As an additional metric of sample diversity, we compute the $L_2$ distance of each generated image with the top-5 nearest neighbor training images.
To reduce computation requirement of searching over the raw pixel space, we use the CLIP ViT-B-16 image encoder [154] to transform images into the feature space before calculating the $L_2$ distance.
This metric measures the distance of generated samples with training images, as a proxy evaluation of diversity and memorization.

TopPR F1 [72].
TopPR is a set of reliable evaluation metrics with statistically consistent estimates of generated data and real data.
We use the F1 score, computed from the TopPR Precision and Recall as an additional metric to evaluate the general quality and diversity of generated images.

CLIP Score [66].
CLIP score measures the cosine similarity between the CLIP embedding of an image-text pair.
It is widely used as a metric to evaluate the fidelity and alignment of the generated images and the conditional text prompts [9].

Memorization Ratio [73].
We compute the memorization ratio as the percentage of generated images whose $L_2$ distances with their nearest neighbor training images are below a pre-defined threshold.
We compute the distances in the feature space of CLIP ViT-B-16 image encoder [154] due to the massive size and resolution of the training images and set the threshold as $0.12$.
Although there are several studies using the distance comparison between the first and second nearest neighbor as a reflection of memorization [155, 156], we found that this metric is not effective for large-scale datasets.

We compute the entropy metric within the latent space of the pre-trained VQ-VAE [58, 56].
Since LDMs (and DiT) learn the data distribution from the latent space of VQ-VAE, we can compute the sample entropy $\mathbb{E}[H(\mathbf{x})]$ using the generated and flatten latent vector $\mathbf{x} \in \mathbb{R}^{HW \times D}$ and the codebook $\mathcal{C} \in \mathbb{R}^{C \times D}$ of VQ-VAE, where $H$ and $W$ are the height and weights of the original latent vectors, $D$ indicates the dimension of the latent space, and $C$ denotes the number of embeddings of the codebook.
We compute the probability of each latent vector as $\softmax \left( \|  {\mathbf{x}} - \mathcal{C} \|_2^2 / \tau \right) $, where $\tau$ is a temperature parameter controlling the sharpness of the probability.  We compute entropy as:

\begin{equation}
\mathbb{E}[H(\mathbf{x})] = \frac{1}{NHW} \sum_i^N \sum_j^{HW} \sum_k^C \softmax \left( \|  \mathbf{x}_{(i,j)} - \mathcal{C} \|_2^2 / \tau \right)     % = - \sum_{i=1}^C \mathbf{p}_i \log \mathbf{p}_i,
\end{equation}

§ FULL RESULTS OF PRE-TRAINING EVALUATION

In this section, we present all results of our pre-training evaluation, over different diffusion models, including LDM-4, DiT-XL/2, and LCM-v1.5, and various types of condition corruption.

§.§ Quantitative Results

We present the full evaluation results of IN-1K class-conditional LDMs and CC3M text-conditional LDMs in <ref> and <ref> respectively.
All the results are computed from using a set of guidance scales.
For IN-1K LDMs, we use $s \in \{1.5, 1.75, 2.0, 2.25, 2.5, 3.0, 3.5, 4.0, 4.5, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 9.0, 10.0\}$.
For CC3M LDMs, we use $s \in \{1.5, 2.0, 2.5, 2.75, 3.0, 3.25, 3.5, 4.0, 5.0, 6.0, 7.0, 7.5, 8.0, 10.0\}$
For all the metrics, including FID, IS, Precision, Recall, sFID, TopPR F1, and CLIP score, we can all observe that slight condition corruption makes LDMs perform better, with improved image quality and diversity.
We also observe that, when there is condition corruption in the dataset, the memorization ratio based on $L_2$ distances actually decreases, in line with observations as in Gu et al. [155].

[TopPR F1]
[Top-$1\%$ RMD]
[Mem. Ratio]
[Avg $L_2$ Dist.]
[CLIP Score]
Qualitative evaluation results of 50K images generated by class-conditional LDMs pre-trained on ImageNet-1K with synthetic corruptions. The images are generated with various guidance scales using 1K class conditions and compared with 50K validation images of ImageNet-1K.

[TopPR F1]
[Top-$1\%$ RMD]
[Mem. Ratio]
[Avg $L_2$ Dist.]
[CLIP Score]
Qualitative evaluation results of 50K images generated by class-conditional LDMs pre-trained on CC3M with synthetic corruptions. The images are generated with various guidance scales using 5K text conditions from MS-COCO and compared with validation images of MS-COCO.

By default we use DPM scheduler for generating the images with 50 inference steps.
But we also study the generation of DDIM scheduler with 250 inference steps, as adopted in [9].
Due to the computation cost of running DDIM scheduler for 250 steps, we only study it with IN-1K LDM-4. The results are shown in <ref>.
One can observe the same trends from the metrics using DPM and DDIM, demonstrating our findings are scheduler agnostic.

Qualitative evaluation results of 50K images generated by class-conditional LDMs pre-trained on ImageNet-1K with synthetic corruptions. The images are generated with various guidance scales using 1K class conditions and compared with 50K validation images of ImageNet-1K.
We use DDIM scheduler with 250 inference steps for these results.

