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arXiv:cond-mat/0401402 · Physical Review A

Theory of spinor Fermi and Bose gases in tight atom waveguides

Marvin D. Girardeau, Maxim Olshanii

Published Aug 23, 2004• 81 citations
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Abstract

Divergence-free pseudo-potentials for spatially even- and odd-wave interactions in spinor Fermi gases in tight atom waveguides are derived. The Fermi-Bose mapping method is used to relate the effectively one-dimensional fermionic many-body problem to that of a spinor Bose gas. Depending on the relative magnitudes of the even- and odd-wave interactions, the $N$-atom ground state may have total spin $S=0$, $S=N∕2$, and possibly also intermediate values, the case $S=N∕2$ applying near a $p$-wave Feshbach resonance, where the $N$-fermion ground state is space-antisymmetric and spin-symmetric. In this case the fermionic ground state maps to the spinless bosonic Lieb-Liniger gas. An external magnetic field with a longitudinal gradient causes a Stern-Gerlach spatial separation of the corresponding trapped Fermi gas with respect to various values of ${S}_{z}$.

DOI
10.1103/physreva.70.023608

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•Feb 5, 2019•0 citations
Divergence-free pseudo-potentials for spatially even- and odd-wave interactions in spinor Fermi gases in tight atom waveguides are derived. The Fermi-Bose mapping method is used to relate the effectively one-dimensional fermionic many-body problem to that of a spinor Bose gas. Depending on the relative magnitudes of the even- and odd-wave interactions, the NNN-atom ground state may have total spin S=0S=0S=0, S=N∕2S=N∕2S=N∕2, and possibly also intermediate values, the case S=N∕2S=N∕2S=N∕2 applying near a ppp-wave Feshbach resonance, where the NNN-fermion ground state is space-antisymmetric and spin-symmetric. In this case the fermionic ground state maps to the spinless bosonic Lieb-Liniger gas. An external magnetic field with a longitudinal gradient causes a Stern-Gerlach spatial separation of the corresponding trapped Fermi gas with respect to various values of Sz{S}_{z}Sz​.
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 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.
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 ground and excited states of a one-dimensional (1D) spin-1/2 Fermi gas (SFG) with both attractive zero-range odd-wave interactions and repulsive zero-range even-wave interactions are mapped exactly to a 1D Lieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, such that the complete SFG and LLH energy spectra are identical. The ground state in the ferromagnetic phase is given exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in the antiferromagnetic phase by a variational method combining Bethe ansatz solutions of the LL and 1D Heisenberg models. There are excitation branches corresponding to LL type I and II phonons and spin waves, the latter behaving quadratically for small wave numbers in the ferromagnetic phase and linearly in the antiferromagnetic phase.
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.
Quantum many-body physics has been studied for many decades. A lot of intriguing phenomena have been observed and theories have been developed. Yet many problems remain unsolved, which is largely due to a lack of general efficient classical computation method. One-dimensional (1D) systems have also drawn much attention over the past few decades. One reason is that many unique strongly correlated quantum phenomena only appear in low dimensions. Another reason is that many exact results can be obtained in 1D for cross benchmarking, such as Bethe Ansatz, Bosonization, Bose-Fermi mapping, etc. In addition, many numerical methods, such as Matrix-Product-State based and Monte Carlo methods, work most efficiently in 1D. In recent years, 1D quantum gases have been realized in many cold atom labs, providing experimental motivation for their studies. One of the most mysterious assumptions about a system of identical quantum particles is that the wavefunction must be symmetric (for bosons) or anti-symmetric (for fermions). This means that no two fermions can occupy the same state, while multiple occupancy is allowed for bosons. Bosons and fermions are therefore generally very different. But in 1D this distinction could become rather subtle, provided that the bosonic multiple occupancy is suppressed, which can happen if strong repulsion exists between them. However, when considering spin degrees of freedom, things become more complicated. In this work, we develop a generalized Bose-Fermi mapping theory, under which, the charge degrees of freedom is mapped to a spinless fermions, while the spin degrees of freedom to an effective spin chain. This mapping works for arbitrary spin, arbitrary trapping potentials, arbitrary spin-dependent interaction potential, and for either spinor bosons or fermions. In the strong interaction limit, the wavefunction of the system is represented by the strong coupling ansatz wavefunction (SCAW), which takes the form of a direct product of spinless fermions wavefunction and spin wavefunction. Using this mapping technique, we study the dynamics and collective modes of the system. Furthermore, we develop a very efficient method to calculate the one-body density matrix, from which we can calculate the momentum distribution of the system.