Enhancing non-Newtonian gravity constraint using a levitated pendulum in vacuum

Graphical abstract

Abstract

The detection of non-Newtonian gravity is crucial for fundamental physics research and our understanding of dark energy. However, conducting an experiment that provides explicit evidence of its existence remains an endeavour. We propose an experiment utilizing a diamagnetically levitated pendulum in vacuum to detect non-Newtonian gravity on a micrometer scale. The pendulum configuration effectively helps to shield electromagnetic force fluctuations in the vacuum levitation system. The structural parameters of the pendulum are intentionally optimized to enhance the constraint on the non-Newtonian gravity strength $\alpha$. The designed pendulum can be stably levitated in the diamagnetic trap thanks to its passive levitation mechanism. By conducting resonance force measurements at room temperature for a duration of ${10}^4$ s, we anticipate a significant improvement in the constraint on the non-Newtonian gravity strength ($\alpha \ge 28$) within the force range of $\lambda =7.6$ µm. This represents an enhancement of over three orders of magnitude compared to the current limit. This study presents a promising tool for investigating short-range forces and exploring frontier physics in tabletop laboratory.

1. Introduction

Although the Newtonian inverse-square law has achieved great success in the macroscopic realm, its validity in the microscopic scale still needs to be explicitly checked [1], [2], [3]. For example, the boundary between gravitation and quantum mechanics remains ambiguous [4], [5], [6], and the unification of gravitation with the other three fundamental forces in the Grand Unification process is yet to be accomplished [7]. Some unification theories, like string theory, propose the existence of extra-dimensional space where gravitational interactions deviate from the Newtonian inverse-square law [8]. Additionally, extensions of the Standard Model suggest the presence of light scalar and pseudoscalar particles, which could lead to novel interactions like the Yukawa potential between point masses [9], [10]. The resulting potential can be expressed as:

$$V\left(r\right)=G\frac{M_1{M}_2}{r}(1+\alpha e^{-r/\lambda })$$ (1)

with $G$ the Newtonian constant of gravitation, $M_1$ and $M_2$ the masses, $r$ their distance, $\lambda$ the interaction range, and $\alpha$ the strength of the non-Newtonian interaction.

The existence of non-Newtonian gravity could potentially offer solutions to the quantum gravitation problem, shed light on the nature of dark matter, and facilitate the Grand Unification [8], [11]. Consequently, numerous experimental approaches have been developed to search for non-Newtonian gravity and establish constraints on its existence [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Over the years, the strongest constraints of non-Newtonian gravity within interaction ranges of tens of nanometers have been obtained from neutron scattering experiments [12], [13], while Casimir experiments employing cantilevers have yielded the most robust constraints in the range from $\lambda =40$ nm to $\lambda =8$ µm [14], [15], [16], [17]. For larger $\lambda$ the gravitation experiments based on the torsional pendulum give the most strong constraints [21], [22], [23]. However, there is still no definitive experimental confirmation of the existence of non-Newtonian gravity. To further investigate the existence of the non-Newtonian gravity, a more precise detection method is required [24].

Levitated oscillators in vacuum have been demonstrated as highly sensitive force sensors due to their excellent isolation from environmental contact [25]. Recently, levitated oscillators in a vacuum have made significant progress in short-range force sensing [26], [27], [28], [29], [30], [31], [32], [33], [34]. They are also predicted to serve as powerful tools for searching for dark energy and detecting gravitational waves [35], [36], [37], [38], [39], [40], [41]. They could provide an ideal method for detecting non-Newtonian gravity. As early as 2010, Andrew’s group proposed detecting non-Newtonian gravity using optically cooled levitated microspheres [42]. Recently, the research team at Stanford University conducted an experiment using a levitated microsphere to investigate non-Newtonian gravity [43]. While levitated oscillator sensors have demonstrated ultrahigh force sensitivity, two issues need to be addressed when used to detect non-Newtonian gravity. Firstly, the non-Newtonian gravity is too weak compared to the minimum detectable force of these levitated oscillator sensors [24], [44], [45]. To enhance the weak force signal, various methods can be implemented, such as deliberate attractor and sensor design [17], [41]. However, the design of the sensor is typically constrained by the fact that a more complex and massive sensor usually requires more energy injection into the levitation system, resulting in increased noise that can impact the sensor’s force sensitivity [46]. Secondly, backgrounds from nearby surfaces can easily overwhelm any force signal of interest. Although periodic modulation of the force under detection can eliminate static force backgrounds, fluctuating forces, particularly electromagnetic forces like Casimir forces and forces induced by patch potentials, continue to pose a concern [42], [47]. One possible solution to mitigate these electromagnetic forces is to place a static metal shield between the attractor and the test mass [7], [17], [22], [23], [48], [49]. However, it is important to note that the shield may interfere with the levitation trap [43], [50].

In this paper, we propose a force sensor based on a diamagnetically levitated pendulum system for detecting non-Newtonian gravity. The diamagnetically levitated oscillator system enables the levitation of large test masses to amplify the non-Newtonian gravity signal, and the pendulum structure of the oscillator helps shield against electromagnetic forces without introducing additional noise. Therefore, a levitated pendulum can potentially address both of the aforementioned issues simultaneously. By using our deliberately designed levitated pendulum as force sensor, we theoretically calculated that for a measurement time of ${10}^4\mathrm{s}$, the non-Newtonian gravity strength $\alpha \ge 28$ is expected in the force range of $\lambda =7.6$ µm at room temperature, which is an improvement of more than 3 orders of magnitude over the existing limit.

2. Force sensor of diamagnetically levitated pendulum

We employ a diamagnetic levitation system to prepare a levitated pendulum sensor to detect non-Newtonian gravity. The use of a diamagnetic levitation system was motivated by its capability to levitate large objects, thus enhancing the signal for non-Newtonian gravity. The diamagnetic levitation system operates based on the principle that the magnetic force acting on a diamagnetic object in an uneven magnetic field counteracts its gravitational force, resulting in a state of balanced levitation. In this study, we considered a diamagnetic object placed within a diamagnetic trap formed by permanent magnets. The force exerted on the diamagnetic object in the presence of magnetic and gravitational fields can be expressed as:

$$F=mg+{\int }_V\frac{\chi }{2\mu }\nabla B^{2}dV$$ (2)

with $m$ the diamagnetic object’s mass, $V$ its geometric volume, $B$ the magnetic induction intensity, $\mu$ the magnetic permeability, and $\chi$ the magnetic susceptibility which is negative for diamagnetic object. Notably, the levitation force is directly proportional to the volume of the diamagnetic object, allowing for levitation of objects with varying sizes and reduced noise levels [51]. Consequently, we designed a pendulum with an appropriate geometric structure to be levitated within the diamagnetic trap. Fig. 1a illustrates the configuration of the levitated pendulum, comprising a thin rigid rod with a sphere attached to each end. Sphere 1, composed of diamagnetic material with a radius of $R_1$, is positioned inside the trap to generate a levitation force proportional to its volume, counteracting the gravity of the entire pendulum. Sphere 2, with a radius of $R_2$, functions as a test mass which protrudes outside the trap to detect any forces of interest. The pendulum oscillates back and forth at an intrinsic frequency under the control of natural gravity. To measure the pendulum’s displacement, a pair of optical fibers are aligned with either sphere 1 or sphere 2 [52].

Fig. 1.

