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Abstract
Diamagnetically levitated micro–nano oscillators play a crucial role in fundamental physics research and the advancement of high-precision sensors. Achieving high sensitivity in acceleration or force sensing is a fundamental requirement within these research domains. The primary limitation in achieving such sensitivity is thermal noise, which is directly proportional to the motion damping of the oscillator. Theoretical modeling suggests the presence of significant damping mechanisms induced by eddy currents. In this study, we validated the theoretical model by optimizing the structure of the magnet trap, confirming the impact of eddy currents on the damping of the oscillators. Additionally, we observed another type of damping caused by static charge in moving levitated dielectrics. Subsequently, we proposed an innovative theoretical model to explain this phenomenon and verified its validity during the charge neutralization process. Through these efforts, we successfully reduced the total damping from 1.6 mHz to 0.15 mHz, resulting in an order of magnitude improvement in performance. Our sensing system achieved the highest sensitivity of acceleration sensing in diamagnetically levitated submillimeter-scale dielectric to date, measuring ${7.6\,\pm \,0.8)} \times \textrm{1}{\textrm{0}^{ - 1\textrm{0}}}\,\textrm{g/}\sqrt {\textrm{Hz}} $. The exploration conducted in this study regarding the analysis and suppression of electromagnetic damping, along with associated thermal noise, holds significant promise for frontier research involving sensing with levitating dielectrics.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Diamagnetically levitated micro–nano scale oscillators play a crucial role in the study of fundamental physics and high-precision sensing applications, such as the realization of macroscopic quantum states [1–4], determination of dark matter [5], establishment of hybrid quantum system [6,7], detection of weak magnetic fields [8], and acceleration sensing [9–12]. The levitated oscillator consists of transparent dielectric materials such as acrylic and polystyrene with laser detection that afford advantages over opaque conductor oscillators such as superconductors and graphite. These advantages include high bias stability and a high-quality factor due to the ultralow absorption of laser and minimal damping of eddy currents when adopting laser detection. It’s a classical and precise laser sensor of measuring the motion of oscillator by analyzing the diffraction pattern of laser after passing through the transparent oscillator. For instance, previous research reported achieving a minimum resolved acceleration of pico-g with a measurement time of 105 s and a quality factor of $\textrm{1}\mathrm{.5\,\times \,1}{\textrm{0}^\textrm{5}}$ using an acrylic sphere levitated on a permanent magnet [12]. Laser radiation force and AC electric field force can also be used to levitate micro-nano scale dielectric. Compared with other levitation method, diamagnetic levitation has the advantage of high sensitivity and low bandwidth sensing applications due to its passive attribution and larger levitation mass.
The sensitivity of acceleration sensing in an oscillator is primarily limited by thermal noise. According to the fluctuation dissipation theorem [13–15], the motion damping of an oscillator represented by γ typically indicates the presence of thermal noise:
Furthermore, there is nearly no theoretical or experimental quantitative research on the relationship between motion damping and the charge carried by levitated dielectrics. Neutralizing charge operation is widely used in dielectric levitation [18–20], yet there are many studies on the design of MEMS sensors that focus on electrostatic damping based on the conductive plate capacitance model [21–23]. However, this common capacitance model is not applicable when dealing with levitating dielectrics near conductors.
