11institutetext:  University of California, Santa Cruz 22institutetext: The
Johns Hopkins University

# 3D-TransUNet for Brain Metastases Segmentation in the BraTS2023 Challenge

Siwei Yang 11 Xianhang Li 11 Jieru Mei 22 Jieneng Chen 22
Cihang Xie 11 Yuyin Zhou 11

###### Abstract

Segmenting brain tumors is complex due to their diverse appearances and
scales. Brain metastases, the most common type of brain tumor, are a frequent
complication of cancer. Therefore, an effective segmentation model for brain
metastases must adeptly capture local intricacies to delineate small tumor
regions while also integrating global context to understand broader scan
features. The TransUNet model, which combines Transformer self-attention with
U-Net’s localized information, emerges as a promising solution for this task.
In this report, we address brain metastases segmentation by training the
3D-TransUNet [6] model on the Brain Tumor Segmentation (BraTS-METS) 2023
challenge dataset. Specifically, we explored two architectural configurations:
the Encoder-only 3D-TransUNet, employing Transformers solely in the encoder,
and the Decoder-only 3D-TransUNet, utilizing Transformers exclusively in the
decoder. For Encoder-only 3D-TransUNet, we note that Masked-Autoencoder pre-
training is required for a better initialization of the Transformer Encoder
and thus accelerates the training process.

We identify that the Decoder-only 3D-TransUNet model should offer enhanced
efficacy in the segmentation of brain metastases, as indicated by our 5-fold
cross-validation on the training set111The code and models are available at
https://github.com/Beckschen/3D-TransUNet. However, our use of the Encoder-
only 3D-TransUNet model already yield notable results, with an average lesion-
wise Dice score of 59.8% on the test set, securing second place in the BraTS-
METS 2023 challenge.

###### Keywords:

Brain Tumor Segmentation Transformer

##  1 Introduction

Tumors, with their subtle intensity variations compared to surrounding
tissues, often pose difficulties, as evidenced by inconsistencies in even
expert-driven manual annotations[8, 24, 9, 13, 1]. Additionally, the wide
variance in tumor appearances and dimensions across patients challenges the
efficacy of traditional shape and location models[15, 19]. Brain metastases,
which are brain tumors that originate from primary cancers elsewhere in the
body, represent the most prevalent malignant tumors in the central nervous
system. With an annual incidence of 24 per 100,000 individuals [11, 3, 17],
brain metastases outnumber the occurrence of all primary brain cancers
combined.

In terms of segmentation methods, Convolutional Neural Networks (CNNs),
especially Fully Convolutional Networks (FCNs)[14], have established their
prominence. Among various architectures, the u-shaped architecture, popularly
known as U-Net [20], stands out for its symmetrical encoder-decoder framework
and skip-connections, excelling at preserving image intricacies. However,
these methods often struggle with modeling long-range dependencies due to
convolution’s inherent locality. To address this, researchers have turned to
Transformers, which rely solely on attention mechanisms, showcasing success in
capturing global contexts [22]. An example is TransUNet [5], a hybrid CNN-
Transformer model, seamlessly blending localized convolution’s efficiency with
global attention’s comprehension. Anchored in the encoder-decoder paradigm,
this innovation leverages and elevates both paradigms, promising a new
frontier in segmentation precision.

This report aims to validate the performance of 3D-TransUNet [6] on the
segmentation of brain metastases in the BraTS 2023 challenge. 3D-TransUNet has
two opted self-attention modules: 1) A _Transformer encoder_ , which tokenizes
image patches extracted from CNN feature maps to capture extensive global
contexts using transformer blocks, and 2) A _Transformer decoder_ , which
innovatively redefines the process of medical image segmentation by treating
it as a mask classification task and dynamically refining organ queries
through cross-attention with multi-scale CNN decoding features. Specifically,
we experiment with two architectures: Encoder-only (CNN encoder + Transformer
encoder + CNN decoder) and Decoder-only (CNN encoder + Transformer decoder +
CNN decoder). Notably, Masked-Autoencoder (MAE) Pre-training can be used to
accelerate the training of the Encoder-only model. To introduce stronger
supervision, we employ deep supervision across all levels of our decoder. Our
model yielded average lesion-wise Dice scores of 59.6% and 59.8%,
respectively, on the validation set and test set of BraTS-METS 2023 datasets.

##  2 Method

In this work, we adopt 3D-TransUNet [6] to segment brain metastases. This
model leverages the advantages of integrating transformers within the encoder
and decoder of the U-Net architecture, as shown in Figure 1. We first studied
the encoder to verify if transformer blocks can extract representative
features. We also explore combining the U-Net pixel decoder with a Transformer
decoder for prediction. The U-Net pixel decoder upsamples the low-resolution
features generated by the image encoder. Simultaneously, the Transformer
decoder enhances these features through a cross-attention mechanism,
effectively refining the final prediction.

Figure 1: Overview of our adaptation of 3D-TransUNet [6] for BraTS-METS 2023
[2].

###  2.1 Transformer as Encoder

####  2.1.1 Image Sequentialization

An input feature map $\bm{\mathrm{x}}$ are first tokenized and reshaped into a
sequence of 3D patches, noted as
{$\bm{\mathrm{x}}^{p}_{i}\in\mathbb{R}^{P^{3}\cdot C}|i=1,..,N\\}$. The size
of each 3D patch is $P\times P\times P$, and the total patch number is
$N=\frac{DHW}{P^{3}}$.

####  2.1.2 Patch Embedding

3D patches $\bm{\mathrm{x}}^{p}$ are linearly projected into a
$d_{enc}$-dimensional embedding space. Learnable positional embeddings are
added to retain spatial information. The final embeddings are formulated as
follows:

$\displaystyle\bm{\mathrm{z}}_{0}$
$\displaystyle=[\bm{\mathrm{x}}^{p}_{1}\bm{\mathrm{E}};\,\bm{\mathrm{x}}^{p}_{2}\bm{\mathrm{E}};\cdots;\,\bm{\mathrm{x}}^{p}_{N}\bm{\mathrm{E}}]+\bm{\mathrm{E}}^{pos},$
(1)

where $\bm{\mathrm{E}}\in\mathbb{R}^{(P^{2}C)\times d}$ and
$\bm{\mathrm{E}}^{pos}\in\mathbb{R}^{N\times d}$ denotes the linear projection
and position embedding accordingly.

The Transformer encoder consists of $L_{\textup{enc}}$ layers of Multi-head
Self-Attention (MSA) Equation. 2 and Multi-Layer Perceptron (MLP) blocks
Equation. 3. Therefore, the final output $\bm{\mathrm{z}}_{\ell}$ of the
$\ell$-th layer is

$\displaystyle\bm{\mathrm{z}}^{\prime}_{\ell}$
$\displaystyle=\textmd{MSA}(\textmd{LN}(\bm{\mathrm{z}}_{\ell-1}))+\bm{\mathrm{z}}_{\ell-1},$
(2) $\displaystyle\bm{\mathrm{z}}_{\ell}$
$\displaystyle=\textmd{MLP}(\textmd{LN}(\bm{\mathrm{z}}^{\prime}_{\ell}))+\bm{\mathrm{z}}^{\prime}_{\ell},$
(3)

where $\textmd{LN}(\cdot)$ is layer normalization operator and
$\bm{\mathrm{z}}_{\ell}$ is the encoded image representation.

