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تسجيل مستخدم جديدTonks-Girardeau and Super Tonks-Girardeau States of a Trapped 1D Spinor Bose Gas
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M.D. Girardeau
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A harmonically trapped ultracold 1D spin-1 Bose gas with strongly repulsive or attractive 1D even-wave interactions induced by a 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 gas 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.
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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 met
astable against collapse in spite of the strong attraction, due to their close connection with 1D hard sphere Bose gases which are highly excited gas-like states. One of these sTG states is a hybrid between an sTG gas with strong $(uparrowdownarrow$ attractions and an ideal Fermi gas with no $(uparrowuparrow)$ or $(downarrowdownarrow)$ 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 $(uparrowdownarrow)$ 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 $(uparrowdownarrow)$ 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 $ll 1$, contrary to a previous suggestion.
Recent theoretical and experimental results demonstrate a close connection between the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gas with hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ of the sTG gas, a h
ighly excited gas-like state with nodes only at interparticle separations $|x_{jell}|=x_{node}approx a_{1D}$. It is shown herein that when the coupling constant $g_B$ in the Lieb-Liniger interaction $g_Bdelta(x_{jell})$ is negative and $|x_{12}|ge x_{node}$, the sTG and HSB wave functions for $N=2$ 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 $|x_{12}|<x_{node}$, whereas for a HSB gas with hard sphere diameter $a_{h.s.}=x_{node}$, the HSB wave function vanishes when all $|x_{12}|<a_{h.s.}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The sTG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for $Nge 2$, $x_{node}$ 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.
We study the local correlations in the super Tonks-Girardeau gas, a highly excited, strongly correlated state obtained in quasi one-dimensional Bose gases by tuning the scattering length to large negative values using a confinement-induced resonance.
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