# SeedBERT: Recovering Annotator Rating Distributions from an Aggregated Label

Aneesha Sampath,1 Victoria Lin, 1 Louis-Philippe Morency 1

###### Abstract

Many machine learning tasks—particularly those in affective computing—are
inherently subjective. When asked to classify facial expressions or to rate an
individual’s attractiveness, humans may disagree with one another, and no
single answer may be objectively correct. However, machine learning datasets
commonly have just one “ground truth” label for each sample, so models trained
on these labels may not perform well on tasks that are subjective in nature.
Though allowing models to learn from the individual annotators’ ratings may
help, most datasets do not provide annotator-specific labels for each sample.
To address this issue, we propose SeedBERT, a method for recovering annotator
rating distributions from a single label by inducing pre-trained models to
attend to different portions of the input. Our human evaluations indicate that
SeedBERT’s attention mechanism is consistent with human sources of annotator
disagreement. Moreover, in our empirical evaluations using large language
models, SeedBERT demonstrates substantial gains in performance on downstream
subjective tasks compared both to standard deep learning models and to other
models that account explicitly for annotator disagreement.

## Introduction

Subjectivity uncertainty is uncertainty that arises from data with subjective
or ambiguous labels—labels on which human annotators themselves can disagree.
Many affective computing tasks, including automatic emotion recognition and
hate-speech detection, are highly subjective due to their heavy reliance on
human opinion. Different people may perceive a speaker’s toxicity level or
emotional intent differently. If asked to rate a subjective sample, a group of
five annotators may very well provide five different labels.

In contrast, machine learning datasets usually provide a single label for each
sample. Labels for machine learning tasks are often crowd-sourced through
survey platforms, where researchers request multiple annotators to rate each
sample. Prior to a dataset’s public release, however, these individual
annotators’ ratings are usually aggregated into one “gold standard” label.
Flattening annotations into a single label results in loss of information
about the subjectivity of the task, and consequently, models trained on
aggregated labels may not be able to achieve their desired level of
performance on subjective tasks. This raises concerns for tasks such as
emotion recognition, where perception is highly dependent on personal opinion
(Zhang, Essl, and Mower Provost 2017; Gordon et al. 2021; Han et al. 2017).

To overcome such concerns and to better represent the nature of subjective
tasks, machine learning models must be able to represent the subjectivity of a
sample. In this paper, we propose models that can infer subjectivity as a
distribution of possible human opinions. Due to the single-label nature of
most standard machine learning datasets, we focus our efforts explicitly on
models that are able to reconstruct these distributions from single-label
data. In our modeling mechanism, we hypothesize that annotator subjectivity
arises in significant part as a function of the portion of the sample on which
the annotator decides to focus.

#### Illustrated Example.

In the remainder of this paper, we take emotion recognition to be an example
of a task with subjectivity uncertainty. Many existing emotion recognition
datasets, including the language datasets CARER (Saravia et al. 2018) and
Stanford Sentiment Treebank (SST) (Socher et al. 2013) provide only one
emotion label.

We consider two sentences from the SST dataset with similar sentiment scores.
In our first example, “You walk out of The Good Girl with mixed emotions –
disapproval of Justine combined with a tinge of understanding for her
actions,” the sentiment rating is $0.61$, or slightly positive. The speaker
expresses both negative and positive attitudes towards the film—“disapproval
of Justine” and “a tinge of understanding for her actions,” respectively—and
even explicitly notes mixed emotions in the sentence. These contrasting
sentiments, however, are aggregated into just one score that leans positive
(where the individual annotators might have considered the sentence to be very
negative, slightly negative, very positive, etc.). On the other hand, in
another sample from the dataset, “A smart, steamy mix of road movie, coming-
of-age story and political satire,” the speaker expresses only positive
emotions (indicating that the individual annotators all considered the
sentence to be positive), but the sentiment score is $0.67$, which similar to
that of the first example.

These score similarities demonstrate how the use of aggregated labels results
in the loss of a nuanced representation of the sentiment. Models trained only
on aggregated scores may not learn a sentiment representation as well as those
that are trained on the full distribution of annotator ratings.

#### Contributions.

This paper’s contributions are threefold:

  * •

We propose a deep language model (SeedBERT) that is capable of recovering
original annotator distributions from a single ground-truth label. This model
uses the recovered distribution to predict a label for the task.

  * •

We introduce a new set of annotations (MOSI-Subjectivity) of human
subjectivity and the mechanisms behind annotator disagreement. These
annotations extend the CMU-MOSI sentiment analysis dataset (Zadeh et al.
2016).

  * •

We present a detailed evaluation with both empirical performance analysis and
a study of human agreement.

## Related Work

#### Inter-Annotator Disagreement.

Annotator disagreement can arise when tasks are inherently subjective.
Affective semantics are particularly difficult to model since people’s biases,
experiences, and knowledge lead them to form different opinions about the same
data. This leads to performance degradation on tasks involving expressive
language (Alm 2011). In subjective contexts, it can therefore be useful to
move away from a single ground-truth label and to instead acknowledge that
multiple answers may be valid.

Several prior works attempt to address this unmet need by explicitly modeling
individual annotators. Models such as Jury Learning (Gordon et al. 2022) and
HuBi-Medium (Kocoń et al. 2021) propose joint training of text embeddings and
annotator embeddings to allow for predictions on the individual annotator-
level. Although these approaches move toward personalized annotator modeling,
they require not only that multiple annotators’ labels are available for each
data sample, but also that annotator-identification information is available.
This requirement contrasts with the typical crowd-sourced machine learning
datasets, in which annotator-level information is not provided.

#### Modeling Perception Uncertainty.

An additional body of work attempts to account for subjectivity by modeling
perception uncertainty directly. Han et al. (2017) propose a multi-task
framework that simultaneously predicts an emotion label and an estimate of
annotator disagreement. The resulting model outperforms single-task emotion
recognition models. Davani, Díaz, and Prabhakaran (2022) also propose a multi-
task framework—one in which each task consists of modeling an individual
annotator. The variance of the resulting annotator distribution is taken as
the inter-annotator disagreement measure. However, both of these methods
require access to annotator-level labels, which are not commonly available in
public datasets, at training time. Moreover, the method proposed by Davani,
Díaz, and Prabhakaran (2022) requires that each annotator label a large
portion of the data.

## MOSI-Subjectivity Dataset

To gain a better understanding of subjectivity uncertainty and the mechanisms
by which it arises, we collected extensive human ratings of sentiment and
subjectivity on an emotion recognition task. We used these ratings to evaluate
the empirical performance and human agreement of our proposed method,
SeedBERT.

#### Dataset.

We used video clips111We initially collected data using transcripts only, but
obtained poor inter-rater reliability on all questions. from the Multimodal
Corpus of Sentiment Intensity dataset (CMU-MOSI), a collection of over 2000
YouTube movie reviews with opinion-level annotations for sentiment and
subjectivity.

#### Data Collection.

We collected new labels for 500 randomly selected audiovisual CMU-MOSI samples
using the Prolific.co222https://www.prolific.co/ crowdsourcing platform. For
each sample, we asked five high-quality annotators (approval rating $\geq
98\%$) to answer the following questions.

  1. 1.

Is the speaker expressing any emotion in the video clip?

  2. 2.

Select the emotion most present in the video.

  3. 3.

Is the person expressing multiple emotions in the video? If so, select all of
the emotions present in the video.

We intended the first question—which has relatively little ambiguity—to be a
calibration question in order to establish the expected level of inter-
annotator agreement in the absence of any subjectivity. This question was
answered using a binary rating scale, with options Yes and No.

