Short-range correlations and entropy in ultracold atomic Fermi gases

We relate short-range correlations in ultracold atomic Fermi gases to the entropy of the system over the entire temperature, $T$, vs. coupling strength, $-1/k_Fa$, plane. In the low temperature limit the entropy is dominated by phonon excitations and the correlations increase as $T^4$. In the 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 $T,-1/k_Fa$ 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.

💡 Research Summary

The paper establishes a quantitative link between the short‑range two‑body correlation (the “contact” C) and the entropy of a homogeneous two‑component ultracold Fermi gas across the entire temperature–interaction plane (T versus –1/k_Fa). Starting from the short‑distance form of the pair correlation function, the authors derive the exact thermodynamic relation C = –(m/4π)∂f/∂a⁻¹, where f is the free‑energy density. Differentiating with respect to temperature yields ∂C/∂T = (m/4π)∂s/∂a⁻¹, showing that the temperature dependence of the contact is directly governed by how the entropy changes with the interaction strength.

In the low‑temperature regime (T ≪ T_c) the system is a superfluid whose excitations are phonons. The phonon contribution to the entropy is s_ph ∝ (T/c_s)³, with the zero‑temperature sound speed c_s increasing monotonically from the BEC side to the BCS side. Because ∂c_s/∂a⁻¹ > 0, the contact grows as C(T) = C(0) + const·T⁴. This T⁴ law holds in the BCS, unitary, and BEC limits, with the prefactor largest near unitarity where the sound speed is maximal.

In the BEC limit the fermions form tightly bound molecules. The authors treat the molecular gas with a Bogoliubov Hamiltonian, defining a molecular contact C_m = –M²π∂f_m/∂a_m⁻¹. By relating C to C_m through the derivative of the scattering lengths, they show that thermal phonons of the molecular condensate also generate a T⁴ increase of the fermionic contact, reproducing Eq. (16). This provides a clear physical picture: phonons modify the short‑distance two‑particle wavefunction, enhancing the contact.

At high temperatures (T ≫ T_F) the fugacity z = e^{μ/T} is small and the free energy can be expanded in the virial series. The second virial coefficient b₂ contains the bound‑state contribution (in the BEC side) and the scattering phase shift. Differentiating the virial expression gives C = n²λ²π ∝ n²/T, i.e. the contact decays as 1/T. Consequently, the low‑T T⁴ rise and the high‑T 1/T fall must intersect, implying a maximum of C at a finite temperature T_max. The authors estimate T_max ≈ T_F; near unitarity the maximum is pronounced (C_max ≈ 2 C(0)), while on the BCS and BEC sides the relative increase is modest (∝|k_F a|).

Using the relation ∂C/∂T = (m/4π)∂s/∂a⁻¹, the authors sketch isentropes in the T–(−1/k_Fa) plane. At low T the isentropes have a positive slope (entropy grows with stronger attraction), whereas at high T they have a negative slope, reflecting the opposite behavior of C. The point where the isentrope slope vanishes coincides with the temperature of the contact maximum.

Finally, the theoretical predictions are compared with recent photo‑association measurements of the contact at temperatures above the superfluid transition. The observed temperature dependence and the existence of a maximum near unitarity agree quantitatively with the calculations.

In summary, the work provides a unified thermodynamic framework that connects short‑range correlations to entropy, explains the T⁴ enhancement by phonons, the 1/T decay by virial physics, and predicts a universal maximum of the contact at finite temperature. This bridges low‑temperature many‑body theory, high‑temperature virial expansions, and experimental observations, offering a powerful tool for future studies of strongly interacting quantum gases.