The pendulation levitodynamics of the levitated pendulum. (a) Diagram of the levitated pendulum. The red and blue parts represent permanent magnet assemblies that create a diamagnetic trap. The pendulum comprises a rigid thin rod and two spheres. The two spheres, Sphere 1 and Sphere 2, are fixed at both ends of the rod and coaxial with the rod. Sphere 1, made of diamagnetic material, is located within the diamagnetic trap, providing a levitated force against the entire pendulum’s gravity. Sphere 2, protruding outside the trap, serves as a test mass. An external force of $f_{\text{ex}}{e}^{\mathrm{i}\omega \mathrm{t}}$ is applied to sphere 2 for testing. A pair of aligned fibers are used to measure the pendulum’s displacement. (b) Diagram of the levitated pendulum’s degrees of freedom. The spatial state of the levitated pendulum is represented with coordinates $(x,y,z,\theta ,\phi )$, with $(x,y,z)$ the center-of-mass Cartesian coordinates of the levitated pendulum, $\theta$ the angle between the rod and $z$ axis, $\phi$ the angle between the projection of the rod on $xy$ plane and $x$ axis. (c) Diagram of the two IPMMs. (d) Displacement response function ${\chi }_{\text{jk}}\left(\omega \right)$ varying as frequency $\omega /2\pi$. ${\omega }_1$ and ${\omega }_2$ are the frequencies at which the peaks of the four response functions occur; ${\omega }_{\mathrm{d}1}$ and ${\omega }_{\mathrm{d}2}$ are the frequencies at which response functions ${\chi }_{11}$ and ${\chi }_{22}$ are sink considerably.

Here provides a simple description of the pendulation levitodynamics of the levitated pendulum. Assuming the pendulum’s rod is an ideal rigid cylinder, coaxial with Sphere 1 and 2, as depicted in Fig. 1b. The levitated pendulum possesses five degrees of freedom, where $(x,y,z)$ represents the Cartesian coordinates of the pendulum’s center-of-mass, $\theta$ denotes the angle between the rod and $z$ axis, and $\phi$ represents the angle between the rod’s projection on the $xy$ plane and the $x$-axis. Thus, the spatial states of the levitated pendulum can be described by $(x,y,z,\theta ,\phi )$. For the sake of simplicity, we neglect motion along the $y$ and $\phi$ directions, focusing solely on the pendulum’s motion in the degrees of freedom $(x,z,\theta )$. In the case of a small angle approximation, with $\sin \theta \approx \theta$ and $\cos \theta \approx 1$, neglecting higher-order infinitesimals, we obtain the dynamic motion equations for the levitated pendulum as follows:

$$\begin{array}{c}m\ddot{\mathrm{x}}=-k_x\Delta x\hfill \\ m\ddot{\mathrm{z}}=-\mathit{mg}-k_{\mathrm{z}}\Delta z\hfill \\ I_{\theta }\ddot{\theta }=k_{\mathrm{z}}\Delta z·L_1\theta -k_{\mathrm{x}}\Delta x·L_1\hfill \end{array}$$ (3)

here $m$ represents the total mass of the levitated pendulum, and $k_x$ and $k_z$ are the elastic coefficients along the $x$ and $y$ directions of the diamagnetic trap, respectively. Additionally, $\Delta x=x+L_1\theta$ and $\Delta z=z-mg/k_{\mathrm{z}}$ denote the deviations of Sphere 1 from the center of the trap along the $x$ and $y$ directions, respectively. By solving Eq. 3, we can determine the intrinsic motion modes of the levitated pendulum:

$$\begin{array}{c}{\omega }_1=\sqrt{(k_{\mathrm{a}}-k_{\mathrm{b}})/m}\hfill \\ {\omega }_2=\sqrt{(k_{\mathrm{a}}+k_{\mathrm{b}})/m}\hfill \\ {\omega }_3=\sqrt{k_{\mathrm{c}}/m}\hfill \end{array}$$ (4)

where ${\omega }_1$ and ${\omega }_2$ represent the two intrinsic pendulation motion modes (IPMMs) of the levitated pendulum, while ${\omega }_3$ signifies the intrinsic translation motion mode along the $z$-axis. The effective stiffness coefficients are denoted as $k_a$, $k_b$, and $k_c$ (for more details, see Appendix A). Fig. 1c illustrates the dynamic motion forms of the two IPMMs, where the ${\omega }_1$ mode corresponds to both Sphere 1 and Sphere 2 oscillating in the same phase, while the ${\omega }_2$ mode refers to them oscillating in opposite phases. Considering our objective is to detect force signals in the pendulation motion mode, any pendulation motion mode can be expressed as a linear combination of the two IPMMs.

Before utilizing the levitated pendulum for non-Newtonian gravity detection, we calculate the response function of the levitated pendulum sensor’s displacement to an external force. Assuming a horizontal external force, denoted as $f_{\text{ex}}^j{e}^{i\omega t}$, is applied to Sphere $j$ ($j=1$ or 2), and we measure the displacement of Sphere $k$ represented by $X_k\left(\omega \right)$. The response function is then defined as:

$$\begin{array}{c}{\chi }_{\text{jk}}\left(\omega \right)=\frac{X_{\mathrm{k}}\left(\omega \right)}{f_{\text{ex}}^{\mathrm{j}}{e}^{\mathrm{i}\omega \mathrm{t}}}(j,k=1\mathit{or}2)\end{array}$$ (5)

To calculate ${\chi }_{jk}\left(\omega \right)$, we incorporate $f_{\text{ex}}^j{e}^{i\omega t}$, damping forces from the environment, and their torques into the pendulum’s dynamic Eq. 3. For more details, please refer to Appendix B. The calculated results of ${\chi }_{jk}\left(\omega \right)$, which vary with frequency, are shown in Fig. 1d. The peak response occurs at the two IPMMs (${\omega }_1$ and ${\omega }_2$). The half-width of each peak is determined by the damping rate. Notably, we discover novel motion modes ${\omega }_{\text{d1}}$ and ${\omega }_{\text{d2}}$, where ${\chi }_{11}\left(\omega \right)$ and ${\chi }_{22}\left(\omega \right)$ respectively exhibit a significant decrease. Thus, we refer to them as elimination modes in this artcle. This phenomenon can be exploited to mitigate certain force noises when detecting signals with frequencies ${\omega }_{\text{d1}}$ or ${\omega }_{\text{d2}}$. Supposing we apply a force signal with a frequency of ${\omega }_{\text{d1}}$, denoted as $F_{\text{signal}}$, to Sphere 2, the resulting displacement response of Sphere 1 is given by $X_{\text{signal}}\left({\omega }_{\text{d1}}\right)=F_{\text{signal}}·{\chi }_{21}\left({\omega }_{\text{d1}}\right)$. However, any force noise $F_{\text{noise}}$ that directly affects Sphere 1 (such as the backaction noise caused by the detection laser or the seismic noise transmitted from the diamagnetic trap to Sphere 1) will also produce a displacement response of $X_{\text{noise}}\left({\omega }_{\text{d1}}\right)=F_{\text{noise}}·{\chi }_{11}\left({\omega }_{\text{d1}}\right)$ on Sphere 1. Fortunately, since ${\chi }_{11}\left({\omega }_{\text{d1}}\right)\ll {\chi }_{21}\left({\omega }_{\text{d1}}\right)$, we can safely neglect the displacement response $X_{\text{noise}}\left({\omega }_{\text{d1}}\right)$ caused by $F_{\text{noise}}$. Therefore, exploiting elimination modes of the levitated pendulum, we can introduce a novel detection method that effectively suppresses certain force noises. In this paper, by utilizing the levitated pendulum, we can select the resonance modes (${\omega }_1$ and ${\omega }_2$) or elimination modes (${\omega }_{\text{d1}}$ and ${\omega }_{\text{d2}}$) to detect non-Newtonian gravity. The former can attenuate displacement noise, while the latter can mitigate force noise. These modes are collectively referred to as detection modes ${\omega }_{\text{det}}$ in the subsequent discussion.

