Neeraj Kumar1, Ankur Narang2, Brejesh Lall1

# Style Description based Text-to-Speech with Conditional Prosodic Layer
Normalization based Diffusion GAN

###### Abstract

In this paper, we present a Diffusion GAN based approach (Prosodic Diff-TTS)
to generate the corresponding high-fidelity speech based on the style
description and content text as an input to generate speech samples within
only 4 denoising steps. It leverages the novel conditional prosodic layer
normalization to incorporate the style embeddings into the multi head
attention based phoneme encoder and mel spectrogram decoder based generator
architecture to generate the speech. The style embedding is generated by fine
tuning the pretrained BERT model on auxiliary tasks such as pitch, speaking
speed, emotion,gender classifications. We demonstrate the efficacy of our
proposed architecture on multi-speaker LibriTTS and PromptSpeech datasets,
using multiple quantitative metrics that measure generated accuracy and MOS.

Index Terms: Diffusion GAN,Speech Synthesis, Normalization, transfer learning

##  1 Introduction

Text to speech (TTS) tries to synthesise a genuine and comprehensible voice
from text and garners significant interest from the machine learning field [1,
2, 3]. TTS models are able to synthesis natural human speech when trained with
a huge quantity of high-quality and single-speaker recordings and this
capability has been extended to multi-speaker settings [4, 5, 6]. Today,
custom voice is gaining popularity in a variety of application situations,
including personal assistant, news broadcast, and audio navigation, and is
extensively supported by commercial speech platforms.

Prior research on TTS has focused on regulating certain style aspects,
including prosody control using word-level prosody tags[7], speaking speed
control[8] using sentence-level speaking-rate, and pitch control using pitch
contours. All previous works require users to input the precise style factor
value with acoustic expertise or choose the reference speech that matches the
requirements, which is time-consuming and not user-friendly. There is a trade
off between fine-tuning parameters and voice quality while converting the
source TTS model to a new voice which is frequently recorded in various
speaking styles, emotions, dialects, and surroundings. Therefore, style
control with natural language text is preferable.

We investigate using a text description(prompt) to guide speech synthesis. The
input prompt consists of a style description and a content description with a
colon in between. “A lady whispers to her friend slowly: everything will go
OK, right?” requires the model to synthesis speech with the content
“everything will go fine, right?” in a female voice, a slow speaking tempo,
and a whispering manner. Users can compose speech from a style text without
acoustic knowledge or reference speech, allowing style freedom.

Some of the existing work, PromptTTS[9]has used style encoder to extract the
style token from style description and content encoder and speech decoder to
generate the final speech. We have proposed the diffusion model based
framework which has the ability to model complex data distribution to solve a
variety of speech synthesis problems[10, 11]. We have used the denosing
diffusion GAN[12] that uses transformer[13] based encoder-decoder based
generator architecture to generate the mel-spectrogram conditioned on
timesteps, intermediate mel-spectrogram and style embeddings. We have
extracted the style tokens from fine tuning the pretrained BERT[14] on
auxiliary task and fed into the denosing diffusion GAN through the proposed
conditional prosodic layer normalization. Our contribution are as follows:

  * •

Partially inspired by the denoising diffusion GAN[12], We model the denoising
distribution using a conditional generator that has been adversarially trained
to match the actual denoising distribution. Prosodic Diff-TTS permits larger
denoising steps at inference, hence drastically reducing the number of
denoising steps and accelerating sampling.

  * •

We have used the pretrained BERT model to learn the 128 dimensional style
token in multi task learning fashion using the cross entropy loss for
optimization.

  * •

To make the generator conditional on style tokens, we have proposed
Conditional Prosodic Layer normalization to inject the style into the
denoising generator model to learn the given style information as well as
other style-agnostic prosodic variations.

  * •

Using extensive experiments on multi-speaker PromptSpeech [9] and LibriTTS[15]
datasets, we show both qualitative and quantitative results along with high-
quality output speech given the input style text and content description.

##  2 Model Architecture

Commonly, diffusion models assume the denoising distribution can be
approximated by Gaussian distributions, necessitating a lengthy reverse
operation. As the denoising step is increased and the non-Gaussian data
distribution is present, the underlying denoising distribution becomes more
complex and multimodal. We have adopted the denosing diffusion GAN to model
the multimodal denoising distribution where the generator produces the mel-
spectrogram($x_{0}$) given the content($y$) and style token through the
conditional prosodic layer normalization. The discriminator distinguishes the
fake and real mel-spectrograms conditional on style embeddings.

Figure 1: Prosodic Diff-TTS architecture Figure 2: Left: Multi Head Attention
block , Centre : Variance Adapter[16] , Right : Variance Predictor

###  2.1 Generator

As illustrated in Figure 1, Prosodic Diff-TTS takes phoneme sequence (denoted
as $\mathbf{y}$) as input to generate mel-spectrogram features
$\mathbf{x}^{\prime}_{0}$ with generator and then uses a HiFi-GAN-based neural
vocoder [17] to produce time-domain waveforms. The generator architecture uses
the phoneme encoder, conditional style layer normalization framework, variance
adaptor, mel-spectrogram decoder along with denoiser conditioned on style
token, timesteps and noisy melspectrogram to generate mel-spectrogram features
$\mathbf{x}_{0}$. The phoneme encoder and mel-spectrogram decoder is based on
a multi-head self-attention network, and position feed-forward network which
consists of two Conv1D and normalization stages. The proposed method stacks
multiple multi-head self-attention[13] blocks with phoneme embedding and
position encoding as an input at the encoder side(Fig. 1), and multiple multi-
head self-attention blocks with position encoding,style token and output from
variance adapter[16](Fig. 2) for the mel-spectrogram generation at the decoder
side.

Figure 3: Style token Generator using pretrained BERT  Figure 4: Conditional
Prosodic Layer Normalization architecture

###  2.2 Style token generator

We have trained the pretrained BERT to generate the 128 dimensional style
embeddings in multi task learning fashion with auxiliary tasks related to
prosodic features from the style text such as pitch , gender, speaking speed,
emotion and volume as shown in Figure 3.The input style text sequence T = [T1,
T2, $\cdots$ , TM] is prepended with a [CLS] token, converted into a word
embedding, and fed into the BERT model, where M refers to the length of style
text.The hidden vector corresponding to the [CLS] token is regarded as the
style representation to guide the content encoder and the speech decoder.

###  2.3 Proportional Prosodic Layer Normalization

Layer normalization could significantly impact the hidden activation and final
prediction through learnable scale and bias in multi head attention block as
given in equations 1 and 2. We have proposed a conditional prosodic style
layer normalization(Fig. 4) which is employed at the phoneme encoder, mel-
spectrogram decoder and denoiser module. The style embeddings are passed into
the linear layer to generate the style related affine parameters namely
$\gamma_{style}$ and $\beta{style}$. The affine parameters of layer
normalization will learn the style agnostic features. Both are combined by
equation 3 and $\rho$ is used to control the amount of information flow
through normalization.The value of $\rho$ is constrained to the range of [0,
1] simply by imposing bounds at the parameter update step. This normalization
will help the variance predictors to predict the duration, energy and
fundamental frequency such that it incorporates the style of the input style
text and content of the input content text.

$\mu\textsubscript{i}=\frac{1}{m}\sum_{j=1}^{m}x_{ij},\sigma\textsubscript{i}^{2}=\frac{1}{m}\sum_{j=1}^{m}(x_{ij}-\mu\textsubscript{i})^{2}$
(1) $\hat{x}_{i,j}=\frac{x_{ij}-\mu\textsubscript{i}}{\sqrt{\sigma_{i}^{2}}}$
(2)
$\hat{y}_{ij}=\rho(\gamma\textsubscript{LN}\hat{x}_{ij}+\beta\textsubscript{LN})+(1-\rho)(\gamma\textsubscript{style}\hat{x}_{ij}+\beta\textsubscript{style})$
(3)

Figure 5: Discriminator architecure of Prosodic Diff-TTS

###  2.4 Discriminator

The discriminator[18] is used to distinguish the fake and real
melspectrogram($x_{t},x_{t-1}$) using unconditional and conditional logits
which is conditioned on time-steps and style embeddings.We have used 1D
convolution based network with leakyRelu as activation function to predict the
unconditional logits and conditional logits based on style tokens as shown in
Figure 5.

###  2.5 Training Loss

We focus on discrete-time diffusion models, where denoising steps are large,
and use a conditional GAN to model the denoising distribution. Prosodic Diff-
GAN trains a conditional GAN-based generator
$p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t})$ to approximate the true
denoising distribution $q(\mathbf{x}_{t-1}|\mathbf{x}_{t})$ with an
adversarial loss that minimizes a divergence $D_{\text{adv}}$ per denoising
step:

$\min_{\theta}\sum_{t\geq
1}\mathbb{E}_{q(\mathbf{x}_{t})}[D_{\text{adv}}(q(\mathbf{x}_{t-1}|\mathbf{x}_{t})||p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x_{t}}))],$
(4)

where we adopt the least-squares GAN (LS-GAN) training formulation [19] to
minimize $D_{\text{adv}}$

The discriminator is trained to minimize the loss

$\begin{split}\mathcal{L}_{D}=\sum_{t\geq
1}\mathbb{E}_{q(\mathbf{x}_{t})q(\mathbf{x}_{t-1}|\mathbf{x}_{t})}[(D_{\phi}(\mathbf{x}_{t-1},\mathbf{x}_{t},t,s)-1)^{2}]\\\
+\mathbb{E}_{p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t})}[D_{\phi}(\mathbf{x}_{t-1},\mathbf{x}_{t},t,s)^{2}].\end{split}$
(5)

The generator is trained adversarially($\mathcal{L}_{adv}$) to minimize the
equation, so that it could generate a realistic mel-spectrogram. The variance
predictor, namely the duration($\mathcal{L}_{duration}$),
energy($\mathcal{L}_{energy}$ and pitch($\mathcal{L}_{pitch}$ use MSE loss to
optimize their network. The mean absolute error is also used on mel-
spectrogram to optimize generator.

$\mathcal{L}_{G}=\mathcal{L}_{adv}+\lambda_{duration}\mathcal{L}_{duration}+\lambda_{energy}\mathcal{L}_{energy}+\\\
\lambda_{pitch}\mathcal{L}_{pitch}+\lambda_{fm}\mathcal{L}_{fm},$ (6)

where $\lambda_{duration}$,$\lambda_{energy}$,$\lambda_{pitch}$,$\lambda_{fm}$
are the training hyperparameters.

