Ultrasensitive Inertial and Force Sensors with Diamagnetically Levitated Magnets

TL;DR

This paper proposes a theoretically designed ultrasensitive force and inertial sensor using a diamagnetically levitated magnet above a superconductor, achieving extremely high sensitivity for various scientific and technological applications.

Contribution

It introduces a novel stable levitation method for magnets without external fields and demonstrates its potential as an ultra-sensitive sensor with unprecedented sensitivity levels.

Findings

Force sensitivity around 10^{-23} N/√Hz for 100 nm magnets

Acceleration sensitivity around 10^{-14} g/√Hz for 10 mm magnets

Potential applications in gravimetry, space industry, and fundamental physics

Abstract

We theoretically show that a magnet can be stably levitated on top of a punctured superconductor sheet in the Meissner state without applying any external field. The trapping potential created by such induced-only superconducting currents is characterized for magnetic spheres ranging from tens of nanometers to tens of millimeters. Such a diamagnetically levitated magnet is predicted to be extremely well isolated from the environment. We therefore propose to use it as an ultrasensitive force and inertial sensor. A magnetomechanical read-out of its displacement can be performed by using superconducting quantum interference devices. An analysis using current technology shows that force and acceleration sensitivities on the order of $10^{-23}\text{N}/\sqrt{\text{Hz}}$ (for a 100 nm magnet) and $10^{-14}g/\sqrt{\text{Hz}}$ (for a 10 mm magnet) might be within reach in a cryogenic environment. Such unprecedented sensitivities can be used for a variety of purposes, from designing ultra-sensitive inertial sensors for technological applications (i.e. gravimetry, avionics, and space industry), to scientific investigations on measuring Casimir forces of magnetic origin and gravitational physics.

Tables

Table 1. Table 1: Summary of the parameters for the readout system.

R ($\mu$m)	$d_h$ ($\mu$m)	$s$ ($\mu$m)	$\eta$ (m-1)
0.1	1	0.85	$7.3\times {10}^5$
1	4	3.5	$4.6\times {10}^7$
10	20	17	$1.8\times {10}^9$
${10}^2$	110	95	$6.0\times {10}^{10}$
${10}^3$	1100	950	$6.0\times {10}^{11}$
${10}^4$	11000	9500	$6.0\times {10}^{12}$

Equations

$$\begin{array}{rl}V(\mathbf{r})\approx & \frac{M}{2}\left({\omega }_x^2{x}^2+{\omega }_y^2{y}^2+{\omega }_z^2{z}^2\right)+\frac{I}{2}{\omega }_{\beta }^2{\beta }^2+\kappa \beta y.\end{array}$$

$$S_{F_i}(\omega )>S_{F_i}^{\star }(\omega )\equiv S_{F_i}^S(\omega )+S_{F_i}^N(\omega ).$$

$$S_{F_i}^S(\omega )=\frac{S_{\Phi }(\omega )}{{\left|{\chi }_{ii}(\omega )\right|}^2{\left({\Phi }_0{\eta }_i\right)}^2},$$

$$\begin{array}{rl}L=& \frac{M}{2}\left({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2\right)\\ & +\frac{I}{2}\left[{\dot{\tilde{\alpha }}}^2+(\dot{\tilde{\gamma }}-{\omega }_{\mu }{)}^2+2(\dot{\tilde{\gamma }}-{\omega }_{\mu })\dot{\tilde{\alpha }}\cos \tilde{\beta }+{\dot{\tilde{\beta }}}^2\right]\\ & -V(\mathbf{r},\beta ),\end{array}$$

$$\begin{array}{rl}\ddot{x}+{\omega }_x^{2}x=& 0,\\ \ddot{y}+{\omega }_y^{2}y+\kappa \beta /M=& 0,\\ \ddot{z}+{\omega }_z^{2}z=& 0,\\ \ddot{\alpha }-{\omega }_{\mu }\dot{\beta }=& 0,\\ \ddot{\beta }+{\omega }_{\beta }^2\beta +{\omega }_{\mu }\dot{\alpha }+\kappa y/I=& 0.\end{array}$$

$$\begin{array}{rl}M\left({\omega }_x^2-{\omega }^2-i\omega {\gamma }_x\right)x=& f_x,\\ M\left({\omega }_y^2-{\omega }^2-i\omega {\gamma }_y\right)y+\kappa \beta =& f_y,\\ M\left({\omega }_z^2-{\omega }^2-i\omega {\gamma }_z\right)z=& f_z,\\ I\left(-{\omega }^2-i\omega {\gamma }_{\alpha }\right)\alpha +i\omega {\omega }_{\mu }I\beta =& {\tau }_{\alpha },\\ \kappa y-i\omega {\omega }_{\mu }I\alpha +I\left({\omega }_{\beta }^2-{\omega }^2-i\omega {\gamma }_{\beta }\right)\beta =& {\tau }_{\beta }.\end{array}$$

$$\mathbf{X}(\omega )=\mathbf{\chi }(\omega )\mathbf{F}(\omega ),$$

$$S_A(\omega )\equiv \frac{1}{2\pi }{\int }_{-\infty }^{\infty }⟨A(t)A(t+\tau )⟩e^{\text{i}\omega \tau }\text{d}\tau ,$$

$$⟨A(t)A(t+\tau )⟩=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_{-T/2}^{T/2}\text{d}tA(t)A(t+\tau ).$$

$$\Phi ({\mathbf{r}}_0+\mathbf{r})\approx \Phi ({\mathbf{r}}_0)+{\Phi }_0{\eta }_{x}x+{\Phi }_0{\eta }_{y}y+{\Phi }_0{\eta }_{z}z,$$

