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arXiv:0801.1523 · Physical Review A

Determination of the equation of state of a polarized Fermi gas at unitarity

Yong-il Shin

Published Apr 10, 2008• 66 citations
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Abstract

We report on the measurement of the equation of state of a two-component Fermi gas of $^{6}\mathrm{Li}$ 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.

DOI
10.1103/physreva.77.041603

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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.
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.
On the basis of the phase diagram of the uniform system, we calculate the density profiles of a trapped polarized Fermi gas at zero temperature using the local density approximation. By varying the overall polarization and the interaction strength, we analyze the appearance of a discontinuity in the profile, signaling a first-order phase transition from a superfluid inner core to a normal outer shell. The local population imbalance between the two components and the size of the various regions of the cloud corresponding to different phases are also discussed. The calculated profiles are quantitatively compared with the ones recently measured by Shin et al. [Phys. Rev. Lett. 101, 070404 (2008)].
We show how to use trapped ultracold atoms to measure the magnetic susceptibility of a two-component Fermi gas. The method is illustrated for a noninteracting gas of 6{}^{6}6Li, using the tunability of interactions around a wide Feshbach resonance. The susceptibility versus effective magnetic field is directly obtained from the inhomogeneous density profile of the trapped atomic cloud. The wings of the cloud realize the high-field limit where the polarization approaches 100%, which is not accessible for an electron gas.
We describe a powerful method for determining the equation of state of an ultracold gas from in situ images. The method provides a measurement of the local pressure of a harmonically trapped gas and we give several applications to Bose and Fermi gases. We obtain the grand-canonical equation of state of a spin-balanced Fermi gas with resonant interactions as a function of temperature (Nascimbène et al 2010 Nature 463 1057). We compare our equation of state with an equation of state measured by the Tokyo group (Horikoshi et al 2010 Science 327 442), which reveals a significant difference in the high-temperature regime. The normal phase, at low temperature, is well described by a Landau Fermi liquid model, and we observe a clear thermodynamic signature of the superfluid transition. In a second part, we apply the same procedure to Bose gases. From a single image of a quasi-ideal Bose gas, we determine the equation of state from the classical to the condensed regime. Finally, the method is applied to a Bose gas in a three-dimensional optical lattice in the Mott insulator regime. Our equation of state directly reveals the Mott insulator behavior and is suited to investigate finite-temperature effects.
We consider population-imbalanced two-component Fermi gases under external harmonic confinement interacting through short-range two-body potentials with diverging sss-wave scattering length. Using the fixed-node diffusion Monte Carlo method, the energies of the ``normal state'' are determined as functions of the population imbalance and the number of particles. The energies of the trapped system follow, to a good approximation, a universal curve even for fairly small systems. A simple parametrization of the universal curve is presented and related to the equation of state of the bulk system.
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.