Theory of ultracold Fermi gases

TL;DR

The paper comprehensively surveys the theory of ultracold Fermi gases, detailing how tunable interactions via s-wave scattering length $a$ drive superfluidity across the BCS–BEC crossover and into the universal unitary regime where $|a|\to\infty$. It surveys equilibrium and dynamical properties in uniform and trapped geometries, employing mean-field theory, quantum Monte Carlo, and beyond-mean-field approaches, and connects predictions to experiments across density profiles, collective modes, expansion, vortices, and lattice physics. A central theme is the emergence of universal, high-$T_c$-like behavior at unitarity, with key quantities scaling with the Fermi energy $E_F$ and the universal parameter $\beta$. The review also discusses spin-polarized gases, mass-imbalanced mixtures, optical lattices, and 1D systems, highlighting both successes and outstanding open questions in thermodynamics, transport, coherence, and phase structure.

Abstract

The physics of quantum degenerate Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions which play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, of the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to non-perturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare the theoretical predictions with the available experimental results.

Figures (36)

• Figure 1: Gallery of molecular BEC experiments. Bimodal spatial distributions were observed for expanding gases at JILA (Greiner, Regal and Jin, 2003) with $^{40}$K, at MIT (Zwierlein et al., 2003) and at ENS (Bourdel et al., 2004) with $^6$Li. They were instead measured in situ at Innsbruck (Bartenstein et al., 2004a) and at Rice University (Partridge et al., 2005) with $^6$Li.
• Figure 2: Transition temperature in units of the Fermi energy $E_F$ as a function of the interaction strength along the BCS-BEC crossover, calculated using BCS mean-field theory (from Sá de Melo, Randeria and Engelbrecht, 1993). The diamond corresponds to the theoretical prediction by Burovski et al. (2006a) based on a Quantum Monte Carlo simulation at unitarity.
• Figure 3: Evidence for quantum degeneracy effects in trapped Fermi gases. The average energy per particle, extracted from absorption images, is shown for two-spin mixtures. In the quantum degenerate regime the data agree well with the ideal Fermi gas prediction (solid line). The horizontal dashed line corresponds to the result of a classical gas. From De Marco, Papp and Jin (2001).
• Figure 4: Magnetic field dependence of the scattering length in $^6$Li, showing a broad Feshbach resonance at $B_0\simeq 834$ G and a narrow Feshbach resonance at $B_0\simeq 543$ G (can not be resolved on this scale). From Bourdel et al. (2003).
• Figure 5: (color online). Energy per particle along the BCS-BEC crossover with the binding-energy term subtracted from $E/N$. Symbols: quantum Monte Carlo results from Astrakharchik et al. (2004a). The dot-dashed line corresponds to the expansion (\ref{['enexpansion']}) and the dashed line to the expansion (\ref{['enerbeyond']}) holding, respectively, in the BCS and in the BEC regime. The long-dashed line (red) refers to the result of the BCS mean-field theory. Inset: enlarged view of the BEC regime $-1/k_Fa\le-1$. The solid line corresponds to the mean-field term in the expansion (\ref{['enerbeyond']}), the dashed line includes the Lee-Huang-Yang correction.