There are six sections in this paper. Section II derives the relevant master equation for two-level atoms tightly confined by an optical lattice, including the dipole-dipole interaction. In Section III we simplify this equation for the case of an infinite one-dimensional optical lattice by transforming to momentum space. In Section IV, we in- troduce a Feynman-like diagrammatic perturbative ex- pansion, and use it to solve for the steady state phonon number distribution to lowest order in the Lamb-Dicke parameters η$_{α}$ . The Appendix does the same expansion using non-diagrammatic methods. In Section V, we eval- uate this steady state phonon distribution for particu- lar lattice setups, and find the parameter settings that minimize the phonon number. Finally, Section VI gives conclusions. In this section, we derive the master equation of our system, including both the dipole-dipole interaction be- tween the atoms and the coupling between their elec- tronic and motional states. Our starting point is the dipole Hamiltonian, which includes energy terms for the electronic and motional (phonon) excitations of the atoms and for free photons, and interaction terms between the atomic dipole mo- ments and the electric fields of the cooling laser and the free photons. We then eliminate the photons to obtain the spontaneous decay and dipole-dipole interaction, us- ing similar methods to Refs. [16–18]. However, they con- sider atoms in fixed positions, while we allow the atoms to move within their lattice sites. We hence obtain addi- tional terms in the dipole-dipole interaction, describing processes that change the motional state as well as the electronic state. We derive explicit expressions for the corresponding coupling constants up to second order in the Lamb-Dicke parameters. between any particles emitting electric dipole radiation (the dipole is the matrix element of the transition), and becomes significant when the distance between them is of the order of the transition wavelength. It modifies the de- cay rate and propagates electronic excitations from atom to atom, and has no effect when no electronic excitations are present. It is hence qualitatively distinct from static interparticle forces, such as the Coulomb interaction be- tween ions [20] or the static-dipole interaction between aligned polar molecules [21], which propagate motional excitations (phonons). Previous treatments of this in- teraction have considered either the superradiance part only [22], the reabsorption part only [23], or a long range approximation [19, 24] which is valid for a moderate den- sity gas but breaks down at the half-wavelength spacing typical of an optical lattice. Our system consists of N two-level atoms in a deep 1D optical lattice with one atom per site, as shown in Figs. 1,2. We denote the two internal levels of atom i by | 0 〉$_{i}$ and | 1 〉$_{i}$ , and the transition energy between them by $_{ℏ}$ω$_{0}$ . We then define the atom raising and lowering operators We assume that the lattice is sufficiently deep that tunneling between neighboring lattice sites is negligible (i.e. atom i stays in site i ), and each site can be ap- proximated by a harmonic well. We denote the trap- ping frequency of this harmonic well by ν$_{α}$ , which we assume to be the same for both internal states, but allow to be anisotropic, where α = x, y, z denotes the direction. In this paper we consider a deep one-dimensional op- tical lattice with one atom per site, as shown in Fig. 2, and include the full dipole-dipole interaction. We solve for the steady state by transforming to momentum space and using a diagrammatic perturbative expansion in the Lamb-Dicke parameter η ≡ $^{√}$E$_{R}$ /ν . We find that the dipole-dipole interaction may increase or decrease the steady state phonon number, depending on the laser pa- rameters. With optimized settings, the phonon number can be made ≈ 30% lower than would be possible without interaction. S † i ≡ | 1 〉$_{ii}$ 〈 0 | and S$_{i}$ ≡ | 0 〉$_{ii}$ 〈 1 | . (1) A. The Schr¨odinger picture FIG. 2: (Color online) Physical layout of our system. We consider a one-dimensional array of tightly confined atoms, one per lattice site. For simplicity we assume only one cool- ing laser, and hence can cool the atomic motion in any one dimension, either along the lattice (which we call x ) or across it ( y or z ), but not all three at once. As in the single parti- cle case, cooling all three dimensions would require multiple lasers. II. THEORETICAL MODEL 2 k$_{L}$ cooling laser  k$_{L}$ cooled motion D$_{01}$ D$_{01}$ θ θ (a) Along-lattice motion (α= x ) (b) Across-lattice motion (α= y ) y x