Thus, this study aims to verify the suggested theoretical model of damping caused by eddy currents through experimental optimization of the magnet trap structure. Furthermore, we propose an innovative theoretical model based on damping caused by static charge and verify it through the charge neutralization process. Ultimately, we successfully suppress the total damping from 1.6 mHz to 0.15 mHz and improve the sensitivity limited by thermal noise to ${(7.6\,\pm \,0.8)\,\times 1}{{0}^{ - 1{0}}}\,\textrm{g}\left/\sqrt {\textrm{Hz}}\right. $ using a levitated 0.4 mm acrylic sphere. To the best of our knowledge, this result represents the best result of sensitivity in diamagnetically levitating submillimeter-scale dielectric to date. The aforementioned exploration into the mechanisms of generating and suppressing electromagnetic damping caused by eddy currents and static charge can significantly impact theoretical and experimental research involving levitated dielectric. Examples include testing the theory of non-Newtonian gravity at the submillimeter scale [24,25] and developing new high-precision relative gravimeters [26,27] that require sensitivities of acceleration sensing of approximately $\textrm{1}{\textrm{0}^{ - 1\textrm{2}}}\,\textrm{g/}\sqrt {\textrm{Hz}} $ and $\textrm{1}{\textrm{0}^{ - 1\textrm{0}}}\,\textrm{g/}\sqrt {\textrm{Hz}} $, respectively. High sensitivity of levitated dielectric oscillator has great potential to contribute to above two fields.
2. Modeling of theory
Altogether, the sources of damping in levitated dielectrics include residual gas [19,28], laser radiation [29–31], eddy currents [17,32,33], and static charge [34], among others. In the following, we present various theoretical models of damping and the measurement method for total damping.
The gas damping of a sphere with radius ${R}$ and density ${\rho }$ under pressure ${P}$ is denoted as ${{\gamma }_{\textrm{gas, inf}}} = \,\textrm{16P}/{(\pi \rho R)}$. However, when a solid block is present near the sphere, the gas damping needs to be modified as follows [28]:
When a laser is vertically incident on a mirror, the laser radiation damping is [30]
where c denotes the light speed in vacuum and I is the light intensity. ${{\gamma }_{\textrm{radiation}}}$ is the only damping in this study that does not introduce thermal noise but shot noise ${\textrm{S}_{aa,\,\textrm{shot}}}({\omega } )\; = \; \textrm{4}\hbar {\nu }{{\gamma }_{\textrm{radiation}}}{/m}$. Here, $\hbar $ is the normalized Planck constant, and ${\nu }$ is the optical frequency.The suggested eddy current model [17] is depicted in Fig. 1(a). The magnetic moment of the dipole in the levitated dielectric sphere, induced by the magnetic field ${\vec{B}}$ of magnet, is expressed as ${\vec{M}} = {{\chi }_\textrm{m}}{V\vec{B}}/{{\mu}_{0}}({{1\; } + \,{{\chi }_\textrm{m}}} )$, where the magnetic moment’s direction is perpendicular to the surface of the magnet. When the sphere moves, the magnetic field emitted by the dipole excites eddy currents in the conductive magnet, resulting in damping. The damping caused by the eddy currents is
Fig. 1. Models of damping caused by eddy current and static charge. (a) Magnetic moment $\vec{M}$ (gray dashed line) in moving dielectric sphere (green ball) excites eddy current (red dashed line) in magnet. (b) Static charge (plus sign) on sphere induces charge (minus sign) on the surface of magnet and the induced charge moving synchronously with the sphere.
Our innovative model of damping caused by static charge is shown in Fig. 1(b). Induced charge ${{Q}_{\textrm{induce}}}$ is generated on the surface of conductive magnet when there is levitated dielectric sphere carrying static charge ${{Q}_{\textrm{static}}}$ near the magnet. The induced charge can be considered to move synchronously with the dielectric sphere while maintaining its spatial distribution. Therefore, the resistance exerted on the induced charge by the magnet is equal to the static force exerted on the sphere. According to the microscopic theory of conductor resistance [35], the damping caused by this interaction can be expressed as follows:
In terms of measuring damping, the ring-down method that analyzes the amplitude of sphere displacement in the time domain, is considered more accurate compared with the fitting method of displacement power spectrum (PSD) in the frequency domain. This is because the nonlinearity of magnetic force and the drift of the elastic coefficient can alter the peak shape of the PSD and introduce errors in the fitting method, whereas the ring-down method is less affected by these factors [2]. In the ring-down method, the amplitude of sphere motion ${X(t)}$ is excited to ${{X}_{\textrm{max}}}$ that is significantly larger than the steady-state amplitude through laser radiation force. Subsequently, the amplitude undergoes an exponential decay:
The measured total damping ${\gamma _{\textrm{mea}}}$ of sphere motion can be obtained by fitting the decay curve.
