Detecting Multiparticle Entanglement of Dicke States

Recent experiments demonstrate the production of many thousands of neutral atoms entangled in their spin degrees of freedom. We present a criterion for estimating the amount of entanglement based on a measurement of the global spin. It outperforms previous criteria and applies to a wider class of entangled states, including Dicke states. Experimentally, we produce a Dicke-like state using spin dynamics in a Bose-Einstein condensate. Our criterion proves that it contains at least genuine 28-particle entanglement. We infer a generalized squeezing parameter of $-11.4(5)\mathrm{dB}$.

Figure 1

Measurement of the entanglement depth for a total number of 8000 atoms. (a) The entanglement depth is given by the number of atoms in the largest nonseparable subset (shaded areas). (b) The spins of the individual atoms add up to the total spin $\mathbf{J}$ whose possible orientations can be depicted on the Bloch sphere. Dicke states are represented by a ring around the equator with an ultralow width $\mathrm{\Delta }{J}_z$ and a large radius $J_{\mathrm{eff}}$. (c) The entanglement depth in the vicinity of a Dicke state can be inferred from a measurement of these values. The red lines indicate the boundaries for various entanglement depths. The experimental result is shown as blue uncertainty ellipses with 1 and 2 standard deviations, proving an entanglement depth larger than 28 (dashed line).

Figure 2

Preparation and detection of a Dicke-like state. (a) A Bose-Einstein condensate in the level $(F,m_F)=(1,0)$ generates clouds with the same number of atoms in the levels $(1,\pm 1)$ (1). A microwave pulse (2) transfers the atoms from $(1,-1)$ to (2,0). Optionally, a microwave pulse (3) can be used to couple the two clouds for the measurement of $J_{\mathrm{eff}}$. Finally, the atoms in the level (1,1) are transferred to (2,2) before detection. (b) The number of atoms is measured by standard absorption imaging (insets). On well-resolved resonances depending on the internal state energy, distinct spatial modes are populated with a large fraction of the total number of atoms. The black line is a Gaussian fit to guide the eye. In our experiments, we use the resonance at $\approx 28\mathrm{Hz}$.

Figure 3

Characterization of the experimentally created Dicke-like state. (a) Measurement of the width $\mathrm{\Delta }{J}_z$ for varying the total number of atoms (red solid line). Each value and its statistical uncertainty (gray shading) is calculated for a 1000-atom interval within the total number of atoms. The measured values of $\mathrm{\Delta }{J}_z$ are well below the shot noise limit (theory: black dashed line, experiment: blue dotted line) and partially explained by a lower limit of the number-dependent detection noise (black dash-dotted line). (b) The measured value of $J_{\mathrm{eff}}$ as a function of the total number of atoms almost reaches its optimal value (black dashed line). The inset shows that the normalized $J_{\mathrm{eff}}$ is slowly reduced during an additional hold time. (c) The recorded data allow for a determination of the generalized squeezing parameter as a function of the total number of atoms. At a total of 8000 atoms, it reaches a value of $-11.4(5)\mathrm{dB}$.

Figure 4

Detection of $k$-particle entanglement based on the total spin. The red solid line marks the boundary for $k$-particle entangled states with $N=8000$ and $k=28$ in the $(⟨{\widehat{J}}_{\text{eff}}/J_{\text{max}}^2⟩,(\mathrm{\Delta }{J}_z{)}^2)$ plane. As a cross-check, random states with $k$-particle entanglement are plotted as blue dots, filling up the allowed region. The criterion of Ref. [33] only detects states that correspond to points below the dashed blue line. An improved linear criterion is gained from calculating a tangent to the new boundary (dash-dotted red line).