We report a measurement of the excitation spectrum omega(k) and the static structure factor S(k) of a Bose-Einstein condensate. The excitation spectrum displays a linear phonon regime, as well as a parabolic single-particle regime. The linear regime provides an upper limit for the superfluid critical velocity, by the Landau criterion. The excitation spectrum agrees well with the Bogoliubov spectrum in the local density approximation, even close to the long-wavelength limit of the region of applicability. Feynman's relation between omega(k) and S(k) is verified, within an overall constant.

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Geometric frustration, the inability of an ordered system to find a unique ground state plays a key role in a wide range of systems. We present a new experimental approach to observe large-scale geometric frustration with 1500 negatively coupled lasers arranged in a kagome lattice. We show how dissipation drives the lasers into a phase-locked state that directly maps to the classical XY spin Hamiltonian ground state. In our system, frustration is manifested by the lack of long range phase ordering. Finally, we show how next-nearest-neighbor coupling removes frustration and restores order.

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