Measurement of Identical Particle Entanglement and the Influence of Antisymmetrization

We explore the relationship between symmetrization and entanglement through measurements on few-particle systems in a multiwell potential. In particular, considering two or three trapped atoms, we measure and distinguish correlations arising from two different physical origins: antisymmetrization of the fermionic wave function and interaction between particles. We quantify this through the entanglement negativity of states, and the introduction of an antisymmetric negativity, which allows us to understand the role that symmetrization plays in the measured entanglement properties. We apply this concept both to pure theoretical states and to experimentally reconstructed density matrices of two or three mobile particles in an array of optical tweezers.

Figure 1

$\mathcal{AN}$ and state reconstruction. (a) The full Hilbert space can be grouped into subsets of normalized density matrices [$\mathrm{Tr}(\sigma )=1$], positive-semidefinite density matrices ($\sigma \ge 0$), and antisymmetric density matrices representing fermionic states. The entanglement of a fermionic state ${\rho }_{\mathcal{A}}$ can be quantified using the $\mathcal{AN}$, which we formulate as a minimization of the standard negativity. The optimization is performed over all states with a fixed projection onto ${\rho }_{\mathcal{A}}$, including those beyond the antisymmetric subset in the full space of normalized, positive-semidefinite density matrices. In an experimental reconstruction, noise may lead to a measured density matrix ${\rho }_{\exp }$ outside the set of physical states, which can be transformed into a positive-semidefinite antisymmetric density matrix via Bayesian maximum likelihood estimation. (b) In the experiment we initialize atoms in an array of optical tweezers and measure single-atom and spin-resolved in situ populations and momentum-space distributions. (c) Example for in situ and momentum correlation functions for two atoms in two tweezers. (d) We decompose momentum correlation functions into a set of trigonometric basis functions, shown for $N=2$ (upper row) and $N=3$ (lower row). The weight of the basis functions is used to constrain off-diagonal entries of the density matrix. The bare experimentally measured density matrix may lie outside the space of physical density matrices. We use Bayesian state estimation to construct a physical state ${\rho }_{\mathcal{A}}$ [cf. (a)].

Figure 2

Momentum correlation measurements for noninteracting fermions, where correlations arise due to antisymmetry of the wave function only. (a) For the state $|{\mathrm{\Psi }}_a⟩=|L\uparrow ⟩|R\uparrow ⟩$, second-order interference during time of flight leads to fermionic antibunching, i.e., strong correlations along the antidiagonal of ${\xi }_{\uparrow \uparrow }^{(2)}$. The fermions are never found in the same momentum state, and avoid each other at characteristic distances. (b) Fermionic antibunching is also present for three particles initialized in the state $|{\mathrm{\Psi }}_b⟩=|L\uparrow ⟩|C\uparrow ⟩|R\uparrow ⟩$. (c) The two-fermion state $|{\mathrm{\Psi }}_c⟩=|L\to ⟩|R\to ⟩$, with both particles polarized in the $x$ direction, exhibits antibunching between all spin combinations. $|{\mathrm{\Psi }}_c⟩$ is created from $|{\mathrm{\Psi }}_a⟩$ by radio-frequency rotation. $P_{\sigma \kappa \tau }$ gives the probability of measuring the respective spin combination.

Figure 3

Entanglement of two spinful particles in a tunnel-coupled double well as a function of interaction strength. The correlation functions shown illustrate the emergence of second-order momentum correlations upon increasing the interaction. The momentum correlation function only factorizes at $U/J=0$. This behavior is reflected in the lower plot, which shows both the standard negativity and the antisymmetric negativity as a function of $U/J$ calculated from experimentally obtained density matrices (circles) and the theoretical expectations from numerical exact diagonalization (solid lines) for the ground state of a double-well potential.

Figure 4

Multipartite generalization of $\mathcal{AN}$. (a) Illustration of investigated three-atom states and measured correlator $C_{\downarrow \uparrow \uparrow }(d_1,d_2)$ for the experimentally realized state (i) at $U/J=16$. (b) Negativity $\mathcal{N}$ for the two states shown in the legend for pure theoretical density matrices and for the experimentally reconstructed state. Shaded regions indicate negativity witnesses as discussed in the main text. (c) Antisymmetric negativity $\mathcal{AN}$ for pure theoretical density matrices of both states. The gray dotted line at 0.3 indicates the $\mathcal{AN}$ obtained from a highly entangled three-particle Dicke state [44].