$$S_{{\Phi }_i}^{\mathrm{sig}}=({\Phi }_0{\eta }_i{)}^2|{\chi }_{ii}{|}^2{S}_{F_i}^{\mathrm{sig}}.$$

$$S_{{\Phi }_i}^{\mathrm{noise}}=S_{\Phi }+({\Phi }_0{\eta }_i{)}^2|{\chi }_{ii}{|}^2\left(S_{F_i}^{\mathrm{g}}+S_{F_i}^{\mathrm{h}}\right).$$

$$S_{F_i}^{\mathrm{sig}}>\frac{S_{\Phi }}{|{\chi }_{ii}{|}^2({\Phi }_0{\eta }_i{)}^2}+(S_{F_i}^{\mathrm{g}}+S_{F_i}^{\mathrm{h}}).$$

$$S_{F_x}^{\mathrm{incl}}=|Mg{|}^2{S}_{\Delta \gamma }.$$

$$S_{F_y}^{\mathrm{incl}}={\left|\frac{{\chi }_{yy}(Mg+\kappa )+{\chi }_{y\beta }I{\omega }_{\beta }^2}{{\chi }_{yy}}\right|}^2{S}_{\Delta \beta }.$$

$$S_{F_x}^{\mathrm{vib}}$$

$$S_{F_y}^{\mathrm{vib}}$$

$$S_{F_z}^{\mathrm{vib}}$$

$$S_{F_i}^{\mathrm{B}}=|\mu {|}^2{S}_{{\partial }_i{B}_y}.$$

$$\mathbf{F}={\int }_S{\mathbf{K}}_{\mathrm{M}}({\mathbf{r}}^{\prime })\times \mathbf{B}({\mathbf{r}}^{\prime })\mathrm{d}{V}^{\prime },$$

$$B_i^{\mathrm{M}}(d,\omega )\approx \sqrt{\frac{\mathcal{C}(\omega )}{d}}$$

$$F_x(d,\omega )=R^2\frac{\mu }{V}{\int }_0^{2\pi }\mathrm{d}\varphi {\int }_0^{\pi }\mathrm{d}\theta B_y^{\mathrm{M}}{\sin }^2\theta \cos \varphi ,$$

$$\begin{array}{rl}\Delta {\Phi }_1(\mathbf{r})& ={\Phi }_0(+{\eta }_{x}x-{\eta }_{y}y+{\eta }_{z}z),\\ \Delta {\Phi }_2(\mathbf{r})& ={\Phi }_0(+{\eta }_{x}x-{\eta }_{y}y-{\eta }_{z}z),\\ \Delta {\Phi }_3(\mathbf{r})& ={\Phi }_0(-{\eta }_{x}x-{\eta }_{y}y+{\eta }_{z}z),\\ \Delta {\Phi }_4(\mathbf{r})& ={\Phi }_0(-{\eta }_{x}x-{\eta }_{y}y-{\eta }_{z}z),\end{array}$$

Full text

Most modern force and inertial sensors are based on the response of a mechanical oscillator to an external perturbation. Such sensors find applications in a wide range of domains: from measuring accelerations in smartphones and automobiles Bogue2013 in present-day technology, to being used on the cutting edge of research for magnetic resonance force microscopy Rugar2004 ; Degen2009 ; bachtold2013 , mass spectroscopy at the single-molecule level Naik2009 , and measuring gravitational and Casimir physics at short distances Geraci2008 ; Geraci2010 ; ArkaniHamed1998 ; BookCasimir ; Klimchitskaya2009 . Most force and inertial sensors are based on microfabricated clamped mechanical oscillators, whose sensitivity is ultimately limited by mechanical dissipation due to material and clamping losses imboden_dissipation_2014 . Levitation offers a clear route to avoiding these loss mechanisms. Indeed, the most precise commercial accelerometers are based on levitated systems: the superconducting gravimeter, which levitates a superconducting centimeter-sized sphere in the mixed superconducting state to achieve acceleration sensitivities of $3.1\times 10^{-10}g/\sqrt{\rm Hz}$ goodkind1999 , and the MicroStar accelerometer, which electrostatically levitates a centimeter-sized cube in space leading to $10^{-11}g/\sqrt{\text{Hz}}$ christophe2015 . In research, different levitated systems are being explored to push into unexplored levels of sensitivity. This includes the demonstration of a record force sensitivity of $4\times 10^{-22}$N/$\sqrt{\rm Hz}$ with an ion crystal Biercuk2010 , the use of optically levitated dielectric nanospheres ORI2010 ; Chang2010 ; Barker2010 ; Raizen2010 ; Gieseler2012 ; Kiesel2013 ; Millen2015 as novel force sensors with promising sensitivities Yin2013 ; Rodenburg2016 ; Ranjit2016 of $2\times 10^{-20}~{}\text{N}/\sqrt{\text{Hz}}$ Gieseler2013 , and matter-wave interferometry using clouds of atoms with a sensitivity of $\sim 10^{-9}g/\sqrt{\rm Hz}$ Hu2013 ; Abend2016 .

