Figure 1

(a) The clock measurement consists in free evolution with the Hamiltonian $\widehat{H}={\widehat{J}}_z$, followed by a $\pi /2$ rotation around the $x$ axis and a measurement of the number of atoms in the two internal states. (b) Sensitivity limits rescaled by the shot noise $N=2j$ for the state $\left|\mathrm{\Psi }\right(t)\rangle$ with $j=25$. We compare the maximal measurement sensitivity (2) obtained by the eigenstates of ${\widehat{J}}_y$ (blue line) to the sensitivity (3) enhanced by additional Hamiltonian measurements (red line). The enhancement is shown as the green dashed line. The upper sensitivity bound is given by the quantum Fisher information [gray line, see Eq. (6)], which reaches the value $j(2j+1)$ (gray dashed line). For comparison, we show the spin squeezing coefficient (orange dot-dashed line). The maximal enhancement is found in oversqueezed states around ${\tau }_{\mathrm{opt}}\simeq 0.94/\sqrt{j}$. (c) Normalized coefficient $c_{\widehat{H}}$, expressing the contribution $\widehat{H}$ as a function of $j$ at the point ${\tau }_{\mathrm{opt}}$ of maximal enhancement. (d) Despite the near-vanishing contribution of $\widehat{H}$, the relative gain at ${\tau }_{\mathrm{opt}}$, expressed by the ratio of Eqs. (2) and (3), increases linearly with $j=N/2$. (e) Coefficients $c_{m_y}$ of the optimal measurement observables with (${\widehat{X}}_{\mathrm{opt}}$, red dots) and without (${\widehat{X}}_{\mathrm{opt}}^{\left(0\right)}$, blue dots) measurement access to $\widehat{H}$ for $j=100$. We have $c_{m_y}=0$ for all $|m_y|\ge 30$ for both ${\widehat{X}}_{\mathrm{opt}}$ and ${\widehat{X}}_{\mathrm{opt},0}$. The normalized contribution of $\widehat{H}$ to ${\widehat{X}}_{\mathrm{opt}}$ is $c_{\widehat{H}}\lesssim 2\times {10}^{-3}$.