Ultralow mechanical damping with Meissner-levitated ferromagnetic microparticles

A. Vinante,1, 2, ∗ P. Falferi,2 G. Gasbarri,1 A. Setter,1 C. Timberlake,1 and H. Ulbricht1, †

1School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, UK2Istituto di Fotonica e Nanotecnologie - CNR and Fondazione Bruno Kessler, I-38123 Povo, Trento, Italy

(Dated: February 7, 2020)

Levitated nanoparticles and microparticles are considered excellent candidates for the realizationof extremely isolated mechanical systems, with a huge potential impact in sensing applications andin quantum physics. Unfortunately, popular active techniques such as optical levitation are knownto be limited in their ultimate performance by internal heating and photon scattering. In contrast,magnetic levitation based on static fields may avoid these drawbacks, due to the unique propertyof being completely passive. Here, we show experimentally that micromagnets levitated above typeI superconductors can indeed feature very low damping at low frequency and low temperature. Inour experiment, we detect 5 out of 6 rigid-body mechanical modes of a levitated ferromagneticmicrosphere, using a dc SQUID (Superconducting Quantum Interference Device) with a single pickup coil. The measured frequencies are in substantial agreement with a finite element simulation basedon ideal Meissner effect. For two specific modes we find further substantial agreement with analyticalpredictions based on the image method. We measure damping times τ exceeding 104 s and qualityfactors Q beyond 107, improving by 2 − 3 orders of magnitude over previous experiments based onthe same principle. The ratio T/τ ≈ 10−4 K/s, which sets the limit for most sensing applications, isalready at the state of the art for micro and nanosystems, and can be further improved by cooling tomillikelvin temperature. We investigate the possible residual loss mechanisms besides gas collisions,and argue that much longer damping time can be achieved with further effort and optimization.Our results open the way towards a new class of force sensors and platforms for testing quantummechanics at mesoscale, with potential application to magnetometry and gravimetry. In particular,we show that our levitated micromagnet can outperform existing magnetometers in terms of fieldresolution normalized over the sensing volume.

Micromechanical and nanomechanical resonators arewidely employed in fundamental and applied physics, dueto the fact that they can be interfaced to virtually anykind of system. Broadly speaking, we can identify twomain groups of applications. A first one is representedby quantum optomechanics [1, 2] and tests of quantumphysics in the large scale limit [3, 4]. A second one isrepresented by mechanical sensing of weak signals. Wellknown examples are force microscopy [5], force-detectedsingle spins [6], detection of small masses at molecularlevel [7], ultrasensitive accelerometry [8] and gravimetry[9, 10], magnetometry [11] and tests of wavefunction col-lapse models [12].

For sensing applications a proper figure of merit tocompare different systems is the ratio T/τ where T isthe temperature and τ is the mechanical damping time.In fact, the ultimate limit for sensing is given by thethermal force noise, with spectral density Sf = 4kBTmΓwhere Γ = ω/Q = 2/τ is the energy dissipation rate, ωis the angular frequency, Q is the quality factor and m isthe mass [13]. It is apparent that, for a given Q factor,lower noise can be achieved at lower frequencies.

To date, very sophisticated clamped mechanical res-onators have been developed in the context of optome-chanics, achieving Q factors up to 109 − 1010 [14, 15].However, these devices are optimized for high frequen-

∗Electronic address: a.vinante@soton.ac.uk†Electronic address: h.ulbricht@soton.ac.uk

cies, ranging from MHz to GHz, exploiting highlystressed membranes and beams. The factor T/τ can beas low as 10−2 K/s for cryogenic micromembranes [14].Similar values have been achieved by aluminum mem-branes cooled to millikelvin temperatures [16]. At kHzfrequencies, the most typical devices are micromachinedcantilevers for force microscopy [5, 17]. For devices cooledto millikelvin temperatures, T/τ can be as low as 10−4

K/s [18]. Finally, microelectromechanical systems opti-mized for seismometry and gravimetry at Hz frequenciesachieve Q ≈ 104 in vacuum [8, 10]. This means thatthey might achieve T/τ ≈ 10−4 K/s at cryogenic tem-peratures, but so far they are usually operated at roomtemperature.

Levitated microparticles and nanoparticles have re-cently emerged as a very promising alternative toclamped mechanical resonators. Besides flexibility andtunability of mechanical parameters, a particularly at-tractive feature of levitated systems is the absence ofclamping losses. As a consequence, very low levels of dis-sipation and decoherence are potentially achievable [19].

Levitation by optical gradient forces is certainly themost popular and developed technique [20], featuringa wide spectrum of studies, which include cavity op-tomechanics [21–23], feedback-cooling techniques [24–26], nonlinear dynamics [27], nonequilibrium thermody-namics [28] and ultrasensitive force sensing [29, 30]. Inspite of remarkable recent results and very high flexi-bility, optical levitation suffers from heating induced byphoton-absorption and scattering, which ultimately leadsto excess thermal noise and decoherence [31]. Indeed,

arX

iv:1

912.

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2v2

[qu

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6 F

eb 2

020

2

the typical floor of the T/τ factor for optically levitatednanoparticles is about 1 K/s.

This suggests that different levitation methods, suchas Paul traps [32–35], or magnetic traps [36–39] couldoutperform optical levitation in fields in which a lownoise level is required, such as force/acceleration sen-sors or mechanical platforms with ultralow quantum de-coherence to test quantum mechanics. In particular,magnetic levitation appears very promising because oftwo unique properties: trapping is completely passive,requiring only static fields, and detection can be per-formed with ultralow power dissipation, such as SQUIDsor microwave cavities. As a consequence it appears as anexcellent candidate to achieve ultralow damping at lowT . Specific implementations involve levitation of diamag-netic [36, 37] or superconducting particles [31] in externalstatic fields, or ferromagnetic particles above supercon-ductors [38, 39]. In addition to ultrasensitive force andinertial sensing [38, 40], these technologies have been pro-posed to test quantum mechanics in currently inacces-sible regimes [4], to enable novel quantum technologiessuch as quantum magnetomechanics [31] and acoustome-chanics [41], to couple to single spins [35] and for ultra-sensitive magnetometry [11]. In particular, it has beensuggested that a levitated micromagnet can overcome thestandard quantum limitations to the resolution per unitvolume of a magnetometer [11, 42].

Unfortunately, to date there is no conclusive exper-imental study demonstrating that magnetic levitationcan indeed achieve ultralow damping at low temperature:levitation of ferromagnetic micromagnets has achieved sofar only T/τ ≈ 10−2 K/s [38, 39]. Diamagnetic levita-tion has demonstrated very low damping but needs op-tical readout and has not yet been demonstrated at lowtemperature [36, 37]. However, very low T/τ ≈ 10−3 K/shave been achieved at room temperature [36].

