0.5 1.0 1.5 2.0 0 1 2 3 4 5 a /λ 〈 m 〉 ss /10 -3 atom -1 still sufficient accuracy for better than non-interacting cooling. The small waves in the contours are N -dependent, decreasing in both amplitude and wavelength as N in- creases. We conjecture that they are an artifact of the periodic boundary conditions. We do not consider lattice spacings less than half a wavelength, as the Brillouin zone then contains wavevec- tors | p | > k$_{0}$ , at which Re c ( p ) is very small (zero in the N → ∞ limit). Such slowly decaying excitations are ad- ditional L$_{0}$ ≈ 0 states, so our perturbative expansion is not valid in this regime. 12 To study how this enhanced cooling happens, we now break down the phonon number into simpler parts, which we plot in Fig. 9. The first step in this process is to break the wavevector-averaged 〈 m 〉$_{ss}$ into the individual modes 〈 m$_{p}$$_{α}$ 〉$_{ss}$ , so all these plots are against wavevector. The rows (a)-(d) are the different parts, while the columns (A)-(E) are the 5 settings that we consider here, given at the top of their column. FIG. 8: (Color online) Steady state phonon number 〈 m 〉$_{ss}$ , av- eraged over wavevectors, against lattice spacing a , for along- lattice motion (black) and across-lattice motion (blue). Thin lines are standard sideband cooling Δ = ν, ϑ = 0, thick lines are Δ , ϑ chosen to minimize 〈 m 〉$_{ss}$ . The dotted horizontal line is the non-interacting minimum. Fig. 9(a) plots the single mode steady state phonon number 〈 m$_{p}$$_{α}$ 〉$_{ss}$ , heating rate R$_{h}$$_{α}$ ( p ) and cooling rate R$_{c}$$_{α}$ ( p ), each of which is scaled so it would be 1 with no dipole-dipole interaction and the same ϑ, Δ. We see that all of these are strongly wavevector dependent, with all 5 settings having both regions that are better than the non-interacting value and regions that are worse. The un- optimized cases (A),(D) have strong heating coinciding in wavevector space with weak cooling, while the opti- mized cases (B),(C),(E) mostly pair strong heating with strong cooling, which reduces the total phonon number (if H > h and C > c , then H/c + h/C > H/C + h/c ). We first consider across-lattice motion with lattice spacing a = 0 . 5 λ (Fig. 6(b)). If we leave the settings at the non-interacting optimum of ϑ = 0 , Δ = 10 . 02Γ, the interaction increases the steady state phonon num- ber from 1 . 62 × 10 − 3 to 2 . 03 × 10 − 3 per atom. However, if we re-optimize ϑ, Δ for the interacting system, we can reach a minimum of 1 . 51 × 10 − 3 phonons per atom at ϑ = ± 0 . 35 , Δ = 10 . 12Γ. This is lower than the non- interacting minimum, i.e. dipole interaction can improve cooling effectiveness. In this case, this requires a non-zero angle ϑ between the laser and the motion to be cooled. For along-lattice motion, the effect of interaction is stronger, as the longest range term of C ( x ) (Eq. (13), Fig. 3) acts across the dipole, which with our assump- tions includes along the laser. At lattice spacing a = 0 . 5 λ (Fig. 6(c)), there is a high peak at ϑ = 0 (4 . 76 × 10 − 3 phonons per atom at ϑ = 0 , Δ = 10 . 02Γ), which slowly gets higher as N increases. This can be avoided by using a non-zero ϑ , with a minimum of 2 . 05 × 10 − 3 per atom at ϑ = ± 0 . 56 , Δ = 9 . 90Γ. While worse than the non- interacting case, this is much better than not optimizing. Fig. 9(b) separates the heating rate into the 6 individ- ual diagrams in Fig. 5, while Fig. 9(c) does the same for the cooling rate. The dominant cooling process in all 5 cases is phonon absorption on laser excitation (blue line in Fig. 9(c)), i.e. ordinary sideband cooling, as this is resonantly enhanced. The strongest heating process is usually phonon emission on decay (recoil heating, black line), but emission on laser excitation (blue) and the in- terference between them (dashed blue/black) are often also significant. In all these cases, the heating and cool- ing rates, and hence the time required to reach the steady state, are of the same order of magnitude as the corre- sponding non-interacting values. Finally, Fig. 9(d) plots the Fourier space dipole in- teraction coefficients. One effect of interaction is that the cooling process cannot be exactly on resonance at all wavevectors, as the atomic excitations have a non- trivial dispersion Im c ( p ) while the phonons have a sin- gle frequency ν . In particular, the dips in cooling rate at p + k$_{L}$ = ± k$_{0}$ , in all cases except ϑ = 0 across-lattice (D), occur because Im c ( p + k$_{L}$ ) is large (divergent as N → ∞ ) there. (At a = 0 . 5 λ (A),(B),(E), these are the same If the lattice spacing is allowed to vary (which can be done by using lattice lasers crossed at an angle in- stead of head-on), it again becomes possible to reach phonon numbers below the non-interacting minimum. The minimum of 1 . 14 × 10 − 3 per atom is reached at a = 0 . 744 λ, ϑ = ± 0 . 05 , Δ = 10 . 04Γ (Fig. 6(d)). There are also local minima at a = 1 . 248 λ and a = 1 . 749 λ (Fig. 8), which are less deep but still go below the non- interacting minimum. In all three interacting cases, the phonon number in- creases a few times faster than in the non-interacting case as we move away from the optimum ϑ . However, ± 5 ◦ is C. Interacting case: analysis