# SuperVortexNet: Reconstructing Superfluid Vortex Filaments Using Deep
Learning

Nick A. Keepfer  0000-0002-7803-3713 Joint Quantum Centre (JQC)
Durham–Newcastle, School of Mathematics, Statistics and Physics, Newcastle
University, Newcastle upon Tyne, NE1 7RU, UK  Thomas A. Flynn
0000-0001-9800-594X Joint Quantum Centre (JQC) Durham–Newcastle, School of
Mathematics, Statistics and Physics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, UK  Nick G. Parker  0000-0002-7238-8492 Joint Quantum Centre
(JQC) Durham–Newcastle, School of Mathematics, Statistics and Physics,
Newcastle University, Newcastle upon Tyne, NE1 7RU, UK  Thomas P. Billam
0000-0003-2456-6204 Joint Quantum Centre (JQC) Durham–Newcastle, School of
Mathematics, Statistics and Physics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, UK

(August 28, 2024)

###### Abstract

We introduce a novel approach to the three-dimensional reconstruction of
superfluid vortex filaments using deep convolutional neural networks.
Superfluid vortices, quantum mechanical phenomena of immense scientific
interest, are challenging to image due to their small dimensions and intricate
topology. Here, we propose a deep-learning methodology that serves as a proof-
of-principle for fully reconstructing the topology of superfluid vortex
filaments. We have trained a convolutional neural network on a large dataset
of simulated superfluid density images obtained by solving the
Gross–Pitaevskii equation at scale, enabling it to learn the complex patterns
and features inherent to superfluid vortex filaments. The network ingests the
integrated density along the axial, coronal, and sagittal directions and
outputs the reconstructed superfluid vortex filaments in three dimensions. We
demonstrate the success of this approach over a range of vortex densities of
simulated isotropic quantum turbulence, enabling access to the characteristic
scaling law of the decaying vortex line length.

Superfluid vortices are the fundamental entities of rotation in a frictionless
superfluid, consisting of individual thin filaments around which the
superfluid rotates either clockwise or anticlockwise with quantised
circulation [1, 2]. The turbulent dynamics of tangled superfluid vortex
filaments — so-called quantum turbulence — has long been an object of study in
liquid Helium [3, 4]. However, the difficult problem of directly imaging the
superfluid vortex filaments themselves, which have a vortex core size on the
order of Angstroms, was only relatively recently solved using solid tracer
particles suspended in the liquid [5].

The advent of ultracold atomic Bose–Einstein condensates has provided another,
highly configurable platform for the realisation of superfluid vortex
filaments and quantum turbulence [6, 7]. In these systems the vortex core size
is typically on the order of microns, and superfluid vortex filaments can be
imaged directly through absorption imaging [8, 9, 10] or other techniques
[11]. In quasi-two-dimensional setups, where the vortex filaments are
constrained to approximate straight lines along the tightly confined direction
[12], this has enabled impressive realisations of phenomena in so-called two-
dimensional quantum turbulence. These include a superfluid von Kármán vortex
street [13] and Onsager vortex clustering [14, 15]. In three dimensions,
absorption imaging of outcoupled atoms has been used to observe the real-time
dynamics of straight vortex filaments [8], and interacting, curved, vortex
filaments in an elongated condensate [10]. The ability to image the tangle of
superfluid vortex filaments in three-dimensional quantum turbulence in similar
detail is highly desirable in order to validate theoretical predictions. For
example, the tangle is predicted to have either a classical or a so-called
ultraquantum form, depending on how the tangle is generated, with different
scaling laws for the decay of the vortex line length in each case [16].
However, with standard imaging techniques for condensates yielding images
projected onto a particular plane, direct observation of a three-dimensional
tangle of superfluid vortex filaments in these systems remains elusive.

In recent years, advances in machine learning, and more specifically
convolutional neural networks (CNNs), have opened up new possibilities for
imaging and analysing complex systems [17]. CNNs have demonstrated remarkable
success in image recognition tasks [18] and were recently applied to detection
of superfluid vortices in two dimensions for both synthetic [19] and
experimental [20] data. They are therefore a promising tool for the more
challenging case of detecting superfluid vortex filaments in three dimensions.
Filament-like structures are ubiquitous in the natural world. In the specific
use-case of identifying filament-like features, CNN’s have found success in
applications such as identifying road-cracks [21, 22], retinal vessel
segmentation [23, 24], root structure analysis in agriculture [25], river and
stream analysis form satellite data [26] and even cosmological filament
detection [27, 28].

In this paper we introduce SuperVortexNet (SVN), a neural network designed to
process two-dimensional integrated density profiles along the axial, saggital
and coronal directions into a three-dimensional binary voxelised array that
indicates the presence or absence of a superfluid vortex filament. We have
trained a variant of SVN (SVN-B) which is robust to noise in its input
integrated density profiles. We demonstrate on simulations of isotropic
quantum turbulence that this recovers the three-dimensional vortex
distribution over a range of vortex densities. This presents a breakthrough
towards experimental imaging of three-dimensional quantum turbulence and opens
the door to combine SVN with experimental two-dimensional absorption images so
as to observe real three-dimensional superfluid vortex tangles in detail.

## Results

Figure 1: Data collection pipeline: Initially, we run a GPE simulation of
decaying ultraquantum turbulence, similar to Ref. [29, 16]. The numerical
wavefunctions then pass through a spectral filter to remove high-frequency
noise. Following this, we run a vortex detection algorithm using a plaquette-
based method [30]. We then use the detected vortex filaments to construct a
density field using an ansatz solution [31] for the vortex cores; we refer to
this as a ‘cleaned’ density field with this specific, realistic configuration
of vortex filaments but no noisy density fluctuations. From this density field
we can obtain the integrated density profiles which serve as the inputs
(images) to our model. We also generate the ground truth (labels) of vortex
filaments in a voxelised grid by using the detected vortex filament positions
to populate a binary discretised grid and performing morphological inflation
(see “Methods”). As an additional aid to train our model, we use the voxelised
filaments to create line-of-sight depthmaps. In these the distance to each
vortex is encoded as a value up to unity along the axial, saggital, and
coronal directions. ∗We stress here that the depthmaps are only used as a
training aid, and are not needed at inference time for the model.

SuperVortexNet consists of a modified U-Net [32] followed by the expanding
path of a custom V-Net [33], constructed and trained using Flux.jl [34, 35]
(see “Methods” for details of the motivation of this deep learning approach,
the model architecture and training, and the loss functions used). Our data
collection pipeline, shown in Fig. 1, provides a point of reference from which
we outline the operation of the network. We gathered a training dataset of
three-dimensional isotropic quantum turbulence from simulations of the weakly-
dissipative Gross–Pitaevskii equation (GPE) simulations starting from highly
non-equilibrium initial conditions. This previously studied approach [29, 16]
is known to simulate a thermal quench of the system and give rise to an
isotropic tangle of vortices which decays over time (see “Methods” for the
details of training data acquisition). Post-processing of the simulation data
yields three datasets: a ground truth, in the form of detected superfluid
vortex filaments; input data, in the form of cleaned integrated density along
the axial, coronal and saggital directions, which are embedded into the three
colour channels of an input image; and additional data, used only as a
training aid and not for inference, in the form of line-of-sight depthmaps
along each direction. We trained two variants of the network. Variant SVN-A
learned directly from cleaned integrated densities, whereas variant SVN-B
learned from integrated densities with added noise. Specifically, we added
independent Gaussian random noise to each integrated density perspective with
variance $\sigma=0.1$, corresponding to $10$% of the background bulk density.
This added noise can obscure or distort the vortex signatures, making it more
difficult for the models to accurately identify and localise vortex filaments.
This provides a valuable test of the model’s capability to maintain high
accuracy and reliability in the presence of disturbances that are
representative of practical experimental conditions.

