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arXiv:cond-mat/0606043 · Physical Review A

Zero-temperature thermodynamics of asymmetric Fermi gases at unitarity

Aurel Bulgac, Michael McNeil Forbes

Published Mar 27, 2007• 84 citations
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Abstract

The equation of state of a dilute two-component asymmetric Fermi gas at unitarity is subject to strong constraints, which affect the spatial density profiles in atomic traps. These constraints require the existence of at least one nontrivial partially polarized (asymmetric) phase. We determine the relation between the structure of the spatial density profiles and the $T=0$ equation of state, based on the most accurate theoretical predictions available. We also show how the equation of state can be determined from experimental observations.

DOI
10.1103/physreva.75.031605

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The equation of state of a dilute two-component asymmetric Fermi gas at unitarity is subject to strong constraints, which affect the spatial density profiles in atomic traps. These constraints require the existence of at least one nontrivial partially polarized (asymmetric) phase. We determine the relation between the structure of the spatial density profiles and the T=0T=0T=0 equation of state, based on the most accurate theoretical predictions available. We also show how the equation of state can be determined from experimental observations.
We report on the measurement of the equation of state of a two-component Fermi gas of 6Li^{6}\mathrm{Li}6Li atoms with resonant interactions. By analyzing the in situ density distributions of a population-imbalanced Fermi mixture reported in a recent experiment [Y. Shin et al., Nature 451, 689 (2008)], we determine the energy density of a resonantly interacting Fermi gas as a function of the densities of the two components. We present a method to determine the equation of state directly from the shape of the trapped cloud, where the fully polarized, noninteracting ideal Fermi gas in the outer region provides the absolute calibration of particle density. From the density profiles obtained at the lowest temperature, we estimate the zero-temperature equation of state.
We investigate the phase diagram of asymmetric two-component Fermi gases at zero temperature as a function of polarization and interaction strength. The equations of state of the uniform superfluid and normal phase are determined using quantum Monte Carlo simulations. We find three different mixed states, where the superfluid and the normal phase coexist in equilibrium, corresponding to phase separation between (a) the polarized superfluid and the fully polarized normal gas, (b) the polarized superfluid and the partially polarized normal gas, and (c) the unpolarized superfluid and the partially polarized normal gas.
We study the zero temperature properties of a trapped polarized Fermi gas at unitarity by assuming phase separation between an unpolarized superfluid and a polarized normal phase. The effects of the interaction are accounted for using the formalism of quasiparticles to build up the equation of state of the normal phase with the Monte Carlo results for the relevant parameters. Our predictions for the Chandrasekhar-Clogston limit of critical polarization and for the density profiles, including the density jump at the interface, are confirmed with excellent accuracy by the recent experimental results at MIT. The role of interaction on the radial width of the minority component, on the gap of spectral functions, and on the spin oscillations in the normal phase is also discussed. Our analysis points out the Fermi liquid nature of these strongly interacting spin-polarized configurations.
Strongly interacting, dilute Fermi gases exhibit a scale-invariant, universal thermodynamic behavior. This is notoriously difficult to understand theoretically because of the absence of a small interaction parameter. Here, we present a systematic comparison of theoretical predictions from different quantum many-body theories with recent experimental data of Nascimbne et al (2010 Nature 463 1057). Our comparisons have no adjustable parameters, either theoretically or experimentally. All the model approximations seem to fluctuate rather than converge on the experimental data. It turns out that a simple Gaussian pair fluctuation theory gives the best quantitative agreement, except at the critical superfluid transition region. In the normal state, we also calculate the equation of state by using a quantum cluster expansion theory and explore in detail its applicability to low temperatures. Using the accurate experimental result for the thermodynamic function S ( T ), we determine the temperature T of a trapped Fermi gas at unitarity as a function of a non-interacting temperature T i , which can be obtained by an adiabatic sweep to the free gas limit. By analyzing the recent experimental data, we find a characteristic temperature ( T / T F ) 0 =0.19±0.02 or ( T i / T F ) 0 =0.16±0.02 in a harmonic trap, below which there are deviations from normal Fermi-liquid-like behavior that may be attributed to pairing effects. Here, T F is the Fermi temperature for a trapped ideal, non-interacting Fermi gas. Our thorough comparison may shed light on the further theoretical development of strongly interacting fermions.
The polarization produced by the relative displacement of the potentials trapping two spin species of a dilute Fermi gas with N=N is calculated at unitarity by assuming phase separation between the superfluid and a polarized phase at zero temperature. Because of the energy cost associated with pair breaking, the dipole polarizability is strongly quenched and exhibits important deviations from the ideal gas behavior even for nonlinear displacements of the order of the size of the atomic cloud. The behavior in the presence of different trapping frequencies (monopole polarization) for the two spin species is also discussed. Our results suggest new experimental perspectives to explore the quantum phases of interacting Fermi gases.
We calculate the mean-field thermodynamics of a spherically trapped Fermi gas with unequal spin populations in the unitarity limit, comparing results from the Bogoliubov--de Gennes equations and the local density approximation. We follow the usual mean-field decoupling in deriving the Bogoliubov--de Gennes equations and set up an efficient and accurate method for solving these equations. In the local density approximation we consider locally homogeneous solutions, with a slowly varying order parameter. With a large particle number these two approximation schemes give rise to essentially the same results for various thermodynamic quantities, including the density profiles. This excellent agreement strongly indicates that the small oscillation of order parameters near the edge of trap, sometimes interpreted as spatially inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov states in previous studies of Bogoliubov--de Gennes equations, is a finite-size effect. We find that a bimodal structure emerges in the density profile of the minority-spin state at finite temperature, as observed in experiments. The superfluid transition temperature as a function of the population imbalance is determined and is shown to be consistent with recent experimental measurements. The temperature dependence of the equation of state is discussed.