$\mathcal{L}_{adv}=\sum_{t\geq
1}\mathbb{E}_{q(\mathbf{x}_{t})}\mathbb{E}_{p_{\theta}(\mathbf{x}_{t-1}|\mathbf{x}_{t})}[(D_{\phi}(\mathbf{x}_{t-1},\mathbf{x}_{t},t,s)-1)^{2}],$
(7)

To avoid the mode collapse, the feature matching loss[20] $\mathcal{L}_{fm}$,
is used on generator by summing l1 distances between every discriminator
feature maps of real and generated samples:

$\mathcal{L}_{fm}=\mathbb{E}_{q(\mathbf{x}_{t})}[\sum_{i=1}^{N}||D_{\phi}^{i}(\mathbf{x}_{t-1},\mathbf{x}_{t},t,s)-D_{\phi}^{i}(\mathbf{x}^{\prime}_{t-1},\mathbf{x}_{t},t,s)||_{1}]$,

(8)

###  2.6 Training and Inference algorithm of Prosodic Diff-TTS

Algorithm 1 Training procedure of Prosodic Diff-TTS

Input: The generator $G_{\theta}$ with parameters $\theta$. The discriminator
$D_{\phi}$ with parameter $\phi$. A pre-calculated variance schedule
$\beta_{1},\cdots,\beta_{T}$ with $T$ diffusion steps with style token s.

1: repeat

2: Sample $\mathbf{x}_{0}\sim q(\mathbf{x})$,
$\epsilon\sim\mathcal{N}(\mathbf{0},\mathbf{I})$, and
$t\sim\mathrm{Unif}(\\{1,\cdots,T\\})$

3: Sample $\mathbf{x}_{t}$ and $\mathbf{x}_{t-1}$ given $\mathbf{x}_{0}$
according to $q(\mathbf{x}_{1:T}|\mathbf{x}_{0})=\prod_{t\geq
1}q(\mathbf{x}_{t}|\mathbf{x}_{t-1}),\quad
q(\mathbf{x}_{t}|\mathbf{x}_{t-1}):=\mathcal{N}(\mathbf{x}_{t};\sqrt{1-\beta_{t}}\mathbf{x}_{t-1},\beta_{t}\mathbf{I})$

4: $\mathbf{x}^{\prime}_{0}$ = $G_{\theta}(x_{t},y,s,t)$

5: Sample $\mathbf{x}^{\prime}_{t-1}\sim$
$q(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{x}^{\prime}_{0})$

6: Do back-propagation with $\mathcal{L}_{G}$ and update $\theta$ one step
with gradient descent;

7: Do back-propagation with $\mathcal{L}_{D}$ and update $\phi$ one step with
gradient descent;

8: until refinement model converged

Algorithm 2 Inference procedure of Prosodic Diff-TTS

Input: A trained generator $G_{\theta}$ and one testing sample
$(\mathbf{y},s)$.

1: Sample $\mathbf{x}_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$;

2: $\mathbf{x}_{t}\leftarrow\mathbf{x}_{T}$;

3: for $t=T,T-1,...,1$ do

4: Forward-propagate $(\mathbf{x}_{t},\mathbf{y},s,t)$ to $G_{\theta}$ to
calculate $\mathbf{x}^{\prime}_{0}$;

5: Sample $\mathbf{x}^{\prime}_{t-1}$ given $\mathbf{x}^{\prime}_{0}$ and
$\mathbf{x}_{t}$ by
$q(\mathbf{x}_{t-1}|\mathbf{x}_{t},\mathbf{x}^{\prime}_{0})$;

6: $\mathbf{x}_{t}\leftarrow\mathbf{x}^{\prime}_{t-1}$;

7: end for

8: Return $\mathbf{x}_{t}$;

##  3 Experiments

###  3.1 Datasets

We train and evaluate the model on two datasets namely PromptSpeech[9] and
LibriTTS[15]. PromptSpeech has 5 different style factors (gender, pitch,
speaking speed, volume, and emotion) and we extracted the audio from the
commercial TTS API 111https://azure.microsoft.com/en-us/services/cognitive-
services/text-to-speech/#overview. LibriTTS has 4 different styles (gender,
pitch, speaking speed and volume). The number of training and test samples are
1.5 lakh and 5k respectively for PromptSpeech and 26k and 1.3k respectively
for LibriTTS.

###  3.2 Training and Preprocesing Steps

We convert the text sequence into the phoneme sequence[21, 22] using open-
source grapheme-to phoneme tool[23]. We extract the phoneme duration with
MFA[24], an open-source system for speech-text alignment to improve the
alignment accuracy. We extracted the pitch contour, F0 using PyWorldVocoder
tool 222https://github.com/JeremyCCHsu/Python-Wrapper-for-World-Vocoder We
transfer the raw waveform into melspectrograms by setting the frame size and
hop size to 1024 and 256 with respect to the sample rate of 22050 Hz. We have
used pretrained HiFi universal vocoder to generate the audio waveforms.

###  3.3 Model Configuration

We employ a pre-trained BERT model with 12 hidden layers and 110M parameters.
On the basis of an auxiliary classification task involving five style
parameters, the BERT model is fine tuned. In the generator architecture's
phoneme encoding and output mel-spectrogram decoding stages, four feed forward
transformer blocks were utilised. The hidden size, number of attention heads,
kernel size, and filter size are set to 256, 2, 9, and 1024, respectively, for
the one-dimensional convolution in the multi head attention block. The number
of attention heads is set to 2. The denoiser module has 20 residual blocks
along with hidden dimension of 512 and dropout set to 0.2. Two blocks of
Conv1D, ReLU, layer normalization, and a dropout layer compose the Variance
predictor. The kernel sizes of the 1D-convolution are set to 3, the
input/output sizes for both layers are 256/256, and the dropout rate is set to
0.5. The mel-spectrogram that is generated is optimised with mean square error
loss. The network topology of the discriminator with unconditional block and
the discriminator with conditional block consists of two 1D convolutional
layers. The convolution channels consist of 64, 128, 512, 128, and 1. The
kernel sizes are 3, 5, 5, 5, 3, and the strides are 1, 2, 2, 1.

###  3.4 Style transfer evaluation on synthesized samples

We have performed the experiment to check whether the input style has been
successfully been transferred to the generated speech. We have used the
PyWorldVocoder tool to compute the accuracy of synthesized speech on various
style tasks such as gender, pitch,speaking speed, volume a classification. We
have trained a neural network classifier for emotion classification with more
than 98$\%$accuracy. Table 1 shows the comparison of Prosodic Diff-TTS with
the prior work, PromptTTSwhich shows better performance of the proposed method
in most style related task which is attributed to conditional prosodic layer
normalization as well as diffusion GAN based architecture which generate high
fidelity speech.

Table 1: The accuracy (%) of PromptTTS and Prosodic Diff-TTS on 1-PromptSpeech and 2-LibriTTS datasets. Model | Gender | Pitch | Speed | Volume | Emotion | Mean
---|---|---|---|---|---|---
PromptTTS-1 | 99.57 | 82.60 | 86.55 | 87.35 | 91.78 | 89.57
Prosodic Diff-TTS-1 | 99.55 | 85.72 | 87.49 | 88.40 | 93.45 | 90.62
PromptTTS-2 | 99.16 | 82.69 | 92.57 | 89.73 | - | 91.04
Prosodic Diff-TTS-2 | 99.28 | 84.17 | 93.32 | 88.91 | - | 92.81

###  3.5 Speech Quality

Twenty samples of speakers are taken for PromptSpeech and LibriTTS test set
are used to perform MOS [25] to evaluate the generated samples in terms of
naturalness(how the synthesized voices sound natural like human),
similarity(how the synthesized voices sound with respect to the input style
description).We compare the MOS of audio samples including: (1) GT, the
ground-truth recordings; (2) GT mel + HiFiGAN, where we first convert ground-
truth speech into mel-spectrogram, and then convert the mel-spectrogram back
to speech using HiFiGAN [17]; (3) PromptTTS; (4) Prosodic Diff-TTS. Both
systems in (3) and (4) use HiFiGAN as vocoder. According to Table 2, it can be
seen that Prosodic Diff-TTS outperforms the PromptTTS slightly in terms of
speech quality, as the proposed Prosodic Diff-TTS is able to model the multi
modal distribution more efficiently than PromptTTS.Synthesized audio samples
are present at this site 333https://sites.google.com/view/prosdifftts

Table 2: The results of speech quality with 95% confidence intervals. Version | Setting | MOS
---|---|---
PromptSpeech | GT |  4.15 $\pm$ 0.08
| GT mel + HiFiGAN |  4.10 $\pm$ 0.08
| PromptTTS |  3.98 $\pm$ 0.06
| Prosdic Diff-TTS |  4.01 $\pm$ 0.02
LibriTTS | GT |  4.21 $\pm$ 0.08
| GT mel + HiFiGAN |  4.13 $\pm$ 0.07
| PromptTTS |  3.82 $\pm$ 0.08
| Prosdic Diff-TTS |  3.85 $\pm$ 0.08

###  3.6 Qualitative Results

We extracted the pitch and energy from the predicted speech and the ground
truth speech using the tool. Figure 6 shows the prosody of generated samples
are similar to that of ground truth samples. Figure 7 shows the similarity of
predicted and ground truth mel-spectrogram at T=4.

Figure 6: Plot of energy and pitch of synthesized samples with real
speech(Content: Your memory must be conveniently short,"" chafed the master ,
Style: Please say a loud girl with a bass) Figure 7: Plot of mel-spectrogram
of synthesized samples with real speech.(Content: Your memory must be
conveniently short,"" chafed the master , Style: Please say a loud girl with a
bass)

###  3.7 Ablation Study

We have done the ablation study by changing the time-steps with values from
T=1,2,4. Table 3 shows the better MOS at T=4 as compared to other time-steps.
The possible reasons include the difficulty of directly generating samples
from a complex distribution in one or two timesteps, and the problem of
overfitting when the discriminator only examines clean samples.

Table 3: The results of speech quality with 95% confidence intervals at different timesteps Version | Setting | MOS
---|---|---
PromptSpeech | T=1 |  3.81 $\pm$ 0.09
| T=2 |  3.95 $\pm$ 0.06
| T=4 |  4.01 $\pm$ 0.08
LibriTTS | T=1 |  3.72 $\pm$ 0.10
| T=2 |  3.79 $\pm$ 0.07
| T=4 |  3.85 $\pm$ 0.08

##  4 Conclusion

In this paper, we have proposed the diffusion GAN based speech synthesis
architecture which can generate realistic speech based on input content and
style text description. We have proposed the conditional prosodic layer
normalization which injects the style into the encoder and decoder of
generator architecture at multiple layers through the affine parameters of
normalization. We have extracted the 128 dimensional style embedding by fine
tuning the pretrained BERT model on multiple auxiliary tasks such as pitch,
gender, volume, speaking speed and emotions. Using extensive experiments on
multi-speaker datasets(PromptSpeech and LibriTTS), we have shown both
qualitative and quantitative results along with high-quality of audio output.

