Surpassing the Energy Resolution Limit with Ferromagnetic Torque Sensors

We discuss the fundamental noise limitations of a ferromagnetic torque sensor based on a levitated magnet in the tipping regime. We evaluate the optimal magnetic field resolution taking into account the thermomechanical noise and the mechanical detection noise at the standard quantum limit. We find that the energy resolution limit, pointed out in recent literature as a relevant benchmark for most classes of magnetometers, can be surpassed by many orders of magnitude. Moreover, similarly to the case of a ferromagnetic gyroscope, it is also possible to surpass the standard quantum limit for magnetometry with independent spins, arising from spin-projection noise. Our finding indicates that magnetomechanical systems optimized for magnetometry can achieve a magnetic field resolution per unit volume several orders of magnitude better than any conventional magnetometer. We discuss possible implications, focusing on fundamental physics problems such as the search for exotic interactions beyond the standard model.

Figure 1

(a) Model and conventions adopted. A ferromagnet with spin $\mathit{S}$ and magnetic moment $\mathit{\mu }=-{\gamma }_0\mathit{S}$ is trapped along the $x$ axis. It can harmonically librate around $z$ by an angle $\alpha$ and around $y$ by an angle $\beta$. Application of an oscillating magnetic field $\mathit{B}$ along $y$ will produce a torque $\mathit{\tau }=\mathit{\mu }\times \mathit{B}$ along $z$, driving the librational motion in the $\alpha$ degree of freedom. (b) Scheme of a ferromagnet above an infinite superconducting plane (gray region) in the Meissner regime. In this implementation, harmonic confinement of the $\beta$ angle arises naturally due to the interaction with an image dipole ${\mathit{\mu }}^{\prime }$.

Figure 2

(a) Magnetic field sensitivity as a function of frequency for a levitated NdFeB magnetic sphere with radius $R=30\mu \mathrm{m}$, $f_{\alpha }=1.88\mathrm{Hz}$: ERL (blue dotted line), thermal noise at $Q={10}^7$, $T=4.2\mathrm{K}$ (red dashed-dotted) and at $T=50\mathrm{mK}$, $Q={10}^9$ (red dashed), SQL on mechanical torque (green solid), spin projection noise for independent spins (thin black dotted). The vertical line represents the Einstein-de Haas frequency $f_I=0.1f_{\alpha }$. (b) Subresonant magnetic field sensitivity as a function of radius $R$, with $f_{\alpha }=10f_I$. Lines corresponding to ERL, thermal, SQL, and spin-projection as in (a). The vertical line marks the radius used in (a). Horizontal lines correspond to effective magnetic fields corresponding to relativistic frame dragging effects, as discussed in Ref. [5]: de Sitter effect (purple dashed), Lense-Thirring effect (orange solid).

Figure 3

Exclusion plot from a hypothetical experiment searching for spin-spin interactions between electrons mediated by an exotic boson. The $x$ axis is the mass of the exchanged boson, and the $y$ axis is the dimensionless spin-spin coupling [6]. We assume a levitated micromagnet with radius $R=0.2\mathrm{mm}$, a rotating polarized sphere actuator of radius 2 mm placed at distance 4 mm underneath the levitated magnet, and measurement time ${10}^6\mathrm{s}$. Blue solid (red dashed) lines: theoretical bound assuming thermal noise at $T=4.2\mathrm{K}$, $Q={10}^7$ (SQL noise). Black line and shaded region: exclusion plot from the best experiment in the literature [25].