*3.2. Modified Quantum Master Equation*

Applying the approximate expression given in Section 3.1, the equation of motion for the quantum density operator can be transformed in the *P*-space representation, where the influence of the *Q*-space is renormalized into the original interaction Hamiltonian, and the creation and annihilation operators in the dissipation terms. Omitting the redundant mathematical transformations, the quantum master equation is modified as follows,

$$\frac{\partial \rho\_{\rm st}(t)}{\partial t} \approx \frac{\partial \rho\_{\rm st}^{\rm p}(t)}{\partial t} = -\frac{i}{\hbar} [H\_{\rm int, \rm st}^{P'} + H\_{\rm ext, \rm st}^{P} \rho\_{\rm st}^{P}(t)] + \mathcal{L}\_{\rm st}^{(\rm nr)'} \rho\_{\rm st}^{P}(t) + \mathcal{L}\_{\rm st}^{(\rm r)'} \rho\_{\rm st}^{P}(t), \tag{14}$$

where the quantum density matrix operator in the *P*-space is *<sup>ρ</sup><sup>P</sup>*st(*t*) = *<sup>P</sup>ρ*st(*t*)*<sup>P</sup>*, and the modified interaction Hamiltonian reads

$$H\_{\rm int,st}^{P'} \equiv PH\_{\rm int,st}P + \sum\_{m \in Q} \frac{PH\_{\rm int,st} |\phi\_m^Q\rangle \langle \phi\_m^Q | H\_{\rm int,st} P}{\langle \phi\_m^Q | H\_{\rm int,st} | \phi\_m^Q \rangle}. \tag{15}$$

In (15), the operator *Q* is rewritten by the intermediate states |*φQm* for clear understanding, i.e.,

$$Q = \sum\_{m \in Q} |\phi\_m^Q\rangle\langle\phi\_m^Q| \,\tag{16}$$

where the summation is applied to the artificially selected basis states in the *Q*-space. (15) means that the interaction Hamiltonian with the coherent dynamics is corrected by the transition between the basis states in the *Q*- and the *P*-spaces.

The dissipation terms in (5) and (6) are similarly rewritten as the following form,

$$\mathcal{L}\_{\rm st}^{\left(\text{r,nr}\right)'} \rho\_{\rm st}^{\mathcal{P}}(t) = \frac{\gamma^{\left(\text{r,nr}\right)}}{2} \sum\_{i,j} \left( 2a\_i^{\mathcal{P}'} \rho\_{\rm st}^{\mathcal{P}} a\_j^{\mathcal{P}' \dagger} - \left\{ a\_i^{\mathcal{P}' \dagger} a\_j^{\mathcal{P}'}, \rho\_{\rm st}^{\mathcal{P}}(t) \right\} \right), \tag{17}$$

where

$$a\_i^{P'} \equiv Pa\_i P + \sum\_{m \in Q} \frac{Pa\_i |\phi\_m^Q\rangle\langle\phi\_m^Q| H\_{\text{int,st}} P}{\langle\phi\_m^Q|H\_{\text{int,st}}|\phi\_m^Q\rangle}. \tag{18}$$

Ideally, the contribution of the dissipation should be renormalized into the relaxation constants *<sup>γ</sup>*(r,nr), and the creation and annihilation operators should be left as the original form of the basis states transformed by a non-equilibrium steady state. However, (17) is only an approximate expression for the operators, since the theoretical formulation has not been completed in this stage. This is a problem to be solved in the future. In (18), the second term means that there are dissipation processes with the energy flow from the *P*-space to the *Q*-space.