3. Search for non-Newtonian gravity

To generate a modulation of any non-Newtonian gravity along $x$, a spherical source mass of radius $R_{\mathrm{s}}$ and mass $m_{\mathrm{s}}$ is mounted on a cantilever beam to undergo a vertical displacement amplitude of $A_{\mathrm{s}}$ at the frequency of ${\omega }_{\mathrm{s}}$, as shown in Fig. 2. By setting ${\omega }_{\mathrm{s}}={\omega }_{\text{det}}$, any non-Newtonian gravity between the test mass and the source mass is modulated to be resonant with the detection mode of the levitated pendulum sensor. Taking Sphere 2 as the test mass with radius $R_{\mathrm{t}}$ and mass $m_{\mathrm{t}}$, we integrate the source mass $m_{\mathrm{s}}$ and the test mass $m_{\mathrm{t}}$ to derive the lateral component of the non-Newtonian gravity acting on the test mass:

$$F_{\mathrm{Y}}\left(r\right)=\alpha G{m}_s^{\prime }{m}_t^{\prime }(\frac{1}{r\lambda }+\frac{1}{r^2})e^{-\frac{r}{\lambda }}·\frac{d}{r}$$ (6)

Here, $r$ represents the center-of-mass distance between the test mass and the source mass, given by $r=\sqrt{A_{\mathrm{s}}^2{\cos }^2\left({\omega }_{\mathrm{s}}t\right)+d^2}$. The parameter $d$ corresponds to the nearest distance between the centers of the two spheres, calculated as $d=d_0+R_{\mathrm{t}}+R_{\mathrm{s}}$, where $d_0$ denotes the nearest surface distance. The effective masses of the source mass and test mass are respectively represented as $m_{\mathrm{s}}^{\prime }$ and $m_t^{\prime }$, with their expressions given by:

$$\begin{array}{c}{m}_t^{\text{'}}=2\pi {\rho }_t{\lambda }^2((R_{\mathrm{t}}-\lambda )e^{\frac{R_{\mathrm{t}}}{\lambda }}+(R_{\mathrm{t}}+\lambda )e^{-\frac{R_{\mathrm{t}}}{\lambda }})\hfill \\ m_s^{\text{'}}=2\pi {\rho }_s{\lambda }^2((R_s-\lambda )e^{\frac{R_s}{\lambda }}+(R_s+\lambda )e^{-\frac{R_s}{\lambda }})\hfill \end{array}$$ (7)

where ${\rho }_t$ and ${\rho }_s$ denote the densities of the test mass and the source mass, respectively.

Fig. 2.

Schematic diagram of the proposed experiment to detect non-Newtonian gravity on a micrometer scale. The pendulum is levitated in a diamagnetic trap in vacuum, with the test mass protruding outside the trap to detect non-Newtonian gravity. A spherical source mass of radius $R_{\mathrm{s}}$ is mounted on a cantilever beam to undergo a vertical motion amplitude of $A_{\mathrm{s}}$ at a frequency of ${\omega }_{\mathrm{s}}$. To make the vertical motion occur, this scheme utilizes the lever principle with an electromagnet below the cantilever beam’s tail, charged and discharged at frequency ${\omega }_{\mathrm{s}}$. A Faraday cage encloses the electromagnet assembly to prevent electromagnetic interference with the test mass. By setting ${\omega }_{\mathrm{s}}={\omega }_{\text{det}}$, any non-Newtonian gravity between the test mass and the source mass is modulated to be resonant with the detection mode of the levitated pendulum sensor. A metal shield of large enough area is placed statically between the test and the source masses to screen the electromagnetic forces between them.

Several backgrounds need to be carefully considered to determine the force sensitivity of this scheme. Firstly we calculated the gravitation $F_{\mathrm{g}}$ between the test mass and the source mass. By integrating the test mass and source mass, we obtain the expression for the horizontal component of the gravitation between them:

$$F_{\mathrm{g}}=G·\frac{m_t{m}_s}{r^2}·\frac{d}{r}$$ (8)

It is noteworthy that the gravitation is modulated with the same frequency as non-Newtonian gravity, which may interfere with the detection of non-Newtonian gravity. Considering gravitation is a long-range force that is typically very weak at the micrometer scale. In contrast, non-Newtonian gravity follows a surface force expression as shown in Eq. 6. Therefore, we can engineer the sensor’s structural parameters to deduct the gravitation, as demonstrated in reference [22], [23].

In addition to the backgrounds from the gravitation, it is crucial to assess the backgrounds from random noise. In this scheme, random noise is dominated by the thermal Langevin force, which results from the gas collisions. To evaluate the force sensitivity constrained by the thermal Langevin force, we employ the test mass as the force probe and utilize the pendulation motion mode as the detection mode. The detailed calculation, based on the fluctuation-dissipation theorem [53], can be found in Appendix C. The calculated force sensitivity constrained by the thermal Langevin force is as follows:

$$\sqrt{S_{\text{ff}}^{\text{th}}}=\sqrt{4m_{\mathrm{t}}{\gamma }_{\mathrm{t}}{k}_{\mathrm{B}}T}$$ (9)

with $k_{\mathrm{B}}$ the Boltzmann constant, $T$ ambient temperature, and ${\gamma }_{\mathrm{t}}$ the damping rate estimated using the proximity force approximation [49], [54]. The expression for ${\gamma }_{\mathrm{t}}$ is given by:

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{t}}=\sqrt{\frac{\pi m_{\mathrm{air}}}{k_{\mathrm{B}}T}}\frac{R_{\mathrm{t}}^2}{m_{\mathrm{t}}}[\sqrt{8}(2+\frac{\pi }{4})+\frac{\pi R_{\mathrm{t}}^2}{\sqrt{2}{d}_{\mathrm{m}}^2\ln (1+\frac{R_{\mathrm{t}}^2}{d_{\mathrm{m}}^2})}]P\end{array}$$ (10)

where $m_{\text{air}}$ is the mass of an air molecule, $P$ is ambient pressure, and $d_{\mathrm{m}}$ is the surface distance between the test mass and nearby shielding membrane.

At the micrometer scale, electromagnetic forces, specifically patch potential-induced force noise and Casimir force noise from nearby surfaces, are usually greater than the thermal Langevin force over several magnitudes [42], [47]. Thus, addressing these noises becomes essential in improving the constraint on the detection of non-Newtonian gravity. To screen them, we designed a static metal shield between the test mass and the source mass, as depicted in Fig. 2. This method is known as equipotential technique, which has been proven to effectively mitigate electromagnetic forces interference in many short-range force detection experiments [7], [17], [22], [23], [48], [49]. To prevent any interference between the shield and the trap, the levitated pendulum operates by protruding the test mass outside the trap. This design allows for a sufficiently large shield that effectively screens out force noises while avoiding any disruption to the trap (see Appendix D).