3. Demonstration of simulation and experiment
3.1 Experiment setup
In our acceleration sensing system, an acrylic sphere was used as the levitated oscillator positioned between four tips made of magnetically soft alloy 1J22, as depicted in Fig. 2(a). It is well-known that a larger mass of the oscillator generally leads to higher sensitivity in acceleration sensing. Through precise magnetic field simulation, we increased the diameter of the sphere in our system from the initial 40 µm to 410.4 ± 2 µm, resulting in a mass ${m} = {43}\mathrm{.5\,\pm \,0}{.7\,\mathrm{\mu}\textrm{g}}$). This quadrupole magnetic trap configuration is compatible with the structure of sphere motion detection using a spatial laser, as shown in Fig. 2(b). This configuration offers advantages in terms of stability and ease of adjustment. In our detection setup, we employed two lenses (Model LA1213, Thorlabs Inc.) and a self-built four-quadrant detector (QPD). All components shown in Fig. 2(b), except for the QPD, were placed inside the vacuum chamber.
Fig. 2. Levitation and motion detection of sphere. (a) The polarity of these magnetic tips are N, S, N, and S, respectively, from the top left clockwise to the bottom left. Structural parameter ${{h}_{\textrm{ud}}}$ represents the distance between top and down magnetic tips, and ${{d}_{\textrm{min}}}$ denotes the minimum distance between sphere and magnetic tips. (b) Laser (1064 nm) is focused on the sphere by an objective and transmitted to quadrant photodetector (QPD) by a condenser. The voltage output by QPD is proportional to the displacement of the levitated sphere. The x-, y-, and z-axis are along the long side of the lower magnetic tip, vertical down direction, and the direction of laser propagation, respectively, and the origin is located at the geometric center of the magnetic trap. (c) PSD of QPD output and it’s Lorentz fitting. The resonant frequency of the sphere moving along the x-axis, about 3.7 Hz when gas pressure equals to 1.2 mbar. (d) QPD output for calibration in time domain.
The x-axis was selected for acceleration detection because of its small elastic coefficient $\textrm{k}$ and its effectiveness in reducing the noise of displacement detection. When the amplitude of sphere motion is within a certain range, the output voltage of QPD is proportional to the displacement of the sphere centroid. By fitting the output voltage PSD of QPD with the Lorentz function at pressure $\textrm{P} = {1}\textrm{.2 mbar}$, we calibrated the voltage–displacement coefficient ${{\xi }_{\textrm{fit}}}$ to $\textrm{(1}\mathrm{.71\,\pm \,0}\mathrm{.02)\,\times \,1}{{0}^{6}}\,\textrm{V/m}$, as the Fig. 3(c) shows.
Fig. 3. Computation of magnetic field and damping caused by eddy current in simulation. (a) and (b) Distribution of magnetic field (Unit in T) in y–z plane and x–y plane, respectively. (c) Combined forces of magnet force and gravity exerted on the sphere along three axes. The position of equilibrium point along each axis is marked with a black ellipse. (d) The simulation results of relation between structural parameter of magnetic trap ${{h}_{\textrm{ud}}}$, ${d}_{\textrm{min}}$, and eddy current damping ${{\gamma }_{\textrm{eddy}}}$. At the beginning, the minimum distance between the surface of sphere and magnet ${{d}_{\textrm{min}}}$ (orange line, triangle) increases with the distance between the upper and lower magnetic tips ${{h}_{\textrm{ud}}}$ and the damping caused by eddy current ${{\gamma }_{\textrm{eddy}}}$ (blue line, circle) also drops. Gradually, the sphere starts to fall and tends to fail to levitate and damping ${{\gamma }_{\textrm{eddy}}}$ begins to increase.