The second question, which requests that annotators mark the most salient
emotion in the video clip, is one commonly asked in emotion recognition data
collection procedures. CMU-MOSEI (Bagher Zadeh et al. 2018), for instance,
used a similar question to collect its labels. As with CMU-MOSEI, the six
Ekman emotions were presented as label options in a multiple-choice format.

The third question is a multiple-select question in which annotators were
instructed to select all of the Ekman emotions that the speaker expresses in
the video clip. This question allows raters to express greater nuance on the
emotion recognition task rather than limiting them to a single response, which
we felt could have implications for the subjectivity of the task.

After collecting a portion of the annotations and examining them, we
hypothesized that disagreement might arise due to annotators attending to
different parts of the sentence, where each part may emphasize a different
emotion. To verify this hypothesis, we collected additional annotations for
another $60$ samples, wherein we asked raters to explain their selections.
Again recruiting five high-quality annotators (approval rating $\geq 98\%$) to
rate each sample, we asked:

  1. 4.

On the previous page, you selected $X$ as the emotion most present in the
clip. Which words or phrases led you to your answer? Please highlight the
portions of the transcript that led you to make your selection.

### Analysis of Annotator (Dis)agreement

#### Quantitative Analysis.

To determine the level of agreement among annotators for each of the questions
asked, we calculated the Cohen’s kappa coefficient (Cohen 1960) using the
agreement software package.333https://github.com/jmgirard/agreement To account
for the multiple-selection setting of the third question, we treated the
responses to each emotion as answers to a single binary question, then
averaged Cohen’s kappa among all emotions.

Question |  Kappa
---|---
Is the speaker expressing any emotion in the video? |  $0.744$
Select the emotion most present in the video. |  $0.410$
Select all of the emotions present in the video. |  $0.710$
Table 1: Cohen’s kappa coefficients for unordered categories for the three
survey questions.

We observe that annotators exhibit similar levels of agreement for the
calibration question and the multi-label question (Table 1). However,
agreement is substantially lower for the second question concerning the
emotion most present in the sample. These results seem to indicate that
annotators can acknowledge and agree upon the presence of other opinions, but
greater subjectivity and disagreement arise when annotators are asked to
select the most salient opinion.

#### Qualitative Analysis.

Observing this discrepancy, we sought to investigate why annotators have high
agreement when selecting all emotions present but low agreement when selecting
the primary emotion. In Table 2, we provide selected examples to illustrate
our analysis.

Sample |  Main Emotions
---|---
these action sequences are certainly a lot better than the incoherent impossible to follow slop that we see in a typical michael bay movie |  Happiness, Anger, Disgust
and i honestly think if we hadnt have seen watchman this would be the best credit sequence of the year |  Happiness, Sadness, Surprise
but there are you know there’re critics of movies that are sometimes too thoughtful in a way and i think that would never let me go |  Disgust, Surprise
Table 2: Selected examples from the first survey. The first column is the
transcript of the video clip shown to the annotators, and the second column is
a set of the emotions selected as most present by the five annotators.

Given the transcript of the sample and the set of annotators’ responses to the
emotion most present in the sample, we observe that different words or phrases
within the sample may elicit different emotions. Taking the first example from
Table 2, we can see that the italicized part of the sentence is characterized
by a happy tone, whereas the bolded part has a disgusted or angry tone. While
all annotators indicated that happiness and disgust or anger were present when
answering the third question, they disagreed as to which of the three was the
primary emotion when answering the second question. This leads us to
hypothesize that annotators attend to different parts of the sentence, which
leads to low agreement when selecting the most salient emotion but high
agreement when selecting all of the emotions present.

Figure 1: Circumplex model of affect.

As mentioned in the Data Collection section, we collected further annotations
to verify this hypothesis, this time asking annotators to explain their
selections for the most salient emotion by highlighting specific words and
phrases in the transcript that led them to their decision. We found that our
hypothesis holds when the emotions present are adjacent on the circumplex
model of affect (Figure 1), in which similar emotions are adjacent to one
another and opposite emotions are across from one another (Russell 1980).

Emotion |  Supporting Evidence |  Annotators
---|---|---
Happiness |  a lot better |  2
Anger |  incoherent impossible to follow slop |  2
Disgust |  incoherent impossible to follow slop |  1
Table 3: Selected example from the second set of surveys. Two annotators
selected Happiness as the main emotion present, two selected Anger, and one
selected Disgust.

Still using the first sample from Table 2 as an illustrative case, we observe
in Table 3 that for the non-adjacent emotion pairs (Happiness, Anger) and
(Happiness, Disgust), annotators cited different portions of the transcript to
support their answer for the main emotion present, suggesting that they attend
to different parts of the sentence. However, this finding did not hold for
adjacent emotions, as the same portion of the sentence is provided as
justification for both the Anger and Disgust labels. This result may suggest
that subjectivity may arise both from the part of the input attended to and
from personal interpretation of the same portion of the input. For the
remainder of this work, we will focus on subjectivity due to the part of the
utterance attended to.

## Methods

Figure 2: SeedBERT architecture. Pre-trained BERT encoders are paired with a
linear layer initialized from a random seed.

#### SeedBERT.

Our proposed method, SeedBERT (Figure 2), is premised on the idea that a
collection of randomly initialized layer in a neural network will (by virtue
of their different starting points) learn to attend to different parts of an
input, even when trained on the same data. When the input is inherently
subjective, the linear layer—in attending to a particular portion—will
represent the opinions and biases of an arbitrary annotator. Following this
reasoning, SeedBERT is a collection of $n$ identical networks consisting of
pre-trained BERT encoders with a task-specific linear layer. Each network is
trained on the same dataset with identical loss functions, with the only
difference being that each network has a different random initialization of
the final linear layer. Each network is meant to reflect one annotator, and
SeedBERT’s final prediction is the majority vote of the networks.

As an ensemble of networks, SeedBERT bears certain methodological similarities
to classical ensemble methods like bagging (Breiman 1996). Rather than using a
collection of weak learners to improve predictive performance, however,
SeedBERT leverages a powerful existing model—pre-trained BERT—and uses its
“ensembling” to induce individual learners to attend to different portions of
the input.

#### Evaluation.

We compare our approach, SeedBERT, against four baseline models, two of which
we describe previously in the Related Work section. These latter two models
are trained on individual annotator ratings, which are not typically available
in machine learning datasets. Due to their use of this additional training
data, we can consider them to be oracle models that provide upper bounds on
the ability of a model to learn annotator distributions. All models use pre-
trained BERT as an encoder (Kenton and Toutanova 2019), with a final linear
layer fine-tuned for the task via the Adam optimizer. We use a learning rate
of $5\times 10^{-6}$ and fine-tune for $3$ epochs. We evaluated our models on
metrics accuracy and $F_{1}$-score using 5-fold cross-validation and repeated
this process 5 times with different splits of the data.

#### Baseline Models.

Our first baseline model (BERT) reflects a system where there is no explicit
attempt to account for subjectivity. This model consists of a single pre-
trained BERT encoder with a task-specific final linear layer.

Our second baseline model incorporates Bayesian neural networks (BNN), which
have previously been used to estimate prediction uncertainty due to their use
of probablistic rather than deterministic weights. Recent work has applied
BNNs to speech emotion recognition in order to directly estimate annotator
distributions (Raj Prabhu et al. 2022). We leverage a similar architecture of
an encoder followed by a BNN, but use a transformer-based BERT encoder.

We also compare SeedBERT against two oracle models trained on individual
annotator ratings, which are not typically available in machine learning
datasets. We use a Label Distribution Learning (LDL) framework (Geng 2016) and
the Multi-Task framework proposed by Han et al. (2017) to determine the upper
bounds of achievable model performance when modeling annotator distributions.

