Temperature of a trapped unitary Fermi gas at finite entropy

We present theoretical predictions for the equation of state of a harmonically trapped Fermi gas in the unitary limit. Our calculations compare Monte Carlo results with the equation of state of a uniform gas using three distinct perturbation schemes. We show that in experiments the temperature can be usefully calibrated by making use of the entropy, which is invariant during an adiabatic conversion into the weakly interacting limit of molecular BEC. We predict the entropy dependence of the equation of state.

Figure 1

(Color online) Chemical potential of a uniform Fermi gas in the unitary limit as a function of temperature, in units of Fermi energy ${ϵ}_F={\hslash }^2{k}_F^2∕2m=k_B{T}_F^0$. The lines plotted are our proposed results (solid lines), NSR $\left(G_0{G}_0\right)G_0$ predictions in Refs. 13, 14 (dashed line), asymmetric $\left(G{G}_0\right)G_0$ scheme (dotted line), and fully self-consistent $\left(GG\right)G$ scheme (dot-dashed line). All these results are contrasted to the Monte Carlo data from Ref. 23 (open circles).

Figure 2

(Color online) Energy of a trapped unitary gas is plotted as a function of temperature (thick solid line), and compared to experimental data (open circles with error bars). The comparison with respect to the prediction of an ideal Fermi gas (dashed line) is also shown. The thin solid line is an empirical power law fit to the measured energies [8], i.e., $E\left(T\right)=E_0[1+97.3(T∕T_F{)}^{3.73}]$ for $T<T_c\simeq 0.27T_F$ and $E\left(T\right)=E_0[1+4.98(T∕T_F{)}^{1.43}]$ above $T_c$. Here $E_0$ is the energy at zero temperature and $T_c=0.27T_F$ is the experimentally extracted superfluid transition temperature [8].

Figure 3

(Color online) Specific heat $C_V=dE∕dT$ of a trapped unitary gas as a function of temperature. For comparison, the predictions of an ideal Fermi gas (dashed line) and of an ideal Bose gas with number of particles $N_B=N∕2$ (dot-dashed line) [27], together with the experimental extracted specific heat (line with open circles) are also presented. It should be noted that a precise determination of the specific heat from experimental data of energies is not permissible. The experimental specific heat shown here is calculated from the empirical power law fit to the measured energies, which is outlined in Fig. 2. The resulting sharp jump at $T_c$ is thereby possibly an artifact of the fit. We note also that the abrupt discontinuity in the specific heat of an ideal trapped Bose gas at $T_c$ will be rounded off quickly with inclusions of boson-boson interactions, see, for example, Fig. 3 in Ref. 28.

Figure 4

(Color online) (a) The temperature as a function of entropy in the unitary limit (solid line) and in the deep BEC limit of $1∕k_{F}a\simeq 4.9$ (dashed line), with $T_c=0.518T_F$. At low temperature, our calculated BEC entropy satisfies the scaling law for a weakly interacting Bose gas [32]: $S^{\prime }\left(T^{\prime }\right)\propto$$(T^{\prime }{)}^{5∕2}$. (b) Calibration curve of unitary temperature versus weakly interacting Bose gas temperature. (c) Unitary energy as a function of BEC temperature $T^{{}^{\prime }}$.