The resonant frequency of the sphere moving along x-axis, y-axis and z-axis are 3.7 Hz, 52 Hz and 47 Hz, respectively. Axial (z-axis) laser force can be approximated as a constant propulsive force, pushing the sphere forward by 0.1 pm and the fluctuation of laser intensity is less than 0.2% in 8 hours. In contrast, the excited the amplitude of sphere motion ${{X}_{\textrm{max}}}$ in experiments is about 0.6 µm. Radial laser force is a kind of linear restoring force and the stiffness induced by light force are 88.3 nN/m. In contrast, stiffnesses induced by magnetic force are 19.5 µN/m and 3.8 mN/m along the x-axis and z-axis, respectively. Thus, the influence of light force is much smaller than that of magnetic force and can be ignored in the measurement of sphere motion.
3.2 Suppressing damping caused by eddy current in simulation
The damping caused by eddy current depends strongly on the distribution of magnetic field. Thus, this type of damping can be suppressed by optimizing the structure of the magnetic trap. As shown in Fig. 3(a) and 3(b), the three-dimensional field of magnetic induction intensity $\vec{B}$ near four magnetic tips is calculated using the module of magnetostatics field analysis of finite element software ANSYS.
The three-dimensional field of magnetic force is computed using equation $\overrightarrow {{{F}_\textrm{m}}} \, = \,\frac{{{{\chi }_\textrm{m}}{V}}}{{{2}{{\mu}_{0}}}}\nabla {|{{\vec{B}}} |^{2}}$, where ${{\chi }_\textrm{m}}$ is magnetic susceptibility, V denotes the volume of levitated sphere, and ${\mu _0}$ indicates permeability in vacuum. The combined forces of magnetic force and gravity along three-axis is shown in Fig. 3(c). The force equilibrium point of sphere centroid can be obtained from the abscissa, where the combined force curve passes through the horizontal axis and has a negative slope. Thus, there are two symmetrical equilibrium points ($\textrm{x} = {\,\pm \,183}.2\,\mathrm{\mu}\textrm{m}$, $\textrm{y} = {78}.9\,\mathrm{\mu}\mathrm{m}$, $\textrm{z} = 0\,\mathrm{\mu}\mathrm{m}$) along the x-axis in our magnetic trap. The theoretical damping caused by eddy current was computed based on magnet field $\vec{B}$ and the position of equilibrium point using Eq. (4).
We found that the region on the sphere that is the closest to two lower magnetic tips plays a major role in exciting eddy current, as damping caused by eddy current ${{\gamma }_{\textrm{eddy}}}$ is inversely proportional to the fourth power of $\textrm{d}$, and the two lower tips are considerably closer to the sphere than the two upper tips. Thus, it is the key to lifting the levitated sphere and increasing ${{d}_{\textrm{min}}}$, under the condition of keeping the sphere levitated. The simulation results of relation between structural parameter of magnetic trap ${{h}_{\textrm{ud}}}$, ${{d}_{\textrm{min}}}$, and damping ${{\gamma }_{\textrm{eddy}}}$ is shown in Fig. 3(d). The damping caused by eddy current can be suppressed from 1.4 mHz to 0.22 mHz when the distance between the upper and lower magnetic tips ${{h}_{\textrm{ud}}}$ adopts the optimal value (∼550 µm).
