  Current magnetic traps can be made so anisotropic that the atomic gas trapped
inside behaves like quasi one or two dimensional system. Unlike the homogeneous
case, quantum phase fluctuations do not destroy macroscopic off-diagonal order
of trapped Bose gases in $d\leq 2$. In the dilute limit, quantum fluctuations
increase, remain constant, and decrease with size for $% 3, 2, 1d$
respectively. These behaviors are due to the combination of a finite gap and
the universal spectrum of the collective mode.