###  2.2 Transformer as Decoder

The segmentation task can be reformulated into a binary mask classification
problem inspired by the set prediction mechanism proposed in DETR [4]. As
shown in Figure. 1, we train the CNN decoder and the Transformer decoder
simultaneously, allowing for the refinement of organ queries and feature maps.
Specifically, in the $t$-th layer of the Transformer decoder, the refined
organ queries are denoted by $\mathbf{P}^{t}\in\mathbb{R}^{N\times d_{dec}}$.
Alongside, an intermediate feature from the U-Net is transformed into a
$d_{dec}$-dimensional feature, represented by $\mathcal{F}$. The number of
upsampling blocks in the CNN decoder aligns with the Transformer decoder
layers, so multi-scale CNN features are effectively projected into the feature
space $\mathcal{F}\in\mathbb{R}^{(D_{t}H_{t}W_{t})\times d_{dec}}$, where
$D_{t}$, $H_{t}$, and $W_{t}$ define the spatial dimensions of the feature map
at the $t$-th upsampling block. Transitioning from the $t$-th to the $t+1$-th
layer, the organ queries $\mathbf{P}^{t}$ are updated through the cross-
attention mechanism as described by the following formula:

$\displaystyle\mathbf{P}^{t+1}=\mathbf{P}^{t}+\text{Softmax}\left((\mathbf{P}^{t}\mathbf{w}_{q})(\mathcal{F}^{t}\mathbf{w}_{k})^{\top}\right)\mathcal{F}^{t}\mathbf{w}_{v},$
(4)

where $\mathbf{w}_{q}\in\mathbb{R}^{d_{dec}\times d_{q}}$,
$\mathbf{w}_{k}\in\mathbb{R}^{d_{dec}\times d_{k}}$, and
$\mathbf{w}_{v}\in\mathbb{R}^{d_{dec}\times d_{v}}$ are the weight matrices
that linearly project the $t$-th query features, keys, and values for the
subsequent layer. This process is repeated, with a residual connection
updating $\mathbf{P}$ after each layer, in line with the previous method
([7]). The final prediction, $\mathbf{Z}^{T}$, is derived through Equation 5,
which details the process of converting $\mathbf{P}^{T}$ into the binarized
segmentation map. It involves a dot product with U-Net’s last block feature,
$\mathbf{F}$, resulting in $\mathbf{Z}^{T}$.

$\displaystyle\mathbf{Z}^{T}=g(\mathbf{P}\times\mathbf{F}^{\top}),$ (5)

where $g(\cdot)$ is sigmoid activation followed by a hard thresholding
operation with a threshold set at 0.5, such to decode region-wise binary brain
tumor masks.

Note that unlike [6], we do not use masked attention here due to the observed
training instability.

###  2.3 3D-TransUNet Variants

Two 3D-TransUNet variants, _i.e._ , Encoder-only and Decoder-only, are
involved in the experiment. Encoder-only 3D-TransUNet is used as the main
architecture for all of our submissions since Decoder-only 3D-TransUNet
requires longer training compared to the Encoder-only 3D-TransUNet. This is
mainly due to its use of high-resolution features in the Transformer decoder.
Additionally, the Hungarian matching process used to match binary masks to
ground truth in the Decoder-only model is slower compared to directly
computing cross-entropy and Dice loss.

####  2.3.1 Encoder-Only

The Transformer encoder along with the CNN encoder compose a CNN-Transformer
hybrid encoder in this variant. Feature maps are first extracted by CNN then
patchified and tokenized before being fed to the Transformer encoder. A
standard U-Net is used as the decoder without the Transformer decoder.

####  2.3.2 Decoder-Only

This variant uses a conventional CNN encoder only for the encoding phase while
both CNN and Transformer are used as the decoder. Before being processed by
the Transformer decoder, they are augmented with learnable positional
embeddings following Eq. (1).

###  2.4 Training Details

Masked-Autoencoder (MAE) Pre-training. For our Encoder-only 3D-TransUNet, we
first pre-train the transformer encoder in an MAE style [12]. Specifically, We
initially tokenized 3D input using a 2D patch embedding layer along the
z-axis, then flattened it into a 1D sequence. We randomly mask out 75% of the
tokens and then utilize just one lightweight decoder block to predict the
masked tokens. Following [12], we calculate the reconstruction loss
$L_{\text{recon}}=\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\hat{x}_{i,\text{masked}})^{2}$,
where $L_{\text{recon}}$ is the mean square error, $x_{i}$ are the original
pixel values, $\hat{x}_{i,\text{masked}}$ denotes the predicted pixel values
for the masked tokens, and $N$ represents the total count of masked tokens.
Following [12], we adopt pixel normalization on each patch. Specifically, we
compute the mean and standard deviation of all pixels in a patch and use them
to normalize this patch. After pre-training, we discard the patch embedding
layer and decoder block and solely initialize all transformer encoder blocks
with the pre-trained weights. We observe that utilizing a pre-trained encoder
significantly speedup model convergence. For instance, with MAE pre-training,
the model attains an average Dice score across five folds of 58.2% with only
300 epochs of training, whereas a model trained from scratch needs 600 epochs
to achieve comparable performance.

Training Loss. Unless indicated otherwise, we mainly follow the training
details of 3D-TransUNet [6]. In addressing the challenge posed by setting the
number of coarse candidates $N$ considerably greater than the class count $K$,
it becomes inevitable that predictions for each class will exhibit false
positives. To mitigate this, we employ a post-processing step to refine the
coarse candidates, drawing on a matching process between predicted and ground
truth segmentation masks. Taking cues from prior work [4, 23], we utilize the
Hungarian matching approach to establish the correspondence between
predictions and ground-truth segments. The resulting matching loss is
formulated as follows:

$\displaystyle\mathcal{L}=\lambda_{0}(\mathcal{L}_{ce}+\mathcal{L}_{dice)}+\lambda_{1}\mathcal{L}_{cls},$
(6)

Here, the pixel-wise losses $\mathcal{L}_{ce}$ and $\mathcal{L}_{dice}$ denote
the binary cross-entropy and dice loss [16] respectively, while
$\mathcal{L}_{cls}$ represents the classification loss computed using the
cross-entropy for each candidate region. The hyper-parameters $\lambda_{0}$
and $\lambda_{1}$ serve to strike a balance between per-pixel segmentation and
mask classification loss.

Deep Supervision. To introduce stronger supervision, every intermediate level
of the 3D-TransUNet decoder produces a prediction map on which the loss
function is applied during training.

##  3 Experiments and results

###  3.1 Experimental Setting

Implementation Details. All experiments are conducted with a single NVIDIA
A5000. Batch size and base learning rate are set as 2 and 2e-3 accordingly.
The learning rate follows polynomial decay with a power factor of 0.9.
Augmentation including random rotation, scaling, flipping, white Gaussian
noise, Gaussian blurring, color jittering, low-resolution simulation, Gamma
transformation. Our main experiments in Table 2 are conducted using Encoder-
only 3D-TransUNet. The architecture combines a 3D nn-UNet with a pre-trained
12-layer Vision Transformer (ViT) as the Transformer encoder, utilizing Masked
Autoencoder (MAE) weights. The latent dimension $d$ is set at 768. For
decoder-only, the number of layers is 3. And $d_{dec}$ is set to 192. For
training loss 6, $\lambda_{0}$ and $\lambda_{1}$ are set as 0.7 and 0.3.
During the testing phase, we train our model on the entire training set for
600 epochs and submitt the predictions on the validation set. During the
testing phase, we applied 10-fold cross-validation, where we trained an
individual model for every fold for 1,000 epochs.

MAE pretraining settings. The input has a shape of $128\times 128\times 128$
after random cropping. We train all data on 8 GPUs distributedly for our MAE
training, with a batch size of 2 on each GPU. We train the model in 4800
epochs, including a warm-up period of 40 epochs. The base learning rate is set
to 1.5e-4, accompanied by a weight decay 0.05. AdamW optimizer is used by
default.

Datasets. We report results on BraTS-MET 2023 [17] which is pivotal for
crafting sophisticated algorithms to detect and segment brain metastases,
aiming for easy clinical integration. This dataset encompasses a collection of
untreated brain metastases mpMRI scans, sourced from multiple institutions and
conducted under regular clinical protocols. It should be noted that we didn’t
use other officially allowed datasets, _e.g._ , NYUMets [18], UCSF-BMSR [21],
BrainMetsShare [10] as these datasets don’t share the same mpMRI modalities
and annotation format as the BraTS-MET 2023.

It should be noted that we only use BraTS-METS 2023 for training as these
datasets share the mpMRI modalities and annotation format.

Evaluation Metrics. In assessing the accuracy of a medical image segmentation
model, various metrics offer insights into different aspects of performance:

  1. 1.

