17 Each of the four operator terms in ˜ H (1) cond takes ρ (0) ss to another eigenstate of L $^{(0)}$; for example, b$_{p}$$_{α}$ ˜ s † p + $_{k$_{L}$}$ρ (0) ss has the eigenvalue − c ( p + k$_{L}$ ) / 2 + i ν$_{α}$ , the first term from | ˜ 1 〉〈 ˜ 0 | = ˜ s † p + $_{k$_{L}$}$| ˜ 0 〉〈 ˜ 0 | and the second term from the b$_{p}$$_{α}$ . We can hence apply $^{[}$L (0) ] − 1 by simply inserting the reciprocal eigenvalues. Using the short hand notation this yields This inverse is not unique: we could add in an arbitrary L (0) = 0 state. However, doing so would have no effect on the final result, as it would be turned into an L (0) = 0 state by the second L (1) then annihilated by P . We now apply the second L $^{(1)}$, and finally apply P , i.e. keep only terms which satisfy Eqs. (A.12) and (A.13). This gives where Ξ is defined by Eq. (A.24) and the Hermitian conjugate comes from $^{[}$L (0) ] − 1 $^{(}$ρ (0) ss ˜ H (1) † cond $^{)}$= Ξ $^{†}$. Using Eq. (23), we find that the first term in this equation is given by Analogously, remembering that P ( | ˜ 1 〉〈 ˜ 1 | ) = | ˜ 0 〉〈 ˜ 0 | , we find that Moreover, we obtain from Eq. (27), PR $^{(1)}$(Ξ) = − ∑ p ,α,β $_{ℏ}$Ω 2 c $^{∗}$( k$_{L}$ ) ( i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( p + k$_{L}$ ) c ( k$_{L}$ ) ) [ Im d$_{β}$ ( p + k$_{L}$ ) ( b$_{p}$$_{α}$ρ (0) ss b † p β − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b † − p $_{α}$ρ (0) ss b$_{−}$$_{p}$$_{β}$ − c ( p + k$_{L}$ ) − 2i ν$_{α}$$^{)}$ − Im d$_{β}$ ( k$_{L}$ ) ( b † p $_{β}$b$_{p}$$_{α}$ − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b$_{−}$$_{p}$$_{β}$b † − p α − c ( p + k$_{L}$ ) − 2i ν$_{α}$$^{)}$ ρ (0) ss ] + ∑ α,β $_{ℏ}$N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | 4 Im d$_{β}$ ( k$_{L}$ ) $^{(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{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 $^{)}$. (A.28) P $^{[}$Ξ ˜ H (1) † cond ] = ∑ p ,α,β ℏ $^{2}$Ω2 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}$$_{α}$ ρ (0) ss b † p β − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b † − p α ρ (0) ss b$_{−}$$_{p}$$_{β}$ − c ( p + k$_{L}$ ) − 2i ν$_{α}$$^{)}$ + ∑ α,β i$_{ℏ}$ $^{2}$N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | $_{4}$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}$$_{α}$ ρ (0) ss b † 0 β − b † 0 α ρ (0) ss b$_{0}$$_{β}$ $^{)}$. (A.27) P $^{[}$˜ H (1) $_{cond}$Ξ ] =   ∑ p ,α,β ℏ $^{2}$Ω2 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}$$_{α}$ − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b$_{−}$$_{p}$$_{β}$b † − p α − c ( p + k$_{L}$ ) − 2i ν$_{α}$$^{)}$ + ∑ α,β i$_{ℏ}$ $^{2}$N Ω 4 ν$_{α}$ | c ( k$_{L}$ ) | $_{4}$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 $_{β}$b$_{0}$$_{α}$ − b$_{0}$$_{β}$b † 0 α )] ρ (0) ss . (A.26) P$^{(}$L (1) $^{[}$L (0) ] − 1 L $^{(1)}$( ρ (0) ss )$^{)}$ = P$^{[}$− 1 ℏ 2 H (1) $_{cond}$Ξ + 1 ℏ 2 Ξ H (1) † cond − i ℏ R $^{(1)}$(Ξ)$^{]}$ + H . c . , (A.25) Ξ = [ ∑ p ,α $_{ℏ}$Ω$^{(}$ i η$_{α}$ ˆ k$_{L}$$_{α}$ − d$_{α}$ ( k$_{L}$ ) − d$_{α}$ ( p + k$_{L}$ ) c ( k$_{L}$ ) ) ( b$_{p}$$_{α}$ − c ( p + k$_{L}$ ) + 2i ν$_{α}$ + b † − p α − c ( p + k$_{L}$ ) − 2i ν$_{α}$$^{)}$ ˜ s † p + k$_{L}$ − ∑ α i$_{ℏ}$ √ N Ω 2 ν$_{α}$ | c ( k$_{L}$ ) | 2 $^{(}$Im d$_{α}$ ( k$_{L}$ ) − η$_{α}$ Re c ( k$_{L}$ ) ˆ k$_{L}$$_{α}$ $^{)(}$b$_{0}$$_{α}$ − b † 0 α ) ] ρ (0) ss . (A.24) Ξ ≡ $^{[}$L (0) ] − 1 $^{(}$˜ H (1) cond ρ (0) ss ) , (A.23)