[Paper Review] Breakdown of the Thomas Fermi approximation for polarized Fermi gases

Rajdeep Sensarma, W. R. Schneider+2 more|ArXiv.org|Jun 12, 2007

Cold Atom Physics and Bose-Einstein Condensates2 references3 citations

TL;DR

This paper demonstrates that the Thomas-Fermi approximation (TFA) breaks down in polarized unitary Fermi gases trapped in anisotropic harmonic potentials, revealing a magnetized superfluid region with oscillating order parameter—unstable in bulk—leading to magnetization profiles that deviate from equipotential contours. The breakdown is more pronounced at high trap anisotropy and low particle number, explaining discrepancies between MIT and Rice experiments.

ABSTRACT

We use Bogoliubov de-Gennes theory to show that the commonly used Thomas-Fermi approximation (TFA) can fail in describing polarized unitary gases in anisotropic harmonic traps. We find a magnetized superfluid region inside the trap, with order parameter oscillations, even though there is no such stable bulk phase. This leads to magnetization profiles that deviate from contours of constant potential energy. We determine how this violation scales with trap anisotropy and number of particles, and show that we are able to account for important differences between the MIT and Rice experiments.

Motivation & Objective

• To investigate the validity of the Thomas-Fermi approximation (TFA) in describing polarized unitary Fermi gases under anisotropic harmonic traps.
• To resolve the discrepancy between experimental observations: MIT finds TFA holds, while Rice observes strong violations of equipotential contour conditions.
• To identify the conditions under which TFA fails due to spatially inhomogeneous superfluid order parameter and magnetization profiles.
• To derive a scaling criterion, δμ/ω₀ ∝ (Nα)¹ᐟ³f(P), for TFA consistency and relate it to experimental parameters.
• To provide a theoretical explanation for the observed magnetization profile differences between MIT and Rice experiments using Bogoliubov-deGennes (BdG) calculations.

Proposed method

• Solving the zero-temperature Bogoliubov-deGennes (BdG) equations self-consistently for a polarized Fermi gas in a 3D anisotropic harmonic trap.
• Using a single-channel Hamiltonian with contact interaction and a trapping potential V(r) = ½mω₀²(r² + α²z²), with α controlling radial anisotropy.
• Computing the order parameter Δ(r), local magnetization m(r) = n↑(r) - n↓(r), and density profiles from the BdG quasiparticle wavefunctions.
• Applying the gap equation Δ(r) = g∑Ei>h u_i(r)v_i*(r) and particle number constraints to ensure consistency with total N and polarization P.
• Deriving a scaling relation δμ/ω₀ = (Nα)¹ᐟ³f(P) from TFA estimates to quantify TFA breakdown conditions.
• Comparing column-integrated magnetization profiles with experimental data to assess deviations from equipotential contours.

Experimental results

Research questions

• RQ1 Why do experimental observations of magnetization profiles in polarized unitary Fermi gases differ between MIT and Rice, despite similar interaction strengths?
• RQ2 In what parameter regimes does the Thomas-Fermi approximation fail for polarized Fermi gases in anisotropic traps?
• RQ3 How does the size of the intermediate magnetized superfluid region depend on trap anisotropy and particle number?
• RQ4 What is the scaling behavior of the TFA breakdown with respect to N and α, and how does it explain experimental differences?
• RQ5 Can the observed violation of equipotential contour conditions in magnetization profiles be quantitatively linked to the presence of FFLO-like oscillations?

Key findings

• The Thomas-Fermi approximation breaks down due to the emergence of a magnetized superfluid region with oscillating order parameter, even though no such stable phase exists in the uniform system.
• The magnetized superfluid region grows significantly with increasing trap anisotropy (1/α) and is larger than kF⁻¹ for strong anisotropy.
• Violation of the equipotential contour condition for magnetization increases with anisotropy but decreases with increasing total particle number N.
• The TFA consistency condition δμ/ω₀ ≫ 1 is best captured by the scaling δμ/ω₀ ∝ (Nα)¹ᐟ³f(P), with f(P) increasing with polarization P.
• The MIT experiment (N ≈ 10⁷, α ≈ 1, (Nα)¹ᐟ³ ≈ 100) observes TFA validity due to large (Nα)¹ᐟ³, while the Rice experiment (N ≈ 10⁵, α ≈ 1/50, (Nα)¹ᐟ³ ≈ 10) shows strong violations.
• Column-integrated magnetization profiles show a transition from axial localization (small N) to elliptical, equipotential-like shapes (large N), matching experimental trends.