Engineering entanglement Hamiltonians with strongly interacting cold atoms in optical traps

We present a proposal for the realization of entanglement Hamiltonians in one-dimensional critical spin systems with strongly interacting cold atoms. Our approach is based on the notion that the entanglement spectrum of such systems can be realized with a physical Hamiltonian containing a set of position-dependent couplings. We focus on reproducing the universal ratios of the entanglement spectrum for systems in two different geometries: a harmonic trap, which corresponds to a partition embedded in an infinite system, and a linear potential, which reproduces the properties of a half partition with open boundary conditions. Our results demonstrate the possibility of measuring the entanglement spectra of the Heisenberg and XX models in a realistic cold-atom experimental setting by simply using gravity and standard trapping techniques.