In this Letter, we aim at exploiting the exquisite isolation from the environment provided by magnetic levitation in a cryogenic environment. In particular, we propose an all-magnetic passively-levitated sensor that can be scaled over a broad range of sizes and is predicted to reach unprecedented ultra-high force and inertial sensitivities of $10^{-23}\text{N}/\sqrt{\text{Hz}}$ and $10^{-14}g/\sqrt{\text{Hz}}$, respectively. We show that a spherical particle with a permanent magnetic moment can be stably trapped on top of a punctured superconducting (SC) plane in the Meissner state, without the application of external magnetic fields. The hole in the SC surface introduces an effective pinning center that, together with the gravitational force, confines the magnet in three dimensions. Since diamagnetic levitation due to superconductivity does not have any associated length scale, as opposed to the light’s wavelength in optical levitation Pflanzer2012 ; Jain2016 , it can be applied to magnets of any size as long as fields in the SC do not prevent superconductivity. The SC surface in the Meissner state (i.e. without superconducting vortices) provides a general lossless levitation mechanism. Furthermore, low frequency magnetic field fluctuations arising from the surface are predicted to be minimized in the Meissner state Skagerstam06 ; Hohenester07 . The position of the magnet can be precisely measured by placing an array of superconducting interference devices (SQUIDs) in the vicinity of the trap center. The displacement of the magnet couples inductively to the SQUIDs. We shall argue below that these features lead to an alternative approach for ultra-sensitive force and inertial sensing.

Let us consider an infinite SC thin film with a circular hole of radius $a$ (whose center defines the origin of coordinates) and thickness $b\ll a$. A spherical magnet with radius $R$ and magnetic moment $\boldsymbol{\mu}\equiv\mu(\cos\alpha\cos\beta\,\mathbf{e}_{x}+\sin\alpha\cos\beta\,\mathbf{e}_{y}+\sin\beta\,\mathbf{e}_{z})$ is situated on top, see Fig. 1(a). The SC is described by the London model, which is valid under the approximation that the coherence length of the SC, $\xi$, is much smaller than its London penetration depth, $\lambda$ ($\lambda\gg\xi$). We assume a thin film, $b\ll\lambda$, and define the two-dimensional Pearl screening length, $\Lambda\equiv 2\lambda^{2}/b$ clem ; pearl . The SC is assumed to be in the complete shielding state, namely $\Lambda/a\ll 1$ Brojeny2003 ; Mawatari2012 . Importantly, we consider that the SC has been cooled in the absence of any external field, namely that no flux is trapped in the hole. In this case, the SC sheet-current density, ${\bf K}^{\text{SC}}$, can be calculated from the London equation as ${\bf K}^{\text{SC}}=-2\mathbb{A}/(\mu_{0}\Lambda)$, where $\mathbf{A}$ is the total magnetic vector potential in the London gauge. Zero-field cooling imposes that the fluxoid tinkham ; cardwell $\Phi^{\prime}=\Phi+\mu_{0}\Lambda\oint_{\rm C}\mathbf{K}^{\rm SC}\text{d}l/2$ is zero for any closed path in the SC, including those enclosing the hole ($\Phi$ is the external magnetic flux crossing the surface defined by the closed path C). ${\bf K}^{\text{SC}}$ is obtained by making a quasi-static approximation assuming that the SC responds on a timescale much faster than the motion of the magnet. This allows us to numerically solve the 3D magnetostatic problem using a finite-element method with the COMSOL Multiphysics software.

The magnetic potential felt by the magnet is approximated by $V_{\rm m}=-\boldsymbol{\mu}\cdot\mathbf{B}_{\rm ind}(\mathbb{r})/2$, where $\mathbf{B}_{\rm ind}$ is the field generated by $\mathbf{K}^{\rm SC}$. This assumes $\mathbf{B}_{\text{ind}}$ to be sufficiently homogeneous within the volume of the sphere 111A thorough analysis of the validity of this approximation together with the investigation of the mass limits in diamagnetic levitation in the Meissner state will be addressed elsewhere.. We remark that the micromagnetic origin of magnetization depends on the size of the magnet. Magnets smaller than a characteristic size, namely the single-domain radius $R_{\text{sd}}$, consist of a single magnetic domain. Below the so-called blocking temperature, which is the case in a cryogenic environment, the domain is fixed to a given direction. Magnets bigger than $R_{\text{sd}}$ have numerous domains and while their micromagnetic description is cumbersome, can be macroscopically characterized through the hysteresis loop. In that case, the magnet is assumed to be in remanence. We consider magnets made of ${\rm Nd}_{2}{\rm Fe}_{14}$B, for which $R_{\text{sd}}\approx 110$nm coey .

The total potential in the presence of gravity reads $V=V_{m}+Mgz$, where $M$ is the mass of the magnet. The normalized magnetic potential $\tilde{V}_{\rm m}=V_{\rm m}/V_{0}$, with $V_{0}\equiv\mu_{0}\mu^{2}/(4\pi a^{3})$, is numerically calculated as a function of the normalized coordinates, $\tilde{\mathbf{r}}=\mathbf{r}/a$. When the magnetic moment of the magnet is parallel to the SC surface ($\alpha=\pi/2$ and $\beta=0$ such that $\boldsymbol{\mu}=\mu\,\mathbf{e}_{y}$), it gives rise to a stable trap on the $z$-axis at some $z=z_{0}$, see further details in Supplemental Material (SM) SM . The closest possible trapping point above the SC is at $\tilde{z}_{0}\approx 1.168$. Due to the direction of $\boldsymbol{\mu}$, the trap frequencies in the $x$ and $y$ directions are different and there is a non-negligible cross coupling between $\tilde{y}$ and $\beta$. The circular hole makes the potential independent on $\alpha$. Alternatively, one could use an ellipsoidal or a polygonal-shaped hole to introduce one or several values of $\alpha$ where energy is minimized. Furthermore, one could consider the use of non-spherical magnets, as recently proposed in the context of magnetometry JacksonKimball2016 . For the spherical case, the total potential around the trapping position $\mathbf{r}_{0}=z_{0}\mathbf{e}_{z}$ and orientation $\alpha=\pi/2$, $\beta=0$, is given by

[TABLE]

