Quasi 1 and 2d Dilute Bose Gas in Magnetic Traps : Existence of Off-Diagonal Order and Anomalous Quantum Fluctuations

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.