Our primary result for SVN-A is shown in Fig. 2. In the absence of density
noise SVN-A is able to reconstruct complicated three-dimensional superfluid
vortex filaments from unseen integrated-density test data with qualitatively
impressive performance. Quantitatively, when evaluating the network on our
test set, we see an average Dice loss coefficient of 0.244 and an average
Intersection over Union (IoU) of 0.538.

Figure 2: Ground truth (red) and SVN-A prediction (blue) for unseen test data.
Both are represented as 3D meshes. The ground truth mesh is simply a
voxelisation of vortex core positions whilst the prediction mesh is generated
by applying a probability threshold of $\epsilon=0.5$ to the 3D probability
field outputted by the neural network. This threshold is used to determine the
presence of a vortex within each voxel of the prediction volume. Voxels with a
probability equal to or greater than $\epsilon$ are considered to contain a
vortex.

Going beyond individual filament configurations, we tested SVN-B’s ability to
extract theoretically important measures of quantum turbulence from noisy data
by computing the line length of the reconstructed vortex filaments on unseen
data from simulations of decaying quantum turbulence. We generated an
additional, unseen test dataset for this purpose using the same simulation
method as outlined above for our training and test data, but run without
dissipation and over a far longer time period than the simulations used to
train the model. This was repeated over an ensemble of five trajectories with
randomised initial conditions. Note that the particular state of quantum
turbulence produced, the so-called ultraquantum turbulence, is characterised
by the vortex line length $L$ decaying according to the scaling law $L\sim
t^{-1}$[16].

Figure 3: Time evolution of the averaged vortex line length $L$ for an
ensemble of five simulations of decaying quantum turbulence. The ground truth
and SVN-B prediction for unseen, noisy, test data are depicted as orange and
purple lines, respectively. A dashed black line, scaling with $t^{-1}$,
illustrates the characteristic decay law of ultraquantum turbulence [16].
Insets show 3D representations of vortex configurations at specific times
$t=500\tau$ (red), $t=775\tau$ (blue) and $t=1000\tau$ (green). In each inset,
the true vortex configuration is shown in red and the prediction in blue. A
shaded grey region indicates the time window over which the neural network
received training data.

The extraction of proxy measurements such as this using SVN provides an
innovative way to quantify and analyse the complex topology of superfluid
vortices. Computing the total vortex line length in the system from the SVN
reconstruction simply involves tallying the lengths of all voxelised segments
that make up the reconstructed vortex filaments, taking the ablation of the
vortices into account using morphological erosion (see “Methods”). To validate
the accuracy of our vortex line length estimation, we compared the estimated
vortex line lengths from SVN-B’s reconstructions with the actual lengths
calculated directly from the GPE simulations in our additional test dataset,
as depicted in Fig. 3. The close correspondence observed in Fig. 3 underscores
the reliability of SVN in capturing this critical aspect of quantum
turbulence. Despite SVN-B being trained on realisations from $500\tau$ to
$1000\tau$ as indicated by the shaded region, we achieve good accuracy beyond
this timescale. That suggests that our model is successfully generalising to
different data outside of the original test set, which in turn suggests that
SVN has become proficient in the task of extracting the topology of vortices
just from the integrated density profiles alone. At times before $500\tau$
there is a clear deviation from the ground truth result. However, we expect
the very high density of vortex filaments in the tangle prior to this time to
provide much occlusion of filament segments by other filament segments along
every direction in the tangle. High numbers of occlusions make inference more
difficult for the model, and we believe the line length deviation at times
before $500\tau$ can be understood on these grounds. It is important to note
that the SVN line length prediction clearly recovers the $L\sim t^{-1}$
scaling behaviour to a similar level of accuracy as the GPE simulations
themselves, highlighting the potential of the SVN approach to measure and
verify decay laws in quantum turbulence experiments.

## Discussion

In this study, we have introduced a novel deep-learning methodology for the
three-dimensional reconstruction of superfluid vortex filaments. The SVN model
was trained on a large dataset of simulated superfluid density images,
generated from large-scale GPE simulations of isotropic quantum turbulence. We
find that SVN meets the challenge of reconstructing three dimensional vortex
filament structure and line length from limited two-dimensional, integrated-
density information from axial, coronal, and sagittal perspectives.
Furthermore, by applying SVN to simulations of decaying, isotropic quantum
turbulence, we demonstrate that it successfully measures the vortex line
length over a range of vortex densities and allows verification of the
expected scaling law for the decay of vortex line length.

We have focused on a clear, proof-of-principle demonstration of the
methodology in this work. However, we note that the integrated-density
information that the present version of SVN ingests is similar to the
information that would be experimentally accessible by absorption-imaging
outcoupled fractions of a Bose–Einstein condensate along multiple axes [8,
10]. Hence, this work is a major step towards experimental observation of a
three-dimensional vortex tangle and of experimentally measuring the decay of
vortex line length, a key probe of the nature of the turbulent state.

As the methodology shows promise, there are several avenues for future work.
Optimising the architecture and hyperparameters of our deep-learning model
could lead to further improvements in the efficiency and accuracy of vortex
reconstruction. A logical next step would also be the integration of real-
world experimental data to validate the efficacy of our approach, although we
expect the details of any experimental imaging system to necessitate some
custom training of the network. Moreover, given the recent advances in
transformer-based [36, 37] neural networks for a variety of complex tasks, we
envisage the adaptation of such architectures as a promising avenue for
enhancing the performance and interpretability of the methodology.

## Methods

### Deep Learning Approach

In the context of deep CNN approaches to identifying filamentary structures
the U-Net architecture has proven to be particularly effective, thanks to its
unique design that emphasises the localisation of features within an image
[32]. The U-Net architecture consists of a contracting path to capture context
and a symmetric expanding path that enables precise localisation. This design
has made it a preferred choice for tasks that require the detection of
intricate patterns and structures, such as the filament-like features
mentioned above. Similarly, the V-Net architecture has emerged as a powerful
tool for volumetric segmentation, particularly in the context of medical
imaging [33]. The V-Net extends the principles of the U-Net to 3D data,
employing a fully convolutional volumetric neural network that excels in
capturing complex spatial relationships. As mentioned above, SuperVortexNet
consists of a modified U-Net followed by the expanding path of a custom V-Net.
Instead of re-using a CNN for each individual view, as in multi-view networks
[38, 39], we instead embed our different views into the colour channels of our
input image. While multi-view networks have shown proficiency in generating 3D
volumes from 2D images of classical objects with relatively fixed topology,
their application to quantum vortices encounters significant challenges.
Unlike typical 3D objects such as chairs or cars, which offer topological and
dimensional regularities for the neural network to generalise from, quantum
vortices are quasi-one-dimensional filaments with a consistent thickness but
non-fixed, fluid topology. This makes it challenging to establish a ‘generic’
shape or structural basis to generalise across instances. The inherent
limitations in capturing such low-dimensional and topologically variable
features make multi-view networks an unsuitable choice for accurate 3D
reconstruction of quantum vortices. Our approach instead allows us to
efficiently process multiple views within a single CNN, reducing the
complexity and computational requirements of the model and allowing contextual
flow between channels. This integration allows the model to simultaneously
capture information from multiple perspectives, enhancing the feature
extraction process. The model architecture is illustrated in Fig. 4 and is
constructed and trained using Flux.jl [34, 35].

### Model Architecture

The architecture of our network is designed to address the unique challenges
of three-dimensional reconstruction of superfluid vortices, particularly
focusing on the intricate filament-like structures inherent to these
phenomena.

Figure 4: Schematic representation of the deep convolutional neural network
designed for the three-dimensional reconstruction of superfluid vortices. Key
modules include convolutional layers with Swish activation, depthwise
convolution, GroupNorm regularisation, and an effective Squeeze-Excite
mechanism. The architecture also integrates max pooling and pyramid pooling
modules to capture spatial hierarchies. H-Loss and D-Loss correspond to the
Huber loss and Dice loss metrics respectively. The bottom insets detail
specific functional blocks and their interactions, while the legend provides a
clear mapping of colours to individual operations or modules.