In this paper we experimentally investigate magneticlevitation of micromagnets using passive superconduct-ing traps, employing SQUIDs for low power position de-tection instead of lasers. Specifically, we levitate, trapand detect individual ferromagnetic microparticles in atype I superconducting trap made of lead, and identifyrigid body modes in the frequency range 1 − 400 Hz.We perform a mechanical characterization at T = 4.2 Kat pressures down to 10−5 mbar, measuring quality fac-tors in excess of 107 and damping time in excess of 104

s. These results improve over similar previous experi-ments by more two orders of magnitude. The value ofT/τ ≈ 10−4 K/s is already at the state of the art formicrosensors, and can be further improved by cooling tomillikelvin temperature. This opens the way towards anew class of levitated microsystems with ultralow noise.We investigate the limiting dissipation mechanisms anddiscuss a number of possible applications, including forceand acceleration sensing, gravimetry, magnetometry aswell as fundamental and quantum physics experiments.As a case study, we demonstrate that the measurement ofa magnetic field through the torque applied to the micro-

magnet features a resolution, normalized over the sensingvolume, better than existing magnetometers.

I. EXPERIMENTAL SETUP AND MODEIDENTIFICATION

The trap consists of a cylindrical well inside a 99.95%-purity Pb block (Figs. 1a and 1b) with 4 mm diameterand 4 mm depth. The Meissner surface currents inducedby the magnetic particle, combined with gravity, providethe vertical confinement, while the lateral surface pro-vides the horizontal one. No special loading procedure isimplemented. The magnetic particle is individually ma-nipulated at room temperature and placed at the bottomof the trap before cooling down. Upon cooldown, Meiss-ner repulsion alone is sufficient to overcome electrical in-teractions and levitate the particle. We have checked thisbehaviour for particle radii down to ∼ 30 µm.

Figure 1: (a) Simplified scheme of the trapping and detectiontechnique. A hard ferromagnetic microparticle is levitated in-side a cylindrical well in a type-I superconductor, in our caselead. Meissner repulsion combined with gravity keeps themagnetic particle trapped in all directions. Both the transla-tional and the rotational motion of the particle are detectedby a superconducting pick-up connected to a dc SQUID. (b)Conventions adopted on the translational and rotational de-grees of freedom. Under weak breaking of cylindrical symme-try, all modes are trapped and detected, except γ. (c) Opticalmicroscope image of a microsphere with radius R = (27 ± 1)µm placed on the bottom of the trap. The polycrystallinelead grain boundaries are visible.

We have experimentally demonstrated stable trappingfor a variety of spherical and cylindrical magnetic parti-cles with characteristic size in the range 30 − 1000 µm.Here, we will focus on the smallest particle we have lev-itated, a microsphere made of a neodymium-based alloy[43] with radius R = (27± 1) µm (Fig. 1c), determinedby optical microscope inspection. The microsphere hasbeen fully magnetized in a 10 T NMR magnet prior tothe experiment, with an expected saturated magnetiza-tion µ0M ≈ 0.7 T.

We detect the motion of the particle using a commer-cial dc SQUID from Quantum Design connected through

3

a single pick-up coil placed above the levitated particle(Fig. 1a). The motion along a given degree of freedominduces a change of the magnetic flux Φ in the pick-up coil produced by the permanent magnetic dipole ~µof the particle. As the pick-up coil connected to theSQUID input coil constitutes a superconducting loopof inductance L, this results in a current I and a fluxΦS = MI = (M/L) Φ into the SQUID, with M the mu-tual inductance between SQUID and input coil. Thepick-up coil consists of 6 loops of NbTi wire, woundaround a cylindrical PVC holder with radius 1.5 mm,coaxial with the trap. The SQUID is enclosed in a sep-arate superconducting shield. The NbTi wires connect-ing the SQUID input coil and the pick-up coil are pair-twisted and shielded by a lead tube. We can drive themicroparticle by means of a second NbTi coil wound onthe same pick-up coil holder, connected to an externalresistive line.

Trap and SQUID are enclosed in a indium-sealed cop-per vacuum chamber which can be dipped in a standardhelium transport dewar. A rough mechanical isolation isobtained by hanging the dewar from the laboratory ceil-ing through a silicone tube. This reduces seismic noisein the bandwidth 1-20 Hz by more than 1 order of mag-nitude. A cryoperm shield reduces the earth field bya factor of about 10 at the lead trap location. Afterevacuating the chamber at room temperature, 1 mbarof helium exchange gas is added for thermalization andpumped out after cooldown. We monitor the pressure inthe vacuum chamber with a Pirani-Penning gauge placedat room temperature. Pressure data is corrected for thegas composition by assuming that the gas is pure helium.

Figure 2: Spectrum of the flux measured by the SQUID attwo representative nominal pressures, P = 1.0 × 10−4 mbar(black continuous line) and P = 5.0× 10−2 mbar (red dottedline). The rigid body mechanical modes of the microsphereare marked by labels.

Fig. 2 shows two typical SQUID spectra at low and

high pressure. The spectra are averaged over a total timeof 4.2 × 104 s. Five modes related with the microsphererigid body motion have been identified, as described be-low. Other peaks and structures are due to seismic andacoustic noise. We can distinguish the levitated particlemodes from spurious peaks thanks to the high qualityfactor Q and its characteristic dependence on pressure,as discussed below.

To identify the observed peaks with specific transla-tional and rotational modes of the microparticle, we needa reliable modeling of the system. We parameterize themotion of the particle with the center of mass coordi-nates x, y, z, the librational angle β and the azimuthalangle α defined by the permanent magnetic moment ~µ,and the rotation angle γ around ~µ (Fig. 1b). The ori-gin is chosen at the center of the bottom surface of thetrap. For ideal Meissner effect and trap axis aligned withthe gravitational field (i.e. with no tilt), the equilibriumposition corresponds to x0, y0, β0 = 0 and a finite equi-librium height z0 depending on the equilibrium betweenMeissner repulsion and gravity. There will be no con-finement in the α and γ angle, so we would expect todetect only 4 modes. In a real implementation any finitetilt with respect to the Earth gravitational field shiftsthe horizontal equilibrium position off center, breakingthe rotational symmetry around α, and shifting all fre-quencies. A trapped torsional α mode will then showup, leading to 5 detectable modes. Because of symmetrybreaking all 5 modes are expected to show some finitecoupling to the pick-up coil.

In the following, we will mostly focus on two specificmodes, namely the z and the β ones. Remarkably, thefrequency of these modes can be analytically predictedwith reasonable accuracy, as a straightforward applica-tion of the image method [44, 45]. This is because thedynamics of these modes depends almost exclusively onthe interaction of the microsphere with the bottom sur-face, as long as the equilibrium height z0 is much smallerthan the trap radius. Therefore, the resonance frequencyis expected to depend very little on the trap tilt. Indeed,we do observe two modes with relatively stable frequencyat (56.5± 0.1) Hz and (377± 5) Hz, while the other 3modes show frequency variations up to 50% upon tiltingthe experimental setup by a few degrees.

