Quasi 1 and 2d dilute bose gas in magnetic traps:: Existence of off-diagonal order and anomalous quantum fluctuations

Current magnetic traps cart be made so anisotropic that dilute Bose gases confined in these traps will occupy the lowest quantum state in the tightly confining direction, while still in the Thomas-Fermi limit in the loosely confining direction, As a result, the trapped Bose gas behaves like a quasi one or two dimensional systems. Unlike the homogeneous case, quantum phase fluctuations do not destroy macroscopic off-diagonal order of trapped Bose gases in d less than or equal to 2 because they are suppressed by the the trapping potential, 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.