Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases

At present, many laboratories are performing experiments to simulate theoretical models of strongly correlated systems using cold atoms in optical lattices, a program referred to as "Quantum Simulation". It is hoped that these experiments will shed light on some long standing problems in condensed matter physics. The goal of Quantum Simulation is to obtain information of homogenous bulk systems. However, experiments are performed in confining traps. The non-uniformity of the trapping potential inevitably introduces different phases in the sample, making it difficult to deduce the properties of a single bulk phase. So far, there are no algorithms to use the experimental data to map out phase diagrams and important thermodynamic quantities. Here, we present an algorithm to achieve this goal. Apart from phase diagram, it also maps out entropy density, superfluid density of superfluids, and staggered magnetizations of anti-ferromagnets. Our scheme is exact within local density approximation.