The experimental setup may be also susceptible to vibration-induced force noise, which can arise from two sources: source mass-driven vibration and seismic vibration. To eliminate source mass-driven vibration force, we vertically drive the source mass to avoid coupling with lateral forces, assuming our focus is on detecting the lateral component of non-Newtonian gravity. Seismic vibration force noise can be reduced using commercial seismic isolation technology, which significantly attenuates noise levels below ambient thermal noise [55]. Further details can be found in Appendix E.

When measuring force at the resonant frequency, it is important to consider the noise caused by laser intensity fluctuations and beam pointing fluctuations, commonly referred to as detection noise [56]. However, in this experimental scheme, this kind of noise can be deemed negligible. This is attributed to the utilization of low laser power in the diamagnetic traps, which can be as low as a few micro-watts. More information can be found in research [34], [52].

Because source mass-driven vibration force and gravitation are interference signals, they are regular and we can always find ways to deduct them in real experiment [23]. Eventually, the force sensitivity of this scheme is subject to thermal Langevin force and seismic vibration noise:

$$\begin{array}{c}\sqrt{S_{\text{ff}}\left(\omega \right)}=\sqrt{S_{\text{ff}}^{\text{th}}\left(\omega \right)+S_{\mathrm{f}\mathrm{f}}^{\text{sei}}\left(\omega \right)}\hfill \end{array}$$ (11)

Here, $S_{\text{ff}}^{\text{th}}\left(\omega \right)$ and $S_{\text{ff}}^{\text{sei}}\left(\omega \right)$ are the force power spectral densities (PSDs) of the thermal Langevin force and seismic vibration noise, respectively. Subsequently, the signal-to-noise ratio (SNR) of this scheme is given by:

$$\text{SNR}=\frac{F_{\mathrm{Y}}}{\sqrt{S_{\text{ff}}\left(\omega \right)/t_{\text{mea}}}}$$ (12)

where $t_{\text{mea}}$ is the measurement time, and $\sqrt{S_{\text{ff}}\left(\omega \right)/t_{\text{mea}}}$ represents the minimum detectable force, referred to as $F_{\text{min}}$ hereafter.

If the condition for detecting non-Newtonian gravity requires $\text{SNR}=1$, combining Eqs. 6 and 12, we can derive the constraint on $\alpha$ as follows:

$$\alpha =\frac{F_{\text{min}}}{G{m}_s^{\prime }{m}_{\mathrm{t}}^{\prime }(\frac{1}{r\lambda }+\frac{1}{r^2})e^{-\frac{r}{\lambda }}·\frac{d}{r}}$$ (13)

To perform an experiment testing non-Newtonian gravity at the micrometer scale, we suggest a set of parameters outlined in Table 1. The scheme is to conduct a resonance measurement and choose ${\omega }_{\text{det}}={\omega }_2=2\pi \times 10.3\text{Hz}$ as the detection frequency. By setting the source mass driven frequency equal to ${\omega }_{\text{det}}$ and using a vibration amplitude of $A_{\mathrm{s}}=60$ µm, along with employing golden test mass and a golden source mass with equal radii ($R_{\mathrm{t}}\sim 30$ µm, $R_{\mathrm{s}}\sim 30$ µm) and a closest surface distance of $d_0\sim 4.2$ µm between them, we estimate the force noises from various backgrounds theoretically. The surface distance between the test mass and the shield is $d_{\mathrm{m}}=0.9d_0=3.8$ µm, resulting in the shield exerting an attractive Casimir force of $F_{\text{cas}}=1.5\times {10}^{-15}\mathrm{N}$ on the test mass. This force causes the test mass to approach the shield by $9.8\mathring{A}$ and induces a tilt of the levitated pendulum of $0.1\mu \text{rad}$ away from the vertical direction. Such an effect is considered negligible in this scheme. Assuming an ambient temperature of $T=300\mathrm{K}$ and a vacuum pressure of $P={10}^{-5}\text{Pa}$, the final force sensitivity of this scheme is dominated by the thermal Langevin force noise, with $\sqrt{S_{\text{ff}}}\approx \sqrt{S_{\text{ff}}^{\text{th}}}\sim 6.3\times {10}^{-18}\mathrm{N}/\sqrt{\text{Hz}}$. Considering a measurement time of $t_{\text{mea}}={10}^4$s, the minimum detectable force is $F_{\text{min}}=6.3\times {10}^{-20}\mathrm{N}$. If the criterion for detecting non-Newtonian gravity is that $\text{SNR}=1$, this minimum detectable force leads to the value of $\alpha =28$ at $\lambda =7.6$ µm, according to Eq. 13. It means that the constraint on $\alpha$ will be improved by more than 3 orders of magnitude compared to the existing best result [17]. By employing the parameters provided in Table 1, we calculate the corresponding values of $\alpha$ for different $\lambda$. As depicted by the solid red line in Fig. 3, the constraint on $\alpha$ is significantly improved within the sub-micron to sub-millimeter range when compared to existing experimental results [17], [22]. Furthermore, we further optimized the parameters $R_{\mathrm{t}}$ ($R_{\mathrm{s}}$) and $d_0$ to minimize the value of $\alpha$ at different $\lambda$. By setting $d_0\left(\lambda \right)=0.55\lambda$ and $R_{\mathrm{t}}\left(\lambda \right)=R_{\mathrm{s}}\left(\lambda \right)=4\lambda$, we achieved optimized minimum values of $\alpha$ as a function of $\lambda$, shown by the dashed red line in Fig. 3. This optimization leads to an improved constraint on $\alpha$ over a broader range of $\lambda$. Such an anticipated performance can be attributed to two factors. Firstly, the utilization of a diamagnetic trap allows for the levitation of a more massive test mass, thereby enhancing the non-Newtonian gravity signal associated with mass. Secondly, the configuration of a levitated pendulum permits the incorporation of a large shield without interfering with the levitation trap, consequently reducing electromagnetic forces and improving force sensitivity.

Table 1.

Suggested parameters for detecting non-Newtonian gravity. The force sensitivity of this scheme is subject to the thermal Langevin force. The minimum detectable force is ultimately determined by the combined influences of the thermal Langevin force and the gravitation. $\alpha$ is obtained by setting $\text{SNR}=1$. namely $F_{\mathrm{Y}}=F_{\text{min}}$.

Parameter	Symbol	Units	Value
Detection frequency	${\omega }_{\text{det}}/2\pi$	$\text{Hz}$	10.3
Ambient pressure	$P$	$\mathrm{Pa}$	$1\times {10}^{-5}$
Ambient temperature	$T$	$\mathrm{K}$	300
Damping rate	${\gamma }_{\mathrm{t}}$	$\mathrm{Hz}$	$1\times {10}^{-6}$
Force sensitivity	$\sqrt{S_{\text{ff}}}$	$\mathrm{N}/\sqrt{\text{Hz}}$	$6.3\times {10}^{-18}$
Measurement time	$t_{\text{mea}}$	$\mathrm{s}$	$1\times {10}^4$
Min detectable force	$F_{\text{min}}$	$\mathrm{N}$	$6.3\times {10}^{-20}$
$\alpha (@\lambda =7.6\mu \mathrm{m})$	$\alpha$	$\mathrm{a}.\mathrm{u}.$	28

Fig. 3.