We then show the pre-training results of DiT-XL/2 and LCM-v1.5 in <ref>, where we primarily compute the FID, IS, Precision, and Recall.
For DiT-XL/2, we use $s \in \{1.5, 1.75, 2.0, 2.25, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5\}$.
For LCM-v1.5, we use $s \in \{1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 9.0, 10.0\}$.
Slight condition corruption also facilitates the performance by using the most suitable guidance scale.

Qualitative evaluation results of 50K images generated by class-conditional DiT-XL/2 pre-trained on IN-1K with synthetic corruptions. The images are generated with various guidance scales using 1K class conditions from IN-1K and compared with validation images of IN-1K.

Qualitative evaluation results of 50K images generated by text-conditional LCM-v1.5 pre-trained on CC3M with synthetic corruptions. The images are generated with various guidance scales using 5K text conditions from MS-COCO and compared with validation images of MS-COCO.

Finally, we present the results of LDMs pre-trained on CC3M with LLM corruption and IN-1K with asymmetric corruption, where similar observations still hold.

Qualitative evaluation results of 50K images generated by text-conditional LDMs pre-trained on CC3M with LLM re-writing corruptions. The images are generated with various guidance scales using 5K text conditions from MS-COCO and compared with validation images of MS-COCO.

Qualitative evaluation results of 50K images generated by class-conditional LDMs pre-trained on IN-1K with asymmetric corruptions. The images are generated with various guidance scales using 1K class conditions from IN-1K and compared with validation images of IN-1K.

§.§ Qualitative Results

We present more visualization results of class-conditional LDM-4 in <ref>, class-conditional DiT-XL/2 in <ref>, text-conditional LDM-4 in <ref>, and text-conditional LCM-v1.5 in <ref>.
One can observe that DMs pre-trained with slight condition corruption in general more visually appealing images.

Visualization of LDMs IN-1K pre-training results.

Visualization of DiT-XL/2 IN-1K pre-training results.

Visualization of LDMs CC3M pre-training results.

Visualization of LCM CC3M pre-training results.

§ FULL RESULTS OF DOWNSTREAM PERSONALIZATION EVALUATION

We present complete results of downstream personalization here.

§.§ Quantitative Results

We show the results of ControlNet IN-1K LDM-4 in <ref>, T2I-Adapter IN-1K LDM-4 in <ref>, ControlNet CC3M LDM-4 in <ref>, and T2I-Adapter CC3M LDM-4 in <ref>.
For all personalization experiments, we compute the results for both Canny and SAM spatial controls.
Guidance scales of $\{1.25, 1.5, 2.0, 2.25, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0\}$ and $\{2.0, 3.0, 4.0, 5.0, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 10.0\}$ are used for IN-1K models and CC3M models, respectively, for all experiments here.

From the results, one can observe that models pre-trained with slight condition corruption also present the best performance in downstream personalization tasks.

[Canny, FID]
[Canny, IS]
[Canny, Precision]
[Canny, Recall]

[SAM, FID]
[SAM, IS]
[SAM, Precision]
[SAM, Recall]

Qualitative evaluation results of 5K images generated by class-conditional LDMs pre-trained on ImageNet-1K and personalized on ImageNet-100 using ControlNet. We personalized the models with different control styles, including canny ((a) - (d)), segmentation mask from SAM ((e) - (h)), and lineart ((i) - (l)). The images are generated using 100 class conditions with various guidance scales, compared with 5K validation images of ImageNet-100.

[Canny, FID]
[Canny, IS]
[Canny, Precision]
[Canny, Recall]

[SAM, FID]
[SAM, IS]
[SAM, Precision]
[SAM, Recall]

Qualitative evaluation results of 5K images generated by class-conditional LDMs pre-trained on ImageNet-1K and personalized on ImageNet-100 using T2I-Adapter. We personalized the models with different control styles, including canny ((a) - (d)), segmentation mask from SAM ((e) - (h)), and lineart ((i) - (l)). The images are generated using 100 class conditions with various guidance scales, compared with 5K validation images of ImageNet-100.

[Canny, FID]
[Canny, IS]
[Canny, Precision]
[Canny, Recall]

[SAM, FID]
[SAM, IS]
[SAM, Precision]
[SAM, Recall]

Qualitative evaluation results of 5K images generated by text-conditional LDMs pre-trained on CC3M and personalized on ImageNet-100 using ControlNet. We personalized the models with different control styles, including canny ((a) - (d))and segmentation mask from SAM ((e) - (h)). The images are generated using text captions annotated from BLIP with various guidance scales, compared with 5K validation images of ImageNet-100.

[Canny, FID]
[Canny, IS]
[Canny, Precision]
[Canny, Recall]

[SAM, FID]
[SAM, IS]
[SAM, Precision]
[SAM, Recall]

Qualitative evaluation results of 5K images generated by text-conditional LDMs pre-trained on CC3M and personalized on ImageNet-100 using T2I-Adapter. We personalized the models with different control styles, including canny ((a) - (d)) and segmentation mask from SAM ((e) - (h)). The images are generated using text captions annotated from BLIP with various guidance scales, compared with 5K validation images of ImageNet-100.

§.§ Qualitative Results

We present the qualitative comparison of ControlNet personalization results here.
Since T2I-Adapter personalization results are similar but visually worse (quantitatively worse too), we skip their results.
The visualizations of ControlNet IN-1K LDM-4 with Canny and SAM conditions are shown in <ref> and <ref>, respectively.
The visualizations of ControlNet CC3M LDM-4 with Canny and SAM conditions are shown in <ref> and <ref>, respectively.
Similarly, models pre-trained with slight condition corruption present the best image quality.

