Phonon Dispersion Relation of an Atomic Bose-Einstein Condensate

We measure the time oscillations of a freely evolving standing wave of phonons in a Bose-Einstein condensate. We present the technique of short Bragg pulses, which stimulates the standing wave. The subsequent oscillations are observed in situ. The frequency of the oscillations gives the dispersion relation, the amplitude gives the static structure factor, and the decay gives the dephasing time. The new technique gives orders of magnitude more sensitivity than Bragg spectroscopy, allowing for the observation of deviations from the local density approximation. Specifically, it is seen that the phonons undergo a transition from three dimensions to one dimension, when their wavelength becomes longer than the transverse radius of the condensate. The one-dimensional regime contains an inflection point in the dispersion relation, a decrease in the superfluid critical velocity, a minimum in the group velocity, and an increase in the lifetime of the standing wave oscillations.