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We determine the exact many-body properties of a bosonic Tonks-Girardeau gasconfined in a harmonic potential with a tunable $\delta$-function barrier atthe trap center. This is done by calculating the reduced single particledensity matrix, the pair-distribution function, and momentum distribution ofthe gas as a function of barrier strength and particle number. With increasingbarrier height we find that the ground state occupation in a diagonal basisdiverges from the $\sqrt N$ behavior that is expected for the case of a simpleharmonic trap. In fact, the scaling of the occupation number depends on whetherone has an even or odd number of particles. Since this quantity is a measure ofthe coherence of our sample we show how the odd-even effect manifests itself inboth the momentum distribution of the Bose gas and interference fringevisibility during free temporal evolution.

Ground State Properties of a Tonks-Girardeau Gas in a Split Trap