10 V. EXAMPLES AND PARAMETER OPTIMIZATION A. Ordinary (non-interacting) sideband cooling independent of p . These are the standard results for single particle sideband cooling [13, 32]. In this case, only the first 2 diagrams in Fig. 5 are nonzero. The first term in each of these equations rep- resents a phonon being created (heating) or destroyed (cooling) when the laser excites an atom, followed by spontaneous decay with no change in phonon state. Cool- ing is resonantly enhanced when Δ ≈ ν$_{α}$ , as expected from Fig. 1. The second term represents laser excitation with no change in phonon state, followed by spontaneous For multiple modes it can then be obtained by simple averaging; in particular, the mean phonon number per atom over all wavevectors is We do not average over motion directions α , as in a single laser setup we expect cooling in one dimension (along the laser) and heating in the other two. In ordinary sideband cooling, the cooled motion α is parallel to the laser wave vector ( ϑ = 0) and hence ˆ k$_{L}$$_{α}$ = 1 and ˆ D$_{01}$$_{α}$ = 0 for this motion. Substituting into Eq. (29), we obtain the heating and cooling rates independent of p , where ˆ D$_{01}$$_{α}$ is the component of the dipole unit vector ˆ D$_{01}$ along α . However, the phonon state as a whole is not thermal because different modes p α have different temperatures. We therefore instead use the mean phonon number as our measure of heat content, as this is defined for any finite energy state, and when multiplied by $_{ℏ}$ν$_{α}$ gives the mean phonon energy. For a single mode it is We finally consider how to measure the heat content of this steady state, as in the next section we will want to minimise this. For each individual phonon mode p α , the state (33) is a thermal state with temperature This leaves only the single atom terms C$_{ii}$ (Eq. (25)) and E αβ ii , as D α ii = 0 by definition (cf. Eq. (25)). The Fourier transforms are hence As expected for a steady state, heating and cooling are in detailed balance: the transition rate from m to m + 1 phonons is equal to the transition rate from m + 1 to m . A normalizable steady state only exists for the modes where the heating rate R$_{h}$$_{α}$ ( p ) is smaller than the corresponding cooling rate R$_{c}$$_{α}$ ( p ). If not, we obtain unbounded heating. and it is unique up to normalization because once c$_{0}$ is chosen the others can be determined iteratively. As a check on our calculation we first consider the case of atoms far enough apart to neglect their dipole-dipole interaction. In this case, we should obtain ordinary side- band cooling of each atom independently. No interaction means In this section we evaluate the steady state phonon number for various parameter values, and in particular consider what settings minimize it. We find that the dipole-dipole interaction has a significant effect on the steady state phonon number, at lattice spacing a = λ/ 2 typically ∼ 100% at the most strongly affected phonon wavevectors and ∼ 20% averaged over all wavevectors. This effect can have either sign depending on the param- eters; in particular, there are settings that give a lower phonon number 〈 m 〉$_{ss}$ than any non-interacting settings with the same lattice strength ν . One solution of this equation is Here the coefficient c$_{m}$$_{p}$$_{α}$ is the probability of having m$_{p}$$_{α}$ phonons in the mode p α . As the state is a product state, the phonon distributions of different modes are indepen- dent. Substituting (31) into (30) and setting ˙ ρ = 0, we obtain the steady state is a mixture (not a superposition) over phonon numbers, ρ (0) phn = ∏ p ,α ∑ m$_{p}$$_{α}$ c$_{m}$$_{p}$$_{α}$ | m$_{p}$$_{α}$ 〉〈 m$_{p}$$_{α}$ | . (31) 0 = m$_{p}$$_{α}$ R$_{h}$$_{α}$ ( p ) c$_{m}$$_{p}$$_{α}$$_{−}$$_{1}$ − [( m$_{p}$$_{α}$ + 1) R$_{h}$$_{α}$ ( p ) + m$_{p}$$_{α}$R$_{c}$$_{α}$ ( p )] c$_{m}$$_{p}$$_{α}$ +( m$_{p}$$_{α}$ + 1) R$_{c}$$_{α}$ ( p ) c$_{m}$$_{p}$$_{α}$$_{+1}$ . (32) c$_{m}$$_{p}$$_{α}$ = $^{[}$R$_{h}$$_{α}$ ( p ) R$_{c}$$_{α}$ ( p )$^{]}$$^{m$_{p}$$_{α}$}$ , (33) C$_{ij}$ = 0 , D α ij = 0 , E αβ ij = 0 ∀ i = j . (37) c ( p ) = C$_{ii}$ = Γ + 2i Δ , d$_{α}$ ( p ) = 0 , e$_{αβ}$ ( p ) = E αβ ii = η$_{α}$η$_{β}$ Γ 10 $^{(}$ˆ D$_{01}$$_{α}$ ˆ D$_{01}$$_{β}$ − 2 δ$_{αβ}$ ) , (38) T$_{p}$$_{α}$ = ν$_{α}$ k$_{B}$ log [ R$_{c}$$_{α}$ ( p ) /R$_{h}$$_{α}$ ( p )] . (34) 〈 m$_{p}$$_{α}$ 〉$_{ss}$ = ∑ m$_{p}$$_{α}$ m$_{p}$$_{α}$ c$_{m}$$_{p}$$_{α}$ ∑ m$_{p}$$_{α}$ c$_{m}$$_{p}$$_{α}$ = R$_{h}$$_{α}$ ( p ) R$_{c}$$_{α}$ ( p ) − R$_{h}$$_{α}$ ( p ) . (35) R$_{h}$$_{α}$ = η 2 $_{α}$Ω $^{2}$Γ [ 1 Γ 2 + 4(Δ + ν$_{α}$ ) 2 + 2 5(Γ 2 + 4Δ $^{2}$)$^{]}$ , R$_{c}$$_{α}$ = η 2 $_{α}$Ω $^{2}$Γ [ 1 Γ 2 + 4(Δ − ν$_{α}$ ) 2 + 2 5(Γ 2 + 4Δ $^{2}$)$^{]}$ , (39) 〈 m 〉$_{ss}$ = 1 N ∑ p 〈 m$_{p}$$_{α}$ 〉$_{ss}$ . (36)