In this lecture, we will present two recent experimental results obtained in

the Bloch group that illustrate the potential of ultracold atomic gases to

understand the emergence of many-body phenomena. In a first part, we

will show that a gas of ultracold atoms in a Mott-insulating state represent

a fairly good realization of the Heisenberg model, with the electronic hyperfine

state playing the role of an effective spin. Using a new experimental technique,

we were able to directly reveal the Heisenberg coupling by monitoring the

real-time dynamics of a single spin impurity introduced in the system in a

controlled way. Beyond this paradigmatic, but analytically solvable problem,

we also explored a regime where real particle hopping becomes important

and the spin impurity gets dressed by the surrounding bath to form a

polaronic quasiparticle.

In the second part of the lecture, we will introduce a different setup that is a

realization of a traverse-field Ising model with long-range interactions. This

time we coupled the atoms to a laser beam driving a transition to a highly-

excited electronic state, a so-called Rydberg state. The enormous van der

Waals interaction between two atoms in such a state gives rise to strong

spatial correlations over distances much larger than the interparticle distance.

Here we could observe the spontaneous formation of well-defined geometric

structures of a few Rydberg excitations and gather some evidence that the

system had been excited to a highly-entangled many-body state.
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