End-point thermodynamics of an atomic Fermi gas subject to a Feshbach resonance

The entropy and kinetic, potential, and interaction energies of an atomic Fermi gas in a trap are studied under the assumption of thermal equilibrium for finite temperature. A Feshbach resonance can cause the fermions to pair into diatomic molecules. The entropy and energies of mixtures of such molecules with unpaired atoms are calculated, in relation to recent experiments on molecular Bose-Einstein condensates produced in this manner. It is shown that, starting with a Fermi gas of temperature $T=0.1T_F^0$, where $T_F^0$ is the noninteracting Fermi temperature, an extremely cold degenerate Fermi gas of temperature $T\lesssim 0.01T_F^0$ may be produced without further evaporative cooling. This requires adiabatic passage of the resonance, subsequent sudden removal of unpaired atoms, and adiabatic return. We also calculate the ratio of the interaction energy to the kinetic energy, a straightforward experimental signal which may be used to determine the temperature of the atoms and indicate condensation of the molecules.

Figure 1

Shown is the ratio of the interaction energy to the kinetic energy as a function of $T∕T_F^0$ for a weakly interacting degenerate Fermi gas. This ratio is an established experimental observable [30] that may be used to measure the degeneracy of the gas. This ratio, called $\beta$ in the unitarity limit, i.e., when $k_F^0\mid a\mid \gtrsim 1$, is thought to be a universal constant in that regime [31, 32]. Here, the temperature dependence of the ratio is illustrated for increasing values of $k_F^0\mid a\mid$.

Figure 2

Shown is the density profile of a Fermi gas (solid curve) in the presence of a molecular condensate (dashed curve), the latter being in the Thomas-Fermi limit with $R_m$ the Thomas-Fermi radius. The atoms occupy a spherical shell outside the molecular region. Note that the parameters of the JILA experiment [6] have been used for $T∕T_{\mathrm{BEC}}^0=0.25$ with the molecule-molecule and atom-molecule scattering lengths derived in Ref. 18.

Figure 3

Atomic Fermi gas. Shown is the ratio of the interaction energy to the kinetic energy as a function of the degeneracy $T∕T_F^0$, all in the presence of the molecular mean field. Since the atoms are pushed to the outside of the molecular mean field in space, the degeneracy cannot be extracted in a simple way from the momentum distribution; this plot serves as an alternative method to determine their degeneracy. The left hand curve is calculated for a Bose-condensed molecular mean field, i.e., $T⪡T_{\mathrm{BEC}}^0$, while the right hand one is calculated in the opposite limit. Note that $T_F^0=1.53{(N_a∕N_m)}^{1∕3}{T}_{\mathrm{BEC}}^0$; for the parameters shown here, $T_F^0=1.11T_{\mathrm{BEC}}^0$.

Figure 4

Molecular Bose gas. Shown is the ratio of the interaction energy to the kinetic energy as a function of $T∕T_{\mathrm{BEC}}^0$. This ratio may be used as a signal of condensation since, for the molecules, the power law of $E_{\mathrm{int}}∕E_{\mathrm{kin}}\propto {(T∕T_{\mathrm{BEC}}^0)}^{\delta }$ changes for $\delta \simeq -5∕2$ above the phase transition to $\delta \simeq -1$ below. The left hand curve is calculated for a Bose-condensed molecular mean field, i.e., $T⪡T_{\mathrm{BEC}}^0$, while the right hand one is calculated in the opposite limit. The parameters of the JILA experiment have been used for plotting purposes [6].

Figure 5

Shown is the total entropy of an atom-molecule mixture (solid curve), as well as the separate contributions of the atoms (plus signs) and molecules (open circles). Clearly, for low temperatures, the atoms carry the majority of the entropy, which is what makes the entropic cooling scheme outlined in Table effective. Note that the parameters of the JILA experiment [6] have been used.