  We demonstrate the existence of phase fluctuations in elongated Bose-Einstein
Condensates (BECs) and study the dependence of those fluctuations on the system
parameters. A strong dependence on temperature, atom number, and trapping
geometry is observed. Phase fluctuations directly affect the coherence
properties of BECs. In particular, we observe instances where the phase
coherence length is significantly smaller than the condensate size. Our method
of detecting phase fluctuations is based on their transformation into density
modulations after ballistic expansion. An analytic theory describing this
transformation is developed.
