Achieving Heisenberg-limited metrology with spin cat states via interaction-based readout

Spin cat states are promising candidates for quantum-enhanced measurement. Here, we analytically show that the ultimate measurement precision of spin cat states approaches the Heisenberg limit, where the uncertainty is inversely proportional to the total particle number. In order to fully exploit their metrological ability, we propose to use the interaction-based readout for implementing phase estimation. It is demonstrated that the interaction-based readout enables spin cat states to saturate their ultimate precision bounds. The interaction-based readout comprises a one-axis twisting, two $\frac{\pi }{2}$ pulses, and a population measurement, which can be realized via current experimental techniques. Compared with the twisting echo scheme on spin squeezed states, our scheme with spin cat states is more robust against detection noise. Our scheme may pave an experimentally feasible way to achieve Heisenberg-limited metrology with non-Gaussian entangled states.

Figure 1

The ultimate precision scalings of different ${|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{M}}$. The circles, squares, diamonds, triangles, and inverted triangles are the numerical results with ${|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{CAT}}$ for $\theta =0,\pi /8,3\pi /16,\pi /4,7\pi /20$, respectively. The solid lines are the analytic results for the spin cat states, which perfectly agree with the numerical ones. The slopes equal $-1$, indicating that the spin cat states exhibit Heisenberg-limited precision scalings. The pentagrams and crosses denote the numerical results for MSSCS ${|\mathrm{\Psi }(15\pi /32)\rangle }_{\mathrm{M}}$ and SCS ${|\mathrm{\Psi }(\pi /2)\rangle }_{\mathrm{SCS}}$, which do not satisfy the analytic expression (12). Therefore, the dotted and dashed lines are the fitting results for ${|\mathrm{\Psi }(15\pi /32)\rangle }_{\mathrm{M}}$ and ${|\mathrm{\Psi }(\pi /2)\rangle }_{\mathrm{SCS}}$, where the slopes are $-0.83$ and $-0.5$, respectively.

Figure 2

The schematic of the interaction-based readout scheme with input state cat states. First, an input spin cat state is prepared. Then, an estimated phase is accumulated during interrogation. To extract the phase, an interaction-based readout protocol is applied, in which a nonlinear evolution is sandwiched by two $\frac{\pi }{2}$ pulses before the half-population difference measurement.

Figure 3

The phase precision obtained with spin cat states via interaction-based readout for the estimated phase in the vicinity of (a) $\varphi \sim 0$ and (b) $\varphi \sim \pi /2$. The standard derivations of the phase $\mathrm{\Delta }\varphi$ versus the nonlinear evolution $\chi t$ for spin cat states ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ (blue), ${|\mathrm{\Psi }(\pi /8)\rangle }_{\mathrm{CAT}}$ (green), ${|\mathrm{\Psi }(\pi /4)\rangle }_{\mathrm{CAT}}$ (cyan), and ${|\mathrm{\Psi }(7\pi /20)\rangle }_{\mathrm{CAT}}$ (magenta) are shown. The minimum phase precision $\mathrm{\Delta }{\varphi }_{\min }$ is obtained by optimizing the nonlinear evolution $\chi t$. In the insets, $\mathrm{\Delta }{\varphi }_{\min }$ and the corresponding optimal evolution time $t_{\mathrm{opt}}$ for different MSSCS ${|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{M}}$ are given. Here, the total particle number $N=100$. (c) The log-log scaling of the minimum phase precision versus the total particle number for spin cat states ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ (blue), ${|\mathrm{\Psi }(\pi /8)\rangle }_{\mathrm{CAT}}$ (green), ${|\mathrm{\Psi }(\pi /4)\rangle }_{\mathrm{CAT}}$ (cyan), and ${|\mathrm{\Psi }(7\pi /20)\rangle }_{\mathrm{CAT}}$ (magenta). The dotted lines with inverted triangles and the dashed lines with squares correspond to the cases of $\varphi \sim 0$ and $\varphi \sim \pi /2$, respectively.

Figure 4

(a) The optimal phase uncertainty under optimal control against detection noise $\sigma$ with spin cat states ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ (blue), ${|\mathrm{\Psi }(\pi /8)\rangle }_{\mathrm{CAT}}$ (green), ${|\mathrm{\Psi }(\pi /4)\rangle }_{\mathrm{CAT}}$ (cyan), and ${|\mathrm{\Psi }(7\pi /20)\rangle }_{\mathrm{CAT}}$ (magenta). The dotted and dashed lines correspond to the cases of $\varphi \sim 0$ and $\varphi \sim \pi /2$, respectively. Here, the total particle number is $N=100$. (b) The relative phase standard deviation $\mathrm{\Delta }\varphi /\mathrm{\Delta }{\varphi }_{\mathrm{CAT}}^Q$ versus $\sigma /\left(\mathrm{\Delta }{\widehat{J}}_z\right)$ for spin cat states ${|\mathrm{\Psi }\left(\theta \right)\rangle }_{\mathrm{CAT}}$. Here, $\mathrm{\Delta }{\varphi }_{\mathrm{CAT}}^Q\approx {\left(2\overline{M}\right)}^{-1}$, $\mathrm{\Delta }{\widehat{J}}_z\approx \overline{M}$, and the relative phase standard deviation begins to blow up when $\sigma \approx 0.5\overline{M}\approx N/2C\left(\theta \right)={\tilde{c}}_D\left(\theta \right)N$.

Figure 5

The phase precision $\mathrm{\Delta }\varphi$ versus the nonlinear evolution $\chi t$ for spin cat states ${|\mathrm{\Psi }\left(0\right)\rangle }_{\mathrm{CAT}}$ (blue), ${|\mathrm{\Psi }(\pi /8)\rangle }_{\mathrm{CAT}}$ (green), ${|\mathrm{\Psi }(\pi /4)\rangle }_{\mathrm{CAT}}$ (cyan), and ${|\mathrm{\Psi }(7\pi /20)\rangle }_{\mathrm{CAT}}$ (magenta) under dephasing. Here, we consider the accumulated phase is $\varphi \sim 0$ and after that, the interaction-based readout process suffers from dephasing with $\gamma$ the dephasing rate. The dotted, dashed, and solid lines denote the cases of $\gamma =0$, $\gamma =2\chi$, and $\gamma =6\chi$, respectively. Here, the total particle number is $N=100$.