In the past decades it has become clear that Landau's theory of phase transitions which involves the appearance of a broken-symmetry order parameter does not apply to a series of phases of matter with so-called topological order. The absence of a local order parameter makes the identification of a topological phase a difficult task. Among the different techniques that have been applied to probe topological phases, entanglement measurements, first introduced in the context of quantum computation, turned out to be very successful. Li and Haldane suggested to use the entanglement spectrum : it is the spectrum of the reduced density matrix obtained when the system is cut into two parts. They found that, for fractional quantum Hall model states, the counting of states of the entanglement spectrum has a universal part which is related to the one of the edge excitations. During my Ph.D thesis, I tried to understand what information the entanglement spectrum could provide when applied to fractional quantum Hall phases. These phases are the typical examples of strongly interacting topological phases. I first studied the entanglement spectrum as proposed by Li and Haldane. I showed that, away from model states, it was possible to define a clear entanglement gap. I also related the structures above the entanglement gap to quasihole-quasiparticule excitations. Then, I defined two other types of entanglement spectrum that rely on different ways of partitioning the system. The particle entanglement spectrum gives access to quasihole excitations whereas the real space entanglement spectrum solves several issues of the original proposal for the entanglement spectrum. Finally, I used these tools to identify phases similar to the one of the fractional quantum Hall effect that emerges in bosonic cold atoms gases in an optical lattice or in fractional Chern insulators.
