  The quantum coherence of a Bose-Einstein condensate is studied using the
concept of quantum fidelity (Loschmidt echo). The condensate is confined in an
elongated anharmonic trap and subjected to a small random potential such as
that created by a laser speckle. Numerical experiments show that the quantum
fidelity stays constant until a critical time, after which it drops abruptly
over a single trap oscillation period. The critical time depends
logarithmically on the number of condensed atoms and on the perturbation
amplitude. This behavior may be observable by measuring the interference
fringes of two condensates evolving in slightly different potentials.