The parameter region of non-Newtonian gravity. The grey region has been excluded by previous experiments (IUPUI 2016 [17] and Eöt-Wash 2020 [22]). The solid and dashed red line represent constraints on $\alpha$ predicted by this work. The solid red line is obtained by using the parameters listed in Table 1. The dashed red line is obtained by using the optimized parameters $R_{\mathrm{t}}$, $R_{\mathrm{s}}$ and $d_0$, which vary with $\lambda$.

4. Conclusion

This work proposes an experimental scheme for detecting non-Newtonian gravity using a diamagnetic levitated pendulum sensor. It takes advantage of the capability of diamagnetic levitation systems to levitate large mechanical oscillators, thereby enhancing non-Newtonian gravity signals. Meanwhile, the pendulum structure is compatible with a large enough electromagnetic shielding film to isolate electromagnetic forces interference. Therefore, the levitated pendulum has the potential to address both major challenges facing levitated mechanical oscillator systems for non-Newtonian gravity detection simultaneously. Employing the levitated pendulum as a sensor, our proposed experimental design anticipates improving constraint on non-Newtonian gravity by $2\sim 3$ orders of magnitude at the micrometer scale. Considering that the force sensitivity of this system is mainly limited by thermal noise, obtaining stronger constraints on non-Newtonian gravity is expected in low-temperature environments. Additionally, by deriving the sensor’s response function, we obtained a novel elimination mode that can suppress force noise during precision measurements. This finding has significant implications for future experiments surpassing the standard quantum-limited position measurement. In conclusion, levitated pendulum proposed by this work shows great promise as a powerful tool for future short-range sensing, basic physics research, and quantum precision measurement.

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Fang Xiong(BRID: 01051.00.86577) is a postdoctor of Research Center for Quantum Sensing of Zhejiang Lab. She received the Ph.D. degree of physics in Nanjing University in 2021. Her major research direction is quantum precision measurement using vacuum levitated mechanical oscillator systems.

Tong Wu(BRID: 00675.00.62966) is an assistant researcher of Research Center for Quantum Sensing of Zhejiang Lab. He received the Ph.D. degree of physics in Nanjing University in 2022. His major research directions are quantum precision measurement and quantum optics.

Huizhu Hu(BRID: 03282.00.82582) is a Qiushi distinguished professor of Zhejiang University, PI of Zhejiang Lab, and director of Fundamental Science on Optical Inertial and Sensing Technology Laboratory. He got his bachelor of science from Xi’an Jiaotong University in 1999, and received his doctor of philosophy from Zhejiang University in 2004. Then, he joined the current university as a faculty and was promoted as a professor in 2012. His current research interests are in optical sensing and precision measurement.

Appendix A. Intrinsic motion modes of the levitated pendulum

Assuming the levitated pendulum is rigid, the coordinates of the center of mass of Sphere 1 in the coordinate system $(x,y,z,\theta ,\phi )$ can be expressed as:

$$\begin{array}{c}{x}_1=x+L_1\sin \theta \cos \phi \hfill \\ y_1=y+L_1\sin \theta \sin \phi \hfill \\ z_1=z+L_1\cos \theta \hfill \end{array}$$ (14)

and the center-of-mass coordinate of Sphere 2 is written as

$$\begin{array}{c}{x}_2=x-L_2\sin \theta \cos \phi \hfill \\ y_2=y-L_2\sin \theta \sin \phi \hfill \\ z_2=z-L_2\cos \theta \hfill \end{array}$$ (15)

$L_1$ ($L_2$) is the center-of-mass distance between Sphere 1 (Sphere 2) and the levitated pendulum, with

$$\begin{array}{c}{\displaystyle L_1=\frac{m_2(R_1+L+R_2)+m_3(R_1+L/2)}{m}}\hfill \\ {\displaystyle L_2=\frac{m_1(R_1+L+R_2)+m_3(R_2+L/2)}{m}}\hfill \end{array}$$ (16)

where $L$ is the total length of the rod. $m=m_1+m_2+m_3$ and $m_1$, $m_2$, $m_3$ are the masses of Sphere 1, Sphere 2, rod respectively. Additionally, Sphere 1 deviates slightly from the center of the diamagnetic trap, with the three-dimensional deviations described as

$$\Delta x=x_1,\Delta y=y_1,\Delta z=z_1-(L_1+\frac{mg}{k_z})$$ (17)

The three-dimensional deviations will determine the magnetic restoring force on Sphere 1. Considering the three-dimensional magnetic forces ($-{\mathrm{k}}_{\mathrm{x}}\Delta x,-{\mathrm{k}}_{\mathrm{y}}\Delta y,-{\mathrm{k}}_{\mathrm{z}}\Delta z$) and pendulum’s gravity $mg$ and their corresponding torques exerted on the pendulum, the dynamic equations for each degree of freedom of the levitated pendulum as show:

$$\begin{array}{c}m\ddot{x}=-k_x\Delta x\hfill \\ m\ddot{y}=-k_{\mathrm{y}}\Delta y\hfill \\ m\ddot{z}=-\mathit{mg}-k_{\mathrm{z}}\Delta z\hfill \\ I_{\theta }\ddot{\theta }=k_{\mathrm{z}}\Delta z·L_1\sin \theta -k_{\mathrm{x}}\Delta x\cos \phi ·L_1\cos \theta -k_{\mathrm{y}}\Delta y\sin \phi ·L_1\cos \theta \hfill \\ I_{\phi }\ddot{\phi }=k_x\Delta x\sin \phi ·L_1\sin \theta -k_y\Delta y\cos \phi ·L_1\sin \theta \hfill \end{array}$$ (18)

$I_{\theta }$ and $I_{\phi }$ represent the rotational inertia of the levitated pendulum in the $\theta$ direction and the $\phi$ direction, respectively, which are wirren as:

$$\begin{array}{c}{I}_{\theta }=m_1{L}_1^2+m_2{L}_2^2+\frac{m_3(L_1^3+L_2^3)}{3(L_1+L_2)}+\frac{2}{5}{m}_1{R}_1^2+\frac{2}{5}{m}_2{R}_2^2\hfill \\ I_{\phi }=(m_1{L}_1^2+m_2{L}_2^2+\frac{m_3(L_1^3+L_2^3)}{3(L_1+L_2)}){\sin }^2\theta +\frac{2}{5}{m}_1{R}_1^2+\frac{2}{5}{m}_2{R}_2^2\hfill \end{array}$$ (19)

In the case of small angle approximation, $\sin \theta \approx \theta ,\cos \theta \approx 1$. Substituting Eqs. 14 and 17 into the dynamics equation system Eq. 18 and ignoring the higher-order infinitesimals, and supposing the levitated pendulum only move in the $xz$ plane ($\phi \equiv 0$), we can simplify Eq. 18 as

$$\begin{array}{c}m\ddot{x}=-k_{x}x-k_x{L}_1\theta \hfill \\ m\ddot{z}=-k_{z}z\hfill \\ I_{\theta }\ddot{\theta }=-\mathit{mg}{L}_1\theta -k_x{L}_{1}x-k_x{L}_1^2\theta \hfill \end{array}$$ (20)