Visualization of LDMs IN-1K ControlNet Canny personalization results.

Visualization of LDMs IN-1K ControlNet SAM personalization results

Visualization of LDMs CC3M ControlNet Canny personalization results.

Visualization of LDMs CC3M ControlNet SAM personalization results

§ FULL RESULTS OF CONDITIONAL EMBEDDING PERTURBATION

§.§ Qualitative Results

More visualizations of CEP, compared with clean and IP pre-trained are shown here.
We present the more results of IN-1K LDM-4 and DiT-XL/2 in <ref> and <ref>, respectively.
We also present more results of CC3M LDM-4 and LCM-v1.5 in <ref> and <ref>, respectively.
CEP generally helps DMs generate more visually appealing and realistic images.
We also show the more personalization visualization in <ref> and <ref>.

[IN-1K LDM-4]
[IN-1K DiT-XL/2]
Visualization of CEP on IN-1K pre-trained LDM-4 and DiT-XL/2

[CC3M LDM-4]
[CC3M LCM-v1.5]
Visualization of CEP on CC3M pre-trained LDM-4 and LCM-v1.5

Visualization of CEP on ControlNet adapted IN-1K pre-trained LDM-4

Visualization of CEP on ControlNet adapted CC3M pre-trained LDM-4

§.§ Ablation Study

In <ref>, we compute the $L_2$ distance of perturbed condition embeddings and the clean ones, as a measurement for the corruption levels (of CEP).
Here, we elaborate how we compute the $L_2$ distances.
For fixed corruption, we calculate the distances as:

\begin{equation}
\frac{1}{N} \sum_{i=1}^N \|\mathbf{c}_{\theta^*}(y^c_i) - \mathbf{c}_{\theta^*}(y_i) \|^2_2,
\end{equation}

where $\theta^*$ is learned from clean data.
For CEP, we directly calculate the $L_2$ norm of sampled noise:

\begin{equation}
\sum_{i=1}^N \| \boldsymbol{\sigma}_i \|^2_2
\end{equation}

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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

# Membrane-in-the-middle optomechanics with a soft-clamped membrane at
milliKelvin temperatures

Eric Planz  Xiang Xi  Thibault Capelle  Eric C. Langman  Albert Schliesser
<EMAIL_ADDRESS>Niels Bohr Institute, University of Copenhagen,
2100 Copenhagen, Denmark  Center for Hybrid Quantum Networks (Hy-Q), Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

###### Abstract

Soft-clamped silicon nitride membrane resonators reach coherence times $\tau$
in excess of 100 ms at milliKelvin bath temperatures. However, harnessing
strong optomechanical coupling in dry dilution refrigerators remains
challenging due to vibration issues and heating by optical absorption. Here,
we propose to address these issues with an actuator-free optical cavity and
mechanical resonator design, in which the cavity is mounted on a simple
vibration-isolation platform. We observe dynamical backaction when the cavity
is driven with a free-space optical beam stabilized close to the red sideband
using a two-beam locking scheme. Finally, we characterize the effect of
absorption heating on the coherence time, and find a scaling with the
intracavity power $P$ as $\tau\propto P^{-\left(0.34\pm 0.04\right)}$

## Introduction

Cavity optomechanics has emerged as a dynamic field over the past few decades,
fuelled by progress in fabrication of integrated, low-loss optical and
mechanical resonators [1]. The ability to couple light to mechanical motion
through radiation pressure effects in optomechanical systems has led to
advances both in fundamental physics and technological applications. Of
particular significance are membrane-in-the-middle (MIM) systems using a
partially reflective membrane resonator inside an optical cavity [2]. They
have been utilized in a wide variety of experiments, ranging from
investigations of the tenets of continuous quantum measurement [3, 4, 5, 6,
7], to topological [8] and parametric [9] energy transfer, and for
applications in quantum information processing [10], gravitational wave
detection [11], and force sensing [12, 13].

The operation of MIM systems in a regime, where the motion of the mechanical
oscillator is dominated by quantum uncertainties requires the demonstration of
optomechanical coupling rates exceeding the thermal decoherence rate. Towards
this goal, a large effort has been made to develop highly coherent mechanical
resonators. Among those, soft-clamped membrane resonators [14] that comprise a
phononic crystal pattern with an isolated defect, have allowed reaching the
quantum regime at moderate ($T\sim 10$K) cryogenic temperatures [5] and even
approached the quantum regime at room temperature [15].

Alternatively to MIM systems, electromechanical coupling has enabled the
operation of such soft-clamped membrane in the quantum regime in a dilution
refrigerator. This has been realized by depositing an aluminum pad on the
membrane, and placing it in the vicinity of the capacitive part of a microwave
cavity [16]. However, optomechanical operation in such dilution refrigerators
imposes several challenges associated with (i) maintaining stability within
the high-finesse cavity, and (ii) optical absorption heating of the membrane
resonator at high intracavity fields.

Challenge (i) involves aligning to and locking the high-finesse cavity.
Various approaches have been explored for alignment to this end. These include
solutions in which misaligned, fiber-coupled cavities at room temperature
achieve high coupling efficiencies at low temperatures [17]. To realize
optical lock, actuators in fiber-coupled cavities [18] and free-space coupling
to optomechanical cavities [10] have been used. However, dry dilution
refrigerators, often used in these procedures, tend to possess significant
vibrations due to the use of a pulse tube system to maintain the Helium 4 at
sufficiently low temperature [19]. These vibrations often result in large
excursions that complicate locking to high-finesse cavities.

