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arXiv:0804.2759 · Physical Review A

Ground-state properties of a few-boson system in a one-dimensional hard-wall split potential

Xiangguo Yin, Yajiang Hao, Shu Chen, Yunbo Zhang

Published Jul 2, 2008• 33 citations
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Abstract

We carry out a detailed examination of the ground-state properties of a few-boson system in a one-dimensional hard-wall potential with a $\ensuremath{\delta}$ split in the center. In the Tonks-Girardeau limit with infinite repulsion between particles, we use the Bose-Fermi mapping to construct the exact $N$-particle ground-state wave function, which allows us to study the correlation properties accurately. For the general case with finite interparticle interaction, the exact diagonalization method is exploited to study the ground-state density distribution, occupation number distribution, and momentum distribution for variable interaction strengths and barrier heights. The secondary peaks in the momentum distribution reveal the interference between particles on the two sides of the split, which is more prominent for large barrier strength and small interaction strength.

DOI
10.1103/physreva.78.013604

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We carry out a detailed examination of the ground-state properties of a few-boson system in a one-dimensional hard-wall potential with a \ensuremathδ\ensuremath{\delta}\ensuremathδ split in the center. In the Tonks-Girardeau limit with infinite repulsion between particles, we use the Bose-Fermi mapping to construct the exact NNN-particle ground-state wave function, which allows us to study the correlation properties accurately. For the general case with finite interparticle interaction, the exact diagonalization method is exploited to study the ground-state density distribution, occupation number distribution, and momentum distribution for variable interaction strengths and barrier heights. The secondary peaks in the momentum distribution reveal the interference between particles on the two sides of the split, which is more prominent for large barrier strength and small interaction strength.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a hard-wall trap or a harmonic oscillator trap with a tunable $\ensuremath{\delta}$-function barrier at the trap center. The mixture consists of a 1D Bose gas with repulsive interactions and of a 1D noninteracting spin-aligned Fermi gas, with both species interacting through hard-core interactions. Using a generalized Bose-Fermi mapping, we calculate the reduced single-particle density matrix and the momentum distribution of the gas as a function of barrier strength and the parity of particle number. The secondary peaks in the momentum distribution show remarkable correlation between particles on the two sides of the split.
We describe the properties of a pair of ultracold bosonic atoms in a one-dimensional harmonic trapping potential with a tunable zero-ranged barrier at the trap center. The full characterization of the ground state is done by calculating the reduced single-particle density, the momentum distribution, and the two-particle entanglement. We derive several analytical expressions in the limit of infinite repulsion (Tonks-Girardeau limit) and extend the treatment to finite interparticle interactions by numerical solution. As pair interactions in double wells form a fundamental building block for many-body systems in periodic potentials, our results have implications for a wide range of problems.
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.
We investigate ground-state properties of interacting two-component Bose gases in a hard-wall trap using both the Bethe ansatz and exact numerical diagonalization method. For equal intra-atomic and interatomic interaction, the system is exactly solvable. Thus the exact ground-state wave-function and density distributions for the whole interacting regime can be obtained from the Bethe ansatz solutions. Since the ground state is a degenerate state with total spin S=N/2S=N/2S=N/2, the total density distributions are the same for each degenerate state. The total density distribution evolves from a Gauss-like Bose distribution to a Fermi-like one as the repulsive interaction increases. The distribution of each component is ${N}_{\ensuremath{\alpha}}/N$ of the total density distribution. This is approximately true even in the experimental situation. In addition the numerical results show that with the increase in interspecies interaction the distributions of two Tonks-Girardeau gases exhibit composite-fermionization crossover with each component developing NNN peaks in the strongly interacting regime.
We study ground state properties of spinless, quasi one-dimensional bosons which are confined in a harmonic trap and interact via repulsive delta-potentials. We use the exact diagonalization method to analyze the pair correlation function, as well as the density, the momentum distribution, different contributions to the energy and the population of single-particle orbitals in the whole interaction regime. In particular, we are able to trace the fascinating transition from bosonic to fermi-like behavior in characteristic features of the momentum distribution which is accessible to experiments. Our calculations yield quantitative measures for the interaction strength limiting the mean-field regime on one side and the Tonks-Girardeau regime on the other side of an intermediate regime.
We study the dynamical properties of a few bosons confined in an one-dimensional split hard wall trap with the interaction strength varying from the weakly to strongly repulsive regime. The system is initially prepared in one side of the double well by setting the barrier strength of the split trap to be infinity and then the barrier strength is suddenly changed to a finite value. Both exact diagonalization method and Bose-Hubbard model (BHM) approximation are used to study the dynamical evolution of the initial system. The exact results based on exact diagonaliztion verify the enhancement of correlated tunneling in the strongly interacting regime. Comparing results obtained by two different methods, we conclude that one-band BHM approximation can well describe the dynamics in the weakly interacting regime, but is not efficient to give quantitatively consistent results in the strongly interacting regime. Despite of the quantitative discrepancy, we validate that the form of correlated tunneling gives an important contribution to tunneling in the large interaction regime. To get a quantitative description for the dynamics of bosons in the strongly interacting regime, we find that a multi-band BHM approximation is necessary.