Exploiting Nonclassical Motion of a Trapped Ion Crystal for Quantum-Enhanced Metrology of Global and Differential Spin Rotations

We theoretically investigate prospects for the creation of nonclassical spin states in trapped ion arrays by coupling to a squeezed state of the collective motion of the ions. The correlations of the generated spin states can be tailored for quantum-enhanced sensing of global or differential rotations of subensembles of the spins by working with specific vibrational modes of the ion array. We propose a pair of protocols to utilize the generated states and demonstrate their viability even for small systems, while assessing limitations imposed by spin-motion entanglement and technical noise. Our work suggests new opportunities for the preparation of many-body states with tailored correlations for quantum-enhanced metrology in spin-boson systems.

Figure 1

(a) Example modes of a 1D chain. Arrows qualitatively indicate the ion participation in each mode. (b) Sequence for NR protocol. (c) Metrological gain $N(\mathrm{\Delta }\theta {)}^2$ as a function of boson squeezing ${\xi }_b^2$. The performance achieved with measurements of $\widehat{M}={\widehat{S}}_{z,+}$ (CM, dot-dashed line), ${\widehat{\mathcal{S}}}_z$ (B, dot-dashed line) and ${\widehat{S}}_{z,-}$ (B, faded dot-dashed line) is compared to the QCRB given by the QFI of the full system, $N(\mathrm{\Delta }\theta {)}^2=N/F_Q$ (solid lines). We also compare against the spin-only QFI via $N/F_{Q,s}$ (dotted lines). (d) Spin-boson entanglement $S_{\mathrm{s}\mathrm{b}}$ [colors same as (c)]. Panels (c) and (d) use $N=6$.

Figure 2

(a) Sequence for SA protocol. The final steps of the IBR depend on the coupled mode and measurement. (b) Metrological gain as a function of boson squeezing ${\xi }_b^2$ using $\widehat{M}={\widehat{S}}_{z,+}$ (CM and B, dot-dashed blue and red lines) and ${\widehat{S}}_{z,-}$ (B, faded dot-dashed red line). We also plot the QCRB $N/F_Q$ (faded blue and red lines). Calculations use $N=6$. (c) Scaling of optimal metrological gain with ion number $N$ for NR (circles) and SA (squares) protocols for the CM (blue) and B (red) modes. We distinguish ${\widehat{S}}_{z,-}$ (faded) and ${\widehat{\mathcal{S}}}_z$ (darker) measurements for the B mode NR protocol. Data for this panel is obtained using a truncated Wigner approximation [43, 52].

Figure 3

Optimal metrological gain as a function of phase noise magnitude $\sigma$ for NR (dot-dashed lines) and SA (solid lines) protocols with $N=6$ and $N=20$ for CM mode. Faded lines are approximate analytic predictions [43].