We study the optical response of a 2D square lattice of atoms using classical electrodynamics. Due to dipole-dipole interactions, the lattice atoms polarize as if the lattice were an atom with up to three resonance frequencies, with cooperatively shifted resonances and altered transition linewidths. We show that when the distance between two 2D lattices is large enough and Bragg reflections are absent, the lattices interact among themselves as if they radiated a plane wave whose amplitude is in accordance with the radiation from a dipole moment continuously distributed in the lattice plane. We employ these results to study light propagation in stacks of 2D lattices, drawing on simple qualitative pictures of the response of a 2D lattice and light propagation in 1D waveguides. We show that a stack of 2D lattices may emulate regularly spaced atoms in a lossless 1D waveguide, and argue that in a suitable geometry the resonance shifts characteristic of 1D and 2D lattice structures may completely cancel to eliminate density dependent resonance shifts of atoms bound to a 3D lattice. A generalization to the case of anisotropic polarizability, such as in the presence of a magnetic field, reveals light frequencies induced by the magnetic field for which the lattice is either completely transparent, or completely opaque.

We model single photon nonlinearities resulting from the dipole-dipole interactions of cold polar molecules. We propose utilizing "dark state polaritons" to effectively couple photon and molecular states; through this framework, coherent control of the nonlinearity can be expressed and potentially used in an optical quantum computation architecture. Due to the dipole-dipole interaction the photons pick up a measurable nonlinear phase even in the single photon regime. A manifold of protected symmetric eigenstates is used as basis. Depending on the implementation, major sources of decoherence result from non-symmetric interactions and phonon dispersion. We discuss the strength of the nonlinearity per photon and the feasibility of this system.

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We use quantum trajectories to simulate Josephson oscillations of atomic condensates between the two sides of a double-well potential. In the simulations the atoms in both wells are monitored using off-resonant light scattering, and the ultimate outcome of our thought experiment is a sequence of photon counts probing the numbers of the atoms in each potential well. We show how to reconstruct the Josephson oscillations from the observed photon counts using Bayesian inference, and study the oscillations quantitatively by averaging the inferred time dependent oscillation amplitude over a large number of realizations. Scaling behaviors that characterize the oscillations are uncovered and related to physics principles such as measurement back-action. It turns out that the scalings hold true for quite small atom numbers, so that in this sense four atoms in a potential well may already make a Bose-Einstein condensate.