  Experiments with ultracold atoms provide a highly controllable laboratory
setting with many unique opportunities for precision exploration of quantum
many-body phenomena. The nature of such systems, with strong interaction and
quantum entanglement, makes reliable theoretical calculations challenging.
Especially difficult are excitation and dynamical properties, which are often
the most directly relevant to experiment. We carry out exact numerical
calculations, by Monte Carlo sampling of imaginary-time propagation of Slater
determinants, to compute the pairing gap in the two-dimensional Fermi gas from
first principles. Applying state-of-art analytic continuation techniques, we
obtain the spectral function, and the density and spin structure factors
providing unique tools to visualize the BEC-BCS crossover. These quantities
will allow for a direct comparison with experiments.