## References

  * [1] S. Ö. Arik, M. Chrzanowski, A. Coates, G. F. Diamos, A. Gibiansky, Y. Kang, X. Li, J. Miller, A. Ng, J. Raiman, S. Sengupta, and M. Shoeybi, ``Deep voice: Real-time neural text-to-speech,'' in _International Conference on Machine Learning_ , 2017.
  * [2] Y. Ren, Y. Ruan, X. Tan, T. Qin, S. Zhao, Z. Zhao, and T.-Y. Liu, ``Fastspeech: Fast, robust and controllable text to speech,'' _ArXiv_ , vol. abs/1905.09263, 2019.
  * [3] J. Shen, R. Pang, R. J. Weiss, M. Schuster, N. Jaitly, Z. Yang, Z. Chen, Y. Zhang, Y. Wang, R. J. Skerry-Ryan, R. A. Saurous, Y. Agiomyrgiannakis, and Y. Wu, ``Natural tts synthesis by conditioning wavenet on mel spectrogram predictions,'' _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 4779–4783, 2017.
  * [4] W. Ping, K. Peng, A. Gibiansky, S. Ö. Arik, A. Kannan, S. Narang, J. Raiman, and J. Miller, ``Deep voice 3: 2000-speaker neural text-to-speech,'' _ArXiv_ , vol. abs/1710.07654, 2017.
  * [5] N. Kumar, S. Goel, A. Narang, and B. Lall, ``Normalization driven zero-shot multi-speaker speech synthesis,'' in _Interspeech_ , 2021.
  * [6] H. Sun, X. Tan, J.-W. Gan, H. Liu, S. Zhao, T. Qin, and T.-Y. Liu, ``Token-level ensemble distillation for grapheme-to-phoneme conversion,'' _ArXiv_ , vol. abs/1904.03446, 2019.
  * [7] Y. Guo, C. Du, and K. Yu, ``Unsupervised word-level prosody tagging for controllable speech synthesis,'' _ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 7597–7601, 2022.
  * [8] J. Bae, H. Bae, Y.-S. Joo, J. Lee, G.-H. Lee, and H.-Y. Cho, ``Speaking speed control of end-to-end speech synthesis using sentence-level conditioning,'' _ArXiv_ , vol. abs/2007.15281, 2020.
  * [9] Z. Guo, Y. Leng, Y. Wu, S. Zhao, and X. Tan, ``Prompttts: Controllable text-to-speech with text descriptions,'' _ArXiv_ , vol. abs/2211.12171, 2022\.
  * [10] R. Huang, M. W. Y. Lam, J. Wang, D. Su, D. Yu, Y. Ren, and Z. Zhao, ``Fastdiff: A fast conditional diffusion model for high-quality speech synthesis,'' in _International Joint Conference on Artificial Intelligence_ , 2022.
  * [11] S. Liu, D. Su, and D. Yu, ``Diffgan-tts: High-fidelity and efficient text-to-speech with denoising diffusion gans,'' _ArXiv_ , vol. abs/2201.11972, 2022.
  * [12] Z. Xiao, K. Kreis, and A. Vahdat, ``Tackling the generative learning trilemma with denoising diffusion gans,'' _ArXiv_ , vol. abs/2112.07804, 2021.
  * [13] A. Vaswani, N. M. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, ``Attention is all you need,'' _ArXiv_ , vol. abs/1706.03762, 2017.
  * [14] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, ``Bert: Pre-training of deep bidirectional transformers for language understanding,'' _ArXiv_ , vol. abs/1810.04805, 2019.
  * [15] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, ``Librispeech: An asr corpus based on public domain audio books,'' _2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , pp. 5206–5210, 2015.
  * [16] Y. Ren, C. Hu, X. Tan, T. Qin, S. Zhao, Z. Zhao, and T.-Y. Liu, ``Fastspeech 2: Fast and high-quality end-to-end text to speech,'' _ArXiv_ , vol. abs/2006.04558, 2020.
  * [17] J. Kong, J. Kim, and J. Bae, ``Hifi-gan: Generative adversarial networks for efficient and high fidelity speech synthesis,'' _ArXiv_ , vol. abs/2010.05646, 2020.
  * [18] J. Yang, J. Lee, Y.-I. Kim, H. Cho, and I. Kim, ``Vocgan: A high-fidelity real-time vocoder with a hierarchically-nested adversarial network,'' _ArXiv_ , vol. abs/2007.15256, 2020.
  * [19] X. Mao, Q. Li, H. Xie, R. Y. K. Lau, Z. Wang, and S. P. Smolley, ``Least squares generative adversarial networks,'' _2017 IEEE International Conference on Computer Vision (ICCV)_ , pp. 2813–2821, 2016.
  * [20] A. B. L. Larsen, S. K. Sønderby, H. Larochelle, and O. Winther, ``Autoencoding beyond pixels using a learned similarity metric,'' _ArXiv_ , vol. abs/1512.09300, 2015.
  * [21] D. Amodei, S. Ananthanarayanan, R. Anubhai, J. Bai, E. Battenberg, C. Case, J. Casper, B. Catanzaro, Q. Cheng, G. Chen, J. Chen, J. Chen, Z. Chen, M. Chrzanowski, A. Coates, G. Diamos, K. Ding, N. Du, E. Elsen, and Z. Zhu, ``Deep speech 2: End-to-end speech recognition in english and mandarin,'' 12 2015\.
  * [22] J. Shen, R. Pang, R. Weiss, M. Schuster, N. Jaitly, Z. Yang, Z. Chen, Y. Zhang, Y. Wang, R. Skerrv-Ryan, R. Saurous, Y. Agiomvrgiannakis, and Y. Wu, ``Natural tts synthesis by conditioning wavenet on mel spectrogram predictions,'' 04 2018, pp. 4779–4783.
  * [23] G2P, ``G2p, https://github.com/kyubyong/g2,'' 10 2017.
  * [24] M. McAuliffe, M. Socolof, S. Mihuc, M. Wagner, and M. Sonderegger, ``Montreal forced aligner: Trainable text-speech alignment using kaldi,'' in _INTERSPEECH_ , 2017.
  * [25] M. Chu and H. Peng, ``An objective measure for estimating mos of synthesized speech,'' in _Interspeech_ , 2001.


# Causes and consequences of ordering and dynamic phases of confined vortex
rows in superconducting nanostripes

Benjamin A. McNaughton
School of Science and Technology, Physics Division
University of Camerino
62032 Camerino, Italy
<EMAIL_ADDRESS>
& Nicola Pinto
School of Science and Technology, Physics Division
University of Camerino
62032 Camerino, Italy
<EMAIL_ADDRESS>
& Andrea Perali
School of Pharmacy, Physics Unit
University of Camerino
62032 Camerino, Italy
<EMAIL_ADDRESS>
& Milorad V. Milošević
Department of Physics
University of Antwerp
Antwerp, Belgium
<EMAIL_ADDRESS>

###### Abstract

Understanding the behaviour of vortices under nanoscale confinement in
superconducting circuits is of importance for development of superconducting
electronics and quantum technologies. Using numerical simulations based on the
Ginzburg-Landau theory for non-homogeneous superconductivity in the presence
of magnetic fields, we detail how lateral confinement organises vortices in a
long superconducting nanostripe, and present a phase diagram of vortex
configurations as a function of the stripe width and magnetic field. We
discuss why average vortex density is reduced and reveal that confinement also
has profound influence on vortex dynamics in the dissipative regime under
sourced electrical current, mapping out transitions between asynchronous and
synchronous vortex rows crossing the nanostripe as the current is varied.
Synchronous crossings are of particular interest, since they cause single-mode
modulations in the voltage drop along the stripe in a high (typically GHz-to-
THz) frequency range.

_K_ eywords Superconducting, nanostripes, vortex, confinement, critical
current, flux.

##  1 Introduction

Superconducting nanostripes (SN) are a fundamental component in
superconducting electronics, crucial for various applications in the field of
quantum technology. For example, superconducting nanostripe single-photon
detectors (SNSPD) are used for quantum communication and applications in
astronomy and spectroscopy [1, 2, 3, 4]. Other superconducting electronics
include prototypical logic devices [5, 6, 7], flux qubits used in quantum
computers [8, 9, 10], diodes [11, 12, 13] and electromagnetic resonators [14,
15, 16]. Narrow SN experience an enhancement of critical parameters [17, 18,
19, 20] due to confinement forces acting on the superconducting condensate
[21, 22, 23, 24, 25, 26]. Such confinement in narrow SNs can cause large
magnetoresistance oscillations [27, 28, 29], where time-averaged
voltage/resistance, as a function of the applied magnetic field, exhibits
pronounced peaks at alternating transitions between static and dynamic vortex
phases. At higher applied fields with multiple rows of vortices, or high
currents, a continuous motion of vortices causes a monotonic background on
which the resistance oscillations due to entries of additional vortices are
superimposed [27, 30]. Commensurate effects between the SN width $w$, and the
number of vortex rows $n$, have also been seen in the critical current as a
function of the out-of-plane magnetic field $H$ (for fixed $w$) or $w$ (for
fixed $H$) [31, 32]. Optimized operation of some of the suggested
superconducting electronics may be achieved on a specific geometry of
vortices. For example, a single row of vortices was found preferable in Ref.
[7], producing a giant non-local electrical resistance from vortices moving
very far (several microns) from the local current drive. This effect appears
important for a feasible long-range information transfer by vortices unaltered
by the passing current.

Moving vortices however exceed in importance the bare transfer of information.
For example, vortices coherently crossing SNs can produce electromagnetic
radiation [15, 16], where higher radiation power is emitted when multiple
vortices exit the SN simultaneously. In narrow SNs, rows of vortices can cross
the SN asynchronously and synchronously [27, 33], depending on competing
forces (confinement, vortex-vortex interaction, Lorentzian forces) but
criteria for synchronous crossings are not yet well understood. In this
respect, a study on the behaviour of vortices in SNs with small widths is
important to reveal favoured geometry of vortices for the static case (no
sourced current), and the relation to the dynamic case (with sourced current).
Understanding how a vortex lattice is affected by the interaction with the
edge confining force and other dynamic forces is important when considering
SNs for applications mentioned above. Studying the dynamic dissipative states
under strong confinement in the 1D-2D crossover regime, can reveal how
vortices cross the SN under different conditions ($w$, $H$, current
intensity). Moreover information on the possible vortex velocity under
confinement [34, 35], can be important to both the fast information transfer
and the frequency of emitted radiation by moving vortices. Therefore, in this
work we investigate how confinement in SNs affects the vortex configurations,
using Ginzburg-Landau simulations [36]. We provide the vortex row phase
diagram as a function of $H$ for a given $w$. Investigation of magnetic field
dependence of the average number of vortices reveals strong confinement
effects. With increasing width, reconfiguration from the vortex rows to the
vortex lattice takes place, offering a criterion to define quasi 1D -to- 2D
dimensional-crossover where SN effectively becomes a nanofilm in terms of the
superconducting properties. Additionally, a commensurate behaviour of the
critical current, $J_{c1}(H)$, has been found when varying $H$, using a time-
dependent GL approach to simulate effects of the sourced current. We show that
the local minima values in $J_{c1}(H)$ (defined as the onset of vortex motion
and corresponding dissipation) directly relate to row transitions shown in our
vortex-row phase diagram.

