Theory of ultracold atomic Fermi gases

The physics of quantum degenerate atomic 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 that 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 broad 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, on 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 nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.

Figure 1

(Color online) Gallery of molecular BEC experiments. Bimodal spatial distributions were observed for expanding gases at JILA (155) with ${}^{40}\mathrm{K}$, at MIT (400), and at ENS (45) with ${}^6\mathrm{Li}$. They were instead measured in situ at Innsbruck (196) and at Rice University (290) with ${}^6\mathrm{Li}$.

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. The diamond corresponds to the theoretical prediction by 61 based on a quantum Monte Carlo simulation at unitarity. From 329.

Figure 3

(Color online) 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 112.

Figure 4

Magnetic field dependence of the scattering length in ${}^6\mathrm{Li}$, showing a broad Feshbach resonance at $B_0\simeq 834\mathrm{G}$ and a narrow Feshbach resonance at $B_0\simeq 543\mathrm{G}$ (cannot be resolved on this scale). From 44.

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 18. The dot-dashed line corresponds to the expansion (32) and the dashed line to the expansion (38), respectively, in the BCS and in the BEC regime. The long-dashed line refers to the result of the BCS mean-field theory. Inset: Enlarged view of the BEC regime $-1∕k_{F}a\le -1$. The solid line corresponds to the mean-field term in the expansion (38) and the dashed line includes the Lee-Huang-Yang correction.

Figure 6

(Color online) QMC results of the momentum distribution $n_{\mathbf{k}}$ for different values of $1∕k_{F}a$ (solid lines). The dashed lines correspond to the mean-field results from Eq. (60). Inset: $n_{\mathbf{k}}$ for $1∕k_{F}a=4$. The dotted line corresponds to the momentum distribution of the molecular state Eq. (61).

Figure 7

(Color online) Condensate fraction of pairs $n_{\mathrm{cond}}$ [Eq. (29)] as a function of the interaction strength: FN-DMC results (symbols), Bogoliubov quantum depletion of a Bose gas with $a_{\mathrm{dd}}=0.60a$ (dashed line), BCS theory including the Gorkov$-$Melik-Barkhudarov correction (dot-dashed line), and mean-field theory using Eq. (53) (solid line).

Figure 8

(Color online) QMC results for the pair correlation function of parallel spins $g_{\uparrow \uparrow }\left(r\right)$ for different values of the interaction strength: solid line, $1∕k_{F}a=0$; dashed line, $1∕k_{F}a=1$; dotted line, $1∕k_{F}a=4$. The thin solid line refers to the noninteracting gas and the thin dotted line is the pair correlation function of a Bose gas with $a_{\mathrm{dd}}=0.60a$ and the same density as the single-component gas corresponding to $1∕k_{F}a=4$.

Figure 9

(Color online) QMC results for the pair correlation function of antiparallel spins $g_{\uparrow \downarrow }\left(r\right)$ for different values of the interaction strength: dashed line, $1∕k_{F}a=-1$; solid line, $1∕k_{F}a=0$; dotted line, $1∕k_{F}a=4$. The thin solid line refers to the noninteracting gas.

Figure 10

(Color online) Experimental results of the double integrated density profiles along the BCS-BEC crossover for a gas of ${}^6\mathrm{Li}$ atoms. The results at $850\mathrm{G}$ correspond to unitarity, while the results at 809 and $882\mathrm{G}$ correspond, respectively, to the BEC and BCS side of the resonance. The continuous curve at unitarity is the best fit based on Eq. (77). The dashed lines correspond to the predictions for a noninteracting gas. From 25.

Figure 11

(Color online) Column integrated momentum distributions of a trapped Fermi gas. The symbols correspond to the experimental results from 317 and the lines to the mean-field results based on Eq. (80) for the same values of the interaction parameter $1∕k_F^{0}a$. From top to bottom: $1∕k_F^{0}a=-71$, $1∕k_F^{0}a=0$, and $1∕k_F^{0}a=0.59$.

Figure 12

Verifying the virial theorem at unitarity in a Fermi gas of ${}^6\mathrm{Li}$: $⟨x^2⟩∕⟨x^2\left(0\right)⟩$ vs $E∕E\left(0\right)$ showing linear scaling. Here $⟨x^2⟩$ is the transverse mean-square radius, proportional to the oscillator energy. $E$ is the total energy evaluated as in 206. $E\left(0\right)$ and $⟨x^2\left(0\right)⟩$ denote ground-state values. From 365.

