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

Zhao Liang Wang, An Min Wang, Xue Chao Li, Yang Yang

Published Jan 29, 2012• 0 citations
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Abstract

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.

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•Jan 1, 2002•1 citation
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 NNN-body system are the same as that of (N+1)(N+1)(N+1)-body system in overlapped areas in both commensurate and incommensurate cases. For large NNN in commensurate case, the occupation number of the lowest natural orbital and momentum distributions (MD) are invariant with NNN that results from the exponential attenuation with NNN of the off-diagonal elements. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to N\sqrt{N}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.
The relations among the occupation number of the lowest natural orbital (ONLNO), momentum distributions (MD) and off-diagonal long-range element (ODLRE) of the reduced single-particle density matrix (RSPDM) are studied while Tonks-Girardeau gas in one dimensional periodic potential is in the ground state. For NNN-body systems of large enough, RSPDM and its lowest natural orbital do not vary with NNN in overlapped areas in commensurate and incommensurate cases correspondingly. In commensurate case, the ODLRE is exponential attenuation with NNN, which results in that the ONLNO and MD are invariant with NNN. While in contrast, in incommensurate case, the off-diagonal elements are inversely proportional to N\sqrt{N}N​, which results in the different behavior of the ONLNO and MD.
In this paper, we investigate the ground-state properties of a bosonic Tonks-Girardeau (TG) gas confined in a one-dimensional periodic potential. The single-particle reduced density matrix is computed numerically for systems up to N=265N=265N=265 bosons. Scaling analysis of the occupation number of the lowest orbital shows that there are no Bose-Einstein condensation (BEC) for the periodically trapped TG gas in both commensurate and incommensurate cases. We find that, in the commensurate case, the scaling exponents of the occupation number of the lowest orbital, the amplitude of the lowest orbital and the zero-momentum peak height with the particle numbers are 0, $\ensuremath{-}0.5$ and 1, respectively, while in the incommensurate case, they are 0.5, $\ensuremath{-}0.5$, and 1.5, respectively. These exponents are related to each other in a universal relation.
We calculate the reduced single-particle density matrix (RSPDM), momentum distribution, natural orbitals and their occupancies, for a strongly repulsive one-dimensional Bose-Fermi mixture in a double-well potential with a large central barrier. We assume that all particles have the same mass, and fermions are spin polarized. For mesoscopic systems, we find that the ground-state properties qualitatively differ for mixtures with even number of particles (both odd-odd and even-even mixtures) in comparison to mixtures with odd particle numbers (odd-even and even-odd mixtures). For even mixtures the momentum distribution is smooth, whereas the momentum distribution of odd mixtures possesses distinct modulations; the differences are observed also in the off-diagonal correlations of the RSPDM, and in the occupancies of natural orbitals. The calculation is based on a formula which enables efficient calculation of the RSPDM for mesoscopic mixtures in various potentials.
The one-particle density matrices for hard-core bosons in a one-dimensional harmonic trap are computed numerically for systems with up to 160 bosons. Diagonalization of the density matrix shows that the many-body ground state is not Bose-Einstein condensed. The ground-state occupation, the amplitude of the lowest natural orbital, and the zero momentum peak height scale as powers of the particle number, and the corresponding exponents are related to each other. Close to its diagonal, the density matrix for hard-core bosons is similar to that of noninteracting fermions.
The exact N-particle ground-state wave function for a one-dimensional condensate of hard-core bosons in a harmonic trap is employed to obtain accurate numerical results for the one-particle density matrix, occupation number distribution of the natural orbitals, and momentum distribution. Our results show that the occupation of the lowest orbital varies as N0.59,{N}^{0.59},N0.59, in contrast to N0.5{N}^{0.5}N0.5 for a spatially uniform system, and N for a true Bose-Einstein condensate.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.