Here $\mathbf{r}$ is the position vector with origin at $\mathbf{r}_{0}$, $\omega_{i}^{2}\equiv\partial_{i}^{2}V(0)/M$ (with $i\in\{x,y,z\}$), $\omega_{\beta}^{2}\equiv\partial_{\beta}^{2}V(0)/I$, $\kappa\equiv\partial_{\beta}\partial_{y}V(0)$, and $I$ is the moment of inertia of the magnet. In Fig. 1(b) we show the trapping position and frequencies as a function of the radius of the magnet assuming constant mass density and magnetization. Whilst $z_{0}\sim R^{3/4}$, trapping frequencies show a slow dependence $\sim R^{-3/8}$. Trap depths, defined as the energy (in Kelvins) required to escape the centre of the trap, grow as $\sim R^{15/4}$ and are of $T\approx 14$K for $R=100$nm. The magnetic field at the SC surface is much smaller than the first critical field of Nb (taken as a reference) for all magnet sizes plotted in Fig. 1(b). The Euler-Lagrange equations describing the motion of the magnet Rusconi2017 in the potential given by Eq. (1) can be written in the frequency domain as $\mathbf{X}(\omega)=\boldsymbol{\chi}(\omega)\mathbf{F}(\omega)$, where $\mathbf{X}=(x,y,z,\alpha,\beta)^{\rm T}$ is the vector of coordinates, $\mathbf{F}=(f_{x},f_{y},f_{z},\tau_{\alpha},\tau_{\beta})^{\rm T}$ is the vector containing external forces ($f$) and torques ($\tau$), and $\boldsymbol{\chi}$ is the susceptibility matrix, see SM .

The position of the magnet can be read out by measuring the magnetic field it creates through a nearby SQUID. The flux in the SQUID can be related to the position of the magnet via magnetomechanical coupling factors defined as $\eta_{i}\equiv\Phi_{0}^{-1}\partial_{i}\Phi(\mathbb{r}_{0})$ (with $i\in\{x,y,z\}$), where $\Phi_{0}$ is the quantum of flux, and $\Phi(\mathbb{r})$ is the flux crossing the SQUID created by the magnet at position $\mathbb{r}$. $\eta_{i}$ depend on the distance, size and arrangement of the SQUID. In order to measure the three coordinates of the center of mass independently, a suitable arrangement of SQUID loops is used. We consider 4 loops arranged in the same plane, e.g. a plane parallel to XY above the magnet or a plane parallel to XZ, see Fig. 1(a). The position of the magnet can be fully determined through an appropriate linear combination of the flux signals in each loop SM .

From a practical point of view, one needs to devise a way to load the magnet and a method to reduce the measurement time of the high-Q oscillator, which is given as a multiple of its ring-down time. A possible loading mechanism can rely on guiding the magnet through a conductive cylinder, whose opening is close to the trapping position. Eddy currents induced in the cylinder would slow down the motion of the magnet, which is trapped magnetically upon leaving the cylindrical guide. Reduction of measurement time can be conveniently achieved by feedback cooling, which simultaneously decreases the mechanical quality factor and the temperature of the oscillator and hence, maintains a constant overall sensitivity Geraci2010 . In particular, parametric feedback cooling Gieseler2012 could be implemented by applying an external field, such as the one created by an infinite wire with current $I_{\text{w}}$, parallel to the $x$-axis, passing through the $z$-axis at $z_{\text{w}}>z_{0}$, see Fig. 1(a). This field modifies the vertical trapping position $z_{0}$, thereby modulating the trapping frequencies, see SM and 222A thorough analysis on how to perform feedback cooling in an optimal way such that the added noise does not compromise the overall sensitivity will be addressed elsewhere..

The power spectral density (PSD) of a force $F_{i}$ ($i=\{x,y,z\}$) acting on the magnet, defined as $S_{F_{i}}(\omega)=(2\pi)^{-1}\int_{-\infty}^{\infty}\langle F_{i}(t)F_{i}(t+\tau)\rangle e^{\text{i}\omega\tau}\text{d}\tau$, is lower bounded by

[TABLE]

$S_{F_{i}}^{\star}(\omega)$ is the PSD of the minimal force that can be measured (i.e. signal-to-noise ratio of 1), which is limited by contributions due to read-out noise ($S^{S}_{F_{i}}$), and to noise forces acting on the magnet ($S^{N}_{F_{i}}$). The read-out noise is given by

[TABLE]

where $S_{\Phi}(\omega)$ is the PSD describing the flux noise in the SQUID and $\chi_{ii}$ is the diagonal element of the susceptibility matrix $\boldsymbol{\chi}$. The contribution $S^{N}_{F_{i}}$ contains stochastic forces due to gas collisions and magnetic losses. The PSDs of accelerations acting on the magnet can be simply obtained as $S_{a_{i}}=S_{F_{i}}/M^{2}$.