## Results & Discussion

Figure 3: Example of SeedBERT predicted annotator distribution of emotions
(left) and BERT predicted distribution (right). The predicted distributions
are in blue, while the true annotator distributions are in red.

In our empirical evaluation, we observe in Table 4 that SeedBERT model
outperforms all other models on the emotion recognition task. Notably, given
only the aggregated label at training time, SeedBERT outperforms not only the
baselines (standard pre-trained BERT and BNN) but also the two models trained
on individual annotator data. These results suggest that SeedBERT’s mechanism
of reconstructing annotator distributions is effective in improving
performance on subjective tasks.

Through qualitative analyses, we find that an individual BERT model tends to
predict uniform distributions of annotator ratings (i.e., uniform across
emotions) irrespective of the input sample (Figure 3), possibly because it
cannot fit well to low-agreement data. As seen in Table 4, this shortcoming
has clear negative implications for its performance on the emotion recognition
task. On the other hand, the SeedBERT approach moves toward reflecting
annotators’ biases, as each random initialization can reflect the preferences
of an arbitrary annotator. As shown in Figure 3, the annotator distribution
predicted by SeedBERT much more closely reflects the true annotator
distribution.

We further find that certain random initializations of the individual SeedBERT
networks tended to bias toward a subset of the Ekman emotions—much in the same
way that a human annotator might have a tendency to favor certain emotions in
their ratings. These observations align with our hypothesis that the random
initializations may serve as proxies for individual annotator perceptions or
biases.

Model | Accuracy |  $F_{1}$-Score
---|---|---
BERT | .204 | .102
LDL | .284 | .0821
Multi-Task | .180 | .0954
BNN | .150 | .132
SeedBERT | .308 | .183
Table 4: SeedBERT outperforms all baselines when averaged across 5 iterations
of 5-fold cross validation.

## Conclusions

In this paper, we analyze human subjectivity and the mechanisms by which
annotator disagreement arises. We propose SeedBERT, a novel method capable of
recovering original annotator distributions from a single ground-truth label
to ultimately improve performance on downstream tasks. Via SeedBERT, we find
that manipulating the random initializations of pre-trained models can serve
as an effective proxy for modeling varying annotator perceptions.

Our findings should be considered in light of several points. First, our
extension to CMU-MOSI consisted of a relatively small number of samples (500)
for fine-tuning a large language model. Second, fewer samples were rated for
anger and fear, which could limit the efficacy of our method when evaluating
these emotions. Third, although CMU-MOSI was designed to be a multimodal
dataset, we implemented SeedBERT and its comparison methods as unimodal
language models, which likely limits their predictive ability. Fourth, the
validation set accuracies and $F_{1}$-scores are relatively low across all
models, which warrants further exploration. Finally, though random
initializations can effectively represent a collection of arbitrary
annotators, there are cases where prior information about the annotators’
preferences is known (for example, an annotator’s favorite movie genre). In
these cases, SeedBERT may be improved by incorporating this prior information
to represent a specific annotator distribution rather than an arbitrary one.

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2022

[1]Tjerk H. Oosterkamp

1]Leiden Institute of Physics, Leiden University, P.O. Box 9504, 2300 RA
Leiden, The Netherlands 2]School of Physics and Astronomy, University of
Southampton, SO17 1BJ, Southampton, UK 3]Istituto di Fotonica e
Nanotecnologie, CNR and Fondazione Bruno Kessler, I-38123 Povo, Trento, Italy

# Measuring gravity with milligram levitated masses

Tim M. Fuchs  Dennis G. Uitenbroek  Jaimy Plugge  Noud van Halteren  Jean-Paul
van Soest  Andrea Vinante  Hendrik Ulbricht<EMAIL_ADDRESS>[  [  [

###### Abstract

Gravity differs from all other known fundamental forces since it is best
described as a curvature of spacetime. For that reason it remains resistant to
unifications with quantum theory. Gravitational interaction is fundamentally
weak and becomes prominent only at macroscopic scales. This means, we do not
know what happens to gravity in the microscopic regime where quantum effects
dominate, and whether quantum coherent effects of gravity become apparent.
Levitated mechanical systems of mesoscopic size offer a probe of gravity,
while still allowing quantum control over their motional state. This regime
opens the possibility of table-top testing of quantum superposition and
entanglement in gravitating systems. Here we show gravitational coupling
between a levitated sub-millimeter scale magnetic particle inside a type-I
superconducting trap and kg source masses, placed approximately half a meter
away. Our results extend gravity measurements to low gravitational forces of
attonewton and underline the importance of levitated mechanical sensors.

##  1 Introduction

Einstein’s theory of general relativity (GR), our widely accepted theory of
gravity, has seen different experimental confirmations Einstein1916 ;
walsh1979  by observing massive astronomical objects and their dynamics, most
recently by the direct observation of gravitational waves from the merger of
two black holes abbott2016  and the imaging of a black hole by the event
horizon telescope akiyama2019 , as well as dedicated satellite missions for
testing the basic principle of GR — the equivalence principle touboul2017  and
frame dragging effects everitt2011 . Laboratory experiments have been
continuously increasing the sensitivity of gravity phenomena, including
general relativistic effects in atom clocks and atom interferometers
bothwell2022 ; asenbaum2017 , tests of the equivalence principle rosi2017 ;
asenbaum2020 , precision measurements of Newton’s constant rosi2014 ;
quinn2013  and tests of the validity of Newton’s law at micrometre-scale
distances geraci2008 ; tan2020 .

However, gravity has never been tested for small masses and on the level of
the Planck mass. Measurements of gravity from classical sources in laboratory
table-top settings is contrasted by an increasing interest to study
gravitational phenomena originating from quantum states of source masses, for
example, in the form of the gravitational field generated by a quantum
superposition state bronstein2012republication ; rickles2011role ; bose2017 ;
marletto2017 ; al2018optomechanical . The effort ultimately aims at directly
probing the interplay between quantum mechanics and general relativity in
table-top experiments. Because quantum coherence is easily lost for increasing
system size, it is important to isolate gravity as a coupling force for as
small objects as possible, which in turn means to measure gravitational forces
and interactions extremely precisely.

At the same time, massive quantum sensors are especially suited for tests in a
regime with appreciable gravitational influences, which is favourable in
probing fundamental decoherence mechanisms related to gravity R1 ; R2  or
proposed physical models of the wave function collapse diosi2015 ; vinante2016
; bassi2013  featuring the system mass explicitly, such as the continuous
spontaneous localization (CSL) model ghirardi1990  and the Diósi-Penrose model
of gravitationally-induced collapse diosi1987 ; penrose1996 ; oosterkamp2013 .

An emerging technology for ultra-sensitive sensing is based on levitated
mechanical systems. These can be used for the mechanical sensing of very weak
forces and to probe quantum physics at increasing scales of mass (and space).
In optical levitation schemes, the heating from trapping lasers is the most
prominent source of noise. Worse, in any quantum experiment, they will provide
a source of decoherence, greatly increasing the difficulty of creating
macroscopic quantum states. In magnetically levitated systems, this pathway of
decoherence is largely removed Romero2021 .

The extremely low damping of magnetic systems, combined with their relatively
high mass and operation in low noise cryogenic environments, makes them well
suited for mesoscopic probes of quantum mechanics and could provide a test to
possible limits of the applicability of quantum mechanics to the macroscopic
world leggett2002 ; arndt2014 . Van Waarde et al. Waarde2016  and Vinante et
al. vinante2020  have previously realised such magnetic levitation of sub-
milligram particles, in which the motional state of the particle is read out
by means of superconducting quantum interference device (SQUID) detection.