Figure 13

(Color online) Measured entropy in a trapped ${}^6\mathrm{Li}$ gas at unitarity vs total energy. The data reveal the occurrence of a characteristic change of behavior in the slope at $[E-E(0\left)\right]∕N=0.41E_F$, which is interpreted as the signature of the phase transition to the superfluid regime. The solid lines are power-law fits below and above the critical point, while the dashed lines show the extended fits. From 233.

Figure 14

(Color online) Aspect ratio as a function of time during the expansion of an ultracold Fermi gas at unitarity. Upper symbols, experiment; upper line, hydrodynamic theory. For comparison, also shown are the results in the absence of interactions. Lower symbols, experiment; lower line, ballistic expansion. From 271.

Figure 15

Aspect ratio after expansion from the trap along the crossover. The solid lines are the hydrodynamic predictions at unitarity and for a gas in the deep BEC regime. The Fermi momentum $k_F$ corresponds to the value (9). From 9.

Figure 16

(Color online) Sound velocity in the center of the trap measured along the BCS-BEC crossover (symbols). The theory curves are obtained using Eq. (91) and are based on different equations of state: BCS mean field (dotted line), quantum Monte Carlo (solid line), and Thomas-Fermi molecular BEC with $a_{\mathrm{dd}}=0.60a$ (dashed line). The Fermi momentum $k_F$ corresponds to the value (9). From 19.

Figure 17

(Color online) Frequency of the radial quadrupole mode for an elongated Fermi gas in units of the radial frequency (upper panel). The dashed line is the hydrodynamic prediction $\sqrt{2}{\omega }_{\perp }$. The lower panel shows the damping of the collective mode. The Fermi momentum $kf$ corresponds to the value (9). From 8.

Figure 18

(Color online) Frequency of the radial compression mode for an elongated Fermi gas in units of the radial frequency. The curves refer to the equation of state based on BCS mean-field theory (lower line) and on Monte Carlo simulations (upper line). The Fermi momentum $k_F$ corresponds to the value (9). From 7.

Figure 19

Spectrum of collective excitations of the superfluid Fermi gas along the BCS-BEC crossover (thick lines). Energy is given in units of the Fermi energy. Left: BCS regime $(k_{F}a=-1)$. Center: unitarity. Right: BEC regime $(k_{F}a=+1)$. The thin lines denote the threshold of single-particle excitations. From 100.

Figure 20

Landau’s critical velocity (in units of the Fermi velocity) calculated along the crossover using BCS mean-field theory. The critical velocity is largest near unitarity. The dashed line is the sound velocity. From 100.

Figure 21

Measured critical velocity along the BCS-BEC crossover. The solid line is a guide to the eye. From 250.

Figure 22

RF spectra for various magnetic fields along the BCS-BEC crossover and for different temperatures. The RF offset $\delta \nu =\nu -{\nu }_{23}$ is given relative to the atomic transition $2\to 3$. The molecular limit is realized for $B=720\mathrm{G}$ (first column). The resonance regime is studied for $B=822$ and $837\mathrm{G}$ (second and third columns). The data at $875\mathrm{G}$ (fourth column) correspond to the BCS side of the crossover. Upper row, signals of unpaired atoms at high temperature (larger than $T_F$); middle row, signals for a mixture of unpaired and paired atoms at intermediate temperature (fraction of $T_F$); lower row, signals for paired atoms at low temperature (much smaller than $T_F$). From 91.

Figure 23

Frequency shift $\delta {\nu }_{\max }$ measured in ${}^6\mathrm{Li}$ at low temperature as a function of the magnetic field for two different configurations corresponding to $T_F=1.2\mathrm{\mu }\mathrm{K}$ (filled symbols) and $3.6\mathrm{\mu }\mathrm{K}$ (open symbols). The solid line shows the value $\delta {\nu }_{\max }$ predicted in the free molecular regime where it is essentially given by the molecular binding energy. From 91.

Figure 24

(Color online) Time evolution of the angle characterizing the scissors mode in a Fermi gas at unitarity (left panel) and on the BCS side of the resonance (right panel). The measured frequencies agree well with the theoretical predictions (see text). From 383.

