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arXiv:cond-mat/0602163 · Physical Review A

Tonks-Girardeau gas in an optical lattice: Effects of a local potential

Hao Fu, Alberto G. Rojo

Published Jul 21, 2006• 6 citations
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Abstract

The physics of a Tonks-Girardeau gas confined in an optical lattice in the presence of a local potential is studied. In order to evaluate the single-particle density matrix (SPDM) of the many-body ground state, the Wigner-Jordan transformation is used. The eigenvector with the largest eigenvalue of the SPDM corresponds to the ``Bose-Einstein condensate''(BEC) state. We find that the BEC-state density at the position of the local potential decreases, as expected, in the case of a repulsive potential. For an attractive potential, it decreases or increases depending on the strength of the potential. The superfluidity of this system is investigated both numerically and perturbatively. An experimental method for detecting the effect of an impurity in a Tonks-Girardeau gas is discussed.

DOI
10.1103/physreva.74.013620

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The physics of a Tonks-Girardeau gas confined in an optical lattice in the presence of a local potential is studied. In order to evaluate the single-particle density matrix (SPDM) of the many-body ground state, the Wigner-Jordan transformation is used. The eigenvector with the largest eigenvalue of the SPDM corresponds to the ``Bose-Einstein condensate''(BEC) state. We find that the BEC-state density at the position of the local potential decreases, as expected, in the case of a repulsive potential. For an attractive potential, it decreases or increases depending on the strength of the potential. The superfluidity of this system is investigated both numerically and perturbatively. An experimental method for detecting the effect of an impurity in a Tonks-Girardeau gas is discussed.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.
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.
The two-component interacting Bose gas confined in a one-dimensional optical lattice with infinite intra-component repulsive interactions and tunable inter-component repulsions is studied. By a generalized Jordan-Wigner transformation, it is shown that this model can be solved exactly. The energy spectrum and explicit wave function are obtained. The physical quantities such as the off-diagonal density matrix, momentum distribution and single-particle von Neumann entropy of the present model display different properties from those of the Fermi-Hubbard model due to the different intrinsic symmetry of particles.
We investigate the effect of a quench from a one-dimensional gas with strong and repulsive local interactions to a strongly attractive one, known as the super-Tonks-Girardeau effect. By incorporating both an optical lattice and nonlocal interactions (specifically nearest neighbor), we discover a previously unexplored phenomenon: the disruption of the state during the quench, but within a specific range of interactions. Our study employs the extended Bose-Hubbard model across various system sizes, starting with analytical results for two atoms and progressing to few-body systems using exact diagonalization, density matrix renormalization group and time-dependent variational principle methods. Finally, we use a numerical implementation of the local density approximation for a macroscopic number of atoms. Consistently, our findings unveil a region where the initially self-bounded structure expands due to the super-Tonks-Girardeau quench. The fast evaporation provides a tool to characterize the phase diagram in state-of-art experiments exploring the physics of the extended Bose-Hubbard model.
We study the Bose-Fermi mixture with infinite boson-boson repulsion and finite boson-fermion repulsion. Using a generalized Jordan-Wigner transformation, we show that the system can be mapped to a repulsive Hubbard model and thus can be solved exactly for the case with equal boson and fermion masses. Using the Bethe-ansatz solutions, we investigate the ground-state properties of the mixture system. Our results indicate that the system with commensurate filling n=1n=1n=1 is a charge insulator but still a superfluid with nonvanishing superfluid density. We also briefly discuss the case with unequal boson and fermion masses.
We theoretically demonstrate features of Anderson localization in a Tonks-Girardeau gas confined in one-dimensional potentials with controlled disorder. That is, we investigate the evolution of the single-particle density and correlations of a Tonks-Girardeau wave packet in such disordered potentials. The wave packet is initially trapped, the trap is suddenly turned off, and after some time the system evolves into a localized steady state due to Anderson localization. The density tails of the steady state decay exponentially, while the coherence in these tails increases. The latter phenomenon corresponds to the same effect found in incoherent optical solitons.