of the density operator ρ , we find the equations The master equation (25) is equivalent to the following equation for the mean value 〈O〉 = Tr ( O ρ ) of an arbitrary operator O : In particular, we find the equations d dt 〈 a 〉 = i Δ$_{c}$ 〈 a 〉 + G 〈 σ 〉 − κ 2 〈 a 〉 + η, d dt 〈 σ 〉 = i Δ$_{a}$ 〈 σ 〉 + G 〈 aσ$_{z}$ 〉 − γ 2 〈 σ 〉 , (28) 〈 ˙ O〉 = i ¯ h 〈 [ H$_{eff}$, O ] 〉 − γ 2 〈O σ $^{†}$σ − 2 σ $^{†}$O σ + σ $^{†}$σ O〉 − κ 2 〈O a $^{†}$a − 2 a $^{†}$O aρ + a $^{†}$a O〉 . (27) ˙ ρ$_{e,n}$$_{;}$$_{e,n}$ ′ = − i Δ$_{c}$( n ′ − n ) ρ$_{e,n}$$_{;}$$_{e,n}$ ′ − G $_{(}$√$_{n}$ ′ + 1 ρ$_{e,n}$$_{;}$$_{g,n}$ ′ +1 $_{+}$√$_{n}$$_{+ 1}$$_{ρ}$$_{g,n}$$_{+1;}$$_{e,n}$ ′ ) + η $_{(}$√$_{n}$ $^{′}$ρ$_{e,n}$$_{;}$$_{e,n}$ $^{′}$− 1 $_{+}$√$_{nρ}$$_{e,n}$$_{−}$$_{1;}$$_{e,n}$ ′ − √$_{n}$ ′ + 1 ρ$_{e,n}$$_{;}$$_{e,n}$ ′ +1 − √$_{n}$$_{+ 1}$$_{ρ}$$_{e,n}$$_{+1;}$$_{e,n}$ ′ ) − γρ$_{e,n}$$_{;}$$_{e,n}$ ′ − κ 2 [ ( n ′ + n ) ρ$_{e,n}$$_{;}$$_{e,n}$ ′ − 2 √ ( n ′ + 1)( n + 1) ρ$_{e,n}$$_{+1;}$$_{e,n}$ ′ +1 ] , ˙ ρ$_{g,n}$$_{;}$$_{g,n}$ ′ = − i Δ$_{c}$( n ′ − n ) ρ$_{g,n}$$_{;}$$_{g,n}$ ′ + G $_{(}$√$_{n}$ $^{′}$ρ$_{g,n}$$_{;}$$_{e,n}$ $^{′}$− 1 $_{+}$√$_{nρ}$$_{e,n}$$_{−}$$_{1;}$$_{g,n}$ ′ ) + η $_{(}$√$_{n}$ $^{′}$ρ$_{g,n}$$_{;}$$_{g,n}$ $^{′}$− 1 $_{+}$√$_{nρ}$$_{g,n}$$_{−}$$_{1;}$$_{g,n}$ ′ − √$_{n}$ ′ + 1 ρ$_{g,n}$$_{;}$$_{g,n}$ ′ +1 − √$_{n}$$_{+ 1}$$_{ρ}$$_{g,n}$$_{+1;}$$_{g,n}$ ′ ) + γρ$_{e,n}$$_{;}$$_{e,n}$ ′ − κ 2 [ ( n ′ + n ) ρ$_{g,n}$$_{;}$$_{g,n}$ ′ − 2 √ ( n ′ + 1)( n + 1) ρ$_{g,n}$$_{+1;}$$_{g,n}$ ′ +1 ] , ˙ ρ$_{g,n}$$_{;}$$_{e,n}$ ′ = − i Δ$_{a}$ ρ$_{g,n}$$_{;}$$_{e,n}$ ′ − i Δ$_{c}$( n ′ − n ) ρ$_{g,n}$$_{;}$$_{e,n}$ ′ − G $_{(}$√$_{n}$ ′ + 1 ρ$_{g,n}$$_{;}$$_{g,n}$ ′ +1 − √$_{nρ}$$_{e,n}$$_{−}$$_{1;}$$_{e,n}$ ′ ) + η $_{(}$√$_{n}$ $^{′}$ρ$_{g,n}$$_{;}$$_{e,n}$ $^{′}$− 1 $_{+}$√$_{nρ}$$_{g,n}$$_{−}$$_{1;}$$_{e,n}$ ′ − √$_{n}$ ′ + 1 ρ$_{g,n}$$_{;}$$_{e,n}$ ′ +1 − √$_{n}$$_{+ 1}$$_{ρ}$$_{g,n}$$_{+1;}$$_{e,n}$ ′ ) − γ 2 ρ$_{g,n}$$_{;}$$_{e,n}$ ′ − κ 2 [ ( n ′ + n ) ρ$_{g,n}$$_{;}$$_{e,n}$ ′ − 2 √ ( n ′ + 1)( n + 1) ρ$_{g,n}$$_{+1;}$$_{e,n}$ ′ +1 ] . (26) 10