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arXiv:1105.0293

Tonks-Girardeau gas, super-Tonks-Girardeau gas, and bound states of one-dimensional bosons in a hard-wall trap

Hua Li, Liu, Taifeng, Yaojiang Hao, Yunbo Zhang

Published May 2, 2011• 0 citations
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Abstract

We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the TG and STG gas show the same profiles at the strongly interacting point $-1/γ=0$, and in the weakly interacting limit $γ=0$ the ground states TG and BS join to each other smoothly.

DOI
10.48550/arxiv.1105.0293

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We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the TG and STG gas show the same profiles at the strongly interacting point −1/γ=0-1/γ=0−1/γ=0, and in the weakly interacting limit γ=0γ=0γ=0 the ground states TG and BS join to each other smoothly.
We investigate the ground state of the system of NNN bosons enclosed in a hard-wall trap interacting via a repulsive or attractive $\ensuremath{\delta}$-function potential. Based on the Bethe ansatz method, the explicit ground state wave function is derived and the corresponding Bethe ansatz equations are solved numerically for the full physical regime from the Tonks limit to the strongly attractive limit. It is shown that the solution takes a different form in different regime. We also evaluate the one body density matrix and second-order correlation function of the ground state for finite systems. In the Tonks limit the density profiles display the Fermi-like behavior, while in the strongly attractive limit the Bosons form a bound state of NNN atoms corresponding to the NNN-string solution. The density profiles show the continuous crossover behavior in the entire regime. Further, the correlation function indicates that the Bose atoms bunch closer as the interaction constant decreases.
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We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
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 provide evidence in support of a recent proposal by Astrakharchik et al for the existence of a super Tonks–Girardeau gas-like state in the attractive interaction regime of quasi-one-dimensional Bose gases. We show that the super Tonks–Girardeau gas-like state corresponds to a highly excited Bethe state in the integrable interacting Bose gas for which the bosons acquire hard-core behaviour. The gas-like state properties vary smoothly throughout a wide range from strong repulsion to strong attraction. There is an additional stable gas-like phase in this regime in which the bosons form two-body bound states behaving like hard-core bosons.
We investigate the ground state of the one-dimensional fermionic system enclosed in a hard-wall trap with attractive contact ppp-wave interactions. Based on the Bethe ansatz method, the explicit wave function is derived by numerically solving the Bethe ansatz equations for the full physical regimes $(\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{c}_{F}\ensuremath{\le}0)$. With the exact wave function some quantities which are important in many-body physics are obtained, including the one-body density matrix and the momentum distribution of the ground state for finite system. It is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic Tonks-Girardeau limit the density distribution shows the same behavior as that of an ideal Bose gas. However, the one-body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts.