Relations Among Some Ground-state properties of a Bose Gas in Periodic Potentials

The relations among the occupation number of the lowest natural orbital, momentum distributions and off-diagonal elements of the reduced single-particle density matrix (RSPDM) are studied while a Bose gas in periodic potentials is in the ground state. For large particle number systems, RSPDM and its lowest natural orbital of $N$-body system are the same as that of $(N+1)$-body system in overlapped areas in both commensurate and incommensurate cases. For large $N$ in commensurate case, the occupation number of the lowest natural orbital and momentum distributions (MD) are invariant with $N$ that results from the exponential attenuation with $N$ of the off-diagonal elements. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to $\sqrt{N}$, which result in the different behavior of the occupation number of the lowest natural orbital and MD. There are no Bose-Einstein condensation (BEC) for periodic trapped TG gas in both commensurate and incommensurate cases.