Further simulations of current-voltage (I-V) characteristics in SNs have
evidenced transitions among different resistive regimes (Meissner, flux-flow,
flux-flow instability, phase slips, normal state). I-V curves showing similar
features to our simulations for SNs have been experimentally measured only for
wider structures [37, 38]. We find that for SN with average vortex density
$\lesssim$ $1/80\xi^{2}$ ($\xi$ being the coherence length) in a flux-flow
regime, vortices cross the SN in a periodic/continuous fashion, causing
modulations of the voltage drop detectable experimentally. Such a periodic
flow may produce electromagnetic radiation [15], and features characteristic
power spectra [39] that we report by performing fast Fourier transformation of
the calculated voltage drop as a function of time during vortex motion. The
recorded average vortex velocity (up to 10’s Km/s) is used to discuss the
washboard frequencies [16, 15], in the flux-flow regime for thin SNs of
niobium [38]. Providing the vortex density is sufficiently high, we that
vortex row crossings transition between quasi-synchronous to synchronous, to
finally asynchronous regimes. Synchronised crossings are desirable for small-
band electromagnetic emitters operating in the GHz or THz range. For typical
ultra thin niobium SNs [18], modulation frequencies range in the microwave
regime between 10-800 GHz. Asynchronous regimes are disruptive for a coherent
emission, but host a number of local dynamic vortical transitions and
transformations that are of fundamental importance for advanced devices, and
unattainable otherwise.

The article is organized as follows. We first introduce the theoretical
framework and methods used for the numerical simulations. We then present
results and discussions of all the above-listed phenomena, using both
stationary and time-dependent Ginzburg-Landau approach. Main conclusions of
our work are emphasized already in the results section, before being
additionally commented on in the conclusions of the article.

##  2 Materials and Methods

The numerical simulations performed in this work are all conducted on SN, such
as exemplified in figure 1. The SN have dimensions with lengths $L$, widths
$w$, and thickness $d\ll\xi,\lambda$, where $\xi$ is the coherence length and
$\lambda$ the magnetic field penetration depth of the superconducting state.
For sufficiently large $H$, vortices form in the sample with a normal core of
radius $\xi$ and penetration of the magnetic field up to a characteristic
length of $\lambda$. In samples of our interest, being very thin, the
effective penetration depth $\Lambda=\lambda^{2}/d$ by far exceeds the
dimensions of the SN, such that the magnetic response of the superconductor is
negligibly small compared to the applied magnetic field. Simulations of such
SN are performed using the stationary (SGL) and time-dependent Ginzburg-Landau
(TDGL) formalism. In the SGL approach we self-consistently solve the coupled
equations

$\left(-i\nabla-\mathbf{A}\right)^{2}\Psi=\Psi\left(1-\left|\Psi\right|^{2}\right),$
(1)
$\vec{j}=-\kappa^{2}\nabla^{2}\mathbf{A}=\frac{1}{2i}\left(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*}\right)-\left|\Psi\right|^{2}\mathbf{A}$
(2)

where $\Psi$ is the superconducting order parameter, $\mathbf{A}$ is the
vector potential, and $\kappa=\Lambda/\xi$ is the effective Ginzburg-Landau
parameter. We work with dimensionless units, where length is given in units of
the temperature-dependent coherence length $\xi(T)=\xi$, the vector potential
$\mathbf{A}$ in units of $c\hbar/2e\xi$, magnetic field $\vec{H}$ in units of
the bulk upper critical field $H_{c2}=c\hbar/2e\xi^{2}$, current in units of
the GL current $j_{GL}=c\Phi_{0}/(8\pi^{2}\lambda^{2}\xi)$, and the order
parameter $\Psi$ is normalized to its value in absence of applied field or
sourced current ($\Psi_{0}$). We impose the Neumann boundary condition at the
superconductor-insulator boundary at the lateral edges of the SN

$\vec{n}\cdot\left(-i\nabla-\mathbf{A}\right)\Psi|_{boundary}=0.$ (3)

Along the length of the SN ($x$-axis) we enforce periodic boundary conditions
for $\mathbf{A}$ and $\Psi$ (for the unit cell length $32\xi$, sufficient to
capture the physics of interest in this work), of the form [40]

$\mathbf{A}(x_{0}+L_{x})=\mathbf{A}(x)+\nabla\chi_{f}(x)$ (4)
$\Psi(x_{0}+L_{x})=\Psi(x)\exp\left[i\frac{2e}{\hbar c}\chi_{f}(x)\right],$
(5)

where $\nabla\chi_{f}$ respects the gauge used for the magnetic field.
Equations (1) and (2) are solved numerically on a discretized Cartesian grid
according to Ref. [36], using the finite-difference method and the link-
variable approach [41], iteratively until convergence within a prespecified
error is achieved. Then the supercurrent is calculated from the value of the
order parameter and the vector potential (nearly entirely provided by the
external magnetic field). With this method we obtain the vortex-row
configuration-transition diagram as a function of $H$ and $w$ of the SN.

Figure 1: Schematic illustration of a superconducting nanostripe having width
$w$, length $L$ and thickness $d$, along the $x$, $y$ and $z$ directions,
respectively, in a homogeneous out-of-plane applied magnetic field,
$\mathbf{H}$. The SN contains an example of a single vortex, with a normal
core of radius $\sim\xi$, and distribution of magnetic field around it
characterized by $\Lambda$. When a current density $\mathbf{J}$ is sourced,
the vortex will experience the Lorentz force, $\mathbf{F}_{L}$.

The generalised time-dependent Ginzburg-Landau formalism [42, 43] should
instead be employed to properly study the dynamical properties of the
superconducting condensate (with order parameter $\Psi(\textbf{r},t)$) in the
presence of an external magnetic field H (with vector potential A) and sourced
current density J, given by

$\displaystyle\tau_{GL}N(0)\frac{u}{\sqrt{1-(\Gamma|\Psi|)^{2}}}\left[\frac{\delta\Psi}{\delta
t}+i\tfrac{e^{*}}{\hbar}\varphi\Psi+\left(\frac{\Gamma}{\sqrt{2}}\right)^{2}\frac{\delta|\Psi|^{2}}{\delta
t}\Psi\right]$ (6)
$\displaystyle=-\left(a+b|\Psi|^{2}\right)\Psi+\frac{\hbar^{2}}{2m^{*}}\left(\nabla-
ie^{*}\mathbf{A}\right)^{*}\Psi,$

$\nabla^{2}\varphi=\nabla\left[Im\left\\{\Psi^{*}\left(\nabla-i\textbf{A}\right)\Psi\right\\}\right],$
(7)

where $a=\frac{\alpha}{2m^{*}\gamma}$, $b=\frac{\beta}{4m^{*2}\gamma^{2}}$,
and $\Gamma=\frac{2\tau_{i}}{\hbar\sqrt{2m^{*}\gamma}}$. The Ginzburg-Landau
order parameter relaxation time is $\tau_{GL}$; $N(0)$ is the density of
states at the Fermi level; the parameter $u=5.79$ in conventional
superconductors; $\varphi$ is the electrostatic potential; $\tau_{i}$ is the
electron-phonon inelastic scattering time and $\alpha,\beta,\gamma$ are
material parameters. Equation (6) is solved coupled with the equation for the
electrostatic potential (eq.(7)), using Neumann boundary conditions at all
sample edges, except for the leads where sourced current is injected, where
$\Psi=0$ and $\nabla\varphi=\pm J$. This theory is derived for dirty gapless
superconductors, where Cooper-pair breaking occurs due to strong inelastic
electron-phonon scattering, and the physical quantities $\Psi$ and $A$ must
relax over a time-scale much longer than $\tau_{i}$. The distance over which
an electric field can penetrate into the superconductor, and the length over
which relaxation processes occur is given by the characteristic inelastic
diffusion length $L_{i}=\sqrt{D\tau_{i}}$, where $D$ is the diffusion
parameter proportional to the electronic mean-free path. In cases where
$L_{i}<<\xi$, our simulations require very fine grid spacing (reflecting in
consequently smaller time step in the used implicit Crank–Nicolson method) to
yield physically correct results. In general, superconducting materials at $T$
close to the superconducting-to-normal transition temperature, $T_{c}$,
satisfy the conditions for slow temporal and spatial variations ideally
required for the applicability of the GL formalism. In the TDGL formalism,
distances are given in units of $\xi(T)=\xi$; time in units of
$\tau_{GL}=\frac{\pi\hbar}{8k_{B}T_{c}(1-T/T_{c})u}$; temperature is in units
of $T_{c}$; the order parameter $\Psi$ in units of
$\Delta(0)=4k_{B}T_{c}u^{1/2}(1-T/T_{c})^{1/2}/\pi$; $\varphi$ in units of
$\varphi_{GL}=\hbar/e^{*}\tau_{GL}$; vector potential A is scaled to
$A_{0}=H_{c2}\xi$ and current density to $J_{0}=\sigma_{n}\varphi_{0}/\xi$.
The simulations are performed irrespective of the temperature $T/T_{c}$, all
physical quantities are scaled and normalised by reference quantities at a
given temperature. Finally, in our analysis we do not consider the heating
effects, which would require coupling of the TDGL formalism to the thermal
balance equation [34]. This is justified for samples that are very efficiently
thermally coupled to the substrate and/or thermal bath (i.e. have large heat
transfer coefficient).

##  3 Results

In what follows, using SGL and TDGL simulations, we study how confinement
forces in narrow SN affect the stationary vortex configurations and their
dynamics, under $H$ and a sourced dc current density, $J$.

###  3.1 Equilibrium vortex configurations

We start by producing the vortex row phase diagram using the SGL approach,
showing the conditions for the formation of a number of vortex rows, $n$, as a
function of $H$ and $w$ of the SN. Each dashed curve in the diagram shown in
figure 2, plotting the width of the SN versus $H$, represents the appearance
of the $n^{th}$ vortex row ($n=1-5$) in the ground state of the system as
magnetic field is increased. Examples of corresponding vortex configurations
for a SN of $w=12\xi$ for different $H$ intensity are shown in figure 3,
corresponding to the pinpointed dots (labelled a-h) in figure 2.

