Superconducting microsphere magnetically levitated in an anharmonic potential with integrated magnetic readout Mart ı Gutierrez Latorre1Gerard Higgins1 2Achintya Paradkar1Thilo Bauch1and Witlef Wieczorek1

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Superconducting microsphere magnetically levitated in an anharmonic potential with

integrated magnetic readout

Mart´ı Gutierrez Latorre,1Gerard Higgins,1, 2 Achintya Paradkar,1Thilo Bauch,1and Witlef Wieczorek1, ∗

1Department of Microtechnology and Nanoscience (MC2),

Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden

2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Vienna, Austria

(Dated: June 5, 2023)

Magnetically levitated superconducting microparticles oﬀer a promising path to quantum ex-

periments with picogram to microgram objects. In this work, we levitate a 700 ng ∼1017 amu

superconducting microsphere in a magnetic chip trap in which detection is integrated. We measure

the particle’s center-of-mass motion using a DC-SQUID magnetometer. The trap frequencies are

continuously tunable between 30 and 160 Hz and the particle remains stably trapped over days in a

dilution refrigerator environment. We characterize motional-amplitude-dependent frequency shifts,

which arise from trap anharmonicities, namely Duﬃng nonlinearities and mode couplings. We ex-

plain this nonlinear behavior using ﬁnite element modelling of the chip-based trap potential. This

work constitutes a ﬁrst step towards quantum experiments and ultrasensitive inertial sensors with

magnetically levitated superconducting microparticles.

I. INTRODUCTION

Systems of levitated nano- and microparticles in vac-

uum [1] oﬀer extreme isolation of the particles from the

environment as well as in-situ tuning of the trapping po-

tential [2,3]. These platforms provide novel opportu-

nities for realizing ultrasensitive force [4–7] and accel-

eration sensors [8–12], for studying thermodynamics in

the underdamped regime [13–15], for exploring many-

body physics with massive objects [16–20], and for ex-

ploiting rotational degrees of freedom [21–24]. Recently,

the center-of-mass (COM) motion of optically levitated

nanospheres was cooled to the motional ground state [25–

29], opening the possibility to perform quantum exper-

iments with levitated nanoparticles [3,30,31]. Electri-

cally levitated nanoparticles have also seen tremendous

progress towards reaching the quantum regime [22,32–

35].

To extend quantum control from nano- to microparti-

cles, magnetic levitation [36] has recently gained renewed

interest [12,37–44]. Magnetic levitation can be used to

levitate objects of diﬀerent shapes [45,46] with masses

ranging from picograms to tons [36]. It oﬀers extreme

isolation from the environment [38–40,43,47,48] and

allows for tunable potential landscapes [49]. This combi-

nation of properties makes magnetically levitated parti-

cles particularly well suited for high precision sensing of

forces and accelerations [5,8–10,12,50], as well as for

fundamental physics experiments with picogram to mi-

crogram objects [47–49]. Recent experimental develop-

ments in this direction include levitating micro-magnets

on top of superconductors [12,38,51–53], diamagnetic

particles in strong magnetic ﬁelds [37,39–41,54], and

superconducting microparticles in millimeter-scale super-

conducting magnetic traps [43,44,55].

∗witlef.wieczorek@chalmers.se

We pursue magnetic levitation of superconductors

in a superconducting magnetic trap, since this ap-

proach promises the least intrinsic mechanical dissipa-

tion [47,48]. Levitated superconductors in the Meissner

state do not suﬀer from magnetic moment drift or in-

trinsic eddy current damping, unlike levitated magnets

[5,12,38,52,53]. By using a persistent current mag-

netic trap [56] the trap can be made perfectly stable, un-

like systems of diamagnetic particles levitated between

strong magnets [39,54,57].

We levitate a superconducting microsphere in a fully

chip-based system. The chip-based approach [44,58,59]

enables higher magnetic ﬁeld gradients and trapping fre-

quencies, as well as the potential to scale up the system to

levitate multiple particles on the same chip. We measure

the particle’s motion using magnetic pickup loops which

are coupled to a SQUID magnetometer. The pickup loops

are integrated in the chip; this allows for precise posi-

tioning and enhanced measurement sensitivity. In the

future we will replace the SQUID by a ﬂux-tunable super-

conducting microwave cavity [60–65] to achieve quantum

control over the COM motion of the levitated micropar-

ticle [10,47–49].