Lesion-wise Dice Score: Dice score represents the similarity between two
binary segmentations. It is given by:

$\text{Dice}(\mathbf{A},\mathbf{B})=\frac{2|\mathbf{A}\cap\mathbf{B}|}{|\mathbf{A}|+|\mathbf{B}|}$
(7)

where $\mathbf{A}$ and $\mathbf{B}$ denote the binary segmentations.

In this challenge, dice scores localized to individual lesions are adopted for
lesion-specific evaluation. The Lesion-wise Dice for a designated lesion is:

$\text{Dice}_{\text{lesion-
wise}}(\mathbf{A}_{i},\mathbf{B}_{i})=\frac{2|\mathbf{A}_{i}\cap\mathbf{B}_{i}|}{|\mathbf{A}_{i}|+|\mathbf{B}_{i}|}$
(8)

This formula assesses the overlap between the $i^{th}$ ground-truth lesion
$\mathbf{B}_{i}$ and all the lesions that overlap with it $\mathbf{A}_{i}$.

  2. 2.

Hausdorff Distance (95%): A metric quantifying the maximum of minimum
distances between two binary images at the $95$-th percentile to mitigate
outlier effects. Mathematically:

$H(A,B)=\max\left\\{\ \sup_{a\in A}\inf_{b\in B}d(a,b),\sup_{b\in B}\inf_{a\in
A}d(a,b)\ \right\\}$ (9)

Model Ensemble and Test-Time Augmentation During testing phase, five models
are randomly chosen from ten models trained on each fold for ensemble.
Predictions from these five models were averaged to ensemble predictions. To
further boost the model’s performance with test-time augmentation, predictions
from augmented views with flipping and rotation (90°, 180°, 270°) are averaged
to produce the final predictions.

Method | Lesion-wise Dice ($\uparrow$) | HD95 ($\downarrow$)
---|---|---
ET | TC | WT | Avg. | ET | TC | WT | Avg.
Encoder-only 3D-TransUNet [6] | 54.79% | 58.96% | 56.05% | 56.60% | 108.9 | 107.6 | 109.5 | 108.7
Decoder-only 3D-TransUNet [6] | 56.80% | 61.12% | 60.09% | 59.34% | 99.4 | 95.9 | 93.97 | 96.4
Table 1: Ablation Performance of 3D-TransUNet on the training set of BraTS-
METS 2023 [17] under 5-fold cross-validation.

###  3.2 Encoder-only v.s. Decoder-only

Dataset Split | Lesion-wise Dice ($\uparrow$) | HD95 ($\downarrow$)
---|---|---
ET | TC | WT | Avg. | ET | TC | WT | Avg.
Validation | 59.2% | 63.4% | 56.5% | 59.6% | 94.8 | 94.8 | 110.9 | 100.1
Test | 57.4% | 62.0% | 59.9% | 59.8% | 103.0 | 99.8 | 99.6 | 100.8
Table 2: Performance of Encoder-only 3D-TransUNet on the validation and test
set of BraTS-METS 2023 [17].

In order to compare the effectiveness between the Encoder-only model and the
Decoder-only model, we apply 5-fold cross-validation on the entire 238
training cases and report the average Dice and HD95 for all testing cases. The
5 models from the 5 folds of BraTS-METS 2023 training set are trained for 200
epochs. In Table 1, we report the comparison between Encoder-only 3D-TransUNet
and Decoder-only 3D-TransUNet on BraTS-METS 2023’s training set with 5-fold
cross-validation are presented in Table 1. Compared to our internal validation
results with the baseline nnUNet, which yielded average Dice scores of 54.90%,
58.67%, and 55.75% for segmenting ET, TC, and WT, respectively, resulting in
an overall average Dice score of 56.44%, it becomes evident that while the
Encoder-only model offers only marginal improvement in segmentation, the
Decoder-only model demonstrates a substantial increase in Dice score by 2.9%.
It is important to note that MAE pretraining was not applied to the Encoder-
only model in this evaluation. However, with MAE pretraining, the advantages
of the Encoder-only model are expected to be more pronounced, albeit still
inferior to the Decoder-only model.

###  3.3 Main Results

Since the Decoder-only 3D-TransUNet requires longer training, due to the time
and computation limit, we were only able to submit results from the Encoder-
only 3D-TransUNet during the validation phase and testing phase, where the
official evaluation results are presented in Table 2. Specifically, we achieve
average lesion-wise Dice scores of 59.6% and 59.8% on the validation and test
set, securing the second place in the BraTS 2023 challenge. We also display a
qualitative example to further demonstrate our method’s effectiveness, as
shown in Fig. 2.

Figure 2: Visual comparison between ground-truths and predictions from
Encoder-only 3D-TransUNet on BraTS-METS 2023. The lesion-wise Dice score of
each category on this patient is also listed (best viewed in color).

##  4 Conclusion

Brain tumors, especially brain metastases, present challenges in segmentation
due to their diverse appearances and sizes. The TransUNet model, combining
Transformer self-attention and U-Net’s features, shows promise for this task.
We trained the 3D-TransUNet model on the BraTS-METS 2023 dataset for brain
metastases segmentation, exploring Encoder-only and Decoder-only
configurations. Pre-training the Encoder-only model with Masked-Autoencoder
improves initialization, facilitating faster training. Although the Decoder-
only model is expected to perform better, the Encoder-only model achieved
notable results in a shorter timeframe, securing second place in the BraTS-
METS 2023 challenge with a 59.8% average lesion-wise Dice score on the test
set.

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# Exploring magnetic fields in merging galaxy: combining polarization and
velocity gradient in the Centaurus Galaxy

Quynh Lan Nguyen  Department of Physics and Astronomy, University of Notre
Dame, Notre Dame, IN 46556, USA  Yue Hu* Institute for Advanced Study, 1
Einstein Drive, Princeton, NJ 08540, USA  Alex Lazarian  Department of
Astronomy, University of Wisconsin-Madison, Madison, WI 53706, USA  *NASA
Hubble Fellow

###### Abstract

In this study, we apply the Velocity Gradient Technique (VGT) to the merging
Centaurus galaxy. We compare gradient maps derived from the PHANGS-ALMA survey
using CO emission lines with magnetic field tracings from dust polarization
data obtained via the HAWC+ instrument. Our analysis reveals a strong
correspondence between the directions indicated by these two tracers across
most of the galactic image. Specifically, we identify jet regions as areas of
anti-alignment, consistent with previous reports that gradients tend to rotate
90 degrees in outflow regions. Statistically, we find that the alignment of
magnetic fields, as revealed by polarization, is most accurate in regions with
the highest signal-to-noise ratios. Our findings underscore the utility of
velocity gradients as a valuable complementary tool for probing magnetic
fields and dynamical processes in merging galaxies.

Extragalactic magnetic fields (507) — Interstellar medium (847) — Galaxy
nuclei (609)

††software: Julia (Bezanson et al., 2012)

##  1 Introduction

Merging plays an important role in the galaxy’s evolution, so tracing the
magnetic field in the merging galaxy is interesting. It can provide valuable
insight into the galaxy’s dynamic and evolution as well as the jet formation
(Whittingham et al., 2021). Through the interaction of the magnetic field in
the merging galaxy and the magnetic field in the interstellar medium and other
galaxies and cluster galaxies, we can understand better the forming and the
shape of the large-scale structure of the universe (Rodenbeck & Schleicher,
2016; Chibueze et al., 2021; Paliya et al., 2020).

The observations in far-infrared and radio for galactic and extragalactic
magnetic fields have significantly developed in the past decades (Kronberg,
1994; Beck, 2015; Lopez-Rodriguez, 2021). Even if it has some initial results,
it still has some limitations. Dust and synchrotron polarization can trace the
plane of the sky (POS) magnetic field orientation. However, dust polarization
is affected by dust alignment efficiency (Lazarian & Hoang, 2007), and
synchrotron polarization is subject to Faraday rotation (Beck, 2015).
Synchrotron polarization provides the method that can trace the magnetic
fields in the warm and high temperatures of the gas (Beck, 2001; Fletcher et
al., 2011; Beck, 2015) and dust polarization contains contribution from
different gas phases, including the one associated with diffuse atomic gas and
dense molecular gas (Lazarian & Hoang, 2007; Andersson et al., 2015).

