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decay with phonon creation or destruction. This is re- coil heating; as it has equal heating and cooling rates, it alone would have an infinite equilibrium temperature, so it has a heating effect on any finite temperature state. 

Fig. 6(a) is a contour plot of the steady state phonon number 〈 m 〉$_{ss}$ as a function of the laser detuning Δ and the angle ϑ between the cooling laser and the motion (cf. Fig. 2). As expected, the lowest steady state phonon number is obtained for ϑ = 0 and Δ close (but not exactly equal) to the phonon frequency ν$_{α}$ . 

Since we consider a one laser setup, the atomic motion β orthogonal to the laser wave vector is not cooled, but is still subject to recoil heating. For this motion we have ˆ k$_{L}$$_{β}$ = 0, and hence 

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Both the cooling and the heating rate are the same for this motion. This means the mean phonon number 〈 m 〉$_{ss}$ 

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for this motion diverges, i.e. unbounded heating, as ex- pected. A diagonally oriented laser could cool all three dimensions, but inefficiently as it can only be in reso- nance with one of them. Alternatively, three lasers could be used, one along each dimension. 

We now move to the case with dipole-dipole interac- tion, and consider varying the lattice spacing a and the laser detuning Δ and orientation ϑ , for both the along- and across-lattice modes (as defined in Fig. 2). We re- strict to the case where the laser wavevector k$_{L}$ is orthog- onal to the atomic dipole moment D$_{01}$ , which requires the least laser power for a given Ω, and both are in the x – y plane. 

Figs. 6(b,c,d) are contour plots of the steady state phonon number 〈 m 〉$_{ss}$ against the laser detuning Δ and angle ϑ , for three choices of lattice spacing and orienta- tion. Fig. 7 plots the Δ = constant cross-section through the minimum of 〈 m 〉$_{ss}$ , for the same cases. Fig. 8 plots steady state phonon number against lattice spacing. 

The numerical results presented here were all obtained at N = 200 with the dipole interaction cut off at that range (i.e. each pair of particles have one interaction term, not an infinite series of image interaction terms) and ν = 10Γ. We have tried other N, ν and found the same qualitative results for N ≳ 5 and ν ≳ 5Γ (along- lattice motion) or ≳ 2Γ (across-lattice motion). Increas- ing ν decreases the steady state phonon number, for large ν/ Γ as approximately 〈 m 〉$_{ss}$ ∼ (Γ /ν ) $^{2}$, as in the non- interacting case. 

## B. Interacting case: results 