The second challenge (ii) is the optical absorption induced heating of the
membrane resonator at high intracavity fields. This phenomenon, which has been
observed in numerous optomechanical and electro-optic experiments [20, 21], is
problematic as it can lead to higher mechanical bath temperatures. Studies
have shown that patterned SiN membranes demonstrate low thermal conductivity
[22], and absorption heating effects can become significant at milliKelvin
temperatures, especially within the near-infrared regime [11]. Plain membranes
with sub-millimeter dimensions have been employed in (wet) dilution
refrigerators before, in which no significant heating was observed when driven
with laser radiation at $\sim 1$-µm wavelenghth [23, 10]. Here, we investigate
soft-clamped membranes resonators, which can offer higher quality factors
($Q>10^{9}$) and coherence times $\tau>140\text{ ms}$ [16]. Their larger
dimensions ($\sim$cm) and patterned structure is expected to lead to larger
heating.

In response to these issues, we here propose a design for a sideband-resolved
optomechanical assembly. It offers a method for effective coupling and locking
a laser to the cavity, within a dilution refrigerator. Figure 1 illustrates
the soft-clamped membrane design with a mechanical resonance of interest at
1.32 MHz. The chip design incorporates additional coupling to a microwave
cavity, although the details of this feature are outside the scope of this
manuscript. We also investigate the absorption heating effect of the membrane
from $805\text{ nm}$ wavelength laser light.

Figure 1: (a) Picture of the ’Lotus’ membrane design used in the noise
thermometry experiments. The defect diameter is designed to be 230 µm,
corresponding to clipping losses low enough to achieve sideband-resolved
optomechanics. (b) Room-temperature spectrum of the membrane showing a bandgap
between the dashed lines and the defect mode at $1.32\text{ MHz}$ that we
address. (c) Bird’s view of the simulated displacement of the mechanical mode
localized at the defect in the phononic crystal patterned into a silicon
nitride (SiN) membrane. False-color indicates displacement amplitude from
negative (blue) over small (gray) to positive (red).

## Mechanical Design

The cavity design shown in Fig. 2(a) employs a plano-convex, over-coupled
Fabry-Pérot cavity with highly reflective mirrors. The wavelength-dependent
reflectivities of these mirrors allows tuning both the finesse of the cavity
and the over-coupling ratio by adjusting the wavelength. These features enable
us to achieve a cavity finesse over 30,000, a cavity linewidth below 300 kHz,
and substantial over-coupling greater than 95%. The cavity assembly is made of
oxygen-free high-conductivity copper. We use short stainless steel screws to
hold its parts together, reducing fluctuations within the cavity due to
differences in thermal contraction. In the assembly, the membrane frame gets
to lie parallel to the flat cavity mirror, separated only by a 500-µm-thick
silicon spacer. With an equally thick membrane chip, the membrane is located
at about 1-mm distance from the flat mirror’s surface [24] The total cavity
length is $\sim 24\text{ mm}$. With the convex mirror’s radius of curvature
around $25\text{ mm}$, the waist of the optical mode at the position of the
membrane is $\sim 43\text{ \textmu m}$.

The light is coupled to the cavity by aiming a free-space laser beam through
windows in the cryostat, onto the more transmissive cavity mirror. Movements
of the cavity assembly in directions orthogonal to the longitudinal cavity
axis can induce rather dramatic fluctuations in the intracavity field. On the
other end, movements of the cavity in the axial direction can lead to a motion
of the mechanical resonator not limited by thermal noise. To address those two
issues, the cavity assembly is affixed to a home-built, simple vibration
isolation platform shown in Fig. 2(b) and (c). It consists of a massive
($1.9\text{ kg}$) copper plate that is suspended from the mixing chamber plate
of a dry dilution refrigerator (LD250 by Bluefors), via thin copper sheets
measuring 24 cm x 5.1 cm x 0.6 mm. This construction is ‘soft’ for
oscillations along the cavity axis, allowing the platform to swing at a low
eigenfrequency of 2.2 Hz, with the aim of mitigate MHz mechanical noise due to
non-thermal, external vibrations. For oscillations orthogonal to the cavity
axis, it is much stiffer, with the first eigenfrequency appearing at 164 Hz
(vertical) and 183 Hz (horizontal). Here, the goal is to avoid large, low
frequency excursions that would entail pointing noise of the cavity axis. This
platform allowed us to lock the laser to cavities with finesse up to $\sim
31,000$ (albeit without an intracavity membrane), with the pulse tube active.

Figure 2: (a) Schematic of the cavity assembly that shows the cavity mirrors
(blue), the SiN membrane chip (red) and a silicon spacer chip (yellow), held
in place by the copper parts of the assembly (gray). In particular, two copper
“mirror holders” press the two mirrors against the bulk cavity spacer using
nitrile O-rings. The curved mirror’s holder can be moved orthogonal to the
cavity axis, which allows centering the cavity field on the membrane defect.
(b) Displacement pattern of the lowest-frequency mode along the cavity axis
(y-direction) at 2.2 Hz. (c) Displacement pattern of the lowest-frequency mode
orthogonal to the cavity axis (z-direction) at 164 Hz.