To separate the magnetic field associated with molecular gas, which is the
most important fueling material for the galactic nuclei, Hu et al. (2022)
proposed the use of the Velocity Gradient Technique (VGT; González-Casanova &
Lazarian 2017a; Lazarian & Yuen 2018; Hu et al. 2018) and CO spectroscopic
observations. Despite that, the study of VGT has been explored for six
individual galaxies M51, NGC 1068, NGC 1097, NGC 3627, and NGC 4826 (Hu et
al., 2022), NGC3627 (Liu et al., 2023), NGC 0628 (Zhao et al., 2024a) but its
application or test to galaxy merger has not been done yet. Centaurus galaxy,
as one of the closest nuclear active galaxy mergers, is the best test-bed for
VGT.

In this work, we trace the magnetic field of the Centaurus galaxy by using the
VGT, particularly its branch, namely, Velocity Channel Gradients (VChGs;
Lazarian & Yuen 2018) and $\rm{}^{12}CO$ (J = 2–1) emission lines from the
PHANGS-ALMA survey (Leroy et al., 2021). On the other hand, the magnetic field
in Centaurus has been studied by (Lopez-Rodriguez, 2021) using HAWC+ dust
polarization at 89$\mu$m. Here we compare the magnetic field traced by VGT to
the one inferred from HAWC+.

However, it should be noted that the dynamics of galaxy mergers are more
complicated than normal galaxies. This makes the interpretation of the
comparison more difficult. For instance, early applications Hu et al. (2022);
Liu et al. (2023); Zhao et al. (2024a) of the VGT have conventionally set the
turbulence injection scale, $L_{\rm inj}$, at approximately 100 pc, a measure
derived from extensive turbulence studies within the Milky Way (Armstrong et
al., 1995; Chepurnov & Lazarian, 2010; Ha et al., 2022). This assumption
aligns with the typical scale at which energy is injected into the
interstellar medium via stellar feedback and other dynamic processes in
relatively stable galactic environments. However, in scenarios involving
galaxy mergers, the dynamics of turbulence injection can be substantially
altered. Mergers not only impact galactic structure and star formation but
also contribute to turbulence on a much larger scale (Whittingham et al.,
2021). The merging process itself can act as a significant driver of
turbulence, likely extending the injection scale well beyond the 100 pc
typically observed in more quiescent galaxies like the Milky Way. Given this
context, it is reasonable to hypothesize that the injection scale $L_{\rm
inj}$ in the Centaurus galaxy, which has experienced dynamic merger events,
could be larger than 100 pc and we expect the $\rm{}^{12}CO$ observation with
an average spatial resolution of $145.5$ pc satisfies the VGT’s requirements
for this pilot study.

The details of the archival data are shown in Section 2. Centaurus galaxy is
one of the closest nuclear active galaxies and we want to use it as a test for
VGT. In Section 3, we introduce the details theory of the VGT, velocity
caustics, and VGT pipeline. In Section 4, we will discuss some results about
the mapping magnetic field of the Centaurus galaxy and compare it with dust
polarization result. In Section 5, we will provide some conclusions about the
magnetic fields of the Centaurus galaxy using VGT and some possibility of the
galaxies merges scenarios.

##  2 Observational Data

###  2.1 Emission Lines

In this study, we utilized the $\rm{}^{12}CO$ (J = 2–1) emission lines
provided by the Physics at High Angular Resolution in Nearby Galaxies
(PHANGS)–ALMA survey (Leroy et al., 2021). The PHANGS-ALMA survey offers
$\rm{}^{12}CO$ (J = 1 - 0) emission line data at a spatial resolution of
approximately $8.13^{\prime\prime}$ (about 145.5 pc) and a velocity resolution
of 2.5 km s-1 for the nearby galaxy Centaurus. This survey achieves a high
signal-to-noise ratio, with an RMS brightness temperature noise level around
$\sim 0.30\pm 0.13$ K km s-1.

###  2.2 Polarization measurement

For polarization measurements, we employed data from the High-resolution
Airborne Wideband Camera Plus (HAWC+) archived in the HAWC+ database (Harper
et al., 2018; Lopez-Rodriguez, 2021). Magnetic field orientations were
determined using the relation $\phi_{B}=\phi+\pi/2$, where $\phi$ represents
the polarization angle. Measurements were taken from band C (89 $\mu$m, FWHM
$\approx 7.8^{\prime\prime}$).

We limited our analysis to pixels where the polarization fraction ($p$)
divided by its uncertainty ($\sigma_{p}$) exceeds 1 ($p/\sigma_{p}>1$). To
compare with the magnetic field measured with VGT, we smooth the polarization
data to FWHM $\approx 40^{\prime\prime}$ matching the spatial resolution of
VGT.

##  3 Methodology

###  3.1 Theoretical consideration: Velocity Gradient Technique

The theoretical underpinning of the Velocity Gradient Technique (VGT) lies in
the anisotropy inherent in MHD turbulence, a concept extensively developed by
Goldreich & Sridhar (1995) and Lazarian & Vishniac (1999). This anisotropy
suggests that turbulent eddies elongate along magnetic field lines,
manifesting notably in local reference frames. Lazarian & Vishniac (1999)
formalized this anisotropy as follows:

$l_{\parallel}=L_{\rm inj}\left(\frac{l_{\bot}}{L_{\rm
inj}}\right)^{\frac{2}{3}}M_{A}^{-4/3},~{}~{}~{}M_{A}\leq 1,$ (1)

where $l_{\bot}$ and $l_{\parallel}$ are the scales perpendicular and parallel
to the local magnetic field, respectively. Here, $M_{A}=v_{\rm inj}/v_{A}$
represents the Alfvén Mach number, with $v_{\rm inj}$ being the injection
velocity and $v_{A}$ the Alfvén speed.

This anisotropy has been demonstrated by numerical simulations (Cho &
Vishniac, 2000; Maron & Goldreich, 2001; Cho & Lazarian, 2003; Kowal &
Lazarian, 2010; Hu et al., 2021) and in situ measurements in the solar wind
(Wang et al., 2016; Matteini et al., 2020; Duan et al., 2021; Zhao et al.,
2024b). From Eq.1, we understand that $l_{\parallel}$ significantly exceeds
$l_{\bot}$, leading to the dominance of the gradient of velocity fluctuations
perpendicular to the local magnetic field:

$\nabla v_{l}\approx\frac{v_{l,\bot}}{l_{\bot}}=\frac{v_{\rm inj}}{L_{\rm
inj}}M_{A}^{1/3}\left(\frac{l_{\bot}}{L_{\rm inj}}\right)^{-2/3},M_{A}\leq 1.$
(2)

This scaling relation ($\propto l_{\bot}^{-2/3}$) is crucial as it reveals
that the amplitude of the turbulent velocity gradient increases at smaller
scales, which is a characteristic not typically observed in non-turbulent
velocity fields such as galactic differential rotation. For effective
application to external galaxies, observations should resolve the scale at
which turbulence’s anisotropy manifests prominently, typically at $L_{\rm
inj}$ for sub-Alfvén and trans-Alfvén turbulence or $L_{\rm inj}M_{A}^{-3}$
for super-Alfvénic turbulence (Lazarian, 2006).

Figure 1: The magnetic field of Centaurus mapped with VGT using the $\rm CO$
(2-1) emission line. The magnetic field segments are laid upon Centaurus’ CO
emission intensity map.  Figure 2: Same as Fig. 1, but the magnetic field is
measured by HAWC+ polarization at $89~{}\mu$m.

###  3.2 Velocity caustics

The anisotropic nature of MHD turbulence can be observed in velocity channel
maps, $p(x,y,v_{\rm los})$, obtained from spectroscopic observations. This
observation is primarily due to the velocity caustics effect, which posits
that real density structures with different velocities may be sampled into the
same velocity channel, thereby significantly altering the observed intensity
structures within the spectroscopic channel maps (Lazarian & Pogosyan, 2000,
2004; Kandel et al., 2016; Hu et al., 2023).