In this work we demonstrate stable levitation of a

48 µm-diameter (700 ng) superconducting microsphere

over days. We smoothly tune the particle’s COM fre-

quencies between 30 and 160 Hz by varying the trap cur-

rent. We observe that the COM frequencies depend on

the motional amplitudes. This arises from trap anhar-

monicities [32,66–70]. The observed behavior is con-

sistent with estimations of the trap anharmonicities ex-

tracted from ﬁnite element modelling (FEM) of our sys-

tem. In the future we will employ cryogenic vibration

isolation [71,72] and feedback cooling [26,27,73] to re-

duce the motional amplitudes, then the eﬀects of trap

anharmonicities will be mitigated.

arXiv:2210.13451v3 [quant-ph] 1 Jun 2023

(b)

180 μm

48 μm

(c)

Trap coil windings

Pick-up loop

SQUID

electronics

50 mK

Dilution refrigerator

SQUID

Magnetometer

Room Temperature

(a)

Oscilloscope

280 μm

FIG. 1. Chip-based magnetic levitation setup. (a) Schematic

of the experimental setup. The superconducting particle is

levitated in a magnetic ﬁeld minimum (magnetic ﬁeld lines

are shown in green). The magnetic trap coils and pickup

loops are patterned on two stacked chips, which are housed

in a dilution refrigerator. The vertical separation between the

top and the bottom coils is 280 µm. Read-out of the particle

motion relies on coupling ﬂux from the pickup loops into a

SQUID magnetometer. (b) Microscope image of the two-chip

trap. Through the 180 µm hole in the middle of the top chip,

we see a 48 µm-diameter lead microsphere resting on the bot-

tom chip’s surface. (c) Scanning electron microscope image

of the top view of the two-chip trap with false coloring.

II. EXPERIMENTAL SETUP

The magnetic trap is produced by a current ﬂowing in

two superconducting coils, which are patterned on two

silicon chips. The chips are stacked on top of each other

to form an anti-Helmholtz-like conﬁguration (see Fig. 1),

for details see Ref. [44]. The superconducting particle

stably levitates near the minimum of the trap’s magnetic

ﬁeld. The particle is conﬁned within a closed container

given by the side walls of the hole in the top chip, the

top surface of the bottom chip, and a glass slide on top

of the hole. The force due to the magnetic trapping ﬁeld

is restoring within the container’s bounds, with a trap

depth larger than 1 ×1010 K. To detect the particle mo-

tion, we use two pickup loops which are integrated on the

same chips, see Fig. 1.

As the particle moves it changes the magnetic ﬂux

threading the pickup loops and thus the current in-

duced in the loops. The pickup loops are connected to

a commercial DC-SQUID magnetometer, which trans-

duces the ﬂux into a measurable voltage. Typical pickup

eﬃciencies are {ηx, ηy, ηz}={1.58,3.3,19.4}mϕ0µm−1

for the three COM modes, in terms of the ﬂux cou-

pled into the SQUID, and where ϕ0is the magnetic

ﬂux quantum. ηz> ηx, ηydue to the system’s geom-

etry (see Appendix S2). The measurement noise ﬂoor

(0.32 mϕ0Hz−0.5) is limited by magnetic ﬁeld ﬂuctua-

tions caused by trap current ﬂuctuations and corresponds

to a noise ﬂoor of {200,97,17}nm Hz−0.5for displace-

ments along the x, y, z directions, respectively. We con-

nect the pickup loops in series to reduce the sensitivity

to these ﬁeld ﬂuctuations. In the future, we will mitigate

this noise by driving the trap using a persistent current

[56], which should enable reaching the intrinsic noise ﬂoor

of the SQUID (∼1µϕ0Hz−0.5).

The magnetic trap and the SQUID magnetometer are

thermally connected to the mixing stage of a dilution

refrigerator. This allows the experiment to operate be-

tween temperatures as low as 50 mK and as high as

the critical temperature of the superconducting particle

(6.2 K for lead). Before levitating, the particle thermal-

izes on the bottom chip surface to the temperature of

the chip substrate [see Fig. 1(b)]. To lift the particle oﬀ

the chip surface, we ramp the trap current up to 0.8 A

to overcome the adhesive force between the particle and

the chip surface. With a current of 0.8 A, the lift force is

≈300 nN, which is usually suﬃcient to lift the particle.

