  Quantum phase fluctuations play a crucial role in low dimensional systems. In
particular they prevent true long range phase order from forming in one
dimensional condensates, even at zero temperature. Nevertheless, by dynamically
splitting the condensate into two parallel decoupled tubes, a macroscopic
relative phase, can be imposed on the system. This kind of setup can serve as a
matter wave interferometer, which relies on the interference between the
displaced condensates as a measure of the relative phase between them. Here we
show how the quantum phase fluctuations, which are so effective in equilibrium,
act to destroy the macroscopic relative phase that was imposed as a non
equilibrium initial condition of the interferometer. We show that the phase
coherence between the two condensates decays exponentially with a dephasing
time that depends on intrinsic parameters: the dimensionless interaction
strength, sound velocity and density. Interestingly, at low temperatures the
dephasing time is almost independent of temperature. At temperatures higher
than a crossover scale T* dephasing gains significant temperature dependance.
In contrast to the usual phase diffusion, which is essentially an effect of
confinement, the dephasing due to fluctuations in one dimension is a bulk
effect that survives the thermodynamic limit.