Solve Eq. 20, we get the intrinsic motion modes of the levitated pendulum as

$$\begin{array}{c}{\omega }_1=\sqrt{(k_{\mathrm{a}}-k_{\mathrm{b}})/m}\hfill \\ {\omega }_2=\sqrt{(k_{\mathrm{a}}+k_{\mathrm{b}})/m}\hfill \\ {\omega }_3=\sqrt{k_{\mathrm{c}}/m}\hfill \end{array}$$ (21)

where $k_{\mathrm{a}}$, $k_{\mathrm{b}}$ and $k_{\mathrm{c}}$ are the effective stiffness coefficients as

$$\begin{array}{c}{\displaystyle k_{\mathrm{a}}=\frac{I_{\theta }{k}_{\mathrm{x}}+m^{2}g{L}_1+m{k}_{\mathrm{x}}{L}_1^2}{2I_{\theta }}}\hfill \\ {\displaystyle k_{\mathrm{b}}=\frac{{(m^2{k}_{\mathrm{x}}^2{L}_1^4+2m^{3}g{k}_{\mathrm{x}}{L}_1^3+m^4{g}^2{L}_1^2+2m{k}_{\mathrm{x}}^2{L}_1^2{I}_{\theta }-2m^{2}g{k}_{\mathrm{x}}{L}_1{I}_{\theta }+k_{\mathrm{x}}^2{I}_{\theta }^2)}^{\frac{1}{2}}}{2I_{\theta }}}\hfill \\ k_{\mathrm{c}}=k_{\mathrm{z}}\hfill \end{array}$$ (22)

Appendix B. Displacement response of the levitated pendulum to external force

When an external force of form $f_{\text{ex}}^{\mathrm{j}=1}{e}^{\mathrm{i}\omega \mathrm{t}}$ is horizontally applied on Sphere 1, the dynamic equation system of the levitated pendulum Eq. 20 becomes

$$\begin{array}{ccc}\hfill m\ddot{x}& =& {\displaystyle -k_{x}x-k_x{L}_1\theta -{\Gamma }_1\dot{x_1}-{\Gamma }_2\dot{x_2}-{\int }_{-L_2}^{L_1}{\Gamma }_3/L(l\dot{\theta }+\dot{x})\mathit{dl}+f_{\text{ex}}^{\mathrm{j}=1}{e}^{i\omega t}}\hfill \\ \hfill I_{\theta }\ddot{\theta }& =& {\displaystyle -\mathit{mg}{L}_1\theta -k_x{L}_{1}x-k_x{L}_1^2\theta -{\Gamma }_1{L}_1\dot{x_1}+{\Gamma }_2{L}_2\dot{x_2}}\hfill \\ & & {\displaystyle -{\int }_{-L_2}^{L_1}{\Gamma }_3/L(l\dot{\theta }+\dot{x})\mathit{ldl}+f_{\text{ex}}^{\mathrm{j}=1}{e}^{i\omega t}{L}_1}\hfill \end{array}$$ (23)

where ${\Gamma }_1,{\Gamma }_2,{\Gamma }_3$ are the air molecule collision damping rates of Sphere 1, Sphere 2, and connecting rod, respectively, which have the following form:

$$\begin{array}{c}{\Gamma }_1={\left(\frac{\pi m_{\mathrm{air}}}{k_{\mathrm{B}}T}\right)}^{1/2}{R}_1^2\sqrt{8}(2+\frac{\pi }{4})P\hfill \\ {\Gamma }_2={\left(\frac{\pi m_{\mathrm{air}}}{k_{\mathrm{B}}T}\right)}^{1/2}{R}_2^2[\sqrt{8}(2+\frac{\pi }{4})+\frac{\pi R_2^2}{\sqrt{2}{d}_{\mathrm{m}}^2\ln (1+R_2^2/d_{\mathrm{m}}^2)}]P\hfill \\ {\Gamma }_3={\left(\frac{2\pi m_{\text{air}}}{k_{\mathrm{B}}T}\right)}^{\frac{1}{2}}{r}_{\mathrm{L}}\mathit{LP}\hfill \end{array}$$ (24)

${\Gamma }_2$ is calculated according to proximity force approximation, as it is in close proximity to the electromagnetic shielding membrane. $d_{\mathrm{m}}$ is the surface distance between the test mass and the nearby shielding membrane. $m_{\text{air}}$ is the mass of a air molecule. $R_1$ and $R_2$ are the radii of Sphere 1 and Sphere 2 respectively. $r_{\mathrm{L}}$ is the radius of the rod and $L$ is its length. $T$ is ambient temperature. $P$ is ambient pressure. Combine Eqs. 5, 14 and 23, the Sphere $k^{\prime }$s ($k=1,or2$) displacement responses to the external force $f_{\text{ex}}^{\mathrm{j}=1}{e}^{\mathrm{i}\omega \mathrm{t}}$ are as follows:

$$\begin{array}{c}{\displaystyle {\chi }_{11}=\frac{x_1}{f_{\text{ex}}^{\mathrm{j}=1}{e}^{\mathrm{i}\omega \mathrm{t}}}=\frac{A_{22}-L_1{A}_{12}}{A_{11}{A}_{22}-A_{12}{A}_{21}}+L_1\frac{L_1{A}_{11}-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \\ {\displaystyle {\chi }_{12}=\frac{x_2}{f_{\text{ex}}^{\mathrm{j}=1}{e}^{\mathrm{i}\omega \mathrm{t}}}=\frac{A_{22}-L_1{A}_{12}}{A_{11}{A}_{22}-A_{12}{A}_{21}}-L_2\frac{L_1{A}_{11}-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \end{array}$$ (25)

with

$$\begin{array}{c}{A}_{11}=-m{\omega }^2+k_x+\mathit{Ai}\omega \hfill \\ A_{12}=A_{21}=k_x{L}_1+\mathit{Bi}\omega \hfill \\ A_{22}=-{\mathrm{I}}_{\theta }{\omega }^2+\mathit{mg}{L}_1+k_x{L}_1^2+\mathit{Ci}\omega \hfill \\ A={\Gamma }_1+{\Gamma }_2+{\Gamma }_3\hfill \\ B={\Gamma }_1{L}_1-{\Gamma }_2{L}_2+{\Gamma }_3\frac{L_1-L_2}{2}\hfill \\ C={\Gamma }_1{L}_1^2+{\Gamma }_2{L}_2^2+{\Gamma }_3\frac{L_1^3+L_2^3}{3(L_1+L_2)}\hfill \end{array}$$ (26)

where $i$ is the imaginary unit.

Similarly, when an external force of form $f_{\text{ex}}^{\mathrm{j}=2}{e}^{\mathrm{i}\omega \mathrm{t}}$ is horizontally applied on Sphere 2, the Sphere $k^{\prime }$s ($k=1,or2$) displacement responses to the external force $f_{\text{ex}}^{\mathrm{j}=2}{e}^{\mathrm{i}\omega \mathrm{t}}$ are as follows:

$$\begin{array}{c}{\displaystyle {\chi }_{21}=\frac{x_1}{f_{\text{ex}}^{\mathrm{j}=2}{e}^{\mathrm{i}\omega \mathrm{t}}}=\frac{A_{22}+L_2{A}_{12}}{A_{11}{A}_{22}-A_{12}{A}_{21}}+L_1\frac{-L_2{A}_{11}-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \\ {\displaystyle {\chi }_{22}=\frac{x_2}{f_{\text{ex}}^{\mathrm{j}=2}{e}^{\mathrm{i}\omega \mathrm{t}}}=\frac{A_{22}+L_2{A}_{12}}{A_{11}{A}_{22}-A_{12}{A}_{21}}-L_2\frac{-L_2{A}_{11}-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \end{array}$$ (27)

Sphere 1 was chosen to be a bismuth sphere with radius $R_1=30\mu$m, sphere 2 was chosen to be a gold sphere with radius $R_2=30\mu$m, the rod has a radius of $r_{\mathrm{l}}=5\mu$m and length $L=10$mm made of silicon, at a pressure of $P={10}^{-5}$Pa, with an surface-to-surface distance between sphere 2 and the membrane of $d_m=3.8\mu$m, the image of ${\chi }_{\text{jk}}\left(\omega \right)$ is shown in Fig. 1d.

