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arXiv:1009.5044 · Physical Review A

Effective super Tonks-Girardeau gases as ground states of strongly attractive multicomponent fermions

Xiangguo Yin, Xi‐Wen Guan, Murray T. Batchelor, Shu Chen

Published Jan 10, 2011• 15 citations
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Abstract

In the strong interaction limit, attractive fermions with $N$-component hyperfine states in a one-dimensional waveguide form unbreakable bound cluster states. We demonstrate that the ground state of strongly attractive SU($N$) Fermi gases can be effectively described by a super Tonks-Girardeau gaslike state composed of bosonic cluster states with strongly attractive cluster-cluster interaction for even $N$ and a Fermi duality of a super Tonks-Girardeau gaslike state composed of fermionic cluster states with weakly interacting cluster-cluster $p$-wave interaction for odd $N$.

DOI
10.1103/physreva.83.013602

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In the strong interaction limit, attractive fermions with NNN-component hyperfine states in a one-dimensional waveguide form unbreakable bound cluster states. We demonstrate that the ground state of strongly attractive SU(NNN) Fermi gases can be effectively described by a super Tonks-Girardeau gaslike state composed of bosonic cluster states with strongly attractive cluster-cluster interaction for even NNN and a Fermi duality of a super Tonks-Girardeau gaslike state composed of fermionic cluster states with weakly interacting cluster-cluster ppp-wave interaction for odd NNN.
Fermi gases confined in tight one-dimensional waveguides form two-particle bound states of atoms in the presence of a strongly attractive interaction. Based on the exact solution of the one-dimensional spin-1/2 interacting Fermi gas, we demonstrate that a stable excited state with no pairing between attractive fermionic atoms can be realized by a sudden switch of interaction from the strongly repulsive regime to strongly attractive regime. Such a state is an exact fermionic analog of the experimentally observed super-Tonks-Girardeau state of bosonic Cesium atoms [Science 325, 1224 (2009)] and should be possible to be observed by the experiment. The frequency of the lowest breathing mode of the fermionic super-Tonks-Girardeau gas is calculated as a function of the interaction strength, which could be used as a detectable signature for the experimental observation.
A significant feature of the one-dimensional super Tonks-Girardeau gas is its metastable gas-like state with a stronger Fermi-like pressure than for free fermions which prevents a collapse of atoms. This naturally suggests a way to search for such strongly correlated behavior in systems of interacting fermions in one dimension. We thus show that the strongly attractive Fermi gas without polarization can be effectively described by a super Tonks-Girardeau gas composed of bosonic Fermi pairs with attractive pair-pair interaction. A natural description of such super Tonks-Girardeau gases is provided by Haldane generalized exclusion statistics. In particular, they are equivalent to ideal particles obeying more exclusive statistics than Fermi-Dirac statistics.
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.
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 provide evidence in support of a recent proposal by Astrakharchik et al for the existence of a super Tonks–Girardeau gas-like state in the attractive interaction regime of quasi-one-dimensional Bose gases. We show that the super Tonks–Girardeau gas-like state corresponds to a highly excited Bethe state in the integrable interacting Bose gas for which the bosons acquire hard-core behaviour. The gas-like state properties vary smoothly throughout a wide range from strong repulsion to strong attraction. There is an additional stable gas-like phase in this regime in which the bosons form two-body bound states behaving like hard-core bosons.
A Fermi-Bose mapping method is used to determine the exact ground states of several models of mixtures of strongly interacting ultracold gases in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). We detail the case of a Bose-Fermi mixture with TG boson-boson (BB) and boson-fermion (BF) interactions. Exact results are given for density profiles in a harmonic trap, single-particle density matrices, momentum distributions, and density-density correlations. Since the ground state is highly degenerate, we analyze the splitting of the ground manifold for large but finite BB and BF repulsions.