After that, the current is ramped down to the operating

trap current.

III. CHARACTERIZATION OF

CENTER-OF-MASS MOTION

Near the trap center, the particle experiences a har-

monic trapping potential. The COM frequencies depend

on the trap geometry, the particle’s density, and the trap

current. The penetration depth (∼40 nm for lead) is

much smaller than the particle radius (24 µm) and so the

particle can be modelled as an ideal diamagnet with mag-

netic susceptibility of −1 and its trapping frequencies ωi

are given by [74]

ωi=∇iBr3

2µ0ρ=ζi

µ0NI

R2r3

2µ0ρ,(1)

where ∇iBis the magnetic ﬁeld gradient along the idi-

rection at the trap center, µ0is the vacuum permeability,

Iis the trap current, Nis the number of trap coil wind-

ings, Ris the trap coil inner radius, ρis the particle’s den-

sity and ζis a geometric factor. At the center of an ideal

anti-Helmholtz conﬁguration 2ζx= 2ζy=ζz= 0.86. In

our system ζx= 0.04, ζy= 0.06 and ζz= 0.12. (our coils

are separated by 280 µm and have inner radii ≈125 µm

as shown in Appendix S2). The trap axes are indicated

in Fig. 1(a).

Fig. 2(a) shows the power spectrum of a levitated

microsphere. Throughout this work, unless otherwise

stated, we levitated a 48 µm-diameter lead sphere using

0.5 A trap current and the cryostat temperature was 4 K.

The peaks corresponding to the COM motion are colored.

40 60 80 100 120 140 160

Pb/2 (Hz)

100

120

140

160

180

SnPb/2 (Hz)

0.4 0.45 0.5 0.55 0.6 0.65 0.7

Frequency (Hz)

100

120

140

160

/2 (Hz)

PSD (Φ0

2Hz-1)

PSD (m2Hz-1)

10_2

70 120 200 30040

ω/2π(Hz)

10_6

10_10

10-14

10-10

10_18

(a) ωx/2πωy/2πωz/2π

Trap current (A) Pb ω/2π(Hz)

0.4 0.5 0.6 0.7

160

120

(b) (c)

50 mK

4 K

FEM

SnPb ω/2π(Hz)

40 80

120 160

180

100

140

slope = ρPb

ρSnPb

Model

Experiment

FIG. 2. Trap frequencies. (a) Power spectrum of the SQUID

signal. The frequencies of the particle’s COM motion along

the x, y and z directions are highlighted. The black lines

show the expected trap frequencies obtained from FEM sim-

ulations. The secondary y-axis (green) shows the z ampli-

tude in units of displacement. To obtain the x and y ampli-

tudes in units of displacement, the secondary y-axis has to

be multiplied by 151 and 35, respectively. (b) The COM fre-

quencies increase linearly with the trap current, as expected

from Eq. (1). The results are similar when the cryostat tem-

perature is at 50 mK and at 4 K. Lines show the simulated

frequencies obtained from FEM. (c) When the same trap set-

tings are used, a tin-lead sphere has a higher trap frequency

than a lead sphere due to its lower density (see [Eq. (1)]).

We identiﬁed these modes by comparing the peak fre-

quencies with predictions from FEM simulations of our

system [44,45]. We ﬁnd good agreement between the

measured and simulated COM frequencies. Peaks at the

second harmonic of the COM frequencies are pronounced,

particularly the second harmonic of the ∼40 Hz peak at

∼80 Hz. These peaks arise from the nonlinear pickup

eﬃciency, rather than due to actual particle motion at

these frequencies. We describe these peaks further in

Appendix S3. In the future we will feedback cool the

particle, then eﬀects of the nonlinear pickup will be neg-

ligible.

The linear relation between the COM frequencies and

the trap current given by Eq. (1) is shown in Fig. 2(b).

We observed no signiﬁcant diﬀerence between measuring

this eﬀect with the cryostat temperature of 50 mK or 4 K.

We ﬁnd good agreement between measurement results

and FEM simulation results for the xand ymodes. We

attribute the 4% discrepancy for the z-mode frequency

to simulation uncertainties in the FEM.

