Density correlations in ultracold Fermi systems within the exact Richardson solution

We discuss the occupation number correlations in an ultracold system of interacting fermionic atoms. For a system with a special energy-level distribution, viz. two multiply degenerate levels, explicit expressions for the correlation functions are derived in a canonical approach using the exact ground state wave function of the reduced BCS Hamiltonian. We evaluate the correlators numerically for different interaction strengths and find analytical expressions in some limiting cases. Due to the underlying fermionic nature of the pairs the occupations are predominantly anticorrelated and their statistics is a multinomial distribution.

Figure 1

(Color online) Development of the ground state and first and second excited state energy vs interaction constant $g$ for a simple two-boson system. The two levels at ${ϵ}_0=0$ and ${ϵ}_1=1$ are both threefold degenerate.

Figure 2

(Color online) Many-body coefficient ${\left[{\phi }^{\left(1\right)}\left(\nu \right)\right]}^2$ as a function of $g$ for the ground state energy ${ϵ}_2^{\left(1\right)}$.

Figure 3

(Color online) Average particle number of one of the upper levels as a function of $G$ for various system sizes. The black solid lines correspond to the solutions obtained from BCS theory. Note that the self-consistency Eqs. (16) in this case only have real solutions for $G>1$. The average particle number of one of the lower levels follows from $⟨{\widehat{n}}_0⟩=1-⟨{\widehat{n}}_1⟩$.

Figure 4

(Color online) $g\left(0,0^{\prime }\right)=g\left(1,1^{\prime }\right)$ (upper plot) and $g\left(0,1\right)$ (lower plot) as a function of $G$ for various system sizes. The inset shows that, in contrast to $g\left(0,0^{\prime }\right)$, $g\left(0,1\right)$ is always negative in the low-interaction regime.

Figure 5

(Color online) Zoom into the region of positive correlations of $g\left(0,0^{\prime }\right)=g\left(1,1^{\prime }\right)$ in the weak-coupling range for different system sizes. A positive peak develops around $G=1$ for small systems and becomes maximal for $N\approx 16$. For large system sizes the absolute values become smaller again, but the overall feature sharpens.

Figure 6

(Color online) Average particle number of one of the upper levels as a function of $G$ for various system sizes in the quarter-filled case $\delta =1/4$. The BCS Eqs. (16) have real solutions for $G>0$ in this case. The average particle number of one of the lower levels follows from $⟨{\widehat{n}}_0⟩=0.5-⟨{\widehat{n}}_1⟩$.

Figure 7

(Color online) $g\left(0,0^{\prime }\right)$, $g\left(0,1\right)$, and $g\left(1,1^{\prime }\right)$ as a function of $G$ for various system sizes at quarter-filling $\delta =1/4$.