Mathematically, the intensity observed in a channel can be expressed as the
sum of two terms (Hu et al., 2023):

$p(x,y,v_{\rm los})=p_{v}(x,y,v_{\rm los})+p_{d}(x,y,v_{\rm los}),$ (3)

where $p_{v}$ represents fluctuations primarily caused by velocity—known as
the velocity caustics effect—while $p_{dc}$ reflects the inhomogeneities
within the actual density field. The relative contribution of these terms is
dependent on the channel width, i.e., the velocity resolution of the
observation. A narrower channel width enhances the contribution of $p_{v}$,
leading to a scenario where intensity fluctuations in thin channels are
predominantly due to velocity fluctuations. Specifically, when the channel
width $\Delta v$ is narrower than the velocity dispersion of the turbulent
eddies under observation, the intensity fluctuations within such a thin
channel will be primarily driven by velocity fluctuations.

Turbulence studies within the Milky Way suggest that the velocity dispersion
at a scale of 100 pc is approximately 10 km/s (Armstrong et al., 1995;
Chepurnov & Lazarian, 2010; Ha et al., 2022). This dispersion can be even
higher in environments such as galaxy mergers. Given that the PHANGS-ALMA
survey offers a velocity resolution of 2.5 km/s, this ensures that the
velocity caustics dominate the observed intensity in the spectroscopic
velocity channels.

###  3.3 VGT pipeline

In this study, we employed the VGT, following a multi-step process detailed in
(Hu et al., 2022; Liu et al., 2023):
Step 1. Each thin channel map, $p(x,y,v_{\rm los})$, undergoes convolution
with 3$\times$3 Sobel kernels $G_{x}$ and $G_{y}$ to compute the gradients:

$\displaystyle\nabla_{x}p(x,y,v_{\rm los})$ $\displaystyle=G_{x}*p(x,y,v_{\rm
los}),$ (4) $\displaystyle\nabla_{y}p(x,y,v_{\rm los})$
$\displaystyle=G_{y}*p(x,y,v_{\rm los}),$

where * denotes convolution. These gradients are used to determine the
pixelized gradient map $\psi_{g}(x,y,v_{\rm los})$:

$\psi_{g}(x,y,v_{\rm los})=\tan^{-1}\left(\frac{\nabla_{y}p(x,y,v_{\rm
los})}{\nabla_{x}p(x,y,v_{\rm los})}\right).$ (5)

Only pixels with a brightness temperature less than three times the RMS noise
are retained for subsequent analysis.

Step 2. The pixelized map $\psi_{g}(x,y,v_{\rm los})$ is segmented into
20$\times$20 pixel sub-blocks—although other sizes have been tested (Yuen &
Lazarian, 2017; Lazarian & Yuen, 2018; Hu et al., 2020a) —and processed as
follows:

  1. 1.

A histogram of gradient orientations within each sub-block is produced and
fitted with a Gaussian distribution.

  2. 2.

The peak of this Gaussian distribution is taken as the dominant gradient
orientation for that sub-block.

Step 3. Following sub-block averaging, the averaged gradient angle map
$\psi_{gs}(x,y,v_{\rm los})$ is compiled for each velocity channel. Repeating
the Steps 1 and 2 for every channel within the velocity range of interest, we
can then construct the pseudo-Stokes parameters $Q_{g}(x,y)$ and $U_{g}(x,y)$:

$\displaystyle Q_{\rm g}(x,y)$ $\displaystyle=\int_{v_{\rm los,min}}^{v_{\rm
los,max}}p(x,y,v_{\rm los})\cos(2\psi_{gs}(x,y,v_{\rm los}))dv_{\rm los},$ (6)
$\displaystyle U_{\rm g}(x,y)$ $\displaystyle=\int_{v_{\rm los,min}}^{v_{\rm
los,max}}p(x,y,v_{\rm los})\sin(2\psi_{gs}(x,y,v_{\rm los}))dv_{\rm los},$

where $v_{\rm los,max}$ and $v_{\rm los,min}$ represent the upper and lower
levels of the velocity range used for integration. The POS magnetic field
orientation is inferred from:

$\psi_{\rm B}(x,y)=\frac{1}{2}\tan^{-1}\left(\frac{U_{\rm g}(x,y)}{Q_{\rm
g}(x,y)}\right)+\frac{\pi}{2}.$ (7)

To quantify the agreement between VGT and the magnetic field inferred from
polarization, we utilize the Alignment Measure (AM; González-Casanova &
Lazarian 2017b), expressed as:

$\displaystyle{\rm AM}=2\left(\cos^{2}\theta_{\rm r}-\frac{1}{2}\right)$ (8)

Here, $\theta_{\rm r}=|\phi_{\rm B}-\psi_{\rm B}|$, where $\phi_{\rm B}$ is
the magnetic field angle inferred from polarization. An AM value of 1 implies
parallel alignment of $\phi_{\rm B}$ and $\psi_{\rm B}$, while -1 indicates
perpendicularity.

Figure 3: Left: H I emission in greyscale overlaid with radio continuum
contours (grey) and H I absorption contours (red). The green box highlights
the region where we are studying the VGT-CO. Reproduced with permission ©ESO,
Struve et al. (2010), A&A, 515, A67. Right: The spatial distributions of the
AM between VGT-CO measurement and HAWC+ polarization with a resolution of FWHM
$40^{\prime\prime}$.

##  4 Result

Figure 4:  A 2D histogram of the dust polarization’s signal-to-noise ratio and
AM between VGT and polarization

Fig. 1 presents the magnetic fields of Centaurus as mapped by VGT-CO. The
image captures the twisting of the magnetic field from the southeast and
northwest filamentary structures, which resemble spiral-arm-like features,
towards the molecular disk and into the nucleus. The magnetic field, as
depicted by VGT-CO, primarily aligns with these two filamentary structures.
However, it exhibits a twist across the central region of approximately 500 pc
radius around the nucleus. Additionally, the magnetic fields inferred from
HAWC+ polarization, illustrated in Fig. 2, exhibit similar morphology and
generally agree with the measurements from VGT-CO. Notably, near the southeast
tail, the polarization-inferred magnetic field diverges from the galactic
structure indicated by CO, instead crossing it. This specific pattern is
absent in the VGT-CO measurements.

The misalignment between the magnetic fields traced by VGT-CO and those
indicated by dust polarization is quantified by the AM, as shown in Fig. 3.
The anti-alignment, indicated by negative AM values, is observed in the two
tails, while a misalignment with AM $\approx 0$ is noted in the northwest of
the central disk. A higher degree of the mean alignment is observed in several
Seyfert galaxies and the Milky Way (Hu et al., 2022; Liu et al., 2023; Zhao et
al., 2024a). We attribute this effect to the fact that the Centaurus is a
merging galaxy with much more complex dynamics of gas. Jets, shocks, and
inflows affect the gradients, especially if the measurements. These effects
were discussed at length in Lazarian & Yuen (2018); Hu et al. (2019, 2020b);
Liu et al. (2024). In particular, anti-alignment is present in the radio jet
regions, which is expected for the outflow regions. Further studies of the
origin of this discrepancy are required. This can elucidate the dynamics of
gas in the merging galaxy.

Another source of the observed misalignment can be the difference in CO and
dust distribution in these regions. Unlike CO, dust is well mixed with all
interstellar phases, including molecular and atomic gas (Andersson et al.,
2015). In Fig. 3, we see the central low-AM region is associated with strong
HI absorption, suggesting the dust polarization in that region might be
dominated by dust associated with H I rather than CO. In addition, HAWC+ dust
polarization at $89~{}\mu$m is anticipated to trace magnetic fields in warmer
media than CO. If magnetic fields in warm and cold gas are very different,
there is a warmer phase preferentially traced by HAWC+; the morphology of
magnetic fields with low-AM could reflect that the magnetic fields in warm and
cold gas are different.

We also note that the most significant misalignment between VGT-CO and dust
polarization measurements is in the low-density regions associated with poor
signal-to-noise ratio. This is illustrated in Fig. 4, where we observe that we
do not observe a negative AM at the high signal-to-noise ratio regions.