We consider the following intrinsic noise sources. Regarding the SQUID noise, we assume a low-$T_{c}$ dc-SQUID mainly affected by white noise SQUID_Bible with a conservative noise floor of $\sqrt{S_{\Phi}}=10^{-6}\Phi_{0}{\rm Hz}^{-1/2}$ for $\mu$SQUIDs schurig_making_2014 . The noise of an optimized SQUID scales with the self-inductance of the loop, $L$, as $\sqrt{S_{\Phi}}\propto L$ SQUID_Bible . Hence, the noise increases as $\sqrt{S_{\Phi}}\propto s\log(s)$ for bigger SQUIDs, where $s$ is the side length of the SQUID loop. The magnet experiences random gas collision events with a rate proportional to the pressure of the gas, $P$. This gives rise to an effective damping in all coordinates gamma_gas approximately given by $\gamma_{g}\approx 15.8R^{2}P/(M\bar{v}_{g})$, where $\bar{v}_{g}$ is the thermal velocity of the gas molecules. The associated stochastic force PSD, whose expression can be obtained from the fluctuation-dissipation theorem kuboFDT , is given by $S_{F}^{\rm g}(\omega)=M\gamma_{g}k_{\rm B}T/\pi$. The magnet fluctuates around its trapping position due to its thermal motion. In the reference frame of the magnet, a time-dependent magnetic field is hence applied. This will cause small fluctuations of the magnetization of the magnet, thereby inducing magnetic losses leading to mechanical damping and a corresponding fluctuating force. In general, one can identify hysteresis losses due to the irreversible relation between the magnetization and the external field as well as eddy-current losses due to induced currents in the magnet. Hysteresis losses can be estimated as follows. The thermally excited amplitude in each center-of-mass direction is given by $A_{i}\approx\sqrt{k_{\rm B}T/(M\omega_{i}^{2})}$. The field created by the SC currents can be approximated by the one created by an image of $\boldsymbol{\mu}$ at $\mathbf{r}=-z_{0}\mathbf{e}_{z}$. The variation of external field at a given point inside the magnet, $\mathbf{r}^{\prime}$, is $\Delta_{i}\mathbf{H}(\mathbf{r}^{\prime})=\mathbf{H}(\mathbf{r}^{\prime}+\mathbf{r}_{0}+A_{i}\mathbf{e}_{i})-\mathbf{H}(\mathbf{r}^{\prime}+\mathbf{r}_{0})$. The variation of magnetization is $\Delta_{i}\mathbf{M}(\mathbf{r}^{\prime})\approx\chi_{\rm m}\Delta_{i}\mathbf{H}(\mathbf{r}^{\prime})$, where $\chi_{\rm m}$ is the magnetic susceptibility of the magnet in remanence. The magnetic energy lost per cycle can be estimated as $\Delta_{i}W_{\rm h}\approx\int_{V}\mu_{0}\Delta_{i}\mathbf{M}(\mathbf{r}^{\prime})\Delta_{i}\mathbf{H}(\mathbf{r}^{\prime})dV^{\prime}=\mu_{0}\chi_{\rm m}\int_{V}\Delta_{i}\mathbf{H}(\mathbf{r}^{\prime})^{2}dV^{\prime}$. This is an overestimation of the hysteresis loss per cycle, both because the three components of the magnetization are assumed to change due to the external field (by using a simple scalar $\chi_{\rm m}$) and because all the energy in the product $\mu_{0}\Delta\mathbf{M}\Delta\mathbf{H}$ is considered to be irreversibly dissipated 333Taking into account the tiny amplitude of the field oscillations and the coercitive field of the magnet, one can expect a rather linear magnetic behaviour with a narrow hysteresis loop that would contain a small fraction of the estimated energy.. The damping rate is then given by $\gamma_{h_{i}}\approx\omega_{i}\,\Delta_{i}W_{\rm h}/(2\pi k_{\rm B}T)$ and, assuming thermal equilibrium, the associated stochastic force is $S_{F_{i}}^{\rm h}(\omega)=M\gamma_{h_{i}}k_{\rm B}T/\pi$. Eddy-current losses can be estimated through a similar procedure. The energy loss per cycle, $\Delta W_{\rm e}$, is proportional to the electrical conductivity of the magnet and the frequency of the field. Considering the poor conductivity of typical magnets, and in particular of ${\rm Nd}_{2}{\rm Fe}_{14}$B, and the small frequencies involved ($<100$ Hz), one can readily show that $\Delta W_{\rm e}\ll\Delta W_{\rm h}$. In the limit of small magnets with a single magnetic domain, the only magnetic dissipative process is related to the alignment of the magnet to a non-parallel external magnetic field, which involves a minimum time scale related to the relaxation of the crystal lattice to the equilibrium orientation (described by the Landau-Lifshitz-Gilbert equation) landau35 ; gilbert2004 . This effect is predicted to be negligible at the low frequencies considered.

The previous analysis can now be applied to magnets with sizes spanning over very different scales, from nanometers to millimeters. Small masses provide high force sensitivities since the mechanical susceptibility scales as $\chi\propto 1/M$. Force noise due to gas collisions ($\propto R$) and magnetic losses are minimized for small masses. On the other limit, large masses provide high sensitivity on the acceleration of the magnet. Larger magnets create stronger magnetic fields leading to bigger couplings to the SQUIDs. However, losses related to magnetic hysteresis become relevant as the volume of magnetic material increases. Fig. 2 shows the noise contributions and the final sensitivity for different sizes of the magnet at the reference temperature of 1K for Nd2Fe14B coey , see SM . The largest force (acceleration) sensitivity at small (large) radii is limited by the SQUID noise (hysteresis losses). Recall that hysteresis losses are overestimated, so one could expect even better acceleration sensitivities. Sensitivities, evaluated at a fraction of the corresponding resonance frequency, reach $5\times 10^{-23}$N/$\sqrt{\rm Hz}$ at $f\sim 18$Hz for a magnet of $R=100$nm (with resonance frequency $f\sim 180$Hz and $Q\sim 10^{9}$) and $7\times 10^{-15}$g/$\sqrt{\rm Hz}$ at $f\sim 1$Hz for a magnet of $R=10$mm (resonance frequency $f\sim 2$Hz and $Q\sim 10^{5}$). Such a force sensitivity is more than an order of magnitude better than the current state-of-the art using trapped ions Biercuk2010 . The acceleration sensitivity is more than three orders of magnitude better than in commercial devices goodkind1999 ; christophe2015 .