Appendix C. Thermal Langevin force

Assuming that the thermal Langevin forces applied to Sphere 1 and Sphere 2 are ${\xi }_1\left(t\right)$ and ${\xi }_2\left(t\right)$ respectively, they satisfy

$$\begin{array}{c}\langle {\xi }_1\left(t\right)\rangle =0,\langle {\xi }_2\left(t\right)\rangle =0\hfill \\ \langle {\xi }_1\left(t\right)\mid {\xi }_1\left(t^{\prime }\right)\rangle =S_{\text{ff}1}^{\text{th}}\delta (t-t^{\prime })\hfill \\ \langle {\xi }_2\left(t\right)\mid {\xi }_2\left(t^{\prime }\right)\rangle =S_{\text{ff}2}^{\text{th}}\delta (t-t^{\prime })\hfill \\ \langle {\xi }_1\left(t\right)\mid {\xi }_2\left(t\right)\rangle =0,\langle {\xi }_1\left(t\right)\mid {\xi }_2\left(t^{\prime }\right)\rangle =0\hfill \end{array}$$ (28)

Substitute ${\xi }_1\left(t\right)$ and ${\xi }_2\left(t\right)$ and corresponding torques into equation of the levitated pendulum, considering only the direction of oscillation of interest ($x$ direction), we have

$$\begin{array}{c}m\ddot{x}=-k_{x}x-k_x{L}_1\theta -A\dot{x}-B\dot{\theta }+{\xi }_1\left(t\right)+{\xi }_2\left(t\right)\hfill \\ I_{\theta }\ddot{\theta }=-k_x{L}_{1}x-(\mathit{mg}{L}_1+k_x{L}_1^2)\theta -B\dot{x}-C\dot{\theta }+{\xi }_1\left(t\right)L_1-{\xi }_2\left(t\right)L_2\hfill \end{array}$$ (29)

Performing a Fourier transform on equation system Eq. 29, we solve the Fourier transform of $x$ and $\theta$, as

$$\begin{array}{c}\hfill \tilde{\mathrm{x}}=\frac{{\mathrm{A}}_{22}({\tilde{\mathrm{\xi }}}_1+{\tilde{\mathrm{\xi }}}_2)-{\mathrm{A}}_{12}({\tilde{\mathrm{\xi }}}_1{\mathrm{L}}_1-{\tilde{\mathrm{\xi }}}_2{\mathrm{L}}_2)}{{\mathrm{A}}_{11}{\mathrm{A}}_{22}-{\mathrm{A}}_{12}{\mathrm{A}}_{21}}={\mathrm{B}}_{1\mathrm{x}}{\tilde{\mathrm{\xi }}}_1+{\mathrm{B}}_{2\mathrm{x}}{\tilde{\mathrm{\xi }}}_2\\ \hfill \tilde{\mathrm{\theta }}=\frac{{\mathrm{A}}_{11}({\tilde{\mathrm{\xi }}}_1{\mathrm{L}}_1-{\tilde{\mathrm{\xi }}}_2{\mathrm{L}}_2)-{\mathrm{A}}_{21}({\tilde{\mathrm{\xi }}}_1+{\tilde{\mathrm{\xi }}}_2)}{{\mathrm{A}}_{22}{\mathrm{A}}_{11}-{\mathrm{A}}_{12}{\mathrm{A}}_{21}}={\mathrm{B}}_{1\mathrm{\theta }}{\tilde{\mathrm{\xi }}}_1+{\mathrm{B}}_{2\mathrm{\theta }}{\tilde{\mathrm{\xi }}}_2\end{array}$$ (30)

where $\tilde{x}$ and $\tilde{\theta }$ are the Fourier transforms of $x$ and $\theta$, respectively, and ${\tilde{\xi }}_1$ and ${\tilde{\xi }}_2$ are the Fourier transforms of the Langevin forces ${\xi }_1$ and ${\xi }_2$ acting on sphere 1 and sphere 2, respectively, with coefficients

$$\begin{array}{c}{\displaystyle {\mathrm{B}}_{1\mathrm{x}}=\frac{A_{22}-A_{12}{L}_1}{A_{11}{A}_{22}-A_{12}{A}_{21}},{\mathrm{B}}_{2\mathrm{x}}=\frac{A_{22}+A_{12}{L}_2}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \\ {\displaystyle {\mathrm{B}}_{1\theta }=\frac{A_{11}{L}_1-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}},{\mathrm{B}}_{2\theta }=\frac{-A_{11}{L}_2-A_{21}}{A_{11}{A}_{22}-A_{12}{A}_{21}}}\hfill \end{array}$$ (31)

According to the theorem of equipartition of energy and Parseval’s theorem and the relationship of PSD of velocity$S_{\text{vv}}\left(\omega \right)={\omega }^2{S}_x\left(\omega \right)$ and PSD of angular velocity $S_{{\omega }_{\theta }}\left(\omega \right)={\omega }^2{S}_{\theta }\left(\omega \right)$, we have:

$$\begin{array}{c}{\displaystyle \frac{m}{2}{\int }_{-\infty }^{\infty }{\omega }^2\langle S_{\mathrm{x}}\left(\omega \right)\rangle d\omega =\frac{4\pi k_{B}T}{2}}\hfill \\ {\displaystyle \frac{I_{\theta }}{2}{\int }_{-\infty }^{\infty }{\omega }^2\langle S_{\theta }\left(\omega \right)\rangle d\omega =\frac{4\pi k_{B}T}{2}}\hfill \end{array}$$ (32)

Substitute Eqs. 28 and 30 into Eq. 32, and combine the definition of the PSD: $S_x\left(\omega \right)=2|\tilde{x}{|}^2/t_{\text{mea}},S_{\theta }\left(\omega \right)=2{\left|\tilde{\theta }\right|}^2/t_{\text{mea}}$, and relational expression: $\langle {\tilde{\xi }}_i|{\tilde{\xi }}_j\rangle ={\delta }_{ij}{S}_{\text{ff}i}^{\text{th}}·t_{\text{mea}}$, we can get:

$$\begin{array}{c}{\displaystyle {\int }_0^{\infty }\frac{2}{t_{\text{mea}}}({\mathrm{B}}_{1\mathrm{x}}^2{S}_{\text{ff}1}^{\text{th}}{t}_{\text{mea}}+{\mathrm{B}}_{2\mathrm{x}}^2{S}_{\text{ff}2}^{\text{th}}{t}_{\text{mea}}){\omega }^{2}d\omega =\frac{4\pi k_{B}T}{m}}\hfill \\ {\displaystyle {\int }_0^{\infty }\frac{2}{t_{\text{mea}}}({\mathrm{B}}_{1\theta }^2{S}_{\text{ff}1}^{\text{th}}{t}_{\text{mea}}+{\mathrm{B}}_{2\theta }^2{S}_{\text{ff}2}^{\text{th}}{t}_{\text{mea}}){\omega }^{2}d\omega =\frac{4\pi k_{B}T}{I_{\theta }}}\hfill \end{array}$$ (33)

By solving the above equations, the PSD of the Langevin forces acting on sphere 1 and sphere 2 can be obtained:

$$\begin{array}{c}{S}_{\text{ff}1}^{\text{th}}\left(\omega \right)=4m_1{\gamma }_1{k}_{B}T\hfill \\ S_{\text{ff}2}^{\text{th}}\left(\omega \right)=4m_2{\gamma }_2{k}_{B}T\hfill \end{array}$$ (34)

where $m_1{\gamma }_1={\Gamma }_1$, and $m_2{\gamma }_2={\Gamma }_2$.

