Entanglement between two spatially separated ultracold interacting Fermi gases

Multiparticle entangled states, essential ingredients for modern quantum technologies, are routinely generated in experiments of atomic Bose-Einstein condensates (BECs). However, the entanglement in ultracold interacting Fermi gases has not been yet exploited. In this work, by using an ansatz of composite bosons, we show that many-particle entanglement between two fermionic ensembles localized in spatially separated modes can be generated by splitting an ultracold interacting Fermi gas in the (molecular) BEC regime. This entanglement relies on the fundamental fermion exchange symmetry of molecular constituents and might be used for implementing quantum applications in oncoming experiments. We show that the generated fermionic ensembles can be highly entangled and exhibit nonlocal quantum correlations. Entanglement-induced suppression of fluctuations in the single-fermion spectral density of the ultracold fermionic gas is also observed.

Figure 1

(a) Purity of the fermionic ensemble 1 (with $M=N/2$ fermion pairs) for different values of the interaction parameter ${\left(k_{F}a\right)}^{-1}$. The gray dots represent the lower bound of the purity $P_1={\left(\genfrac{}{}{0pt}{}{N}{M}\right)}^{-1}$. For the colored circles, squares, stars, and triangles, respectively, the distribution of the purity with the population imbalance of the condensates ($1-2M/N$) is the same and it is shown in (b). (c) Schmidt coefficients ${\lambda }_{nl}$ for $N=360$ and ${\left(k_{F}a\right)}^{-1}=0.5$, 1, and 2. We label single-particle states $(n,l)$ (with degeneracy $g_l=2l+1$) using a single index $j$. All depicted quantities are dimensionless.

Figure 2

(a) Mean population of the $t$ lowest energetic states of the single-particle spectrum (${\langle N_t\rangle }_N=N{\sum }_{j=1}^t{D}_j$) of an interacting Fermi gas with $N={10}^3$ fermion pairs and interaction parameter ${\left(k_{F}a\right)}^{-1}=2$, 1, and 0.5 (green, orange, and blue, respectively). Suppression of particle fluctuations in this spectral region ${\tilde{\mathrm{\Lambda }}}_t$ is shown in (b), where the probability $\mathcal{P}\left(n\right)$ is plotted for $t=56$. Dashed areas are Poissonian distributions and connected gray dots are binomial distributions. All depicted quantities are dimensionless.

Figure 3

Particle fluctuations in the states ${\tilde{\mathrm{\Lambda }}}_t$ of the fermionic ensemble 1 $[{\mathcal{P}}_1\left(n_1\right)]$ after splitting the system into two balanced ensembles of $M=N/2=0.5\times {10}^3$ fermion pairs. When decreasing ${\left(k_{F}a\right)}^{-1}$, ${\mathcal{P}}_1\left(n_1\right)$ approaches the binomial distribution $\left(\genfrac{}{}{0pt}{}{t}{n_1}\right)/2^t$ of two maximally entangled fermionic ensembles with perfectly correlated fluctuations. All depicted quantities are dimensionless.

Figure 4

Violation of the CHSH inequality ($\mathcal{M}\le 2$) for local theories. The represented states $j=1,5,21,57,121$ are the first nondegenerate states $\left(nl\right)$ of Fig. 1 with ${\left(k_{F}a\right)}^{-1}=0.5$, $l=0$, and $n=1,2,3,4,5$. All depicted quantities are dimensionless.

Figure 5

Occupation probability of the lowest energetic state $N{D}_1\left[N\right]$, normalized to $N{\lambda }_1$, for a Fermi gas with ${\left(k_{F}a\right)}^{-1}=0.5$ and a mean number of particles $\overline{N}$. Red lines are the function $1/[1+{\lambda }_j(N-1)]$. All depicted quantities are dimensionless.