According to the image method [44, 45], the potentialenergy of a permanent magnet with magnetic moment ~µand mass m, placed above an horizontal infinite super-conducting plane, is given by:

U =µ0µ

2

64πz3

(1 + sin2β

)+mgz (1)

Upon minimization one finds that the equilibrium posi-tion is achieved at β = β0 = 0, i.e. with the magnetpositioned horizontally, and z = z0, with:

z0 =

(3µ0µ

2

64πmg

) 14

(2)

4

The angular resonance frequencies of the z and β modesare then easily calculated as ωz =

√kz/m and ωβ =√

kβ/I, where I is the moment of inertia and kz,kβ arethe spring constants:

kz :=

[d2U

dz2

](z,β)=(z0,β0)

, (3)

kβ :=

[d2U

dβ2

](z,β)=(z0,β0)

(4)

Assuming our particle is a homogeneous sphere, themoment of inertia is I = 2

5mR2 and we finally obtain:

ωz =

√4g

z0(5)

ωβ =

√5z0g

3R2(6)

Interestingly, both frequencies can be written as thesquare root of the ratio of g with an effective length,as in a simple pendulum. Furthermore, by multiplyingEq. (5) with Eq. (6) we derive the following remarkablerelation:

R =

√20

3

g

ωzωβ(7)

which provides the microsphere radius R as a functionof the two frequencies, independent of mass density andmagnetization.

Plugging the experimental values of the two sta-ble frequencies into Eq. (7), we obtain a radius R =(30.1± 0.5) µm, which is compatible within 10% withthe radius R = (27± 1) µm measured by optical inspec-tion. Further use of Eqs. (5,6) allows identification ofthe z-mode at ωz/2π = (56.5± 0.1) Hz and the β-modeat ωβ/2π = (377± 5) Hz. The resonance frequencies arecorrectly reproduced by using the values ρ = 7430 kg/m3

for the mass density and µ0M = 0.71 T for the magne-tization, which are consistent with the values providedby the manufacturer. The derived equilibrium heightz0 = 311 µm is much smaller than the hole radius, whichvalidates the assumption of infinite plane.

A finite element analysis is required to account for thefull dynamics and to predict the frequency of all modes.Results from a full 5D finite element simulation of oursystem based on FEniCS software [46] are shown in Ap-pendix 1. The simulation provides the equilibrium po-sition and the resonant frequencies of all 5 modes, as afunction of a finite tilt θ applied along the x-axis. Themode frequencies are in reasonable agreement with theexperimental results by assuming a tilt of about 3 de-grees. In particular, the z and β mode are in substantialagreement with the image method estimation, and showindeed a weak dependence on the tilt, in contrast withthe other modes. The low frequencies peaks in the spec-trum (Fig. 2) at about 4 and 10 Hz are identified as they and x mode. The remaining peak is identified as the

α mode. For the latter, a somewhat larger discrepancybetween experiment and simulation is observed, with apredicted value of ∼ 100 Hz and measured value of about∼ 150 Hz.

The simulation also provides an estimation of the equi-librium values of the 5 independent variables. Assumingthe tilt along the x axis, the equilibrium configurationis (x, y, z, β, α) ≈ (x0, 0, z0, 0, π/2), with z0 in agreementwithin 10% with the image method estimation and x0

which is strongly dependent on the tilt.

II. CHARACTERIZATION OF MECHANICALMODES

We measure the amplitude decay time τ of a givenmode at frequency f = ω/2π by means of standard ring-down measurements. The mode is selectively excited atlarge amplitude by sending a current signal into the exci-tation coil, and the ringdown is acquired by a lock-in am-plifier with reference tuned on the resonance frequency.τ is inferred from an exponential fit of the amplitude ver-sus time. The quality factor is calculated as Q = πfτ .At the lowest measured pressures P ∼ 10−5 mbar, τ is ofthe order of several hours and the measurement becomesvery difficult, due to the interplay between environmen-tal vibrational noise and the onset of nonlinearities whichrestrict the available linear dynamic range. Nevertheless,we could perform reliable ringdown measurements of theβ mode at all pressures, and of the z mode at almost allpressures.

Figure 3: (a) Example of ringdown measurement of the βmode at the lowest nominal pressure P = 5.7 × 10−5 mbar.(b) Frequency shift of the β mode as a function of the uncal-ibrated square amplitude, measured during a ringdown.

As first example, we show in Fig. 3a a ringdownmeasurement of the β mode at nominal pressure P =5.7× 10−5 mbar, featuring a decay time τ = 1.13× 104 sand a quality factor Q = 1.34×107. The ringdown is ex-ponential and does not feature significant damping non-linearities. Instead, we observe a clear spring softening

5

effect, appearing as a negative frequency shift propor-tional to the square amplitude, as shown in Fig. 3b. Thisbehaviour is entirely expected, as the trapping potentialis intrinsically nonlinear. Based on the analytical po-tential derived from the image method, we can calculateexplicitly the expected frequency shift as a function ofthe angular amplitude (see Appendix 2 for details). Thisprovides an indicative calibration of the absolute valueof the angular motion. The absolute motion for the ring-down of the β mode in Fig. 3a corresponds to a rangeβ = 3 − 20 × 10−3 rad, while the typical amplitude ofthe undriven mode after the exponential ringdown cor-responds to about 10−4 rad. This is significantly largerthan the mean thermal motion, which can be estimatedas βth =

√kBT/kβ ≈ 6 × 10−6 rad, suggesting that the

β mode is dominated by ambient vibrational noise, forinstance seismic noise.

In Figs. 4a and 4b we show a similar ringdown and fre-quency shift measurement of the mode z. The ringdownis acquired at nominal pressure P = 1.0×10−4 mbar, andshows a decay time τ = 1.17× 104 s and a quality factorQ = 2.1 × 106. The frequency shift measurement showsa behaviour similar to the β mode, proportional to thesquare of the oscillation amplitude. Using the expectedfrequency shift from the image method, we find that thethe absolute motion in the ringdown measurement cor-responds to a range z = 10 − 20 µm, while the typicalamplitude of the undriven mode corresponds to about 0.2µm. Similarly to the β mode, the motion of the undrivenmode is much larger than the expected mean thermal mo-tion zth =

√kBT/kz ≈ 0.7 nm, showing that the z mode

is also dominated by ambient vibrations. A further me-chanical isolation by at least 3 orders of magnitude wouldbe needed to suppress excess noise well below the thermalnoise.

Figure 4: (a) Ringdown measurement of the z mode at thenominal pressure P = 1.0×10−4 mbar. (b) Frequency shift ofthe z mode as a function of the uncalibrated square amplitude,measured during a ringdown.

Let us discuss the behaviour of damping as a functionof the gas pressure. For high pressure, P > 10−4 mbar,

we find very similar τ for all 5 modes, regardless of fre-quency. This is in agreement with standard gas damp-ing models, which predict for translational and rotational(amplitude) damping in molecular regime the followingexpressions [47, 48]:

1

τt=

1

2

(1 +

8

π

)P

ρRvth(8)

1

τr=

5

π

P

ρRvth(9)

where P is the gas pressure, vth =√

8kBT/πmg themean velocity of the gas molecules, with mg gas molec-ular mass, and ρ and R density and radius of the mi-crosphere. The two expressions are indeed frequency-independent, depend linearly on pressure and differ onefrom each other by ∼ 10%.

