Short-range correlations and entropy in ultracold-atom Fermi gases

We relate short-range correlations in ultracold-atom Fermi gases to the entropy of the system over the entire temperature, $T$, vs coupling strength, $-1/k_{F}a$, plane. In the low-temperature limit the entropy is dominated by phonon excitations and the correlations increase as $T^4$. In the Bose-Einstein condensate (BEC) limit, we calculate a boson model within the Bogoliubov approximation to show explicitly how phonons enhance the fermion correlations. In the high-temperature limit, we show from the virial expansion that the correlations decrease as $1/T$. The correlations therefore reach a maximum at a finite-temperature. We infer the general structure of the isentropes of the Fermi gas in the $\left(T,-1/k_{F}a\right)$ plane, and the temperature dependence of the correlations in the unitary, BEC, and BCS limits. Our results compare well with measurements of the correlations via photoassociation experiments at higher temperatures.

Figure 1

(Color online) The correlation strength $C\left(T\right)$ in the (a) BCS, (b) unitary, and (c) BEC limits. The dashed lines at high $T$ are the virial results, and the dashed line at low $T$ in (b) is from Eq. (17). The solid lines are qualitative interpolations between the increasing low $T$ results and the decreasing high $T$ results. The maximum in $C\left(T\right)$ for $T=T_{\text{max}}$ is prominent in the strong-coupling limit, but in the BCS and BEC limits are too small to be visible here.

Figure 2

(Color online) Sketches of isentropes of a balanced two-component Fermi gas, from which one can infer the temperature dependence of $C\left(T\right)$.

Figure 3

(Color online) Loss of ${}^6\text{L}\text{i}$ atoms in the trap through photoassociation vs the probe duration at $B=865\text{G}$. The solid curve is the numerical evaluation of Eq. (33). The circles are the $T=0.75T_F$ experimental data from Ref. [9].