A magnetically levitated conducting rotor with ultra-low rotational damping circumventing eddy loss

Abstract

Levitation of macroscopic objects in a vacuum is key towards the development of high-precision inertial sensors and pressure sensors, as well as towards the fundamental studies of quantum mechanics and its relation to gravity. Diamagnetic levitation offers a passive method at room temperature to isolate macroscopic objects in vacuum environments, yet eddy current damping remains a critical limitation for electrically conductive materials. We show that there are situations where the motion of conductors in magnetic fields does not, in principle, produce eddy damping, and demonstrate an electrically conducting rotor diamagnetically levitated in an axially symmetric magnetic field in a high vacuum. Experimental measurements and finite-element simulations reveal gas collision damping as the dominant loss mechanism at high pressures, while residual eddy damping, which arises from symmetry-breaking factors such as platform tilt or material imperfections, dominates at low pressures. The conclusion is supported by an analytic proof and an analytic example of zero steady current density for a rotating conductor in a magnetic field. This demonstrates a macroscopic levitated rotor with extremely low rotational damping and paves the way to fully suppress rotor damping, enabling ultra-low-loss rotors for gyroscopes, pressure sensing, and fundamental physics tests.

Figures (5)

• Figure 1: Diamagnetic levitation and rotation of a pyrolytic graphite (PG) disk in an axially symmetric magnetic trap. a: Lateral view of the levitation setup. An axially symmetric magnetic array, consisting of cylindrical and surrounding ring magnets arranged with alternating vertical magnetization, levitates a PG disk. The magnets are fixed to a breadboard, whose initial inclination is adjusted to be as horizontal as possible using screws inside the vacuum chamber (reaching $P\sim$ 5 $\times 10^{-5}$ Pa). The chamber itself is mounted on an aluminum plate atop an optical table, with its inclination adjusted away from the horizontal using the optical table and screw jacks. The top surface of the levitating disk is viewed via a flat mirror and two convex lenses (L1, L2), forming a Keplerian telescope for magnification, and its motion is tracked using an event-based camera (EBC). The inset shows a photograph of the levitated PG disk, marked with a white ink dot for orientation. b: EBC [Prophesee EVK-V3HD], detections of the tracking dot on the rotating PG disk. Twenty consecutive detections (squares) are shown for illustration. c: Schematic of the PG disk fabrication. A square PG plate (3 cm sidelength) is roughly milled using a desktop CNC machine [SnapMaker]. It is secured in a Petri dish half-filled with water, which disperses cut material, cools the disk, and eliminates dust formation. The disk is then precisely shaped and polished using a lapidary faceting machine [Ultratec V5], producing the final 10 mm diameter PG disk (right image).
• Figure 2: Spin-down of an initially rotating levitated disk. The disk is first spun up using an applied torque. Once it reaches a sufficiently high angular velocity $\omega$, the torque is removed, and its spin-down is monitored over time. (a, b): Experimentally measured values of $\ln(\omega)$ as a function of time for high and low gas pressures $P$, along with best-fit lines whose slopes correspond to $\gamma(P)$. The data closely follow $\omega(t)= \omega_0 \exp(-\gamma t)$, indicating that angular velocity damping is well-approximated by viscous drag. c: Rotational damping rate $\gamma(P)$ as a function of gas pressure $P$ for a fixed, non-zero inclination of supporting platform. The horizontal dashed line represents the damping rate estimated via FEM-COMSOL simulation in the high-pressure regime, where the continuum assumption of fluid dynamics holds. The diagonal solid line represents the theoretical prediction of $\gamma(P)$ in the free molecular flow regime Cavalleri2010GasReservoir. The experimentally determined values of $\gamma(P)$ from spin-down measurements are shown as red circles. The associated error, obtained from the error in the linear fit, is smaller than the marker size. For $P<10^{-1}$ Pa, a plateau in $\gamma(P)$ suggests a dominant damping mechanism beyond gas friction, likely eddy damping due to deviations from perfectly axial symmetry. Potential sources include material imperfections in the disk or magnets, or a slight inclination of the setup, shifting the PG disk's center of mass off-axis.