We have measured τ for the z and β modes at severalpressures. The data is shown in Fig. 5. In order to com-pare the data with the gas damping model, we have toidentify the pressure P in Eqs. (8,9) with the pressurePc of the gas in the cold side of the vacuum chamberat Tc = 4.2 K, where the microparticle is levitated. Ingeneral, Pc differs from the nominal pressure Pw mea-sured by the pressure gauge at the warm side (room)temperature Tw, due to the so called thermomoleculareffect [49]. This phenomenon was studied in detail forthe case of helium gas between 4 K and 300 K by Weberand Schmidt [50, 51] and interpolating formulae are re-ported in Ref. [49]. In Fig. 5 the pressure on the x-axisis the cold pressure Pc obtained from the Weber-Schmidtmodel by specifying the measured warm pressure Pw andthe inner radius r = 0.97 cm of the stainless steel tubewhich connects the cold chamber to the warm gauge pres-sure. In the limit of low pressure, the thermomolecular

pressure ratio scales as Pc/Pw = (Tc/Tw)12 and becomes

independent of geometric factors.The data for both the β mode and the z modes in Fig. 5

are fitted by a second-order polynomial fit significantlybetter then by a linear fit. We attribute the deviationfrom the expected linear behaviour to an imperfect esti-mation of Pc. The Weber-Schmidt formulae are obtainedfor ideal cylindrical connecting tubes, but in our experi-ment this is a crude approximation, due to the presenceof the wiring inside the tube. However, as the low pres-sure limit of Weber-Schmidt formulae is independent ofform factors of the connecting tube, we expect Eqs. (8,9)to be valid in this limit. Indeed, the linear terms obtainedfrom the fits are respectively (5.6± 0.3) s−1/mbar for theβ mode and (6.2± 0.5) s−1/mbar for the z mode. Theseestimations are in good agreement with the theoreticalpredictions 5.35 s−1/mbar and 5.96 s−1/mbar obtainedfrom Eqs. (8,9), using the measured values of R and thenominal density of the material ρ = 7430 kg/m3 pro-vided by the manufacturer. The agreement provides anindirect support to the Weber-Schmidt model. A notablefeature of the gas damping model which is reproduced bythe experiment is the ratio of translational to rotational

6

damping of the order of 1.1.For the β mode the intercept of the fit 1/τβ0 =

(4.3± 0.2) × 10−5 s−1 corresponds to a quality factorQ = 2.7× 107 at vanishing gas pressure. For the z modethe intercept of the fit 1/τz0 = (−5± 6) × 10−6 s−1 isinstead compatible with zero. This indicates that theresidual damping time of the z mode is longer than 1day, while the Q factor is at least of the order of that ofthe β mode at some 107.

Figure 5: Measured amplitude damping time as a functionof the effective pressure at T = 4.2 K calculated using theWeber-Schmidt model. The data refer to the β mode (blacksquares) and z mode (red circles). Solid and dashed lines aresecond order polynomial fits to the β and z mode data.

Finally, we have performed single mode noise measure-ments, focusing on the α mode, which corresponds toa horizontal twisting motion as in a torsion pendulum(Fig. 1b). Compared to other modes, the α one ap-pears less sensitive to ambient vibrational noise. Fig. 6shows the same spectra of Fig. 2, zoomed in around theα mode. The high pressure data can be accurately fittedby a Lorentzian curve:

SΦ = A0 +A1f4

0

(f2 − f20 )

2+ (ff0/Q)

2(10)

which gives Q = 421 and A1 = 3.82× 10−14 Φ20/Hz. The

parameter A1 is proportional to the torque noise actingon the microsphere. In the thermal noise limit, the latteris given by the expression:

ST = 4kBTIω0

Q(11)

so we expect A1 to drop at low pressure due to increasingQ. Indeed, the data at low pressure feature a much lowervalue of A1. The curve superimposed on the low pressuredata in Fig. 6 is a fit restricted to the tails of the peak.In fact, we have removed 9 bins around resonance which

are affected by spectral leakage, due to the peak beingnarrower than the spectrum resolution. From the fit weget A1 = (1.6± 0.1)× 10−16 Φ2

0/Hz.The fact that A1 is much lower at low pressure than at

high pressure is clearly indicating that the latter dataare dominated by thermal noise. Based on this as-sumption, we estimate the torque noise using Eq. (11),√ST = 1.00× 10−20 Nm/

√Hz. The torque noise at low

pressure is then calculated through the scaling of A1 as√ST = (6.4± 0.2)× 10−22 Nm/

√Hz.

Figure 6: Power spectrum of the flux measured by the SQUIDaround the frequency of the α mode, at two representativenominal pressures, P = 1.0 × 10−4 mbar (narrow curve) andP = 5.0 × 10−2 mbar (broad curve). Lorentzian fits are alsoshown.

III. DISCUSSION

The comparison between β mode and z mode suggeststhat when the pressure is reduced towards 0 the dissipa-tion moves from a gas damping regime, where the damp-ing 1/τ is independent of frequency, to a regime in which1/τ grows with frequency. This may correspond for in-stance to a loss angle 1/Q either constant or increasingwith frequency. Under this regime the modes at low fre-quency should then achieve a Q of the order of 107 orhigher. For a mode at 3 − 4 Hz, as the y mode in thepresent experiment, this would translate to a dampingtime τ ≈ 106 s.

Besides the obvious damping related with the gas,which can be in principle reduced to negligible levels byoperating at lower pressure and temperature, we stillneed to discuss the origin of the residual dissipation,which is apparent at least in the β mode data. We con-sider magnetic and eddy current losses in the ferromag-netic sphere and losses in the superconductor as possibleoptions.

7

Estimations of the first two sources are reported inAppendix 3. For our particle, the order of magnitude ofeddy current losses is Q ≈ 1011, so we can reasonably rulethem out as the dominant effect. On the other hand mag-netic hysteresis losses in the ferromagnetic particle couldgive the right order of magnitude, assuming a magneticsusceptibility with imaginary part of order 10−3.

A hint on the limiting loss mechanism could be givenby the dependence of the Q factor on the microparticlesize. We have performed additional measurements on asecond similar particle with radius a = 39 µm finding aresidualQ = 6×106 for the β mode. Other measurementsperformed with a different type of magnetic particle, suchas cylinders with size larger of 200×400 µm have shownQfactors below 106. The general trend is therefore towardsan improvement of Q by lowering the particle size. Asshown in Appendix 3, magnetic losses in the particle arein principle consistent with this picture.