Figure 2: Equilibrium vortex-row phase diagram, plotting the SN width (in
units of $\xi$) as a function of the applied magnetic field intensity (in
units of $H_{c2})$, for different numbers of formed vortex rows($n$).
Simulations have been done using the SGL approach with periodic boundary
conditions along the length, with unit cell length of $L=32\xi$. The dashed
lines denote the threshold for the formation of an additional vortex row (here
shown up to $n=5$). The coloured regions represent the approximated regions
for $n>1$, delimited by solid lines given by expression $H_{row}/H_{c2}$ =
$\frac{\pi n^{2}\xi^{2}}{\sqrt{3}w^{2}}$. Circles, labeled $a-h$, relate to
the vortex configurations shown in figure 3. Black dotted line corresponds to
the analytical expression $H_{0}/H_{c2}=K\frac{\pi^{2}\xi^{2}}{2w^{2}}$ [44],
with $K=1.7$ [45].

To identify the threshold $H$ for the transition to the vortex row
configuration with a higher $n$, the ground states were first obtained for
each SN at different $H$; then the spatial distribution of the superconducting
order parameter $|\Psi|^{2}$ has been plotted (similar to figure 3) and
carefully analysed, focusing on the geometrical interpretation of the vortex
configuration. In the SGL approach adopted in our simulations, the SN was
considered periodic along its length, with a unit cell of $L=32\xi$. Several
checks, carried out by extending the unit cell length till $80\xi$, have
confirmed all following results.

In a SN, the early theoretical works [46, 44] have shown that the magnetic
field at which the surface barrier is suppressed and a single vortex can be
stable in the SN is $H_{0}/H_{c2}=\pi^{2}\xi^{2}/2w^{2}$. The subsequent
experimental observations of vortex penetration fields by Stan et al. [45]
have shown a very good agreement with latter expression, up to a multiplying
constant $K$. Our numerical data (black dots in figure 2) reconfirm that
finding, as vortex penetration fields were found to nearly ideally match the
same functional dependence on $w$, with a multiplying constant $K=1.7$.

The approximate criteria for further reconfiguration of the vortex states and
appearance of additional vortex rows can be obtained in the following way. We
consider the Abrikosov triangular lattice, with the lattice parameter
$a=1.075\sqrt{\phi_{0}/H}$. The vortices are arranged in a body-centered
hexagonal lattice, and so the Wigner-Seitz unit cell is hexagonal with a unit
area per flux quantum of $A=\frac{\sqrt{3}}{2}a^{2}$. For a narrow SN, to
accommodate $n$ rows of vortices, the spacing, $w_{v}$, between vortex rows
must obey the inequality $w_{v}\leq w/n$. Using the previous expression for
the Abrikosov vortex density, we substitute
$A=\frac{\sqrt{3}}{2}w_{v}^{2}=\frac{\sqrt{3}w^{2}}{2n^{2}}$ to obtain the
zeroth order approximation for the threshold magnetic field required for the
formation of new rows, yielding $H_{row}/H_{c2}$ = $\frac{\pi
n^{2}\xi^{2}}{\sqrt{3}w^{2}}$. Those approximate threshold $H$ values are
shown in figure 2 by the solid lines delimiting different coloured regions,
indicating transitions between states with different number of vortex rows. In
general, the behavior of threshold $H$ found using SGL simulations agrees well
with the formula prediction. The values were however mostly higher than the
approximate ones, which is attributed to the role played by the edge barriers
for vortex entry and exit (varying, depending on $w$ and $H$). In addition,
the rearrangement of the vortex lattice with every vortex penetration is not
taken into account in latter basic analytical formula. Note that such effects
of the vortex-vortex interactions and interactions with the edge Meissner
currents (causing the confinement force), dominate the formation of the vortex
configurations in narrow SNs and present the main point of interest in this
work.

Figure 3: Calculated vortex configurations plotted as Cooper-pair density for
the ground state of a SN of width $w=12\xi$, in a periodic cell of $L=64\xi$,
at different applied $H/H_{c2}$ values: (a) 0.08; (b) 0.20; (c) 0.42; (d)
0.43; (e) 0.45; (f) 0.77; (g) 0.80; (h) 0.87 (cf. figure 2). Panels (b), (d)
and (g) depict the vortex states at the nucleation of a second, third and
fourth row, respectively. Panels (c), (e) and (h) show the most lattice-like
packing conditions for two, three and four vortex rows, respectively. White
lines, connecting the cores of three neighbouring vortices, illustrate the
deformation of the Abrikosov lattice [47] in the SN. Color bar denotes the
values of the Cooper-pair density shown in the panels. Each depicted
configuration is indicated in figure 2 with an open dot and is labelled
accordingly. Figure 4: Exemplified re-entrant transition between the one- and
two-row vortex state in a SN of width $w=20\xi$, caused by the competition
between the confinement imposed by Meissner currents and the vortex density,
while both changing with increasing $H$ (value indicated inside the panels).

For a SN of $w=12\xi$, we show different vortex-row configurations in figure
3, as formed in the ground state at different $H$ values (marked by open dots
in figure 2). After formation and growing of the first vortex row population
(figure 3-a), increasing $H$ vortices rearrangement into a closely packed
“zig-zag" state (figure 3-b). This close packing is emphasised by a white
triangle progressively deviating from the equilateral shape expected in the
Abrikosov vortex lattice, with rise of $H$. Obviously, in this state the
Meissner currents will exert a strong repulsive and confining force on the
vortices from the SN edges (i.e. strong Bean-Livingston edge barrier [25]),
resulting in a vortex spacing far smaller than the above rough analytical
estimates (leading to the solid lines in figure 2).

Starting from the one row configuration (figure 3-a), raising $H$ and $n$
further, strengthens the relevance of the vortex-vortex interaction forces on
the resulting vortex configuration, which will increase separation between the
two rows (figure 3-c). At this point, we observe that additional vortices in
the SN cannot uniformly balance the aforementioned competing force in the
entire SN, leading to a local rearrangement of the vortex lattice to three
rows (figure 3-d). Only with further increasing field and having enough
vortices in the SN the full three-row state is formed (figure 3-e; notice a
nearly ideal triangular lattice formed). For the considered width of the SN,
the state with 3 vortex rows persists to a much larger field due to quantum
confinement, such that vortices very strongly overlap in a closely packed
structure (figure 3-f). Nevertheless, in the vicinity of the bulk upper
critical field a fourth row forms, first locally (figure 3-g) and eventually
in the entire SN (figure 3-h), before superconductivity is destroyed. No
further rows of vortices can form at higher field and the existing vortex rows
increasingly overlap until the normal state is established.

Figure 5: Area of the Wigner-Seitz unit cell containing a single vortex, as a
function of the applied magnetic field, for SN of width $w=6-30\xi$, and
$w=40-80\xi$ in the inset for small $H$ values. The analytical expression for
the Abrikosov vortex lattice (AVL) area, $\frac{A}{\xi^{2}}$ =
$2\pi\frac{H_{c2}}{H}$, is plotted as a black line. Open dots in each curve
indicate the $H$ intensity for formation of the third vortex row, above which
the curves progressively approach the AVL expression, upon increasing the $w$.

We reiterate that the transitions between rows of vortices and the final
arrangement of vortices in the lattice are strongly affected by competition of
the two forces, both dependent on $H$. As the magnetic field is increased the
edge Meissner current also increases, up to the penetration of new vortices,
while every new vortex changes the landscape of the vortex-vortex interactions
in the SN. As exemplified in figure 4 for $w=20\xi$, this nontrivial balance
of competing forces can lead to a re-entrant behavior in terms of the number
of vortex rows formed. In such cases, the zig-zag instability of the vortex
row can be "cured" back into a single row by the increasing Meissner currents,
as the lateral confinement forces grow with increasing magnetic field. As $H$
is increased further the additional penetrating vortices tip the scale in
favor of vortex interactions and a definite reconfiguration into a state with
two rows form. This re-entrant behavior has been observed for nearly all
considered SN widths in the range $w=20-60\xi$. In such cases we have taken
the first onset of the zig-zag instability to mark the $n\rightarrow n+1$
transition in figure 2. Moreover, this range of widths, where such strong edge
effects are detected, marks the crossover from the quasi-1D to a 2D film-like
behavior.

As the magnetic field is increased, vortices penetrate the SN of different
width and vortex rows are formed; and a gradual evolution from a quasi-1D row
pattern into a 2D vortex lattice is expected. To evaluate such crossover, we
calculated the average area occupied by a single vortex as a function of $H$
in all the states found, and compared to the expected behavior of the
Abrikosov vortex lattice area. The strong confinement in narrowest SN [31, 25]
dominates the vortex-vortex interaction, leading to compression of vortices
into fewer vortex rows and consequently larger average area per vortex. This
can be seen in figure 5 for $w\leq 8\xi$. As the width of the SN is made
larger, the confining force from the edge current (at a given $H$) becomes
less dominant with respect to the vortex-vortex interaction, resulting into
progressively closer agreement with the expected behaviour of a triangular
vortex lattice [47]. This tendency is clearly visible upon formation of the
third vortex row (cf. figure 5).

Figure 6: First critical current density normalised to $J_{DP}=0.385J_{GL}$ as
a function of the applied magnetic field normalised to $H_{c2}$, for a SN of
width $w=12\xi$, obtained using the TDGL approach. Vertical red lines mark the
transition to 1, 2 and 3 vortex-row states, at a magnetic field of $H/H_{c2}$
= 0.09, 0.23, 0.42, respectively. Insets illustrate the vortex row
configurations at selected magnetic fields (marked by open dots) for applied
current density just under the critical one.

