Enhanced Magnetic Sensitivity with Non-Gaussian Quantum Fluctuations

The precision of a quantum sensor can overcome its classical counterpart when its constituents are entangled. In Gaussian squeezed states, quantum correlations lead to a reduction of the quantum projection noise below the shot noise limit. However, the most sensitive states involve complex non-Gaussian quantum fluctuations, making the required measurement protocol challenging. Here we measure the sensitivity of nonclassical states of the electronic spin $J=8$ of dysprosium atoms, created using light-induced nonlinear spin coupling. Magnetic sublevel resolution enables us to reach the optimal sensitivity of non-Gaussian (oversqueezed) states, well above the capability of squeezed states and about half the Heisenberg limit.

Figure 1

(a) Scheme of the experimental setup. Starting with a coherent state of the electronic spin of dysprosium atoms aligned with the south pole (b), we induce nonlinear dynamics using an off-resonant laser beam (c). We then perform a spin rotation (d) followed by a projective measurement along $z$ using a magnetic field gradient (e). A typical absorption image is shown in (f).

Figure 2

(a) Projection probabilities ${\mathrm{\Pi }}_m(\widehat{\mathbf{n}})$ along $\widehat{\mathbf{n}}=\widehat{\mathbf{z}}$ and $\widehat{\mathbf{n}}\perp \widehat{\mathbf{z}}$, for an interaction time $t=0.83(1)\tau$, with $\tau =(\sqrt{2J}\chi {)}^{-1}$. The solid (dotted) red line indicates the magnetization $m_{\widehat{\mathbf{n}}}$ (values of $m_{\widehat{\mathbf{n}}}\pm \mathrm{\Delta }{J}_{\widehat{\mathbf{n}}}$). (b) Magnetization $m_z$ as a function of the interaction time $t$. (c) Maximum and minimum spin projection variances $\mathrm{\Delta }{J}_{\max }^2$ and $\mathrm{\Delta }{J}_{\min }^2$ (blue dots and red squares, respectively). (d) Comparison between the uncertainty product $\mathrm{\Delta }{J}_{\max }\mathrm{\Delta }{J}_{\min }$ and the half mean spin length $|m_z|/2$. The solid lines in (b)–(d) correspond to the one-axis twisting model predictions. In all figures of this Letter error bars represent the $1\text{−}\sigma$ statistical uncertainty determined using a bootstrap sampling method.

Figure 3

(a),(b) Evolution of the projection probabilities ${\mathrm{\Pi }}_m$ upon a Larmor rotation of angle $\theta$ around the direction ${\widehat{\mathbf{n}}}_{\max }$ of maximum sensitivity, for a coherent state (a), a squeezed state (b) [interaction time $t=0.58(2)\tau ]$ and an oversqueezed state [c, $t=2.01(1)\tau$]. The solid (dotted) red line corresponds to the magnetization $m_z$ (values of $m_z\pm \mathrm{\Delta }{J}_z$) computed from the ${\mathrm{\Pi }}_m$ values. (d) Usual metrological gain $\overline{G}$ and value of $1/{\xi }_{\mathrm{R}}^2$ deduced from the Figs. 2 and 2 data as a function of the interaction time $t$. The solid line corresponds to the one-axis twisting model prediction.

Figure 4

(a),(b) Projection probabilities ${\mathrm{\Pi }}_m$ measured for small rotation angles $\theta$ around $\widehat{\mathbf{b}}=\cos \varphi \widehat{\mathbf{x}}+\sin \varphi \widehat{\mathbf{y}}$, with $\varphi =0.10(2)\pi \simeq {\varphi }_{\min }$ and $0.56(2)\pi \simeq {\varphi }_{\max }$, respectively, for an interaction time $t=0.835(5)\tau$. Each probability is the average of three independent experiments. (c) Hellinger distances $d_{\mathrm{H}}^2(\theta ,0)$ deduced from (a),(b), together with a quadratic fit of the $\text{small}\text{−}\theta$ data. (d) Metrological gain $G$ deduced from the curvature of the Hellinger distance as a function of the azimutal angle $\varphi$ (gray circles). The black line is a sine fit of the data. Gray dots correspond to the upper bound $2\mathrm{\Delta }{J}_{\widehat{\mathbf{b}}}^2/J$ extracted from the Fig. 2 data. (e) Measured metrological gain $G$ (blue dots) as a function of the interaction time $t$. The gray diamonds correspond to the upper bound $2\mathrm{\Delta }{J}_{\max }^2/J$ [from Fig. 2], and the red squares are the $\overline{G}$ values from Fig. 3. The solid blue and dashed red lines correspond to the gains $G$ and $\overline{G}$ expected from the one-axis twisting model.

Figure 5

(a)–(c) Husimi $Q$ function measured for a coherent, squeezed, and oversqueezed spin states (a)–(c), achieved after evolution times $t/\tau =0$, 0.48(2) and 2.2(1), respectively. The Bloch sphere is parametrized by the spherical angles $(\mathrm{\Theta },\mathrm{\Phi })$ associated with the frame $(y,z,x)$. The red stars in (c) indicate the fitted zeros of the Husimi function. (d),(f) Husimi (d) and Wigner (f) functions of the quantum state expected from the one-axis twisting model for an interaction time $t=2.2\tau$. (e) Wigner function reconstructed from the same data used in (c).