Finally let us consider possible effects arising from non-idealities in the superconducting lead trap. Under idealMeissner effect we do not expect significant losses fromthe superconductor, due to oscillation frequency beingmany orders of magnitude smaller than the supercon-ducting gap frequency [40]. However, it is notoriouslydifficult to observe ideal Meissner effect in real type-I su-perconducting samples [52]. Flux penetration and mixedstates can arise even in very pure lead polycrystallinesamples. Normal metal regions arising from partial fieldpenetration may in principle cause additional eddy cur-rent dissipation [39].

In order to investigate a possible role of flux penetra-tion in our experiment, we have performed additionalmeasurements by applying an external field to the leadtrap during the superconducting transition. The fieldis applied through a superconducting coil placed outsidethe lead trap. Flux penetration appears as a shift of the zand β modes with respect to the typical values obtainedin absence of applied field. We find that flux penetra-tion shows up at fields down to 50 µT, comparable withthe Earth magnetic field and much lower than the nom-inal critical field of lead Hc = 80 mT. The amount offlux penetration and frequency shift is not reproducibleupon thermal cycling of the superconductor. For largefrequency shifts, above 20%, we also observe a strongincrease of dissipation, up to 2-3 orders of magnitude.This finding indicates that a strong attenuation of anyexternal field is crucial to achieve high Q. In our exper-iment the magnetic shielding reduces the Earth field tothe level of a few µT. This is one order of magnitudelower than the field necessary to cause significant fluxpenetration, indicating that residual losses due to fluxpenetration are likely negligible in the current setup. Aninteresting corollary is that our experimental techniquecan be used to probe deviations from the ideal Meissnereffect in a macroscopic sample of a type I superconductor,and to assess the conditions under which ideal Meissnereffect can effectively show up.

Let us compare our results with other mi-

cro/nanomechanical techniques and experiments.While higher quality factors have been reported inseveral systems, both in optically levitated nanoparticles[27] and in clamped ultrastressed membranes [14], valuesexceeding 107 are not easily obtained. Compared toprevious attempts of magnetically levitating a ferromag-netic particle above a superconductor [38, 39] we obtainquality factors 3 orders of magnitude higher.

As discussed in the introduction, the most proper fig-ure of merit for force sensing is actually the ratio T/τ . Inthis respect our magnetically levitated particles achieveT/τ ≈ 10−4 K/s, which is already at the state of the artfor micro/nanosystems. Comparable values have beenachieved so far in ultracold cantilevers [18]. However,we stress that our approach has a substantial potentialfor improvement by fully suppressing gas damping andfurther cooling to millikelvin temperature, with valuesbelow 10−6 K/s which appear to be within reach. In or-der to fully exploit this remarkable potential, one needsto suppress technical excess noise due to external vibra-tions, which is ubiquitous at Hz frequency, and is stillsignificant in our experiment. This can be done usingstandard mechanical isolation techniques.

To illustrate the potential of our approach, we con-sider as a case study the use of our particle as magne-tometer. We limit our discussion to the α mode, andassume the magnetic moment µ is oriented along the yaxis. By applying a magnetic field B along x, one gen-erates a torque T = µB directly driving the α mode.From the torque noise of the α mode estimated from datashown in Fig. 6, we infer an effective magnetic field noise√SB =

√ST /µ = 14 fT/

√Hz. This figure is compara-

ble with that of the best known SQUID [53] or atomic[54] magnetometers with much larger characteristic sizeof the order of 1 mm. Therefore, our levitated particleappears to outperform existing state-of-the-art magne-tometers in terms of field resolution normalized over thesensed magnetic field volume. This finding can be statedmore precisely in terms of the so called quantum limitof magnetometry [42], which can be defined through therelation:

SB2µ0

=~V

(12)

where V is the volume. Eq. (12) expresses the fact thatthe magnetic field resolution scales inversely with the av-eraging volume. By taking the physical volume of ourmicrosphere as reference, the quantum limited magneticfield resolution can be calculated for our experiment as√SB,QL = 57 fT/

√Hz. This means that, in contrast

with other magnetometers [42], our micromagnet is ac-tually beating the quantum limit by a factor of 4. Ifwe could achieve the theoretical thermal motion of theβ mode with the measured Q factor, we would obtain aresolution

√SB ≈ 1 fT/

√Hz almost two orders of magni-

tude better than the quantum limit. A more detailed dis-cussion of this unexpected achievement goes beyond thescope of this paper. However, we point out that schemes

8

employing levitated ferromagnets to overcome quantumlimits on magnetometry have been theoretically proposedin literature [11]. In addition to genuine magnetometry,this outstanding sensitivity suggests a possible use of lev-itated micromagnets to probe pseudomagnetic fields suchthose arising from exotic spin-dependent interactions ofelectrons [11] or from interactions mediated by axion-likeparticles [55].

IV. OUTLOOK AND POSSIBLEAPPLICATIONS

In addition to magnetometry, in this section we dis-cuss a number of other possible applications of magneti-cally levitated ferromagnetic microparticles with ultralowdamping.

The most direct use is as ultrasensitive force sensors.For a thermal noise limited operation in the low pres-sure limit at T = 4.2 K, the microparticle described inthis work would feature a force noise spectral density√Sf ≈ 1 aN/

√Hz for the z mode, assuming a frequency

independent residual Q of 3 × 107. For a very low fre-quency mode at 1 Hz and operation at 100 mK the noisewould drop to 0.03 aN/

√Hz. These figures can be cur-

rently achieved only with much smaller masses, such ascarbon nanotubes [56] or optically levitated nanoparti-cles [29] with diameter below 100 nm. However, if theforce to be measured scales with the size of the particle,such as in the case of gravitational or magnetic forces,the signal to noise ratio for our magnetic microparticle isdramatically higher.

Potential use as subresonant wideband gravimeter canbe considered. Indeed, for operation at T = 4.2 K anda lowest mode at 1 Hz with Q = 3 × 107, we estimatea thermal noise limited acceleration noise of 3 × 10−10

m/s2/√

Hz, which would be competitive with the bestknown gravimeters [57] despite the much smaller size.Due to the compactness of our setup, an attractive possi-

bility is the realization of gravity gradiometers composedof several pairs of identical trapped particles [58]. A spe-cific design optimized to suppress frequency drifts will beneeded to realize practical devices. Moreover, feedbackcooling of all normal modes is likely necessary to avoidnonlinear effects.

Other applications can be envisioned in the contextof fundamental and quantum physics. Ultralow damp-ing 1/τ and temperature are key ingredients to performstrong noninterferometric tests of wavefunction collapsemodels [12, 18]. Our fully passive levitation techniqueoffers the unique option of combining ultralow dampingwith the operation at very low temperature. Ultralowdamping alone can be used to set significant bounds onso called dissipative extensions of collapse models [33].In the context of quantum technologies, ultraisolated mi-cromagnets can be strongly coupled to single spins [35],superconducting circuits [31] or acoustomagnonic modes[41].

