Entanglement-Based dc Magnetometry with Separated Ions^*

We demonstrate sensing of inhomogeneous dc magnetic fields by employing entangled trapped ions, which are shuttled in a segmented Paul trap. As sensor states, we use Bell states of the type $|\uparrow \downarrow ⟩+{\mathrm{e}}^{\mathrm{i}\phi }|\downarrow \uparrow ⟩$ encoded in two ${}^{40}{\mathrm{Ca}}^{+}$ ions stored at different locations. The linear Zeeman effect leads to the accumulation of a relative phase $\phi$, which serves for measuring the magnetic-field difference between the constituent locations. Common-mode magnetic-field fluctuations are rejected by the entangled sensor state, which gives rise to excellent sensitivity without employing dynamical decoupling and therefore enables accurate dc sensing. Consecutive measurements on sensor states encoded in the $S_{1/2}$ ground state and in the $D_{5/2}$ metastable state are used to separate an ac Zeeman shift from the linear dc Zeeman effect. We measure magnetic-field differences over distances of up to 6.2 mm, with accuracies down to 300 fT and sensitivities down to $12\mathrm{pT}/\sqrt{\mathrm{Hz}}$. Our sensing scheme features spatial resolutions in the 20-nm range. For optimizing the information gain while maintaining a high dynamic range, we implement an algorithm for Bayesian frequency estimation.

Popular Summary

Precise magnetic-field sensors are essential for many applications in modern technology as well as applied and fundamental research. With recent advances in quantum technology, highly sensitive magnetometers based on the spins of single electrons have become available. In an external magnetic field, the spin orientation wobbles at a frequency that is set by the field’s strength. This frequency can be determined with high accuracy by creating a superposition of the spin-up and spin-down states—states where quantum mechanics allows the spin to point up and down at the same time. Being highly sensitive to magnetic fields, these states are fragile and easily destroyed by noise. We have designed a magnetometer that is resilient to specific sources of noise by using quantum entanglement.

We entangle the valence electrons of two calcium ions such that their spins are always aligned in opposite directions, while the state of each individual spin is completely undetermined. This shields the quantum state from magnetic noise acting on both ions so that the difference in the magnetic field between the two locations can be measured with high precision. We use a segmented linear ion trap to move entangled ions freely along the trap axis, separating them by macroscopic distances of up to 6.2 mm. Our measurement scheme enables us to measure inhomogeneous direct-current magnetic fields with spatial resolution at the 20-nm level, reaching accuracies down to 310 fT.

For future work, our device could be extended to measure magnetic fields in any direction (not just along the trap axis) as well as to probe the magnetic properties of miniscule objects such as additional ions or single-molecule magnets.

Figure 1

Experimental procedure for measurements of inhomogeneous magnetic fields. After creation of the sensor state at the LIZ, the two constituent ions are separated and shuttled to the desired trap segments $L$ and $R$. In order to measure the accumulated phase during the interrogation time $T$, the ions are individually shuttled to the LIZ to perform basis rotations that allow for state read-out via electron shelving and fluorescence detection in either the ${\widehat{X}}_1{\widehat{X}}_2$ or ${\widehat{X}}_1{\widehat{Y}}_2$ basis. For basis rotations, electron shelving, and fluorescence detection, the relevant energy levels are shown.

Figure 2

Incremental measurement of the phase accumulation rate $\mathrm{\Delta }\omega$ at an ion distance of $d=6.2\mathrm{mm}$. A linear fit to measurements of the accumulated phase $\phi$ at predefined interrogation times (top part), and the fit residuals $\delta \phi$ for each phase measurement are shown (bottom part). For each point, measurements of both operators have been repeated 50 times.

Figure 3

(a) Sensor state contrast $C$ versus interrogation time $T$ at the maximum ion distance of $d=6.2\mathrm{mm}$ (red dots) and at an ion distance of $d=4.2\mu \mathrm{m}$ (blue squares). For illustration, the black curve and gray region indicate a third-order polynomial fit to a separate read-out fidelity measurement and its confidence bands. (b) Simultaneous drift of the measured frequency difference for ion distances $d=6.2\mathrm{mm}$ (blue circles) and $d=3.2\mathrm{mm}$ (purple triangles) over a duration of about 6 hours with an interrogation time of $T=150\mathrm{ms}$. For $d=3.2\mathrm{mm}$, the measured drift is suppressed by a factor of about 1.6 as compared to the maximum ion distance.

Figure 4

Bayesian evaluation of a measurement at an ion distance of $d=800\mu \mathrm{m}$. (a) Update functions (colored bars) in $(\mathrm{\Delta }\omega ,{\phi }_0)$-parameter space of the first four phase measurements with $N=M=50$ repetitions of the experimental procedure and the posterior PDF (purple ellipse) after these iterations. (b) Interrogation times (blue bars) and error of the frequency determination (green points) for each phase measurement versus elapsed measurement time. (c) The posterior PDFs at maximum interrogation time for subsequent phase measurements are visualized by open ellipses, corresponding to the 39.4%-credible regions. One can see that in this regime, the measurement is tracking the drift of the magnetic-field difference (see text). The posterior PDF pertaining to the last measurements is indicated as a density plot.

Figure 5

Frequency differences from the LIZ. A sensor state is prepared, and the probe ion is moved to an arbitrary desired position. At the same time, the reference ion is either moved to (a) segment 1 or (b) segment 32 to maximize the ion distance. The results for each segment are shown in panel (c). In diagram (d), high-precision measurements close to the LIZ yield standard errors for the ion position and frequency difference of about 10 nm and $2\pi \times 10\mathrm{mHz}$, respectively.

Figure 6

(a) Relevant transitions for creating the $D_{5/2}$ sensor state. (b) Theoretical shot-noise limited sensitivity of magnetic-field measurements versus interrogation time $T$ for both $S_{1/2}$ and $D_{5/2}$ sensor states, showing how spontaneous decay limits the sensitivity of the $D_{5/2}$ sensor state for long interrogation times (see text).

Figure 7

Frequency shift of the $S_{1/2}$ sensor state due to the ac Zeeman effect along the trap axis. Within the error bars of about $2\pi \times 0.2\mathrm{Hz}$, the ac Zeeman shift is similar for both $S_{1/2}$ and $D_{5/2}$ sensor states.

Figure 8

Overview of recently developed magnetic-field sensors in a size-sensitivity diagram [2, 3, 5, 6, 9, 10, 11, 14, 15, 16, 20, 21, 24, 60]: The shaded area represents the accessible regime for dc and quasistatic $<1\mathrm{Hz}$ (blue) or ac (red) fields. The regime for dc sensing is substantially extended by this work.