• Figure 3: Simulation of rotational damping in the continuum flow regime (high pressure). A finite element method (FEM) COMSOL fluid dynamic model is developed to predict rotational damping in the regime where the continuum approximation for the gas dynamics holds. a: Cross-sectional view of the simulated laminar flow around a rotating disk (gray rectangle) with perfect coaxial symmetry. The disk rotates about its vertical axis $r=0$ at $\omega= 2\pi\times2$$\rm{rad}\cdot s^{-1}$ within a cylindrical gas-filled tank in the continuum regime. Sliding wall boundary conditions are applied to the disk surfaces. Contours and arrows indicate axial flow, while the colormap represents the magnitude of total (axial and azimuthal) flow. b: Dependence of the angular damping torque $|\mathcal{T}|$ on the disk for various angular velocities $\omega$ (light blue circles) at $P\sim 10^5$ Pa, along with the best-fit line (blue). The slope of this line corresponds to ${\it{\Gamma}}(P)=\gamma(P)I$. c: Comparison of experimental data (red) and FEM simulation results (blue) at high pressure, with no fitted parameters. The associated error for the simulation results, obtained from the error in the linear fit, is smaller than the marker size. The simulation aligns well with experimental data at high pressures (unshaded region) but deviates for $P\le 10^2$ Pa (shaded region), where the continuum approximation begins to break down. The solid black line represents the theoretical estimate for $\gamma(P)$ in the molecular flow regime.
• Figure 4: Studying the dependence of rotational damping rate $\gamma$ on the inclination of the plate to the horizontal: $\gamma(\theta_x,\theta_y)$. As the setup tilts, the center of mass (COM) of the PG disk shifts off axial symmetry, increasing eddy damping. a: Tilting of the supporting platform by angles $(\theta_x, \theta_y)$, about the $(x,y)$ axes. b: Force diagram of the levitated disk's COM. The PG disk is trapped by a balance of perpendicular forces: $F_{\rm M\perp}=F_{{\textrm{g}}}\cos\theta$, and lateral forces: $F_{\rm M \parallel}\sim m\omega_{{\textrm{L}}}^2\,d=F_{{\textrm{g}}}\sin\theta$, where $\omega_{{\textrm{L}}}$ is the lateral oscillation frequency of the PG plate, and $d$ is the lateral COM displacement from the axis of symmetry of the magnets, which increases with $\theta$. c: Measured dependence of $\gamma(\theta_x,\theta_y)$. Red circles and blue squares indicate two measurement sequences, showing $\min \gamma(\theta_x,\theta_y)$ at $(\theta_x,\theta_y)=(\theta_x^0, \theta_y^0)\ne (0,0)$ due to a small initial misalignment. The associated error (SE) is smaller than the marker size. d: Radial fit to $\gamma(\delta\theta_x+\theta_x^0,\delta\theta_y+\theta_y^0)\equiv \tilde{\gamma}({\it{\Delta}} \theta)$, where ${\it{\Delta}}\theta^2=\delta\theta_x^2+\delta\theta_y^2$, yielding a 1D collapsed plot of the 2D data from c. The associated vertical and horizontal errors for the measurements, obtained from the error in the linear fit and SE, are smaller than the marker size. The dashed curve is a FEM-COMSOL estimate assuming only eddy damping, matching experiments for ${\it{\Delta}} \theta > 0.1^\circ$ but breaking down for ${\it{\Delta}} \theta < 0.1^\circ$ (see e). e: Numerical FEM study of $\gamma(d)$. From symmetry arguments, we expect $\gamma(d\rightarrow 0)\rightarrow 0$, but due to the non-axially symmetric nature of the FEM mesh and numerical errors, the simulation yields $\gamma(d\rightarrow 0)\ne 0$. For $d> 0.05$ mm, simulations show a near-perfect power-law dependence (Eq. \ref{['eq:power_law_fit']}), supporting $\gamma(d=0)=0$ for perfect axial symmetry.
• Figure 5: Comparison between the simulated magnetic field generated by a typical axially symmetric magnetic array and the analytical example. The simulated magnetic field $\mathbf{B}_{\rm{ex}}$ is calculated with an exact analytical expression CACIAGLI2018423 with the parameters listed in the Table. \ref{['table:parameter_values']} and \ref{['table:simulation_parameters']}. The analytical resembling magnetic field, $\mathbf{B}_{\rm{res}}$$=B_r(r) \:\hat{r} +B_z(r,z)\:\hat{z}$, is built from Eq. \ref{['eq:analytical_B_field_example']}, with $\alpha=\pi/6$, $\beta=0.1$, and $z_0=-4$.