We conﬁrm the inverse relation between trapping fre-

quency and particle density of Eq. (1) by comparing the

trapping frequencies of a lead particle and a tin-lead par-

ticle as we vary the trap current, see Fig. 2(c). The ratio

of the frequencies is given by the square root of the ratio

of the material densities.

Time (s)

050 100 150 200 250

Time (h)

05 10 15 20 25 30 35

(b)

0.8

71.0

70.0

118.6

119.0

40.5

39.5

0.6

0.4

0.2

Energy correlation

(a)

Mode frequency ω/2π(Hz)

0 50 100 150 200 250

Tim e (s )

0. 0

0. 2

0. 4

0. 6

0. 8

1. 0

Au to -correlation

τ = 87 sx

τ = 78 s

τ = 79 s

39.5

40.0

40.5

41.0

70.5

71.0

71.5

72.0

118.4

118.6

118.8

119.0

FIG. 3. Trap stability. (a) The particle stably levitates over

35 hours and its COM mode frequencies do not drift over that

time. We explain the ﬂuctuations of the frequencies by ﬂuctu-

ations of the COM mode amplitudes together with frequency

pulling. (b) The correlations of the COM mode energies decay

over around 80 s, because of ﬂuctuations of the mode ampli-

tudes over this timescale. The lines represent exponential ﬁts.

We can stably levitate the superconducting sphere for

days in the chip-based magnetic trap. Fig. 3(a) shows

the ﬂuctuations of the COM frequencies of a levitated

sphere over a 35 h measurement. We have yet to observe

an upper limit to the levitation time, provided that the

particle is not illuminated. When the particle is illumi-

nated, as in Ref. [44], it heats up, loses superconductivity

and falls on the bottom chip. When the particle is kept

in the dark, as in this work, we have not yet observed

any upper limit to the levitation time; we have measured

up to 48 h.

Around once per day, we observe a sudden jump of all

the COM frequencies of around 1 Hz (see Appendix S5).

We attribute these jumps to changes in the magnitude

or orientation of trapped ﬂux in the particle. Fig. 3(a)

shows 35 h of data in which we do not discern any such

frequency jumps. The trapped ﬂux will be the topic of a

dedicated future investigation.

The particle’s COM motion does not thermalize at the

cryostat temperature, since it is strongly driven by the

cryostat’s vibrations. We estimate the mean amplitudes

of the three modes to be 24, 10, and 7 µm for the x, y, and

z modes, respectively, corresponding to an eﬀective tem-

perature of about 1 ×109K. The energy in each mode

(mω2

i⟨r2

i⟩) ﬂuctuates on a time scale ∼80 s, as shown

by the autocorrelation functions in Fig. 3(b). The mode

energy ﬂuctuations are due to the mode amplitude ﬂuc-

tuations, which occur over this time scale. By ﬁtting the

autocorrelation functions to exponential decay functions,

we extract quality factors 3400, 4500, and 9300 for the x,

y, and z modes, respectively [39]. We expect the quality

factors to be limited by eddy current damping caused by

normal-conducting metals in the vicinity of the levitated

particle. In the future, we will mitigate this damping

mechanism by surrounding the particle with a supercon-

ducting shield, with no normal-conducting metal within

the shield.

IV. FREQUENCY PULLING

We can explain the COM frequency ﬂuctuations of

Fig. 3(a) as resulting from the ﬂuctuating mode ampli-

tudes [Fig. 3(b)] together with frequency pulling. Fre-

quency pulling describes the dependence of the COM fre-

quencies on the mode amplitudes. It arises from quartic

terms in the trapping potential of the form

Upull =X

mγij r2

ir2

j(2)

which cause the motional frequencies to be shifted de-

pending on the motional amplitudes according to [75]

∆ωi=3γii

ωi

⟨r2

i⟩+X

j̸=i

2γij

ωi

⟨r2

j⟩.(3)

The γii terms in the potential are called Duﬃng nonlin-

earities, while the γij terms describe couplings between

the modes.

Experimental data showing frequency pulling is shown

in Fig. 4. The spectral peak corresponding to the x-

mode (y-mode) shifts depending on the amplitude of the

x-mode motion in Fig. 4(a) [(b)]. This is described by

Eq. (3). The remaining seven graphs showing the depen-

dence of ωion ⟨r2

j⟩are included in Appendix S4.