Finally, we mention the potential use of ultraisolatedmagnetically levitated particles for even more ambitiousexperiments, such as measuring the gravitational con-stant G with small test masses [59], fundamental directtests of the quantum superposition principle and gravi-tationally induced collapse [4], gravity-induced entangle-ment and tabletop tests of quantum gravity [60, 61].

Note: when finalizing this work we became aware of arelated work by Gieseler et al [62].

Acknowledgments

We thank V. Sglavo, G. Guella, P. Connell and M.Utz for technical help. We acknowledge financial sup-port from the EU H2020 FET project TEQ (GrantNo. 766900), the Leverhulme Trust (RPG-2016-046), theCOST Action QTSpace (CA15220), INFN and the Foun-dational Questions Institute (FQXi).

[1] M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, CavityOptomechanics, Rev. Mod. Phys. 86, 1391 (2014).

[2] A.D. O’Connell et al., Quantum ground state and single-phonon control of a mechanical resonator, Nature 464,697 (2010).

[3] W. Marshall, C. Simon, R. Penrose, and D.Bouwmeester, Towards Quantum Superpositions of aMirror, Phys. Rev. Lett. 91, 130401 (2003).

[4] H. Pino, J. Prat-Camps, K. Sinha, B. P. Venkatesh,and O. Romero-Isart, On-chip quantum interference ofa superconducting microsphere, Quant. Sci. Techn. 3,025001(2018).

[5] G. Binnig, C.F. Quate, and Ch. Gerber, Atomic ForceMicroscope, Phys. Rev. Lett. 56, 930 (1986).

[6] D. Rugar, R. Budakian, H.J. Mamin, and B.W. Chui,Single spin detection by magnetic resonance force mi-croscopy, Nature 430, 329 2004.

[7] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali,and A. Bachtold, A nanomechanical mass sensor withyoctogram resolution, Nature Nanotechn. 7, 301 (2012).

[8] H. Liu, W.T. Pike, C. Charalambous, A.E. Stott, PassiveMethod for Reducing Temperature Sensitivity of a Micro-electromechanical Seismic Accelerometer for MarsquakeMonitoring Below 1 Nano-g, Phys. Rev. Appl. 12, 064057(2019).

[9] R.P. Middlemiss, A. Samarelli, D.J. Paul, J. Hough, S.Rowan, and G.D. Hammond, Measurement of the Earthtides with a MEMS gravimeter, Nature 531, 614 (2016).

[10] S. Tang, H. Liu, S. Yan, X. Xu, W. Wu, J. Fan, J. Liu,C. Hu, L. Tu, A high-sensitivity MEMS gravimeter witha large dynamic range, Microsyst. and Nanoeng. 5, 45(2019).

[11] D.F. Jackson Kimball, A.O. Sushkov, and Dmitry Bud-ker, Precessing Ferromagnetic Needle Magnetometer,

9

Phys. Rev. Lett. 116, 190801 (2016).[12] A. Vinante, M. Bahrami, A. Bassi, O. Usenko, G. Wi-

jts, and T.H. Oosterkamp, Upper Bounds on SpontaneousWave-Function Collapse Models Using Millikelvin-CooledNanocantilevers, Phys. Rev. Lett. 116, 090402 (2016).

[13] Note that we use here the symbol τ for the amplitudedamping time, which is the quantity physically measuredin the experiment, rather than the energy damping timewhich is a factor 2 smaller.

[14] D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A.Schliesser, Continuous force and displacement measure-ment below the standard quantum limit, Nature Phys. 15,745 (2019).

[15] G.S. MacCabe, H. Ren, J. Luo, J.D. Cohen, H. Zhou,A. Sipahigil, M. Mirhosseini, O. Painter, Phononicbandgap nano-acoustic cavity with ultralong phonon life-time, arXiv:1901.04129.

[16] J.D. Teufel, T. Donner, D. Li, J.W. Harlow, M.S. All-man, K. Cicak, A.J. Sirois, J.D. Whittaker, K.W. Lehn-ert, and R.W. Simmonds, Sideband cooling of microme-chanical motion to the quantum ground state, Nature 475,359 (2011).

[17] H.J. Mamin and D. Rugar, Sub-attonewton force detec-tion at millikelvin temperatures, Appl. Phys. Lett. 79,3358 (2001).

[18] A. Vinante, R. Mezzena, P. Falferi, M. Carlesso, andA. Bassi, Improved Noninterferometric Test of CollapseModels Using Ultracold Cantilevers, Phys. Rev. Lett. 119,110401 (2017).

[19] D.E. Chang, C.A. Regal, S.B. Papp, D.J. Wilson, J.Yeb, O. Painter, H.J. Kimble, and P. Zoller, Cavity opto-mechanics using an optically levitated nanosphere, PNAS107, 1005 (2010).

[20] A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, and S. Chu,Observation of a single-beam gradient force optical trapfor dielectric particles, Opt. Letters 11, 288 (1986).

[21] N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek,and M. Aspelmeyer, Cavity cooling of an optically levi-tated submicron particle, PNAS 110, 14180 (2013).

[22] D. Windey, C. Gonzalez-Ballestero, P. Maurer, L.Novotny, O. Romero-Isart, R. Reimann, Cavity-based 3Dcooling of a levitated nanoparticle via coherent scattering,Phys. Rev. Lett. 122, 123601 (2019).

[23] U. Delic, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletic,M. Aspelmeyer, Cavity cooling of a levitated nanosphereby coherent scattering, Phys. Rev. Lett. 122, 123602(2019).

[24] T. Li, S. Kheifets, and M.G. Raizen, Millikelvin coolingof an optically trapped microsphere in vacuum, NaturePhys. 7 527-530 (2011).

[25] J. Gieseler, B. Deutsch, R.Quidant, and L. Novotny, Sub-kelvin Parametric Feedback Cooling of a Laser-TrappedNanoparticle, Phys. Rev. Lett. 109, 103603 (2012).

[26] J. Vovrosh, M. Rashid, D. Hempston, J. Bateman, M.Paternostro, and H. Ulbricht, Parametric feedback cool-ing of levitated optomechanics in a parabolic mirror trap,J. Opt. Soc. Am. B 34, 1421 (2017).

[27] J. Gieseler, L. Novotny, R. Quidant, Thermal nonlinear-ities in a nanomechanical oscillator, Nature Phys. 9, 806(2013).

[28] J. Gieseler, R. Quidant, C. Dellago, L. Novotny, Dy-namic relaxation of a levitated nanoparticle from a non-equilibrium steady state, Nature Phys. 9, 358 (2014).

[29] D. Hempston, J. Vovrosh, M. Toros, G. Winstone, M.

Rashid, H. Ulbricht, Force sensing with an optically levi-tated charged nanoparticle, Appl. Phys. Lett. 111, 133111(2017).

[30] M. Rashid, M. Toros, A. Setter, H. Ulbricht, PrecessionMotion in Levitated Optomechanics, Phys. Rev. Lett.121, 253601 (2018).

