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arXiv:0812.1277 · Physical Review A

Ground-state properties of a Tonks-Girardeau gas in a periodic potential

Bo-Bo Wei, Shi-Jian Gu, Hai-Qing Lin

Published Jun 24, 2009• 5 citations
arXivHTMLPDF

Abstract

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 $N=265$ 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.

DOI
10.1103/physreva.79.063627

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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\ensuremath{-}0.5\ensuremath−0.5 and 1, respectively, while in the incommensurate case, they are 0.5, \ensuremath−0.5\ensuremath{-}0.5\ensuremath−0.5, and 1.5, respectively. These exponents are related to each other in a universal relation.
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.
We present a simulation study of the properties of the Tonks-Girardeau gas, a system of NNN hard-core bosons confined in a one-dimensional harmonic trap. For this system the ground-state wave function is exactly known based on a Bose-Fermi mapping theorem. We employ Nelsons' interpretation of quantum mechanics in which this NNN-body wave function gives rise to a set of stochastic differential equations for the positions of the NNN particles, which can be simulated by standard methods. In particular, real space densities and momentum distributions can simply be determined by time averages along particle trajectories. Employing this approach we are able to significantly extend the range of particle numbers NNN treated numerically compared to earlier approaches, while reproducing all exactly known results for this model. We also show that for the bosons in a harmonic trap, contrary to what has been assumed so far, the momentum distribution reflects the system size scaling of the occupation numbers of the natural orbitals; that is, it can be used to decide on the presence of a Bose-Einstein condensate.
We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.
We determine the exact many-body properties of a bosonic Tonks-Girardeau gas confined in a harmonic potential with a tunable $\ensuremath{\delta}$-function barrier at the trap center. This is done by calculating the reduced single-particle density matrix, the pair-distribution function, and the momentum distribution of the gas as a function of barrier strength and particle number. With increasing barrier height we find that the ground-state occupation in a diagonal basis diverges from the N\sqrt{N}N​ behavior that is expected for the case of a simple harmonic trap. In fact, the scaling of the occupation number depends on whether one has an even or odd number of particles. Since this quantity is a measure of the coherence of our sample we show how the odd-even effect manifests itself in both the momentum distribution of the Bose gas and interference fringe visibility during free temporal evolution.
With Girardeau's Fermi–Bose mapping, we have constructed the eigenstates of the Tonks–Girardeau (TG) gas in an external magnetic field. When the number of bosons N is commensurate with the number of potential cycles M , the probability of this TG gas in the ground state is bigger than the TG gas raised by Girardeau in 1960. Through the comparison of properties between this TG gas and Fermi gas, we find that the following issues are always the same: their average value of particle coordinates and potential energies, system total momenta, single-particle densities and pair distribution functions. But the reduced single-particle density matrices and their momentum distributions between them are different.