ$_{i}$ 〉 = Tr { ˆ σ$_{i}$ρ$_{d}$ } . Using this notation, one has and thus To determine the correlation function G ( ± s ), one first calculate the Laplace transform ˜ V ( s ) of the interaction term V ( t ). For convenience, we introduce the vector op- erator ˆ σ for the TQD whose components are defined as where s = iω$_{m}$ . Also, ˜ V ( s ) is the Laplace transform of V ( t ) and the Liouvillians L TQD is given in Eq. (16). A$_{±}$ = 2 Re[ G ( ± iω$_{m}$ )] + η $^{2}$Γ$_{1}$ ρ st $_{00}$. (A12) G ( s ) = − Tr$_{d}$ ∞ ∫ 0 dtV (0) e L$_{TQD}$ $^{t}$V (0) e st = − ∞ ∫ 0 dτ 〈 V ( t ) V (0) 〉 e $^{st}$= −〈 ˜ V ( s ) V (0) 〉 , (A13) L 0 P ρ = 0 . (A6) P ˙ ρ ( t ) = P L 2 P ρ ( t ) + ∞ ∫ 0 dτ P L $_{1}$e L$_{0}$ $^{τ}$L 1 P ρ ( t ) . (A7) Tr$_{d}$ {P L 2 P ρ ( t ) } = 1 2 [ γn ( ω$_{m}$ ) + η $^{2}$Γ$_{1}$ ρ st $_{00}$] D [ b ] µ + 1 2 { γ [ n ( ω$_{m}$ ) + 1] + η $^{2}$Γ$_{1}$ ρ st 00 } D [ b $^{†}$] µ, (A8) ˆ σ$_{1}$ = | 1 〉〈 1 | , ˆ σ$_{2}$ = | 2 〉〈 2 | , ˆ σ$_{3}$ = | 3 〉〈 3 | , ˆ σ$_{4}$ = 1 2 , ˆ σ$_{5}$ = 2 1 , ˆ σ$_{6}$ = 1 3 , ˆ σ$_{7}$ = | 3 〉〈 1 | , ˆ σ$_{8}$ = | 2 〉〈 3 | , ˆ σ$_{9}$ = | 3 〉〈 2 | , (A14) Tr$_{d}$ { ∞ ∫ 0 dτ P L $_{1}$e L$_{0}$ $^{τ}$L 1 P ρ ( t ) } = − i ( ω$_{m}$ + δ$_{m}$ )[ b $^{†}$b, µ ] , +Re[ G ( iω$_{m}$ )] D [ b ] µ + Re[ G ( − iω$_{m}$ )] D [ b $^{†}$] µ, (A9) V = 2 η Ω$_{1}$(ˆ σ$_{7}$ − ˆ σ$_{6}$ ) + η Ω$_{2}$(ˆ σ$_{9}$ − ˆ σ$_{8}$ ) . (A15) G ( s ) = − 2 η Ω$_{1}$[ S$_{7}$ ( s ) − S$_{6}$ ( s )] − η Ω$_{2}$[ S$_{9}$ ( s ) − S$_{8}$ ( s )] , (A16) δ$_{m}$ = Im[ G ( iω$_{m}$ ) + G ( − iω$_{m}$ )] . (A10) ˙ µ = − i ( ω$_{m}$ + δ$_{m}$ )[ b $^{†}$b, µ ] + 1 2 [ γn ( ω$_{m}$ ) + A$_{+}$ ] D [ b ] µ + 1 2 { γ [ n ( ω$_{m}$ ) + 1] + A$_{−}$ } D [ b $^{†}$] µ, (A11) d 〈 ˆ σ ( t ) 〉 dt = M 〈 ˆ σ ( t ) 〉 + B, (A17)