In a recent publication, Westphal et al. have demonstrated gravitational
coupling between two $90\text{\,}\mathrm{m}\mathrm{g}$,
$1\text{\,}\mathrm{m}\mathrm{m}$ radius, gold spheres, achieved off resonance
at millihertz frequencies in a torsion balance-type geometry aspelmeyer2021 .
Recent work by Brack et al. brack2022  has shown the dynamical detection of
gravitational coupling between two parallel beams of a meter in size in the
hertz regime. In this paper we present work with a
$2.4\text{\,}\mathrm{k}\mathrm{g}$ source mass and a magnetically levitated
sub-milligram test mass, giving a coupling of
$1030\text{\,}\mathrm{a}\mathrm{N}$ with a force noise of
$0.5\text{\,}\mathrm{f}\mathrm{N}\mathrm{/}\sqrt{Hz}$. This work provides an
intermediate step towards an experiment where a small test mass senses the
gravity sourced by a small source mass.

##  2 Results

The core of the setup is a type-I superconducting trap with a magnetic
particle levitated therein, as shown in Fig. 1b. The trap is made of tantalum
with a critical temperature of $T_{c}=$4.48\text{\,}\mathrm{K}$$. We perform
the experiment at temperatures below $100\text{\,}\mathrm{m}\mathrm{K}$. The
trap has an elliptical shape ($4.5\text{\,}\mathrm{m}\mathrm{m}$$~{}\times$
$3.5\text{\,}\mathrm{m}\mathrm{m}$, with the height from bottom of the trap to
the coil being $4.7\text{\,}\mathrm{m}\mathrm{m}$) to confine the modes of the
levitated magnetic particle to the axial system of the trap. The particle is
composed of a set of three
$$0.25\text{\,}\mathrm{m}\mathrm{m}$\times$0.25\text{\,}\mathrm{m}\mathrm{m}$\times$0.25\text{\,}\mathrm{m}\mathrm{m}$$
Nd2Fe14B magnets that are magnetically attached North-to-South as also shown
in Fig. 1b, and a spherical glass bead, $0.25\text{\,}\mathrm{m}\mathrm{m}$
radius, that is attached to the middle magnet using stycast. This bead is
added to break the rotational symmetry of the zeppelin around the x-axis
(angle $\gamma$). Typical remnant magnetisation of Nd2Fe14B magnets is in the
order of $1.4\text{\,}\mathrm{T}$. The estimated mass of the full particle, as
depicted in Fig. 1d, is $0.43\text{\,}\mathrm{m}\mathrm{g}$. Using the
infinite plane approximation of Vinante et al. vinante2020 , we calculate an
expected z-mode frequency of $27\text{\,}\mathrm{H}\mathrm{z}$ in this
geometry.

The motion of the particle results in a change of flux through a loop at the
top of the trap (the pick-up loop), which is detected using a two-stage biased
SQUID coupled inductively to the pick-up loop. The loop is positioned off-
center so that the symmetry is broken, and all modes couple to the loop. A
third loop positioned halfway between the SQUID input loop and the pick-up
loop is coupled inductively to a calibration loop. This transformer is used to
calibrate the energy coupling $\beta^{2}$ between the detection circuit and
the degrees of motion of the zeppelin, providing calibrated motion of the
zeppelin from the measured flux signal. This procedure is further described in
Supplementary Materials C.

The set-up is suspended from springs in a multi-stage mass spring system to
shield the experiment from external vibrations, both vertical and lateral. The
bottom three masses (one aluminum, two copper) are similar in weight to the
experimental setup, with a lowest resonance frequency at
$0.9\text{\,}\mathrm{H}\mathrm{z}$. Above that is a millikelvin mass spring
system with a lowest resonance frequency of
$4.8\text{\,}\mathrm{H}\mathrm{z}$. We refer to Ref. wit2019  for more details
on a near identical mass-spring system and its performance in a similar
dilution refrigerator. This combination is suspended from the 1K plate by a
long spring. Thermalisation of the experiment is provided by a flattened
silver wire, which is mechanically soft while providing a good thermal link.
This entire system is depicted in Fig. 1A.

The cryostat as a whole is rigidly attached to a 25 metric ton concrete block,
which is again placed on pneumatic dampers to limit vibrations coupling in
from the building. The pulse tube cooler and the vacuum pumps for the
circulation of the mixture are rigidly attached to the building through a
second frame, and attached to the cryostat only by edge welded bellows and
soft copper braiding to further limit external excitations from reaching the
particle.

To demonstrate the force sensitivity of the system and as a proof of concept
for gravitational coupling in levitated magnetic systems, we utilized an
electrically driven wheel with a set of three
$2.45\text{\,}\mathrm{k}\mathrm{g}$ brass masses, placed equally spaced along
the outer rim. This wheel was used to create a time dependent gravitational
gradient at the resonant frequency of a selected mode of the zeppelin, in an
effort to drive the motion gravitationally. The frequency of the masses was
read out optically using a laser and photodiode, in which the masses act as a
mechanical shutter.

Figure 1: Schematic depiction of the experimental setup.
_A_ : Multi-stage mass spring system to isolate from external vibrations, as
discussed in the text. Electromagnetic shielding of the trap is discussed in
the Supplementary Material A. _B_ : Conventions for degrees of freedom adopted
from Vinante et al. vinante2020 . Detection by SQUID as discussed in the text.
Calibration loop as discussed in the text and Supp.Mat. C. _C_ : An image of:
the dilution refrigerator used for the experiments, including the multi-stage
mass spring system. _D_ : The magnetic particle, composed of three
$$0.25\text{\,}\mathrm{m}\mathrm{m}$\times$0.25\text{\,}\mathrm{m}\mathrm{m}$\times$0.25\text{\,}\mathrm{m}\mathrm{m}$$
Nd2Fe14B magnets (SuperMagnetMan, C0005-10) magnetically attached end-to-end
and a single spherical glass bead with a $0.25\text{\,}\mathrm{m}\mathrm{m}$
radius attached using Stycast to the middle of the magnets, which is used to
break the symmetry of the $\gamma$ mode. _E_ : The trap, as placed in the
aluminium holder without the shielding cylinder. The aluminium foil envelope
provides additional electro-magnetic shielding between the calibration
transformer and the pick-up loop. Further details and images of the setup are
shown in the Supplementary Material A.

In figure 2B. we show the uncalibrated spectrum. We clearly observe the six
different modes corresponding to the three translational modes, and three
rotational modes respectively. Furthermore, we see a distinct peak at 27 Hz,
which we attribute to the z-mode as discussed in the beginning of this
section. These modes were validated and calibrated by performing a magnetic
drive, excited by a flux injected through the calibration transformer, as
shown in figure 1B.

Figure 2: Levitated Particle Spectra.
_A_ : Force noise of the $27\text{\,}\mathrm{H}\mathrm{z}$ mode when
gravitationally driven at $1.3\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$
detuning, overnight. _B_ : Typical resonator power spectrum. In this figure we
have greyed out the regular $50\text{\,}\mathrm{H}\mathrm{z}$ European
electrical noise. This noise typically has a similar power to the particle
resonances.

Using this magnetic drive, we determine the decay time of the modes during the
subsequent ringdown. For the $26.7\text{\,}\mathrm{H}\mathrm{z}$ mode, we find
a lower bound $\tau=$1.09\text{\cdot}{10}^{5}\text{\,}\mathrm{s}$$, or a Q
factor of $Q=$9.13\text{\cdot}{10}^{6}\text{\,}$$. This procedure is further
discussed in the Supplementary Material B. For the other modes, we arrive at Q
factors that are about an order of magnitude lower.

