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arXiv:cond-mat/0605197

Calogero-Sutherland-Lieb-Liniger gas in one-dimensional cold atoms

Yue Yu

Published May 8, 2006• 0 citations
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Abstract

We study an array of cigar-like Bose atom condensates confined in a cylinder and examine the competition between the dipole-dipole and the short range interactions. The system is effectively reduced to a one-dimensional boson one with a contact and inverse square interactions. We call this system the Calogero-Sutherland-Lieb-Liniger gas. The universal properties of the ground state are analyzed by the renormalization group theory. By using the bosonization techniques to the excluson gas, we calculate the non-universal exponent depending on the microscopic parameters. This exponent may be experimentally measurable.

DOI
10.48550/arxiv.cond-mat/0605197

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We study an array of cigar-like Bose atom condensates confined in a cylinder and examine the competition between the dipole-dipole and the short range interactions. The system is effectively reduced to a one-dimensional boson one with a contact and inverse square interactions. We call this system the Calogero-Sutherland-Lieb-Liniger gas. The universal properties of the ground state are analyzed by the renormalization group theory. By using the bosonization techniques to the excluson gas, we calculate the non-universal exponent depending on the microscopic parameters. This exponent may be experimentally measurable.
We show that the Calogero-Sutherland (C-S) gas, a famous exact soluble one-dimensional system with an inverse square long range interaction, can be realized by dimension reduction in a cold Bose atom system with a dipole-dipole interaction. Depending on the orientation of the dipoles, the effective interaction is either attractive or repulsive. The low-lying effective theory may be a Luttinger liquid when the exclusion statistics parameter λλλ may be well-defined. We hope that the C-S gas can be realized experimentally and the Luttinger liquid character can be observed.
We show that the dipole-dipole coupling between Wannier modes in cigar-shaped Bose-Einstein condensates (BECs) is significantly enhanced while the short-range coupling is strongly suppressed. As a result, the dipole-dipole interaction can become the dominant interaction between ultracold alkali Bose atoms. In the long length limit of a cigar-shaped BEC, the resulting effective one-dimensional models possess an effective inverse squared interacting potential, the Calogero-Sutherland potential, which plays a fundamental role in many fields of contemporary physics; but its direct experimental realization has been a challenge for a long time. We propose to realize the Calogero-Sutherland model in ultracold alkali Bose atoms and study the effects of the dipole-dipole interaction.
A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue λ 0 of the one-body-density matrix scales as λ 0 ∼ N , where N is the total number of particles.Putting λ 0 ∼ N C to define the scaling exponent C, then C = 1 corresponds to ODLRO and C = 0 to the single-particle occupation of the density matrix orbitals.When 0 < C < 1, C can be used to quantify deviations from ODLRO.In this paper we study the exponent C in a variety of one-dimensional bosonic and anyonic quantum systems at T = 0.For the 1D Lieb-Liniger Bose gas we find that for small interactions C is close to 1, implying a mesoscopic condensation, i.e. a value of the zero temperature "condensate" fraction λ 0 /N appreciable at finite values of N (as the ones in experiments with 1D ultracold atoms).1D anyons provide the possibility to fully interpolate between C = 1 and 0. The behaviour of C for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.[2] A. Colcelli, J. Viti, G. Mussardo, and A. Trombettoni, Universal off-diagonal longrange-order behavior for a trapped Tonks-Girardeau gas, Phys.Rev. A 98, 063633 (2018).
We compute the Tan's contact of a weakly interacting Bose gas at zero temperature in a cigar-shaped configuration. Using an effective one-dimensional Gross-Pitaevskii equation and Bogoliubov theory, we derive an analytical formula that interpolates between the three-dimensional and the one-dimensional mean-field regimes. In the strictly one-dimensional limit, we compare our results with Lieb-Liniger theory. Our study can be a guide for actual experiments interested in the study of Tan's contact in the dimensional crossover.