# Label Distribution Learning with Valence Arousal for Facial Expression
Recognition

###### Abstract

Facial expression recognition is a crucial task to understand human emotions.
Most recent works focus on improving recognition accuracy, while real-world
problems such as noisy labels or ambiguity are not fully investigated. In this
paper, we propose a new label distribution learning approach to improve the
robustness of facial expression recognition methods when dealing with noisy or
ambiguous data. Our work utilizes the valence-arousal to represent human
emotions in continuous space, then establishes the relationship between image
instances to construct emotion distributions. The constructed distribution can
be integrated into a deep network to provide more supervision training signals
and improve the recognition accuracy. Intensive experiments on large-scale
datasets under various settings show that our method outperforms recent state-
of-the-art approaches, while being robust against noisy and inconsistent
labels.

##  I Introduction

Facial expression recognition (FER) plays an important role in understanding
people’s feelings and interactions between humans. Recently, automatic emotion
recognition has gained a lot of attention from the research community
[academia_industry] due to its applications in healthcare
[application_health_care], surveillance [application_surveillance], or human-
robot interaction [application_hci]. Most recent FER methods utilize deep
learning [cnn] and achieve better results than handcrafted features approaches
[hog, lbp]. The success of deep networks can be attributed to large-scale FER
datasets such as AffectNet [affectnet], EmotioNet [emotionet], and RAF-DB
[raf_db]. Some datasets describe emotion in terms of Action Units (AUs)
following the Facial Action Coding System [facs] or quantify affection over
continuous scales, such as valence and arousal [dim_model], while most of them
classify facial expressions into basic universal emotions [6emotions,
contempt] and the neutral state.

Unfortunately, large-scale FER datasets often suffer from the problem of label
uncertainty and annotation ambiguity [ipa2lt, ldl_alsg, dmue]. People with
different backgrounds might perceive and interpret facial expressions
differently, which can lead to inconsistent and uncertain labels [ipa2lt,
dmue]. In addition, real-life facial expressions usually manifest a mixture of
feelings [edl, ldl_alsg] rather than a single exaggerated emotion often found
in the lab-controlled setting. For example, Figure 3 shows that people may
have different opinions about the expressed emotion, particularly in ambiguous
images. Consequently, a distribution over emotion categories is better than a
single label because it takes all sentiment classes into account and can cover
various interpretations, thus mitigating the effect of ambiguity [deepldl].
However, most current large-scale FER datasets only provide a single label for
each sample instead of a label distribution, which means we do not have a
comprehensive description for each facial expression. This can lead to
insufficient supervision during training and pose a big challenge for many FER
systems.

Figure 1:  User study results by 50 volunteers on three random images from
RAF-DB dataset. The expression on the right image are more ambiguous, which
leads to high uncertainty in the emotion label. Labels at the bottom denote
the provided annotation from the dataset. (Su=Surprise, Fe=Fear, Di=Disgust,
Ha=Happy, Sa=Sad, An=Angry, Ne=Neutral)

To overcome annotation ambiguity in FER, this paper proposes a new
uncertainty-aware label distribution learning method that constructs emotion
distributions for training samples. Specifically, for each instance, we
leverage valence-arousal information to identify a set of neighbors and
calculate their corresponding contributions using our adaptive similarity
mechanism. We then aggregate neighborhood information with the provided single
label, adjusted by its learnable uncertainty factor, to generate the target
label distribution. Finally, we use the constructed distribution as
supervision signals to optimize the model via label distribution learning. We
also introduce a discriminative loss that reduces intra-class variations and
encourages inter-class differences to improve the model’s robustness against
ambiguous features. Note that the distribution construction only occurs during
training while the inference process remains intact. In summary, our
contributions are as follows:

  1. 1.

We propose a new method, namely Label Distribution Learning with Valence-
Arousal (LDLVA), for FER with ambiguous annotation by exploiting neighborhood
information in the valence-arousal space.

  2. 2.

Our uncertainty-aware label distribution construction provide more accurate
and richer supervision for training deep FER networks, allowing them to learn
from ambiguous data effectively in an end-to-end manner.

  3. 3.

