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Science

Observation of a One-Dimensional Tonks-Girardeau Gas

Toshiya Kinoshita, Trevor Wenger, David S. Weiss

Published Jul 29, 2004• 1,413 citations
Publisher

Abstract

We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.

DOI
10.1126/science.1100700

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We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.
We investigate the Bose gas with repulsive or attractive interactions between atoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typical quantum phases in the current research of 1D interacting cold atoms are clarified in terms of the energy spectrum, single particle density distribution and two-particle correlation function. We identify two matching points in the phase diagram, i.e. the TG and STG gas show the same profiles at the strongly interacting point −1/γ=0-1/γ=0−1/γ=0, and in the weakly interacting limit γ=0γ=0γ=0 the ground states TG and BS join to each other smoothly.
A harmonically trapped, ultracold, one-dimensional (1D) spin-1 Bose gas with strongly repulsive or attractive 1D even-wave interactions induced by a three-dimensional (3D) Feshbach resonance is studied. The exact ground state, a hybrid of Tonks-Girardeau (TG) and ideal Fermi gases, is constructed in the TG limit of infinite even-wave repulsion by a spinor Fermi-Bose mapping to a spinless ideal Fermi gas. It is then shown that in the limit of infinite even-wave attraction this same state remains an exact many-body eigenstate, now highly excited relative to the collapsed generalized McGuire-cluster ground state, showing that the hybrid TG state is completely stable against collapse to this cluster ground state under a sudden switch from infinite repulsion to infinite attraction. It is shown to be the TG limit of a hybrid super-Tonks-Girardeau (STG) state, which is metastable under a sudden switch from finite but very strong repulsion to finite but very strong attraction. It should be possible to create it experimentally by a sudden switch from strongly repulsive to strongly attractive interaction, as in the recent Innsbruck experiment on a spin-polarized bosonic STG gas. In the case of strong attraction, there should also exist another STG state of much lower energy, consisting of strongly bound dimers, a bosonic analog of a recently predicted STG state which is an ultracold gas of strongly bound bosonic dimers of fermionic atoms, but it is shown that this STG state cannot be created by such a switch from strong repulsion to strong attraction.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
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.
A harmonically trapped ultracold 1D spinor Fermi gas with a strongly attractive 1D even-wave interaction induced by a 3D Feshbach resonance is studied. It is shown that it has two different super Tonks-Girardeau (sTG) energy eigenstates which are metastable against collapse in spite of the strong attraction, due to their close connection with 1D hard sphere Bose gases which are highly excited gaslike states. One of these sTG states is a hybrid between an sTG gas with strong $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ attractions and an ideal Fermi gas with no $(\ensuremath{\uparrow}\ensuremath{\uparrow})$ or $(\ensuremath{\downarrow}\ensuremath{\downarrow})$ interactions, the sTG component being an exact analog of the recently observed sTG state of a 1D ultracold Bose gas. It should be possible to create it experimentally by a sudden switch of the $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ interaction from strongly repulsive to strongly attractive, as in the recent Innsbruck experiment on the bosonic sTG gas. The other is a trapped analog of a recently predicted sTG state which is an ultracold gas of strongly bound $(\ensuremath{\uparrow}\ensuremath{\downarrow})$ fermion dimers which behave as bosons with a strongly attractive boson-boson interaction leading to sTG behavior. It is proved that the probability of a transition from the ground state for strongly repulsive interaction to this dimer state under a sudden switch from strongly repulsive to strongly attractive interaction is $\ensuremath{\ll}$111, contrary to a previous suggestion.
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.