Such unprecedented sensitivities could be used, among others, to measure inclinations, vibrations, and magnetic field fluctuations. For magnets of $R=10$mm, inclinations on the order of $7\times 10^{-15}$rad/$\sqrt{\rm Hz}$ and vibrations on the order of $2\times 10^{-16}$m/$\sqrt{\rm Hz}$ could be detected. Magnetic gradients of up to $5\times 10^{-16}$T/(m$\sqrt{\rm Hz}$) at $f\sim 1$Hz would also be detectable. The latter could be used to detect magnetic fields created by fluctuating currents in nearby solids, i.e.  to detect magnetic Casimir forces henkel1999 . For a magnet with $R=10\mu$m close to a silver surface, this force falls within the detectability threshold for separations of up to $\approx 15\mu$m from the surface, see SM . Electric Casimir forces could be detected by coating the magnet with a non-magnetic dielectric material and approach a dielectric surface to it. Further, with appropriate shielding from Casimir forces, the device could also be used to test corrections to the gravitational force at short distances Geraci2008 ; Geraci2010 ; ArkaniHamed1998 . A more ambitious goal would be to use the extreme acceleration sensitivity of our device to detect gravitational forces between small masses and accurately characterize Newtons’s constant $G$, see Schmole2016 and references therein. Note that the gravitational interaction between a magnet of $R\approx 5~{}\text{mm}$ and another sphere of the same mass separated by a gap of $3~{}\text{mm}$ could be in principle detected. Finally, using our device as an inertial sensor could have relevant applications in avionics and space industry. The detection of small variations of gravitation force could also be applied to geological exploration or mining, among others.

In conclusion, we have presented an alternative approach for force and inertial sensing based on diamagnetic levitation of magnets. Remarkably, the concept is rather general and can be applied to magnets with sizes ranging from nanometers to millimeters, spanning over 6 orders of magnitude. The underlying mechanism behind such an astonishing broad window is the diamagnetic levitation provided by the superconductor in the Meissner state. The use of a magnet with a strong magnetic moment gives rise to a simple passive trapping scheme, and provides direct ways to read and feedback cool its motion. Our analysis, including current technologies and realistic assumptions, indicates very promising sensitivities over a wide range of scales, which we hope will motivate its experimental implementation.

This work is supported by the European Research Council (ERC-2013-StG 335489 QSuperMag) and the Austrian Federal Ministry of Science, Research, and Economy (BMWFW). We acknowledge discussions with J. Hofer, G. Kirchmair, C. Navau, A. Sanchez, and C. Schneider.

SUPPLEMENTAL MATERIAL

I Numerical characterization of the trapping potential

The normalized magnetic potential $\tilde{V}_{\rm m}=V_{\rm m}/V_{0}$, with $V_{0}\equiv\mu_{0}\mu^{2}/(4\pi a^{3})$, is numerically calculated for a magnet with magnetic moment parallel to the SC surface ($\boldsymbol{\mu}=\mu\,\mathbf{e}_{y}$). This orientation of the magnet gives rise to a stable trap on the z-axis, as shown in the inset of Fig. 3(a), which is plotted in terms of dimensionless coordinates, $\tilde{\mathbf{r}}=\mathbf{r}/a$. On the $z$-axis, the potential shows a non-monotonic dependence with $\tilde{z}_{0}$ [Fig. 3(a)]. In the limit $\tilde{z}_{0}\gg 1$ it agrees, as expected, with the potential created by an infinite ideal superconductor (SC), whose analytical expression obtained from the image method is $\tilde{V}_{\rm m}^{\rm imag}=1/(16\tilde{z}_{0}^{3})$. It can be shown that all second derivatives with respect to crossed spatial coordinates on the $z$-axis are zero, demonstrating that there is no coupling between them. Since the magnetic moment is oriented along the $y$-axis, axial symmetry is broken and second derivatives with respect to $\tilde{x}$ and $\tilde{y}$ have different values (see Fig. 3b). The second derivative with respect to $\tilde{z}$ is zero at $\tilde{z}_{0}\approx 1.168$, determining the closest possible trapping point above the SC. Alternatively, one could also trap at $\tilde{z}_{0}\approx 0$; in that case gravity would shift the final trapping position slightly below the SC. Cross derivatives with respect to $\beta$ show an interesting property of the system; whilst symmetry ensures that $\tilde{x}\beta$ and $\tilde{z}\beta$ derivatives are zero on the $z$-axis, derivatives with respect to $\tilde{y}\beta$ are large. This is also related to the symmetry-breaking direction of the magnetic moment and leads to a coupling between these two coordinates.

II Derivation of the equations of motion

The Lagrangian of the magnet trapped in the potential we characterized reads Rusconi2017

[TABLE]

where $V$ is the trapping potential given in Eq. (1) of the main text, $\tilde{\alpha}$, $\tilde{\beta}$ and $\tilde{\gamma}$ are the Euler angles in the ZYZ convention, and $\omega_{\mu}=\hbar\mu/(Ig_{e}\mu_{\rm B})$, where $g_{e}$ is the gyromagnetic ratio of the electron. This Lagrangian assumes an ideal hard-magnet with infinite magnetic anisotropy energy such that the magnetic moment is perfectly clamped to the anisotropy axis of the crystal Rusconi2017 . In order to express it in terms of the angle $\beta$ defined in the main text, one can make the following change of variables $\tilde{\alpha}=\alpha$, $\tilde{\beta}=\pi/2-\beta$, and $\tilde{\gamma}=\gamma$. The Euler-Lagrange equations are obtained as $d(\partial L/\partial\dot{q})/dt-\partial L/\partial q=0$ . After linearising them around the trapping position for a non-spinning magnet they read

[TABLE]

All parameters are defined in the main text. One can now introduce fluctuating forces ($f$) and torques ($\tau$) acting on each coordinate, as well as the corresponding damping rates ($\gamma$) assuming the fluctuation-dissipation theorem in thermal equilibrium. Rewriting these equations in the frequency domain one obtains

[TABLE]

This system of linear equations can then be simply solved as

[TABLE]

where $\mathbf{X}\equiv(x,y,z,\alpha,\beta)^{\rm T}$, $\mathbf{F}\equiv(f_{x},f_{y},f_{z},\tau_{\alpha},\tau_{\beta})^{\rm T}$, and $\boldsymbol{\chi}$ is the mechanical susceptibility matrix given by the inverse of the matrix giving the system of linear equations in Eq. (6).

