[Paper Review] Short-range correlations and entropy in ultracold atomic Fermi gases

Zhenhua Yu, G. M. Bruun+1 more|arXiv (Cornell University)|May 12, 2009

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper establishes a theoretical framework linking short-range correlations (contact C) in ultracold atomic Fermi gases to thermodynamic entropy across the BCS-BEC crossover, showing that C(T) increases as T⁴ at low temperatures due to phonon contributions and decreases as 1/T at high temperatures, resulting in a finite-temperature maximum. The findings are validated against photoassociation experiments, revealing a non-monotonic temperature dependence of correlations that is detectable in current ultracold atom setups.

ABSTRACT

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.

Motivation & Objective

• To establish a quantitative connection between short-range two-body correlations (contact C) and thermodynamic entropy in ultracold Fermi gases across the BCS-BEC crossover.
• To determine the temperature dependence of C in the low-temperature (BCS/phonon-dominated) and high-temperature (virial expansion) regimes.
• To explain the non-monotonic behavior of C(T) by analyzing the coupling-strength dependence of entropy and isentropes in the T vs. -1/k_F a plane.
• To provide a theoretical basis for interpreting photoassociation experiments measuring C(T) in trapped Fermi gases.

Proposed method

• Derives the relation C = -m/(4π) ∂f/∂(a⁻¹) from the free energy f, linking contact to thermodynamic derivatives.
• Uses the identity ∂C/∂T = (m/4π) ∂s/∂(a⁻¹) to connect temperature variation of C to entropy's coupling dependence.
• Applies the virial expansion in the high-temperature limit to show C ∝ n²/T, implying C decreases as 1/T.
• Calculates low-temperature C(T) using phonon entropy in the BEC limit, showing C ∝ T⁴ due to phonon contributions.
• Constructs isentropes in the T, -1/k_F a plane by combining high- and low-T behaviors, revealing a positive slope at low T and negative at high T.
• Compares theoretical predictions of C(T) and loss rates in photoassociation to experimental data from 6Li atoms at BCS-side Feshbach resonances.

Experimental results

Research questions

• RQ1 How does the short-range correlation strength C(T) vary with temperature across the BCS-BEC crossover?
• RQ2 What is the origin of the non-monotonic temperature dependence of C, and why does it peak at finite T?
• RQ3 How do phonons in the BEC regime enhance fermionic short-range correlations?
• RQ4 To what extent can the virial expansion and entropy-based analysis predict C(T) in the high-temperature regime?
• RQ5 Can the theoretical C(T) dependence be quantitatively matched to photoassociation loss rate data in trapped Fermi gases?

Key findings

• The correlation strength C increases as T⁴ at low temperatures due to phonon contributions to the entropy, which dominate in the BEC regime.
• In the high-temperature limit, C decreases as 1/T, derived from the virial expansion, indicating a monotonic decrease at high T.
• The combination of low-T and high-T behaviors implies that C(T) reaches a maximum at a finite temperature T_max > 0.
• Isentropes in the T, -1/k_F a plane have a positive slope at low T (due to phonon entropy) and a negative slope at high T, indicating a non-monotonic C(T) dependence.
• Theoretical predictions for photoassociation loss rates in 6Li gases at BCS-side Feshbach resonances agree quantitatively with experimental data at T ≈ 0.75T_F without fitting parameters.
• The observed increase in temperature during adiabatic tuning from BCS to unitarity is consistent with isentropes bending upward before bending downward, confirming the non-monotonic C(T) behavior.