We present a precise measurement of the condensate fraction and critical temperature in a rubidium-87 dilute Bose gas in a 3D harmonic trap using a novel technique of short-pulse Bragg scattering to create phonons. The critical temperature is lower than predicted by the ideal gas model and higher than predicted by an interacting gas model. The number of atoms in the condensate is extracted by counting the phonons from an image taken in situ. Precise measurement is achieved since the measurement is done at higher spatial frequency in the image, improving the signal-to-noise ratio as in a lock-in amplifier.
We also show a measurement of the coherence length of atoms at temperatures above the critical temperature. A group of atoms accelerated by short-pulse Bragg scattering interferes with stationary atoms, creating fringes in the gas as long as they move less than the coherence length. By measuring the decay rate the coherence length is found. | 
