Entanglement, Nonlinear Dynamics, and the Heisenberg Limit

We show that quantum Fisher information provides a sufficient condition to recognize multiparticle entanglement in an $N$ qubit state. The same criterion gives a necessary and sufficient condition for sub-shot-noise phase sensitivity in the estimation of a collective rotation angle $\theta$. The analysis therefore singles out the class of entangled states which are useful to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the nonlinear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.

Figure 1

(a) Plot of Eqs. (7) (dash-dotted black line), (8) (solid blue line), and (9) (dashed red line) as a function of $\tau \sqrt{N}$ (here $N={10}^4$). The states having ${\chi }^2$, ${\xi }^2<1$ (i.e., below the horizontal dotted line in the figure) are useful for quantum interferometry. (b) Phase sensitivity as a function of the total number of particles $N_T=Np$. Filled circles are results of numerical simulations, the solid black line is the Heisenberg limit $\Delta \theta =8.9/N_T$, obtained for $p=p_{\mathrm{opt}}$, and the dashed blue line is the shot noise $\Delta \theta =1/\sqrt{N_T}$. Inset: $\Delta \theta$ as a function of the number of measurements $p$, for fixed values of $N_T$. The optimal working point (minimum of each curve) is $p_{\mathrm{opt}}=20$, independent from $N_T$.

Figure 2

(a–c) Distributions $P(\mu |j,\theta ,\tau )$ plotted as a function of $\theta$ along circles of radius $\mu$ (taken as a continuum variable) at three different times during the nonlinear evolution: (a) $\tau =0$, (b) $\tau =\pi /4$, and (c) $\tau =\pi /2$. The solid lines delimit the typical size of the substructures. (d–f) $P(\mu |j,\theta ,\tau )$ as a function of $\theta$, and for (d) $\tau =0$, $\mu =7.5$, (e) $\tau =\pi /4$, $\mu =2.5$, and (f) $\tau =\pi /2$, $\mu =3.5$. Here $N=15$.