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arXiv:cond-mat/0208376 · Journal of Physics B Atomic Molecular and Optical Physics

Coupling constants for a degenerate Fermi gas confined to a quasi-one-dimensional harmonic trap

F. Gleisberg, W. Wonneberger

Published Mar 24, 2004• 8 citations
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Abstract

A theory for the coupling constants between spin-polarized degenerate fermionic atoms is developed for the case of atoms confined to a quasi-one-dimensional harmonic trap. Exploiting the presence of the Fermi edge and the large number of fermions, it is shown that the resulting effective one-dimensional interaction can be parametrized by three sets of interaction coefficients, two for forward scattering and one for backward scattering. In the case of identical fermions, backscattering dominates, because the contact part of the effective one-dimensional interaction must be subtracted. Analytic expressions for the interaction coefficients for the effective contact interaction, which is relevant for the inter-component interactions, and for the 'p-wave' interaction appearing in next order, are given. As an example, we calculate and discuss in detail the effective coupling constants for the marginally long ranged dipole–dipole interaction.

DOI
10.1088/0953-4075/37/7/053

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A theory for the coupling constants between spin-polarized degenerate fermionic atoms is developed for the case of atoms confined to a quasi-one-dimensional harmonic trap. Exploiting the presence of the Fermi edge and the large number of fermions, it is shown that the resulting effective one-dimensional interaction can be parametrized by three sets of interaction coefficients, two for forward scattering and one for backward scattering. In the case of identical fermions, backscattering dominates, because the contact part of the effective one-dimensional interaction must be subtracted. Analytic expressions for the interaction coefficients for the effective contact interaction, which is relevant for the inter-component interactions, and for the 'p-wave' interaction appearing in next order, are given. As an example, we calculate and discuss in detail the effective coupling constants for the marginally long ranged dipole–dipole interaction.
An asymptotically exact many-body theory for spin-polarized interacting fermions in a one-dimensional harmonic atom trap is developed using the bosonization method and including backward scattering. In contrast to the Luttinger model, backscattering in the trap generates one-particle potentials which must be diagonalized simultaneously with the two-body interactions. Inclusion of backscattering becomes necessary because backscattering is a dominant interaction process between confined identical one-dimensional fermions. The bosonization method is applied to the calculation of one-particle matrix elements at zero temperature. A detailed discussion of the validity of the results from bosonization is given, including a comparison with direct numerical diagonalization in fermionic Hilbert space. A model for the interaction coefficients is developed along the lines of the Luttinger model with only one coupling constant K. With these results, particle densities, the Wigner function, and the central pair-correlation function are calculated and displayed for large fermion numbers. It is shown how interactions modify these quantities. The anomalous dimension of the pair-correlation function in the center of the trap is also discussed and found to be in accord with the Luttinger model.
A many-body theory for a two-component system of spin-polarized interacting fermions in a one-dimensional harmonic trap is developed. The model considers two different states of the same fermionic species and treats the dominant interactions between the two using the bosonization method for forward scattering. Asymptotically exact results for the one-particle matrix elements at zero temperature are given. Using them, occupation probabilities of oscillator states are discussed. Particle and momentum densities are calculated and displayed. It is demonstrated how interactions modify all these quantities. An asymptotic connection with Luttinger liquids is suggested. The relation of the coupling constant of the theory to the dipole-dipole interaction is also discussed.
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We employ ab initio methods of quantum chemistry to investigate spin-1/2 fermions interacting via a two-body contact potential in a one-dimensional harmonic trap. The convergence of the total energy with the size of the one-particle basis set is analytically investigated for the two-body problem and the same form of the convergence formula is numerically confirmed to be valid for the many-body case. Benchmark calculations for two to six fermions with the full configuration interaction method equivalent to the exact diagonalization approach, and the coupled cluster (CC) method including single, double, triple, and quadruple excitations are presented. The convergence of the correlation energy with the level of excitations included in the CC model is analyzed. The range of the interaction strength for which single-reference CC methods work is examined. Next, the CC method restricted to single, double, and noniterative triple excitations, CCSD(T), is employed to study a two-component Fermi gas composed of 6–80 atoms in a one-dimensional harmonic trap. The density profiles of trapped atomic clouds are also reported. Finally, a comparison with experimental results for few-fermion systems is presented. Upcoming possible applications and extensions of the presented approach are discussed.
Summary form only given.The recent success of atomic Bose-Einstein condensation (BEG) experiments has opened up new possibilities to probe statistical effects of trapped atoms. While most work so far has focused on bosonic atoms, not much attention has been paid to statistical signatures of fermions. A distinctive feature of a degenerate Fermi system is the so-called Fermi-edge singularity which appears as a consequence of interactions and has profound effects on transport and optical properties of the system. We analyze a quasi one-dimensional trap coupled to a single excited state via resonant dipole-dipole interactions.
The main topic of these lectures is the physics of quantum degenerate atomic Fermi gases both under uniform confinement and in harmonic traps. To begin with, a general introduction on ideal Fermi gases is proposed. This "academic" limit is relevant in experiments where ultracold fermionic atoms are all polarized in a single spin state: the low atom momenta together with the fermionic nature of the atoms forbid atomic s-wave scattering and the atoms can be considered as free. Interactions are then introduced for the case of a gas with two spin components. Interactions can be tuned via Feshbach resonances and control the microscopic features of the superfluid phase, ranging from the BCS limit (formation of Cooper pairs) to the BEC limit (Bose-Einstein condensation of dimers). Various theoretical approaches for the description of the BCS-BEC crossover and their predictions in comparison with the experimental results are discussed.