3.3 Suppressing damping caused by static charge in simulation
The operation of neutralizing charge has been widely used in dielectric levitation to avoid disturbing the ambient electric field or detecting free fractional charge. we usually make the sphere carry positive net charge by arc generator [6] and then the ultraviolet shining on four magnetic tips [20] slowly neutralizes the net charge. However, the experiments have revealed that the static charge carried by levitated dielectric sphere also plays a crucial role in causing additional damping and thermal noise. Although the method of experimentally suppressing the impact of this damping by neutralizing static charge on sphere ${{Q}_{\textrm{static}}}$ is mature, it is still necessary to develop a model that quantitatively explains the relation between damping caused by static charge ${{\gamma }_{\textrm{charge}}}$ and static charge on sphere ${{Q}_{\textrm{static}}}$. Based on our model in Eq. (5), ${{R}_\textrm{c}}\; = \; {{Q}_{\textrm{induce}}}/\,{{Q}_{\textrm{static}}}$ needs to be calculated in the simulation stage because other parameters in the equation are all constants that are related to the material of levitated dielectric and magnet.
The distribution of the surface density of induce charge ${{\rho }_{Q,\; \textrm{surf}}}$ can be analyzed based on the ANSYS module of electrostatic field, as show in Fig. 4(a). A positive static charge ${{Q}_{\textrm{static}}}$ of −1 C was assumed to be evenly distributed on the surface of the levitated sphere. the total amount of induced charge ${{Q}_{\textrm{induced}}}$ was obtained by the surface integral of ${{\rho }_{Q,\; \textrm{surf}}}$ on all faces of four magnetic tips, and we obtained ${{R}_\textrm{c}} = \,{0}\textrm{.30}$. Thus, the amount of static charge on sphere ${{Q}_{\textrm{static}}}$ and induced charge on four magnetic tips ${{Q}_{\textrm{induced}}}$ are of the same order of magnitude. This is reasonable because the minimum distance between sphere and magnetic tips ${{d}_{\textrm{min}}}$ (37.3 µm) is considerably less than the radius of sphere (205.2 µm). Thus, the surfaces of sphere and magnetic tips can be approximated as two planes of a capacitor. The electron concentration ${n}$ and the surface resistivity of magnetic conductor ${\sigma _{\textrm{surf}}}$ are ${8}\mathrm{.5\,\times \,1}{{0}^{{28}}}\,{\textrm{m}^{\textrm{ - 3}}}$ and ${1}\mathrm{.4\,\times \,1}{{0}^{6}}\,\textrm{S/m}$, respectively.
Fig. 4. Computation of the surface density of induced charge and damping caused by static charge in simulation. (a) Distribution of the surface density of induced charge ${{\rho }_{Q,\; \textrm{surf}}}$ (base-10 logarithm, Unit in $\textrm{C/}{\textrm{m}^{2}}$). As an example, ${{\rho }_{Q,\; \textrm{surf}}}$ on the upper surface of the down and right magnetic tip is shown on the right side in more detail. (b) Linear relation between damping caused by static charge ${{\gamma }_{\textrm{charge}}}$ and static charge on sphere ${{Q}_{\textrm{static}}}$.
Finally, the linear relation between damping caused by static charge ${{\gamma }_{\textrm{charge}}}$ and the amount of static charge on sphere ${{Q}_{\textrm{static}}}$ was observed, as shown in Fig. 4(b). Thus, ${{Q}_{\textrm{static}}}$ must be less than approximately $\mathrm{1\,\times \,1}{{0}^{4}}\,{e}$ to ensure that the damping caused by static charge is smaller than the minimum of damping of eddy current (0.22 mHz) in our sensing system.
3.4 Suppressing two types of damping in experiment
To measure the total damping ${{\gamma }_{\textrm{mea}}}$ in the experiment, another green laser (532 nm) was incident on the levitated sphere. The intensity of the green laser is proportional to the velocity of the sphere; this is also known as differential heating. It will continuously injects energy into the motion of sphere. The heating was abruptly deactivated, allowing for the measurement of total damping ${{\gamma }_{\textrm{mea}}}$ with the data of amplitude of sphere motion, as expressed in Eq. (6).