## Resonator Design

This work utilizes a variant of Lotus-class soft-clamped membranes, which have
demonstrated quality factors exceeding 1 billion in electromechanical
experiments [16]. Our experiment specifically employs a phononic dimer
membrane containing two coupled defects leading to a pair of hybridized
mechanical modes, one symmetric and the other antisymmetric [13]. The released
membrane extends on a rectangle of size $5.4\text{ mm}\times 4.8\text{ mm}$.
In our case, only one mechanical mode, at a frequency of $\Omega_{m}/2\pi\sim
1.32$MHz will be considered. Fig. 3(a) shows a zoom-in of the mechanical
defect of a ’Lotus’ membrane described in Fig. 1.

Figure 3:  (a) Zoom-in to the geometry of a ’Lotus’ defect. (b) Microscope
picture of one of the 200 nm thick clipping test membranes used in the test
assemblies. The hexagonal pattern of holes simulates the hole structure in
soft-clamped membranes. (c) Scan of the in-coupling laser beam along x-axis
with one cavity mirror removed. The back-reflected power shows interference
between light being reflected off the membrane and the cavity mirror behind.
(d) Scan of the in-coupling laser beam after minimizing the tilt. The scan
shows a clear feature stemming from the hole structure around the defect
between 7.8 and 8.1 mm. These graphs are used to align the in-coupling laser
beam to the center of the defect. (e) Optomechanical coupling at various
wavelengths, as determined by measuring the frequencies of 24 subsequent
optical resonances. (f) Cavity loss seen in optomechanical cavity assemblies
with various defect sizes. The data point at $\infty$ has been performed using
a plane membrane, i.e., without hole structure.

If the cavity mode diameter at the membrane position exceeds the defect size
of a patterned membrane, additional optical losses occur. They are mainly due
to clipping along the edges of the silicon nitride (SiN) material and the
phase difference between the light field that travels through the material and
that outside the defect. This results in altering the cavity wavefront and
promotes coupling into higher-order modes. Minimizing these additional cavity
losses is crucial to attain sideband resolution, a regime characterized by a
mechanical frequency much larger than the optical loss rate, which is a
prerequisite for the sideband cooling to the quantum regime of a mechanical
resonator. To gain insights into this cavity loss effect, we carried out a
series of tests using optomechanical cavity assemblies with varied defect
sizes. Figure 3(b) shows the hexagonal hole pattern etched into 200 nm thick
SiN membranes, which we used in these tests to simulate a soft-clamped
membrane defect.

In the assembly procedure for these tests, we first clamp the membrane and
flat cavity mirror together. Then, we scan a laser beam across the membrane,
and measure its back-reflected power. These scans typically show interference
between the light reflected directly from the membrane and off the flat cavity
mirror behind the membrane (Fig. 3(c)). With scans in the two directions of
the membrane plane, we can calculate the tilt between the membrane and cavity
mirror from the interference fringe. In the example shown in Fig. 3(c), this
leads to an angle of approx. $1\text{ mrad}$ for one interference fringe
across 0.8 mm with a 830 nm laser beam. For all test assemblies, the angle is
kept $<$1 mrad to ensure tilt is not a dominant source of cavity loss.

The same scan method is then used to find the center of the membrane defect
with the laser beam. In the example shown in Fig. 3(d), it occurs at a
position around 7.93 mm. Once we have found the center position in the
membrane plane, we position and fix the curved mirror in the assembly, forming
the membrane-in-the-middle cavity.

The longitudinal position of the membrane with respect to the standing wave of
the optical cavity determines the optomechanical coupling, among other
factors. To identify positions of high coupling, we measure a series of
subsequent fundamental cavity resonance frequencies. The position-dependent
perturbation of the intracavity field through the dielectric SiN material
causes a deviation of the cavity spectrum from equidistant modes separated by
a constant free spectral range (FSR). This deviation
$\Delta\omega_{\text{FSR}}$ can be converted into a coupling point using the
model provided in [25]. It shows a periodicity of $2kz=2\pi Nz/L$, where $N$
is the mode number, $z$ the distance to the closest cavity mirror (here 1 mm),
and $L$ is the cavity length (here 24 mm). We therefore record the resonance
frequency of 24 subsequent cavity modes, in order to observe all coupling
points and obtain the graph in Fig. 3(e). Then we perform cavity loss
measurements via cavity ringdowns at various coupling points, yielding the
data presented in Fig. 3(f). The findings demonstrate that the defect size
does impact the cavity loss. We find that too small defect sizes lead to
significant enough cavity losses that sideband resolution becomes
unattainable. Based on these results, we use Lotus-class defects with an
innermost diameter of 230 µm for relevant optical cavities.

## Noise Thermometry

For the optomechanical measurements at milliKelvin temperatures, we clamp the
cavity assembly to the vibration isolation platform described in Fig. 2(b). We
shine laser light towards the cavity, through the cryostat’s windows, from
fiber couplers that are mounted on a breadboard, which is itself clamped to
the outer shield of the dilution refrigerator. Figure 4 shows a simplified
version of the optical setup, which uses two orthogonally polarized beams
derived from the same laser using two acousto-optic modulators (AOMs) [10]. In
our experiments we use a widely tuneable, low-noise Ti:sapphire laser SolsTiS
from MSquared at a wavelength around 800 nm. The first, weak “lock” beam is
parked at resonance with the cavity and its reflection is used to derive a
Pound-Drever-Hall (PDH) error signal to lock the laser frequency to the
cavity. The second, stronger, red-detuned, “science” beam probes (and cools)
the mechanical resonance. Its detuning from the cavity resonance is set by the
frequencies at which the two free-space AOMs from CSRayzer are driven. We
record mechanical noise spectra by direct detection of the science beam using
avalanche photodiodes of type APD410A2 from Thorlabs.