We test the force sensitivity of our mode by driving the
$26.7\text{\,}\mathrm{H}\mathrm{z}$ mode using the brass masses of the mass-
wheel. The resulting excitation at one position of the wheel in the force
spectrum, is shown in figure 2A. The calibration of this spectrum is discussed
in the supplementary materials. The resulting force noise of this z-mode is
approximately $0.5\text{\,}\mathrm{f}\mathrm{N}\mathrm{/}\sqrt{Hz}$, or
equivalently, a displacement noise of
$60\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\sqrt{Hz}$, in an
$8\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$ bandwidth centered around the
orange dotted line that indicates the frequency of the resonance.
Equivalently, we can determine the motion of the trap in which the particle is
levitated by dividing the force noise by the spring constant of the
confinement potential that keeps the particle around its equilibrium height.
The spring constant for the z-mode was determined to be k =
$12e{-3}\text{\,}\mathrm{N}\mathrm{/}\mathrm{m}$, resulting in a trap
displacement noise of $30\text{\,}\mathrm{f}\mathrm{m}\mathrm{/}\sqrt{Hz}$.
This vibrational noise is not yet thermally limited, but rather corresponds to
a mode temperature of 3 K, which we attribute to the limits of the vibration
isolation inside the cryostat.

In figure 3 we show the measured gravitational interaction for different
displacements of the wheel, using the method described in the Supplementary
Material E. We also plot the phase of the masses along the wheels rotation for
which the particle experiences maximal force, see figure S2. For the
longitudinal displacement, the vertical displacement was held at $48\text{\,}$
$\pm$ $4\text{\,}$ centimeter. In the run of vertical positions, the wheel was
kept centered with respect to the trap. Included is the expected gravitational
signal at the location of the magnetic particle for the z-mode of the
particle, which was calculated from an analytical simulation where the mass
was taken to consist of multiple point masses. From this same simulation, a
systematic error bound was derived, based on an estimated systematic error,
for the longitudinal run, of $\pm$5 centimeter longitudinal, $\pm$3 centimeter
lateral, and $\pm$4 centimeter vertical, which were estimated from the
geometry of the wheel, the mass spring system and the systems used to measure
the displacements. For the vertical run, the bounds are $\pm$2 centimeter in
each principal direction, based on the increased stability of the system under
vertical displacement.

Figure 3: Response to gravitational drive as function of separation.
_A_ : Force experienced by the mechanical resonator as a result from drive
using the mass wheel, for different lateral displacement of the mass wheel
relative to the particle. The dashed blue line represents the simulated
gravitational force at the position of the magnetic particle as discussed in
the main text. The blue area denotes the systematic uncertainty discussed in
the text. A second, dashed line is plotted, which has a scaling factor of 0.35
applied, that seems to agree with the data more closely, as discussed in the
main text. _B_ : Similar to A, but now for a vertical displacement of the mass
wheel relative to the particle, when the wheel is kept centered below the
particle. Systematic bounds as discussed in the text. _C_ : Here we see the
wheel phase at which the magnetic particle experiences the strongest force,
plotted against longitudinal displacement. _D_ : Similar to C, but now for a
vertical displacement.

The observed signal agrees with the simulated force signal to within a factor
0.35 with a standard error of 0.02, which was determined by means of an
orthogonal distance regression fit to the data.

##  3 Discussion

We attribute this constant factor to the effect of the wheel on the motion of
the trap and its holder. The trap and its platform also experience a
gravitational pull. For the sake of its vibration isolation, the trap and its
platform is suspended by springs and thus the platform is set in motion by the
gravitational pull on it. The amplitude of this motion is extra small,
however, because the frequency of the gravitational drive is approximately a
factor of 10 above the resonance of the suspension. This naturally gives rise
to a 180° phase shift in the response of the trap’s motion. The small motion
of the trap cause the walls of the trap to exert a small force on the
particle. Because of the 180 degree phase shift, this force is out of phase
with the gravitational pull that the particle experiences. Thus this leads to
a suppression of the particle response. The suppression depends on the total
force on the platform, the gravitational gradient due to the wheel, the angle
which the platform makes with respect to the horizontal etc. Additional
complicating factors are the changing stiffness of the SQUID cable when
cooling down, which can lead to a significant tilt of the platform.

We demonstrate the detection of a $30\text{\,}\mathrm{a}\mathrm{N}$
gravitational signal at $27\text{\,}\mathrm{H}\mathrm{z}$ and a damping
linewidth as low as $\gamma/2\pi=$2.9\text{\,}\mu\mathrm{H}\mathrm{z}$$, with
a $0.43\text{\,}\mathrm{m}\mathrm{g}$ test mass, paving the way for future
experiments in which both source and test mass are in this regime. This work
could be used to derive a more stringent bound on dissipative collapse models.
Furthermore it provides a promising platform to test for possible deviations
from inverse square force laws and fifth force models Blakemore2021 ;
smullin2005 , theories of modified Newtonian dynamics milgrom1983 ;
bekenstein2004  and other extensions of the standard model carney2021 .

By ensuring that the pick-up loop is placed off-centre with respect to the
trap and by breaking the rotational symmetry in we demonstrate detection of
all six mechanical modes, in comparison to earlier work. As we will discuss in
future work, this is critical to the stability of the mode under test due to
non-linear mixing between the different modes.

With a mode temperature of 3 kelvin compared to an operating temperature of 30
millikelvin, we are currently not yet thermally limited. It would require
another 20 dB of vibration isolation to reach thermal motion.

By using a second particle in a different trap as source mass, or a similar
construction, this work paves the way towards easily scalable measurements of
gravitational coupling in the hertz regime and with source masses at Planck
mass level, ultimately allowing for testing gravity in a yet unexplored low-
mass regime and pushing into the quantum controlled domain. Coupling of the
detection SQUIDs in this scheme to an superconducting LC circuit would provide
a means of inserting single microwave photons, providing access to the toolbox
of quantum state manipulation. This would further extend this work towards
truly macroscopic superposition measurements and possibly gravitationally-
induced entanglement.

Acknowledgments We thank K. Heeck, B. Hensen, P. Numberi, G. Koning, C.
Timberlake, E. Simcox and M. Camp for useful discussions and experimental
help.

Funding This work was supported by the NWO grant OCENW.GROOT.2019.088., the
QuantERA grant LEMAQUME, funded by the QuantERA II ERA-NET Cofund in Quantum
Technologies implemented within the European Union’s Horizon 2020 Programme.
Further, we would like to thank for support the UK funding agency EPSRC under
grants EP/W007444/1, EP/V035975/1 and EP/V000624/1, the Leverhulme Trust (RPG-
2022-57), the EU Horizon 2020 FET-Open project TeQ (766900) and the EU Horizon
Europe EIC Pathfinder project QuCoM (10032223).

Author Contributions TMF, DGU, JP, NvH, JPvS, AV, HU and THO developed the
experimental hardware and procedure and contributed to the manuscript; TMF,
DGU, JP, NvH and THO collected the data and contributed to data analysis.

Competing Interests All authors declare that they have no competing interests.

Data Availability All data needed to evaluate the conclusions in the paper are
present in the paper and/or the Supplementary Materials. Raw data (timetraces
of lock-in output) is available on Zenodo: https://zenodo.org/records/10300430

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##  Appendix A Further Schematics and Photographs of the Setup

Since the gravitational coupling is small, great care was taken to shield the
experiment from magnetic and electric forces. The heat- and vacuum shields of
the cryostat are made of several millimeters of gold plated copper, providing
a high level of shielding with respect to electrical forces. Several layers of
aluminium foil were wound around the heat shields that achieve temperatures
below the critical temperature of aluminium (namely, the
$1\text{\,}\mathrm{K}$ shield, and the $50\text{\,}\mathrm{m}\mathrm{K}$
shield). This was done in an effort to provide some basic shielding from stray
magnetic forces. Further magnetic shielding was incorporated in the holder of
the experiment, as shown in figure S1.

