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arXiv:cond-mat/0304141 · Physical Review Letters

Three-Dimensional Quasi-Tonks Gas in a Harmonic Trap

P. Pedri, Luis Santos

Published Sep 9, 2003• 17 citations
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Abstract

We analyze the macroscopic dynamics of a Bose gas in a harmonic trap with a superimposed two-dimensional optical lattice, assuming a weak coupling between different lattice sites. We consider the situation in which the local chemical potential at each lattice site can be considered as that provided by the Lieb-Liniger solution. Because of the weak coupling between sites and the form of the chemical potential, the three-dimensional ground-state density profile and the excitation spectrum acquire remarkable properties different from both 1D and 3D gases. We call this system a quasi-Tonks gas. We discuss the range of applicability of this regime, as well as realistic experimental situations where it can be observed.

DOI
10.1103/physrevlett.91.110401

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We analyze the macroscopic dynamics of a Bose gas in a harmonic trap with a superimposed two-dimensional optical lattice, assuming a weak coupling between different lattice sites. We consider the situation in which the local chemical potential at each lattice site can be considered as that provided by the Lieb-Liniger solution. Because of the weak coupling between sites and the form of the chemical potential, the three-dimensional ground-state density profile and the excitation spectrum acquire remarkable properties different from both 1D and 3D gases. We call this system a quasi-Tonks gas. We discuss the range of applicability of this regime, as well as realistic experimental situations where it can be observed.
We calculate the dynamic structure factor $S(q,\ensuremath{\omega})$ of a one-dimensional (1D) interacting Bose gas confined in a harmonic trap. The effective interaction depends on the strength of the confinement enforcing the (1D) motion of atoms; interaction may be further enhanced by superimposing an optical lattice on the trap potential. In the compressible state, we find that the smooth variation in the gas density around the trap center leads to softening of the singular behavior of $S(q,\ensuremath{\omega})$ at the first Lieb excitation mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response $\overline{S}(q,\ensuremath{\omega})$ remains a nonanalytic function of qqq and $\ensuremath{\omega}$ at the first Lieb excitation mode in the limit of weak trap confinement. The exponent of the power-law nonanalyticity is modified due to the inhomogeneity in a universal way and thus bears unambiguously the information about the (homogeneous) Lieb-Liniger model. A strong optical lattice causes formation of Mott phases. Deep in the Mott regime, we predict a semicircular peak in $S(q,\ensuremath{\omega})$ centered at the on-site repulsion energy, $\ensuremath{\omega}=U$. Similar peaks of smaller amplitudes exist at multiples of UUU as well. We explain the suppression of the dynamic response with entering into the Mott regime, observed recently by Cl\'ement et al. [Phys. Rev. Lett. 102, 155301 (2009)], based on an fff-sum rule for the Bose-Hubbard model.
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.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
We investigate a dilute Bose gas confined in a tight one-dimensional (1D) optical lattice plus a superimposed random potential at zero temperature. Accordingly, the ground-state energy, quantum depletion, and superfluid density are calculated. The presence of the lattice introduces a crossover to the quasi-two-dimensional (2D) regime, where we analyze asymptotically the 2D behavior of the system, particularly the effects of disorder. We thereby offer an analytical expression for the ground-state energy of a purely 2D Bose gas in a random potential. The obtained disorder-induced normal fluid density nn{n}_{n}nn​ and quantum depletion nd{n}_{d}nd​ both exhibit a characteristic 1/ln(1/n2Da2D2)1/\mathrm{ln}(1/{n}_{2\mathrm{D}}{a}_{2\mathrm{D}}^{2})1/ln(1/n2D​a2D2​) dependence. Their ratio nn/nd{n}_{n}/{n}_{d}nn​/nd​ increases to 222 compared to the familiar 4/34/34/3 in lattice-free three-dimensional (3D) geometry, signifying a more pronounced contrast between superfluidity and Bose-Einstein condensation in low dimensions. The conditions for possible experimental realization of our scenario are also proposed.
Several experimental groups are currently working towards realizing quasi-one-dimensional (1D) atom waveguides and loading them with ultracold atoms. The dynamics becomes truly 1D in a regime (Tonks gas) of low temperatures and densities and large positive scattering lengths for which the transverse mode becomes frozen, in which case the many-body Schrodinger dynamics becomes exactly soluble via a Fermi-Bose mapping theorem. In this paper we review our recent work on the exact ground state and quantum dynamics of 1D Tonks gases and assess the possibility of approaching the Tonks regime using Bessel beam optical dipole traps.
Possible signatures are discussed of a super-Tonks-Girardeau gas in bosonic systems of trapped quasi-one-dimensional dipoles at zero temperature. We provide an estimation of the frequency of the lowest compressional mode and compare it to analytical results derived using the harmonic approach in the high-density regime. We construct an exact mapping of the ground-state wave function of a one-dimensional dipolar system of bosons, fermions, and a Bose-Fermi mixture and conclude that the local properties and energy are the same at zero temperature. The question as to what extent the dipolar potential can be treated as long or short range is discussed.