Entanglement and extreme planar spin squeezing

We introduce an entanglement-depth criterion optimized for planar quantum-squeezed (PQS) states. It is connected with the sensitivity of such states for estimating a phase generated by rotations about an axis orthogonal to its polarization. We compare numerically our criterion with the well-known extreme spin-squeezing condition of Sørensen and Mølmer [Phys. Rev. Lett. 86, 4431 (2001)] and show that our condition detects a higher depth of entanglement when both planar spin variances are squeezed below the standard quantum limit. We employ our theory to monitor the entanglement dynamics in a PQS state produced via quantum nondemolition measurements using data from a recent experiment [Phys. Rev. Lett. 118, 233603 (2017)].

Figure 1

(a) Bloch-sphere representation of: (yellow horizontal spheroid) spin-coherent states, including uncertainty in $J_y$ arising from Poissonian fluctuations in the preparation; (blue vertical disk) a SSS produced by squeezing the $J_z$ variance; (red vertical spheroid) a PQS state produced by squeezing both $J_z$ and $J_y$; (green horizontal disk) a Dicke state. (b) Sensitivity advantage of a PQS state compared to a SSS in detecting an unknown phase $\varphi$. The dashed black circles indicate the shot-noise limit $\mathrm{\Delta }\varphi =1/\sqrt{N}$. The SSS provides enhanced sensitivity for detecting phases around $\varphi \simeq 0$ but reduced sensitivity for phases around $\varphi \simeq \pm \pi /2$. In contrast, although the sensitivity of the PQS state is slightly worse around $\varphi \simeq 0$, it provides enhanced sensitivity for all phases $0\le \varphi \le 2\pi$.

Figure 2

Lower bounds to ${\left(\mathrm{\Delta }{L}_{\parallel }\right)}^2/kj$ as functions of $\langle L_{\parallel }\rangle /kj$ for a system of $k$ spin-$j$ particles. The cases of $k=4$ and $j=1$ are shown. (The dashed curve) The function $G_4^{\left(1\right)}\left(X\right)$. (The solid line) The function $G_4^{\mathrm{sy}}\left(X\right)$ computed on the symmetric subspace. The convex hull of $G_4^{\left(1\right)}\left(X\right)$, denoted by ${\mathcal{G}}_4^{\left(1\right)}\left(X\right)$ is a linear function for $X\le X_{min}=\mathrm{argmin}[G_4^{\mathrm{sy}}\left(X\right)/X]$, whereas for $X>X_{min}$ it coincides with $G_4^{\mathrm{sy}}\left(X\right).M$ denotes the point of the curve for which $X=X_{min}$. The straight line provides a lower bound on ${\mathcal{G}}_4^{\left(1\right)}\left(X\right)$. (The inset) The parameter ${\zeta }_J^2$ as a function of $J$.

Figure 3

Lower bound on ${\left(\mathrm{\Delta }{J}_z\right)}^2$ given by the (blue) extreme spin-squeezing and (red) planar-squeezing criteria taken for $k=5$ and $j=1$ as a function of the ratio between the two planar spin variances and the in-plane polarization. Our criterion detects a depth of entanglement higher than that of Sørensen-Mølmer for the parameter values for which the red plot is above the blue one.

Figure 4

Top: The shaded curves: lower bound for the number of atoms in fully entangled groups of at least $(k+1)$ particles. From top to bottom: $k=1\text{–}4$. Bottom: The squeezing parameter ${\xi }_{\parallel }^2$ as a function of the number of photons $N_L$ used in the QND measurement. The orange shaded area represents the $\pm 1\sigma$ confidence interval.