The gas pressure ${P}$ around the sphere was dropped to ${4}\mathrm{.1\,\times 1}{{0}^{ - 7}}\,\textrm{mbar}$, and the corresponding theoretical gas damping ${{\gamma }_{\textrm{gas}}}$ was ${6}\mathrm{.0\,\times 1}{{0}^{ - 6}}\,\textrm{Hz}$ according to Eq. (2). Pertaining to the theoretical laser radiation damping ${{\gamma }_{\textrm{radiation}}}$, it is only ${5}\mathrm{.5\,\times 1}{{0}^{ - 1{4}}}\,\textrm{Hz}$ when the intensity of detection laser is 50 µW, as shown by Eq. (3). Thus, damping caused by gas or laser was considerably less than electromagnetic damping in our sensing system. We will not consider this in the following analysis.
Figure 5(a) presents the attenuation curves of the amplitude of sphere motion $X(\textrm{t})$ in ${1}{{0}^{4}}\,{s}$ with different static charge amounts ${{Q}_{\textrm{static}}}$ on the sphere. Thus, the damping of sphere motion decreased significantly when the sphere carried less static charge. Semi-logarithmic coordinate was adopted to check the exponential decay from a linear perspective.
Fig. 5. Theoretical calculation and experimental measurement of various damping. (a) Attenuation curves of sphere motion amplitude (solid line) with different charge amounts and their exponential fitting (dashed line). (b) Measured damping with different charge amounts and their comparison with theoretical-damping-caused static charge ${{\gamma }_{\textrm{charge}}}$ (black line) and eddy current ${{\gamma }_{\textrm{eddy}}}$ (blue line). The total damping ${{\gamma }_{\textrm{total}}}$ (green line) equals ${{\gamma }_{\textrm{static}}}\, + \,{{\gamma }_{\textrm{eddy}}}\, + \,{{\gamma }_{\textrm{gas}}}\, + \,{{\gamma }_{\textrm{radiation}}}$. The theoretical gas damping ${{\gamma }_{\textrm{gas}}}$ is ${6}\mathrm{.0\,\times 1}{{0}^{ - 6}}\,\textrm{Hz}$ and radiation damping ${{\gamma }_{\textrm{radiation}}}$ is ${5}\mathrm{.5\,\times 1}{{0}^{ - 1{4}}}\,\textrm{Hz}$.
Figure 5(b) provides a comparison of the measured total damping ${{\gamma }_{\textrm{mea}}}$ with theoretical damping caused by eddy current ${{\gamma }_{\textrm{eddy}}}$ and static charge ${{\gamma }_{\textrm{charge}}}$ in cases of different amounts of static charge on sphere ${{Q}_{\textrm{static}}}$. The distance between the upper and lower magnetic tips ${{h}_{\textrm{ud}}}$ was set to 554.2 µm, as measured by a microscope, with the optimal value being ∼550 µm. Measured total damping ${{\gamma }_{\textrm{mea}}}$ was denoted by red dots, and they were also marked with error bar to show the relative uncertainty (10.7%) in damping measurement. The uncertainty was calculated in the simulation of sphere motion. The damping caused by static charge ${{\gamma }_{\textrm{charge}}}$ was greater than any other damping when ${{Q}_{\textrm{static}}}$ ranged from ${1}{{0}^{4}}\,{e}$ to ${1}{{0}^{6}}\,{e}$. ${{\gamma }_{\textrm{mea}}}$ was observed up to 1.6 mHz when ${{Q}_{\textrm{static}}}$ was ${5}\mathrm{.8\,\times \,1}{{0}^{5}}\,{e}$, and it could be reduced to ${{\gamma }_{\textrm{mea, min}}}\, = \,$ 0.15 mHz when ${{Q}_{\textrm{static}}}$ was neutralized to ${7}\mathrm{.5\,\times \,1}{{0}^{2}}\,{e}$. At this time, eddy current dominated the remaining damping as the measured damping remained almost unchanged. Thus, the sensitivity of acceleration in our sensing system limited by thermal noise can be improved to $\textrm{S}_{aa,\; \textrm{min}}^{{1/2}}\,\textrm{= (4}{{k}_\textrm{B}}{T}{{\gamma }_{\textrm{mea, min}}}/{m)}{\,^{{1/2}}} = \,\textrm{(7}\mathrm{.6\,\pm \,0}\mathrm{.8)\,\times 1}{{0}^{ - 1{0}}}\,\textrm{g/}\sqrt {\textrm{Hz}} $ after the static charge is after the static charge is reduced to a certain extent where ${{\gamma }_{\textrm{charge}}}$ is far less than ${{\gamma }_{\textrm{eddy}}}$.