[31] O. Romero-Isart, L. Clemente, C. Navau, A. Sanchez,and J.I. Cirac, Quantum Magnetomechanics with Lev-itating Superconducting Microspheres, Phys. Rev. Lett.109, 147205 (2012).

[32] W. Paul, Electromagnetic traps for charged and neutralparticles, Rev. Mod. Phys. 62, 531 (1990).

[33] A. Pontin, N.P. Bullier, M. Toros, P.F. Barker, An ultra-narrow line width levitated nano-oscillator for testing dis-sipative wavefunction collapse, arXiv:1907.06046.

[34] A. Vinante, A. Pontin, M. Rashid, M. Toros, P.F.Barker, H. Ulbricht, Testing collapse models with levi-tated nanoparticles: Detection challenge, Phys. Rev. A100, 012119(2019).

[35] P. Huillery, T. Delord, L. Nicolas, M. Van Den Boss-che, M. Perdriat, G. Hetet, Spin-mechanics with levitat-ing ferromagnetic particles, arXiv:1903.09699.

[36] B.R. Slezak, C.W. Lewandowski, J-F. Hsu and B.D’Urso, Cooling the motion of a silica microsphere ina magneto-gravitational trap in ultra-high vacuum, NewJ. Phys. 20, 063028, (2018).

[37] D. Zheng et al, Room temperature test of wave-function collapse using a levitated microoscillator,arXiv:1907.06896.

[38] C. Timberlake, G. Gasbarri, A. Vinante, A. Setter, H.Ulbricht, Acceleration sensing with magnetically levitatedoscillators, Appl. Phys. Lett. 115, 224101 (2019).

[39] T. Wang, S. Lourette, S.R. O’Kelley, M. Kayci, Y.B.Band, D.F. Jackson Kimball, A.O. Sushkov, and D. Bud-ker, Dynamics of a Ferromagnetic Particle Levitated overa Superconductor, Phys. Rev. Applied 11, 044041 (2019).

[40] J. Prat-Camps, C. Teo, C.C. Rusconi, W. Wieczorek,and O. Romero-Isart, Ultrasensitive Inertial and ForceSensors with Diamagnetically Levitated Magnets, Phys.Rev. Appl. 8, 034002 (2017).

[41] C. Gonzalez-Ballestero, J. Gieseler, O. Romero-Isart, Quantum Acoustomechanics with a Micromagnet,arXiv:1907.04039.

[42] M.W. Mitchell, S.P. Alvarez, Quantum limits tothe energy resolution of magnetic field sensors,arXiv:1905.00618.

[43] Magnequench, MQP-S-11-9-20001 isotropic powder,URL https://mqitechnology.com/

[44] J.D. Jackson, Classical Electrodynamics, 3rd ed. Wiley,New York, 1998.

[45] Q.G. Lin, Theoretical development of the image methodfor a general magnetic source in the presence of a su-perconducting sphere or a long superconducting cylinder,Phys. Rev. B 74, 024510 (2006).

[46] M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B.Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes,and G.N. Wells, The FEniCS Project Version 1.5, Arch.Numer. Software 3, 9 (2015).

[47] P.S. Epstein, On resistance experienced by spheres intheir motion through gases, Phys. Rev. 23, 710 (1924).

[48] A. Cavalleri, G. Ciani, R. Dolesi, M. Hueller, D. Nicolodi,D. Tombolato, S. Vitale, P.J. Wass, W.J. Weber, Gasdamping force noise on a macroscopic test body in aninfinite gas reservoir, Phys. Lett. A 374, 3365 (2010).

10

[49] T.R. Roberts and S.G. Sydoriak, Thermomolecular Pres-sure Ratios for He3 and He4, Phys. Rev. 102, 304 (1956).

[50] S. Weber, Leiden Comm. No. 264b (1936); 264d (1936);and Suppl. 71b (1932).

[51] S. Weber and G. Schmidt, Leiden Comm. No 246c (1936).[52] W. Treimer, O. Ebrahimi, and N. Karakas, Observation

of partial Meissner effect and flux pinning in supercon-ducting lead containing non-superconducting parts, Appl.Phys. Lett. 101, 162603 (2012).

[53] M. Schmelz, R. Stolz, V. Zakosarenko, T. Schonau,S. Anders, L. Fritzsch, M. Muck, M. Meyer, H.-G.Meyer, Sub-fT/Hz1/2 resolution and field-stable SQUIDmagnetometer based on low parasitic capacitance sub-micrometer cross-type Josephson tunnel junctions, Phys-ica C: Superconductivity and its Applications, 482, 27(2012).

[54] H.B. Dang, A.C. Maloof, and M.V. Romalis, Ultrahighsensitivity magnetic field and magnetization measure-ments with an atomic magnetometer, Appl. Phys. Lett.97, 151110 (2010).

[55] N. Crescini, C. Braggio, G. Carugno, P. Falferi, A. Or-tolan, G. Ruoso, The QUAX-gpgs experiment to searchfor monopole-dipole Axion interaction, Nucl. Instrum.Meth. Phys. Res. A 842, 109 (2017).

[56] J. Moser, J. Guttinger, A. Eichler, M.J. Esplandiu, D.E.Liu, M.I. Dykman, A. Bachtold, Ultrasensitive force de-tection with a nanotube mechanical resonator, NatureNanotechnology 8, 493 (2013).

[57] John. M. Goodkind, The superconducting gravimeter,Rev. Sci. Instrum. 70, 4131 (1999).

[58] C.E. Griggs, M.V. Moody, R.S. Norton, H.J. Paik, and K.Venkateswara, Sensitive Superconducting Gravity Gra-diometer Constructed with Levitated Test Masses, Phys.Rev. Appl. 8, 064024 (2017).

[59] J. Schmole, M. Dragosits, H. Hepach and M. As-pelmeyer, A micromechanical proof-of-principle experi-ment for measuring the gravitational force of milligrammasses, Class. Quantum Grav. 33, 12 (2016).

[60] S. Bose, A. Mazumdar, G.W. Morley, H. Ulbricht, M.Toros, M. Paternostro, A.A. Geraci, P.F. Barker, M.S.Kim, and G. Milburn, Spin Entanglement Witness forQuantum Gravity, Phys. Rev. Lett. 119, 240401 (2017).

[61] C. Marletto and V. Vedral, Gravitationally Induced En-tanglement between Two Massive Particles is SufficientEvidence of Quantum Effects in Gravity, Phys. Rev. Lett.119, 240402 (2017).

[62] J. Gieseler, A. Kabcenell, E. Rosenfeld, J.D. Schae-fer, A. Safira, M.J.A. Schuetz, C. Gonzalez-Ballestero,C.C. Rusconi, O. Romero-Isart, M.D. Lukin, arXiv:arXiv:1912.10397.

[63] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, Wi-ley Classic Library Classic Edition Published, 1995, pag.50.

[64] L.D. Landau, E.M. Lifshitz, Electrodynamics of contin-uous media, 2nd Edition, Pergamon Press, Oxford, pag.205.