Overall, the global agreement between the magnetic field orientations measured
with VGT-CO and those inferred from dust polarization, especially in the
central disk, underscores the coherence of magnetic fields across different
gas phases. This includes the cold molecular phase traced by CO and warmer
phases indicated by HAWC+. The observed coherence across these phases suggests
that the magnetic fields threading the multiphase gas, spanning a range of
densities, are integrally involved in the galactic dynamic evolution. This
supports the notion that molecular clouds are part of a unified magnetic
ecosystem in galaxies and that magnetic fields in multiphase gas maintain an
overall coherent structure even in a merging galaxy. Additional input from
synchrotron polarization could be useful for getting more information. In the
aforementioned studies of Seyfert galaxies with VGT (Hu et al., 2022; Liu et
al., 2023; Zhao et al., 2024a), the alignment between the velocity gradients
in CO and synchrotron could be better than the alignment with dust, which
elucidated the 3D magnetic field structure.

##  5 Conclusion

Velocity gradients have demonstrated efficacy in tracing magnetic fields
within magnetohydrodynamic (MHD) turbulence environments. This capability has
been validated across various regions of the Milky Way and several Seyfert
galaxies. In this paper, we extend the application of velocity gradient
techniques (VGT) to the more complex scenario of a merging galaxy.
Specifically, we measure velocity gradients in the Centaurus galaxy and
evaluate their effectiveness in tracing magnetic fields, using dust
polarization as a comparative measure. Our findings indicate that VGT results
generally concur with dust polarization on a statistical basis. Additionally,
we identify localized regions of significant misalignment and anti-alignment.
Drawing on prior studies, we hypothesize that outflows and the presence of
shocked gas may account for these discrepancies. If corroborated, this
comparison of gradients and polarization could elucidate the underlying
physics of galactic mergers. Furthermore, our study aids in pinpointing
regions within disturbed galaxies where velocity gradients can reliably map
magnetic fields in the absence of other tracers. Future research, utilizing
higher signal-to-noise data and incorporating synchrotron polarization, is
essential to further explore these findings.

A.L. acknowledges the support of NASA ATP AAH7546, NSF grants AST 2307840, and
ALMA SOSPADA-016. Financial support for this work was provided by NASA through
award 09_0231 issued by the Universities Space Research Association, Inc.
(USRA). Y.H. acknowledges the support for this work provided by NASA through
the NASA Hubble Fellowship grant No. HST-HF2-51557.001 awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555. This work used SDSC Expanse CPU at SDSC through allocations
PHY230032, PHY230033, PHY230091, and PHY230105 from the Advanced
Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS)
program, which is supported by National Science Foundation grants #2138259,
#2138286, #2138307, #2137603, and #2138296. Q.L.N. acknowledges the support in
part by grant NSF PHY-1748958 to the Kavli Institute for Theoretical Physics
(KITP), the Institute for Cosmic Ray Research-International University
Research Program, Notre Dame College of Science Internal Grant.

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# Characterization of NbTiN films with thicknesses below 20 nm for low power
kinetic inductance amplifiers

A. Giachero, M.R. Vissers, J.D. Wheeler, M. Malnou, J.E. Austermann,
J. Hubmayr, A. Nucciotti, J.N. Ullom, J. Gao A. Giachero is with Dipartimento
di Fisica, Università di Milano-Bicocca, , also with INFN - Sezione di Milano
Bicocca, Milano, I-20126, Italy, also with Department of Physics, University
of Colorado, Boulder, Colorado 80309, USA, and also with National Institute of
Standards and Technology, Boulder, Colorado 80305, USA (e-mail:
andrea.giachero@nist.gov). M. Malnou, J.N. Ullom, J. Gao are with Department
of Physics, University of Colorado, Boulder, Colorado 80309, USA, also with
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
M.R. Vissers, J.D. Wheeler, J.E. Austerman, J. Hubmayr are with National
Institute of Standards and Technology, Boulder, Colorado 80305, USA A.
Nucciotti is with Dipartimento di Fisica, Università di Milano-Bicocca, and
also with INFN - Sezione di Milano Bicocca, Milano, I-20126, Italy

###### Abstract

A quantum-limited amplification chain is a fundamental advantage for any
application that may benefit from the detection of very faint signals. Reading
out arrays of superconducting detectors (TESs or MKIDs), resonant cavities, or
qubits, calls for large bandwidth amplifiers in addition to having the lowest
possible noise. At millikelvin temperatures, Kinetic Inductance Traveling-Wave
Parametric Amplifiers (KI-TWPAs) working in 3-wave-mixing (3WM) and fabricated
from a 20 nm thick NbTiN film have shown promising noise performances, as they
can operate close to the quantum limit [1]. However, they still require fairly
high pump power. Devices that would require lower pump power would be easier
to implement in readout chains, could reach the quantum limit and they would
be compatible with qubit readout. A possible solution for obtaining this
optimal configuration is to use a thinner superconducting film. In this work
we explore the properties of NbTiN films with a thickness less than 20 nm and
we report the obtained experimental characterizations in terms of critical
temperature, normal resistivity, and kinetic inductance. A new design for a
3WM KI-TWPA amplifier, based on these developed superconducting films, is
introduced and discussed.

##  I Introduction

Ultra-sensitive detection schemes play a central role in many advanced
applications, mostly for fundamental physics experiments, such as: detection
of axionic dark matter [2], dark photons [3], astrophysical measurements [4],
fundamental physics experiments [5] but also are key factors for the
deployment of quantum technologies, and in particular for the qubit readout
[6, 7], quantum key distribution [8], and microwave quantum illumination [9].
In many of these applications, the necessity of reading a large array of
detectors, cavities, qubits, calls for large bandwidth amplifiers with the
lowest possible noise. One of the leading proposals for achieving these
requirements is through the use of a broadband traveling wave parametric
amplifier (TWPA) such as Josephson JTWPA [10] or kinetic inductance KI-TWPA
[11]. These newly developed devices can reach the noise level near the quantum
limit with high gain over a larger bandwidth (a few GHz). TWPA amplifier
generally consists of a transmission

KI-TWPAs achieve amplification through wave-mixing processes induced by the
film’s intrinsic nonlinear superconducting kinetic inductance. When a strong
pump current propagates with a weak signal current along a line, energy is
transferred from the pump to the signal, achieving a parametric amplification.
For these films, the kinetic inductance per unit length $L_{k}$ of the
transmission line can be expanded as a series of even powers of line current
$I$ as follows:

$L_{k}(I)=L_{0}\left[1+\cfrac{I^{2}}{I_{*}^{2}}+\mathcal{O}(I^{4})\right]$ (1)

where $L_{0}$ is the kinetic inductance per unit length in the absence of
inductive non-linearity ($I=0$) and $I_{*}$ controls the scale of the non-
linearity. From the Mattis-Bardeen [12] $L_{0}=\hbar\,R_{n}/(\pi\Delta)$ with
$\Delta=1.76\,k_{\scalebox{0.6}{\mbox{B}}}\,T_{c}$, where $R_{n}$ is the
normal state resistance, $T_{c}$ is the critical temperature and
$k_{\scalebox{0.6}{\mbox{B}}}$ the Boltzmann constant. $I_{*}$ is of order of
the critical current $I_{c}$ of the film ($I_{*}=3\sqrt{3}/2\,I_{c}$ [13]) and
depends on the superconducting material and geometry.

By the introduction of a DC bias applied to the input of the amplifier the
inductance equation becomes:

$L_{k}(I)=L_{d}\left(1+\varepsilon I+\xi
I^{2}\right)\,\,\mbox{with}\,\,L_{d}=L_{0}\left(1+\cfrac{I_{dc}^{2}}{I_{*}^{2}}\right)$
(2)

where $I_{dc}$ is the applied current, $L_{0}$ is the NbTiN kinetic
inductance, $L_{d}$ is the amplifier inductance per unit under DC bias at zero
RF current ($I=0$), $\varepsilon=2I_{dc}/(I_{*}^{2}+I_{dc}^{2})$, and
$\xi=1/(I_{*}^{2}+I_{dc}^{2})$ [14]. The term $\xi I^{2}$ permits the input
pump and signal to 4-wave-mix (4WM), producing an idler at
$f_{i}=2f_{p}-f_{s}$, while the term $\varepsilon I$ depends on the DC bias
and permits the 3-wave-mix (3WM), producing an idler at $f_{i}=f_{p}-f_{s}$.
In 4WM operating mode $L_{k}$ depends exclusively on the RF pump power
($P\propto I^{2}$) while in 3WM $L_{k}$ depends on both the RF pump power and
the DC bias ($I_{dc}^{2}$). An important consequence of this is that in 3WM
less RF pump power is required to generate RF mixing [15].