In Fig. 4 we detail the key components and innovative aspects of our
architecture. The initial part of the network, modelled after a U-Net
structure, consists of several 2D convolutional layers designed to map
integrated density profiles into line-of-sight depthmaps for each of the
axial, saggital and coronal planes simultaneously. The operational cycle of
our network is systematically structured, beginning with a convolution layer
that extracts feature maps from the input data. Following this, the “swish”
[40] activation function introduces non-linearity, allowing the network to
learn complex patterns. To prevent over-fitting and enhance generalisation, a
Dropout [41] layer randomly deactivates a subset of neurons during training.
Subsequent normalisation, specifically group normalisation [42], then
stabilises the learning process by normalising the input features. Finally, we
utilise Effective Squeeze-Excitation blocks, as described in [43], to allow
the network to re-calibrate the channel-wise feature responses, fine-tuning
the network’s focus on relevant features. This operational cycle is present
throughout the entirety of the network. In addition, both ResNet-style Skip-
Connections [44] and element-wise addition Skip-Connections are employed,
serving to mitigate the vanishing gradient problem by improving gradient flow,
thereby enhancing the training stability and model performance.

The contracting path of the U-Net utilises Pyramid-Pooling modules [45, 46,
47] which help to encode contextual spatial information into feature maps at
different scales. Specifically, for a feature map with dimensions [height,
width, channels], we employ pooling operations with kernels corresponding to
powers of $2$, up to $2^{N_{s}-1}$, where $N_{s}$ denotes the height or width
of the feature map at that specific depth in the network. The MaxPool
operation [48] is also utilised at this point in our network to reduce the
spatial dimensions of the feature maps. This process, by selecting the maximum
value over a specified window size, effectively condenses the feature
information. It not only helps in reducing computational load and controlling
overfitting but also aids in retaining the most significant features within
each pooling region. In contrast, the expanding path of the U-Net features
upsampling operations which perform bilinear interpolation on the feature maps
to upscale the dimensionality to match the corresponding image size on the
contracting path. At the final layer of the U-Net we apply a Softplus [49]
activation function to ensure positive outputs.

The V-Net section of our network focuses on converting the generated line-of-
sight depthmaps into a 3D voxel grid that represents the existence or absence
of vortex lines. This part of the architecture is made up of 3D upsampling
deconvolutional layers [50]. At each upsampling block, we add a skip-
connection path and an additional depth-wise convolutional operational block,
which helps in gradient flow, preserving the spatial hierarchies and feature
learning capabilities of the network respectively. At the final layer of SVN
we apply a Sigmoid activation function to map the output to a probability
distribution signifying the likelihood of a vortex being present at a
particular voxel position.

### Loss functions

We utilise a multi-task learning framework for our architecture, where our
main components, the U-Net and V-Net expanding path, are trained concurrently
but optimised for distinct objectives. For the first portion of the network we
aim to convert integrated density profiles into depthmaps, making use of the
Huber loss function [51]. The Huber loss provides a balanced compromise
between the Mean Squared Error and the Mean Absolute Error, effectively
handling outliers without suffering from large gradients that could
potentially destabilise the training process. The Huber loss is formulated as:

$\mathcal{L}_{\text{Huber}}(\theta)=\sum_{\text{batch},c,x,y}\phi(P_{c,x,y}-G_{c,x,y})\,.$
(1)

Here, $\phi(z)$ represents the Huber function and is defined as:

$\phi(z)=\begin{cases}\frac{1}{2}z^{2}&\text{if }|z|\leq\delta\\\
\delta(|z|-\frac{\delta}{2})&\text{otherwise.}\end{cases}$

with $\delta$ as a hyperparameter we set to unity. The summation
$\sum_{\text{batch},c,x,y}$ covers all batches, channels, and pixels. $P$ and
$G$ signify the predicted and ground truth values, respectively, for each
channel $c$ at pixel $(x,y)$. We use the calculated depthmaps to serve as
training labels for this loss function but stress that these depthmaps are
only neccessary during training and not at inference. The second part of the
network, tasked with projecting the depthmaps into a 3D voxelised grid,
utilises the Dice loss function [33], defined as:

$\mathcal{L}_{\text{Dice}}(\theta)=1-\frac{2\sum_{\text{batch},x,y,z}P_{x,y,z}\cdot
G_{x,y,z}}{\sum_{\text{batch},x,y,z}P_{x,y,z}^{2}+G_{x,y,z}^{2}}.$ (2)

This loss function is particularly effective for segmentation tasks involving
imbalanced classes, as it ensures an equitable representation of both the
foreground and background within the 3D grid. The combined use of Huber and
Dice loss functions at different points of the network, each tailored to its
respective task, contributes to the overall robustness and accuracy of our
model. Therefore, the complete loss function for SVN follows as

$\mathcal{L}=\mathcal{L}_{\text{Huber}}+\lambda\mathcal{L}_{\text{Dice}}\,,$
(3)

where we have introduced the hyperparameter $\lambda=0.01$ to balance the loss
contributions.

### Training Data Acquisition

Neural networks require vast amounts of data to effectively train and
generalise to new, unseen data. The quality and diversity of the training
dataset significantly influences the performance of the model. In this section
we introduce our methodology for gathering a suitable dataset for SVN.

We begin by modelling the dynamics of a superfluid system using the
Gross–Pitaevskii equation (GPE) [52, 53, 54]. Using this, we can parameterise
a weakly-interacting superfluid Bose-Einstein condensate via a mean-field
complex wavefunction, $\Psi(\mathbf{r},t)$, governed by the equation

$i\hbar\frac{\partial\Psi}{\partial
t}=\left(1-i\gamma\right)\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},t)+g|\Psi|^{2}\right]\Psi.$
(4)

Here we have introduced the reduced Planck constant $\hbar$, the interatomic
interaction strength $g$ to model binary inelastic collisions, and the atomic
mass $m$. We also introduce a weak dimensionless dissipation parameter
$\gamma=0.01$ to accelerate the dynamics of the system (although we set
$\gamma=0$ when generating the additional test dataset for the decay of
ultraquantum turbulence). Limiting our scope to the case of repulsive inter-
atomic interactions, we assign $g>0$ as is typical in experiments. The atomic
density follows as $n(\mathbf{r},t)=\lvert\Psi(\mathbf{r},t)\rvert^{2}$. We
then characterise the natural length scale, or healing length, as
$\xi=\hbar/\sqrt{m\mu}$ and the natural timescale as $\tau=\hbar/\mu$, where
$\mu$ is the chemical potential, where time-independent solutions of the GPE
satisfy $i\hbar\Psi_{t}=\mu\Psi$. We may then write the natural units of
density and corresponding speed of sound as $n_{0}=\mu/g$, and
$c=\sqrt{n_{0}g/m}$ respectively.

We now detail our data acquisition pipeline, which is depicted graphically in
Fig. 1. First, we initialise our GPE simulation with a highly non-equilibrium
state as described in [16] to emulate a thermal quench. We then solve the GPE
on a $128\times 128\times 128$ homogeneous periodic grid and run for
$1000\tau$, taking samples when the vortex tangles are less complicated past
$500\tau$ once every $2.5\tau$. We do this $50$ times to generate $10000$
samples and then use standard data augmentation techniques such as random
translations and flips to double this amount. The resulting dataset is then
randomly split into a training and test set with $18000$ and $2000$ samples,
respectively.