The theoretical damping caused by eddy current ${{\gamma }_{\textrm{eddy}}}$ (0.22 mHz) in Fig. 3(d) is slightly larger than the minimum remaining damping after neutralization (0.15 mHz) in the experiments in Fig. 5(b). We suppose that the magnetic field in the region on the sphere between two downward tips is not completely perpendicular to the surface of magnetic tips. The perpendicularity is a basic assumption in our eddy current model shown in Fig. 1(a). This discrepancy between theory and experiment may be responsible for the above overestimation.
Moreover, there is a difference between the measured and theoretical motion damping in Fig. 5(b), particularly when the levitated sphere carries a large amount of charge. The difference is attributed to the fact that only the net charge on the sphere can be measured, whereas the local charge on the lower side of sphere near four magnetic tips mainly contributes to the damping cause by static charge ${{\gamma }_{\textrm{charge}}}$, as shown in Fig. 1(b). Furthermore, the above experiments prove that the operation of neutralization can still suppress the damping caused by static charge because it also reduces the local charge on the lower side of the sphere.
4. Discussion and summary
In recent years, levitated micro–nano scale oscillators have gained significant attention in frontier research areas of fundamental physics and high-precision sensing systems. Achieving higher sensitivity of sensing is a fundamental requirement and a common pursuit in these research fields. However, the presence of thermal noise caused by electromagnetic damping poses a formidable obstacle to further improving sensitivity.
In this study, through the optimization of the magnet trap structure (adopting optimal distance ${{h}_{\textrm{ud}}}$ of approximately 550 µm between the top and bottom magnetic tips) and charge neutralization (reducing charge amount ${{Q}_{\textrm{static}}}$ on the sphere from ${5}\mathrm{.8\,\times \,1}{{0}^{5}}\,{e}$ to ${7}\mathrm{.5\,\times \,1}{{0}^{2}}\,{e}$), we experimentally reduced the electromagnetic damping caused by eddy currents and static charge from 1.6 mHz to 0.15 mHz using a levitated 0.4 mm acrylic sphere. This reduction aligns with two theoretical damping models, including the suggested eddy current model and our innovative static charge model. Finally, we improved the sensitivity of acceleration sensing, limited by thermal noise, to $\textrm{(7}\mathrm{.6\,\pm \,0}\mathrm{.8)\,\times 1}{{0}^{ - 1{0}}}\,\textrm{g/}\sqrt {\textrm{Hz}} $; this represents the best result achieved thus far in diamagnetically levitating submillimeter-scale dielectric. Our exploration into the mechanisms of generating and suppressing electromagnetic damping in the magnet trap will be valuable for research involving diamagnetically levitated dielectric. Additionally, our original model for static charge damping can be applied in other types of levitation traps, such as optical traps and electric traps, if a conductor such as a displacement positioning platform is present in proximity to the charged levitated dielectric.
Funding
National Key Research and Development Program of China (2022YFB3203402); Natural Science Foundation of Zhejiang Province (LD22F050002); Major Scientific Project of Zhejiang Laboratory (2019MB0AD01); Center-initiated Research Project of Zhejiang Laboratory (2021MB0AL02, 2022MB0AL02); China Postdoctoral Science Foundation (2021TQ0307, 2022M712925).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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