Experimental Proposal on Scalable Radio-Frequency Magnetometer with Trapped Ions

Abstract

Quantum magnetometry represents a fundamental component of quantum metrology, where trapped-ion systems have achieved $\rm{pT}/\sqrt{\rm{Hz}}$ sensitivity in single-ion radio-frequency magnetic field measurements via dressed states based dynamical decoupling. Here we propose a scalable trapped-ion magnetometer utilizing the mixed dynamical decoupling method, combining dressed states with periodic sequences to suppress decoherence and spatial magnetic field inhomogeneity. With numerical simulations for a $10^4$ ion system with realistic experimental parameters, we demonstrate that a sensitivity of 13 $\rm{fT}/\sqrt{\rm{Hz}}$ for the radio-frequency field could be reached. Such a sensitivity could be obtained via robust resilience to magnetic field drift noise and inhomogeneity, where coherence time could be extended to the order of several minutes on average. This method enables scalable trapped-ion magnetometry, demonstrating its potential as a robust and practical solution for advancing quantum sensing applications.

Figures (5)

• Figure 1: $^{171}$Yb$^+$ level scheme and time sequences for dressed states and MDD methods in RF magnetometry. (a) Hyperfine levels of the $^{171}$Yb$^+$ ground states used as the atomic sensor. Two resonant MW fields with identical Rabi frequency $\Omega_{mw}$ are applied, with the signal RF field (Rabi frequency $\Omega_s$) and echo RF field (Rabi frequency $\Omega_\pi$) driving the transition from $\left| 0 \right\rangle$ to $\left| +1 \right\rangle$. (b) The corresponding RF transition is equal to the transition from $\left| 0 \right\rangle$ to the dressed state $\left| D \right\rangle$ with Rabi frequency $\Omega_s/\sqrt{2}$ (or $\Omega_\pi/\sqrt{2}$) in the dressed states basis {$\left| 0 \right\rangle$, $\left| D \right\rangle$, $\left| u \right\rangle$, $\left| d \right\rangle$}. (c) For the dressed states method, both resonant MW fields are continuously applied during the period when the signal RF field drives the transitions. (d) In the MDD method, a CPMG-concatenated sequence is implemented. Specifically, several RF $\pi$-pulses are introduced during the RF signal intervals to induce a population inversion between the $\left| 0 \right\rangle$ and $\left| D \right\rangle$ states. Both (c) and (d) omit the state preparation at the beginning and the detection process at the end of the sequence.
• Figure 2: Coherence time $T_2$ of single atomic sensor under dressed states and MDD methods. (a) Coherence time $T_2$ of a single atomic sensor under varying magnetic field noise amplitudes $\Delta B$ and RF signal strengths $\Omega_s$ for the MDD method, showing an obvious enhancement compared to the dressed states method. The Rabi frequency of the $\pi$-pulse $\Omega_\pi$ is set to satisfy $t_\pi=6.3$ ms and the effective evolution time is set to $t_s=20$ ms here to achieve the optimal coherence time. (b) Coherence time $T_2$ for the dressed states method, where the Rabi frequencies of two dressed microwave fields are maintained at $\Omega_{mw}=2\pi\times25$ kHz, consistent with that in (a). (c) For the single ion, MDD method demonstrates a better robustness compared to dressed states method under magnetic field drifts. (d) Optimal coherence time $T_2$ for the MDD method as a function of the $\pi$-pulse duration $t_\pi$ (or equivalently, the Rabi frequency $\Omega_\pi$) under fixed effective signal strength $\Omega_0=2\pi\times$1 Hz and magnetic field noise magnitude $\Delta B=0.05$ µ T. The coherence time $T_2$ reaches an optimal value for specific durations of the $\pi$-pulse, illustrating the impact of the duty cycle of the sequence on coherence enhancement.
• Figure 3: Axial magnetic field inhomogeneity of a Coulomb crystal with 10,000 ions in a Paul trap. (a) A Coulomb crystal consisting of 10,000 ions trapped in a Paul trap, sized 1.45 mm $\times$ 0.15 mm $\times$ 0.10 mm, with assumed trap frequencies of $\{ \omega_x, \omega_y, \omega_z \}=2\pi\times\{$0.7, 0.58, 0.12$\}$ MHz. The color describes the inhomogeneous magnetic field experienced by each ion. (b) Axial magnetic field inhomogeneity near the x=0 plane, and the crystal experiences magnetic field non-uniformity of about $10^{-4}$ along the axial direction. (c) Axial magnetic field inhomogeneity near the z=0 plane, corresponding to a radial inhomogeneity of $10^{-6}$.
• Figure 4: Effect of magnetic field inhomogeneity with shot-to-shot temporal noise on measurement. (a) Rabi oscillations under magnetic field inhomogeneity with temporal noise using the dressed states method. The coherence time of the single ion at the center of the crystal (blue curve) is $T_2=0.62$ s, while the coherence time of average 10,000 ions (orange curve) is $T_2=0.36$ s with effective signal strength $\Omega_0=2\pi\times 1$ Hz. (b) Similar to (a), but for the MDD method. The contrast of the Rabi oscillations remains perfect, and the coherence time of the single ion increases to $T_2=620$ s, with the average coherence time of $10,000$ ions reaching $T_2=250$ s. (c)(d) Same as (a)(b), except with different effective signal strength $\Omega_0=2\pi\times 2.5$ Hz. The coherence times in different cases become $T_2=1.4$ s (single ion) and $T_2=1.2$ s (10,000 ions) when using dressed states method, and to be $T_2=150$ s (single ion) and $T_2=75$ s (10,000 ions) when using MDD method. Here we apply $\Omega_{mw}=2\pi\times 25$ kHz, $t_s=20$ ms and $t_\pi=6.3$ ms.
• Figure 5: Sensitivity of RF magnetometer with different methods. Enhancements to the standard quantum limit sensitivity are demonstrated through measurement time optimization and scalable atomic sensor configurations. The scatters are plotted according to simulation from sampling, and the solid curves represent the fitting result assuming the decoherence can be approximated by an exponential power law $e^{-(t/T_2)^n}$. As a result, an RF magnetometer using dressed states method with a single ion achieves an optimal sensitivity of 8.3 pT/$\sqrt{\text{Hz}}$ , while the MDD method yields 0.67 pT/$\sqrt{\text{Hz}}$ for a single sensor configuration. In a scalable system with 10,000 ions, both two methods exhibit improved sensitivity, although the coherence times are slightly reduced. Specifically, Dressed states method reaches 0.14 pT/$\sqrt{\text{Hz}}$, and the MDD method achieves 13 fT/$\sqrt{\text{Hz}}$ under spatial inhomogeneity and temporal noise.