jector P selects terms which fulfill Eqs. (A.12). We also only need to consider phonon operators with an equal number of phonon creation and annihilation operators, as only these keep the total phonon energy diagonal, as described by Eq. (A.13). Using Eqs. (23) and (27), we see that the relevant terms for the calculation of the P L (2) term are given by In the following we summarise the terms which con- tribute to the evaluation of the first term in Eq. (A.18). Only terms which are different from zero when applied to the vacuum state | ˜ 0 〉 of the atoms have to be taken into account, since L (2) is applied to this state. Furthermore, the final state of the atoms should be | ˜ 0 〉 , since the pro- 16 4. The P L (2) term in Eq. (A.18) H (2) cond = ∑ p ,α,β i$_{ℏ}$Ω 2 | c ( k$_{L}$ ) | 2 [ − i 2 Im c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ − e$_{αβ}$ ( k$_{L}$ ) + e$_{αβ}$ ( k$_{L}$ + p )$^{]}$ ( b$_{p}$$_{α}$b † p β + b † − p $_{α}$b$_{−}$$_{p}$$_{β}$ ) + . . . , R $^{(2)}$( ρ ) = ∑ p ,α,β Ω 2 | c ( k$_{L}$ ) | 2 $^{[}$Re e$_{αβ}$ ( k$_{L}$ )( b$_{p}$$_{α}$b † p β + b † − p $_{α}$b$_{−}$$_{p}$$_{β}$ ) ρ − 2Re e$_{αβ}$ ( k$_{L}$ + p )( b$_{p}$$_{α}$ ρ b † p β + b † − p α ρ b$_{−}$$_{p}$$_{β}$ ) +Re e$_{αβ}$ ( k$_{L}$ ) ρ ( b$_{p}$$_{α}$b † p β + b † − p $_{α}$b$_{−}$$_{p}$$_{β}$ ) $^{]}$+ . . . , (A.19) PL $^{(2)}$( ρ (0) ss ) = ∑ p ,α,β Ω 2 | c ( k$_{L}$ ) | 2 [ − i 2 Im c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ − i Im e$_{αβ}$ ( k$_{L}$ ) + e$_{αβ}$ ( k$_{L}$ + p )$^{]}$ ( b$_{p}$$_{α}$b † p β + b † − p $_{α}$b$_{−}$$_{p}$$_{β}$ ) ρ (0) ss +H . c . − ∑ p ,α,β 2Ω 2 | c ( k$_{L}$ ) | $_{2}$Re e$_{αβ}$ ( k$_{L}$ + p )( b$_{p}$$_{α}$ ρ (0) ss b † p β + b † − p α ρ (0) ss b$_{−}$$_{p}$$_{β}$ ) , (A.20) 5. The P L (1) $^{[}$L (0) ] − $^{1}$L (1) term in Eq. (A.18) H (1) cond = ∑ p ,α $_{ℏ}$Ω 2 ( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( p + k$_{L}$ ) c ( k$_{L}$ ) ) ( b$_{p}$$_{α}$ + b † − p $_{α}$) s † p + k$_{L}$ − ∑ α $_{ℏ}$η$_{α}$ √ N Ω 2 2 c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ ( b$_{0}$$_{α}$ + b † 0 $_{α}$) + . . . , R $^{(1)}$( ρ ) = ∑ α i √ N Ω 2 | c ( k$_{L}$ ) | 2 Im d$_{α}$ ( k$_{L}$ ) $^{[}$ρ ( b$_{0}$$_{α}$ + b † 0 $_{α}$) − ( b$_{0}$$_{α}$ + b † 0 $_{α}$) ρ $^{]}$+ . . . , (A.21) ˜ H (1) cond = ∑ p ,α $_{ℏ}$Ω 2 ( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( p + k$_{L}$ ) c ( k$_{L}$ ) ) ( b$_{p}$$_{α}$ + b † − p $_{α}$)˜ s † p + k$_{L}$ + ∑ α ℏ √ N Ω 2 | c ( k$_{L}$ ) | 2 $^{(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ $^{)}$( b$_{0}$$_{α}$ + b † 0 $_{α}$) + . . . (A.22) and ˜ R $^{(1)}$( ρ ) = 0. The master equation is in both cases exactly the same. where . . . stands for terms which give zero when applied to the vacuum state | ˜ 0 〉 of the atoms, and hence can be discarded. As each term of R $^{(1)}$( ρ ) in Eq. (A.21) contains operators acting on either the left or the right of ρ , not both, these terms can be absorbed into the conditional Hamiltonian, giving We next calculate P $^{(}$L (1) $^{[}$L (0) ] − 1 L $^{(1)}$( ρ (0) ss ) ) by applying these operators one at a time. For the first application of L $^{(1)}$, again using Eqs. (23) and (27), we find These operators do not contain atomic operators, and are quadratic in phonon operators, with every product being diagonal in phonon wavevector p . In an isotropic lattice ( ν$_{x}$ = ν$_{y}$ = ν$_{z}$ ) α and β can be different, but in a fully anisotropic lattice ( ν$_{x}$ , ν$_{y}$ , ν$_{z}$ all different) we must have α = β to stay diagonal in phonon energy (Eq. (A.13)). where the dots stand for terms which do not contribute to Eq. (A.18), either because they are zero when acting on ρ (0) ss , or because the projection P removes them. Putting these together,