= + + + + + + O(η$^{4}$) R$_{hα}$ ( p ) = ν$_{α}$ ,  p ν$_{α}$ ,  p 9 asymmetrical diagrams such as Fig. 4(c) give the inter- ference term between their left and right processes. In the case of coherent processes (i.e. not involving any re- set terms), the left and right sides are disconnected, and the transition rate is simply the product of the left and right amplitudes. As already explained, amplitudes on the right are complex conjugated, so this is consistent with rate = | amplitude | $^{2}$. In this section we use these diagrams to calculate the heating and cooling rates. The lowest order (in the Lamb-Dicke parameters η$_{α}$ ) processes that change the phonon number increase or de- crease it by 1 on both sides of ρ , as changing it on only one side is forbidden by energy conservation. This involves 2 external phonon lines (one on each side) which must both end on vertices, and is hence at least 2nd order in η$_{α}$ . In each case there are 6 such diagrams which must be summed to obtain the rate, shown in Fig. 5 for creating a phonon (heating); for destroying a phonon (cooling), they are the same except with all phonon lines reversed. For simplicity we restrict to the fully anisotropic case (three different phonon frequencies ν$_{x}$ , ν$_{y}$ , ν$_{z}$ ), where en- ergy conservation requires that α be the same on both sides. Evaluating these diagrams gives with no external lines. These are evaluated in the Ap- pendix (Eqs. (A.34), (A.35)). Together, these give an effective master equation for the phonons, FIG. 5: (Color online) Feynman diagrams for the heating rate R$_{h}$$_{α}$ ( p ), as evaluated in Eq. 29. The cooling rate diagrams are the same except that all phonon arrows are reversed. As in Fig. 4, solid orange lines are atomic excitations, dotted green lines are phonons, blue circles are laser driving, blue stars are spontaneous decay. The black dotted circle denotes any al- lowed diagram with the given external lines. The line under each diagram is the color used to plot its value in Fig. 9(b). The symmetrical diagrams (first 3) are rate terms; the asym- metrical diagrams (last 3) are interference terms, and are to be read as including both mirror-image versions. B. Heating and cooling rates R$_{h}$$_{α}$ ( p ) , R$_{c}$$_{α}$ ( p ) = − 2Ω $^{2}$Re e$_{αα}$ ( k$_{L}$ ∓ p ) | c ( k$_{L}$ ) | 2 +$^{(}$ η$_{α}$ Ω ˆ k$_{L}$$_{α}$ ) 2 Re c ( k$_{L}$ ∓ p ) | c ( k$_{L}$ ∓ p ) ± 2i ν$_{α}$ | 2 + Ω $^{2}$| d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( k$_{L}$ ∓ p ) | $^{2}$Re c ( k$_{L}$ ∓ p ) | c ( k$_{L}$ ) | $^{2}$| c ( k$_{L}$ ∓ p ) ± 2i ν$_{α}$ | 2 +2Re$^{{}$ − iΩ $^{2}$[ d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( k$_{L}$ ∓ p )]Im d$_{α}$ ( k$_{L}$ ∓ p ) | c ( k$_{L}$ ) | $^{2}$[ c ( k$_{L}$ ∓ p ) ± 2i ν$_{α}$ ] + iΩ $^{2}$η$_{α}$ ˆ k$_{L}$$_{α}$ [ d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( k$_{L}$ ∓ p )]Re c ( k$_{L}$ ∓ p ) c ( k$_{L}$ ) | c ( k$_{L}$ ∓ p ) ± 2i ν$_{α}$ | 2 − Ω $^{2}$η$_{α}$ ˆ k$_{L}$$_{α}$ Im d$_{α}$ ( k$_{L}$ ∓ p ) c $^{∗}$( k$_{L}$ )[ c ( k$_{L}$ ∓ p ) ± 2i ν$_{α}$ ]$^{}}$ + O ( η $^{4}$) , (29) ˙ ρ = V ρ + ∑ p ,α $^{[}$(Σ$_{α}$ ( p ) − i ν$_{α}$ ) b † p $_{α}$b$_{p}$$_{α}$ ρ +(Σ ∗ $_{α}$( p ) + i ν$_{α}$ ) ρ b † p $_{α}$b$_{p}$$_{α}$ + R$_{h}$$_{α}$ ( p ) b † p α ρ b$_{p}$$_{α}$ + R$_{c}$$_{α}$ ( p ) b$_{p}$$_{α}$ ρ b † p α ] . (30) C. Steady state phonon distribution This conserves probability as Re Σ$_{α}$( p ) = − 1 $_{2}$( R$_{h}$$_{α}$ ( p ) + R$_{c}$$_{α}$ ( p ) and V = − ∑ p ,α R$_{h}$$_{α}$ . It also agrees with the steady state equation (A.33) calculated by direct pertur- bative expansion in the Appendix. We now use these equations to calculate the steady state of the phonons. We also introduce the mean phonon number as a measure of heat content, as the steady state does not have a definite temperature. Eq. (30) does not couple different phonon modes p α , so the phonon steady state ρ (0) phn is a product state over modes. As Re Σ$_{α}$( p ) < 0, phonon coherences decay, so At the same order in the η$_{α}$ , there are also a self-energy correction Σ$_{α}$( p ) to the phonon line given by diagrams with one ingoing and one outgoing phonon line on the same side of ρ , and a vacuum term V given by diagrams where p is the phonon momentum, which must be the same on both sides of ρ by conservation, and the up- per sign applies to phonon creation (i.e. heating) and the lower sign to phonon destruction (cooling). The 2Re on the 3 asymmetric diagrams is because we also need to include their mirror images, which have the complex conjugate value.