ρ$_{ss}$ = ρ (0) ss + ρ (1) ss + ρ (2) ss + . . . , L = L (0) + L (1) + L (2) + . . . . (A.2) 14 We then expand both in powers of η$_{α}$ , point mod the Brillouin zone and there is hence one dip; at other a (C) there are two.) The along-lattice locally optimal spacings (Fig. 8) a = 0 . 744 λ , a = 1 . 248 λ and a = 1 . 749 λ are all close to zeros of Im C ( X ) (Fig. 3(b)), which reduce this effect by removing the nearest neighbor (normally largest) term, reducing the size of these dips. These dips are accompanied by a sudden change in the decay rate Re c ( p ). The optimal spacings are also all near an odd number of quarter wavelengths, where ± k$_{0}$ = ± π/ 2 a (not necessarily respectively) mod the Brillouin zone, dividing it into a low Re c ( p ) half and a high Re c ( p ) half. This gives them a particularly good matching of strong cooling to strong heating ( a = 0 . 744 λ shown in Fig. 9(C), the other two are similar), which further reduces their average phonon number. As in the previous section, the superscripts indicate the order of the term with respect to η$_{α}$ . This technique has previously been applied to single particle sideband cooling [4, 32]. It is a form of adiabatic elimination [4, 32], but does not neglect the spontaneous decay, which is important as cooling requires a decay process to carry away entropy. Substituting for L from the master equation (11) and equating the zeroth order terms, we obtain To summarize, we have considered resolved sideband cooling in a deep 1D optical lattice, and in particular the effect of the dipole-dipole interaction that exists between any dipole radiating particles. At the typical optical lattice spacing of half a wave- length, this interaction is not a small perturbation, but is of the same order of magnitude as the single particle spontaneous decay. We handle this by transforming to Fourier space and working in the Lamb-Dicke (deep lat- tice) limit, which makes the dominant dipole-dipole term into a dispersion of collective atomic excitations. We introduce an open system version of Feynman dia- grams, and use them to calculate the steady state density matrix to lowest order in the Lamb-Dicke parameter. We find that the steady state phonon number may be higher or lower than its non-interacting value, depending on the parameter settings, with a minimum of ≈ 30% lower. and its Hermitian conjugate ˜ s † $_{p}$. This eliminates the Ω terms from H (0) $_{cond}$, giving with H (0) cond and R (0) given in Eqs. (23) and (27). To solve this equation, we define In terms of Feynman diagrams, this transformation is equivalent to cutting off the one-line (zeroth order laser) vertices and their associated lines, and absorbing their value (which is constant, as the lines always have wavevector k$_{L}$ and energy 0) into the coefficient of the vertex they are attached to (Fig. 10(a)). We now neglect the hardcore constraint of no more than one excitation per atom and treat the ˜ s † p excitations as bosons. This is valid in the limit of low excitation densities [26, 27], or equivalently weak laser driving. The zeroth order master equation (Eq. (A.3)) is then simply a sum over atom and phonon modes, In this Appendix, we solve for the steady state density matrix by directly expanding the master equation in the Lamb-Dicke parameters η$_{α}$ , and verify that this gives the same result as earlier obtained using Feynman-like dia- grams. In the limit η$_{α}$ → 0 (an infinitely deep lattice), we find that the atomic excitations are in a coherent state, while the phonons are stable free particles and hence can be in any state. As we shall see below, to obtain a unique phonon steady state, we need to expand the master equa- tion up to at least second order in η$_{α}$ . In the following, we use the notation ρ$_{ss}$ for the steady state density matrix and L for the time evolution super- operator, i.e. with the atom and phonon operators given by Appendix: Direct perturbative expansion L (0) = ∑ p L (0) at , p + ∑ p ,α L (0) phn , p α (A.6) L (0) at , p = − 1 2 c ( p ) ˜ s † $_{p}$˜ s$_{p}$ρ − 1 2 c $^{∗}$( p ) ρ ˜ s † $_{p}$˜ s$_{p}$ +Re c ( p ) ˜ s$_{p}$ρ ˜ s † $_{p}$, L (0) phn , p α = − i ν$_{α}$ ( b † p $_{α}$b$_{p}$$_{α}$ ρ − ρb † p $_{α}$b$_{p}$$_{α}$ ) . (A.7) ˙ ρ$_{ss}$ ≡ L ( ρ$_{ss}$ ) = 0 . (A.1) H (0) cond = − ∑ p i 2 $_{ℏ}$c ( p ) ˜ s † $_{p}$˜ s$_{p}$ + ∑ p ,α $_{ℏ}$ν$_{α}$ b † p $_{α}$b$_{p}$$_{α}$ , R $^{(0)}$( ρ ) = ∑ p Re c ( p ) ˜ s$_{p}$ρ ˜ s † $_{p}$. (A.5) ˜ s$_{k}$$_{L}$ ≡ s$_{k}$$_{L}$ + i √ N Ω c ( k$_{L}$ ) and ˜ s$_{p}$ ≡ s$_{p}$ ∀ p = k$_{L}$ , (A.4) − i ℏ $^{[}$H (0) cond ρ (0) ss − ρ (0) ss H (0) † cond $^{]}$+ R $^{(0)}$( ρ (0) ss ) = 0(A.3) 1. Zeroth order VI. CONCLUSIONS