The resultant numerical wavefunctions following the quench contain a high
amount of noisy density fluctuations. We use a spectral filtering technique to
filter out high-frequency noise. We then use a vortex filament detection
algorithm based on the plaquette method [30] to locate the vortex filaments.
We generate a new three-dimensional density field based on the identified
structures using an analytic ansatz for the vortex core profile [31]; this
produces a ‘cleaned’ density field that contains all of the vortex filament
information present in the original state, without the density noise. Previous
studies have demonstrated resiliency for CNNs against noise; however in the
context of CNNs for quantum vortex detection [19] we expect each magnitude of
noise requires a full retraining of the network, which would be costly for our
comparatively more complex network. From the ‘cleaned’ density field we take
integrated density profiles from the axial, coronal, and saggital directions.
These clean profiles are used directly as inputs to the SVN-A variant of the
model, and are used after adding Gaussian random noise with variance
$\sigma=0.1$ as inputs to the SVN-B variant of the model. Our label data is
generated by converting the vortex positions into a voxelised array, wherein
we map the continuous points that form the core of each vortex onto the
corresponding voxels within a uniformly structured grid. This process
effectively converts the spatial information of vortex cores into a discrete,
grid-based representation suitable for use with SVN.

To address the class imbalance inherent in our data — where vortices occupy a
relatively small proportion of the spatial domain compared to the non-vortex
space — we apply a morphological dilation to each vortex filament. This
dilation process involves expanding the voxels that represent the vortex
filaments by a small, predefined amount. The effect is similar to ‘inflating’
the filament, thus increasing the number of voxels representing vortex
presence. In calculating the line length of the vortex filaments we
subsequently utilise the technique of morphological erosion. This process is
the inverse of dilation, where voxels at the boundaries of an object are
removed, effectively ‘shrinking’ the representation. This mitigates any
adverse effects from the dilation process, enabling a more accurate measure of
the vortex line length on the voxelised grid.

### Training and evaluation

In training SVN we use a small batch size of 4 due to computational
constraints. We use the Adam Optimiser with a learning rate of $\eta=0.001$
and momentum decay rates $\beta_{1}=0.9$, $\beta_{2}=0.999$. Implementing the
loss function as defined in Eq. 3, we train SVN-A for approximately 100
epochs, using the early stopping technique to prevent overfitting. After SVN-A
has finished training, we utilise transfer learning, using this model as the
starting point to train SVN-B, where Gaussian random noise is added to each
channel of the input images. To evaluate the performance of the two variants,
we consider the average dice loss coefficient, as defined in Eq. 2 as well as
introducing the Intersection over Union metric

$\text{IoU}=\frac{\sum_{\text{batch},x,y,z}I(P_{x,y,z}>\epsilon)\cdot
I(G_{x,y,z})}{\sum_{\text{batch},x,y,z}I[I(P_{x,y,z}>\epsilon)+I(G_{x,y,z})]}\,,$
(5)

where $P_{x,y,z}$ denotes the predicted probability at the voxel located at
coordinates $(x,y,z)$ in the prediction volume, while $G_{x,y,z}$ represents
the corresponding ground truth value at the same voxel location. The threshold
$\epsilon$ is a value that determines whether a voxel is classified as
containing a vortex or not, based on the predicted probability. The function
$I$ is an indicator function that returns $1$ if its argument is true, and $0$
otherwise. We calculate the mean value of these metrics over each element of
the validation dataset, as presented in Table 1. Importantly, we notice here
the expected superior performance of SVN-A over SVN-B, due to the added
complexity the added noise brings.

Model Variant | Mean Dice Coefficient | Mean IoU
---|---|---
SVN-A | 0.2444 | 0.5387
SVN-B | 0.3983 | 0.3632
Table 1: Comparison of Mean Dice Coefficient and IoU with $\epsilon=0.5$ for
SVN-A and SVN-B over each sample in the unseen validation dataset.

All simulations, post-processing, visualisation, neural network construction
and training were performed using Julia [55, 56, 57, 34], and we utilise CUDA
[58] to offload calculations to a graphics processing unit.

## Acknowledgments

The authors acknowledge support from the UK Engineering and Physical Sciences
Research Council (Grants No. EP/T015241/1, and No. EP/T01573X/1). T. A. F.
also acknowledges support from the UK Engineering and Physical Sciences
Research Council (Grant No. EP/T517914/1). This work made use of the Rocket
High Performance Computing service at Newcastle University. This work made use
of the facilities of the N8 Centre of Excellence in Computationally Intensive
Research (N8 CIR) provided and funded by the N8 research partnership and EPSRC
(Grant No. EP/T022167/1). The Centre is co-ordinated by the Universities of
Durham, Manchester and York.