III PSD definition, sensitivity and signals analysis

III.1 PSD definition

The power spectral density (PSD) of a variable $A(t)$ is defined as

[TABLE]

where the autocorrelation function is

[TABLE]

III.2 Sensitivity

By considering that the stable trapping point of the magnet is $\mathbf{r}_{0}$, the flux in the SQUID when the magnet is at $\mathbf{r}_{0}+\mathbf{r}$ can be approximated to

[TABLE]

where $\mathbf{r}=(x,y,z)$ and $\eta_{i}\equiv\Phi_{0}^{-1}\partial_{i}\Phi(\mathbb{r}_{0})$ for $i=\{x,y,z\}$. Consider the PSD $S_{F_{i}}^{\rm sig}$ of the force acting on the magnet that we want to measure. Eq. (7) gives us the position of the magnet as a result of this force. The PSD of the position of the magnet as a result of this force signal is $S_{r_{i}}^{\rm sig}=|\chi_{ii}|^{2}S_{F_{i}}^{\rm sig}$ and, thus, the PSD of the flux signal in the SQUID is

[TABLE]

There are two types of noise that will limit the sensitivity of this signal; the noise affecting the SQUID (whose PSD is $S_{\Phi}$) and the noises coming from the stochastic forces acting on the magnet (gas and magnetic, with PSDs $S_{F_{i}}^{\rm g}$ and $S_{F_{i}}^{\rm h}$, respectively). The PSD of the position of the magnet due to these stochastic forces is $S_{r_{i}}^{N}=|\chi_{ii}|^{2}(S_{F_{i}}^{\rm g}+S_{F_{i}}^{\rm h})$. The PSD of the flux signal in the SQUID due to all these noises is

[TABLE]

The sensitivity condition is given by $S_{\Phi_{i}}^{\rm sig}>S_{\Phi_{i}}^{\rm noise}$ which can be rewritten as

[TABLE]

This expression corresponds to Eq. (2) in the main text, where the second term on the right hand side is defined as $S_{F_{i}}^{N}$.

III.3 Signal analysis

Apart from the three noise sources analysed in the last section, the system will be affected by other signals such as inclinations and vibrations of the SC sheet as well as magnetic fields produced by nearby objects. Depending on the operating mode of the system, these signals can be considered as part of the noise or, on the contrary, they can be the signals one is interested to measure. In this section we analyse the signals produced by inclinations, vibrations of the SC sheet, and magnetic fields.

Inclinations. Inclinations of the SC surface with an angle $\Delta\gamma$ around the $y$-axis result in a force $F_{x}=Mg\,\Delta\gamma$ which Eq. (7) converts into a position signal as $S_{x}^{\rm incl}=|\chi_{xx}Mg|^{2}S_{\Delta\gamma}$. This position signal can also be interpreted as a result of a force signal such that $S_{F_{x}}^{\rm incl}=S_{x}^{\rm incl}/|\chi_{xx}|^{2}$ so

[TABLE]

Inclinations with an angle $\Delta\beta$ around the $x$-axis result in a force $F_{y}=(Mg+\kappa)\Delta\beta$ and a torque $\tau_{\beta}=I\omega_{\beta}^{2}\Delta\beta$. Using Eq. (7), we find that they only couple to the $y$-coordinate of the magnet, so the PSD of the position signal is $S_{y}^{\rm incl}=|\chi_{yy}(Mg+\kappa)+\chi_{y\beta}I\omega_{\beta}^{2}|^{2}S_{\Delta\beta}$. The force signal corresponding to it is $S_{F_{y}}^{\rm incl}=S_{y}^{\rm incl}/|\chi_{yy}|^{2}$, namely

[TABLE]

Vibrations. In general, vibrations of the SC surface result in forces and torques acting on the magnet, which are converted into position signals through Eq. (7). The PSDs of position signals read $S_{x}^{\rm vib}=|\chi_{xx}M\omega_{x}^{2}+1|^{2}S_{\Delta x}$, $S_{y}^{\rm vib}=|\chi_{yy}M\omega_{y}^{2}+\chi_{y\beta}\kappa+1|^{2}S_{\Delta y}$, and $S_{z}^{\rm vib}=|\chi_{zz}M\omega_{z}^{2}+1|^{2}S_{\Delta z}$. The last term in these expressions accounts for the change of distance between the readout system and the magnet as a result of the vibration. The corresponding force signals are

[TABLE]

Magnetic fields. Gradients of external magnetic fields result in forces acting on the magnet. Considering the magnet as a point particle, with a magnetic moment that points to the $y$-direction, these forces read $F_{i}=\mu\,\partial_{i}B_{y}$. The PSD of the position signal resulting from these forces is $S_{r_{i}}^{\rm B}=|\chi_{ii}\,\mu|^{2}S_{\partial_{i}B_{y}}$ and the PSD of the corresponding force is thus

[TABLE]

When the source of magnetic field is near to the magnet, the point-particle approximation may not be valid. In this case the total force acting on the magnet can be calculated as an integral over its surface $S$