Figure 7: Simplified scheme of the trap with a tilt angle θ.

Appendix A: Numerical study of the System andFrequencies Characterization

The potential energy of a ferromagnetic particle in thepresence of a superconductor is described by

U = −1

2

∫V

dxM(x) ·Bind(x)−mgz, (A1)

where M , m and V are the magnetization, the mass andthe volume of the particle, and Bind is the magnetic fieldinduced by the presence of the superconductor. Deter-mining the induced magnetic field for a dynamical systemis not an easy task, however as the motion of the magnetis much slower then the time scale of the super-currentsand the London penetration depth is much smaller thanthe characteristic length scale of the system, we can relyon a quasi-static approximation and assume the Londonpenetration depth to be zero. Under this approximationthe induced magnetic field Bind can be described by thefollowing differential equation

∇×Bind = 0 x ∈ Ωn ·Bind = −n ·Bf x ∈ ∂Ω

where ∂Ω represents the surface of the superconductorand Ω the space outside the superconductor, allowing usto numerically solve the magneto-static problem using afinite-element method with FEniCS software [46].

The simulation was carried out to find the equi-librium positions (x0, y0, z0, β0, α0) and the frequencies(ωx, ωy, ωz, ωβ , ωα) in the experimental setup with the zaxis tilted from 0o to 3o with respect to the gravitationalaxis, as shown in figure 7. We extract the frequencies byfitting the data from the simulation around the minimawith a quadratic function. Plots of these frequencies andminima are shown in figure 8.

11

0.0 0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

2

4

6

8x

freq

(Hz)

0.0 0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

1000

500

0

x 0 (

m)

(a) (b)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

2

3

y fre

q (H

y)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

0

20

40

y 0 (

m)

(c) (d)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

40

50

60

70

80

z fre

q (H

z)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

280.0

282.5

285.0

287.5

290.0

z 0 (

m)

(e) (f)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

370

380

freq

(Hz)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

0.01

0.00

0.01

0 (ra

d/)

(g) (h)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

40

60

80

100

freq

(Hz)

0.5 1.0 1.5 2.0 2.5 3.0tilt angle (°)

0.50

0.55

0.60

0 (ra

d/)

(i) (j)

Figure 8: Dependence of the frequencies (ωx, ωy, ωz, ωβ , ωα)and equilibrium positions (x0, y0, z0, β0, α0) on the angle oftilt (θ) with respect to gravity.

Appendix B: Calculation of the frequency shift

The frequency shift as a function of the motion ampli-tude for an unforced mechanical resonator due to nonlin-ear restoring terms can be calculated using various ways,such as the Lindstedt-Poincare method or the multiplescales method, see for instance ref. [63] for an extensivereview.

When the equation of motion is written in the followingform, where dissipation terms are neglected and nonlin-ear terms up to the third order are retained:

x+ ω20x+ α2x

2 + α3x3 = 0 (B1)

the resulting effective resonance frequency is found to be:

ω0 = ω0

[1 +

(3

8

α3

ω0− 5

12

α22

ω40

)x2A

](B2)

where xA is the amplitude of the oscillation, meaningthat the oscillation is written as x = xAcos (ω0t+ φ0)

For the β and z modes in the present experimentwe can start from the potential predicted by the imagemethod model, and Taylor expand up to fourth order.We obtain the following equations of motion:

z + ω2zz −

5

2

ω2z

z20

z2 + 5ω2z

z30

z3 = 0 (B3)

β + ω2ββ −

2

3ω2ββ

3 = 0 (B4)

where for each equation we have assumed that only thatspecific mode is excited at large amplitude, so that cross-coupling terms such as β2z2 can be neglected.

Combining Eq. (B2) with Eqs. (B3,B4) we finally ob-tain:

ωz = ωz

(1− 35

48

z2A

z20

)(B5)

ωβ = ωβ

(1− 1

4β2A

)(B6)

where zA and βA are the oscillation amplitudes. In bothcases the nonlinear terms produce a softening effect pro-portional to the square of the amplitude of oscillation.The frequency shift observed in the experiment can thenbe used to provide an approximate estimation of the ab-solute oscillation amplitude.

Appendix C: Estimation of magnetic dissipation

To estimate dissipation from eddy currents, we assumethe microparticle has a finite electrical conductivity σ andrelative permeability µr ≈ 1. The latter condition is ex-pected for a hard ferromagnetic particle fully saturated.Under these assumptions the calculation of eddy currentlosses for a spherical geometry is a standard problem inelectromagnetism [64]. Defining the complex polarizabil-

ity α with the relation ~M = α ~H, where ~M is the magne-

tization induced by eddy currents and ~H magnetic field,

the induced magnetic moment is ~µi = (α′ + iα′′)V ~H.Following Landau-Lifshitz [64] in the low frequency limit

δ a, where δ =√

2/σµ0ω is the penetration depth,condition largely fulfilled in our system:

α′′ =1

5

(aδ

)2

(C1)

This leads to a power dissipation:

Wec =1

2µ0ωα′′B2V =

π

15σω2a5B2. (C2)

where V = 4πa3/3 is the particle volume. The same re-sult can be derived by a direct integration of the powerdissipation, under the condition of complete field pene-tration and dividing the sphere in elementary conductiverings.

In order to calculate the mechanical energy dissipatedby eddy currents in the β mode, we note that a rotation of

12

the magnetic particle by an angle β from the equilibriumposition is equivalent to a change of the image magneticmoment ∆~µ′ = −µβz, which produces a change of the Bfield orthogonal to ~µ:

∆B =µ0

4π

µβ

(2z0)3 (C3)

For a sinusoidal oscillation of β we can identify ∆B withB in Eq. (C2), and taking into account that the mechan-ical energy of the mode is E = 1

2kββ2 we can finally

calculate the loss angle as:

1

Qec=Wec

ωE(C4)

Inserting the parameters of our experiment and assuminga typical value σ ≈ 106 Ω−1m−1 for NdFeB alloys, weobtain Qec ≈ 1011. This is clearly too high to explainthe measured residual loss.

Magnetic hysteresis losses can be estimated follow-ing the same scheme, but now replacing the eddy cur-

rent polarizability with a complex magnetic susceptibil-ity χ = χ′ + iχ′′. As we are using a fully saturated hardferromagnet, we expect |χ| 1. The total power dissi-pation will be given by:

Wm =1

2µ0ωχ′′B2V. (C5)

and we can calculate 1/Qm using Eq. (C4) with Wm inplace of Wec. Carrying out the calculation we find:

1

Qm=

1

24χ′′(a

z0

)3

(C6)

Assuming that χ′′ is frequency independent and takinginto account that z0 ∝ a

34 , this implies 1/Qm ∝ a

34 ,

which means that we expect a slightly sublinear increaseof Q when reducing the radius a. However, if χ′′ ∝ ωand taking into account that ωβ ∝ a−

58 we find a nearly

size-independent behaviour 1/Qm ∝ a18 .