Figure S1: Close-up of the experiment holder in Fig. 1. The SQUID detection
chip is housed in a niobium can (respectively, Magnicon CAR-1 Two-Stage SQUID
and NC-1 Can) that provides shielding from AC magnetic fields through the
Meissner effect. The niobium can is screwed into the larger aluminium holder,
which similarly provides AC magnetic shielding to the trap through the
Meissner effect. The tantalum trap is capped with a PEEK coil-shape, placed
offset from the center of the trap, around which the pick-up loop is wound.
Additional shielding from DC magnetic fields is provided by several layers of
mu-metal foil wrapped around the aluminium holder. This shielding was added
under the assumption that stray magnetic fields influence the position of the
magnetic particle within the trap and otherwise would get ‘frozen-in’ to the
superconductors as they cool down.

In figure S2 we show a schematic to further elucidate the axial system used in
the gravitational measurements, showing the longitudinal direction, the
vertical direction and the wheel phase over which the wheel was displaced. The
factor $n$ follows from the detection by the laser-photodiode combination not
distinguishing between masses.

Figure S2: Schematic depicting the phase angle of the wheel, as used to define
the phase of maximal response in figure 3. Here, $n$ refers to the amount of
masses on the wheel, in this case three. This factor follows from the
detection not distinguishing between the different masses.

In figure S3 are some photographs of critical elements of the experiment.
Namely, (part of) the vibration isolation, the coil used to detect the motion
of the particle, the transformer used to calibrate the coupling, the tantalum
trap, and the wheel used to source the gravitational signal.

Figure S3: Photos of the the experimental apparatus of Fig.1.
_A_ : The mass-spring system and the silver wire used for thermalisation of
the masses and the experiment, with at the bottom the holder on a small
triangular platform to attach the springs. _B_ : The coil after the first two
layers of five loops were wound around it, with the rest of the cap above. The
final pick-up loop consisted of four layers. _C_ : A close-up of the
calibration transformer, before aluminium shielding was added around it. _D_ :
The tantalum trap, with the elliptical shape and milling marks visible. _E_ :
The ‘mass-wheel’ as used in the experiments to show gravitational coupling.
The three brass masses are placed in an equilateral triangle. Not shown is the
laser-photodiode combination used to read out the frequency of the masses. The
wheel is surrounded with one centimeter thick steel plates and is attached to
a bridge made out of MK-profiles, which is used to control the elevation and
positioning of the wheel in the lab relative to the trap inside the cryostat.

##  Appendix B Determination of decay time, damping factor, and the quality
factor of the mechanical modes

The exponential decay time () of each mode was determined through ringdown
measurements at high amplitude. At low vibrational amplitude, the measured
increases, as is expected from non-linearities in the trapping potential. High
amplitude measurements were taken to provide a lower bound on with a high
signal-to-noise ratio. We focus on the $27\text{\,}\mathrm{H}\mathrm{z}$ mode,
as this is the mode we used to detect our gravitational signal. As discussed
in section 2, this mode corresponds most closely to the theoretically expected
z-mode frequency. Furthermore, it showed the highest degree of vibrational
isolation, which is also as expected given our implemented vibration isolation
functions predominantly in z. From the response in phase and amplitude shown
in section 2 this is further verified, matching closely to the expected
scaling of the z-mode gravitational coupling (both amplitude and phase) under
translation of the gravitational source.

Figure S4: A typical ringdown of the $27\text{\,}\mathrm{H}\mathrm{z}$ mode,
as performed after a magnetic drive through the calibration transformer. The
decay time () is extracted from an exponential fit.

The high amplitude provides a lower bound to the low amplitude of the system.
We observe a significant difference in for high and low amplitude, which is
explained by a duffing non-linearity in the equations of motion of the
resonator. From ringdown measurements, we obtain a lower bound to the decay
time of $\tau=$1.09\text{\cdot}{10}^{5}\text{\,}\mathrm{s}$$. The error on the
fit of this value is
$14.7\text{\,}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}$.
At the moment, our understanding of this decay time of individual modes is
that it depends strongly on the amplitudes in other modes due to a coupling of
the non-linearities. This understanding is limited currently, and a further
understanding would require further measurements. To minimise the effect of
this false sense of precision (versus accuracy) in the current publication, we
will instead truncate the values at three decimals. With the quality factor
(Q) of the resonator defined as

$Q=\pi f\tau$ (S1)

and a frequency of the mode $f=$26.7\text{\,}\mathrm{H}\mathrm{z}$$, we obtain
a quality factor of $Q=$9.06\text{\cdot}{10}^{6}\text{\,}$$. From we can also
determine the damping coefficient of the resonator, which is defined as

$\gamma=\frac{2}{\tau}$ (S2)

which gives a damping in the $27\text{\,}\mathrm{H}\mathrm{z}$ mode of
$\gamma=$1.84e{-5}\text{\,}\mathrm{s}^{-1}$$, or a linewidth of
$\gamma/2\pi=$2.92\text{\,}\mu\mathrm{H}\mathrm{z}$$.

frequency [Hz] | tau [s] | Q factor
---|---|---
15.9 |  3.65$\cdot 10^{4}$ | $1.82\cdot 10^{6}$
26.7 |  1.09$\cdot 10^{5}$ | $9.13\cdot 10^{6}$
40.6 |  1.43$\cdot 10^{4}$ | $1.82\cdot 10^{6}$
55.1 |  3.37$\cdot 10^{4}$ | $5.84\cdot 10^{6}$
129 |  0.214$\cdot 10^{4}$ | $8.70\cdot 10^{5}$
147 |  0.152$\cdot 10^{4}$ | $6.98\cdot 10^{5}$
Table S1: Tabulated values of the resonator mode parameters. Again, these
values are truncated at at three decimals, since the fit significance of these
values is much higher than the actual stability of these values with respect
to mode amplitude, as touched upon in the texts. The spring stiffness of the
$26.7\text{\,}\mathrm{H}\mathrm{z}$ mode was calculated to be
$12\text{\,}\mathrm{m}\mathrm{N}\mathrm{/}\mathrm{m}$.

##  Appendix C Analysis and Calibration of the Single-Stage Particle Readout
Circuit and Energy Coupling

The motion of the particle, and the resulting force noise of the particle
modes, can be calibrated from flux to RMS motion by injecting a magnetic drive
through the calibration transformer. The amount of magnetic flux injected is
then directly quantised by the SQUID. By measuring the flux induced by the
particle response and using the inductance in the circuit and the spring
stiffnes, we find a calibration of magnetic flux to motion.

Figure S5: Schematic circuit of the single-stage readout circuit for the
magnetic particle . The transformer used to couple in the external flux used
to calibrate the circuit is depicted at the top, with the particle and pick-up
loop on the left, and the input coil and SQUID on the right.

This calibration can be intuitively derived from the energy coupling,
$\beta^{2}$. The energy coupling is defined as the fraction of the energies in
two coupled oscillating systems, i.e. the amount of energy that couples from
one system to the other. This energy coupling is similar to the quality factor
of a single system, which is defined as the fraction of energy lost per cycle
with respect to the total energy in the system, or equivalently, a measure for
the damping of the system. Conversely, the quality factor provides a measure
for the maximal energy stored in the resonator as a fraction of the input
energy when driven at resonance for a time $T\gg\tau$, that is to say: the
energy at which the fractional energy loss per cycle is equal to the energy
put in each cycle by the resonant drive.