When taking the Sphere 2 as the test mass, thus $m_{\mathrm{t}}=m_2$, and ${\gamma }_{\mathrm{t}}={\gamma }_2$, the thermal Langevin force acting on the test mass is

$$F_{\text{th}}=\sqrt{\frac{S_{\text{ff}}^{\text{th}}}{t_{\text{mea}}}}=\sqrt{\frac{4m_{\mathrm{t}}{\gamma }_{\mathrm{t}}{k}_{\mathrm{B}}T}{t_{\text{mea}}}}$$ (35)

With the necessary parameters listed in Table 1, we calculated that $\sqrt{S_{\text{ff}}^{\text{th}}}=6.3\times {10}^{-18}\mathrm{N}/\sqrt{\text{Hz}}$ and $F_{\text{th}}=6.3\times {10}^{-20}\mathrm{N}$, according to Eqs. 34 and 35.

Appendix D. Electromagnetic force noise

In this scheme, electromagnetic force noises mainly include the Casimir force noise and patch potential-induced force noise. Since our detection frequency domain is approximately $10\text{Hz}$ and a metal shielding membrane has the potential to shield low-frequency electromagnetic noise, we simulated the electrostatic shielding effect of a $300\text{nm}$ thick metal membrane. To elaborate, let us assume that the mass source has a space charge distribution of ${10}^{-4}\mathrm{C}/{\mathrm{m}}^3$ and a radius of 30 µm (There are approximately 70 elementary charges on the sphere), and the shielding membrane has a surface area of 600 µm $\times$ 600 µm $\times$ 0.3 µm. With the metal shield in place, we can observe that the spatial electric field distribution is shown in Fig. D.4a. The electromagnetic field on the right side of the shielding membrane significantly decreases, as shown in Fig. D.4b, indicates that a metal membrane with a thickness of several hundred nanometers can prevent the test mass from being affected by the electric field generated by the source mass. Therefore, the electromagnetic force noises between the source mass and the test mass in this scheme can be screened by a metal membrane of several hundred nanometers thick.

Fig. D1.

Simulation of electromagnetic shielding. (a) Electric field distribution around the source mass and the test mass in the presence of a metal shield. (b) The absolute value of the electric field varies with $x$.

On the other hand, the metal shield between the source mass and the test mass will produce an attractive Casimir force on the test mass. Based on the proximity force approximation [57], the Casimir force between the test mass and the metal shield is

$$F_{\text{cas}}=-\frac{\partial V_{\text{cas}}}{\partial d_{\mathrm{m}}}=\frac{\hslash c{\pi }^2}{720d_{\mathrm{m}}^3}2\pi R_{\mathrm{t}}$$ (36)

It will make the test mass approach the shield at a distance of

$$\Delta d_{\mathrm{m}}=\frac{F_{\text{cas}}}{m_{\mathrm{t}}·\text{Min}({\omega }_1^2,{\omega }_2^2)}$$ (37)

Take the necessary parameters as listed in Table 1, we calculated that ${\omega }_1=2\pi \times 4.2\text{Hz}$, ${\omega }_2=2\pi \times 10.3\text{Hz}$, and $F_{\text{cas}}=1.5\times {10}^{-15}\text{Hz}$. Therefore, $\Delta d_{\mathrm{m}}\sim F_{\text{cas}}/\left(m_{\mathrm{t}}{\omega }_1^2\right)=9.8\times {10}^{-10}m$, and the levitated pendulum will tilt from the vertical direction at an angle of $\Delta \theta \sim d_{\mathrm{m}}/l=0.1\mu \text{rad}$.

Appendix E. Vibration force noise

The vibration in this scheme contains two types, one is source mass-driven vibration force, and the other is seismic vibration noise. They all act on Sphere 1, and impact Sphere 2 through the levitated pendulum’s connecting rod. If a force applying on Sphere 1 is of form $f_1$, it is equivalent to an effective force of $f_2=f_1·{\chi }_{12}/{\chi }_{22}$ applying on Sphere 2.

Because we are going to detect the lateral force, we only focus on the vibration force noise that horizontally acting on Sphere 1. The source mass-driven vibration force is equivalent to an effective force acting on Sphere 2 reads

$$F_{\mathrm{d}2}={\eta }_{\mathrm{c}}·(\frac{m_{\mathrm{s}}{\omega }_s^2{A}_{\mathrm{s}}}{M}·m_1)·\frac{{\chi }_{12}}{{\chi }_{22}}$$ (38)

where $m_{\mathrm{s}}$ is the mass of source mass, ${\omega }_s$ is the source mass-driven frequency, $A_{\mathrm{s}}$ is the vertical vibration amplitude of the source mass, $M$ is the mass that is firmly connected to the source mass, $m_1$ is the mass of Sphere 1, ${\eta }_{\mathrm{c}}$ is the cross-talk factor between horizontal direction and the vertical direction.

The seismic vibration force noise is a white noise. The PSD of its effective force acting on Sphere 2 is

$$S_{\text{ff}2}^{\text{sei}}=m_1^2·S_{\text{aa}1}^{\text{sei}}·{\left|\frac{{\chi }_{12}}{{\chi }_{22}}\right|}^2$$ (39)

$S_{\text{aa}1}^{\text{sei}}$ is the acceleration PSD of the seismic vibration, which is directly acting on Sphere 1. Accordingly, the seismic vibration force noise is equivalent to an effective force on Sphere 2 as

$$F_{\text{sei}2}=\sqrt{\frac{S_{\text{ff}2}^{\text{sei}}}{t_{\text{mea}}}}$$ (40)

Taking Sphere 2 as the test mass, we used the parameters listed in Table 1. Assuming $M=350\text{kg}$, $\sqrt{S_{\text{aa}1}^{\text{sei}}}=0.01\text{ng}/\sqrt{\text{Hz}}$, and ${\eta }_{\mathrm{c}}={10}^{-3}$, we calculated that the effective source mass-driven vibration force on the test mass is $F_{\mathrm{d}}=3.4\times {10}^{-24}\mathrm{N}$. The power spectral density (PSD) of the effective seismic vibration force noise acting on the test mass is $\sqrt{S_{\text{ff}}^{\text{sei}}}=2.2\times {10}^{-19}\mathrm{N}/\sqrt{\text{Hz}}$. Therefore, the effective seismic vibration force noise acting on the test mass is $F_{\text{sei}}=2.2\times {10}^{-21}\mathrm{N}$, as derived from Eqs. (38)–(40).