## References

  * Onsager [1953] L. Onsager, in  _International Conference of Theoretical Physics_ (Science Council of Japan, Kyoto and Tokyo, 1953) p. 877.
  * Feynman [1955] R. P. Feynman, in  _Progress in Low Temperature Physics_, Vol. 1, edited by C. J. Gorter (North Holland, Amsterdam, 1955) p. 17.
  * Donnelly and Swanson [1986] R. J. Donnelly and C. E. Swanson, Journal of Fluid Mechanics 173, 387 (1986).
  * Barenghi  _et al._ [2001] C. F. Barenghi, R. J. Donnelly, and W. F. Vinen,  _Quantized Vortex Dynamics and Superfluid Turbulence_ (Springer, Berlin, New York, 2001).
  * Guo  _et al._ [2014] W. Guo, M. La Mantia, D. P. Lathrop, and S. W. Van Sciver, Proceedings of the National Academy of Sciences 111, 4653 (2014).
  * Wilson  _et al._ [2013] K. E. Wilson, E. C. Samson, Z. L. Newman, T. W. Neely, and B. P. Anderson, Annual Review of Cold Atoms and Molecules 1, 261 (2013).
  * Tsatsos  _et al._ [2016] M. C. Tsatsos, P. E. Tavares, A. Cidrim, A. R. Fritsch, M. A. Caracanhas, F. E. A. dos Santos, C. F. Barenghi, and V. S. Bagnato, Physics Reports 622, 1 (2016).
  * Freilich  _et al._ [2010] D. V. Freilich, D. M. Bianchi, A. M. Kaufman, T. K. Langin, and D. S. Hall, Science 329, 1182 (2010).
  * Lamporesi  _et al._ [2013] G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo, and G. Ferrari, Nature Physics 9, 656 (2013).
  * Serafini  _et al._ [2017] S. Serafini, L. Galantucci, E. Iseni, T. Bienaimé, R. N. Bisset, C. F. Barenghi, F. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev. X 7, 021031 (2017).
  * Wilson  _et al._ [2015] K. E. Wilson, Z. L. Newman, J. D. Lowney, and B. P. Anderson, Phys. Rev. A 91, 023621 (2015).
  * Rooney  _et al._ [2011] S. J. Rooney, P. B. Blakie, B. P. Anderson, and A. S. Bradley, Phys. Rev. A 84, 023637 (2011).
  * Kwon  _et al._ [2016] W. J. Kwon, J. H. Kim, S. W. Seo, and Y. Shin, Phys. Rev. Lett. 117, 245301 (2016).
  * Gauthier  _et al._ [2019] G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely, Science 364, 1264 (2019).
  * Johnstone  _et al._ [2019] S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, and K. Helmerson, Science 364, 1267 (2019).
  * Stagg  _et al._ [2016] G. W. Stagg, N. G. Parker, and C. F. Barenghi, Physical Review A 94, 053632 (2016).
  * LeCun  _et al._ [2015] Y. LeCun, Y. Bengio, and G. Hinton, Nature 521, 436 (2015).
  * Alzubaidi  _et al._ [2021] L. Alzubaidi, J. Zhang, A. J. Humaidi, A. Al-Dujaili, Y. Duan, O. Al-Shamma, J. Santamaría, M. A. Fadhel, M. Al-Amidie, and L. Farhan, Journal of Big Data 8, 53 (2021).
  * Metz  _et al._ [2021] F. Metz, J. Polo, N. Weber, and T. Busch, Machine Learning: Science and Technology 2, 035019 (2021).
  * Kim  _et al._ [2023] M. Kim, J. Kwon, T. Rabga, and Y.-I. Shin, Vortex detection in atomic Bose-Einstein condensates using neural networks trained on synthetic images (2023), arXiv:2308.08405 [cond-mat].
  * Zhang  _et al._ [2016] L. Zhang, F. Yang, Y. Daniel Zhang, and Y. J. Zhu, in  _2016 IEEE International Conference on Image Processing (ICIP)_ (IEEE, Phoenix, AZ, USA, 2016) pp. 3708–3712.
  * Di Benedetto  _et al._ [2023] A. Di Benedetto, M. Fiani, and L. M. Gujski, Infrastructures 8, 90 (2023).
  * Hajabdollahi  _et al._ [2018] M. Hajabdollahi, R. Esfandiarpoor, K. Najarian, N. Karimi, S. Samavi, and S. Reza-Soroushmeh, in  _2018 25th IEEE International Conference on Image Processing (ICIP)_ (IEEE, Athens, 2018) pp. 2785–2789.
  * Chala  _et al._ [2021] M. Chala, B. Nsiri, M. H. El Yousfi Alaoui, A. Soulaymani, A. Mokhtari, and B. Benaji, Expert Systems with Applications 184, 115459 (2021).
  * Atanbori  _et al._ [2019] J. Atanbori, M. E. Montoya-P, M. G. Selvaraj, A. P. French, and T. P. Pridmore, Frontiers in Plant Science 10 (2019).
  * Ling  _et al._ [2019] F. Ling, D. Boyd, Y. Ge, G. M. Foody, X. Li, L. Wang, Y. Zhang, L. Shi, C. Shang, X. Li, and Y. Du, Water Resources Research 55, 5631 (2019).
  * Gheller  _et al._ [2018] C. Gheller, F. Vazza, and A. Bonafede, Monthly Notices of the Royal Astronomical Society 480, 3749 (2018).
  * Aragon-Calvo [2019] M. A. Aragon-Calvo, Monthly Notices of the Royal Astronomical Society 484, 5771 (2019).
  * Berloff and Svistunov [2002] N. G. Berloff and B. V. Svistunov, Phys. Rev. A 66, 013603 (2002).
  * Foster  _et al._ [2010] C. J. Foster, P. B. Blakie, and M. J. Davis, Phys. Rev. A 81, 023623 (2010).
  * Fetter [2009] A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009).
  * Ronneberger  _et al._ [2015] O. Ronneberger, P. Fischer, and T. Brox, U-Net: Convolutional Networks for Biomedical Image Segmentation (2015), arXiv:1505.04597 [cs].
  * Milletari  _et al._ [2016] F. Milletari, N. Navab, and S.-A. Ahmadi, V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation (2016), arXiv:1606.04797 [cs].
  * Innes [2018] M. Innes, Journal of Open Source Software 3, 602 (2018).
  * Innes  _et al._ [2018] M. Innes, E. Saba, K. Fischer, D. Gandhi, M. C. Rudilosso, N. M. Joy, T. Karmali, A. Pal, and V. Shah, Fashionable Modelling with Flux (2018), arXiv:1811.01457 [cs].
  * Vaswani  _et al._ [2017] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, in  _Advances in Neural Information Processing Systems_, Vol. 30 (Curran Associates, Inc., 2017).
  * Dosovitskiy  _et al._ [2021] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, J. Uszkoreit, and N. Houlsby, An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale (2021), arXiv:2010.11929 [cs].
  * Choy  _et al._ [2016] C. B. Choy, D. Xu, J. Gwak, K. Chen, and S. Savarese, in  _Computer Vision – ECCV 2016_, Lecture Notes in Computer Science, edited by B. Leibe, J. Matas, N. Sebe, and M. Welling (Springer International Publishing, Cham, 2016) pp. 628–644.
  * Xie  _et al._ [2019] H. Xie, H. Yao, X. Sun, S. Zhou, and S. Zhang, in  _Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)_ (2019) pp. 2690–2698.
  * Ramachandran  _et al._ [2017] P. Ramachandran, B. Zoph, and Q. V. Le, Searching for Activation Functions (2017), arXiv:1710.05941 [cs] version: 2.
  * Srivastava  _et al._ [2014] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, J. Mach. Learn. Res. 15, 1929 (2014).
  * Wu and He [2018] Y. Wu and K. He, Group Normalization (2018), arXiv:1803.08494 [cs].
  * Lee and Park [2020] Y. Lee and J. Park, in  _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ (2020) pp. 13906–13915.
  * He  _et al._ [2015a] K. He, X. Zhang, S. Ren, and J. Sun, Deep Residual Learning for Image Recognition (2015a), arXiv:1512.03385 [cs].
  * Lazebnik  _et al._ [2006] S. Lazebnik, C. Schmid, and J. Ponce, in  _2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06)_, Vol. 2 (2006) pp. 2169–2178.
  * He  _et al._ [2015b] K. He, X. Zhang, S. Ren, and J. Sun, IEEE Transactions on Pattern Analysis and Machine Intelligence 37, 1904 (2015b).
  * Zhao  _et al._ [2017] H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, in  _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ (2017) pp. 2881–2890.
  * Scherer  _et al._ [2010] D. Scherer, A. Müller, and S. Behnke, in  _Artificial Neural Networks – ICANN 2010_ , edited by K. Diamantaras, W. Duch, and L. S. Iliadis (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010) pp. 92–101.
  * Zheng  _et al._ [2015] H. Zheng, Z. Yang, W. Liu, J. Liang, and Y. Li, in  _2015 International Joint Conference on Neural Networks (IJCNN)_ (2015) pp. 1–4, iSSN: 2161-4407.
  * Zeiler  _et al._ [2010] M. D. Zeiler, D. Krishnan, G. W. Taylor, and R. Fergus, in  _2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition_ (IEEE, San Francisco, CA, USA, 2010) pp. 2528–2535.
  * Huber [1964] P. J. Huber, The Annals of Mathematical Statistics 35, 73 (1964).
  * Pethick and Smith [2008] C. J. Pethick and H. Smith,  _Bose–Einstein Condensation in Dilute Gases_ (Cambridge University Press, 2008).
  * Pitaevskii and Stringari [2016] L. Pitaevskii and S. Stringari,  _Bose-Einstein Condensation and Superfluidity_ (Oxford University Press, 2016).
  * Barenghi and Parker [2016] C. Barenghi and N. G. Parker,  _A Primer on Quantum Fluids_, SpringerBriefs in Physics (Springer International Publishing, Cham, 2016).
  * Bezanson  _et al._ [2015] J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A Fresh Approach to Numerical Computing (2015), arXiv:1411.1607 [cs].
  * Omlin  _et al._ [2021] S. Omlin, L. Räss, N. Keepfer, G. Kwasniewski, B. Malvoisin, and Y. Y. Podladchikov, Solving Nonlinear Partial Differential Equations on GPU Supercomputers Using Julia (2021), Conference Presentation, PASC21 conference.
  * Danisch and Krumbiegel [2021] S. Danisch and J. Krumbiegel, Journal of Open Source Software 6, 3349 (2021).
  * NVIDIA Corporation [2023] NVIDIA Corporation, CUDA Toolkit Documentation, Version 12.3 (2023).