For our system, with a mechanical resonator in the form of the trapped
magnetic particle, and a driving magnetic field coupled in through the
calibration transformer and coupled to the particle through the pick-up loop,
we get

$\beta^{2}\equiv\frac{L_{total}I^{2}}{kx^{2}}$ (S3)

with $x$ any spatial coordinate. Going to the limit of infinitesimal
displacement, and using $k=m\omega^{2}$ and using
$L_{total}I^{2}=\Phi^{2}/L_{total}$, we obtain

$\beta^{2}=\frac{\left(\frac{d\Phi}{dx}\right)^{2}}{L_{total}\cdot
m\omega^{2}}$ (S4)

from which it is evident that the energy coupling can be used to determine the
absolute motion of the magnetic particle from the flux measured in the SQUID.

To measure $\beta^{2}$, we further consider the motion of the resonator as a
simple resonantly driven harmonic oscillator, with the _Ansatz_
$x(t)=A(t)e^{i\omega t}\mathchar 24635\relax\;\ A(t)=\frac{F}{2m\omega}\cdot
t$, from which we obtain a damping force

$F_{damping}\left(=2m\zeta\omega_{0}\frac{dx}{dt}=\gamma_{m}v\right)=\gamma_{m}\cdot\omega\cdot
x$ (S5)

The quality factor Q is equivalently defined as $Q=\frac{1}{2\zeta}$, thus
$\gamma_{m}=\frac{m\omega}{Q}$. In our calibration procedure, we inject a flux
through the calibration transformer. This flux then results in a current
through the detection circuit. This current is detected as a flux by the
SQUID, with known $\frac{dI}{d\phi}$ calibration from the SQUID parameters. We
refer to this current as the the crosstalk current $I_{crosstalk}$. This
current also leads to a flux through the pick-up loop, which gives rise to a
magnetic force on the particle of the form $F_{drive}=\alpha\cdot
I_{crosstalk}$. Here, $\alpha$ is a coupling constant of units
$\mathrm{[}\mathrm{N}\mathrm{/}\mathrm{A}\mathrm{]}$ that contains a
geometrical factor determined by both the relative positioning of the particle
with respect to the pick-up coil, and the physical sizes of this coil and
particle. As the particle is driven, the motion of the particle will in turn
induce a current in the detection circuit. This induced current has the form
$I_{induced}=\delta\cdot x$ where $\delta$ is a coupling constant with units
$\mathrm{[}\mathrm{A}\mathrm{/}\mathrm{m}\mathrm{]}$, which is by symmetry
effected through the pick-up coil-particle geometry in the same way that
$\alpha$ is.
Combining these results, and noting that for the steady state solution the
driving force must be equal to the damping force

$\frac{I_{induced}}{I_{crosstalk}}=\frac{\delta\cdot
x}{F_{drive}/\alpha}=\frac{\delta\cdot\alpha\cdot
x}{\gamma_{m}\cdot\omega\cdot
x}=Q\cdot\frac{\alpha\cdot\delta}{m\omega^{2}}=Q\cdot\frac{\alpha\cdot\delta}{k}$
(S6)

Since some of our decay times are rather long, in our experiments we work with
a $Q_{eff}=\pi fT$, where $T$ is the time during which the drive is applied.
Whilst this is not the steady state, this discrepancy is fully absorbed in our
definition of $Q_{eff}$:

$\frac{I_{induced}}{I_{crosstalk}}=\frac{\delta\cdot
x}{F_{drive}/\alpha}=\frac{\delta\cdot\alpha\cdot
x}{\gamma_{eff}\cdot\omega\cdot
x}=Q_{eff}\cdot\frac{\alpha\cdot\delta}{m\omega^{2}}=Q_{eff}\cdot\frac{\alpha\cdot\delta}{k}$
(S7)

From this derivation, we have found that the fraction of the currents in our
detection circuit contains all the coupling constants in our system. In fact,
looking at the units, we find

$\frac{\alpha\cdot\delta}{k}=\frac{$\left\mathrm{missing}{[}\frac{N}{A}\right\mathrm{missing}{]}$\cdot$\left\mathrm{missing}{[}\frac{A}{m}\right\mathrm{missing}{]}$}{$\left\mathrm{missing}{[}\frac{N}{m}\right\mathrm{missing}{]}$}$
(S8)

Furthermore, for small displacements in $x$, the flux change through the pick-
up loop can be treated as linear. The resultant current in the detection
circuit must then be $I_{induced}=\frac{d\Phi}{dx}\cdot
x\cdot\frac{1}{L_{total}}$ or
$\delta=\frac{d\Phi}{dx}\cdot\frac{1}{L_{total}}\hat{=}$\left\mathrm{missing}{[}\frac{A}{m}\right\mathrm{missing}{]}$$.

As noted before, $\delta$ contains the same geometric scaling as $\alpha$.
From the symmetry, we recognise that $\alpha=\frac{d\Phi}{dx}$
($\left[$\text{\,}\mathrm{W}\mathrm{b}\mathrm{/}\mathrm{m}$\right]=\left[$\text{\,}\mathrm{N}\mathrm{/}\mathrm{A}$\right]$).
Combining these results, we find

$\frac{\alpha\cdot\delta}{k}=\frac{\left(\frac{d\Phi}{dx}\right)^{2}}{L_{total}\cdot
m\omega^{2}}=\beta^{2}$ (S9)

This final results gives us a measurable result for $\beta^{2}$, or,
conversely, the proportionality constant $\frac{d\Phi}{dx}$ that equates our
measured flux to an absolute motion. The calibration measure is then found
from

$\frac{d\Phi}{dx}=\sqrt{L_{total}\cdot
m\omega^{2}\cdot\beta^{2}}=\sqrt{L_{total}\cdot
m\omega^{2}\cdot\frac{I_{induced}}{Q\cdot I_{crosstalk}}}$ (S10)

Noting that both $I_{induced}$ and $I_{crosstalk}$ are converted equally by
the SQUID, the measured voltage can be inserted for the current through the
detection circuit, which provides our sensitivity as

$\frac{d\Phi}{dx}=\sqrt{L_{total}\cdot m\omega^{2}\cdot\frac{\Delta
V_{drive}}{Q\cdot V_{crosstalk}}}$ (S11)

Here $\Phi$ is the flux through the pick-up loop. Since we detect this using
the SQUID, we need to convert this value to the flux through the SQUID, which
is done by dividing by the total inductance of the circuit and multiplying by
the mutual inductance of the SQUID input loop and the SQUID loop. As a final
step, this value can be related directly to the voltage measures made by
taking the SQUID gain value as measured.

$\frac{dV}{dx}=\frac{dV}{d\Phi_{SQ}}\cdot\frac{d\Phi_{SQ}}{dI}\cdot\frac{dI}{d\Phi}\cdot\frac{d\Phi}{dx}=\frac{dV}{d\Phi_{SQ}}\cdot
M_{in,SQ}\cdot\frac{1}{L_{total}}\cdot\frac{d\Phi}{dx}$ (S12)

The values for the inductances in
$L_{total}=L_{PU}+L_{TP}+L_{input}+L_{calibration}$ are:
$L_{PU}=$2.9e{-7}\text{\,}\mathrm{H}$$ the inductance of the pick-up loop,
$L_{TP}=$1e{-7}\text{\,}\mathrm{H}$$ the inductance of the
$10\text{\,}\mathrm{c}\mathrm{m}$ superconducting twisted-pair connecting the
SQUID input to the pick-up loop, $L_{input}=$4e{-7}\text{\,}\mathrm{H}$$ the
inductance of the input coil of the two-stage SQUID device, and
$L_{calibration}=$2e{-9}\text{\,}\mathrm{H}$$ the inductance of the small
calibration transformer coil. For the mutual inductance, the SQUID is
calibrated to a value of
$M_{in,SQ}^{-1}=$0.5\text{\,}\mu\mathrm{A}\mathrm{/}\Phi_{\mathrm{0}}$$. The
SQUID voltage gain is measured as
$dV/d\Phi_{0}=$0.43\text{\,}\mathrm{V}\mathrm{/}\Phi_{\mathrm{0}}$$. These
values, combined with measures of these ring-up’s, gives us a value of
$\frac{dV}{dx}=$0.16\text{\,}{\frac{V}{\mu m}}$$ for the
$27\text{\,}\mathrm{H}\mathrm{z}$ mode, with a relative error of 7%, which is
dominated by the error in the voltage calibration measurement.

