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arXiv:1004.2925 · Physical Review A

Two super-Tonks-Girardeau states of a trapped one-dimensional spinor Fermi gas

Marvin D. Girardeau

Published Jul 20, 2010• 63 citations
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Abstract

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}$$1$, contrary to a previous suggestion.

DOI
10.1103/physreva.82.011607

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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↑\ensuremath↓)(\ensuremath{\uparrow}\ensuremath{\downarrow})(\ensuremath↑\ensuremath↓) attractions and an ideal Fermi gas with no (\ensuremath↑\ensuremath↑)(\ensuremath{\uparrow}\ensuremath{\uparrow})(\ensuremath↑\ensuremath↑) or (\ensuremath↓\ensuremath↓)(\ensuremath{\downarrow}\ensuremath{\downarrow})(\ensuremath↓\ensuremath↓) 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↑\ensuremath↓)(\ensuremath{\uparrow}\ensuremath{\downarrow})(\ensuremath↑\ensuremath↓) 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↑\ensuremath↓)(\ensuremath{\uparrow}\ensuremath{\downarrow})(\ensuremath↑\ensuremath↓) 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}$$1, 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.
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.
Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length a1D{a}_{1\mathrm{D}}a1D​ of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant gB{g}_{B}gB​ in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for N=2N=2N=2 particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region ∣x12∣<xnode|{x}_{12}|<{x}_{\mathrm{node}}∣x12​∣<xnode​, whereas for a HSB gas with hard-sphere diameter ahs=xnode{a}_{\mathrm{hs}}={x}_{\mathrm{node}}ahs​=xnode​, the HSB wave function vanishes when ∣x12∣<ahs|{x}_{12}|<{a}_{\mathrm{hs}}∣x12​∣<ahs​. Arguments are given suggesting that the same theorem holds also for N>2N>2N>2. The STG and HSB wave functions for N=2N=2N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$, xnode{x}_{\mathrm{node}}xnode​ is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.
The ground state of a one-dimensional (1D) quantum gas of dipoles oriented perpendicular to the longitudinal axis, with a strong 1/x(3) repulsive potential, is studied at low 1D densities n. Near contact the dependence of the many-body wave function on the separation x(jℓ) of two particles reduces to a two-body wave function Ψ(rel)(x(jℓ)). Immediately after a sudden rotation of the dipoles so that they are parallel to the longitudinal axis, this wave function will still be that of the repulsive potential, but since the potential is now that of the attractive potential, it will not be stationary. It is shown that as nd(2)→0 the rate of change of this wave function approaches zero. It follows that for small values of nd(2), this state is metastable and is an analog of the super Tonks-Girardeau state of bosons with a strong zero-range attraction. The dipolar system is equivalent to a spinor Fermi gas with spin z components σ(↑)=[perpendicular] (perpendicular to the longitudinal axis) and σ(↓)=[parallel] (parallel to the longitudinal axis). A Fermi-Fermi mapping from spinor to spinless Fermi gas followed by the standard 1960 Fermi-Bose mapping reduces the Fermi system to a Bose gas. Potential experiments realizing the sudden spin rotation with ultracold dipolar gases are discussed, and a few salient properties of these states are accurately evaluated by a Monte Carlo method.
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.
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.