Precessing Ferromagnetic Needle Magnetometer

A ferromagnetic needle is predicted to precess about the magnetic field axis at a Larmor frequency $\mathrm{\Omega }$ under conditions where its intrinsic spin dominates over its rotational angular momentum, $N\hslash \gg I\mathrm{\Omega }$ ($I$ is the moment of inertia of the needle about the precession axis and $N$ is the number of polarized spins in the needle). In this regime the needle behaves as a gyroscope with spin $N\hslash$ maintained along the easy axis of the needle by the crystalline and shape anisotropy. A precessing ferromagnetic needle is a correlated system of $N$ spins which can be used to measure magnetic fields for long times. In principle, by taking advantage of rapid averaging of quantum uncertainty, the sensitivity of a precessing needle magnetometer can far surpass that of magnetometers based on spin precession of atoms in the gas phase. Under conditions where noise from coupling to the environment is subdominant, the scaling with measurement time $t$ of the quantum- and detection-limited magnetometric sensitivity is $t^{-3/2}$. The phenomenon of ferromagnetic needle precession may be of particular interest for precision measurements testing fundamental physics.

Figure 1

A ferromagnetic needle with spin $S=N\hslash$ along its long axis precesses at a frequency $\mathrm{\Omega }$ in a magnetic field $B\ll B^{*}$ [Eq. (4)].

Figure 2

Solid black line: Magnetic field uncertainty as a function of measurement time $t$ for a precessing cobalt needle magnetometer as described in the text ($\ell =10\mu \mathrm{m}$ and $r=1\mu \mathrm{m}$, corresponding to $N\approx 3\times {10}^{12}$ spins) under conditions of cryogenic vacuum ($T\approx 0.1\mathrm{K}$, background gas density $\approx {10}^3{\mathrm{cm}}^{-3}$). For $t\lesssim 1\mathrm{s}$ the noise is dominated by the SQUID detector [$\mathrm{\Delta }{B}_{\det }$, Eq. (6)], and for $t\gtrsim 1\mathrm{s}$ the noise is dominated by collisions with background gas molecules. Dashed black line: Standard quantum limit on magnetic field uncertainty, $\mathrm{\Delta }{B}_{\mathrm{SQL}}$, for $N\approx 3\times {10}^{12}$ spins with ${\mathrm{\Gamma }}_{\mathrm{rel}}=0$. Dotted gray line: Thermally averaged quantum noise limit on magnetic field uncertainty for a precessing cobalt needle magnetometer of given dimensions at $T\approx 0.1\mathrm{K}$ [$\mathrm{\Delta }{B}_Q$, Eq. (14)].