Topological physics with atoms and with photons*

In this talk I will give a parallel review of the latest developments in the theoretical and experimental study of topological and magnetic effects in ultracold gases of neutral atoms and in optical systems. In spite of the apparent differences, these two families of systems share in fact the same basic concepts and are benefitting from a continuous cross-fertilization effect.

This close parallelism is specially intriguing in view of realizing high (d>3) dimensional systems by exploiting some internal degrees of freedom of the basic constituent as an extra synthetic dimension, namely the spin state for atoms or the mode index for photons in multimode cavities. After presenting first applications of this idea to integer quantum Hall effects, I will then outline the present challenge of realizing fractional quantum Hall physics in strongly interacting gases.

I will conclude with a general discussion of the novel features that atomic and optical systems offer in view of the preparation, the manipulation, and the diagnostic of strongly correlated states of matter and I will outline the new perspectives opened by the non-equilibrium nature of optical systems.