KI-TWPA amplifier working in 3WM and fabricated at NIST from a 20 nm thick
NbTiN film has shown promising performance [1, 14]. The developed design is
composed of a long transmission line consisting of several elementary cells.
Each cell consists of a coplanar waveguide (CPW), with inductance $L_{d}$, and
two interdigitated capacitor (IDC) stubs, one for both sides of the center
line, that form the capacitance to ground $C$ (figure 1). The stub length
$\ell$ is tuned in order to obtain a $C$ value such that
$Z_{0}=\sqrt{L_{d}/C}=50\,\Omega$ [16]. In order to generate the exponential
parametric amplification, a dispersion engineering has been accomplished using
periodic loadings created by decreasing the length of the stubs such that
$Z_{0}=\sqrt{L_{d}/C}=80\,\Omega$, as explained in [16] and [1]

Figure 1: Schematic of the KIT amplifier artificial transmission line with
three cells in series (not to scale) are arranged in a CPW topology.

Characterization results showed an average gain of 16.5 dB from 3.5 and 5.5
GHz, and a typical 1 dB compression power of -63 dBm at the amplifier input
within that bandwidth [1]. The measured system-added noise resulted in
$N_{\Sigma}=3.1\pm 0.6$ quanta (equivalent to $T_{N}=0.66\pm 0.15$ K),
suggesting that the KI-TWPA amplifier alone operates near the quantum limit.
This performance has been obtained with an intrinsic scaling current of
$I_{*}=7$ mA, a bias current of $I_{dc}=1.5$ mA, and a pump signal with a
power of -28 dBm at the input of the device. Despite these very promising
results, the developed device still requires a fairly high pump power to reach
the maximum gain. The pump must be isolated from the device under test by
components that unavoidably insert loss, thereby degrading the noise
performance of the chain. An amplifier functioning with a lower pump power may
necessitate fewer of these nefarious isolating components and may improve the
read out performance.

One solution to overcome this problem is to use thinner superconducting films.
In fact, decreasing the thickness $t$ of the film increases both the kinetic
inductance $L_{0}$ along with the inductive non-linearity. In addition, since
$I_{*}$ is proportional to the cross-sectional area of the film $A=t\cdot w$,
where $w$ is the inductor width, this means that by lowering thickness $t$ the
scaling current $I_{*}$ decreases. Considering equation 2 it is possible to
see that with a higher $L_{0}$ and a lower $I_{*}$ it is possible to obtain
the same inductive non-linearity with a lower pump power $P\propto I^{2}$ and
lower bias current $I_{dc}$.

Before designing a KI-TWPA amplifier based on thinner NbTiN films the complete
knowledge of the material and of its intrinsic parameters is a fundamental
requirement. To investigate the thickness dependence of the properties of the
film, we fabricated NbTiN films with different thicknesses $t=5,10,20$ nm. The
$20$ nm thickness, commonly used for the development of KI-TWPA amplifiers at
NIST, was used as a comparison since its intrinsic properties are widely
known. Following the developed fabrication procedure, the superconducting
films are deposited on a high-resistivity, intrinsically doped, float-zone
Silicon wafer. Immediately before loading into the deposition chamber, the
wafer is etched with hydrofluoric acid to remove the native oxide [17]. The
NbTiN is reactively co-sputtered from Nb and Ti targets in an Ar-N2 atmosphere
at 500 ∘C. Similarly to the thicker parametric amplifier films, the deposition
rates of the reactively sputtered NbN and TiN are tuned to maximize the
$T_{c}$ for the co-deposited alloy [18]. After deposition, the films are
lithographically patterned using an i-line stepper and are dry-etched in an
ICP-RIE with a CHF3 based plasma. This etch leaves an etched Si surface that
is compatible with high-Q microwave devices but with minimal trenching into
the Si that could result in a spatially varying impedance across the wafer.

For each thickness, we extrapolated the fundamental material properties by
performing DC measurements on superconducting test structures, and RF
measurements on superconducting resonators.

Figure 2: Normalized Sheet resistance vs temperature indicating a transition from a metallic normal state to the superconducting state for the different film thicknesses. Each transition curve has been measured in a different cool-down in a range of temperatures around the film’s critical temperature. TABLE I: Kinetic inductance and sheet resistance result from 4-terminal sensing measurements  $\bm{t}$ | $\bm{T}_{c}$ | $\bm{R}_{\scalebox{0.6}{\mbox{n}}}$ | $\bm{\rho}$ | $\bm{L}_{k}$
---|---|---|---|---
[nm] | [K] | [$\Omega/$sq] | [$\Omega\cdot$nm] | [pH/sq]
5 | 9.5 | 817 | 4089 | 118.6
10 | 11.6 | 277 | 2772 | 32.9
20 | 13.1 | 100 | 2002 | 10.5

##  II Kinetic Inductance extracted from resistances measurements

From the Mattis-Bardeen relationship, as discussed in the preceding section,
it is clear that the kinetic inductance increases as the sheet resistance
increases (or the critical temperature decreases). In order to measure the
critical temperature and the sheet resistance, the electrical resistances of
the film’s test structure were measured with the 4-terminal sensing
techniques. The transition curves as a function of the temperature are
reported in figure 2 and the obtained results for the extrapolated
characteristic parameters are reported in table I. From these measurements, it
is possible to see that as the film becomes thinner, $T_{c}$ decreases and
$R_{n}$ increases. This means that also $L_{k}$ increases, as expected. The
kinetic inductance measured for the 20 nm film is compatible with previous
results ($\sim 10$ pH/sq [1]) while for 10 nm and 5 nm the obtained values are
around 3 times and one order of magnitude larger, respectively.

##  III Kinetic Inductance extracted from resonator characterizations

To explore the thickness dependence, we have also fabricated lumped element LC
resonators from NbTiN films. For each resonator, the LC resonance is defined
by an interdigitated capacitor with $10\,\mu$m wide fingers and gaps and by a
strip inductor 1 or $2\,\mu$m wide, depending on the design. Starting from the
$L_{k}$ values from the DC characterization, we designed and tested two
different versions of resonator arrays, the former optimized for 20 and 10 nm
while the latter optimized for 5 nm. The resonator chip arrays are designed
with individual resonance frequencies distributed in the 1-4 GHz range. For
each resonator, we estimated the resonant frequency position as a function of
the film kinetic inductance by exploiting Sonnet EM simulations [19]. This
allows us to obtain a set of $f_{\scalebox{0.6}{\mbox{res}}}=f(L_{k})$
calibration curves [20] (figure 3, solid lines). Then $L_{k}$ values were
extrapolated by comparing the measured resonance frequencies with the
calibration curves (figure 3, black markers). The obtained results are
reported in table II and in figure 3. The resonance frequencies were measured
using a vector network analyzer (VNA) at a bath temperature of 50 mK. Each
acquired resonance was fitted by using a python framework developed for
microwave kinetic inductance detectors (MKID) for mm/submm astronomy [21].

Figure 3: Calibration curves for the simulated resonant frequencies (solid colored lines with filled circle points) as a function of the kinetic inductance compared with the measured values for the different resonators and film thicknesses (hexagon, square, and triangle filled points). The vertical dashed black line shows the average kinetic inductance values for every thickness. TABLE II: Kinetic inductance and internal qualities factor extrapolated for the different film thicknesses at 50 mk $\bm{t}$ | $\bm{L}_{k}$ | $\bm{Q}_{i}$
---|---|---
[nm] | [pH/sq] | [$10^{3}$]
5 | 93.3 | 30
10 | 31.5 | 50
20 | 10.6 | 70

The difference between the kinetic inductance values obtained with the RF
characterizations and the values calculated from the local Mattis–Bardeen’s
relationship (section II) is 1% for 20 nm the 4% for 10 nm, and the 21% for 5
nm. In the case of 20 nm the test structures and resonator came from the same
wafer and the difference is negligible. This is not true for 10 nm and 5 nm
and the larger difference can be justified by the different fabrication runs.
From past fabrications, we found a spread on the same wafer of 1-2% in the
film parameters (film thickness, critical temperature, and normal
resistivity). In separate wafers but using the same process steps and device
designs, the deviation was found to be less than 10%. Further improvements on
the fabrication procedure are in progress in order to minimize this effect.
The obtained values for the kinetic inductance $L_{k}$ have the same order of
magnitude and this is sufficient for designing a parametric amplifier, since
the line finger length is designed to have less than $Z_{0}=50\,\Omega$”,
target value that is then achieved by tuning the current bias.

