≡ + ≡ + + + ≡ + + + + + ≡ ≡ + + + ; ; (a) (b) (c) Σ$_{α}$( p ) = + O(η$^{4}$) V = + O(η$^{4}$) - = ν$_{α}$ ,  p ν$_{α}$ ,  p 19 FIG. 10: (Color online) (a) Feynman diagram transformations corresponding to s$_{p}$ → ˜ s$_{p}$ : the laser vertices are replaced by the vertices introduced here, the one-line (zeroth order laser) vertex being removed completely, while the hopping and decay vertices are unchanged. All except the one at the top right have both 1-phonon and 2-phonon versions, with the latter having both phonon lines attached to the same vertex. The reset vertices (the 2 in the top right, with ρ lines through them) always have at least one line on each side of ρ , and hence the phonon-only reset vertex (top right) has no 1-phonon version. (b) Diagrams for the phonon self-energy Σ$_{α}$( p ). (c) Diagrams for the vacuum bubble V . The diagrams for Σ$_{α}$( p ) are all on the left of the ρ line (the same diagrams on the right give Σ ∗ $_{α}$( p )), while those for V are summed over both sides of ρ . As in Fig. 4, solid orange lines are atomic excitations, dotted green lines are phonons (with either direction allowed if there is no arrow), blue circles are laser driving, blue stars are spontaneous decay. In the fully anisotropic case (three different phonon frequencies ν$_{x}$ , ν$_{y}$ , ν$_{z}$ ), Eq. (A.30) simplifies to In this case each phonon mode p α evolves independently, with a heating (phonon creation) rate Re M$_{h}$$_{αα}$ ( p ), a cooling (phonon decay) rate Re M$_{c}$$_{αα}$ ( p ), and an energy dispersion Im M$_{h}$$_{αα}$ ( p ) + Im M$_{c}$$_{αα}$ ( p ). Comparing Eq. (A.31) with Eq. (29), we see that both methods give the same heating and cooling rates, i.e. Re M$_{h}$$_{αα}$ = R$_{h}$$_{α}$, Re M$_{c}$$_{αα}$ = R$_{c}$$_{α}$ . 7. Consistency with diagrammatic method 0 = ∑ p ,α [ − 1 2 M$_{h}$$_{αα}$ ( p ) b$_{p}$$_{α}$b † p $_{α}$ρ (0) ss − 1 2 M ∗ h $_{αα}$( p ) ρ (0) ss b$_{p}$$_{α}$b † p α + Re M$_{h}$$_{αα}$ ( p ) b † p $_{α}$ρ (0) ss b$_{p}$$_{α}$ − 1 2 M$_{c}$$_{αα}$ ( p ) b † p $_{α}$b$_{p}$$_{α}$ ρ (0) ss − 1 2 M ∗ c $_{αα}$( p ) ρ (0) ss b † p $_{α}$b$_{p}$$_{α}$ + Re M$_{c}$$_{αα}$ ( p ) b$_{p}$$_{α}$ ρ (0) ss b † p α ] . (A.33)