###  3.2 Vortex dynamics under sourced current

All above results have been obtained in the stationary case, where no current
is sourced to the SN. A sourced current may change the stationary states or
induce vortex dynamics specific to the nanoconfined regime. In what follows,
we examine those non-equilibrium effects, through TDGL simulations of time-
dependent processes.
When a small transport current is applied along a SN under an applied magnetic
field $H\geq H_{c1}$, the present vortices experience a push across the SN due
to the Lorentz-type of force ($\propto\mathbf{J}\times\mathbf{H}$). As the
applied current density is increased, vortices will continue to shift across
the SN, finally leaving the stripe for sufficiently large Lorentz force. This
defines the first critical current density ($J_{c1}$), for which vortices are
able to overcome the edge barrier [31, 25] and start to cross the SN
continuously, nucleating on one side, moving across the SN, and exiting at the
opposing edge. The critical current depends on magnetic field $H$ for a given
width $w$ of the SN. The first critical current density as a function of $H$,
$J_{c1}(H)$, for a SN of $w=12\xi$, is shown in figure 6. A commensurate
effect is observed between $H$ and $n$, where minima in the curve correspond
to the transition to a state with an additional vortex row. Previous works
have reported similar behaviour using different theoretical approaches [48,
49], including a comprehensive study using the TDGL approach [32] and
revealing the relation between $n$, $w$ and applied magnetic field [31]. The
increase in $J_{c1}$ from local minima as the applied $H$ field is further
increased, is caused by the competition between vortex-vortex interactions and
the confinement from the SN edge. After a local minimum, when a new row is
formed, the vortex-vortex interactions are strong and the confining edge
currents producing an entry/exit barrier are reduced and more easily overcome
with lower applied currents. As $H$ is increased, the induced Meissner
currents at the edge increase [50], reinforcing the edge barrier. The vortex-
row phase diagram in figure 2 can be used to predict the transition field
value, where local minima occur in $J_{c1}(H)$ curves - which is an
experimentally verifiable feature. Note however that the threshold fields for
formation of new rows in presence of applied current are somewhat different
from the ones presented in figure 2, since the Lorentz push exerted by the
current effectively increases the confinement experienced by vortices prior to
the onset of their motion.

Figure 7: Normalised voltage drop as a function of the normalised current
density, for SNs of width $w$ = 6, 9, 12, and 18$\xi$, at magnetic field
$H=0.25H_{c2}$. The black and red dots (for $w$ = 6 and 12$\xi$ respectively)
mark the values of current density at which analysis of the modulation
frequency spectra is presented in figures 9 and 11. Inset: Snapshots of the
Cooper-pair density for SN of $w$ = 12$\xi$, numbered 1-10, indicated by open
red squares.

The TDGL approach has allowed us to also simulate the voltage-current density
(V-J) characteristics of SNs, presented in figure 7 for stripes with $w$ = 6,
9, 12, 18$\xi$, under an applied magnetic field $H=0.25H_{c2}$. Analysis of
the V-J characteristics reveals a number of features related to different
resistive regimes in each curve. At low $J$ values, stationary vortices are
shifted to a new position across the SN due to the Lorentz force produced by
the sourced current, so the resulting voltage drop and resistance remain zero.
An example of such can be seen in figure 7 (for $w=12\xi$) from the states
labelled 1 and 2. When $J_{c1}$ is reached, vortices cross the SN and their
perpetuous motion leads to a finite resistivity value. Snapshots of this flux-
flow regime can be seen from the states labelled 3 and 4 in figure 7. With
further increasing $J$ and in presence of vortex-vortex interaction forces, a
SN in the dissipative state exhibits flux-flow instability, where vortex cores
interact during dynamics and ordered lattice structure is lost during motion
(state labelled 5 in figure 7). At even higher $J$, vortices align during
motion, in a slip-streamed geometry (vortices tailgate, i.e. subsequent
vortices, crossing the SN, move in the wake of the previous vortex [33, 35]),
before a Langer-Ambegaokar phase slip [51] occurs across the SN. The normal
area covered by the phase slip grows laterally with further increasing $J$,
and additional steps in the V-J curve appear with every slip-stream being
merged with the growing phase-slip, as seen in the states labelled 6-10 in
figure 7. When $J$ reaches roughly 0.65$J_{DP}$, the SN transitions to a fully
normal state, with linear ohmic behavior. Similar V-I curves have been
observed both numerically [52] and experimentally for Nb-C microstrips,
fabricated using focused-ion-beam-induced deposition [38].

Figure 8: Normalised voltage drop as a function of time (normalised to
$\tau_{GL}$), for a SN of $w=6\xi$ under a magnetic field of $H=0.25H_{c2}$
sourced with different current densities of: (A) 0.348$J_{DP}$, (B)
0.366$J_{DP}$, (C) 0.387$J_{DP}$, (D) 0.406$J_{DP}$, (E) 0.444$J_{DP}$, and
(F) 0.655$J_{DP}$. Each panel contains an illustrative snapshot of the spatial
distribution of the Cooper-pair density during the dynamics. Figure 9: Spectra
of modulation frequencies $\nu$ (normalised to $\tau_{GL}^{-1}$) of the
temporal voltage signals shown in figure 8.

Next we discuss how the observed vortex crossings modulate the voltage drop
across the SN, and how synchronous and asynchronous crossings affect the
spectrum of frequencies as a consequence of those modulations. In figures 8-11
we show the voltage as a function of time, $V(t)$, for different applied
current $J$, and their corresponding spectra of frequencies (obtained by
Fourier transform of $V(t)$) for SNs of $w=6\xi$ and $w=12\xi$, under an
applied field $H=0.25H_{c2}$. In each case, we have first used the TDGL
approach to find the ground states for each SN at the given magnetic field,
then we swept the $J$ from 0 up to $\simeq J_{DP}$, in sufficiently small
steps (typically $\simeq 0.025J_{DP}$). At each current step the simulation
has been left to run for sufficiently long time so that a dynamic equilibrium
has been reached (typically up to $t=5\times 10^{3}$ $\tau_{GL}$), before
recording data. The so obtained $V(t)$ and the spatial distribution of the
superconducting order parameter at each time step, are used to produce figures
8 and 10. In the dissipative state, $V(t)$ raises as vortices move across the
SN, with maxima corresponding to the exit of a vortex and minima to an entry
of a vortex [53, 34], leading to modulations of $V(t)$ for both SN considered
(figures 8 and 10). Considering the SN of $w=6\xi$, sourced with the lowest
current causing the vortex crossing ($J=0.348J_{DP}$ in this case), $V(t)$
shows evidence of asynchronous vortex dynamics, with several distinct features
having a periodicity of 486 $\tau_{GL}$. Even though the vortices are not
crossing in synchronised rows, there is a quasi-synchronised behaviour
manifesting in the repetition of vortex crossings in a given dynamic
configuration. As $J$ is increased from 0.384$J_{DP}$ to 0.406$J_{DP}$ (panels
A-D in figure 8) the modulations in the voltage evolve, and the number of
modulations caused by quasi-synchronous crossings reduces. Finally beyond
$J=0.444J_{DP}$ (panel E) there is only one mode that repeats periodically,
i.e. vortex dynamics becomes fully synchronous, and accelerates with further
increasing the current (panel F). The relative spectra of frequencies for
$V(t)$ are shown in figure 9, in panels labelled correspondingly to panels of
figure 8. The repetitive modes of vortex crossings within the particular
dynamic configuration lead to peaks at specific frequencies. As the current
density is increased, the spectra show the evolution to a single peak,
corresponding to the frequency of $0.03\tau_{GL}^{-1}$.

Figure 10: Normalised voltage drop as a function of time (normalised to
$\tau_{GL}$), for a SN of $w=6\xi$ under a magnetic field of $H=0.25H_{c2}$
sourced with different current densities: (A) 0.231$J_{DP}$, (B)
0.252$J_{DP}$, (C) 0.270$J_{DP}$, (D) 0.327$J_{DP}$, (E) 0.387$J_{DP}$, and
(F) 0.504$J_{DP}$. Each panel contains an illustrative snapshot of the spatial
distribution of the Cooper-pair density during the dynamics. Figure 11:
Spectra of modulation frequencies $\nu$ (normalised to $\tau_{GL}^{-1}$) of
the temporal voltage signals shown in figure 10.

A similar analysis for the SN with $w=12\xi$ is shown in figure 10, for the
reason that a wider nanostripe allows formation of multiple vortex rows in the
ground state. Panel A of figure 10 shows $V(t)$ at $J=0.231J_{DP}$, when
vortices start dissipatively crossing the SN, in a quasi-synchronous fashion.
As $J$ is increased to $0.327J_{DP}$, the vortex crossings become increasingly
synchronised (panels B-D). However, at $J=0.387J_{DP}$ the flux-flow
instability sets in (panel E), and causes an increasingly chaotic behavior as
$J$ is increased to $J=0.504J_{DP}$ (panel F). In this regime, the apparent
chaotic behavior is caused by the competition between the standard vortex-
vortex repulsion and the effective attractive core-core interaction due to
preferential tailgating at large vortex velocities, interchanging their
dominance on each vortex during the collective dynamics. For $J>0.52J_{DP}$ a
phase slip occurs that will grow as applied current density is raised (shown
in figure 7, states labelled 6-10), and the remaining vortices are crossing
the stripe in tailgated rows. Such case of tailgated vortices causes periodic
modulations $V(\tau)$, however, at such high $J$ this regime is unstable and
therefore not considered in the proceeding discussion. So we only consider the
region of strictly flow-flow during the discussion of synchronised vortex
crossings.
The spectra of frequency modulations (figure 11) show analogous behaviour to
that of the narrower SN discussed previously. At low $J$, when crossings are
quasi-synchronous, we see many mode contributions (i.e. few dominant peaks
accompanied with many additional smaller peaks). As $J$ is increased and
synchronicity improves, the smaller contributions disappear, and the frequency
component with the largest contribution is strengthened. However, at the onset
of the flux-flow instability ($J=0.387J_{DP}$) we observe a broad contribution
centred around frequency $\nu=0.06\tau_{GL}^{-1}$ (corresponding to the median
frequency of crossing of the vortex lattice as a whole, with many individual
asynchronous crossings superimposed). At $J=0.504J_{DP}$ (panel F) the
spectrum loses any order, corresponding to the chaotic behaviour of vortex
crossings. Vortices continuously crossing the SN, will cause oscillations in
the electric and magnetic fields, leading to detectable emission of electro-
magnetic radiation [16, 15, 54]. The crossing of a single vortex releases a
very small amount of energy, whereas multiple vortices moving coherently will
emit a significant (and more easily detectable) amount of energy [54]. In a
coherently moving lattice of vortices, periodic vortex crossing in the SN will
cause emission of radiation at a frequency of $\omega=2\pi v/a$ (washboard
frequency) and at harmonics $\omega=2\pi mv/a$ ($m=2,3...$), where $v$ is the
vortex speed and $a$ is the lattice spacing (i.e., the distance between two
parallel adjacent rows in our case) along the direction of motion [15]. The
highest frequency emitted cannot exceed $\Delta/\hbar$, where $\Delta$ is the
superconducting gap of the SN. The theoretically predicted existence of
radiation has been experimentally confirmed [16].

Figure 12: Panel a - Normalised average vortex velocity in units of
$\xi/\tau_{GL}$ versus the normalised current density applied to SN of width
$6\xi$ and $12\xi$ under an applied magnetic field of $H=0.25H_{c2}$. Panel b
- Histogram of vortex velocities for different values of $J$ relating to
different vortex crossing regimes.

