18 Putting these three equations together, we finally obtain Again, α and β are only independent in the isotropic case; in the fully anisotropic case we must have α = β . Substituting Eqs. (A.20) and (A.29) into Eq. (A.18), and collecting terms acting on the same phonon mode by relabelling − p → p where necessary, we obtain for the phonon steady state 6. The heating and cooling rates of the different phonon modes P$^{(}$L (1) $^{[}$L (0) ] − 1 L $^{(1)}$( ρ (0) ss )$^{)}$ = ∑ p ,α,β Ω 2 2 ( i η$_{β}$ ˆ k$_{L}$$_{β}$ − d$_{β}$ ( k$_{L}$ + p ) − d ∗ $_{β}$( k$_{L}$ ) c $^{∗}$( k$_{L}$ ) )( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( p + k$_{L}$ ) c ( k$_{L}$ ) ) ×$^{(}$ b † p $_{β}$b$_{p}$$_{α}$ ρ (0) ss − b$_{p}$$_{α}$ ρ (0) ss b † p β − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b$_{−}$$_{p}$$_{β}$b † − p α ρ (0) ss − b † − p α ρ (0) ss b$_{−}$$_{p}$$_{β}$ − c ( p + k$_{L}$ ) − 2i ν$_{α}$ ) + ∑ α,β i N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | 4 $^{(}$Im d$_{β}$ ( k$_{L}$ ) − η$_{β}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{β}$ $^{)(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ ) × $^{(}$− b$_{0}$$_{α}$ρ (0) ss b † 0 β + b † 0 $_{β}$b$_{0}$$_{α}$ ρ (0) ss + b † 0 α ρ (0) ss b$_{0}$$_{β}$ − b$_{0}$$_{β}$b † 0 α ρ (0) ss $^{)}$+ H.c. (A.29) 0 = 1 2 ∑ p ,α,β $^{[}$M$_{h}$$_{αβ}$ ( p )( b † p α ρ (0) ss b$_{p}$$_{β}$ − b$_{p}$$_{β}$b † p α ρ (0) ss ) + M ∗ h $_{αβ}$( p )( b † p β ρ (0) ss b$_{p}$$_{α}$ − ρ (0) ss b$_{p}$$_{α}$b † p $_{β}$) + M$_{c}$$_{αβ}$ ( p )( b$_{p}$$_{α}$ ρ (0) ss b † p β − b † p $_{β}$b$_{p}$$_{α}$ ρ (0) ss ) + M ∗ c $_{αβ}$( p )( b$_{p}$$_{β}$ ρ (0) ss b † p α − ρ (0) ss b † p $_{α}$b$_{p}$$_{β}$ ) ] . (A.30) M$_{h}$$_{αβ}$ ( p ) = − Ω 2 c ( k$_{L}$ − p ) + 2i ν$_{α}$ ( i η$_{β}$ ˆ k$_{L}$$_{β}$ − d$_{β}$ ( k$_{L}$ − p ) − d ∗ $_{β}$( k$_{L}$ ) c $^{∗}$( k$_{L}$ ) )( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( k$_{L}$ − p ) c ( k$_{L}$ ) ) − Ω 2 | c ( k$_{L}$ ) | $_{2}$[ − iIm c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ − 2iIm e$_{αβ}$ ( k$_{L}$ ) + 2 e$_{αβ}$ ( k$_{L}$ − p )] , M$_{c}$$_{αβ}$ ( p ) = − Ω 2 c ( k$_{L}$ + p ) − 2i ν$_{α}$ ( i η$_{β}$ ˆ k$_{L}$$_{β}$ − d$_{β}$ ( k$_{L}$ + p ) − d ∗ $_{β}$( k$_{L}$ ) c $^{∗}$( k$_{L}$ ) )( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( k$_{L}$ + p ) c ( k$_{L}$ ) ) − Ω 2 | c ( k$_{L}$ ) | $_{2}$[ − iIm c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ − 2iIm e$_{αβ}$ ( k$_{L}$ ) + 2 e$_{αβ}$ ( k$_{L}$ + p )] . (A.31) M$_{h}$$_{αβ}$ ( 0 ) = η$_{α}$ ˆ k$_{L}$$_{α}$ Ω 2 c ( k$_{L}$ ) + 2i ν$_{α}$ ( η$_{β}$ ˆ k$_{L}$$_{β}$ − 2Im d$_{β}$ ( k$_{L}$ ) c $^{∗}$( k$_{L}$ ) ) − Ω 2 | c ( k$_{L}$ ) | $_{2}$[ − i Im c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ + 2Re e$_{αβ}$ ( k$_{L}$ )] − i N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | 4 $^{(}$Im d$_{β}$ ( k$_{L}$ ) − η$_{β}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{β}$ $^{)(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ ) , M$_{c}$$_{αβ}$ ( 0 ) = η$_{α}$ ˆ k$_{L}$$_{α}$ Ω 2 c ( k$_{L}$ ) − 2i ν$_{α}$ ( η$_{β}$ ˆ k$_{L}$$_{β}$ − 2Im d$_{β}$ ( k$_{L}$ ) c $^{∗}$( k$_{L}$ ) ) − Ω 2 | c ( k$_{L}$ ) | $_{2}$[ − i Im c ( k$_{L}$ ) η$_{α}$η$_{β}$ ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ + 2Re e$_{αβ}$ ( k$_{L}$ )] + i N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | 4 $^{(}$Im d$_{β}$ ( k$_{L}$ ) − η$_{β}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{β}$ $^{)(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ ) . (A.32) However, these extra terms (the last line of each equation) are pure imaginary, symmetric in α, β and of opposite signs in M$_{h}$$_{αβ}$ and M$_{c}$$_{αβ}$ . As b$_{p}$$_{α}$b † p β − b † p $_{β}$b$_{p}$$_{α}$ = δ$_{αβ}$ is a number so commutes with ρ , such terms cancel out in Eq. (A.30), so Eq. (A.31) can also be used for p = 0 . For p = 0 , there appear to be additional terms As this does not contain atomic operators, we can cancel the atomic part | ˜ 0 〉〈 ˜ 0 | , which replaces ρ (0) ss by ρ (0) $_{phn}$. It is also quadratic in phonon operators and diagonal in phonon wavevector. For p = 0 , the coefficients M$_{h}$$_{αβ}$ ( p ) and M$_{c}$$_{αβ}$ ( p ) are given by