Figure 25

(Color online) Aspect ratio vs expansion time of a unitary gas for different values of the initial angular velocity ${\Omega }_0$ (in units of the trap frequency ${\omega }_z$ of the long axis) and of the gas temperature (parametrized by the energy per particle $E∕N$). Squares, no angular velocity; solid circles, ${\Omega }_0∕{\omega }_z=0.40$, $E∕N{E}_F=0.56$; open circles, ${\Omega }_0∕{\omega }_z=0.40$, $E∕N{E}_F=2.1$; triangles, ${\Omega }_0∕{\omega }_z=1.12$, $E∕N{E}_F=0.56$. The dashed, solid, and dotted lines are the results of calculations based on irrotational hydrodynamics. From 96.

Figure 26

Experimental observation of quantized vortices in a superfluid Fermi gas along the BCS-BEC crossover (396).

Figure 27

(Color online) Qualitative phase diagram as a function of the interaction strength $-1∕k_{F}a$ and the polarization $P$. The circle at unitarity corresponds to the critical value $P_c=0.39$ discussed in Sec. 9B. The Fermi wave vector corresponds to the total average density: $k_F=\left[3{\pi }^2\right(n_{\uparrow }+n_{\downarrow }){]}^{1∕3}$. Note that the possible occurrence of the FFLO phase is not considered here.

Figure 28

Equation of state of a normal Fermi gas as a function of the concentration $x$. The solid line is a polynomial best fit to the QMC results (circles). The dashed line corresponds to the expansion (119). The dot-dashed line is the coexistence line between the normal and unpolarized superfluid states and the arrow indicates the critical concentration $x_c$ above which the system phase separates. For $x=1$, the energy of both the normal and superfluid (diamond) states is shown.

Figure 29

Radii of the three phases in the trap in units of the radius $R_{\uparrow }^0=a_{\mathrm{ho}}(48N_{\uparrow }{)}^{1∕6}$ of a noninteracting fully polarized gas.

Figure 30

(Color online) Phase separation in an interacting Fermi gas at unitarity. Normalized central density difference (circles) and condensate fraction (triangles). From both sets of data, one can extract the same value $\sim 0.75$ for the critical polarization. From 341.

Figure 31

(Color online) Double integrated density difference measured at unitarity in an ultracold trapped Fermi gas of ${}^6\mathrm{Li}$ with polarization $P=0.58$ (from 341). The theoretical curves correspond to the theory of Secs. 9B, 9C based on the local-density approximation (solid line) and predictions of a noninteracting gas with the same value of $P$ (dashed line). From 315.

Figure 32

Time-of-flight images obtained after adiabatically ramping down the optical lattice. Image (a) is obtained with $N_{\sigma }=3500$ and $s=5E_R$. Images (b)–(e) are obtained with $N_{\sigma }=15000$ and correspond to $s=5E_R$ (b), $s=6E_R$ (c), $s=8E_R$ (d), and $s=12E_R$ (e). The images show the optical density (OD) integrated along the vertically oriented $z$ axis after $9\mathrm{ms}$ of ballistic expansion. 209.

Figure 33

Dipole oscillations of a Fermi gas of ${}^{40}\mathrm{K}$ atoms at $T=0.3T_F$ in the presence (solid symbols and solid line) and absence (open symbols and dashed line) of a lattice with height $s=3$. The lines correspond to the theoretical predictions and the symbols to the experimental results. The horizontal dot-dashed line represents the trap minimum. From 300.

Figure 34

(Color online) Density correlation function $C\left(d\right)$ measured after expansion as a function of the distance $d=\mid z-z^{\prime }\mid$ in a Fermi gas of ${}^{40}\mathrm{K}$ atoms released from an optical lattice. The difference $C\left(d\right)-1$ of the correlation function with respect to its uncorrelated value shows clear evidence of the antibunching effect at $d=\ell \equiv 2\hslash Kt∕m$. From 327.

Figure 35

Binding energy of molecules measured with ${}^{40}\mathrm{K}$ atoms in a 3D optical lattice. The data correspond to different intensities of the optical lattice: $s=6E_R$ (triangles), $10E_R$ (stars), $15E_R$ (circles), and $22E_R$ (squares). The solid line corresponds to Eq. (138) with no free parameters. At the position of the Feshbach resonance $(a\to \pm \infty )$, the binding energy takes the value ${ϵ}_b=-\hslash {\omega }_{\mathrm{opt}}$. From 354.

Figure 36

(Color online) Molecular binding energy measured with ${}^{40}\mathrm{K}$ in 1D (solid symbols) and in 3D (open symbols) configurations. The lower line corresponds to the theoretical prediction of 30 without free parameters. The upper line corresponds to the law $-{\hslash }^2∕m{a}^2+$ finite-range corrections. The vertical dashed line represents the position of the Feshbach resonance. From 260.