# Vortex to Rotons Transition in Dipolar Bose-Einstein Condensates

Alberto Villois  Dipartimento di Fisica, Università degli Studi di Torino, Via
Pietro Giuria 1, 10125 Torino, Italy  School of Engineering, Mathematics, and
Physics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ,
United Kingdom  Centre for Photonics and Quantum Science, University of East
Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom  Miguel
Onorato  Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro
Giuria 1, 10125 Torino, Italy  INFN, Sezione di Torino, Via Pietro Giuria 1,
10125 Torino, Italy  Davide Proment  School of Engineering, Mathematics, and
Physics, University of East Anglia, Norwich Research Park, Norwich, NR4 7TJ,
United Kingdom  Centre for Photonics and Quantum Science, University of East
Anglia, Norwich Research Park, Norwich, NR4 7TJ, United Kingdom  ExtreMe
Matter Institute EMMI, GSI Helmholtzzentrum fuer Schwerionenforschung,
Planckstrasse 1, 64291 Darmstadt, Germany

###### Abstract

Recent experimental advancements enabled the creation of various systems
exhibiting superfluid behavior, with one notable achievement being the
development of Dipolar Bose-Einstein condensates (dBECs) characterized by
long-range dipole-dipole interactions. When confined along one or more spatial
dimensions, these condensates exhibit a low-dimensional dispersion relation
reminiscent of superfluid liquid helium, featuring a roton minimum. This study
investigates the existence of two-dimensional solitary waves characterized by
topological defects (quantized vorticity) and their transition to vortex-free
Jones-Roberts solitons. The explicit connection between solitary waves and
roton excitations is demonstrated for the first time, providing support for
Feynman’s hypothesis regarding the nature of rotons as fading vortex
excitations.

Introduction. Solitary waves stand out as a primary feature in nonlinear
physics, finding application across diverse fields including fluid dynamics,
optics, and plasma physics, while offering profound insights into the
fundamental behaviour of complex systems. These self-sustaining wave patterns
defy the typical dispersion expected in wave motion, maintaining their shape
and velocity as they propagate through a medium. Their resilience against
dispersion stems from a delicate balance between nonlinear and dispersive
effects, making them stable entities within nonlinear systems.

Solitary waves characterized by the presence of vorticity hold particular
significance within the field of quantum fluids, first discovered in the
context of superfluid liquid helium [1]. Due to the topological nature of
quantized vorticity, such structures play a fundamental role in the study of
vortex dynamics and quantum turbulence [2]. Notable examples are the vortex
ring cascade hypothesis in three dimensions superfluids [3, 4] and the
Berezinskii-Kosterlitz-Thouless transition in two spatial dimensions [5].
Moreover, the study of such solitary waves have significantly contributed to
the understanding of the nature of the superfluid. Specifically, the
nucleation of solitary waves with quantized vorticity is believed, alongside
nucleation of rotons, to be one of the possible mechanisms for breaking
superfluidity in liquid helium, particularly in the low-pressure regime [6,
7].

The relationship between roton excitations and solitary waves has long been a
subject of fascination. Particularly, Feynman postulated the origin of drag in
a superfluid as the generation of roton excitations from a vanishing vortex
ring [3], sparking numerous studies aimed at determining whether the energy
and momentum of vortex rings were comparable to roton excitations. However,
this hypothesis was disproved in the seminal work of Jones & Roberts [8]: With
a weakly interacting model aimed to capture the key features of superfluid
liquid helium, they conducted an investigation into the existence of a family
of axisymmetric solitary waves. These waves exhibit quantized vorticity at low
speeds, taking the form of vortex rings in three dimensions and vortex dipoles
in two dimensions. As their speed increases, they transition into vortex-free
density dips, reaching speeds comparable to speed of phonons, the fastest
excitations in the system before superfluidity breaks down [9].

Over the years, numerous experimental advancements have facilitated the
generation of various systems exhibiting superfluid behavior, ranging from
single and multicomponent Bose-Einstein condensates (BECs) to spinor and Fermi
gases. One of the most notable achievements was the development of BECs
characterized by long-range dipole-dipole interactions, known as dipolar BECs
(dBECs) [10]. When one or more spatial dimensions are strongly confined, these
condensates have elementary excitations which display a dispersion relation
that, akin to superfluid liquid helium, features a roton minimum [11]. Such
elementary excitations have been recently experimentally measured in dBECs
[12] and enabled the generation of a new supersolid state of matter [13, 14,
15]. Unlike liquid helium, cold gases allow for the direct visualization of
small-scale vortex structures, as demonstrated recently [16, 17]. Moreover,
they can be accurately described by mean-field models based on the Gross-
Pitaevskii equation. These properties make such systems particularly
compelling for exploring the interplay between vortical structures and rotons.

In this Letter, we investigate the existence of two-dimensional solitary waves
characterized by topological defects and their transition to vortex-free
Jones-Roberts (JR) solitons in dBECs. The explicit connection between solitary
waves and roton excitations is demonstrated for the first time, providing
support for Feynman’s hypothesis regarding the nature of rotons as fading
vortex excitations.

Figure 1:  In all plots we take $\alpha=\pi/4$ and $\beta=0.9$. (a) Sketch of
the orientation of the dipole moment. (b) Non-dimenional Bogoliubov dispersion
relation along the $k_{x}$ (red) and $k_{y}$-axis (blue). The phonon limit is
shown in dashed lines. (c) Non-dimensional wave speed of the Bogoliubov
excitations along the $k_{x}$ (red) and $k_{y}$-axis (blue). The minimum value
of each curve indicate the maximum speed allowed before breaking
superfluidity.

Theoretical model. In the limit where the condensate is strongly confined
along the $z$-direction with a characteristic confining length scale $l_{z}$
the two-dimensional Gross-Pitaevskii equation for dBEC in the $x$-$y$ plane
reads as

$i\hbar\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+g_{\rm c}|\psi|^{2}\psi+g_{\rm
d}\Phi\psi\,.$ (1)

Here, $\psi(x,y,t)$ is the two-dimensional order parameter,
$\nabla^{2}=\partial_{xx}+\partial_{yy}$, $g_{\rm
c}=2\sqrt{2\pi}\hbar^{2}a_{\rm s}/(ml_{z})$ is the contact (local) interaction
coefficient, $m$ and $a_{\rm s}$ are the mass and the s-wave scattering length
of the boson, respectively, and $g_{\rm d}=|{\bf d}|^{2}/\sqrt{2\pi}l_{z}$ is
the dipole-dipole (nonlocal) interaction coefficient where $\bf d$ is the
dipole moment. The nonlocal dipolar interaction operator is defined as

$\Phi=\frac{4\pi}{3}\mathcal{F}^{-1}\left[\mathcal{F}\left(|\psi|^{2}\right)F\left({\bf
q}\right)\right]$ (2)

where $\mathcal{F}(\cdot)$ is the spatial two-dimensional direct Fourier
transform operator and ${\bf q}={\bf k}l_{z}/\sqrt{2}$ is the non-dimensional
wave vector on the $x$-$y$ plane. Without any loss of generality, we consider
the dipole moment aligned with the $x$-axis and define $\alpha$ being the
angle between ${\bf d}$ and the $z$-axis, see Fig. 1 (a). With this choice the
function $F(\cdot)$, that represents the Fourier transform of the two-
dimensional dipole-dipole interaction [18, 19], results in

$F({\bf q})=\cos^{2}\alpha F_{\perp}(q)+\sin^{2}\alpha
F_{\parallel}(q,q_{x})\,,$ (3)

where $q=|{\bf q}|$, $q_{x}$ being the $x$-component of ${\bf q}$, and

$\displaystyle F_{\perp}(q)=2-3\sqrt{\pi}qe^{q^{2}}{\rm erfc}(q)$
$\displaystyle
F_{\parallel}(q,q_{x})=-1+3\sqrt{\pi}\frac{q_{x}^{2}}{q}e^{q^{2}}{\rm
erfc}(q)\,.$

By assuming the homogeneous solution in the form

$\psi(t)=\sqrt{n}e^{-i\frac{\mu t}{\hbar}}$ (4)

where $|\psi|^{2}=n$ is the average particle number density per unit of
volume, one finds the chemical potential

