Momentum Distribution of a Fermi Gas of Atoms in the BCS-BEC Crossover

We observe dramatic changes in the atomic momentum distribution of a Fermi gas in the crossover region between the BCS theory superconductivity and Bose-Einstein condensation (BEC) of molecules. We study the shape of the momentum distribution and the kinetic energy as a function of interaction strength. The momentum distributions are compared to a mean-field crossover theory, and the kinetic energy is compared to theories for the two weakly interacting limits. This measurement provides a unique probe of pairing in a strongly interacting Fermi gas.

Figure 1

Theoretical column integrated momentum distributions of a trapped Fermi gas $n(k)$ calculated from a mean-field theory [21]. $N$ is the total particle number; $k_F^0$ is the noninteracting Fermi wave vector. The normalization is given by $2\pi \int n(k)kdk=N$, and we note that the normalization is strongly affected by the tail of the distribution. The lines correspond to various values of $1/k_F^{0}a$, with the tallest curve (blue online) representing $a=0$ and the broadest curve (black online) representing $1/k_F^{0}a=0.59$. The middle curves correspond to $1/k_F^{0}a=-0.66$ (green online) and $1/k_F^{0}a=0$ (red online). Inset: Corresponding distributions for a homogeneous system.

Figure 2

Experimental, azimuthally averaged, momentum distributions of a trapped Fermi gas normalized such that the area under the curves is the same as in Fig. 1. The curves, in order of highest to lowest maximum OD, correspond to the following values of $1/k_F^{0}a$: $-71$ (blue online), $-0.66$ (green online), 0 (red online), and 0.59 (black online). Error bars represent the standard deviation of the mean of averaged pixels. Inset: Curves for $1/k_F^{0}a=-71$ (top) and 0 (bottom) weighted by $k^3$. The lines are the results of a fit to Eq. (1).

Figure 3

The points (●) show the measured energy $E_{\mathrm{kin}}$ normalized to $E_{\mathrm{kin}}^0=0.25k_b\mu \mathrm{K}$ ($E_{\mathrm{kin}}^0=\frac{3}{8}{E}_F$ for a harmonically trapped gas at $T=0$). The solid curve above the points (red online) is the expected energy ratio from a calculation only valid in the weakly interacting regime ($1/k_F^{0}a<-1$). In the strongly interacting regime pairing due to many-body effects strongly increases $E_{\mathrm{kin}}$. The vertical bar (green online) represents the expected value of $E_{\mathrm{kin}}/E_{\mathrm{kin}}^0$ at unitarity just due to density profile rescaling (see text). In the molecule limit ($1/k_F^{0}a>1$) we calculate the expected energy for an isolated molecule [curve below the points (blue online)]. Inset: A focus on the weakly interacting regime with the same axes definitions as the main graph. The dashed lines show the uncertainty in the calculation based upon the uncertainty in the Feshbach resonance parameters [4, 18].

Figure 4

Shape of the momentum distribution as described through the parameter $\zeta$ [Eq. (1)]. $\ln (1+\zeta )=0$ corresponds to a Gaussian distribution, and for an ideal Fermi gas $\ln (1+\zeta {)}^{-1}\approx T/T_F$ in the limit of low $T/T_F$. The dashed lines show the uncertainty in the Feshbach resonance position [4].