Figure 4: Experimental Setup. Optical lock and science beams are derived from
the same laser, frequency-offset by two acousto-optic modulators. The two
beams are prepared in orthogonal polarization with linear polarizers (Pol.),
and subsequently combined using a polarizing beam splitter (PBS). In a
combination with a set of waveplates ($\lambda/4$,$\lambda/2$), the same PBS
separates the beams reflected from the cavity again. Non-polarizing beam
splitters (BS) reroute fractions of the lock and science input powers towards
photodetectors PD4 and PD2, respectively. Their signals are used to stabilize
the two beams’ powers by feeding back to an electrooptic amplitude modulator
(EOM/AM) and to the power of the radio-frequency signal driving the AOMs,
respectively. The laser frequency is locked to the cavity via an error signal
generated by demodulating the photocurrent detected on photodetector PD3, with
feedback to a piezoelectric actuator that adjusts the frequency of the laser.
The optomechanical cavity is mounted on the vibration isolation platform
inside the dilution refrigerator together with two in- and out-coupling
lenses. The photodetector PD1 records the intensity of the detuned science
beam. The photocurrent spectra, as recorded with an R&S FSW26 spectrum
analyser, therefore also contain the thermomechanical noise of the membrane
resonances. In addition, the cavity mode can be monitored in transmission via
a camera and PD5.

We initially measure the thermal noise spectrum at room temperature, revealing
the fundamental mode of the soft-clamped membrane at approximately 1.32 MHz
(see Fig. 1(b)). As the setup cools, thermal contraction of the silicon frame
triggers an approx. 2% down-shift in mechanical frequency. At milliKelvin
temperatures we observe the same resonance close to 1.30 MHz. As we increase
the science beam power, we observe dynamical backaction damping as a
broadening mechanical linewidth [1], as shown in Fig. 5(a).

For a systematic laser cooling series, we maintain the dilution refrigerator
temperature at 20 mK and perform measurements at the science beam detunings
[-1.0, -1.5, -2.0] MHz. The science beam’s input power is swept up to
$10\,\mu$W, at an estimated mode-matching efficiency of $0{.}8$. A fit to the
PDH error signal gives a cavity linewidth $\kappa/2\pi=\left(2.0\pm
0.2\right)\text{ MHz}$, corresponding to a finesse of $\left(3.1\pm
0.3\right)\times 10^{3}$.

Figure 5:  Noise Thermometry. (a) Mechanical spectrum for different optical
input powers showing cooling of the mechanical resonance. For example,
$P_{\text{in}}=5\text{ \textmu W}$ at $-1.5\text{ MHz}$ detuning corresponds
to an intracavity photon number of $2.2\times 10^{6}$. (b) Shift in the
mechanical frequency at various optical powers. (c) Noise spectrum background
scaling linearly, thereby indicating a shot-noise limited measurement. (c)
Mechanical linewidth at various powers showing dynamical backaction. (d)
Phonon occupation at cooling power sweeps indicating power-law scaling.

Figures 5(b-d) show the results of Lorentzian fits to these spectra. We note
resonance frequency shifts of different signs for distinct detunings,
attributed to the anticipated optical spring effect. Moreover, the linewidth
increase directly relates to the dynamical backaction effect, from which we
infer a vacuum optomechanical coupling rate [1] of $g_{0}/2\pi\approx
1.2\text{ Hz}$. For comparison, the maximum achievable coupling rate in a
membrane-in-the-middle configuration can be approximated as
$g_{0}^{\text{max}}\approx 2(\omega_{\text{c}}/{L})|r|x_{\text{zpf}}\xi$,
where $\omega_{\text{c}}/2\pi$ is the cavity resonance frequency, $L$ is the
cavity length, $r$ the optical field reflectivity of the membrane,
$x_{\text{zpf}}$ is the zero-point fluctuation, and $\xi$ is the mode overlap
between membrane displacement and optical field [15]. With the $50\text{ nm}$
thick membrane and $24\text{ mm}$ long optical cavity, we anticipate
$g_{0}^{\text{max}}/2\pi\approx 8\text{ Hz}$ at perfect mode-overlap. We
attribute the discrepancy with the measured $g_{0}$ to the unoptimized
positioning of the membrane along the cavity axis and a potentially imperfect
transverse overlap ($\xi<1$).

