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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 

<img src='content_image/381202.jpg'>

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 = 

## F. The effect of spontaneous emission 

<img src='content_image/381195.jpg'>

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. 

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]. 

