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arXiv:0707.1884

Soluble Models of Strongly Interacting Two-level Ultracold Gases in Tight Waveguides with Coupling to the Quantized Electromagnetic Field

Marvin D. Girardeau

Published Jul 12, 2007• 2 citations
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Abstract

A generalized Fermi-Bose mapping method is used to determine the exact ground states of six models of strongly interacting ultracold gases of two-level atoms 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). Three of these models exhibit a quantum phase transition in the presence of an external magnetic field, associated with a cooperative ground state rearrangement wherein Fermi energy is traded for internal excitation energy. After investigation of these models in the absence of an electromagnetic field, one is generalized to include resonant interactions with a single photon mode, leading to a possible thermal phase transition associated with Dicke superradiance.

DOI
10.48550/arxiv.0707.1884

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A generalized Fermi-Bose mapping method is used to determine the exact ground states of six models of strongly interacting ultracold gases of two-level atoms 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). Three of these models exhibit a quantum phase transition in the presence of an external magnetic field, associated with a cooperative ground state rearrangement wherein Fermi energy is traded for internal excitation energy. After investigation of these models in the absence of an electromagnetic field, one is generalized to include resonant interactions with a single photon mode, leading to a possible thermal phase transition associated with Dicke superradiance.
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.
The Fermi-Bose mapping method for one-dimensional Bose and Fermi gases with zero-range interactions is generalized to an anyon-fermion mapping and applied to exact solution of several models of ultracold gases with anyonic exchange symmetry in tight waveguides: anyonic Calogero-Sutherland model, anyons with point hard-core interaction (anyonic Tonks-Girardeau gas), and spin-aligned anyon gas with infinite zero-range odd-wave attractions (attractive anyonic Tonks-Girardeau, or AATG, gas). It is proved that for even N>or=4 there are states of the AATG gas on a ring, with anyonic phase slips which are odd integral multiples of pi/(N-1), of energy lower than that of the corresponding fermionic ground state. A generalization to a spinor Fermi gas state with anyonic symmetry under purely spatial exchange enables energy lowering by the same mechanism.
The two- and three-body correlation functions of the ground state of an optically trapped ultracold spin-12\frac{1}{2}21​ Fermi gas (SFG) in a tight waveguide [one-dimensional (1D) regime] are calculated in the plane of even- and odd-wave coupling constants, assuming a 1D attractive zero-range odd-wave interaction induced by a 3D ppp-wave Feshbach resonance, as well as the usual repulsive zero-range even-wave interaction stemming from 3D sss-wave scattering. The calculations are based on the exact mapping from the SFG to a ``Lieb-Liniger-Heisenberg'' model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, and indicate that the SFG should be stable against three-body recombination in a large region of the coupling constant plane encompassing parts of both the ferromagnetic and antiferromagnetic phases. However, the limiting case of the fermionic Tonks-Girardeau gas, a spin-aligned 1D Fermi gas with infinitely attractive ppp-wave interactions, is unstable in this sense. Effects due to the dipolar interaction and a Zeeman term due to a resonance-generating magnetic field do not lead to shrinkage of the region of stability of the SFG.
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.
A K-matrix for waveguide confined spin-polarized fermionic atoms recently computed by Granger and Blume is identified, in the low-energy domain, with a contact condition for one-dimensional (1D) spinless fermions. Difficulties in consistently formulating the contact conditions in terms of interaction potentials are discussed and a rigorous alternative variational reformulation is constructed. A duality between 1D fermions and bosons with zero-range interactions suggested by Cheon and Shigehara is shown to hold for the effective 1D dynamics of a spin-polarized Fermi gas with 3D p-wave interactions and that of a Bose gas with 3D s-wave interactions in a tight waveguide. This generalizes the mapping from impenetrable bosons (TG gas) to free fermions and is used to derive the equation of state of an ultracold spin-polarized fermionic vapor in a tight waveguide. Near a 1D confinement-induced resonance one has a "fermionic TG gas" which maps to an ideal Bose gas.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.