4 E. Lamb-Dicke expansion D. The no-photon time evolution where Δ ρ = ρ ( t$_{0}$ +Δ t ) − ρ ( t$_{0}$ ). In the limit ω$_{0}$ ≫ 1 / Δ t ≫ ν, Δ , Ω , Γ, the above integrals can be evaluated and give a time-independent master equation, which can be con- veniently written as [16, 17] Eq. (12) is difficult to use as it stands because it con- tains transcendental functions of the operator x$_{i}$ , namely exp( − i k$_{L}$ · x$_{i}$ ) and C ( x$_{j}$ − x$_{i}$ ). In this section we intro- duce the Lamb-Dicke expansion about the site centers, which allows these to be handled perturbatively in the deep lattice limit. As already mentioned above, we approximate each lattice site by a harmonic well with phonon operators B$_{iα}$, B † $_{iα}$. In terms of these operators, the position oper- ator x$_{i}$ is where H$_{cond}$ comes from the terms of (10) in curly brack- ets and R ( ρ ) comes from the last term. where X$_{i}$ is the equilibrium position (site center), α = x, y, z labels vector components, and the Lamb-Dicke pa- rameter η$_{α}$ is defined by The term H$_{cond}$ in Eq. (11) describes processes where no photons are lost from the system, either because none are emitted, or because one is emitted but is reabsorbed before time Δ t [16, 17]. As it appears in the master equa- tion (11) in a similar form to the − i $_{ℏ}$[ H, ρ ] of a Hamilto- nian, but describes only evolution under the condition of no photon loss, this term is called the conditional Hamil- tonian. Unlike a normal Hamiltonian, it is not Hermi- tian, but norm decreasing; this represents the decreasing probability that this condition will continue to hold, and in the master equation is balanced by R ( ρ ). where m is the mass of the atom. In the case of a deep lattice η$_{α}$ is small, and we can hence use Eq. (14) to approximate exp( − i k$_{L}$ · x$_{i}$ ) and C ( x$_{j}$ − x$_{i}$ ) in Eq. (12) by Taylor expansion. Up to second order in η$_{α}$ , we find Evaluating (10) using the methods of [16, 17], we find The terms of (12) describe respectively laser driving, detuning and spontaneous decay, phonon energy, and dipole-dipole interaction. All except the last are iden- tical to standard laser cooling. The last term contains the dipole coupling operator where ˆ k$_{L}$ ≡ k$_{L}$ / | k$_{L}$ | and ˆ k$_{L}$$_{α}$ is the component of this unit vector in the α direction, and Here the dipole coupling constants C$_{ij}$ , D α $_{ij}$, and E αβ ij are obtained by differentiating C ( r ) (and absorbing the η$_{α}$/k$_{0}$ factors), where r$_{α}$ are the components of r , then substituting the equilibrium positions X$_{i}$ for the atoms, where r ≡ ‖ r ‖ , ˆr ≡ r /r , k$_{0}$ ≡ ω$_{0}$/c and ˆ D$_{01}$ ≡ D$_{01}$ / ‖ D$_{01}$ ‖ . The imaginary part of C ( x$_{j}$ − x$_{i}$ ) describes photons being emitted by atom i and reabsorbed by atom j , while the real part describes modifications to the de- cay rate due to interference between their emitted waves. Eq. (13) is exactly the same expression as the dipole cou- pling constants in Refs. [16–18], but they fix the atom positions so their r is a classical vector, while we allow motion within each lattice site and our r is hence an op- erator. C$_{ij}$ ≡ C ( X$_{j}$ − X$_{i}$ ) , D α ij ≡ D $^{α}$( X$_{j}$ − X$_{i}$ ) , E αβ ij ≡ E $^{αβ}$( X$_{j}$ − X$_{i}$ ) . (19) D $^{α}$( r ) ≡ η$_{α}$ k$_{0}$ · ∂C ( r ) ∂r$_{α}$ , E $^{αβ}$( r ) ≡ η$_{α}$η$_{β}$ 2 k 2 0 · ∂ $^{2}$C ( r ) ∂r$_{α}$∂r$_{β}$ , (18) C ( r ) = 3 2 Γ e i k$_{0}$r [ 1 i k$_{0}$r$^{(}$ 1 − | ˆ D$_{01}$ · ˆr | 2 ) +$^{(}$ 1 ( k$_{0}$r ) 2 − 1 i( k$_{0}$r ) 3 ) $^{(}$1 − 3 | ˆ D$_{01}$ · ˆr | 2 ) ] , (13) C ( x$_{j}$ − x$_{i}$ ) = C$_{ij}$ + ∑ α = x,y,z D α ij $^{(}$B$_{jα}$ + B † jα − B$_{iα}$ − B † iα ) + ∑ α,β = x,y,z E αβ ij $^{(}$B$_{jα}$ + B † jα − B$_{iα}$ − B † iα ) × $^{(}$B$_{jβ}$ + B † jβ − B$_{iβ}$ − B † iβ $^{)}$. (17) H$_{cond}$ = N ∑ i =1 $^{[}$1 2 $_{ℏ}$Ω e − i k$_{L}$ · x$_{i}$ S$_{i}$ + H . c . +$_{ℏ}$ $^{(}$Δ − i 2 Γ $^{)}$S † $_{i}$S$_{i}$ + ∑ α = x,y,z $_{ℏ}$ν$_{α}$ B † $_{iα}$B$_{iα}$ − ∑ j = i i 2 $_{ℏ}$C ( x$_{j}$ − x$_{i}$ ) S † j S$_{i}$$^{]}$ . (12) e − i k$_{L}$ · x$_{i}$ = e − i k$_{L}$ · X$_{i}$ [ 1 − i ∑ α = x,y,z ˆ k$_{L}$$_{α}$η$_{α}$ $^{(}$B$_{iα}$ + B † iα ) − ∑ α,β = x,y,z η$_{α}$η$_{β}$ 2 ˆ k$_{L}$$_{α}$ ˆ k$_{L}$$_{β}$ $^{(}$B$_{iα}$ + B † iα $^{)(}$B$_{iβ}$ + B † iβ ) ] , (16) η$_{α}$ ≡ $^{√}$$_{ℏ}$k 2 $_{0}$/ 2 mν$_{α}$ , (15) x$_{iα}$ = X$_{iα}$ + η$_{α}$ k$_{0}$$^{(}$ B$_{iα}$ + B † iα $^{)}$, (14) ˙ ρ = − i ℏ $^{[}$H$_{cond}$ ρ − ρ H † cond $^{]}$+ R ( ρ ) , (11)