(a) R e C 0 λ 2λ -Γ 0 Γ (b) Im C 0 λ 2λ -Γ 0 Γ (c) R e D $^{x}$, R e D y 0 λ 2λ -Γ η$_{α}$ /2 0 Γ η$_{α}$ /2 (d) Im D $^{x}$,  Im D y 0 λ 2λ -Γ η$_{α}$ /2 0 Γ η$_{α}$ /2 (e)R e  E $^{xx}$, R e E yy 0 λ 2λ -Γ η$_{α}$ $^{2}$/4 0 Γ η$_{α}$ $^{2}$/4 | X$_{j}$ - X$_{i}$ | 5 Because they contain only equilibrium positions X$_{i}$ and not position operators x$_{i}$ , these are numbers rather than operators. Finally we remark that C$_{ij}$ = C$_{ji}$ , and hence D α ij = − D α ji and E αβ ij = E αβ ji . In Fig. 3 we plot these coupling constants against the distance | X$_{j}$ − X$_{i}$ | between the atoms, for a few relevant angles between X$_{j}$ − X$_{i}$ and the dipole moment ˆ D$_{01}$ . Substituting Eqs. (16) and (17) back into the condi- tional Hamiltonian (12), we see that in zeroth order in η$_{α}$ , the dipole-dipole coupling constants C$_{ij}$ are the same as for atoms in fixed positions [16–18], while the motional states are unaffected. The first and higher order terms couple the electronic and motional states of the atoms: when atoms exchange a photon, they may also create or destroy one or more phonons, with coupling constants D α ij and E αβ ij . The term R ( ρ ) in Eq. (11) describes processes where photon loss does occur; it is called the reset operator as it “resets” the system state after such a loss [25]. To evaluate this expression we proceed as in Refs. [16, 17], and again obtain something similar to the fixed-positions expression Re C ( x$_{j}$ − x$_{i}$ ) S$_{i}$ ρ S † j with the positions be- coming operators. However, here Re C ( x$_{j}$ − x$_{i}$ ) does not simply act from the left, but becomes a superop- erator (i.e. contains operators acting from both sides). In Lamb-Dicke expansion, each phonon operator acts from the same side of ρ as the same atom’s electronic (de)excitation operator, i.e. B$_{iα}$ + B † iα on the left and B$_{jα}$ + B † jα on the right. Up to second order in η$_{α}$ , this gives F. The effect of spontaneous emission FIG. 3: (Color online) Dipole coupling constants C ( X ), D $^{α}$( X ), and E $^{αα}$( X ), as a function of distance | X | , for along- lattice motion (black) and across-lattice motion (blue). Solid lines are ϑ = 0 ( ˆ D$_{01}$ · ˆ X = sin ϑ for along-lattice motion or cos ϑ for across-lattice motion, cf. Fig. 2). Dotted lines are ϑ = 0 . 56 (along-lattice) or 0 . 35 (across-lattice), which we will see later are the optimum values at lattice spacing a = 0 . 5 λ . For D and E the component plotted is the relevant one for cooling that motion direction: α = x for along-lattice motion or α = y for across-lattice motion. The vertical lines mark the lattice positions for a = 0 . 5 λ (blue dotted) and a = 0 . 744 λ (black solid). R ( ρ ) = N ∑ i =1 Γ S$_{i}$ ρ S † i + ∑ j = i Re C$_{ij}$ S$_{i}$ ρ S † i + ∑ j = i ∑ α = x,y,z Re D α ij $^{[}$S$_{i}$ ρ S † j $^{(}$B$_{jα}$ + B † jα ) − $^{(}$B$_{iα}$ + B † iα $^{)}$S$_{i}$ ρ S † j ] + N ∑ i,j =1 ∑ α,β = x,y,z Re E αβ ij $^{[ (}$B$_{iα}$ + B † iα $^{)(}$B$_{iβ}$ + B † iβ $^{)}$S$_{i}$ ρ S † j − 2 $^{(}$B$_{iα}$ + B † iα $^{)}$S$_{i}$ ρ S † j $^{(}$B$_{jβ}$ + B † jβ ) + S$_{i}$ ρ S † j $^{(}$B$_{jα}$ + B † jα $^{)(}$B$_{jβ}$ + B † jβ $^{) ]}$. (20) As in the case of fixed positions, the spontaneous emis- sion rate can be higher (superradiance) or lower (sub- radiance) than its non-interacting value of Γ per ex- cited atom, depending on the distance between the atoms. With motion included, there is also the possi- bility that a spontaneous emission creates or destroys phonons, at rates Re D α $_{ij}$, Re E αβ ij . The single particle recoil heating is included as the i = j term Re E αβ ii = In the limit of infinitely strong confinement ( η$_{α}$ → 0, and hence D α $_{ij}$, E αβ ij → 0), Eqs. (12), (20) reduce to the known master equation for fixed dipole interacting atoms [16–18]. 1 $_{10}$η$_{α}$ η$_{β}$ Γ $^{(}$ˆ D$_{01}$$_{α}$ ˆ D$_{01}$$_{β}$ − 2 δ$_{αβ}$ $^{)}$, where δ$_{αβ}$ is the Kro- necker delta.