Interference of two independently prepared ideal Bose gases is discussed, onthe basis of the idea of measurement-induced interference: even if the numberof each gas is individually fixed finite and the symmetry of the system is notbroken, an interference pattern is observed on each single snapshot. The keyrole is played by the Hanbury Brown and Twiss effect, which leads to anoscillating pattern of the cloud of identical atoms. Then, how essential is theBose-Einstein condensation to the interference? We describe the ideal Bosegases trapped respectively in two spatially separated 3D harmonic traps at afinite temperature as canonical ensembles with fixed numbers of atoms, andcompute the full statistics of the snapshot profiles of the expanding andoverlapping gases released from the traps. We obtain a simple formula, whichshows that the average fringe spectrum (average fringe contrast) is given bythe purity of each gas. The purity is known to be a good measure ofcondensation, and this result clarifies the relevance of the condensation tothe interference. The fluctuation of the interference spectrum is also studied,and it is shown that the fluctuation is vanishingly small below the criticaltemperature, while it is nonvanishing above. This implies that interferencepattern is certainly observed on every snapshot below the critical temperature.The fact that the number of atoms is fixed in the canonical ensemble is crucialto this vanishing fluctuation.
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