Dynamics of a Ferromagnetic Particle Levitated over a Superconductor

Under conditions where the angular momentum of a ferromagnetic particle is dominated by intrinsic spin, applied torque is predicted to cause gyroscopic precession of the particle. If the particle is sufficiently isolated from the environment, a measurement of spin precession can potentially yield sensitivity to torque beyond the standard quantum limit. Levitation of a micron-scale ferromagnetic particle above a superconductor is a possible method for near-frictionless suspension enabling observation of ferromagnetic particle precession and ultrasensitive torque measurements. We experimentally investigate multiple instances of a micron-scale ferromagnetic particle levitated above a superconducting niobium surface. We find that the levitating particle is trapped in a potential minimum associated with residual magnetic flux pinned by the superconductor and, using an optical technique, characterize the dynamics of the particle in such a trap.

Figure 1

Experimental setup, The microscopy cryostat is pumped by the mechanical pump and turbo pump. The motion of the particle is recorded with the scientific CMOS (sCMOS) camera above the cryostat. The coil is outside the cryostat. A detailed view of the niobium well is shown at the left. PC, personal computer.

Figure 2

comsol Multiphysics simulation result for the shielding of the magnetic field from the coil near the surface of the superconductor well where the ($\mathrm{Pr}$,$\mathrm{Fe}$)$\mathrm{B}$ particle is located. The applied magnetic field from the coil is approximately 100 G. The niobium is set to be ideally diamagnetic to simulate the Meissner effect [16]. The dashed and solid blue lines are the magnetic field distributions along the vertical symmetry axis when the superconductor is in the normal state or the superconducting state, respectively. The height is the distance of the particle from the bottom of the well. The shielding factor at $50\mu \mathrm{m}$ can be estimated on the basis of the dotted red line, and is on the order of ${10}^6$. The shielding factors at $200$ and $300\mu \mathrm{m}$ can be estimated in the same way, and are approximately ${10}^5$ and ${10}^4$, respectively. These heights are indicated by the dotted blue and green lines, respectively.

Figure 3

Measurements of the spectrum of the motion of a particle whose diameter is $25\mu \mathrm{m}$ and whose levitation height is approximately $80\mu \mathrm{m}$. The objective is mounted on a Thorlabs three-axis stage with differential micrometers for manual adjustment; the drive resolution is $0.5\mu \mathrm{m}$. The objective has a high axial resolution. The objective focuses the image on the particle on the surface before it is levitated. When the particle is levitated, we refocus the image on the particle by precisely moving the stage vertically. The levitation height can be roughly measured in this way. In the upper plot, the color represents the angle of the particle’s vibration. The intensity of the lines is proportional to the spectral density of the signal. The constant-frequency lines at 20, 40, 47, 95, and 110 Hz are determined to be due to environmental vibrations and are not related to the particle motion (confirmed by our observing the system with no ferromagnetic particle present). Near the eighth and 36th time steps, the modes starting from 27 and 35 Hz exhibit an avoided crossing. The lower plot shows the externally applied magnetic field felt by the particle with our taking into account the shielding due to the Meissner effect. Simulations of the motion of the particle above a superconductor, using the model in Fig. 5, are overlain on the data plot. The vibrational modes are strongly coupled. The new eigenfrequencies [$x$ (solid blue line), $y$ (solid green line), and $z$ (solid red line)] are plotted by our decoupling the original modes. The modes near 35, 62, and 104 Hz are reproduced by the simulation. The free parameters in the simulation are $m_0$, $m_f$, and $\theta$.

Figure 4

Measurements of the position of the centroid of the particle (left) and the trajectory of the particle (right). The black sphere is the levitated ferromagnetic particle, whose diameter is roughly $25\mu \mathrm{m}$.

Figure 5

A model of the levitation mechanism of the ferromagnetic particle above a superconductor (SC). The blue circle is the ferromagnetic particle, whose magnetization is $m_0$. The dashed blue circle is the initial position of the ferromagnetic particle during the cooling stage, and the frozen image $m_f$ (dashed red circle) is produced with height $h$ inside the superconductor. The dashed green circle is the diamagnetic image, which is the mirror image of the particle.

Figure 6

Measurements of the spectrum of motion of a ferromagnetic particle whose diameter is approximately $30\mu \mathrm{m}$. The levitation height of the particle is approximately $200\mu \mathrm{m}$. Again, the 60 and 120-Hz signals in the spectrum are the line frequency and its harmonic.

Figure 7

Trajectory of the 30-$\mu \mathrm{m}$-diameter particle. Note the larger path length relative to the 25-$\mu \mathrm{m}$-diameter particle shown in Fig. 4 (right). The greater levitation height ($200\mu \mathrm{m}$ vs $50\mu \mathrm{m}$) results in reduced screening of the external applied field, and thus a greater offset of the horizontal equilibrium position.

Figure 8

The frequency of the oscillating magnetic field is swept from 5 to 60 Hz, and the 26- and 40-Hz modes and their harmonic modes are excited. The diagonal line is the response to the swept ac magnetic field. Here the pumps are not disconnected from the cryostat. The 29.5-Hz mode is excited by vibration caused by the scroll pump, whose rotation speed is 1770 rpm (equivalent to 29.5 Hz). The color bar on the right side indicates the amplitude of the motion.

Figure 9

Measurements of the $x$ coordinate (a) and the $y$ coordinate (b) of the particle’s position as a function of time. The particle's motion is induced at times of approximately 28 and 47 s.

Figure 10

Measurements of the amplitude of the vibrations for each mode as a function of the sweeping time. The scroll pump is on, and the 29.5-Hz signal from the scroll pump (1770 rpm) can still be seen, and induced the particle's motion. We calculate the decay time by fitting the amplitude curve exponentially. The $Q$ factor is approximately $1\times {10}^3$ for the 26-Hz mode and approximately $4\times {10}^3$ for the 40-Hz.