[TABLE]

where $\mathbf{K}_{\rm M}\equiv\nabla\times\mathbf{M}$ is the magnetization sheet current density. For the case of magnetic fields arisen from fluctuating currents in a neutral surface (magnetic Casimir forces) henkel1999 , these magnetic field fluctuations are

[TABLE]

where $d$ is the distance to the surface, $i=\{x,y,z\}$, and $\mathcal{C}(\omega)=\mu_{0}^{2}\omega^{2}\hbar\varepsilon_{0}{\rm Im}[\varepsilon(\omega)]/(16\pi)$, where $\epsilon(\omega)$ is the dielectric constant in the spectral representation of the magnetic source, and $\epsilon_{0}$ the vacuum permittivity. Consider a surface parallel to the plane ZY at $x=d$. The magnet will experience a force in $x$-direction given by the surface integral of Eq. (20), namely

[TABLE]

with $B_{y}^{\rm M}=\sqrt{\mathcal{C}(\omega)/(d-R\sin\theta\cos\phi)}$. The force is evaluated using the low-frequency limit for the dielectric function, ${\rm Im}[\varepsilon(\omega)]=1/(\varepsilon_{0}\omega\rho)$, being $\rho$ the electric resistance of the surface henkel1999 . For a magnet of $R=10\mu$m and a surface made of silver with $\rho=1.6\times 10^{-8}\Omega$m the force as a function of the distance $d$ is shown in Fig. 4 [at a frequency $\omega/(2\pi)\approx 25$Hz]. Considering that the force sensitivity for this size of magnet at this same frequency is $\sqrt{S_{F_{x}}}\approx 3.5\times 10^{-21}$N/$\sqrt{\rm Hz}$, forces fall within the detectability threshold for distances up to $d\approx 25\mu$m, corresponding to separations of around $15\mu$m from the magnet.

IV Readout and feedback cooling

For the readout system we consider 4 identical adjacent square loops in a XZ plane. We label them through the position of their centers $\mathbf{r}^{c}$: (1) $z^{c}_{1}>z_{0}$ and $x^{c}_{1}>0$, (2) $z^{c}_{2}<z_{0}$ and $x^{c}_{2}>0$, (3) $z^{c}_{3}>z_{0}$ and $x^{c}_{3}<0$, (4) $z^{c}_{4}<z_{0}$ and $x^{c}_{4}<0$. Taking into account Eq. (10) we can now write an equation for each loop

[TABLE]

where $\Delta\Phi(\mathbf{r})=\Phi(\mathbf{r}_{0}+\mathbf{r})-\Phi(\mathbf{r}_{0})$ corresponds to the variation of magnetic flux measured by the $i$th loop. Notice that the absolute value of the coupling factors is the same for all the loops due to their symmetric arrangement. Only their signs change. The position of the magnet can be then determined by solving this system of equations. Also notice that the signal of the four loops is added up to determine the position of the magnet. For this reason, the coupling factors provided in the next section already contain the contribution of the four loops.

We now show how the trapping position of the magnet and the trapping frequencies can be modified by feeding current to a wire parallel to the $x$-axis at a given $z_{\rm w}$. We consider a particular example with $R=0.1\mu$m, for which the magnet is trapped at $z_{0}\approx 5\mu$m. The wire is set at $z_{\rm w}=8\mu$m and a given intensity $I_{\rm w}$ circulates in the direction defined by $-\mathbf{e}_{x}$. As shown in Fig. 5, both the trapping position and the frequencies are modified by changing the intensity in the wire. This modulation could be used to perform parametric feedback cooling of the levitated magnet. A thorough analysis on how to perform it in an optimal way such that the added noise does not compromise the overall sensitivity will be addressed elsewhere.

V Study case

Noise results presented in Fig. 2 of the main text have been calculated assuming the material parameters of Nd2Fe14B coey . We used $\mu=\rho_{\mu}V$ and $M=\rho_{M}V$, with $V$ being the volume of the magnet and $\rho_{\mu}=1.07\times 10^{6}$A/m and $\rho_{M}=7300$Kg/m3. We also considered a magnetic susceptibility $\chi_{\rm m}=0.05$ and an electrical conductivity $\sigma=6.67\times 10^{5}$A/(V$\cdot$m). For the environment, we considered a pressure of $P=10^{-10}$mbar of a gas with molar mass of $28.97$u at a temperature of $T=1$K. For the SC sheet we assume it to be made of Niobium with a critical temperature $T_{c}=9.26$ K. Below the first critical field $H_{c1}$, Nb behaves as a superconductor in the Meissner state provided it is cooled in zero-field. The normalized trapping position of the magnet is always set to $z_{0}/a=1.8$ (as in Fig. 1 of the main text). For the readout system of SQUIDS, we adapted the distance to the magnet and their size as a function of the radius of the magnet. For simplicity, we considered four identical adjacent square loops of side length $s$. They were placed on the same plane, parallel to the plane XZ at a distance $d_{h}$ and with centers at positions $(\pm s/2,-d_{h},z_{0}\pm s/2)$. The side length of the loops was chosen such that the three coupling factors have similar values $\eta_{x}\approx\eta_{y}\approx\eta_{z}$. Table 1 summarizes the parameters.

Finally, for a radius of the magnet between $0.1$ and $100\mu$m we evaluated the noises at a frequency of $\omega=0.1\omega_{i}$, being $\omega_{i}$ the corresponding resonance frequency. For radii bigger than $100\mu$m, noises were evaluated at $\omega=0.5\omega_{i}$. In Fig. 6 force noises ($x$-component) are plotted for three different radii of the magnet as a function of the frequency. Vertical dashed lines indicate the frequency at which noises have been evaluated to make Fig. 2 of the main text.