The linear relation between the mode frequencies and

the mean-square displacements [Eq. (3)] are shown in

Fig. 4(c) and (d), in which the spectral peak frequencies

ωxand ωyare plotted against the spectral peak area

corresponding to the x-mode (the remaining graphs of

the same form are included in Appendix S4). The slopes

of these lines depend on the values of γxx and γyx as well

as the pickup eﬃciency ηx. We extract estimates of γij

from FEM simulations of our system (see Appendix S1);

this allows us to use the nine gradients (i.e. the nine linear

relations between ωiand ⟨x2

j⟩, for i∈ {x, y, z}and j∈

{x, y, z}) to estimate the three eﬃciencies ηi(the values

are quoted earlier). The estimated eﬃciencies are used

to convert the lower x-axes of Fig. 4(c)-(d) into the upper

x-axes, and yield the secondary y-axis of Fig. 2(a).

The estimation of the three eﬃciencies is an over-

constrained problem, it yields fairly consistent results for

the nine graphs. For instance, the estimation of ηxto-

gether with the FEM estimations of γxx and γyx describes

well both the slope in Fig. 4(c) and Fig. 4(d).

To observe the frequency pulling eﬀect presented in

Fig. 4we did not control the particle’s motional ampli-

tudes, instead, we ﬁltered a long 35 h dataset in which

the motional amplitudes randomly ﬂuctuated: We sepa-

rated the data into 10 s chunks and extracted the mode

frequencies and mode areas from each chunk.

70 7 1 7 2

10 6

10 5

10 4

10 3

38 3 9 4 0 41 42

10 7

10 6

10 5

10 4

10 3

10 2

ωx/2π(Hz)

ωx/2π(Hz) ωy/2π(Hz)

ωx/2π(Hz) ωy/2π(Hz) ωz/2π(Hz)

ωy/2π(Hz)

PSD (Φ0

2Hz-1)

PSD (Φ0

2Hz-1)

Counts

(a) (b)

(e) (f) (g)

71 72 118.5 119.0

200

400

200

400

200

0 0 0

41 42

Experiment Frequency pulling model

0 500

<x2> ( μ m2)

39.5

40.0

40.5

0 500

<x2> ( μ m2)

70.5

71.0

71.5

72.0

0 1000 2000

<x2> (mΦ0

2)<x2> (mΦ0

0 1000 2000

Increasing <x2>

Experiment

Frequency

pulling model

Increasing <x2>

FIG. 4. Frequency pulling. The frequencies of the (a) x-

mode and of the (b) y-mode depend on the amplitude of the

x-mode; spectra represented in lighter colors have higher x-

mode amplitudes. (c) and (d) The x-and-y mode frequen-

cies change linearly with the mean-square amplitude of the

x-mode. The slope of each model line is given by estimation

of the trap anharmonicity obtained from FEM and by the

estimated pickup eﬃciency ηx. (e)-(g) The ﬂuctuating mode

frequencies of Fig. 3(a) are described by asymmetric distribu-

tions. These distributions are reproduced well by using the

same frequency pulling model as in (c) and (d), together with

the distribution of mode amplitudes from the experimental

data.

To investigate the dependence of the mode frequencies

on, e.g., the x-mode amplitude, we ﬁltered out the data

in which the y- and z-mode amplitudes were high. Each

point in Fig. 4(c)-(d) corresponds to one 10 s chunk. To

produce the seven spectra in Fig. 4(a) and (b) we binned

the power spectrum of each 10 s-long dataset into seven

bins, based on the x-peak areas, then we averaged the

power spectra for each of the seven bins.

The mode amplitude ﬂuctuations together with fre-

quency pulling describe the frequency ﬂuctuations of

Fig. 3(a). We represent the frequency distributions by

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SuperconductingmicrospheremagneticallylevitatedinananharmonicpotentialwithintegratedmagneticreadoutMart´ıGutierrezLatorre,1GerardHiggins,1,2AchintyaParadkar,1ThiloBauch,1andWitlefWieczorek1,∗1DepartmentofMicrotechnologyandNanoscience(MC2),ChalmersUniversityofTechnology,SE-41296G¨oteborg,Sweden2Insti...

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Superconducting microsphere magnetically levitated in an anharmonic potential with integrated magnetic readout Mart ı Gutierrez Latorre1Gerard Higgins1 2Achintya Paradkar1Thilo Bauch1and Witlef Wieczorek1

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