Loschmidt echo in the Bose-Hubbard model: turning back time in an optical lattice

I show how to perform a Loschmidt echo (time reversal) in the Bose-Hubbard model implemented with cold bosonic atoms in an optical lattice. The echo is obtained by applying a linear phase imprint on the lattice and a change in magnetic field to tune the boson-boson scattering length through a Feshbach resonance. I discuss how the echo can measure the fidelity of the quantum simulation, the intensity of an external potential (e.g. gravity), or the critical point of the superfluid-insulator quantum phase transition.