Finally, in Fig. 5(e) we plot the area $A$ of the mechanical peak over the
intracavity power $P$ squared. In a direct-detection measurement, as in this
setup, $A/P^{2}$ is proportional to the steady-state phonon occupation number
$\bar{n}_{\text{f}}$. In our experiment’s regime, quantum backacktion is
negligible and the occupation is approximately given by

$\displaystyle\bar{n}_{\text{f}}=\frac{\Gamma_{\text{m}}\bar{n}_{\text{th}}}{\Gamma_{\text{opt}}+\Gamma_{\text{m}}}\approx\frac{\Gamma_{\text{m}}\bar{n}_{\text{th}}}{\Gamma_{\text{opt}}}\text{
,}$ (1)

where $\Gamma_{\text{m}}$ is the mechanical damping rate,
$\bar{n}_{\text{th}}$ occupation of the mechanical bath, and
$\Gamma_{\text{opt}}$ is the optomechanical damping rate [1]. As
$\Gamma_{\text{opt}}\propto P$ [1], we would expect a power law
$A/P^{2}\propto\bar{n}_{\mathrm{f}}\propto P^{s}\propto n_{\mathrm{cav}}^{s}$
with slope $s=-1$ to emerge in the log-log-scale in Fig. 5(e)—were there no
additional heating effects. However, we obtain a slope of $s=-0.67\pm 0.04$.
Combined with eq. (1), this suggests a power scaling of the decoherence rate
$\Gamma_{\text{m}}\bar{n}_{\text{th}}\propto P^{\alpha}$ with
$\alpha=(1-s)=0.33\pm 0.04$, and consequently the (heating-limited) coherence
time $\tau\propto P^{-\alpha}$. This is in rough agreement with published
results of the optical absorption heating and mechanical linewidth broadening
of soft-clamped membranes. Indeed, reference [11] (Fig. S2) suggests that the
mechanical temperature scales approximately $\propto P^{0.33}$, whereas the
articles [16] and [26] find a scaling of the damping with the mechanical bath
temperature similar to $\Gamma_{\text{m}}\propto n_{\mathrm{th}}^{0{.}66}$.
Taken together, a crude approximation based on these reports would then
suggest that
$\Gamma_{\text{m}}\cdot\bar{n}_{\text{th}}\propto(P^{0.33})^{0.66}\cdot
P^{0.33}$, that is $\alpha_{\text{lit.}}\approx 0.55$.

## Conclusion

We have presented a detailed experimental procedure to construct and analyze
an optomechanical setup with a Lotus-class soft-clamped membrane within an
over-coupled Fabry-Pérot cavity. The setup operates in a dry dilution
refrigeration thanks to a stable mechanical design and a vibration isolation
platform. We have explored how the finite size of the membrane defect can
introduce additional cavity losses. We have developed an efficient procedure
of assembling MIM cavities and derived design restrictions on soft-clamped
membranes, here being a defect size $>200\mu m$.

The thermal noise spectra observed during the noise thermometry experiments
shows clear dynamical backaction cooling. From the noise thermometry
measurement, we derived the power-law scaling of the decoherence rate with the
optical power with an exponent of $\alpha=0.33\pm 0.04$, in reasonable
agreement with existing literature. This suggests that soft-clamped membranes
are subject to optical absorption heating and mechanical linewidth broadening.

Our findings provide insights for researchers in quantum optomechanics to
optimize their experimental procedures. The techniques and approaches
developed here could facilitate the realization of optomechanical systems that
operate in the quantum regime. In the future, coupling the second defect of
the phononic dimer to a microwave resonator could enable quantum transduction
experiments, with long-term intermediate storage of the quantum state, opening
new possibilities in quantum communication and computation. Other applications
include compact setups for manipulating optical quantum noise [27, 11], as
well as the search for unconventinoal decoherence [28] and dark matter [29].

## Acknowledgements

This work was supported by the European Research Council project PHOQS (grant
no. 101002179), the Novo Nordisk Foundation (grant no. NNF20OC0061866) and the
Danish National Research Foundation (Center of Excellence “Hy-Q”). This
project has furthermore received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreements No. 801199 and 101107341.