$\mu=g_{\rm
c}n\left[1+\left(3\cos^{2}\alpha-1\right)\frac{4\pi}{3}\beta\right]\,,\quad\text{with
{\hbox{ \beta=\frac{g_{\rm d}}{g_{\rm c}} }},}$ (5)

and its infinitesimal wave-like perturbations of wavelength ${\bf
k}=(k_{x},k_{y})$, and angular frequency $\omega_{\rm Bog}$ to satisfy the
Bogoliubov dispersion relation [18]

$\omega_{\rm Bog}({\bf k})=\pm\frac{|{\bf
k}|}{\sqrt{m}}\sqrt{\frac{\hbar^{2}|{\bf k}|^{2}}{4m}+g_{\rm
c}n\left[1+\beta\frac{4\pi}{3}F\left(\frac{l_{z}{\bf
k}}{\sqrt{2}}\right)\right]}\,.$ (6)

Finally, by introducing the healing length, $\xi=\hbar/\sqrt{2m\mu}$, one
finds the phonon speed (long wave perturbation) as $c_{\rm ph}=\sqrt{\mu/m}$.
One notices that, when $\alpha\neq 0$, Eq. (3) is anisotropic in ${\bf k}$,
hence the Bogoliubov dispersion. Figure 1 (b) illustrates the two dispersion
relation branches along the $k_{x}$ and $k_{y}$ directions, choosing
$\alpha=\pi/4$ and $\beta=0.9$. For such values, used for all the results
presented in this Letter, the dispersion relation along the $k_{x}$ direction
shows a point where the quantity $\omega(k_{x},0)/k_{x}$ possesses a global
minimum and corresponds to an analogue of the roton minimum measured in
superfluid liquid helium, therefore called here roton-like minimum. Note that
this minimum is present for a large range of $(\alpha,\beta)$, and our choice
is achievable in experimental setups [10].

Figure 2:  The blue line shows the non-dimensional energy-momentum plot of
solitary wave family moving along the $x$ and $y$-directions, given in plots
(a) and (b), respectively. Each point on the curve is a particular solitary
wave whose speed is given by the slope of the tangent to the curve and density
profile $n=|\psi|^{2}$ is shown in the respective inset. The points (II)
indicate where the transition between a vortex dipole to a JR soliton takes
place. The black and red dashed lines represent the phonon and roton speed,
respectively.

In this Letter we investigate the existence of solitary wave solutions in two-
dimensional dBECs and revisit Feynman’s hypothesis, originally formulated in
superfluid liquid helium, on the vortex-roton transition. Jones & Roberts have
demonstrated the existence of axisymmetric solitary waves within the Gross-
Pitaevskii equation characterised only by contact interactions [8]. Their
results were reproduced experimentally in [16]. In two dimensions, these
solutions exhibit counter-rotating quantised vortices at low speeds that
transition to density pulses free of topological defects at speeds comparable
to the phonon speed. It is pedagogical to identify first the range of possible
speeds at which a solitary travelling wave can propagate. To do so, one relies
on the so-called band gap analysis introduced in [20]. The idea behind the
band gap analysis is the following: a localised solitary wave (fully
nonlinear) excitation moving at velocity ${\bf v}=(v_{x},v_{y})$ can exist on
the top of the homogeneous solution (4) only if

${\bf v}\cdot{\bf k}\neq\omega_{\rm Bog}({\bf k})$ (7)

This is to prevent the resonant transfer of energy from the solitary wave to a
Bogoliubov excitation. By using this criterion, it is straightforward to
identify the range of speed values that the solitary wave is allowed, as shown
in Fig. 1 (c). Note that Eq. (7) is equivalent to Landau’s criterion for the
breakdown of superfluidity, demonstrating that the along the $x$-direction
superfluid excitations cannot move faster than the phonon speed, while along
the $y$-direction the limiting speed is given by the roton speed $c_{\rm
rot}={\rm min}\left(\omega_{\rm Bog}(0,k_{y})/k_{y}\right)$.

Numerical methods. The solitary wave solutions are computed numerically by
solving a time-independent Gross-Pitaevskii equation written in the reference
frame moving with velocity ${\bf v}$, see the SM for more details. The system
is spatially discretized on a regular lattice with $nx,ny=2^{9}$ collocation
points over a box with size $L_{x}/\xi,L_{y}/\xi=80\pi$. Periodic boundary
conditions are considered in order to compute differential operators making
use of Fourier spectral decomposition. The time-independent solution is found
via a Newton-Raphson method, and the JR branch is followed by slowly varying
the solitary wave speed. We also performed time evolution of the time-
dependent Gross-Pitaevskii equation to analyse qualitatively the stability of
the solitary waves; the time operator is resolved using a standard RK4 method.
A thorough linear stability analysis is performed by looking at the spectral
eigenvalues of the Bogoliubov-de Gennes (linearised) perturbations.

Numerical results. First, we investigate the family of solitary waves
propagating along the direction of the dipole polarisation ($x$-direction),
that is choosing $v_{y}=0$. Each point on the blue line in Fig. 2 (a)
represents the energy-momentum coordinates of a solitary wave solution. The
$x$-component of the velocity of each solitary wave is determined by the slope
of this curve, as given analytically by $v_{x}=\partial\epsilon/\partial
p_{x}|_{v_{y}=0}$, being $\epsilon$ and $p_{x}$ the energy and $x$-component
of the linear momentum densities of the solitary wave, respectively (see SM
for more details). Density profiles about the localised wave are shown in the
insets. Point (I) illustrates a vortex dipole moving along the y-direction,
the transition between a solution with and without vorticity is indicated by
point (II), point (III) shows a density dip. The vortex core density profiles
in the dipole solution are elongated along the polarisation direction, in
agreement with previous results [21, 22]. Here we show that such elongation is
still present in the absence of vorticity. Although, it is lost in the limit
for speeds approaching the phonon speed, see dashed black line in Fig. 2 (a),
with a JR soliton resembling the typical profile of lump soliton solutions for
the so-called Kadomtsev-Petviashvili equation [23].

We then explore the family of solitary waves moving perpendicularly to the
direction of the dipole polarisation ($y$-direction), that is choosing
$v_{x}=0$, whose results are plotted in Fig. 2 (b). Here, the $y$-component of
the velocity of each solitary wave is determined by the slope of this curve,
resulting in $v_{y}=\partial\epsilon/\partial p_{y}|_{v_{x}=0}$ where $p_{y}$
is the $y$-component of the linear momentum density of the solitary wave.
Density profiles about the localised waves are shown in the insets. Similarly
to the previous case, both vortex dipole solutions and JR solitons have
density profiles elongated along the polarisation direction, see point (I) and
point (II) in Fig. 2 (b), respectively. However, unlike the previous case, the
JR solitons cannot approach the phonon speed, resembling lump soliton
solutions, as Landau’s criterion for superfluidity breaking along the
y-direction is determined by the roton speed, shown by a dashed red line in
Fig. 2 (b). While approaching the roton speed, a transition from localised JR
solitons to delocalised roton excitations appearing as stripes becomes
qualitatively visible, see point (III) in Fig. 2 (b).

Figure 3:  (a) Spatial spectra of JR solitary wave solutions moving along the
$y$-direction plotted for ${\bf k}=(0,k_{y})$ with different $v_{y}$ tending
to the roton speed. (b) Spatio-temporal spectrum,
$n(k_{y},\omega)|_{k_{x}=0}=|\mathcal{F}_{y,t}(\psi)|^{2}$, of the solitary
wave solution associated to point (III) in Fig. 2 (b). Dynamics of such
solitary waves cover a distance $L=10L_{y}$. Spectrum is plotted using
logarithmic scale. White noise is present in the initial condition. Signal is
made periodic in time using a double $\tanh^{2}$ filter.

