  Entanglement, and, in particular the entanglement spectrum, plays a major
role in characterizing many-body quantum systems. While there has been a surge
of theoretical works on the subject, no experimental measurement has been
performed to date because of the lack of an implementable measurement scheme.
Here, we propose a measurement protocol to access the entanglement spectrum of
many-body states in experiments with cold atoms in optical lattices. Our scheme
effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian and
is based on the ability to produce several copies of the state under
investigation together with the possibility to perform a global swap gate
between two copies conditioned on the state of an auxiliary qubit. We show how
the required conditional swap gate can be implemented with cold atoms, either
by using Rydberg interactions or coupling the atoms to a cavity mode. We
illustrate these ideas on a simple (extended) Bose-Hubbard model where such a
measurement protocol reveals topological features of the Haldane phase.
