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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] 

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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] 

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and the second term gives 

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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 

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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 

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where s = iω$_{m}$ . Also, ˜ V ( s ) is the Laplace transform of V ( t ) and the Liouvillians L TQD is given in Eq. (16). 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 the average value of each component is 〈 ˆ σ$_{i}$ 〉 = Tr { ˆ σ$_{i}$ρ$_{d}$ } . Using this notation, one has 

and thus 

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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 

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