For all the measured resonators, the internal quality factors $Q_{i}$ resulted
larger than $3\cdot 10^{4}$ and this means an internal loss $1/Q_{i}$ below
$3.3\cdot 10^{-5}$. As reported in [14] Appendix B, considering a KI-TWPA
amplifier composed by $\sim 6\cdot 10^{6}$ cells, any internal quality factor
larger than $5\cdot 10^{3}$ ($1/Q_{i}<2\cdot 10^{-4}$) produces an insertion
loss below $1$ dB. This means that for KI-TWPA amplifiers based on the films
characterized here, the device contribution to the total insertion loss would
be negligible.

In figure 4 the obtained $L_{k}$ values as a function of the film thicknesses
$t$ are shown for both the DC and RF characterizations. The obtained trends
allow us to roughly calibrate the fabrication process for obtaining the wanted
$L_{k}$ by tuning the $t$.

Figure 4: Kinetic inductance as a function of the film thickness, for both the
DC and RF characterizations. The dashed lines represent the best fits
performed with a $\propto 1/t^{\alpha}$ model.

##  IV Amplifier design for thinner thickness

The previous KI-TWPA amplifiers developed at NIST with films 20 nm NbTiN
thick, presented a kinetic inductance around $L_{k}=10$ pH/sq [18, 15, 1].
Considering this value of $L_{k}$, and considering the transmission line
design presented in Section I (figure 1), to have a capacitance $C$ to ground
such that $Z_{0}=50\,\Omega$ the stub length had to be $102\,\mu$m long on
both sides. If the kinetic inductance rises the $C$ value needed to match the
$Z_{0}=50\,\Omega$ condition is necessarily obtained with larger $\ell$. By
keeping the central line $w=1\,\mu$m and considering $L_{k}$ values of around
$30$ pH/sq ($t=10\,$nm) and $100$ pH/sq ($t=5\,$nm) here obtained, this means,
to first order, a stub length of 3 times and one order of magnitude longer,
respectively, with respect to the 20 nm version. Such stub lengths would not
be compatible with the compact design introduced in [1] in which $\sim 6\cdot
10^{6}$ cells fit in a $2\times 2$ cm2 chip by exploiting an optimized double
spiral topology. In particular, longer stub lengths would mean less space
available on the chip to pattern the device line, a larger radius for the
curved part of the line, a larger arc in the spiral corners, and less
repetition of the spiral patterns. All of these restrictions are translated
into a shorter KI-TWPA transmission line that would produce a lower gain.

Figure 5: The new KI-TWPA design with the central line is implemented as a
series of wider and narrower parts. In the three different configurations, the
wider part of the line covers $(a)$ one, $(b)$ two, and $(c)$ three stubs. The
heat map shown in each configuration is related to the RF current density in
the line during the simulations.

One way to keep short the stub lengths would be to enlarge the central line
width $w$. This would reduce the number of squares in the central line,
keeping low the inductance per cell, but, at the same time, since
$I_{*}\,\propto\,w\cdot t$, This would result in an increase in the scaling
current $I_{*}$, requiring a larger pump power to operate the KI-TWPA
amplifier. This runs counter to the aim of the study presented here. Another
possible solution is to pattern the central line as a series of wider and
narrower sections (figure 5). The narrower sections keep $I_{*}$ low while the
wider sections decrease the number of squares per cell which, in turn,
decreases the inductance per cell. Under this condition, following equation 1,
the total kinetic inductance is composed of two contributions,
$L_{k}=L_{k_{W}}+L_{k_{N}}$ (with $W$ for wide and $N$ for narrow). However,
since the current $I_{*}$ is proportional to the line width, if $w_{W}>w_{N}$,
then $I_{*W}^{2}>>I_{*N}^{2}$ and as a result, $L_{k}\simeq L_{k_{N}}$ is the
dominant contribution. Preliminary measurements performed with a tunable
resonator [22] showed a scaling current of around 3 mA for $w=1\,\mu$m and
around 40 mA for $w=13\,\mu$m. These values largely satisfy the above-
mentioned condition.

With this solution, the $Z_{0}=50\,\Omega$ condition can be satisfied with
shorter stub lengths. In order to find the best configuration we simulated
three different configurations, as reported in figure 5. These configurations
differ by the length of the wider parts: in the configuration $(a)$ this
length is $4\,\mu$m and covers one stub, in $(b)$ the length is $8\,\mu$m and
covers two stubs, and, finally, in $(c)$ the length is $13\,\mu$m and covers
three stubs. The goal of the performed simulations was to vary $w$ and $\ell$
for each configuration in order to find the proper $L$ and $C$ values per cell
such that $Z_{0}=50\,\Omega$, but with stub length $\ell$ around $100\,\mu$m.
The width of the narrower sections was fixed at $1\,\mu$m while the width of
the wider sections varied in $7-23\,\mu$m range.

Figure 6: Simulation results of line characteristic impedance $Z_{0}$ as a
function of the stub’s length $\ell$ for the configurations $(a)$, $(b)$, and
$(c)$ reported in figure 5.

In figure 5 the three different configurations with the current density map
are shown. The characteristic impedance $Z_{0}$ as a function of $w$ and
$\ell$ obtained from simulations is reported in figure 6. For the
configuration $(a)$ we found that no value of $\ell$ and $w$ satisfies the
$Z_{0}$ matching condition. In fact, despite the larger trace width $w$,
simulations show that the current remains concentrated in the central part of
the conductor as is shown in in figure 5$(a)$. Hence the effective width of
the central conductor remains smaller than drawn and the kinetic inductance
contribution of the line is too high to compensate. The resulting inductance
per cell remains high and the stub length needed for matching $50\,\Omega$
falls outside our range of interest (figure 6). For the configurations $(b)$
and $(c)$ the current density spreads more uniformly in the wider sections
(figure 5$(b)$ and 5$(c)$), the inductance per cell decreases, and it is
possible to find a set of stub lengths with a capacitance $C$ per cell such
that $Z_{0}=50\,\Omega$. Considering these simulation results we selected the
configuration $(c)$ with stub length $\ell=103\,\mu$m. This length value is
very close to the ones used for the previous 20 nm version. This allowed us to
design the new amplifiers using the same double spiral topology within the
same chip size and with a similar number of cells. The designed devices have
been recently produced at the NIST microfabrication facilities and are
currently under test.

##  V Conclusion

In this work we produced and characterized different NbTiN films in order to
investigate the thickness dependence of their intrinsic properties as the
critical temperature $T_{c}$, the normal resistivity $R_{n}$, the
superconducting kinetic inductance $L_{k}$, and the material internal loss
$1/Q_{i}$. By exploiting DC and RF characterizations we found that the kinetic
inductance increase with lowering the thickness. Starting from the results
obtained for NbTiN film 10 nm thick we have designed and simulated a new KI-
TWPA layout that would provide the same gain, bandwidth, and noise
performances of the previous 20 nm version but with lower pump power, thanks
to the larger kinetic inductance non-linearity. These new devices have been
recently produced and are currently under test at the novel sensor laboratory
at NIST.

## Acknowledgment

This work is supported by DARTWARS, a project funded by the European Union’s
H2020-MSCA Grant Agreement No. 101027746 and by the Italian Institute of
Nuclear Physics (INFN) within the Technological and Interdisciplinary Research
Commission (CSN5). This work is also supported by NIST through the Program on
Scalable Superconducting Computing.

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