Mean-field thermodynamics of a spin-polarized spherically trapped Fermi gas at unitarity

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.

Figure 1

(Color online) Density profiles of spin-up and spin-down atoms for a spin-polarized unitary Fermi gas at different population imbalances and temperatures, as indicated in the figures. The number of total atoms is $N=20000$. The density profiles are normalized by the Thomas-Fermi center density of an ideal symmetric Fermi gas with the same number of total atoms, $n_{\mathit{TF}}\left(0\right)={\left(24N\right)}^{1∕2}{a}_{\mathit{ho}}^{-3}∕\left(3{\pi }^2\right)$, while the length is renormalized by the corresponding Thomas-Fermi radius $r_{\mathit{TF}}={\left(24N\right)}^{1∕6}{a}_{\mathit{ho}}$. The solid lines and dashed lines refer to the BdG results and LDA results, respectively.

Figure 2

(Color online) Gap functions at different temperatures $T=0.01T_F$ (a) and $T=0.20T_F$ (b) for various population imbalances as labeled. The number of total atoms is $N=20000$. The solid lines are the BdG predictions, while the dashed lines are the LDA results. The value of gap is renormalized by the noninteracting Fermi energy of an ideal symmetric Fermi gas $E_F={\left(3N\right)}^{1∕3}\hslash \omega$. Note that the small oscillation in the gap function at the superfluid-normal interface at low temperature $T=0.01T_F$ vanishes when the temperature becomes high enough.

Figure 3

(Color online) Radial (a) and axial (b) column density profiles for $N=20000$ and $P=0.84$ at $T=0.01T_F$. The difference between the BdG results (solid lines) and LDA results (dashed lines) becomes extremely small due to the integration over the extra dimensions.

Figure 4

(Color online) Dependence of the gap function on the number of total atoms at $T=0.01T_F$ and $P=0.48$. The solid line is the LDA result, and the others are BdG results with $N={10}^5$ (dash-dotted line), $N=20000$ (dotted line), and $N=4000$ (dashed line).

Figure 5

(Color online) Chemical potentials as a function of the population imbalance at two temperatures $T=0.01T_F$ and $T=0.20T_F$.

Figure 6

(Color online) Total energy per particle as a function of the population imbalance $P$ at two temperatures $T=0.01T_F$ and $T=0.20T_F$. The inset shows the first-order derivation with respect to the population imbalance at $T=0.20T_F$. The jump at $P\sim 0.9$ marks the phase transition to the normal state.

Figure 7

(Color online) Entropy per particle as a function of the population imbalance at various temperatures.

Figure 8

(Color online) Average order parameter defined in Eq. (5.2) as a function of the population imbalance at three temperatures $T=0.01T_F$, $T=0.20T_F$, and $T=0.25T_F$. The position where the average order parameter vanishes determines the critical population imbalance towards the normal state at a fixed temperature.

Figure 9

(Color online) (a) BCS mean-field critical temperature as a function the population imbalance, determined from the LDA calculations (solid line). Open symbols are available experimental points from Zwierlein et al. [10, 11] and Kinast et al. [7]. The dashed line shows a rescaled BCS critical temperature as described in the text. (b) The critical entropy (the entropy at the critical temperature) against the population imbalance.