content_image/399293.jpg content_image/399297.jpg content_image/399291.jpg content_image/399294.jpg content_image/399292.jpg content_image/399296.jpg content_image/399295.jpg content_image/399290.jpg content_image/399289.jpg
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] 

where H$_{cond}$ comes from the terms of (10) in curly brack- ets and R ( ρ ) comes from the last term. 

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 ). 

Evaluating (10) using the methods of [16, 17], we find 

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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 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. 

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## D. The no-photon time evolution 

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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 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 

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 

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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 

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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, 

## E. Lamb-Dicke expansion 

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