We perform extensive experiments under various synthetic and real-world
ambiguity settings and achieve state-of-the-art results on RAF-DB, AffectNet,
and SFEW datasets.

##  II Related Work

Most recent methods [deexpression_res, cake, fsn, dacl_net, ldl_alsg, scn,
ipa2lt, caer, zhang2022eac, psr, glamor_net] categorize facial expression into
discrete classes corresponding to basic universal emotions [6emotions,
contempt], which is easy to interpret and intuitive to humans. Other
approaches [joint_AU, unmask_FAC] attempted to represent human emotion using
Action Units (AUs) [facs] or continuous scales such as valence and arousal
[dim_model]. In this work, we leverage the auxiliary information of continuous
scales to mitigate the effect of uncertainty and ambiguity in existing FER
datasets when predicting the discrete emotion of a given facial expression.

One challenging problem of FER is that ambiguous facial expressions can make
it difficult to correctly identify the expressed emotions, which might lead to
noisy and uncertain annotations [ldl_alsg, dmue]. Empirical studies also show
that neural networks are sensitive to noise and can easily overfit noisy data
[dnn_overfit, ldl_joint_optim, ldl_pencil]. To overcome this challenge,
previous approaches model the noise by a transition matrix [sukhbaatar, nal,
patrini, ipa2lt]. In [air], precise image features are extracted from a pre-
trained model to regulate the learning process with noisy labels. The authors
in [robust_loss1, robust_loss2] use noise-tolerant loss functions to increase
noise robustness. Other methods [meta_cleaner, scn] measure the uncertainty of
each sample and utilize a sample-weighting strategy to help the network
tolerate noisy samples. Recently, Zhang _et al_ [rul] propose to quantify the
uncertainties from the relative difficulty of samples by feature mixup.
However, these methods only focus on improving the accuracy on mislabelled
data and do not handle the ambiguous nature of facial expressions.

An alternative approach to address label noise and ambiguity is label
distribution learning (LDL) [ldl]. In other domains, previous works
[head_pose, facial_age, deepldl] leverage prior knowledge to transfer logical
labels into discretized bivariate Gaussian label distribution. The authors in
[ldl_joint_optim, ldl_pencil] utilize network’s predictions as label
distributions to correct noise, which can be unstable and hard to optimize.
Instead, our method not only adaptively utilizes the model’s predictions but
also exploits domain knowledge of the valence-arousal space to construct
target distributions. In FER literature, Zhou _et al_ [edl] introduces a
framework to map a facial expression to multiple emotions with corresponding
intensities. Jia _et al_ [local_ldl] proposes to learn emotion distributions
by exploiting label correlations at a local level. Zhao _et al_
[lightweight_ldl] uses a pre-trained label distribution generator to produce
emotion distribution. Other works create the label distribution by computing
the membership degrees to the labels [soft_membership, fuzzy_membership,
soft_facial_landmark, labeling_importance, manifold_learning]. Recently, Chen
_et al_ [ldl_alsg] leverages the topology in facial landmarks and action units
spaces to acquire more information for label distribution learning. She _et
al_ [dmue] proposed to leverage multiple branches to obtain the latent
distribution. However, these methods either rely heavily on good features with
local linearity to work properly [soft_membership, do2022fine,
fuzzy_membership, soft_facial_landmark, labeling_importance, nguyen2021graph,
manifold_learning] or only use the mined label distributions to regularize the
model’s training process instead of directly learning from them [ldl_alsg,
dmue].

Unlike previous works, our method constructs emotion distributions for
training instances and directly uses them as supervision information, thus
reducing the effects of annotation ambiguity. We do not need to be provided
with label distributions to train the network since they can be accurately
estimated using our adaptive similarity mechanism and learnable uncertainty
factors. We experimentally show that our approach is more effective as the
network is trained end-to-end with label distributions, which brings more
meaningful information to the training process.

##  III Methodology

Figure 2: An overview of our Label Distribution Learning with Valence-Arousal
(LDLVA) for facial expression recognition under ambiguity. Dotted lines denote
components used in training only while solid lines denote components used in
both training and testing.

