Many-Body Entanglement in Short-Range Interacting Fermi Gases for Metrology

We explore many-body entanglement in spinful Fermi gases with short-range interactions, for metrology purposes. We characterize the emerging quantum phases via density-matrix renormalization group simulations and quantify their entanglement content for metrological usability via quantum Fisher information (QFI). Our study establishes a method, promoting QFI to be an order parameter. Short-range interactions reveal to build up metrologically promising entanglement in the $XY$-ferromagnetic and cluster ordering, the cluster physics being unexplored so far.

Figure 1

System concept. Top. The $tUJ$ Hamiltonian (1): $t$ drives the hopping, $U$ the on-site interaction, and $J$ the spin-exchange coupling. Bottom. Qualitative phase diagram at quarter filling in the $U/t-J/t$ parameter space, including the following phases: Luttinger liquid, superfluid, charge-density-wave-like, spin-density wave, $XY$ ferromagnetic, clusters with internal $XY$-FM or antiferromagnetic spin ordering, and hemmed clusters (see text for descriptions). Simulations have been performed along the solid lines. Thick solid straight lines: $|J/U|=1$. Thick curves: Guidelines delimiting cluster phases. As $U\to +\infty$, $J_c\simeq 3.8$ separates AFM-like ${\mathrm{SDW}}^{x,y}$ and $XY$-AFM cluster phases, while $J_c\simeq -3.8$ separates $XY$-FM and $XY$-FM clusters. Dot-dashed straight lines: studies from Refs. [35] (tilted) and [37] (horizontal) (see text). We explore the metrological usability of these phases, finding $XY$-FM and $XY$-FM cluster phases especially convenient (see text).

Figure 2

Density profiles for $J=-0.1U$ and different $U$ values. Top: typical profiles in the SF and CDW (left), SDW and $XY$-FM phases (right). Friedel oscillations are present [41]. Bottom: Density profile for $U=60$ with cluster formation. Inset: same profile with $J=0$.

Figure 3

Quantum fermionic correlated phases and metrological usability in a single-shot phase diagram. The QFI (red) vs $U$ gets along the order parameters $C{C}_x(0)$ (green) and ${\sum }_i(\mathrm{\Delta }{n}^2{)}_i$ (blue) describing the building up of $XY$-FM and cluster correlations, respectively (see text).

Figure 4

QFI density $\mathrm{QFI}/N$ ($N$ number of atoms) vs $U$ for $\nu =1/4$ (red) and $1/2$ (blue), and different $J/U$ (see legend). Table: exponents fitted from $\mathrm{QFI}=k{N}^{\gamma }$ for $\nu =1/2$, $1/4$ at $J/U=-0.1$. $U$ values are chosen to correspond to the QFI maximum in the $XY$-FM phase ($U=11$) and in the large-$U$ limit for the cluster phase ($U=60$), at $\nu =1/4$. Inset figure: example of QFI scaling for $\nu =1/4$, $U=60$, and $J/U=-0.1$.