Improved Quantum Magnetometry beyond the Standard Quantum Limit

Under ideal conditions, quantum metrology promises a precision gain over classical techniques scaling quadratically with the number of probe particles. At the same time, no-go results have shown that generic, uncorrelated noise limits the quantum advantage to a constant factor. In frequency estimation scenarios, however, there are exceptions to this rule and, in particular, it has been found that transversal dephasing does allow for a scaling quantum advantage. Yet, it has remained unclear whether such exemptions can be exploited in practical scenarios. Here, we argue that the transversal-noise model applies to the setting of recent magnetometry experiments and show that a scaling advantage can be maintained with one-axis-twisted spin-squeezed states and Ramsey-interferometry-like measurements. This is achieved by exploiting the geometry of the setup that, as we demonstrate, has a strong influence on the achievable quantum enhancement for experimentally feasible parameter settings. When, in addition to the dominant transversal noise, other sources of decoherence are present, the quantum advantage is asymptotically bounded by a constant, but this constant may be significantly improved by exploring the geometry.

Popular Summary

Precisely estimating parameters is fundamental in science: measurements of tiny movements in detectors looking for gravitational waves, atomic frequencies used as time standards, and magnetic fields for brain imaging. The precision in such tasks can be significantly enhanced by harnessing quantum effects, providing an advantage that grows with the number of probe particles. However, this enhancement is very sensitive to noise, which is always present in practice. It has therefore been unclear what advantages can be gained in practically relevant scenarios. In this work, we show that a scaling quantum advantage can be maintained when measuring magnetic fields using a realistic atomic magnetometer.

We demonstrate that a scaling quantum advantage can be recovered in the setting of a recent experiment using entangled states of cesium atoms to measure a weak magnetic field; the required measurement techniques have already been implemented experimentally. We show that the setup geometry (the orientations of the atomic spin and the magnetic field) and the time during which the probe particles sense the magnetic field play crucial roles and that their optimization makes it possible to obtain robust measurements even in the face of the dominant noise. Moreover, we also show that even when other noise sources are present, the quantum advantage can still be large even though it no longer grows indefinitely with the number of probe particles; it is limited by a constant.

Our results, which show that quantum-enhanced metrology maintains its relevance even in the presence of noise, may have applications to magnetometry with other systems such as nitrogen vacancy centers in diamond.

Figure 1

Atomic magnetometry setup. An ensemble of atoms is placed in a strong magnetic field $B$ which induces a level splitting between the magnetic sublevels. The atoms are used to sense a weak field $B_{\mathrm{rf}}$ in the plane perpendicular to $B$, which rotates in this plane with a frequency matched to the Larmor precession induced by $B$. We consider two cases for the state preparation and readout. Scenario (a) corresponds to the geometry of the experiment [3]. All atoms are initially pumped to an extreme magnetic sublevel $m=F$ creating a coherent spin state aligned with $B$. The state is then squeezed to make it more sensitive to the evolution induced by $B_{\mathrm{rf}}$. In a frame rotating around $B$ at the Larmor frequency, the state can be depicted as shown in the lower part. $B_{\mathrm{rf}}$ points then along $z$ and induces a rotation around the $z$ axis. The state is squeezed in $y$ and $B_{\mathrm{rf}}$ is estimated from a measurement of the collective-spin component ${\widehat{J}}_y$. Scenario (b) is similar, but the state is initially perpendicular to $B$. In the rotating frame, it is squeezed in $x$ and ${\widehat{J}}_x$ is measured. The dominant noise in both cases comes from individual atomic motion causing variations in the effective magnetic field and hence the energy splitting. This results in uncorrelated dephasing noise in the direction of $B$; the impact on the collective spin is schematically illustrated by the inner prolate spheroids. Importantly, the noise preserves the spin along $B$ but shrinks it in the perpendicular directions.

Figure 2

Mean-squared error of estimation as a function of (a) squeezing and (b) interrogation time, for parameters corresponding to Ref. [3], $N={10}^{11}$, $\gamma =67\mathrm{Hz}$, $\omega ={3.6\times 10}^{-3}\mathrm{Hz}$. In (a) the interrogation time is fixed at $t=1\mathrm{ms}$, while in (b) the squeezing is fixed at $-8\mathrm{dB}$ (indicated by the dotted lines, these values correspond to the numbers discussed in the text). The results for scenario Fig. 1 (yellow line), scenario Fig. 1 (blue line), and for CSS (dashed gray line) are shown.

Figure 3

The mean-squared error of estimation rescaled by the particle number such that SQL-like scaling is horizontal. The curves for scenario (a) (yellow line) and (b) (red line) with 5% of parallel noise are shown, and for scenario (b) under purely transversal noise (blue line), along with their asymptotes (dashed line), respectively, $2\gamma (1-\epsilon )N$ from (25), $2\gamma \epsilon$ from the parallel noise component, and $2\omega /3N^{5/4}$ from Eq. (24). We also show the performance Eq. (26) of a CSS without squeezing (gray dashed line). The locations where the OATSS strategies reach 90% of their asymptotic gain over this CSS strategy are indicated for scenarios (a) (open triangle) and (b) (closed triangle).

Figure 4

Circles: Results of minimizing $M$ in Eq. (e2) over $\mu$, $t$ versus $N$ for $\{\gamma ,\omega \}=\{10,0.03\}$, $\{1.0,0.3\}$, and $\{0.1,0.03\}$ (top to bottom). Dashed lines: Plots of $2\gamma /N$ for the same values of $\gamma$.