We first introduce a list of notations that will be used throughout this
paper. Let $\bm{x}\in\mathcal{X}$ be the instance variable in the input space
$\mathcal{X}$ and $\bm{x}^{i}$ be the particular $i$-th instance. The label
set is denoted as $\mathcal{Y}=\\{y_{1},y_{2},...,y_{m}\\}$ where $m$ is the
number of classes and $y_{j}$ is the label value of the $j$-th class. The
logical label vector of $\bm{x}^{i}$ is indicated by $\bm{l}^{i}$ =
$(l^{i}_{y_{1}},l^{i}_{y_{2}},...,l^{i}_{y_{m}})$ with
$\l^{i}_{y_{j}}\in\\{0,1\\}$ and $\|\bm{l}\|_{1}=1$. We define the label
distribution of $\bm{x}^{i}$ as $\bm{d}^{i}$ =
$(d^{i}_{y_{1}},d^{i}_{y_{2}},...,d^{i}_{y_{m}})$ with $\|\bm{d}\|_{1}=1$ and
$d^{i}_{y_{j}}\in[0,1]$ representing the relative degree that $\bm{x}^{i}$
belongs to the class $y_{j}$. A neural network with parameters $\theta$
followed by a softmax layer is denoted as $f(\bm{x};\theta)$. The
corresponding feature vector of $\bm{x}^{i}$ extracted by a CNN backbone model
is indicated by $\bm{v}^{i}\in\mathbb{R}^{V}$.

###  III-A Overview

Most existing FER datasets assign only a single class or equivalently, a
logical label $\bm{l}^{i}$ for each training sample $\bm{x}^{i}$. In
particular, the given training dataset is a collection of $n$ samples with
logical labels $D_{l}$ = $\\{(\bm{x}^{i},\bm{l}^{i})|1\leq i\leq n\\}$.
However, as depicted in Figure 3, a label distribution $\bm{d}^{i}$ is a more
comprehensive and suitable annotation for the image than a single label.
Inspired by the recent success of label distribution learning (LDL) in
addressing label ambiguity [deepldl], we aim to construct an emotion
distribution $\bm{d}^{i}$ for each training sample $\bm{x}^{i}$, thus
transform the training set $D_{l}$ into $D_{d}$ =
$\\{(\bm{x}^{i},\bm{d}^{i})|1\leq i\leq n\\}$, which can provide richer
supervision information and help mitigate the ambiguity issue. Consequently,
our goal is to optimize the parameters $\theta$ of the neural network
$f(\bm{x};\theta)$ such that it can learn an appropriate mapping function for
the instance $\bm{x}^{i}$ from the input space to the target label
distribution $\bm{d}^{i}$. Mathematically, we use cross-entropy to measure the
discrepancy between the model’s prediction and the constructed target
distribution [deepldl]. Hence, the solution can be obtained by minimizing the
following classification loss:

$\mathcal{L}_{cls}=\sum_{i=1}^{n}\text{CE}\left(\bm{d}^{i},f(\bm{x}^{i};\theta)\right)=-\sum_{i=1}^{n}\sum_{j=1}^{m}\bm{d}_{j}^{i}\log
f_{j}(\bm{x}^{i};\theta).$ (1)

An overview of our method is presented in Figure 2. To construct the label
distribution for each training instance $\bm{x}^{i}$, we leverage its
neighborhood information in the valence-arousal space. Particularly, we
identify $K$ neighbor instances for each training sample $\bm{x}^{i}$ and
utilize our adaptive similarity mechanism to determine their contribution
degrees to the target distribution $\bm{d}^{i}$. Then, we combine the
neighbors’ predictions and their corresponding contribution degrees with the
provided label $\bm{l}^{i}$ and $\bm{l}^{i}$’s uncertainty factor to obtain
the label distribution $\bm{d}^{i}$. The constructed distribution $\bm{d}^{i}$
will be used as supervision information to train the model via label
distribution learning. It is worth noting that these steps occur only during
training, thus no extra costs are introduced at inference time.