To further characterise such a transition, we analyse the spectral properties
of the solitary wave branch presented in Fig. 2 (b). Figure 3 (a) shows the
spatial spectra along the $k_{y}$-axis of the solitary waves found in Fig. 2
(b) for speeds, $v_{y}/c_{\rm ph}$, varying from $0.38$ to $0.54$. It is
evident that a peak around the roton-like minimum wave number $k_{\rm rot}$
emerges and grows for growing speeds; this is a first indication that the
travelling wave solution approaches the roton-like Bogoliubov excitation when
its speed tends to the roton speed. To corroborate this claim we perform a
time evolution of the fastest solitary wave solution, see point (III) in Fig.
2 (b). White noise is added in the initial condition (noise to uniform
solution ratio is order of $10^{-2}$) to test the robustness of the localised
wave. After the solitary wave has evolved covering a distance 10 times larger
than the system size, a spatio-temporal spectrum,
$n(k_{y},\omega)|_{k_{x}=0}=|\mathcal{F}_{y,t}(\psi)|^{2}$, is measured. The
spectral content associated with the solitary wave corresponds to a straight
signal in Fig. 3 (b), having slope equal to the solitary wave speed
$v_{y}/c_{\rm ph}$, see dotted white line. Elementary excitations introduced
by the presence of the noise in the initial condition, evolve according to the
Bogoliubov dispersion, see solid white line. The Bogoliubov excitation branch
almost meets the solitary wave signal at the roton-like wave number $(k_{\rm
rot}$, indicating that a resonance will occur in the limit of the solitary
wave speed tending to the roton speed. An animation in the SM shows the
evolution of the density and phase fields, showing visually that the solitary
wave is preserved by the time evolution in presence of the added noise. A full
Bogoliubov-de Gennes stability analysis, shown in the SM, has been carried out
in this case demonstrating that the noise perturbations are linearly stable.

Conclusions and outlook. In this Letter we demonstrated the existence of the
Robert-Jones travelling wave solutions in two-dimensional dipolar condensates,
showing that the solution branch is formed by a vortex dipole that
continuously transits into a topologically-free density dip when exceeding a
critical speed. The JR branch strongly depends on the alignment between the
solitary wave velocity and the polarisation direction. When they are fully
aligned, the JR soliton maximum speed tends to the phonon speed, like in
standard (non-dipolar) condensates, while when they are completely orthogonal
the limiting speed is the one given by the roton-like excitation. Note that in
both cases the solitary wave excitation is subsonic and does not break the
superfluidity as its velocity cannot be larger than the Landau’s critical
speed, which depends on the polarisation direction in dipolar condensates.
Finally, we have investigated the fate of the JR soliton moving
perpendicularly to the polarisation direction: at its speed increases the
solution builds a spectral peak that tends to the roton-like Bogoliubov
excitation, validating the analogy of Feynman’s hypothesis regarding the
nature of rotons as fading vortex excitations. Preliminary investigations on
the stability of such JR solitons, both dynamically adding small-amplitude
noise and semi-analytically within the linearised Bogoliubov-de Gennes
framework, have shown that these solitary waves appear to be stable and should
be within the parameter range of feasibility in state-of-the-art dipolar gas
experiments [24]. Open questions like how to generate such excitations, for
example by moving an external potential, their scattering interaction, and the
characterisation of their sonic limit in terms of Kadomtsev-Petviashvili-like
solitons [23] will be the scope of the next publications.

###### Acknowledgements.

M.O. was funded by Progetti di Ricerca di Interesse Nazionale (Grants No.
2020X4T57A and No. 2022WKRYNL) and by the Simons Foundation (Grant No. 652354)
D.P. would like to thank the Isaac Newton Institute for Mathematical Sciences
for support and hospitality during the programme Dispersive hydrodynamics:
mathematics, simulation and experiments when the final part of the work on
this paper was undertaken. This work was supported by EPSRC Grant Number
EP/R014604/1. This research was supported in part by the ExtreMe Matter
Institute EMMI at the GSI Helmholtzzentrum fuer Schwerionenphysik, Darmstadt,
Germany.

## References

  * Donnelly [1991] R. J. Donnelly,  _Quantized vortices in helium II_ , Vol. 2 (Cambridge University Press, 1991).
  * Walmsley and Golov [2008] P. M. Walmsley and A. I. Golov, Phys. Rev. Lett. 100, 245301 (2008).
  * Feynman [1955] R. Feynman (Elsevier, 1955) pp. 17–53.
  * Nemirovskii [2014] S. K. Nemirovskii, Low Temperature Physics 40, 1116 (2014), http://dx.doi.org/10.1063/1.4902230 .
  * Hadzibabic and Dalibard [2011] Z. Hadzibabic and J. Dalibard, La Rivista del Nuovo Cimento 34, 389 (2011).
  * Nancolas  _et al._ [1985] G. G. Nancolas, T. Ellis, P. V. E. McClintock, and R. M. Bowley, Nature 316, 797 (1985).
  * Ellis  _et al._ [1985] T. Ellis, P. V. E. McClintock, and W. F. Vinen, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 315, 259 (1985), https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1985.0041 .
  * Jones and Roberts [1982] C. A. Jones and P. H. Roberts, Journal of Physics A: Mathematical and General 15, 2599 (1982).
  * Pitaevskii and Stringari [2016] L. Pitaevskii and S. Stringari,  _Bose-Einstein condensation and superfluidity_ , Vol. 164 (Oxford University Press, 2016).
  * Chomaz  _et al._ [2022] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Reports on Progress in Physics 86, 026401 (2022).
  * Godfrin  _et al._ [2021] H. Godfrin, K. Beauvois, A. Sultan, E. Krotscheck, J. Dawidowski, B. Fåk, and J. Ollivier, Phys. Rev. B 103, 104516 (2021).
  * Petter  _et al._ [2019] D. Petter, G. Natale, R. M. W. van Bijnen, A. Patscheider, M. J. Mark, L. Chomaz, and F. Ferlaino, Phys. Rev. Lett. 122, 183401 (2019).
  * Tanzi  _et al._ [2019] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Phys. Rev. Lett. 122, 130405 (2019).
  * Böttcher  _et al._ [2019] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Phys. Rev. X 9, 011051 (2019).
  * Chomaz  _et al._ [2019] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Phys. Rev. X 9, 021012 (2019).
  * Meyer  _et al._ [2017] N. Meyer, H. Proud, M. Perea-Ortiz, C. O’Neale, M. Baumert, M. Holynski, J. Kronjäger, G. Barontini, and K. Bongs, Phys. Rev. Lett. 119, 150403 (2017).
  * Serafini  _et al._ [2015] S. Serafini, M. Barbiero, M. Debortoli, S. Donadello, F. Larcher, F. Dalfovo, G. Lamporesi, and G. Ferrari, Phys. Rev. Lett. 115, 170402 (2015).
  * Ticknor  _et al._ [2011] C. Ticknor, R. M. Wilson, and J. L. Bohn, Phys. Rev. Lett. 106, 065301 (2011).
  * Mulkerin  _et al._ [2014] B. C. Mulkerin, D. H. J. O’Dell, A. M. Martin, and N. G. Parker, Journal of Physics: Conference Series 497, 012025 (2014).
  * Villois and Skryabin [2019] A. Villois and D. V. Skryabin, Opt. Express 27, 7098 (2019).
  * Prasad  _et al._ [2024] S. B. Prasad, N. G. Parker, and A. W. Baggaley, Phys. Rev. A 109, 063323 (2024).
  * Mulkerin  _et al._ [2013] B. C. Mulkerin, R. M. W. van Bijnen, D. H. J. O’Dell, A. M. Martin, and N. G. Parker, Phys. Rev. Lett. 111, 170402 (2013).
  * Baronio  _et al._ [2016] F. Baronio, S. Wabnitz, and Y. Kodama, Phys. Rev. Lett. 116, 173901 (2016).
  * Kwon  _et al._ [2021] W. J. Kwon, G. Del Pace, K. Xhani, L. Galantucci, A. Muzi Falconi, M. Inguscio, F. Scazza, and G. Roati, Nature 600, 64 (2021).