## References

  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Reviews of Modern Physics 86, 1391 (2014).
  * Thompson  _et al._ [2008] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane, Nature 452, 72 (2008).
  * Purdy  _et al._ [2013] T. P. Purdy, R. W. Peterson, and C. A. Regal, Observation of Radiation Pressure Shot Noise on a Macroscopic Object, Science 339, 801 (2013).
  * Møller  _et al._ [2017] C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, Quantum back-action-evading measurement of motion in a negative mass reference frame, Nature 547, 191 (2017).
  * Rossi  _et al._ [2018] M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, Measurement-based quantum control of mechanical motion, Nature 563, 53 (2018).
  * Mason  _et al._ [2019] D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, Continuous force and displacement measurement below the standard quantum limit, Nature Physics 15, 745 (2019).
  * Rossi  _et al._ [2019] M. Rossi, D. Mason, J. Chen, and A. Schliesser, Observing and Verifying the Quantum Trajectory of a Mechanical Resonator, Physical Review Letters 123, 163601 (2019).
  * Xu  _et al._ [2016] H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topological energy transfer in an optomechanical system with exceptional points, Nature 537, 80 (2016).
  * Hälg  _et al._ [2022] D. Hälg, T. Gisler, E. C. Langman, S. Misra, O. Zilberberg, A. Schliesser, C. L. Degen, and A. Eichler, Strong parametric coupling between two ultracoherent membrane modes, Physical Review Letters 128, 094301 (2022).
  * Brubaker  _et al._ [2022] B. Brubaker, J. Kindem, M. Urmey, S. Mittal, R. Delaney, P. Burns, M. Vissers, K. Lehnert, and C. Regal, Optomechanical ground-state cooling in a continuous and efficient electro-optic transducer, Physical Review X 12, 021062 (2022).
  * Page  _et al._ [2021] M. A. Page, M. Goryachev, H. Miao, Y. Chen, Y. Ma, D. Mason, M. Rossi, C. D. Blair, L. Ju, D. G. Blair, A. Schliesser, M. E. Tobar, and C. Zhao, Gravitational wave detectors with broadband high frequency sensitivity, Communications Physics 4, 10.1038/s42005-021-00526-2 (2021).
  * Fischer  _et al._ [2019] R. Fischer, D. P. McNally, C. Reetz, G. G. T. Assumpção, T. Knief, Y. Lin, and C. A. Regal, Spin detection with a micromechanical trampoline: towards magnetic resonance microscopy harnessing cavity optomechanics, New Journal of Physics 21, 043049 (2019).
  * Catalini  _et al._ [2020] L. Catalini, Y. Tsaturyan, and A. Schliesser, Soft-clamped phononic dimers for mechanical sensing and transduction, Physical Review Applied 14, 014041 (2020).
  * Tsaturyan  _et al._ [2017] Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, Ultracoherent nanomechanical resonators via soft clamping and dissipation dilution, Nature Nanotechnology 12, 776 (2017).
  * Saarinen  _et al._ [2023] S. A. Saarinen, N. Kralj, E. C. Langman, Y. Tsaturyan, and A. Schliesser, Laser cooling a membrane-in-the-middle system close to the quantum ground state from room temperature, Optica 10, 364 (2023).
  * Seis  _et al._ [2022] Y. Seis, T. Capelle, E. Langman, S. Saarinen, E. Planz, and A. Schliesser, Ground state cooling of an ultracoherent electromechanical system, Nature Communications 13, 10.1038/s41467-022-29115-9 (2022).
  * Fedoseev  _et al._ [2022] V. Fedoseev, M. Fisicaro, H. van der Meer, W. Löffler, and D. Bouwmeester, Realignment-free cryogenic macroscopic optical cavity coupled to an optical fiber, Review of Scientific Instruments 93, 013103 (2022).
  * Doeleman  _et al._ [2023] H. M. Doeleman, T. Schatteburg, R. Benevides, S. Vollenweider, D. Macri, and Y. Chu, Brillouin optomechanics in the quantum ground state (2023), arXiv:2303.04677 [quant-ph] .
  * Olivieri  _et al._ [2017] E. Olivieri, J. Billard, M. D. Jesus, A. Juillard, and A. Leder, Vibrations on pulse tube based dry dilution refrigerators for low noise measurements, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 858, 73 (2017).
  * Mirhosseini  _et al._ [2020] M. Mirhosseini, A. Sipahigil, M. Kalaee, and O. Painter, Superconducting qubit to optical photon transduction, Nature 588, 599 (2020).
  * Hease  _et al._ [2020] W. Hease, A. Rueda, R. Sahu, M. Wulf, G. Arnold, H. G. Schwefel, and J. M. Fink, Bidirectional electro-optic wavelength conversion in the quantum ground state, PRX Quantum 1, 020315 (2020).
  * Leivo and Pekola [1998] M. M. Leivo and J. P. Pekola, Thermal characteristics of silicon nitride membranes at sub-kelvin temperatures, Applied Physics Letters 72, 1305 (1998).
  * Peterson  _et al._ [2016] R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P.-L. Yu, K. W. Lehnert, and C. A. Regal, Laser Cooling of a Micromechanical Membrane to the Quantum Backaction Limit, Physical Review Letters 116, 063601 (2016).
  * Nielsen  _et al._ [2016] W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, Multimode optomechanical system in the quantum regime, Proceedings of the National Academy of Sciences 114, 62 (2016).
  * Dumont  _et al._ [2019] V. Dumont, S. Bernard, C. Reinhardt, A. Kato, M. Ruf, and J. C. Sankey, Flexure-tuned membrane-at-the-edge optomechanical system, Optics Express 27, 25731 (2019).
  * Zhou  _et al._ [2019] X. Zhou, D. Cattiaux, R. R. Gazizulin, A. Luck, O. Maillet, T. Crozes, J.-F. Motte, O. Bourgeois, A. Fefferman, and E. Collin, On-chip thermometry for microwave optomechanics implemented in a nuclear demagnetization cryostat, Physical Review Applied 12, 044066 (2019).
  * Qin  _et al._ [2014] J. Qin, C. Zhao, Y. Ma, X. Chen, L. Ju, and D. G. Blair, Classical demonstration of frequency-dependent noise ellipse rotation using optomechanically induced transparency, Physical Review A 89, 041802 (2014).
  * Nimmrichter  _et al._ [2014] S. Nimmrichter, K. Hornberger, and K. Hammerer, Optomechanical Sensing of Spontaneous Wave-Function Collapse, Physical Review Letters 113, 020405 (2014).
  * Carney  _et al._ [2021] D. Carney, G. Krnjaic, D. C. Moore, C. A. Regal, G. Afek, S. Bhave, B. Brubaker, T. Corbitt, J. Cripe, N. Crisosto, A. Geraci, S. Ghosh, J. G. E. Harris, A. Hook, E. W. Kolb, J. Kunjummen, R. F. Lang, T. Li, T. Lin, Z. Liu, J. Lykken, L. Magrini, J. Manley, N. Matsumoto, A. Monte, F. Monteiro, T. Purdy, C. J. Riedel, R. Singh, S. Singh, K. Sinha, J. M. Taylor, J. Qin, D. J. Wilson, and Y. Zhao, Mechanical quantum sensing in the search for dark matter, Quantum Science and Technology 6, 024002 (2021), publisher: IOP Publishing.