Results of our simulations of the vortex velocity as a function of applied
current density are shown in figure 12a. They evidence a linear dependence at
lower values of $J$, for both above-considered narrow SNs (similar to
behaviour seen in [52, 55]). However, when $J$ is increased to intermediate
values, we find a deviation from the linear dependence, which is due to the
increasingly facilitated vortex tailgating. We use these values of velocities
and the frequency spectra to further discuss the potential for coherent
radiation of vortices crossing the SN. In detail, considering the SN of width
$w=6\xi$, at $J=0.348J_{DP}$, the average velocity is v $\approx
0.03\xi\tau_{GL}^{-1}$. The corresponding spectrum of modulations (figure 9 \-
panel A) shows a number of contributions, the first five occurring at
$\nu_{0}$ = 0.0021, $\nu_{1}$ = 0.0041, $\nu_{2}$ = 0.0062, $\nu_{3}$ =
0.0083, and $\nu_{4}$ = 0.0104 $\tau_{GL}^{-1}$, which are harmonics of the
fundamental mode ($\nu_{0}$). The period of the cycle of repeating vortex
crossings in this case is T = 486 $\tau_{GL}$, while the wavelength is
14.6$\xi$ (obtained from $\lambda=vt$, using the value of vortex velocity in
figure 12). As the vortices move in a quasi-synchronous manner, the washboard
frequency [15] is not applicable.
As the applied current is increased the vortices cross the SN in a more
synchronized manner. In this case we can apply the relation for the washboard
frequency to the values of the average vortex velocity $v$ and the frequency
of first harmonic $\nu_{0}$ and obtain the value for the (virtual) lattice
spacing $a$. Increasing $J$ from $0.366J_{DP}$ to $0.655J_{DP}$, the values of
such lattice spacing decreases from 6.2$\xi$ to 2.8$\xi$, where values were
extrapolated from the washboard frequency relation. The combination of
increasing Lorentz force and edge confining forces, cause the reduction of $a$
and raising of the vortex density during the dynamics. However, in the wider
SN ($w=12\xi$) we do not observe the same behaviour. In this case the value of
$a$ remains constant ($\simeq 3.6\xi$, obtained using the washboard frequency
relation) as the current is increased in the dissipative state, until
transitioning to asynchronous crossings for high $J$ values. This suggests
that in the wider SN the vortex-vortex repulsion within the lattice is more
deterministic for the resulting lattice spacing $a$ than the interactions with
confining edges during the vortex dynamics.

Figure 13: Temporal evolution of the normalized voltage drop, at increasing
(indicated) values of the applied current density for two SNs, of width
12$\xi$ and 24$\xi$. All plots exhibit voltage modulations caused by vortex
crossing. Panels a.i-iv: $w=12\xi$, $H=0.20H_{c2}$ (single row of vortices).
Panels b.i-iv: $w=24\xi$, $H=0.12H_{c2}$ (two vortex rows). Panels c.i-iv:
$w=12\xi$, $H=0.50H_{c2}$ (three rows). Panels d.i-iv: $w=24\xi$,
$H=0.15H_{c2}$ (three rows). The first panel in each row corresponds to the
onset of the dissipative state; the second and third belong to the
synchronous/quasi-synchronous regime; the fourth is the onset of the flux-flow
instability regime.

The synchronisation of vortex crossing in a fixed lattice at large sourced
currents was observed and discussed in ref. [33], albeit without identifying
an exact regime as such. To understand better the regime in which synchronous
lattice crossings can occur, we performed a number of additional simulations.
Figure 13 shows other four examples of the modulations in $V(t)$,
corresponding to a SN with $w=12\xi$ and $H=0.2H_{c2}$ (panels a.i-iv), which
realises a single row of vortices in the static case (figure 2). At
$J=0.28J_{DP}$ the SN becomes dissipative with asynchronous vortex-crossing
behaviour (a.i), until at $J\simeq 0.36J_{DP}$ quasi-synchronous crossings
begin (a.ii). The latter continues (a.iii) until at $J=0.48J_{DP}$ the flux-
flow instability sets in (a.iv), achieving the normal state at
$J_{c2}=0.66J_{DP}$. Panels b.i-iv of figure 13 show similar behaviour for a
SN of width $w=24\xi$ in applied magnetic field $H=0.12H_{c2}$ (realising two
vortex rows), that will not transition to synchronised crossings as applied
current is increased. However, after increasing the magnetic field applied to
either SN, synchronous crossings will take place (panels c.i-iv and d.i-iv).
For SN of $w$ = 12 and 24$\xi$ at $H$ = 0.50 and 0.15$H_{c2}$ respectively,
the vortex configuration comprises three parallel rows. For $w=12\xi$
($w=24\xi$) synchronous crossings start at $J\simeq 0.20J_{DP}$
($J=0.28J_{DP}$) and continue up to the onset of flux-flow instability at
$J=0.28J_{DP}$ ($J=0.42J_{DP}$).

One concludes that the frequency of the radiation stemming from coherent
vortex crossings can be tuned by $H$ or/and by $J$. The applied magnetic field
$H$ changes the vortex density (affecting number of rows) and, hence, $a$;
while transport current directly changes the vortex velocity. Both factors
influence the behaviour of vortex crossings, which in turn affect the
electromagnetic radiation emitted at frequencies $\nu=v/a$ [33]. For an
insight into values expected in experiment, we consider the parameters
measured for Nb thin films by Pinto et al. [18]. For example, in a Nb film of
thickness $d$ = 20 nm, $\xi(0)\simeq$ 8.0 nm and $T_{c}=8$K, providing a
Ginzburg-Landau time of $\tau_{GL}\simeq 65$ fs, our results show an average
velocity of vortices crossing the Nb SN of thickness 20nm and width 50-100 nm
to be in the range 1-10 km/s, with first harmonic frequencies in the range
1-50 GHz. These values are similar to those reported by Dobrovolskiy et al.
[38, 16] and to those of Embon et al. [35]. A thin and narrow superconductor
with high $T_{c}$ with a small value of $\tau_{GL}$ ($\simeq 1-10$fs), where
faster vortex crossings could be realized and used as a terahertz radiation
source. Such sources are highly sought for a variety of applications [56],
including clinical [57] and terahertz time-domain spectroscopy [58].

##  4 Discussion

This study has revealed a consistent theme, where the narrowest SNs exhibit
stronger confinement forces. A vortex row phase diagram (figure 2) showed how
the narrowest stripes support a lower number of vortex rows, and had a lower
average vortex density (for given $H$) that deviated more from the theoretical
value (figure 5). It was found a range of SN widths, 20-60$\xi$, where vortex
rows would transition from a single to two-rows, and back to a single row (see
inset figure 5). This is due to an interplay of vortex interaction and the
edge barrier strengths as the field is increased, below and above the range of
width quoted the confinement forces are too strong or weak respectively. When
studying the critical current as a function of magnetic field (figure 6),
after a new row of vortices emerges $J_{c1}$ is at a minima, but increases as
the field is increased due to stronger confinement forces acting a vortex
pinning potential. It increases to a maxima before vortex density increases
too high that the confinement forces at the edge overpower the vortex-vortex
interaction. Vortices crossing the SN cause modulations in the voltage drop
across the stripe (figures 8, 10, 13), the same process also causes EM
radiation to be emitted [15]. A single vortex crossing the stripe emits a
photon with energy proportional to the inverse of the crossing time. More
vortices crossing in synchronicity causes more power to be emitted for that
frequency. Therefore, vortices moving in synchronous rows will provide greater
radiation power. SNs were found to exhibit an evolution for how the vortex
rows crossed the stripe as the applied current density was increased.
Depending on the width and magnetic field, the SN would start dissipation with
vortices crossing slowly quasi-synchronously (figure 8,A) or with little
repetition (figure 13,b). As $J$ increased the crossings progress to quasi-
synchronous (figure 13,b) or synchronous (figure 10,D), until proceeding to
flux-flow instability. Not every example showed the evolution to synchronised
vortex crossings, however, a regime for the occurrence of vortex rows crossing
in a fixed lattice was observed for states with average vortex density
$A\lesssim 81\xi^{2}$ (in figure 5). This is also related to confinement, a
large vortex density is required so the edge confining forces a can adequately
act on the vortex rows and effectively lock them in a dynamic lattice. The
more synchronised the vortex crossings, the few contributions to the spectrum
of frequencies (figures 9, 11), which is beneficial for generation of EM
radiation.

