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arXiv:cond-mat/0404746 · Physical Review A

Transition from three dimensions to one dimension in Bose gases at zero temperature

Luca Salasnich, Alberto Parola, L. Reatto

Published Jul 13, 2004• 56 citations
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Abstract

We investigate the effects of dimensional reduction in Bose gases induced by a strong harmonic confinement in the transverse cylindric radial direction. By using a generalized Lieb-Liniger theory, based on a variational treatment of the transverse width of the Bose gas, we analyze the transition from a three-dimensional Bose-Einstein condensate to the one-dimensional Tonks-Girardeau gas. The sound velocity and the frequency of the lowest compressional mode give a clear signature of the regime involved. We also study the case of negative scattering length deriving the phase diagram of the Bose gas (uniform, single soliton, multisoliton, and collapsed) in toroidal confinement.

DOI
10.1103/physreva.70.013606

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We investigate the effects of dimensional reduction in Bose gases induced by a strong harmonic confinement in the transverse cylindric radial direction. By using a generalized Lieb-Liniger theory, based on a variational treatment of the transverse width of the Bose gas, we analyze the transition from a three-dimensional Bose-Einstein condensate to the one-dimensional Tonks-Girardeau gas. The sound velocity and the frequency of the lowest compressional mode give a clear signature of the regime involved. We also study the case of negative scattering length deriving the phase diagram of the Bose gas (uniform, single soliton, multisoliton, and collapsed) in toroidal confinement.
We investigate the effects of dimensional reduction in atomic Bose-Einstein condensates (BECs) induced by a strong harmonic confinement in the cylindric radial direction or in the cylindric axial direction. The former case corresponds to a transition from three dimensions (3D) to 1D in cigar-shaped BECs, while the latter case corresponds to a transition from 3D to 2D in disk-shaped BECs. We analyze the first sound velocity in axially homogeneous cigar-shaped BECs and in radially homogeneous disk-shaped BECs. We consider also the dimensional reduction in a BEC confined by a harmonic potential both in the radial direction and in the axial direction. By using a variational approach, we calculate monopole and quadrupole collective oscillations of the BEC. We find that the frequencies of these collective oscillations are related to the dimensionality and to the repulsive or attractive interatomic interaction.
The expansion of a one-dimensional Bose gas after releasing its initial harmonic confinement is investigated employing the Lieb-Liniger equation of state within the local-density approximation. We show that during the expansion the density profile of the gas does not follow a self-similar solution, as one would expect from a simple scaling ansatz. We carry out a variational calculation, which recovers the numerical results for the expansion, the equilibrium properties of the density profile, and the frequency of the lowest compressional mode. The variational approach allows for the analysis of the expansion in all interaction regimes between the mean-field and the Tonks-Girardeau limits, and in particular shows the range of parameters for which the expansion violates self-similarity.
We analyze a recent experiment on a Tonks-Girardeau gas of 87Rb^{87}\mathrm{Rb}87Rb atoms [T. Kinoshita, T. Wenger, and D.S. Weiss, Science 305, 1125 (2004)]. We find that the experimental data are compatible with the one-dimensional theory of Lieb, Seiringer, and Yngvason [Phys. Rev. Lett. 91, 150401 (2003)] but are better described by a theory that takes into account variations in the transverse width of the atomic cloud. By using this theory we investigate also the free axial expansion of the 87Rb^{87}\mathrm{Rb}87Rb gas in different regimes, Tonks-Girardeau gas, one-dimensional Bose-Einstein condensate and three-dimensional Bose-Einstein condensate.
In this paper we develop a variational theory to study the dynamic properties of ultracold Bose gas in a funnel external potential. We obtain one-dimensional nonlinear equation which describes the dynamics of transverse tight confined bosonic gas from three-dimension to one-dimension, and find one-dimensional s-wave scattering length which depends on the shape of transverse confining potential. If the funnel trapping potential is strong enough at zero temperature, all transverse excitations are frozen. We find the dynamic equation which describes the Tonks–Girardeau gas and present a qualitative analysis of the experimental accessibility of the Tonks–Girardeau gas with funnel-trapped alkalic atoms.
Abstract This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its one-dimensional-like geometrical shape, and the deep one-dimensional regime where the motion is frozen in the radial direction. The cigar configuration is well suited to investigating the novel features exhibited by quantized vortices and solitons. In the deep one-dimensional regime Bose gases exhibits a very different behaviour with respect to usual three-dimensional BECs as a consequence of the quantum and thermal fluctuations of the phase. The Lieb–Liniger transition between the one-dimensional mean-field and the Tonks–Girardeau regimes is discussed, and important features concerning the frequency of the collective oscillations and the superfluid behaviour are pointed out. Finally, the chapter discusses some key features exhibited by one-dimensional Fermi gases.