Density correlations in ultracold atomic Fermi gases

We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.

Figure 1

(Color online) (a) Sketch of a typical atomic number density. The measured observable is the atom number $N$ given by $n\left(x\right)$ integrated over the bin volume (the bins are indicated by the dashed lines). (b) Histogram of probabilities to find $N$ particles in a bin.

Figure 2

(Color online) Second cumulant $C_2$ as a function of $\xi =1∕k_{F}a$. For the strongly confined, quasi-two-dimensional system (2D), the Fermi vector can be approximated by the inverse of the ground-state size: $k_F=\pi ∕{\ell }_0$. The dotted line corresponds to ${\xi }_{\mu =0}$. The inset shows $C_2$ normalized to $\Delta ∕{ϵ}_F$.

Figure 3

(Color online) Third cumulant $C_3$ as a function of $\xi$ as defined in Fig. 2. The dotted line corresponds to ${\xi }_{\mu =0}$. The inset shows $C_3$ normalized to $\Delta ∕{ϵ}_F$.