## References

  * [1] Robert H Hadfield.  Single-photon detectors for optical quantum information applications.  Nature photonics, 3(12):696–705, 2009.
  * [2] Chandra M Natarajan, Michael G Tanner, and Robert H Hadfield.  Superconducting nanowire single-photon detectors: physics and applications.  Superconductor science and technology, 25(6):063001, 2012.
  * [3] GN Gol’Tsman, O Okunev, G Chulkova, A Lipatov, A Semenov, K Smirnov, B Voronov, A Dzardanov, C Williams, and Roman Sobolewski.  Picosecond superconducting single-photon optical detector.  Applied physics letters, 79(6):705–707, 2001.
  * [4] Eric A Dauler, Bryan S Robinson, Andrew J Kerman, Joel KW Yang, Kristine M Rosfjord, Vikas Anant, Boris Voronov, Gregory Gol’tsman, and Karl K Berggren.  Multi-element superconducting nanowire single-photon detector.  IEEE Transactions on Applied Superconductivity, 17(2):279–284, 2007\.
  * [5] VK Vlasko-Vlasov, F Colauto, Alexandre I Buzdin, D Rosenmann, T Benseman, and W-K Kwok.  Magnetic gates and guides for superconducting vortices.  Physical Review B, 95(14):144504, 2017.
  * [6] VK Vlasko-Vlasov, F Colauto, Alexandre I Buzdin, D Rosenmann, T Benseman, and W-K Kwok.  Manipulating abrikosov vortices with soft magnetic stripes.  Physical Review B, 95(17):174514, 2017.
  * [7] Rosa Córdoba, Pablo Orús, Željko L Jelić, Javier Sesé, Manuel Ricardo Ibarra, Isabel Guillamón, Sebastián Vieira, Juan José Palacios, Hermann Suderow, Milorad V Milosević, et al.  Long-range vortex transfer in superconducting nanowires.  Scientific reports, 9(1):1–10, 2019.
  * [8] J Brooke, David Bitko, Rosenbaum, and Gabriel Aeppli.  Quantum annealing of a disordered magnet.  Science, 284(5415):779–781, 1999.
  * [9] MW Johnson, P Bunyk, F Maibaum, E Tolkacheva, AJ Berkley, EM Chapple, R Harris, J Johansson, T Lanting, I Perminov, et al.  A scalable control system for a superconducting adiabatic quantum optimization processor.  Superconductor Science and Technology, 23(6):065004, 2010.
  * [10] Frederick W Strauch, Philip R Johnson, Alex J Dragt, CJ Lobb, JR Anderson, and FC Wellstood.  Quantum logic gates for coupled superconducting phase qubits.  Physical review letters, 91(16):167005, 2003.
  * [11] Charles Reichhardt and CJ Olson Reichhardt.  Jamming and diode effects for vortices in nanostructured superconductors.  Physica C: Superconductivity, 470(19):722–725, 2010.
  * [12] JF Wambaugh, C Reichhardt, CJ Olson, F Marchesoni, and Franco Nori.  Superconducting fluxon pumps and lenses.  Physical Review Letters, 83(24):5106, 1999.
  * [13] Akito Daido, Yuhei Ikeda, and Youichi Yanase.  Intrinsic superconducting diode effect.  Physical Review Letters, 128(3):037001, 2022.
  * [14] Nodar Samkharadze, A Bruno, Pasquale Scarlino, G Zheng, DP DiVincenzo, L DiCarlo, and LMK Vandersypen.  High-kinetic-inductance superconducting nanowire resonators for circuit qed in a magnetic field.  Physical Review Applied, 5(4):044004, 2016.
  * [15] LN Bulaevskii and EM Chudnovsky.  Electromagnetic radiation from vortex flow in type-ii superconductors.  Physical review letters, 97(19):197002, 2006.
  * [16] OV Dobrovolskiy, VM Bevz, M Yu Mikhailov, OI Yuzephovich, VA Shklovskij, RV Vovk, MI Tsindlekht, R Sachser, and M Huth.  Microwave emission from superconducting vortices in mo/si superlattices.  Nature communications, 9(1):1–9, 2018.
  * [17] Andrea Perali, Antonio Bianconi, Alessandra Lanzara, and Naurang L Saini.  The gap amplification at a shape resonance in a superlattice of quantum stripes: A mechanism for high tc.  Solid State Communications, 100(3):181–186, 1996.
  * [18] Nicola Pinto, S Javad Rezvani, Andrea Perali, Luca Flammia, Milorad V Milošević, Matteo Fretto, Cristina Cassiago, and Natascia De Leo.  Dimensional crossover and incipient quantum size effects in superconducting niobium nanofilms.  Scientific reports, 8(1):1–12, 2018.
  * [19] Andrea Guidini, Luca Flammia, Milorad V Milošević, and Andrea Perali.  Bcs-bec crossover in quantum confined superconductors.  Journal of Superconductivity and Novel Magnetism, 29(3):711–715, 2016.
  * [20] TT Saraiva, PJF Cavalcanti, Alexei Vagov, AS Vasenko, A Perali, L Dell’Anna, and AA Shanenko.  Multiband material with a quasi-1d band as a robust high-temperature superconductor.  Physical Review Letters, 125(21):217003, 2020.
  * [21] VV Moshchalkov, Lut Gielen, C Strunk, Rik Jonckheere, X Qiu, C Van Haesendonck, and Yvan Bruynseraede.  Effect of sample topology on the critical fields of mesoscopic superconductors.  Nature, 373(6512):319–322, 1995.
  * [22] Supriyo Datta.  Electronic transport in mesoscopic systems.  Cambridge university press, 1997.
  * [23] Tristan Cren, Denis Fokin, François Debontridder, Vincent Dubost, and Dimitri Roditchev.  Ultimate vortex confinement studied by scanning tunneling spectroscopy.  Physical review letters, 102(12):127005, 2009.
  * [24] N Marrocco, GP Pepe, A Capretti, L Parlato, V Pagliarulo, G Peluso, A Barone, R Cristiano, M Ejrnaes, A Casaburi, et al.  Strong critical current density enhancement in nicu/nbn superconducting nanostripes for optical detection.  Applied Physics Letters, 97(9):092504, 2010.
  * [25] CP Bean and JD Livingston.  Surface barrier in type-ii superconductors.  Physical Review Letters, 12(1):14, 1964.
  * [26] L Flammia, L-F Zhang, L Covaci, A Perali, and MV Milošević.  Superconducting nanoribbon with a constriction: A quantum-confined josephson junction.  Physical Review B, 97(13):134514, 2018.
  * [27] GR Berdiyorov, XH Chao, FM Peeters, HB Wang, VV Moshchalkov, and BY Zhu.  Magnetoresistance oscillations in superconducting strips: A ginzburg-landau study.  Physical Review B, 86(22):224504, 2012.
  * [28] PW Anderson and AH Dayem.  Radio-frequency effects in superconducting thin film bridges.  Physical Review Letters, 13(6):195, 1964.
  * [29] Rosa Córdoba, TI Baturina, J Sesé, A Yu Mironov, JM De Teresa, MR Ibarra, DA Nasimov, AK Gutakovskii, AV Latyshev, I Guillamón, et al.  Magnetic field-induced dissipation-free state in superconducting nanostructures.  Nature communications, 4(1):1–7, 2013.
  * [30] GR Berdiyorov, MV Milošević, ML Latimer, ZL Xiao, WK Kwok, and FM Peeters.  Large magnetoresistance oscillations in mesoscopic superconductors due to current-excited moving vortices.  Physical review letters, 109(5):057004, 2012.
  * [31] Gregory J Kimmel, Andreas Glatz, Valerii M Vinokur, and Ivan A Sadovskyy.  Edge effect pinning in mesoscopic superconducting strips with non-uniform distribution of defects.  Scientific reports, 9(1):1–12, 2019.
  * [32] DY Vodolazov.  Vortex-induced negative magnetoresistance and peak effect in narrow superconducting films.  Physical Review B, 88(1):014525, 2013.
  * [33] D Yu Vodolazov and FM Peeters.  Rearrangement of the vortex lattice due to instabilities of vortex flow.  Physical Review B, 76(1):014521, 2007.
  * [34] ŽL Jelić, MV Milošević, and AV Silhanek.  Velocimetry of superconducting vortices based on stroboscopic resonances.  Scientific reports, 6(1):1–8, 2016.
  * [35] Lior Embon, Yonathan Anahory, Željko L Jelić, Ella O Lachman, Yuri Myasoedov, Martin E Huber, Grigori P Mikitik, Alejandro V Silhanek, Milorad V Milošević, Alexander Gurevich, et al.  Imaging of super-fast dynamics and flow instabilities of superconducting vortices.  Nature communications, 8(1):1–10, 2017.
  * [36] MV Milošević and R Geurts.  The ginzburg–landau theory in application.  Physica C: Superconductivity, 470(19):791–795, 2010.
  * [37] G Carapella, P Sabatino, C Barone, S Pagano, and M Gombos.  Current driven transition from abrikosov-josephson to josephson-like vortex in mesoscopic lateral s/s’/s superconducting weak links.  Scientific reports, 6(1):1–12, 2016.
  * [38] OV Dobrovolskiy, D Yu Vodolazov, F Porrati, R Sachser, VM Bevz, M Yu Mikhailov, AV Chumak, and M Huth.  Ultra-fast vortex motion in a direct-write nb-c superconductor.  Nature communications, 11(1):1–10, 2020.
  * [39] SE Hebboul, DP Johnson, and M Rokhlin.  Radio-frequency oscillations in two-dimensional superconducting in- ino x: a possible evidence for vortex density waves.  Physical review letters, 82(4):831, 1999.
  * [40] Mauro M Doria, JE Gubernatis, and D Rainer.  Virial theorem for ginzburg-landau theories with potential applications to numerical studies of type-ii superconductors.  Physical Review B, 39(13):9573, 1989.
  * [41] Ryuzo Kato, Yoshihisa Enomoto, and Sadamichi Maekawa.  Effects of the surface boundary on the magnetization process in type-ii superconductors.  Physical Review B, 47(13):8016, 1993.
  * [42] BI Ivlev and NB Kopnin.  Electric currents and resistive states in thin superconductors.  Advances of Physics, 33(1):47–114, 1984.
  * [43] Lorenz Kramer and RJ Watts-Tobin.  Theory of dissipative current-carrying states in superconducting filaments.  Physical Review Letters, 40(15):1041, 1978.
  * [44] JR Clem.  Paper k36. 06.  Bull. Am. Phys. Soc, 43:411, 1998.
  * [45] Gheorghe Stan, Stuart B Field, and John M Martinis.  Critical field for complete vortex expulsion from narrow superconducting strips.  Physical review letters, 92(9):097003, 2004.
  * [46] GM Maksimova.  Mixed state and critical current in narrow semiconducting films.  Physics of the Solid State, 40(10):1607–1610, 1998.
  * [47] Aleksej A Abrikosov.  Nobel lecture: Type-ii superconductors and the vortex lattice.  Reviews of modern physics, 76(3):975, 2004.
  * [48] Y Mawatari and K Yamafuji.  Critical current density in thin films due to the surface barrier.  Physica C: Superconductivity, 228(3-4):336–350, 1994.
  * [49] Gilson Carneiro.  Equilibrium vortex-line configurations and critical currents in thin films under a parallel field.  Physical Review B, 57(10):6077, 1998.
  * [50] D Yu Vodolazov, IL Maksimov, and EH Brandt.  Vortex entry conditions in type-ii superconductors.: Effect of surface defects.  Physica C: Superconductivity, 384(1-2):211–226, 2003.
  * [51] James S Langer and Vinay Ambegaokar.  Intrinsic resistive transition in narrow superconducting channels.  Physical Review, 164(2):498, 1967.
  * [52] Leonardo Rodrigues Cadorim, Alexssandre de Oliveira Junior, and Edson Sardella.  Ultra-fast kinematic vortices in mesoscopic superconductors: the effect of the self-field.  Scientific reports, 10(1):1–8, 2020.
  * [53] ŽL Jelić, MV Milošević, Joris Van de Vondel, and AV Silhanek.  Stroboscopic phenomena in superconductors with dynamic pinning landscape.  Scientific reports, 5(1):1–10, 2015.
  * [54] OV Dolgov and N Schopohl.  Transition radiation of moving abrikosov vortices.  Physical Review B, 61(18):12389, 2000.
  * [55] GR Berdiyorov, MV Milošević, and FM Peeters.  Kinematic vortex-antivortex lines in strongly driven superconducting stripes.  Physical Review B, 79(18):184506, 2009.
  * [56] HA Hafez, X Chai, A Ibrahim, S Mondal, D Férachou, X Ropagnol, and T Ozaki.  Intense terahertz radiation and their applications.  Journal of Optics, 18(9):093004, 2016.
  * [57] Joo-Hiuk Son, Seung Jae Oh, and Hwayeong Cheon.  Potential clinical applications of terahertz radiation.  Journal of Applied Physics, 125(19):190901, 2019.
  * [58] A Gowen, Creidhe O’Sullivan, and CP O’Donnell.  Terahertz time domain spectroscopy and imaging: Emerging techniques for food process monitoring and quality control.  Trends in Food Science & Technology, 25(1):40–46, 2012.