##  Appendix D Comparison to basic Optomechanical Quantities

Given the vast previous work in the field of optomechanics, we aim to provide
a rough outline of how the work presented in this paper compares to quantities
in optomechanical experiments, in this appendix. For a more complete treatise
of optomechanics and specifically a comparison to magneto-mechanics, we refer
to Romero2021 .

We note that the resonance frequency of our levitated particle corresponds to
the frequency of the mechanical oscillator in an optomechanics experiment. At
present, our experiment lacks the equivalent of the optical cavity and the
light sent to the cavity. It could be added to the experiment, by changing the
readout of our experiment. Currently, we use a SQUID based read-out, that
relies on DC signals. We could read out the SQUID at higher frequencies (using
a GHz cavity). In that case the inductance of the SQUID, which depends on the
flux in the SQUID, would provide frequency tuning to a LC resonator, with
typical resonance frequencies of 2-20 GHz.

We adopted $\omega$ to be the frequency of the mechanical resonator, this in
contrast to optomechanics, in which $\Omega$ is nominally adopted for the
mechanical frequency, with $\omega$ reserved for the optical frequency. In the
rest of this appendix, we will refer to the optical frequency as used in
optomechanics as $\omega_{opt}$ to avoid confusion. In the context of
optomechanics, the optomechanical coupling, g, is given by the infinitesimal
change in wavelength as a function of the change in position of the mechanical
oscillator:

$g_{opt}=\frac{d\omega_{opt}}{dx}$ (S13)

This is an important quantity that provides a measure on how well the
mechanical and optical systems are coupled. Often the value is quoted in terms
of the zero-point-motion ($x_{zpm}$) of the resonator, as $g_{0}$. In the case
of magnetic levitation, a translation from $\frac{d\Phi}{dx}$ to
$\frac{d\omega_{opt}}{dx}$ can be made through the realization that the
inductance of the SQUID $L_{J}=\frac{\Phi_{0}}{2\pi I_{c}(\Phi)}$ depends on
the flux through the SQUID. This change in inductance changes the resonance
frequency of the LC resonator, $\omega_{LC}$, that could be used to readout
our SQUID in the future. In the same way that the resonance frequency of an
optical cavity depends on the position of a membrane, we also see that the
resonance frequency of the LC circuit that the SQUID is part of, depends on
the position of our levitated magnetic particle.

Combining this with equation S10, we find:

$g_{mag}=\frac{d\omega_{LC}}{d\Phi}\frac{d\Phi}{dx}=\frac{d\omega_{LC}}{d\Phi}\sqrt{L_{total}\cdot
m\omega^{2}\cdot\beta^{2}}$ (S14)

Evidently, this is the equivalent to $g_{opt}$ for our SQUID detected system.
Implicit in the term $L_{total}$ is the coupling loss from parasitic
inductances. Since the detection is not resonant, like it would be in an
optical cavity system, this remains a negligible loss. Expressing $g$ in terms
of the resonator zero-point-motion can be done similar to how it is often
derived in optomechanical systems, from equating the energy of the bosonic
system at zero occupation to the kinetic energy at zero point motion:

$x_{zpm}=\sqrt{\frac{\hbar}{2\cdot m_{eff}\cdot\omega_{m}}}$ (S15)

Where $m_{eff}$ is the effective mass of the mechanical oscillator and
$\omega_{m}$ is the frequency of the mechanical oscillator. In our case the
$m_{eff}$ is just the mass of the levitated particle and $\omega_{m}$ is the
eigenfrequency of the measurement. Filling in the mass and frequency used in
the gravity measurements, a zero point motion of
$x_{zpm}=$0.86\text{\,}\mathrm{f}\mathrm{m}$$ can be calculated. Combining
this with our measured sensitivity, we find $\frac{d\Phi}{dx}\cdot
x_{zpm}=$63\text{\,}\mathrm{n}\Phi_{\mathrm{0}}$$ as a zero point flux for our
system. For a slope, $\frac{d\omega_{LC}}{d\Phi}$, of a $\lambda/4$ resonator
terminated with a SQUID of
$1\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}\mathrm{/}\Phi_{\mathrm{0}}$, this
results in a $g_{0,mag}=g_{mag}\cdot
x_{zpm}=$63\text{\,}\mathrm{H}\mathrm{z}$$. We conclude, however, that it will
be a considerable challenge to make the Q factor of such a LC resonator high
enough to reach the sideband resolved limit.

##  Appendix E Correction and Conversion of the Data to Units of
$\text{\,}\mathrm{N}\mathrm{/}\sqrt{Hz}$

The data used in our gravitational experiments comes from the photodiode
detector which measures the passages of the masses on the wheel, and the SQUID
signal. Both these signals are filtered by our lock-in amplifiers. Since our
gravity experiments were performed at differing amplitudes in the mode, each
time trace had a different slope, as discussed in Supp. B. To account for
this, we have subtracted an individual exponential decay in the form of
$Ae^{t/\tau_{fit}}$ from each.

Figure S6: Ringdown subtraction as performed on the time trace measurement of
each wheel position. Shown here for a separation between particle and driving
mass at wheel phase zero of $48.1\text{\,}\mathrm{c}\mathrm{m}$ vertical, and
laterally displaced by $3.5\text{\,}\mathrm{c}\mathrm{m}$. Residual plotted in
the lower half, where we see the fluctuations as result of the drive up and
drive down effected by the mass-wheel, which is slightly detuned in frequency
with respect to the mode. We drive detuned since we wish to stay away from the
non-linear driving and frequency shift of the mode that happens at large
amplitude. Furthermore, detuned driving enables us to substract the absolute
amplitude.

After subtracting the ringdown, we apply a phase factor to the time traces.
This phase factor is determined based on the detuning of the resonator
frequency with respect to the lock-in center frequency. This change ensures
that the central peak of the resonator mode will fall fully in a single bin of
the Fourier transform we perform next, this ensures that the transfer function
of the resonator mode is fully symmetrical. Before performing the Fourier
transform, we also cut out a section of the time trace in which there is an
integer amount of phase cycles for the mass-passage signal, which ensures
smooth periodic boundary conditions for our FFT. This results in a spectrum
giving us the motion of the particle per root hertz, which can be converted to
RMS motion in a specific frequency band by integrating over that frequency
bandwidth.

Figure S7: Spectrum of the time trace converted to motional noise, with the
transfer function plotted overlaid. The orange vertical line indicates the
resonance frequency of the mode.

By then subtracting the transfer-function of the mechanical mode from the
particle signal, and applying our conversion factor to go from $\Phi_{0}$ to
displacement, and using a spring stiffness $k=m\omega^{2}$ and a mode
bandwidth of $df=Q/f$, we arrive at our force noise spectrum in
$\text{\,}\mathrm{N}\mathrm{/}\sqrt{Hz}$

Figure S8: Force noise spectrum. Spectrum of the time trace converted to force
noise, the final product of this procedure. The orange vertical line indicates
the resonance frequency of the mode.

The noise floor of this measurement corresponds to a mode temperature of
$3\text{\,}\mathrm{K}$, from $T_{mode}=kx_{RMS}^{2}/k_{B}$.