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arXiv:cond-mat/0102340 · Physical Review A

Light scattering from a degenerate quasi-one-dimensional confined gas of noninteracting fermions

Patrizia Vignolo, Anna Minguzzi, M. P. Tosi

Published Jul 18, 2001• 34 citations
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Abstract

We evaluate the scattering functions of a gas of spin-polarized, noninteracting fermions confined in a quasi-one-dimensional harmonic trap at zero temperature. The main focus is on the inelastic scattering spectrum and on the angular distribution of scattered light from a mesoscopic atomic cloud as probes of its discrete quantum levels and of its shell structure in this restricted geometry. The dynamic structure factor is calculated and compared with the results of a local-density approximation exploiting the spectrum of a one-dimensional homogeneous Fermi gas. The elastic and inelastic contributions to the static structure factor are separately evaluated: the inelastic term becomes dominant as the momentum transfer increases, masking a small peak at twice the Fermi momentum in the elastic term but still reflecting the discrete quantum levels of the cloud. A fast and accurate numerical method is developed to evaluate the full pair distribution function for mesoscopic inhomogeneous clouds.

DOI
10.1103/physreva.64.023421

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We evaluate the scattering functions of a gas of spin-polarized, noninteracting fermions confined in a quasi-one-dimensional harmonic trap at zero temperature. The main focus is on the inelastic scattering spectrum and on the angular distribution of scattered light from a mesoscopic atomic cloud as probes of its discrete quantum levels and of its shell structure in this restricted geometry. The dynamic structure factor is calculated and compared with the results of a local-density approximation exploiting the spectrum of a one-dimensional homogeneous Fermi gas. The elastic and inelastic contributions to the static structure factor are separately evaluated: the inelastic term becomes dominant as the momentum transfer increases, masking a small peak at twice the Fermi momentum in the elastic term but still reflecting the discrete quantum levels of the cloud. A fast and accurate numerical method is developed to evaluate the full pair distribution function for mesoscopic inhomogeneous clouds.
We develop a Green's function method to evaluate the exact equilibrium particle-density profiles of noninteracting Fermi gases in external harmonic confinement in any spatial dimension and for arbitrary trap anisotropy. While in a spherically symmetric configuration the shell effects are negligible in the case of a large number of particles, we find that for very anisotropic traps the quantum effects due to single-level occupancy and the deviations from the Thomas-Fermi approximation are also visible for mesoscopic clouds.
We extend to finite temperature a Green's-function method that was previously proposed to evaluate ground-state properties of mesoscopic clouds of noninteracting fermions moving under harmonic confinement in one dimension. By calculations of the particle and kinetic-energy density profiles, we illustrate the role of thermal excitations in smoothing out the quantum shell structure of the cloud and in spreading the particle spill out from quantum tunnel at the edges. We also discuss the approach of the exact density profiles to the predictions of a semiclassical model often used in the theory of confined atomic gases at finite temperature.
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.
We present a detailed theoretical study of light scattering off superfluid trapped Fermi gas of atoms at zero temperature. We apply Nambu-Gorkov formalism of superconductivity to calculate the response function of superfluid gas due to stimulated light scattering taking into account the final state interactions. The polarization of light has been shown to play a significant role in response of Cooper-pairs in the presence of a magnetic field. Particularly important is a scheme of polarization-selective light scattering by either spin-component of the Cooper-pairs leading to the single-particle excitations of one spin-component only. These excitations have a threshold of 2Δ2Δ2Δ where ΔΔΔ is the superfluid gap energy. Furthermore, polarization-selective light scattering allows for unequal energy and momentum transfer to the two partner atoms of a Cooper-pair. In the regime of low energy (<<2Δ<< 2Δ<<2Δ) and low momentum (<2Δ/(ℏvF)<2Δ/(\hbar v_F)<2Δ/(ℏvF​), vFv_FvF​ being the Fermi velocity) transfer, a small difference in momentum transfers to the two spin-components may be useful in exciting Bogoliubov-Anderson phonon mode. We present detailed results on the dynamic structure factor (DSF) deduced from the response function making use of generalized fluctuation-dissipation theorem. Model calculations using local density approximation for trapped superfluid Fermi gas shows that when the energy transfer is less than 2Δ02Δ_02Δ0​, where Δ0Δ_0Δ0​ refers to the gap at the trap center, DSF as a function of energy transfer has reduced gradient compared to that of normal Fermi gas.
We study stimulated scattering of polarized light in a two-component Fermi gas of atoms at zero temperature. Within the framework of Nambu-Gorkov formalism, we calculate the response function of superfluid gas taking into account the final state interactions. The dynamic structure factor deduced from the response function provides information about the pairing gap and the momentum distributions of atoms. Model calculations using local density approximation indicates that the pairing gap of trapped Fermi gas may be detectable by Bragg spectroscopy due to stimulated scattering.
Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms.