# Investigating the Vision Transformer Model for Image Retrieval Tasks

Socratis Gkelios  Department of Electrical and
Computer Engineering
Democritus University of Thrace
Xanthi, Greece.
<EMAIL_ADDRESS>Yiannis Boutalis  Department of Electrical and
Computer Engineering
Democritus University of Thrace
Xanthi, Greece.
<EMAIL_ADDRESS>Savvas A. Chatzichristofis *Corresponding author:
<EMAIL_ADDRESS>Intelligent Systems Laboratory
Department of Computer Science
Neapolis University Pafos
Pafos, Cyprus
<EMAIL_ADDRESS>
###### Abstract

This paper introduces a plug-and-play descriptor that can be effectively
adopted for image retrieval tasks without prior initialization or preparation.
The description method utilizes the recently proposed Vision Transformer
network while it does not require any training data to adjust parameters. In
image retrieval tasks, the use of Handcrafted global and local descriptors has
been very successfully replaced, over the last years, by the Convolutional
Neural Networks (CNN)-based methods. However, the experimental evaluation
conducted in this paper on several benchmarking datasets against 36 state-of-
the-art descriptors from the literature demonstrates that a neural network
that contains no convolutional layer, such as Vision Transformer, can shape a
global descriptor and achieve competitive results. As fine-tuning is not
required, the presented methodology’s low complexity encourages adoption of
the architecture as an image retrieval baseline model, replacing the
traditional and well adopted CNN-based approaches and inaugurating a new era
in image retrieval approaches.

###### Index Terms:

Vision Transformer, Image Retrieval, CBIR

##  I Introduction

Content-based image retrieval (CBIR) is one of the fundamental scientific
challenges that the multimedia, computer vision, and robotics communities have
thoroughly researched for decades. Three eras that vary in how researchers
export the different features that define an image’s visual content
characterize content-based image retrieval [1]. The literature strongly
focuses on global low-level descriptors during the early years, shaping the
first CBIR era. A single vector was used to represent different aspects of an
image, such as color, shape, and texture. In the sequel, the community started
to concentrate on representations focused on the extraction and use of local
features. From 2003 onwards, new approaches adopt local image descriptors to
search for salient image patches and points-of-interest, such as edges,
corners, and blobs. Since 2015, image retrieval research strongly relies on
approaches focused on Deep Learning (DL) and Convolutional Neural Networks
(CNN). Although DL-based techniques require a large amount of data for
training, the pre-trained networks have been demonstrated in several studies
to be particularly useful as feature extractors and achieve high retrieval
performance.

Meanwhile, in natural language processing (NLP), the self-attention-based
architecture, particularly Transformers, is now considered as the new standard
[2]. The Transformer is a type of deep-neural network mainly based on self-
attention mechanism [3]. Recently, researchers have expanded transformers for
computer vision tasks inspired by the influence of the Transformer in NLP [4,
5, 6]. For example, the authors in [7] have trained a sequence transformer to
auto-regressively predict pixels and achieve competitive results with CNNs on
an image classification task. Recently, the authors in [8] have attempted to
apply a typical transformer directly to images, with the fewest adjustments
possible shaping an image recognition network. Vision Transformer network
(ViT) provides better performance than a traditional convolutional neural
network in image recognition tasks.

This paper builds upon the Vision Transformer architecture to shape a global
descriptor for image retrieval tasks. Following the procedure that the vast
majority of the deep-learning-based image retrieval models adopt, we discard
the fully connected layers of the Transformer’s network and use the last layer
as a feature to describe the visual content of the image. Best we can tell,
this is the first attempt to evaluating the performance of the transformers on
image retrieval tasks.

The remainder of this paper is structured as follows. Section II briefly shows
how the vision transformer’s architecture operates and presents the process of
shaping the image retrieval descriptor. Section III offers a thorough
experimental evaluation of the descriptor complemented by a discussion of
results. Finally, Section IV concludes the paper, and discusses potential
opportunities for future work.

Figure 1: Graphical Abstract of the ViT Extraction Procedure.

##  II ViT Descriptor

The Transformer architecture was first introduced in [2] for neural machine
translation exhibiting state-of-the-art performance and replacing models, such
as Long short-term memory artificial recurrent neural networks and Gated
recurrent units (GRUs) in several Natural Language Processing (NLP) tasks. The
architecture of the Transformer usually consists of stacked Transformer
layers, each of which takes as input a sequence of vectors and outputs a new
sequence of vectors of the same shape. The critical element of the Transformer
architectures resides in the attention mechanism. The transformer
development’s primary motivation was to overcome the limitations of the
recurrent neural networks (RNN) and mainly to capture long-term global
dependencies. Based on this foundation, many noteworthy approaches emerged,
for example, Bidirectional Encoder Representations from Transformers (BERT)
[9], and GPT-3 [10], both pre-trained in an unsupervised manner from the
unlabeled text. These methods demonstrated rich representation capability
without fine-tuning. Both methods are conceptually simple and empirically
powerful.

The application of Transformers in computer vision tasks poses a significant
challenge for two main reasons:

  * •

Images contain much more information compared to words or sentences.

  * •

Attention to every pixel is computationally exhaustive.

The authors in [8] proposed a novel approach called Vision Transformer (ViT)
to tackle these challenges. ViT borrows the encoder part of the NLP
Transformer. The standard Transformer receives a 1D token embedding sequence
as input. In this case, the images are split into fixed-sized patches and feed
into the model. A learnable positional embedding vector is assigned to every
patch to utilize the order of the input sequence. Besides, a special token is
added, just like in the case of BERT. In a nutshell, a set of patches is
constructed for each image multiplied with an embedding matrix, which is
finally fused with a positional embedding vector, forming the Transformer
input. In all its layers, the Transformer uses constant latent vector size, so
with a trainable linear projection, the patches are flattened to map these
dimensions. This projection’s output is referred to as patch embedding.

Each encoder consists of two sub-layers. In the first sub-layer, the inputs
flow through a self-attention module, while in the second, the outputs of the
self-attention operation are passed to a position-wise feed-forward neural
network. Furthermore, skip connections [11] are incorporated around each sub-
layer that undergo layer normalization. The architecture of the encoder is
depicted in Figure 2. Every layer contains a constant linear projection
mapping of the flattened patches of dimension D refined during training. D is
equal to 768 for ViT-B architectures, whereas in ViT-L, the corresponding
value is 1024. Even though the encoder layers are composed of identical
structural elements, they do not share weights.

The self-attention module consists of three learnable components, Query (Q),
Key (K), and Value (V). A score matrix is assembled by Q and K dot product
multiplication that portrays the attention of each ‘word’ to each other ‘word’
(in our approach, each ‘word’ corresponds to one of the image patches). The
score matrix is scaled accordingly and is passed to a softmax layer to convert
scores into probabilities. In the sequel, the softmax output is multiplied
with the V vector to highlight the important words. Multi-headed attention
improves the performance of the Transformer. In particular, the model performs
multiple parallel attention functions to different linear projections of the
vectors Q, V, and K instead of utilizing a single one. Thus, this procedure
enables the model to focus on different positions and representation
subspaces.

Position-wise neural network module is a fully connected feed-forward neural
network (FFNN), which consists of two linear transformations with a ReLU
activation in between. Layer Layernorm (LN) is applied before each module, and
residual connections after every block, followed by layer normalization [12].

Figure 2:  Modelling the Transformer Encoder Architecture.

In this paper, we adopted the pre-trained architecture of [8]. More
specifically, this paper adopts the ‘Base’ models (ViT-B) and the ‘Large’
models (ViT-L) from BERT. Two variants of each architecture regarding their
input patch size are applied. ViT-L16 corresponds to $16\times 16$ input batch
size, while ViT-L32 to $32\times 32$ patch size. The same notations are used
for the ViT-B model accordingly. The models with smaller patch sizes are more
resource-intensive due to the inversely proportional relationship between the
Transformer’s length sequence and the patch size square.

The adopted architectures have been trained on 21k-ImageNet (21k classes/14
million images) and have been finetuned on the ILSVRC-2012 ImageNet (1k
classes/1.3 million images) [13]. ViT-B features 12 encoder layers with an
FFNN of size 3072. On the other hand, ViT-L contains 24 encoder layers and an
FFNN of size 4096.

In the case of ViT descriptor, initially, all the images are resized to
$384\times 384$ pixel size. A pre-preprocessing unit subtracts 127.5 to all
pixels and scales by 255 to have pixel values ranging [-1, 1]. The pre-
processed images flow through the Transformer Encoder. The last softmax layer
of the models is removed, leaving the normalized last encoder’s output of
dimension D as the final layer. This procedure shapes a representation vector
for each image. The final step is comprised of a normalization procedure to
shape the ViT descriptor. Figure1 illustrates the proposed CBIR architecture.

##  III Experimental Setup

This section provides details about the experiments conducted for the
evaluation of the ViT descriptor. The presented image representation is
evaluated on four well-known image datasets:

  * •

INRIA Holidays [14]: This dataset consists of 1491 cellphone-taken pictures
with a wide range of holiday scenes and objects. The number of images varies
from 2 to 13 images per group. The INRIA Holidays dataset provides many query
images instead of the UKBench database. The ground truth comprises images of a
visual definition identical to the query image, without indicating the same
object’s co-occurrence.

  * •

UKBench [15]: The UKBench dataset consists of 10,200 images arranged in 2250
categories dataset. There are four pictures of a single object in each
category, taken from various views and under varying lighting conditions. The
top-4 candidate (NS) score [15] is used for this dataset during the evaluation
process to calculate the retrieval accuracy.

  * •

Paris6K [16]: 6412 photographs representing unique Paris landmarks are
included in the Paris6k Dataset. This collection consists of 55 pictures of
buildings and monuments from requests. There is more variety in the landmarks
in Paris6k than those in Oxford5k.

  * •

Oxford5k [17]: This building’s dataset is made up of 5062 Flickr images. The
set has been manually annotated to produce a detailed ground truth for 11
distinct landmarks, each represented by five potential queries. The index,
overall, consists of 55 requests. In this dataset, completely different views
of the same building are labeled by the same name, making the collection
challenging for image retrieval tasks [18].

For the INRIA, Oxford5k, and Paris6K datasets, the mean Average Precision
(mAP) [19] is used as an evaluation metric. In the perfect retrieval case, mAP
is equal to 100, while as the number of the nonrelevant images ranked above a
retrieved relevant image increases, the mAP approaches the value 0. On the
other hand, the retrieval performance on the UKBench dataset is evaluated
using the recall rate for the top-4 candidates (NS) as an evaluation metric.
In the case of perfect retrieval, NS is equal to 4.

To calculate the similarity between the descriptors, we use multiple distance
metrics. There are specific advantages and drawbacks of each similarity
metric, each one being more appropriate fro particular data types [20]. This
paper assesses the use of Manhattan, Euclidean, Cosine, Bray-Curtis, Canberra,
Chebyshev, and Correlation distance metrics in the current analysis. The
Manhattan and Euclidean distance metrics are considered well-known, and
therefore, no further details are given in this paper. Here we only note that
the Manhattan Distance (also known as city block distance) is preferred over
the Euclidean distance metric as the data’s dimension increases.

The Cosine distance counts the image descriptors’ inner-product space,
considering their orientation and not their magnitude.

$d_{Cosine}({\bf p},{\bf
q})={{p}{q}\over\|{p}\|\|{q}\|}=\frac{\sum_{i=1}^{n}{{p}_{i}{q}_{i}}}{\sqrt{\sum_{i=1}^{n}{({p}_{i})^{2}}}\sqrt{\sum_{i=1}^{n}{({q}_{i})^{2}}}}$
(1)

Bray-Curtis and Canberra are mutated from Manhattan, where, as seen in the
following equation, the sum of the differences between the coordinates of the
vectors is normalized:

$d_{Canberra}({\bf p},{\bf
q})=\sum_{i=1}^{n}\frac{|p_{i}-q_{i}|}{|p_{i}|+|q_{i}|}$ (2)

$d_{Bray-Curtis}({\bf p},{\bf
q})=\frac{\sum_{i=1}^{n}|p_{i}-q_{i}|}{\sum_{i=1}^{n}(p_{i}+q_{i})}$ (3)

Furthermore, the Chebyshev distance between two feature vectors is estimated
as the maximum variation along any coordinate dimension:

$d_{Chebyshev}({\bf p},{\bf
q})=\max_{i}(\left|\begin{matrix}p_{i}-q_{i}\end{matrix}\right|)$ (4)

Finally, the Correlation distance measures the dependency between the two
feature vectors:

$d_{Correl.}({\bf p},{\bf
q})=\frac{\sum_{i=1}^{n}(p_{i}-\bar{p})(q_{i}-\bar{q})}{\sqrt{\sum_{i=1}^{n}(p_{i}-\bar{p})^{2}}\sqrt{\sum_{i=1}^{n}(q_{i}-\bar{q})^{2}}}$
(5)

In all metrics, $p$ and $q$ are the corresponding image descriptors.

Additionally, this section also evaluates the appropriate normalization
technique. The batch normalization technique promotes training by specifically
normalizing each layer’s inputs to have zero mean and unit variance. Weight
normalization reparameterizes the weight vectors in a neural network that
decouples the length of those weight vectors from their direction [21],
inspired by batch normalization. In other words, by their L2-norm or L1-norm
[22], weight normalization reparameterizes incoming weights.

The L1-norm uses the sum of all the values providing equal penalty to all
parameters, enforcing sparsity:

$X_{i}^{\prime}=\frac{X_{i}}{\sum_{k=0}^{n}X_{k}}$ (6)

Similarly, the L2-norm uses the square root of the sum of all the squared
values.

$X_{i}^{\prime}=\frac{X_{i}}{\sum_{k=0}^{n}X_{k}^{2}}$ (7)

In all cases, $X_{i}$ is the value to be normalized, and $X_{i}^{\prime}$ is
the normalized score.

The post-processing normalization method consists of the independent
normalization of each derived descriptor (L1-norm and L2-norm Axis-1), the
normalization of each characteristic (L1-norm and L2-norm Axis-0), and the
scaling of ROBUST.

$X_{i}^{\prime}=\frac{X_{i}-q_{1}(X_{i})}{q_{2}(X_{i})-q_{1}(X_{i})}$ (8)

where $q_{1}$ and $q_{2}$ are quantiles.

The scaling normalization of ROBUST eliminates the median. It scales the data
according to the quantile range, meaning that the sample mean and variance are
not affected by the outliers.

###  III-A Retrieval Results

Tables I, II, III and IV list the image retrieval performance on the INRIA,
UKBench, Paris6K and Oxford5K datasets respectively. The first conclusion that
emerges by observing the results is that the ViT descriptor performs
remarkably well with almost all the different similarity metrics and all the
normalization techniques in all four image datasets. In all experiments, the
Chebyshev metric significantly lags behind the effectiveness of the other
distance metrics.

A more in-depth analysis of the results also reveals that the ViT-B16
descriptor performs robustly well in all databases. Of course, in the UKBench
dataset, the ViT-L16 descriptor manages to exceed the performance of the
ViT-B16 descriptor slightly, but the difference is not significant to
generalize a conclusion. The ViT-B16 descriptor presents high retrieval
accuracy, promoting robustness in all datasets and almost all the different
similarity metrics.

Regarding the normalization techniques, ROBUST scaling appears to be the
optimal solution. Moreover, one quickly concludes that the ROBUST
normalization technique, together with the Cosine distance as the similarity
metric, shape the descriptor’s most effective setup. This combination manages
to outperform any other design in all performed experiments. Thus, it is safe
to conclude that the combination of the ViT-B16 model, with ROBUST
normalization and Cosine distance as similarity metric, shapes a robust plug-
and-play solution for effective image retrieval.

TABLE I: Experimental results on INRIA database  | INRIA
---|---
Model | Manh. | Eucl. | Cos | BC | Canb. | Cheb. | Correl.
ViT-L16 |
L2 Axis=1 | 84.95 | 84.98 | 84.98 | 84.88 | 84.62 | 76.24 | 84.97
L2 Axis=0 | 85.02 | 85.24 | 85.36 | 85.07 | 84.54 | 77.67 | 85.36
L1 Axis=1 | 84.77 | 84.88 | 84.98 | 84.91 | 84.67 | 76.26 | 84.97
L1 Axis=0 | 85.02 | 85.23 | 85.53 | 85.08 | 84.54 | 78.02 | 85.52
ROBUST | 84.98 | 85.10 | 86.31 | 86.08 | 85.50 | 76.71 | 86.31
ViT-L32 |
L2 Axis=1 | 86.84 | 86.56 | 86.56 | 86.77 | 86.32 | 75.90 | 86.56
L2 Axis=0 | 86.65 | 86.61 | 86.96 | 86.92 | 86.30 | 76.34 | 86.96
L1 Axis=1 | 86.88 | 86.57 | 86.56 | 86.78 | 86.32 | 75.45 | 86.56
L1 Axis=0 | 86.65 | 86.64 | 86.97 | 86.92 | 86.30 | 76.52 | 86.97
ROBUST | 86.61 | 86.61 | 87.03 | 87.05 | 86.80 | 76.83 | 87.03
ViT-B16 |
L2 Axis=1 | 87.16 | 87.09 | 87.09 | 87.17 | 86.57 | 75.77 | 87.09
L2 Axis=0 | 86.79 | 86.53 | 86.76 | 87.26 | 86.55 | 77.95 | 87.67
L1 Axis=1 | 87.18 | 86.97 | 87.09 | 87.18 | 86.58 | 75.30 | 87.09
L1 Axis=0 | 86.78 | 86.51 | 87.67 | 87.29 | 86.55 | 77.31 | 87.67
ROBUST | 86.42 | 86.33 | 87.99 | 87.81 | 87.32 | 77.07 | 87.99
ViT-B32 |
L2 Axis=1 | 85.07 | 84.80 | 84.80 | 85.19 | 84.61 | 62.52 | 84.80
L2 Axis=0 | 84.97 | 85.04 | 85.61 | 85.38 | 84.74 | 77.96 | 85.61
L1 Axis=1 | 85.31 | 84.96 | 84.80 | 85.19 | 84.76 | 61.84 | 84.80
L1 Axis=0 | 84.98 | 84.93 | 85.63 | 85.38 | 84.74 | 77.25 | 85.63
ROBUST | 84.96 | 85.07 | 86.11 | 86.22 | 85.32 | 76.85 | 86.11
TABLE II: Experimental results on UKBench database  | UKBench
---|---
Model | Manh. | Eucl. | Cos | BC | Canb. | Cheb. | Correl.
ViT-L16 |
L2 Axis=1 | 3.782 | 3.785 | 3.785 | 3.781 | 3.764 | 3.561 | 3.785
L2 Axis=0 | 3.778 | 3.78 | 3.785 | 3.782 | 3.764 | 3.575 | 3.785
L1 Axis=1 | 3.781 | 3.783 | 3.785 | 3.781 | 3.764 | 3.56 | 3.785
L1 Axis=0 | 3.778 | 3.781 | 3.785 | 3.782 | 3.764 | 3.574 | 3.785
ROBUST | 3.777 | 3.779 | 3.784 | 3.781 | 3.763 | 3.567 | 3.784
ViT-L32 |
L2 Axis=1 | 3.75 | 3.752 | 3.752 | 3.75 | 3.73 | 3.416 | 3.752
L2 Axis=0 | 3.735 | 3.737 | 3.754 | 3.751 | 3.731 | 3.455 | 3.754
L1 Axis=1 | 3.752 | 3.751 | 3.752 | 3.75 | 3.731 | 3.412 | 3.752
L1 Axis=0 | 3.735 | 3.737 | 3.754 | 3.751 | 3.731 | 3.453 | 3.754
ROBUST | 3.735 | 3.737 | 3.749 | 3.747 | 3.729 | 3.455 | 3.749
ViT-B16 |
L2 Axis=1 | 3.745 | 3.746 | 3.746 | 3.745 | 3.728 | 3.345 | 3.746
L2 Axis=0 | 3.733 | 3.737 | 3.758 | 3.752 | 3.729 | 3.496 | 3.758
L1 Axis=1 | 3.746 | 3.746 | 3.746 | 3.746 | 3.729 | 3.34 | 3.746
L1 Axis=0 | 3.733 | 3.737 | 3.758 | 3.752 | 3.729 | 3.496 | 3.758
ROBUST | 3.733 | 3.737 | 3.759 | 3.752 | 3.733 | 3.496 | 3.759
ViT-B32 |
L2 Axis=1 | 3.712 | 3.702 | 3.702 | 3.712 | 3.692 | 2.971 | 3.702
L2 Axis=0 | 3.709 | 3.714 | 3.725 | 3.718 | 3.693 | 3.442 | 3.725
L1 Axis=1 | 3.711 | 3.701 | 3.702 | 3.712 | 3.693 | 2.952 | 3.702
L1 Axis=0 | 3.709 | 3.714 | 3.725 | 3.717 | 3.693 | 3.439 | 3.725
ROBUST | 3.709 | 3.714 | 3.726 | 3.722 | 3.697 | 3.45 | 3.726
TABLE III: Experimental results on Paris6k database  | Paris6k
---|---
Model | Manh. | Eucl. | Cos | BC | Canb. | Cheb. | Correl.
ViT-L16 |
L2 Axis=1 | 86.24 | 86.23 | 86.23 | 86.25 | 85.82 | 77.34 | 86.25
L2 Axis=0 | 86.27 | 86.24 | 86.63 | 86.46 | 85.82 | 76.16 | 86.63
L1 Axis=1 | 86.17 | 86.15 | 86.23 | 86.25 | 85.82 | 77.17 | 86.25
L1 Axis=0 | 86.25 | 86.23 | 86.64 | 86.47 | 85.82 | 75.97 | 86.64
ROBUST | 86.18 | 86.13 | 86.74 | 86.71 | 86.38 | 76.16 | 86.74
ViT-L32 |
L2 Axis=1 | 85.29 | 85.34 | 85.34 | 85.41 | 85.09 | 72.56 | 85.34
L2 Axis=0 | 84.48 | 84.47 | 85.68 | 85.65 | 85.11 | 69.76 | 85.68
L1 Axis=1 | 85.37 | 85.42 | 85.34 | 85.41 | 85.1 | 72.6 | 85.34
L1 Axis=0 | 84.45 | 84.45 | 85.69 | 85.65 | 85.11 | 69.57 | 85.69
ROBUST | 84.4 | 84.35 | 85.62 | 85.61 | 85.18 | 69.04 | 85.62
ViT-B16 |
L2 Axis=1 | 86.94 | 87.07 | 87.07 | 86.85 | 86.25 | 75.96 | 87.07
L2 Axis=0 | 86.28 | 86.23 | 87.21 | 86.97 | 86.27 | 77.26 | 87.21
L1 Axis=1 | 86.93 | 87.06 | 87.07 | 86.84 | 86.25 | 75.72 | 87.07
L1 Axis=0 | 86.27 | 86.2 | 87.23 | 86.99 | 86.27 | 77.06 | 87.23
ROBUST | 86.43 | 86.46 | 87.83 | 87.63 | 87.06 | 76.61 | 87.83
ViT-B32 |
L2 Axis=1 | 85.53 | 85.29 | 85.29 | 85.56 | 85.24 | 64.04 | 85.28
L2 Axis=0 | 85.34 | 85.29 | 85.95 | 85.83 | 85.23 | 75.96 | 85.94
L1 Axis=1 | 85.5 | 85.19 | 85.29 | 85.56 | 85.23 | 63.55 | 85.28
L1 Axis=0 | 85.34 | 85.27 | 85.96 | 85.83 | 85.23 | 75.77 | 85.96
ROBUST | 85.29 | 85.24 | 86.32 | 86.35 | 85.9 | 75.51 | 86.32
TABLE IV: Experimental results on Oxford5k database  | Oxford5k
---|---
Model | Manh. | Eucl. | Cos | BC | Canb. | Cheb. | Correl.
ViT-L16 |
L2 Axis=1 | 60.84 | 60.82 | 60.82 | 60.78 | 59.87 | 51.49 | 60.83
L2 Axis=0 | 60.9 | 61.20 | 61.50 | 61.15 | 59.85 | 53.26 | 61.50
L1 Axis=1 | 60.73 | 60.74 | 60.82 | 60.78 | 59.85 | 51.21 | 60.83
L1 Axis=0 | 60.91 | 61.13 | 61.58 | 61.2 | 59.85 | 53.12 | 61.58
ROBUST | 60.82 | 60.86 | 62.37 | 62.09 | 61.49 | 52.45 | 62.37
ViT-L32 |
L2 Axis=1 | 55.17 | 55.3 | 55.30 | 55.01 | 53.84 | 41.26 | 55.28
L2 Axis=0 | 54.92 | 55.14 | 55.55 | 55.21 | 53.84 | 41.45 | 55.55
L1 Axis=1 | 55.35 | 55.43 | 55.3 | 55.01 | 53.86 | 41.39 | 55.28
L1 Axis=0 | 54.93 | 55.18 | 55.56 | 55.23 | 53.84 | 41.31 | 55.56
ROBUST | 54.93 | 55.19 | 56.30 | 55.88 | 55.16 | 41.78 | 56.30
ViT-B16 |
L2 Axis=1 | 63.24 | 63.59 | 63.59 | 63.22 | 62.17 | 52.44 | 63.59
L2 Axis=0 | 62.74 | 63.16 | 64.19 | 63.53 | 62.12 | 56.79 | 64.19
L1 Axis=1 | 63.13 | 63.68 | 63.59 | 63.26 | 62.13 | 52.3 | 63.59
L1 Axis=0 | 62.79 | 63.14 | 64.22 | 63.55 | 62.12 | 56.48 | 64.22
ROBUST | 62.72 | 63.05 | 64.68 | 64.09 | 62.84 | 56.02 | 64.68
ViT-B32 |
L2 Axis=1 | 63.18 | 62.49 | 62.49 | 63.08 | 62.84 | 41.78 | 62.49
L2 Axis=0 | 63.34 | 63.01 | 63.56 | 63.52 | 62.82 | 53.96 | 63.56
L1 Axis=1 | 63.19 | 62.41 | 62.49 | 63.1 | 62.8 | 41.33 | 62.49
L1 Axis=0 | 63.35 | 62.99 | 63.59 | 63.54 | 62.82 | 53.78 | 63.59
ROBUST | 63.35 | 63.09 | 64.43 | 64.28 | 63.35 | 53.99 | 64.43

###  III-B Comparison with the State-of-the-art

This subsection compares the ViT descriptor’s performance with the state-of-
the-art local, global, and deep convolutional-based approaches from the recent
literature. The proposed descriptor was compared to 36 descriptors from the
literature. The choice of the descriptors used for experimentation was
primarily based on their reported performance and overall popularity. The
descriptors are listed in Table V and sorted based on their retrieval
performance on the INRIA dataset. The reason for choosing this sorting method
is related to the fact that most of the descriptors/approaches reported in
this subsection have been evaluated on the INRIA dataset and their retrieval
results are publicly available or reproducible.

TABLE V: Retrieval score per image collection. This paper uses mAP (Mean Average Precision) to evaluate retrieval accuracy on the INRIA (Holidays), Paris6k, and Oxford5k datasets. The N-S score is specifically used on the UKBench dataset. (†) refers to results that are reported in [23], [24] Retrieval Method | INRIA | UKBen. | Paris | Oxford
---|---|---|---|---
MMF-HC [25] | 94.1 | 3.87 | - | -
MR Spatial search [26] | 89.6 | - | 87.9 | 84.3
GatedSQU [23] | 88.8 | 3.74 | 81.3 | 69.4
ViT-16B | 88.0 | 3.76 | 87.8 | 64.7
ASMK+SA (large) [27] | 88.0 | - | 78.7 | 82.0
ASMK+MA (large) [27] | 86.5 | - | 80.5 | 83.8
Zheng Et al. (PPS) [28] | 85.2 | 3.79 | - | -
R-mac [29] | 85.2(†) | 3.74(†) | 83.0 | 66.9
R-mac-RPN [24] | 86.7 | - | 87.1 | 83.1
CroW [30] | 85.1 | 3.63(†) | 79.7 | 70.8
MMF-SIFT [25] | 84.4 | 3.94 | - | -
CNNaug-ss [31] | 84.3 | - | 46.0 | 36.0
SMK+SA (large) [27] | 84.0 | - | 73.2 | 78.5
HE+WGC [32] | 81.3 | 3.42 | 61.5 | 51.6
GoogleNet [31] | 83.6 | - | 58.3 | 55.8
ReDSL.FC1 [33] | - | - | 94.7 | 78.3
R101-DELG [34] | - | - | 82.9 | 78.5
Varga et al. [35] | - | - | - | 74.4
SMK+MA (large) [27] | 82.9 | - | 74.2 | 79.3
CVLAD [36] | 82.7 | 3.62 | - | 51.4
OxfordNet [31] | 81.6 | - | 59.0 | 59.3
Local CoMo [37] | 81.1 | - | - | -
CNN+BoVW [38] | 80.2 | - | - | -
MOP-CNN [39] | 80.2 | - | - | -
Patch-CKN [40] | 79.3 | 3.76 | - | 56.5
Neural codes [41] | 79.3 | 3.29 | - | 54.5
Alexet-conv3 [42] | 79.3 | 3.76 | - | 43.4
LBOW [15] | 78.9 | 3.50 | - | -
Alexet-conv4 [42] | 77.1 | 3.73 | - | 34.3
SaCoCo [43] | 76.1 | 3.33 | - | -
Alexet-conv5 [42] | 75.3 | 3.69 | - | 47.7
PhilippNet [44] | 74.1 | 3.66 | - | -
CEDD [45] | 72.6 | 3.06 | - | -
CoMo [37] | 72.6 | - | - | -
Alexet-conv2 [42] | 62.7 | 3.19 | - | 12.5
Alexet-conv1 [42] | 59.0 | 3.33 | - | 18.8
BoVW [15] | 57.2 | 2.95 | - | -

Experimental findings show that the ViT descriptor performs robustly and
provides remarkable results in all benchmarking datasets. The results in Table
V demonstrate that the presented approach performs as good as the state-of-
the-art approaches on all datasets. Moreover, the acquired retrieval results
are among the leading published ones and are directly comparable with the
literature’s best reported.

In the INRIA and UKBench datasets, the shaped solution presents the fourth-
best reported result. In the case of the Paris6K dataset, the ViT descriptor
reports the third-best performance. Finally, in the Oxford5K dataset, the ViT
descriptor’s effectiveness is comparable with the CNN-based image retrieval
approaches. It is noteworthy that the full image is used as a query in both
Oxford5K and Paris6K datasets instead of the annotated region of interest.

In a nutshell, the ViT descriptor significantly outperforms all the
handcrafted features and most machine-crafted, CNN-based ones in all the
datasets. Of course, the literature contains more sophisticated methods and
algorithms that exceed the shaped descriptor’s retrieval accuracy. But once
again, it is worth noting that the ViT descriptor is ready to use, plug-and-
play descriptor, and no training or parameter fitting is needed. The vast
majority of the listed deep-learning-based approaches are considerably more
complicated and demanding than the ViT solution.

For example, in [25], a multi-index fusion scheme for image retrieval was
proposed based on AlexNet and ResNet50 networks. The method is called MMF and
inherits the core idea of Collaborative Index Embedding to fuse different
visual representations on an index level. Furthermore, MMF explores the high-
order information assumed by considering the sparse index structure for
retrieval, an inverted index structure’s intrinsic property. The impressive
performance reported by [25] requires the creation and learning of an index-
specific functional matrix to propagate similarities and the application of an
Augmented Lagrange Multiplier (ALM) to optimize multi-index fusion. By taking
into account various factors such as geometric invariance, layers, and search
efficiency, an optimized CNN architecture was created and trained. This
architecture is known as MR Spatial search[26]. It is important to note that
the MR Spatial search performs a costly spatial verification at test-time. The
authors in [23] focuses on global feature pooling over CNN activations for
image instance retrieval. The descriptor is called GatedSQU, and it contains
the design of a channel-wise pooling and learning with triplet Loss. The
authors in [35] combined a semantic-level similarity and a feature-level
similarity to efficiently calculate the similarity distance. The authors
perform a comprehensive series of analytical studies in [33], aimed at forming
more efficient representations of features. Finally, it is worth noting that
ReDSL.FC1 method significantly outperforms all reported methods on Paris and
Oxford datasets as the authors fine-tuned the pre-trained CNN architecture
directly on these landmarks using similarity learning.

The experiments were conducted on a workstation with AMD Ryzen 7 1700, 32GB
RAM, and GTX 1080 GPU using the KERAS framework [46], a well-known high-level
Python neural network library that runs on top of TensorFlow or Theano. Vision
Transformer’s implementation is based on the open-source release of the
network available on GitHub111https://github.com/faustomorales/vit-keras.

##  IV Conclusions

This paper presents a fully unsupervised, parameter-free image retrieval
descriptor. Overall, the shaped solution appears to be suitable for content-
based image retrieval tasks. The experimental evaluation findings on several
different datasets clearly shown that the proposed approach managed to
outperform many state-of-the-art, more sophisticated, and complex solutions
from the literature. Although the ViT descriptor does not supersede all the
other methods, it is safe to conclude that it brings a new approach to the
image retrieval domain. A critical direction to improve image retrieval
performance is to revisit the well-established CNN-based methods in the
literature and replace the backbone pre-trained network with the vision
transformers. A new chapter in the image retrieval scientific subject is
looming.

## References

  * [1] L. Zheng, Y. Yang, and Q. Tian, “Sift meets cnn: A decade survey of instance retrieval,” _IEEE transactions on pattern analysis and machine intelligence_ , vol. 40, no. 5, pp. 1224–1244, 2018.
  * [2] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,” _Advances in neural information processing systems_ , vol. 30, pp. 5998–6008, 2017.
  * [3] X. Wang, R. Girshick, A. Gupta, and K. He, “Non-local neural networks,” in _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 2018, pp. 7794–7803.
  * [4] K. Han, Y. Wang, H. Chen, X. Chen, J. Guo, Z. Liu, Y. Tang, A. Xiao, C. Xu, Y. Xu _et al._ , “A survey on visual transformer,” _arXiv preprint arXiv:2012.12556_ , 2020.
  * [5] F. Locatello, D. Weissenborn, T. Unterthiner, A. Mahendran, G. Heigold, J. Uszkoreit, A. Dosovitskiy, and T. Kipf, “Object-centric learning with slot attention,” _arXiv preprint arXiv:2006.15055_ , 2020.
  * [6] H. Hu, J. Gu, Z. Zhang, J. Dai, and Y. Wei, “Relation networks for object detection,” in _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 2018, pp. 3588–3597.
  * [7] M. Chen, A. Radford, R. Child, J. Wu, H. Jun, D. Luan, and I. Sutskever, “Generative pretraining from pixels,” in _International Conference on Machine Learning_. PMLR, 2020, pp. 1691–1703.
  * [8] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly _et al._ , “An image is worth 16x16 words: Transformers for image recognition at scale,” _arXiv preprint arXiv:2010.11929_ , 2020.
  * [9] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [10] T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, S. Agarwal, A. Herbert-Voss, G. Krueger, T. Henighan, R. Child, A. Ramesh, D. M. Ziegler, J. Wu, C. Winter, C. Hesse, M. Chen, E. Sigler, M. Litwin, S. Gray, B. Chess, J. Clark, C. Berner, S. McCandlish, A. Radford, I. Sutskever, and D. Amodei, “Language models are few-shot learners,” 2020.
  * [11] Q. Wang, B. Li, T. Xiao, J. Zhu, C. Li, D. Wong, and L. Chao, “Learning deep transformer models for machine translation,” 01 2019, pp. 1810–1822.
  * [12] J. L. Ba, J. R. Kiros, and G. E. Hinton, “Layer normalization,” _arXiv preprint arXiv:1607.06450_ , 2016.
  * [13] J. Deng, W. Dong, R. Socher, L. Li, Kai Li, and Li Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in _2009 IEEE Conference on Computer Vision and Pattern Recognition_ , 2009, pp. 248–255.
  * [14] H. Jegou, M. Douze, and C. Schmid, “Hamming embedding and weak geometric consistency for large scale image search,” in _Computer Vision – ECCV 2008_ , D. Forsyth, P. Torr, and A. Zisserman, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008, pp. 304–317.
  * [15] C. Wengert, M. Douze, and H. Jégou, “Bag-of-colors for improved image search,” in _Proceedings of the 19th ACM International Conference on Multimedia_ , ser. MM ’11. New York, NY, USA: ACM, 2011, pp. 1437–1440. [Online]. Available: http://doi.acm.org/10.1145/2072298.2072034
  * [16] J. Philbin, O. Chum, M. Isard, J. Sivic, and A. Zisserman, “Lost in quantization: Improving particular object retrieval in large scale image databases,” in _Computer Vision and Pattern Recognition, 2008. CVPR 2008\. IEEE Conference on_. IEEE, 2008, pp. 1–8.
  * [17] ——, “Object retrieval with large vocabularies and fast spatial matching,” in _2007 IEEE conference on computer vision and pattern recognition_. IEEE, 2007, pp. 1–8.
  * [18] V. R. Chandrasekhar, D. M. Chen, S. S. Tsai, N.-M. Cheung, H. Chen, G. Takacs, Y. Reznik, R. Vedantham, R. Grzeszczuk, J. Bach _et al._ , “The stanford mobile visual search data set,” in _Proceedings of the second annual ACM conference on Multimedia systems_ , 2011, pp. 117–122.
  * [19] S. Chatzichristofis, C. Iakovidou, Y. Boutalis, and E. Angelopoulou, “Mean normalized retrieval order (mnro): a new content-based image retrieval performance measure,” _Multimedia Tools and Applications_ , pp. 1–32, 2012\. [Online]. Available: http://dx.doi.org/10.1007/s11042-012-1192-z
  * [20] S. Patil and S. Talbar, “Content based image retrieval using various distance metrics,” in _Data Engineering and Management_ , R. Kannan and F. Andres, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 154–161.
  * [21] T. Salimans and D. P. Kingma, “Weight normalization: A simple reparameterization to accelerate training of deep neural networks,” in _Advances in neural information processing systems_ , 2016, pp. 901–909.
  * [22] S. Wu, G. Li, L. Deng, L. Liu, D. Wu, Y. Xie, and L. Shi, “$l1$-norm batch normalization for efficient training of deep neural networks,” _IEEE transactions on neural networks and learning systems_ , vol. 30, no. 7, pp. 2043–2051, 2018.
  * [23] Z. Chen, J. Lin, V. Chandrasekhar, and L.-Y. Duan, “Gated square-root pooling for image instance retrieval,” in _2018 25th IEEE International Conference on Image Processing (ICIP)_. IEEE, 2018, pp. 1982–1986.
  * [24] A. Gordo, J. Almazán, J. Revaud, and D. Larlus, “Deep image retrieval: Learning global representations for image search,” in _European conference on computer vision_. Springer, 2016, pp. 241–257.
  * [25] Z. Zhang, Y. Xie, W. Zhang, and Q. Tian, “Effective image retrieval via multilinear multi-index fusion,” _IEEE Transactions on Multimedia_ , 2019\.
  * [26] A. S. Razavian, J. Sullivan, S. Carlsson, and A. Maki, “Visual instance retrieval with deep convolutional networks,” _ITE Transactions on Media Technology and Applications_ , vol. 4, no. 3, pp. 251–258, 2016.
  * [27] G. Tolias, Y. S. Avrithis, and H. Jégou, “Image search with selective match kernels: Aggregation across single and multiple images,” _International Journal of Computer Vision_ , vol. 116, no. 3, pp. 247–261, 2016.
  * [28] L. Zheng, S. Wang, and Q. Tian, “Coupled binary embedding for large-scale image retrieval,” _Image Processing, IEEE Transactions on_ , vol. 23, no. 8, pp. 3368–3380, 2014.
  * [29] G. Tolias, R. Sicre, and H. Jégou, “Particular object retrieval with integral max-pooling of cnn activations,” _arXiv preprint arXiv:1511.05879_ , 2015.
  * [30] Y. Kalantidis, C. Mellina, and S. Osindero, “Cross-dimensional weighting for aggregated deep convolutional features,” in _European conference on computer vision_. Springer, 2016, pp. 685–701.
  * [31] J. Yue-Hei Ng, F. Yang, and L. S. Davis, “Exploiting local features from deep networks for image retrieval,” in _Proceedings of the IEEE conference on computer vision and pattern recognition workshops_ , 2015, pp. 53–61.
  * [32] H. Jégou, M. Douze, and C. Schmid, “Improving bag-of-features for large scale image search,” vol. 87, no. 3, 2010, pp. 316–336. [Online]. Available: https://doi.org/10.1007/s11263-009-0285-2
  * [33] J. Wan, D. Wang, S. C. H. Hoi, P. Wu, J. Zhu, Y. Zhang, and J. Li, “Deep learning for content-based image retrieval: A comprehensive study,” in _Proceedings of the 22nd ACM international conference on Multimedia_ , 2014, pp. 157–166.
  * [34] B. Cao, A. Araujo, and J. Sim, “Unifying deep local and global features for image search,” in _European Conference on Computer Vision_. Springer, 2020, pp. 726–743.
  * [35] D. Varga and T. Szirányi, “Fast content-based image retrieval using convolutional neural network and hash function,” in _2016 IEEE international conference on systems, man, and cybernetics (SMC)_. IEEE, 2016, pp. 002 636–002 640.
  * [36] W.-L. Zhao, H. Jégou, and G. Gravier, “Oriented pooling for dense and non-dense rotation-invariant features,” 2013.
  * [37] S. A. Vassou, N. Anagnostopoulos, K. Christodoulou, A. Amanatiadis, and S. A. Chatzichristofis, “Como: a scale and rotation invariant compact composite moment-based descriptor for image retrieval,” _Multimedia Tools and Applications_ , pp. 1–24, 2018.
  * [38] A. Sharif Razavian, H. Azizpour, J. Sullivan, and S. Carlsson, “Cnn features off-the-shelf: an astounding baseline for recognition,” in _Proceedings of the IEEE conference on computer vision and pattern recognition workshops_ , 2014, pp. 806–813.
  * [39] Y. Gong, L. Wang, R. Guo, and S. Lazebnik, “Multi-scale orderless pooling of deep convolutional activation features,” in _Computer Vision - ECCV 2014 - 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part VII_ , 2014, pp. 392–407.
  * [40] M. Paulin, M. Douze, Z. Harchaoui, J. Mairal, F. Perronnin, and C. Schmid, “Local convolutional features with unsupervised training for image retrieval,” in _2015 IEEE International Conference on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015_ , 2015, pp. 91–99.
  * [41] A. Babenko, A. Slesarev, A. Chigorin, and V. Lempitsky, “Neural codes for image retrieval,” in _European conference on computer vision_. Springer, 2014, pp. 584–599.
  * [42] M. Paulin, M. Douze, Z. Harchaoui, J. Mairal, F. Perronin, and C. Schmid, “Local convolutional features with unsupervised training for image retrieval,” in _Proceedings of the IEEE international conference on computer vision_ , 2015, pp. 91–99.
  * [43] C. Iakovidou, N. Anagnostopoulos, M. Lux, K. Christodoulou, Y. S. Boutalis, and S. A. Chatzichristofis, “Composite description based on salient contours and color information for CBIR tasks,” _IEEE Trans. Image Processing_ , vol. 28, no. 6, pp. 3115–3129, 2019. [Online]. Available: https://doi.org/10.1109/TIP.2019.2894281
  * [44] P. Fischer, A. Dosovitskiy, and T. Brox, “Descriptor matching with convolutional neural networks: a comparison to sift,” _arXiv preprint arXiv:1405.5769_ , 2014.
  * [45] S. A. Chatzichristofis and Y. S. Boutalis, “Cedd: Color and edge directivity descriptor: A compact descriptor for image indexing and retrieval,” in _ICVS_ , 2008, pp. 312–322.
  * [46] A. Gulli and S. Pal, _Deep learning with Keras_. Packt Publishing Ltd, 2017.


# Quantum entanglement transfer assisted via Duffing nonlinearity

D. R. Kenigoule Massembele<EMAIL_ADDRESS>Department of Physics,
Faculty of Science, University of Ngaoundere, P.O. Box 454, Ngaoundere,
Cameroon  P. Djorwé<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon
Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre
at Stellenbosch University, Stellenbosch 7600, South Africa  Amarendra K.
Sarma<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Guwahati, Guwahati 781039, India  S. G. Nana Engo  serge.nana-
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, University
of Yaounde I, P.O. Box 812, Yaounde, Cameroon

###### Abstract

We propose a scheme to enhance quantum entanglement in optomechanical system
that is based on Duffing nonlinearity. Our benchmark system consists of an
electromagnetic field that is driving two mechanically coupled mechanical
resonators. One of the mechanical resonators support a Duffing nonlinear term,
while the other is free of it. The phonon hopping rate is $\theta$-phase-
dependent that induces a synthetic magnetism, which triggers Exceptional
Points (EPs) singularities in the system. Without the Duffing nonlinear term,
the entanglement between the electromagnetic field and the mechanical
resonators is generated. This entanglement features the sudden death and
revival phenomenon, where the peaks happen at the multiple of
$\theta=\frac{\pi}{2}$. As the Duffing nonlinearity is accounted, the
bipartite entanglement involving the nonlinear resonator vanishes. However,
there is an entanglement transfer from the resonator supporting the nonlinear
term towards the one that is mechanically coupled to it. This nonlinearly
induced entanglement is robust again thermal fluctuation, and more stable
compared to what is generated without the nonlinear term. This work paves a
way to a generation of quantum entanglement using nonlinear resources,
enabling quantum technology such as quantum information processing, quantum
sensing, and quantum computing in complex systems.

Optomechanics, entanglement, exceptional point, Duffing nonlinearity,
synthetic magnetism

###### pacs:

42.50.Wk, 42.50.Lc, 05.45.Xt, 05.45.Gg

##  I Introduction

Optomechanics is a research field that deals with the interactions between
electromagnetic field and mechanical motion. Recently, this field became an
interesting platform to investigate plethora of physical phenomena both in
classical Pokharel et al. (2022); Golokolenov et al. (2021); Foulla et al.
(2017); Walter and Marquardt (2016); Djorwé et al. (2013) and quantum regime
Lemonde et al. (2016); Barzanjeh et al. (2021); Aspelmeyer et al. (2010);
Djorwé et al. (2012). In the classical domain, numerous phenomena including
synchronization Colombano et al. (2019); Rodrigues et al. (2021); Djorwé et
al. (2020); Li et al. (2022), nonlinear dynamics Djorwe et al. (2019);
Navarro-Urrios et al. (2017); Roque et al. (2020); Djorwe et al. (2022), and
chaos-like behaviors Zhu et al. (2023); Stella et al. (2023) have been
reported. In the quantum regime, interesting properties of quantum
correlations Purdy et al. (2017); Bemani et al. (2019); Riedinger et al.
(2016) and nonclassical states Zivari et al. (2022); Fiaschi et al. (2021);
Djorwé et al. (2013) were revealed. Such quantum effects constitute a
hallmarks of quantum physics, particularly the squeezed Banerjee et al. (2023)
and entangled states Riedinger et al. (2018); Kotler et al. (2021); Mercier de
Lépinay et al. (2021). Indeed, quantum entanglement is a prerequisite for
quantum technology, such as quantum information processing Meher and Sivakumar
(2022); Slussarenko and Pryde (2019), quantum computing Arute et al. (2019),
and sensing Xia et al. (2023).

Owing to the important role played by entanglement in number of quantum
technology, there is an urgent quest to come up with new mechanisms to
generate entangled states which are robust enough again decoherence and
thermal fluctuation. For this purpose, some nonlinearities such as cross-Kerr
effect and parametric amplifier have been used to enhance quantum entanglement
in optomechanics for instance. Exceptional Points (EPs), which are Non-
Hermitian degeneracies, have been also recently used as a tool to engineer
stable and robust entanglement Chakraborty and Sarma (2019). Despite the fact
that EPs improve the generation of entanglement, it is therefore not an easy
task to synthesize them in a given physical system. In fact, the system
parameters most fulfill some requirements such as balanced gain-loss for
instance, which is usually not a trivial condition to meet. Recently, this
requirement has been relaxed through the synthetic magnetism, which allows the
engineering of EPs in a lossy system. This approach has led to a generation of
noise-tolerant optomechanical entanglement via a dark-mode breaking Lai et al.
(2022).

The aim of the current work is to combine the effect of nonlinearity together
with the synthetic magnetism effect to generate robust quantum entanglement.
Our benchmark system consists of an electromagnetic field that is driving two
mechanical resonators, which are mechanically coupled through a phase-
dependent phonon-hopping. This phase-dependency is reminiscent of the
synthetic magnetism, and it triggers EPs singularities in the system. To
account for nonlinearity in our proposal, we assume that one of the mechanical
resonators supports a Duffing nonlinear term Venkatachalam and Zhou (2023). In
our investigation, we pointed out that bipartite entanglement vanishes
whenever the resonator supporting this nonlinear is involved. However, there
is an entanglement transfer assisted by this Duffing nonlinearity towards the
other connected resonator. Moreover, the nonlinear term enhances stability of
the system, enabling further entanglement for strong driving strength.
Furthermore, the nonlinearly induced entanglement is robust again thermal
fluctuation compared to what is generated without the nonlinear term. This
work shows how the Duffing nonlinearty can be used to enhance quantum
entanglement through a transfer phenomenon. Our proposal sheds light on the
exploration of nonlinear effects to generate entanglement, and it constitutes
a step towards information processing and quantum computing in complex
systems.

The rest of the work is organized as follows. In section II, we describe the
model, the related dynamical equations, and presents the linear stability of
the system. The entanglement and the effect of the Duffing nonlinearity are
carried out in Section III, while the work is concluded in Section IV.

##  II MODEL AND DYNAMICS

Our benchmark system is sketched in Fig.1, where an electromagnetic field
(mode $a$) is driving two mechanical resonators (modes $b_{j}$), which are
mechanically coupled through a phonon hopping $J_{m}$. The phonon hopping term
exhibits a synthetic magnetism through the phase modulation $\theta$, and the
first resonator supports a Duffing nonlinearity caputered by its amplitude
$\eta$.

Figure 1: Sketch of our benchmark optomechanical system. An electromagnetic
field (mode $a$) is driving two mechanically coupled mechanical resonators
($b_{j}$). The phonon hopping ($J_{m}$) between the two mechanical resonators
is modulated through the phase $\theta$ that induces a synthetic magnetism.
The optomechanical coupling between the electromagnetic field and each
mechanical resonator is denoted by $g$.

In the frame rotating at the driving frequency $\omega_{p}$, our system is
described by the following Hamiltonian ($\hbar=1$):

$\displaystyle H$ $\displaystyle=-\Delta
a^{\dagger}a+\sum_{j=1,2}\omega_{j}b_{j}^{\dagger}b_{j}-ga^{\dagger}a(b_{j}+b_{j}^{\dagger})$
$\displaystyle+{J_{m}}({e^{i\theta}}{b_{1}^{\dagger}}{b_{2}}+{e^{-i\theta}}{b_{1}}{b_{2}^{\dagger}})+iE^{in}(a^{\dagger}+a)$
$\displaystyle+\frac{\eta}{2}(b_{1}+b_{1}^{\dagger})^{4},$ (1)

where $a$($a^{\dagger}$) and $b_{j}$($b_{j}^{\dagger}$) are respectively the
annihilation (creation) bosonic operators of the cavity field and for the
$j^{th}$ mechanical resonator (having frequency $\omega_{j}$). The detuning is
defined as $\Delta=\omega_{p}-\omega_{c}$, where $\omega_{c}$ is the cavity
frequency. The other parameters are the optomechanical coupling $g$, for each
resonator, and the amplitude of the driving field $E^{in}$. Using the
Heisenberg equation, the Quantum Langevin Equations (QLEs) from the
Hamiltonian Eq.II yield,

$\displaystyle\left\\{\begin{array}[]{ll}\dot{a}&=\left(\begin{array}[]{lll}i\left(\begin{array}[]{rcr}\Delta+\sum_{j=1,2}g(b_{j}^{\dagger}+b_{j})\end{array}\right)-{\frac{\kappa}{2}}\end{array}\right)a\\\
&+\sqrt{\kappa}\alpha^{in}+\sqrt{\kappa}a^{in},\\\
\dot{b_{1}}&=-({\frac{\gamma_{1}}{2}}+i{\omega_{1}})b_{1}-i{J_{m}}e^{i\theta}b_{2}+i{g}a^{\dagger}{a}\\\
&-2i\eta(b_{1}+b_{1}^{\dagger})^{3}+\sqrt{\gamma_{1}}b_{1}^{in},\\\
\dot{b_{2}}&=-({\frac{\gamma_{2}}{2}}+i{\omega_{2}})b_{2}-i{J_{m}}e^{-i\theta}b_{1}+i{g}a^{\dagger}{a}\\\
&+\sqrt{\gamma_{2}}b_{2}^{in},\end{array}\right.$ (10)

where $E^{in}$ has been substituted by $\sqrt{\kappa}\alpha^{in}$ with
$\alpha^{in}$ being related to the input power $P^{in}$ as
$\alpha^{in}=\sqrt{\frac{P^{in}}{\hbar\omega_{p}}}$. The cavity and mechanical
dissipations are captured by $\kappa$ and $\gamma_{j}$, respectively.
Moreover, $a^{in}$ and $b_{j}^{in}$ are the zero mean valued noise operators
characterized by their autocorrelations functions,

$\displaystyle\langle a^{in}(t)a^{in\dagger}(t^{\prime})\rangle$
$\displaystyle=\delta(t-t^{\prime}),$ (11) $\displaystyle\langle
a^{in\dagger}(t)a^{in}(t^{\prime})\rangle$ $\displaystyle=0,$ (12)
$\displaystyle\langle b_{j}^{in}(t)b_{j}^{in\dagger}(t^{\prime})\rangle$
$\displaystyle=(n_{th}^{j}+1)\delta(t-t^{\prime}),$ (13) $\displaystyle\langle
b_{j}^{in\dagger}(t)b_{j}^{in}(t^{\prime})\rangle$
$\displaystyle=n_{th}^{j}\delta(t-t^{\prime}),$ (14)

with
$n_{th}^{j}=\left[exp\left(\frac{\hbar\omega_{j}}{k_{b}T}\right)-1\right]^{-1}$
represents the number of phonons at temperature $T$ and $k_{B}$ the Boltzmann
constant. To derive the quadratures of our system, the nonlinear set of
equations given in Eq.10 can be linearized through the standard linearization
process. Indeed operators are splitted over their mean values and
fluctuations, $\mathcal{O}=\langle\mathcal{O}\rangle+\delta\mathcal{O}$, where
$\mathcal{O}\equiv$ ($a$,$b_{j}$). By setting $\alpha=\langle a\rangle$, and
$\beta_{j}=\langle b_{j}\rangle$), the dynamical mean values of our system
yields,

$\displaystyle\left\\{\begin{array}[]{ll}\dot{\alpha}&=\left(i\tilde{\Delta}-\frac{\kappa}{2}\right)\alpha+\sqrt{\kappa}\alpha^{in},\\\
\dot{\beta}_{1}&=-({\frac{\gamma_{1}}{2}}+i{\omega_{1}}){\beta}_{1}-i{J_{m}}e^{i\theta}{\beta}_{2}+i{g}|\alpha|^{2}\\\
&-2i\eta(\beta_{1}+\beta_{1}^{*})^{3},\\\
\dot{\beta_{2}}&=-({\frac{\gamma_{2}}{2}}+i{\omega_{2}}){\beta}_{2}-i{J_{m}}e^{-i\theta}{\beta}_{1}+i{g}|\alpha|^{2},\end{array}\right.$
(19)

and the fluctuation dynamics reads,

$\displaystyle\left\\{\begin{array}[]{ll}\delta\dot{a}&=\left(i\tilde{\Delta}-\frac{\kappa}{2}\right)\delta
a+i\sum_{j=1,2}G({\delta}b_{j}^{\dagger}+{\delta}b_{j})\\\
&+\sqrt{\kappa}a^{in},\\\
\delta\dot{b_{1}}&=-({\frac{\gamma_{1}}{2}}+i{\omega_{1}})\delta{b}_{1}-i{J_{m}}e^{i\theta}\delta{b}_{2}+i(G^{*}\delta
a\\\ &+G\delta a^{\dagger})-i\Lambda(\delta b_{1}+\delta
b_{1}^{\dagger})+\sqrt{\gamma_{1}}b_{1}^{in},\\\
\delta\dot{b_{2}}&=-({\frac{\gamma_{2}}{2}}+i{\omega_{2}})\delta{b}_{2}-i{J_{m}}e^{-i\theta}\delta{b}_{1}+i(G^{*}\delta
a\\\ &+G\delta a^{\dagger})+\sqrt{\gamma_{2}}b_{2}^{in},\end{array}\right.$
(26)

where both effective detuning and optomechanical coupling are defined as
$\tilde{\Delta}=\Delta+2\sum_{j=1,2}g\rm Re(\beta_{j})$ and $G=g\alpha$. From
now on, we will assume that $\alpha$ is real, meaning that $G$ is real as
well. Moreover, $\Lambda=24\eta(\Re(\beta_{1}))^{2}$ captures the nonlinear
process. The fluctuation dynamics in Eq.26 can be put in its compact form,

$\dot{x}=\rm{A}x+y,$ (27)

where ${x}=(\delta a,\delta a^{\dagger},\delta b_{1},\delta
b^{\dagger}_{1},\delta b_{2},\delta b^{\dagger}_{2})^{T}$, and the noise
vector
$y=(\sqrt{\kappa}a^{in},\sqrt{\kappa}a^{in\dagger},\sqrt{\gamma_{1}}b_{1}^{in},\sqrt{\gamma_{1}}b^{in\dagger}_{1},\sqrt{\gamma_{2}}b_{2}^{in},\sqrt{\gamma_{2}}b^{in\dagger}_{2})^{T}$,
and the matrix $\rm{A}$ is given by:

$\rm{A}=\begin{pmatrix}A_{1}&A_{12}&A_{13}\\\ A_{12}^{T}&A_{2}&A_{23}\\\
A_{13}^{T}&A_{23}^{T}&A_{3}\end{pmatrix},$ (28)

with

$A_{1}=\begin{pmatrix}i\tilde{\Delta}-\frac{\kappa}{2}&0\\\
0&-i\tilde{\Delta}-\frac{\kappa}{2}\end{pmatrix},$ (29)

$A_{2}=\begin{pmatrix}-(\frac{\gamma_{1}}{2}+i(\omega_{1}+\Lambda))&-i\Lambda\\\
i\Lambda&-(\frac{\gamma_{1}}{2}-i(\omega_{1}+\Lambda))\end{pmatrix},$ (30)

$A_{3}=\begin{pmatrix}-(\frac{\gamma_{2}}{2}+i\omega_{2})&0\\\
0&-(\frac{\gamma_{2}}{2}-i\omega_{2})\end{pmatrix}$ (31)

$A_{12}=\begin{pmatrix}iG&iG\\\ -iG^{*}&-iG^{*}\end{pmatrix},$ (32)

$A_{13}=\begin{pmatrix}iG&iG\\\ -iG^{*}&-iG^{*}\end{pmatrix},$ (33)

and

$A_{23}=\begin{pmatrix}-iJ_{m}e^{i\theta}&0\\\
0&iJ_{m}e^{-i\theta}\end{pmatrix}.$ (34)

The steady-state equation of our system is obtain by assuming that the mean
values in Eq.(19) are not time dependent ($\dot{\alpha}=\dot{\beta_{j}}=0$).
Following the Routh-Hurwitz criterion, and by using the steady-state values in
matrix $\rm{A}$, we get the stability diagrams shown in Fig.2.

Figure 2: Stability diagrams. (a) Driving strength $\alpha^{in}$ vs the non-
linear amplitude $\eta$ for $\kappa=2\times 10^{-1}$ and $J_{m}=1\times
10^{-2}$. (b) Driving strength $\alpha^{in}$ vs the cavity decay rate $\kappa$
for $\eta=1\times 10^{-5}$ and $J_{m}=1\times 10^{-2}$. The dark color is
stable, while the light color is unstable. The other parameters are
$\omega_{1}=\omega_{2}=\omega_{m}$, $\gamma_{1}=\gamma_{2}=1\times
10^{-5}\omega_{m}$, $\Delta=-\omega_{m}$, $g=5\times 10^{-4}\omega_{m}$ and
$\theta=\frac{\pi}{2}$.

Fig.2(a) shows the stability diagram in ($\eta,\alpha^{in}$) space, and it can
be seen that for a wide range of the Duffing nonlinearity, the system is
stable for a driving strength up to $\alpha^{in}\sim 1500\omega_{m}^{1/2}$. It
can be also seen that as the nonlinearity increases, the stability of the
system is slightly improved. For $\eta=10^{-5}\omega_{m}$, Fig.2(b) exhibits
diagram stability of the system in ($\kappa,\alpha^{in}$) space. Similarly as
in Fig.2(a), the system is stable for a wide range of $\kappa$. These two
diagrams are plotted for a synthetic phase of $\theta\sim\pi/2$, and that
corresponds to the optimal value of $\theta$ around which the logarithmic
negativity is strong enough as it will be seen later on.

##  III stationary entanglement transfer

To investigate quantum entanglement, we need to define the quadratures and the
system must fulfill stability conditions depicted in Fig.2. The quantum
quadratures are defined as,
$\delta{{x}}=\frac{\delta{a}^{\dagger}+\delta{a}}{\sqrt{2}}$,
$\delta{{y}}=i\frac{\delta{a}^{\dagger}-\delta{a}}{\sqrt{2}}$,
$\delta{q}_{j}=\frac{\delta{b}_{j}^{\dagger}+\delta{b}_{j}}{\sqrt{2}}$,
$\delta{{p}}_{j}=i\frac{\delta{{b}}_{j}^{\dagger}-\delta{b}_{j}}{\sqrt{2}}$,
with their corresponding noise operators
$\delta{{x}}^{in}=\frac{\delta{a}^{in\dagger}+\delta{a}^{in}}{\sqrt{2}}$,
$\delta{{y}}^{in}=i\frac{\delta{a}^{in\dagger}-\delta{a}^{in}}{\sqrt{2}}$,
$\delta{q}_{j}^{in}=\frac{\delta{b}_{j}^{in\dagger}+\delta{b}_{j}^{in}}{\sqrt{2}}$
and
$\delta{{p}}_{j}^{in}=i\frac{\delta{{b}}_{j}^{in\dagger}-\delta{b}_{j}^{in}}{\sqrt{2}}$.
By using Eq.(26), the system quadratures can be written in its compact form
as,

$\dot{u}=\rm{M}u+z,$ (35)

where the quadratures vector is
${u}^{T}=(\delta{x},\delta{y},\delta{q}_{1},\delta{p}_{1},\delta{q}_{2},\delta{p}_{2})$,
with the corresponding noise vector set as
$z^{T}=(\sqrt{\kappa}x^{in},\sqrt{\kappa}y^{in},\sqrt{\gamma_{1}}q_{1}^{in},\sqrt{\gamma_{1}}p_{1}^{in},\sqrt{\gamma_{2}}q_{2}^{in},\sqrt{\gamma_{2}}p_{2}^{in})$,
and the drift matrix is:

$\rm{M}=\begin{pmatrix}M_{1}&M_{12}&M_{13}\\\ M_{12}^{T}&M_{2}&M_{23}\\\
M_{13}^{T}&M_{23}^{T}&M_{3}\end{pmatrix},$ (36)

where the $M_{i}$ and $M_{ij}$ are blocs of $2\times 2$ matrices defined as,

$M_{1}=\begin{pmatrix}-\frac{\kappa}{2}&-\tilde{\Delta}\\\
\tilde{\Delta}&-\frac{\kappa}{2}\end{pmatrix},$ (37)

$M_{2}=\begin{pmatrix}-\frac{\gamma_{1}}{2}&\omega_{1}\\\
-(2\Lambda+\omega_{1})&-\frac{\gamma_{1}}{2}\end{pmatrix},$ (38)

$M_{3}=\begin{pmatrix}-\frac{\gamma_{2}}{2}&\omega_{2}\\\
-\omega_{2}&-\frac{\gamma_{2}}{2}\end{pmatrix},$ (39)

$M_{12}=\begin{pmatrix}-2\rm Im(G)&0\\\ -2\rm Re(G)&0\end{pmatrix},$ (40)

$M_{13}=\begin{pmatrix}-2\rm Im(G)&0\\\ -2\rm Re(G)&0\end{pmatrix},$ (41)

and

$M_{23}=\begin{pmatrix}J_{m}\sin\theta&J_{m}\cos\theta\\\
-J_{m}\cos\theta&J_{m}\sin\theta\end{pmatrix}.$ (42)

Figure 3: (a) Entanglement between the electromagnetic field and the nonlinear
resonator (mode $b_{1}$) versus the synthetic phase $\theta$. (b) Entanglement
between the electromagnetic field and the linear resonator (mode $b_{2}$)
versus the synthetic phase $\theta$. The solid line is without Duffing
nonlinearity ($\eta=0$), while the dash-dotted and dashed lines are for
$\eta=5\times 10^{-6}\omega_{m}$ and $\eta=5\times 10^{-5}\omega_{m}$,
respectively. We have used $\kappa=2\times 10^{-1}\omega_{m}$, $J_{m}=2\times
10^{-1}\omega_{m}$, $\alpha^{in}=1500$ and $n_{th}=100$. The other parameters
are as in Fig.2.

The diagonal blocs matrices $M_{1}$, $M_{2}$ and $M_{3}$ represent the optical
mode, the first and second mechanical mode, respectively. The blocs $M_{12}$
and $M_{13}$ capture the correlations between the driving field with the first
and second mechanical resonators, while $M_{23}$ describes the correlations
between the mechanical modes. Any bipartite entanglement can be evaluated
through the logarithmic negativity by tracing out the non-necessary third
mode. The logarithmic negativity $\rm{E_{N}}$ is defined as,

$\displaystyle E_{N}=\rm max[0,-\ln(2\nu^{-})],$ (43)

where $\nu^{-}=\frac{1}{\sqrt{2}}\sqrt{\sum(V)-\sqrt{\sum(V)^{2}-4\rm detV}}$.
$V$ is the covariance matrix with elements defined as,

$\displaystyle V_{ij}=\frac{\left(\langle
u_{i}u_{j}+u_{j}u_{i}\rangle\right)}{2}.$ (44)

When the stability condition is satisfied, the covariance matrix fulfills the
following Lyapunov equation,

$MV+VM^{T}+D=0,$ (45)

where
$D=Diag[\frac{\kappa}{2},\frac{\kappa}{2},\frac{\gamma_{1}}{2}(2n_{th}+1),\frac{\gamma_{1}}{2}(2n_{th}+1),\frac{\gamma_{2}}{2}(2n_{th}+1),\frac{\gamma_{2}}{2}(2n_{th}+1)]$
is the diffusion matrix.

Fig.3 captures the bipartite entanglement between the driving field and the
mechanical resonators versus the synthetic phase $\theta$. The interaction
with the first mechanical resonator is depicted in Fig.3(a), while
entanglement involving the second mechanical resonator is shown in Fig.3(b).
In these figures, the solid line represents the case without the Duffing
nonlinearity ($\eta=0$), while the dash-dotted and the dashed lines are for
$\eta=5\times 10^{-6}\omega_{m}$ and $\eta=5\times 10^{-5}\omega_{m}$,
respectively. For $\eta=0$, these figures show how the entanglement is
modulated over the synthetic phase. Indeed, the entanglement $\rm{E_{N}}$
drops around $\theta=n\pi$ for $n\in\mathbb{N}$, and it reaches its optimal
value at the vicinity of $\theta=m\pi/2$ where $m$ is and odd number. This
dynamical behavior of the entanglement in Fig.3 is reminiscent of sudden death
and revival of entanglement induced through exceptional points in Chakraborty
and Sarma (2019). We would like to underligned that the synthetic magnetism
phase may also leads to exceptional points in our system for $\theta=m\pi/2$,
however, we did not focused our interest on it here. For $\eta\neq 0$, one
observes that the overall entanglement becomes weak in Fig.3(a), while it
slightly becomes stronger in Fig.3(b). This reveals how the nonlinear term
tends to suppress the entanglement on the resonator supporting it, and this
entanglement is transfered towards the connected nonlinear-free resonator.

Figure 4: Entanglement between the electromagnetic field with the nonlinear
resonator (a), and with the linear resonator (b) versus the driving field
$\alpha^{in}$. Entanglement between the electromagnetic field with the
nonlinear resonator (c), and with the linear resonator (d) versus the cavity
decay rate $\kappa$. In all these figures, the solid line is without Duffing
nonlinearity ($\eta=0$), while the dash-dotted and dashed lines are for
$\eta=5\times 10^{-6}\omega_{m}$ and $\eta=5\times 10^{-5}\omega_{m}$,
respectively. The phonon number is $n_{th}^{1,2}=100$ for all figures, while
in (c,d) $\alpha^{in}=1000$. The rest of the parameters are as in Fig.3.

To further highly this effect of the Duffing nonlinearity, we have plotted
Fig.4, where the entanglement is displayed versus the driving strength
(Fig.4(a-b)) and over the cavity decay rate (Fig.4(c-d)). In these figures,
the solid line is without nonlinearity ($\eta=0$), and the other lines are for
$\eta\neq 0$ as highlighted on the insets. The overall observation is that,
the entanglement decreases and seems to switch out on the nonlinear resonator
as the nonlinearity is increasing (see Fig.4(a,c)). In the mean time, there is
an enhancement of the entanglement on the resonator that is free of the
nonlinear term (see Fig.4(b,d)). Beside this sort of entanglement transfer
induced by the Duffing nonlinearity, there is also a stability improvement
assisted by this nonlinear feature as aforementined in Fig.2(a). This
nonlinear-induced stability triggers more entanglement by extending it over
the parameter ranges precluded when the resonators are free of the Duffing
term. Therefore, these results shed light on the nonlinear effects which can
be used to switch and to enhance entanglement in coupled resonators.

Figure 5: Entanglement between the electromagnetic field with the nonlinear
resonator (a), and with the linear resonator (b) versus the nonlinear term
$\eta$. Entanglement between the electromagnetic field with the nonlinear
resonator (c), and with the linear resonator (d) versus the thermal phonon
number $n_{th}$. In (a,b), the solid line is for $n_{th=0}$ and the dashed
line is for $n_{th=100}$. In (c,d), the solid line is without Duffing
nonlinearity ($\eta=0$), while the dash-dotted and dashed lines are for
$\eta=5\times 10^{-6}\omega_{m}$ and $\eta=5\times 10^{-5}\omega_{m}$,
respectively. The driving field is $\alpha^{in}=1500$ in (a,b) and
$\alpha^{in}=1000$ in (c,d). The rest of the parameters are as in Fig.3.

To have a wider view of the Duffing nonlinearity on the entanglement, and to
get insight into how this induced entanglement is robust and stable against
thermal fluctuation, we refer to Fig.5. The behavior of the entanglement over
the Duffing term is depicted in Fig.5(a) for the resonator supporting the
nonlinearity, and Fig.5(b) displays the same information of the nonlinear free
resonator. The solid line in these figures is for $n_{th}=0$, and the dashed
line is when the thermal phonon number is $n_{th}=100$. By tracking the solid
lines, we can confirm that the nonlinearity enhance the entanglement. As the
thermal fluctuation is accounted, the strength of this enhanced entanglement
decreases over the nonlinear term. However, the transfered entanglement is
robust enough against thermal fluctuation compared to what is generated on the
resonator supporting the nonlinearity (compare both lines in Fig.5(a) and
Fig.5(b)). The robustness of the transfered entanglement is further
highlighted in Figs.5(c,d), where the logarithmic negativity is displayed over
the thermal phonon number. By comparing the dashed and dash-dotted lines in
these figures, it appears that the transfered entanglement features a better
robustness against thermal fluctuation as the nonlinearity is strong enough.
This reveals how Duffing nonlinearity can be used as a requirement to foster
strong and stable entanglement against thermal fluctuation, leading to an
improvement of quantum information processing and quantum computing in complex
systems supporting nonlinearities.

##  IV CONCLUSION

We have investigated stationary entanglement in an optomechanical system made
of one electromagnetic field that is driving two mechanically coupled
mechanical resonators. The phonon hopping rate between the two resonators is
modulated in order to induce a synthetic magnetism. Moreover, one of the
resonator supports a Duffing nonlinearity. By using experimentally feasible
parameters, we have shown that the stability of the system is slightly
improved, which is a good requirement to enhance stationary entanglement. When
the system is free of the Duffing nonlinearity, we have shown that the
logarithmic negativity features a phenomenon of sudden death and revival of
entanglement. The optimal values of the entanglement appear around the
synthetic phase of $\theta=m\frac{\pi}{2}$, and they switch out as this phase
meets $\theta=n\pi$, where $m$ is an odd number and $n$ is an integer. After
appropriately setting the synthetic phase, we observed that the entanglement
between the electromagnetic field and the resonator supporting the Duffing
nonlinearity becomes weak and tends to switch out for strong enough
nonlinearity. However, the entanglement is switched towards the resonator that
is free of the Duffing term, leading to a sort of nonlinear-assisted transfer
of entanglement. This transfered entanglement has been shown to be robust
enough against thermal fluctuation. Our work paves a way towards nonlinear
transfer entanglement phenomenon in complex system. These results can be used
to generate robust and stable entanglement in nonlinear system, which is
interesting for quantum information processing and quantum computing purposes.

## Acknowledgments

This work has been carried out under the Iso-Lomso Fellowship at Stellenbosch
Institute for Advanced Study (STIAS), Wallenberg Research Centre at
Stellenbosch University, Stellenbosch 7600, South Africa. P. Djorwe
acknowledges the receipt of a grant from the APS-EPS-FECS-ICTP Travel Award
Fellowship Programme (ATAP), Trieste, Italy.

## References

  * Pokharel et al. (2022) A. Pokharel, H. Xu, S. Venkatachalam, E. Collin, and X. Zhou, Nano Letters 22, 7351 (2022), ISSN 1530-6992.
  * Golokolenov et al. (2021) I. Golokolenov, D. Cattiaux, S. Kumar, M. Sillanpää, L. Mercier de Lépinay, A. Fefferman, and E. Collin, New Journal of Physics 23, 053008 (2021), ISSN 1367-2630.
  * Foulla et al. (2017) D. P. Foulla, P. Djorwé, S. T. Kingni, and S. G. N. Engo, Physical Review A 95 (2017).
  * Walter and Marquardt (2016) S. Walter and F. Marquardt, New Journal of Physics 18, 113029 (2016), ISSN 1367-2630.
  * Djorwé et al. (2013) P. Djorwé, J. H. Talla Mbé, S. G. Nana Engo, and P. Woafo, The European Physical Journal D 67, 1 (2013), URL https://doi.org/10.1140/epjd/e2013-30581-0.
  * Lemonde et al. (2016) M.-A. Lemonde, N. Didier, and A. A. Clerk, Nature Communications 7 (2016), ISSN 2041-1723, URL https://doi.org/10.1038/ncomms11338.
  * Barzanjeh et al. (2021) S. Barzanjeh, A. Xuereb, S. Gröblacher, M. Paternostro, C. A. Regal, and E. M. Weig, Nature Physics 18, 15 (2021), ISSN 1745-2481.
  * Aspelmeyer et al. (2010) M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, Journal of the Optical Society of America B 27, A189 (2010), ISSN 1520-8540.
  * Djorwé et al. (2012) P. Djorwé, J. H. T. Mbé, S. G. N. Engo, and P. Woafo, Physical Review A 86 (2012).
  * Colombano et al. (2019) M. Colombano, G. Arregui, N. Capuj, A. Pitanti, J. Maire, A. Griol, B. Garrido, A. Martinez, C. Sotomayor-Torres, and D. Navarro-Urrios, Physical Review Letters 123, 017402 (2019), ISSN 1079-7114.
  * Rodrigues et al. (2021) C. C. Rodrigues, C. M. Kersul, A. G. Primo, M. Lipson, T. P. M. Alegre, and G. S. Wiederhecker, Nature Communications 12 (2021), ISSN 2041-1723.
  * Djorwé et al. (2020) P. Djorwé, Y. Pennec, and B. Djafari-Rouhani, Physical Review B 102, 155410 (2020).
  * Li et al. (2022) J. Li, Z.-H. Zhou, S. Wan, Y.-L. Zhang, Z. Shen, M. Li, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Physical Review Letters 129, 063605 (2022), ISSN 1079-7114.
  * Djorwe et al. (2019) P. Djorwe, Y. Pennec, and B. Djafari-Rouhani, Scientific Reports 9, 1684 (2019), URL https://api.semanticscholar.org/CorpusID:59617258.
  * Navarro-Urrios et al. (2017) D. Navarro-Urrios, N. E. Capuj, M. F. Colombano, P. D. García, M. Sledzinska, F. Alzina, A. Griol, A. Martínez, and C. M. Sotomayor-Torres, Nature Communications 8 (2017), ISSN 2041-1723.
  * Roque et al. (2020) T. F. Roque, F. Marquardt, and O. M. Yevtushenko, New Journal of Physics 22, 013049 (2020), ISSN 1367-2630.
  * Djorwe et al. (2022) P. Djorwe, J. Yves Effa, and S. G. Nana Engo, Nonlinear Dynamics 111, 5905 (2022), ISSN 1573-269X.
  * Zhu et al. (2023) G.-L. Zhu, C.-S. Hu, Y. Wu, and X.-Y. Lü, Fundamental Research 3, 63 (2023), ISSN 2667-3258.
  * Stella et al. (2023) M. T. Stella, P. Djorwe, T. Murielle, and N. E. Serge Guy, Physica Scripta (2023), ISSN 1402-4896.
  * Purdy et al. (2017) T. P. Purdy, K. E. Grutter, K. Srinivasan, and J. M. Taylor, Science 356, 1265 (2017), ISSN 1095-9203.
  * Bemani et al. (2019) F. Bemani, R. Roknizadeh, A. Motazedifard, M. H. Naderi, and D. Vitali, Physical Review A 99, 063814 (2019), ISSN 2469-9934.
  * Riedinger et al. (2016) R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, and S. Gröblacher, Nature 530, 313 (2016), ISSN 1476-4687.
  * Zivari et al. (2022) A. Zivari, R. Stockill, N. Fiaschi, and S. Gröblacher, Nature Physics 18, 789 (2022), ISSN 1745-2481.
  * Fiaschi et al. (2021) N. Fiaschi, B. Hensen, A. Wallucks, R. Benevides, J. Li, T. P. M. Alegre, and S. Gröblacher, Nature Photonics 15, 817 (2021), ISSN 1749-4893.
  * Djorwé et al. (2013) P. Djorwé, S. N. Engo, J. T. Mbé, and P. Woafo, Physica B: Condensed Matter 422, 72 (2013).
  * Banerjee et al. (2023) P. Banerjee, S. Kalita, and A. K. Sarma, Journal of the Optical Society of America B 40, 1398 (2023), ISSN 1520-8540.
  * Riedinger et al. (2018) R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, Nature 556, 473 (2018), ISSN 1476-4687.
  * Kotler et al. (2021) S. Kotler, G. A. Peterson, E. Shojaee, F. Lecocq, K. Cicak, A. Kwiatkowski, S. Geller, S. Glancy, E. Knill, R. W. Simmonds, et al., Science 372, 622 (2021), ISSN 1095-9203.
  * Mercier de Lépinay et al. (2021) L. Mercier de Lépinay, C. F. Ockeloen-Korppi, M. J. Woolley, and M. A. Sillanpää, Science 372, 625 (2021), ISSN 1095-9203.
  * Meher and Sivakumar (2022) N. Meher and S. Sivakumar, The European Physical Journal Plus 137 (2022), ISSN 2190-5444.
  * Slussarenko and Pryde (2019) S. Slussarenko and G. J. Pryde, Applied Physics Reviews 6 (2019), ISSN 1931-9401.
  * Arute et al. (2019) F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, et al., Nature 574, 505 (2019), ISSN 1476-4687.
  * Xia et al. (2023) Y. Xia, A. R. Agrawal, C. M. Pluchar, A. J. Brady, Z. Liu, Q. Zhuang, D. J. Wilson, and Z. Zhang, Nature Photonics 17, 470 (2023), ISSN 1749-4893.
  * Chakraborty and Sarma (2019) S. Chakraborty and A. K. Sarma, Physical Review A 100, 063846 (2019), ISSN 2469-9934.
  * Lai et al. (2022) D.-G. Lai, J.-Q. Liao, A. Miranowicz, and F. Nori, Physical Review Letters 129, 063602 (2022).
  * Venkatachalam and Zhou (2023) S. Venkatachalam and X. Zhou, Nanotechnology 34, 215202 (2023), ISSN 1361-6528.


# Droplets bouncing over a vibrating fluid layer

Pablo Cabrera-Garcia1 and Roberto Zenit2
1 Facultad de Ciencias
2 Instituto de Investigaciones en Materiales
Universidad Nacional Autónoma de México
Cd. Universitaria, México D.F., 04510
MÉXICO

###### Abstract

This is an entry for the Gallery of Fluid Motion of the 65st Annual Meeting of
the APS-DFD ( fluid dynamics video ). This video shows the motion of levitated
liquid droplets. The levitation is produced by the vertical vibration of a
liquid container. We made visualizations of the motion of many droplets to
study the formation of clusters and their stability.

##  1 Introduction

If a liquid drop is deposited over a liquid surface, the drop will, first,
rebound, then arrest and eventually coalesce. Couder et al. [1] reported a
technique to retard indefinitely the coalescence phase. By making the liquid
container, the drop can be made to ‘sit’ on top of the surface for a long time
period. We built a similar experiment to study how several droplets cluster.

##  2 Experimental Conditions

A short glass container was mounted on top of a commercial loudspeaker. The
loudspeaker was fed with an amplified signal from a function generator. The
frequency and amplitude of the signal were chosen such that large surface
instabilities were not observed (Faraday waves). The liquid used was tap
water. A small amount of liquid soap was used; we found that in this manner
the drops where more stable. The process was filmed with a high speed camera.

##  3 Videos

Our video contributions can be found at:

  * •

Video 1, mpeg4, full resolution

  * •

Video 2, mpeg2, low resolution

## References

  * [1] Y. Couder, E. Fort, C.-H. Gautier, and A. Boudaoud, From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath, Phys. Rev. Lett. 94, 177801 (2005).