###  III-B Adaptive Similarity Measuring

As in previous works [smoothness_assumption, le_ldl, ldl_alsg], we assume that
facial images should have similar emotions to their neighbors in an auxiliary
or supporting space. Therefore, the label distribution of an instance can be
constructed using the information of its neighbors. Since our goal is to
reconstruct the target label distribution with high fidelity, the chosen
supporting space should highly correlate with the emotion space to transfer as
much information as possible. Although information such as facial landmarks
and action units can be utilized as the supporting space, we find that
valence-arousal values are more closely associated with discrete emotions and
thus particularly suitable to be the auxiliary space. In practice, the
valence-arousal has been widely used to represent the human emotional
spectrum, with valence describing how positive or negative an expression is
and arousal indicating the intensity or activation degree of the expression
[valence_arousal].

Similar to the smoothness assumption [smoothness_assumption], we assume that
the label distribution of the main instance $\bm{x}^{i}$ can be computed as a
linear combination of its neighbors’ distributions. To determine the
contribution of each neighbor, we propose an adaptive similarity mechanism
that not only leverages the relationships between $\bm{x}^{i}$ and its
neighbors in the auxiliary space but also utilizes their feature vectors
extracted from the backbone. In particular, we first use the $K$-Nearest
Neighbor algorithm to identify $K$ closest points for each training sample
$\bm{x}^{i}$, denoted as $N(i)$, based on the distance between training
instances in the valence-arousal space. We then compute a local similarity
score between $\bm{x}^{i}$ and each of its $K$ neighbors using the following
formula:

$s^{i}_{k}=\exp\left(-\frac{\|\bm{a}^{i}-\bm{a}^{k}\|^{2}_{2}}{\delta^{2}}\right),\quad\forall\bm{x}^{k}\in
N(i),$ (2)

where $\bm{a}$ is the corresponding auxiliary valence-arousal vector of
$\bm{x}$, and $\delta$ is a hyperparameter controlling similarity measurement.
Intuitively, the higher $s^{i}_{k}$ is, the more $\bm{x}^{k}$ contributes to
the label distribution of $\bm{x}^{i}$.

However, since valence-arousal values are not always available in practice, we
leverage an existing method [va_prediction] to generate pseudo-valence-
arousal. Consequently, these values can be inaccurate and lead to incorrect
calculation of $s^{i}_{k}$. Therefore, we proposed to correct these potential
errors with our adaptive similarity mechanism. Specifically, we calculate a
calibration score for each $(\bm{x}^{i},\bm{x}^{k})$ pair using the feature
vectors $(\bm{v}^{i},\bm{v}^{k})$ extracted by the CNN backbone of
$\bm{x}^{i}$ and its neighbor instance $\bm{x}^{k}\in N(i)$ as follows:

$\zeta^{i}_{k}=\text{Sigmoid}\left(g([\bm{v}^{i},\bm{v}^{k}];\phi)\right),$
(3)

where $[\bm{\cdot},\bm{\cdot}]$ is the concatenation operator, $g$ is a three-
layer perceptron (MLP) with parameter $\phi$. The dimensionality of each layer
is 512, 256, and 1, respectively. We also apply layer normalization and ReLU
non-linearity in the first two layers.

The final contribution degrees of neighbor instances are calculated as the
product of the local similarity and the calibration score:

$c^{i}_{k}=\begin{cases}\zeta^{i}_{k}s^{i}_{k},&\text{for }\bm{x}^{k}\in
N(i),\\\ 0,&\text{otherwise}.\end{cases}$ (4)

###  III-C Uncertainty-aware Label Distribution Construction

After obtaining the contribution degree of each neighbor $\bm{x}^{k}\in N(i)$,
we can now generate the target label distribution $\bm{d}^{i}$ for the main
instance $\bm{x}^{i}$. The target label distribution is calculated using the
logical label $\bm{l}^{i}$ and the aggregated distribution
$\tilde{\bm{d}}^{i}$ defined as follows:

$\displaystyle\tilde{\bm{d}^{i}}$
$\displaystyle=\frac{\sum_{k}c^{i}_{k}f(\bm{x}^{k};\theta)}{\sum_{k}c^{i}_{k}},$
(5) $\displaystyle\bm{d}^{i}$
$\displaystyle=(1-\lambda^{i})\bm{l}^{i}+\lambda^{i}\tilde{\bm{d}^{i}},$ (6)

where $\lambda^{i}\in[0,1]$ is the uncertainty factor for the logical label.
It controls the balance between the provided label $\bm{l}^{i}$ and the
aggregated distribution $\tilde{\bm{d}^{i}}$ from the local neighborhood.
Intuitively, a high value of $\lambda^{i}$ indicates that the logical label is
highly uncertain, which can be caused by ambiguous expression or low-quality
input images as illustrated in Figure 7, thus we should put more weight
towards neighborhood information $\tilde{\bm{d}^{i}}$. Conversely, when
$\lambda^{i}$ is small, the label distribution $\bm{d}^{i}$ should be close to
$\bm{l}^{i}$ since we are certain about the provided manual label. In our
implementation, $\lambda^{i}$ is a trainable parameter for each instance and
will be optimized jointly with the model’s parameters using gradient descent.

Mathematically speaking, consider Equation 1 and 6, the derivative of
$\mathcal{L}_{cls}$ with respect to $\lambda^{i}$ can be computed as:

$\displaystyle\frac{\partial\mathcal{L}_{cls}}{\partial\lambda^{i}}$
$\displaystyle=\frac{\partial\text{CE}\left(\bm{d}^{i},f(\bm{x}^{i};\theta)\right)}{\partial\lambda^{i}}$
(7) $\displaystyle=-\sum_{j}\tilde{\bm{d}^{i}_{j}}\log
f_{j}(\bm{x}^{i};\theta)+\sum_{j}\bm{l}^{i}_{j}\log f_{j}(\bm{x}^{i};\theta)$
(8)
$\displaystyle=\text{CE}(\tilde{\bm{d}^{i}},f(\bm{x}^{i};\theta))-\text{CE}(\bm{l}^{i},f(\bm{x}^{i};\theta)).$
(9)

If $\text{CE}(\bm{l}^{i},f(\bm{x}^{i};\theta))$ is smaller than
$\text{CE}(\tilde{\bm{d}^{i}},f(\bm{x}^{i};\theta))$, the derivative of
$\mathcal{L}_{cls}$ with respect to $\lambda^{i}$ is positive, which leads to
a negative update for $\lambda^{i}$ following gradient descent optimization
scheme. This is desirable because in this case, the network output is in more
agreement with the logical label than the aggregated neighborhood
distribution. In other words, it is more confident about the provided label
and thus, we should decrease the value of the uncertainty factor
$\lambda^{i}$. The same reasoning can be applied in the opposite situation.

###  III-D Loss Function

Recent literatures have shown the benefits of learning discriminative features
in FER [island_loss, affwild2, dda_loss, dacl_net]. Inspired by this, we
believe it is beneficial to encourage the network to learn good facial
descriptions because it can help improve the model’s ability to discriminate
between ambiguous emotions. We find that the center loss [center_loss] is
suitable for our purpose because of its simplicity and efficacy in reducing
the intra-class variations of the learned representations. Nevertheless, in
the traditional formulation of the center loss [center_loss], the features of
a sample are “blindly” pulled towards its corresponding class center given its
label. This means when the provided label is incorrect, it can cause the
network to learn imprecise features. We propose to overcome this problem by
incorporating the label uncertainty factor $\lambda^{i}$ to adaptively
penalize the distance between the sample and its corresponding center. For
instances with high uncertainty, the network can effectively tolerate their
features in the optimization process. Furthermore, we also add pairwise
distances between class centers to encourage large margins between different
classes, thus enhancing the discriminative power. Our discriminative loss is
calculated as follows:

$\displaystyle\mathcal{L}_{D}$
$\displaystyle=\frac{1}{2}\sum_{i=1}^{n}(1-\lambda^{i})\|\bm{v}^{i}-\bm{\mu}_{y^{i}}\|_{2}^{2}$
$\displaystyle+\sum_{j=1}^{m}\sum_{\begin{subarray}{c}k=1\\\ k\neq
j\end{subarray}}^{m}\exp\left(-\frac{\|\bm{\mu}_{j}-\bm{\mu}_{k}\|_{2}^{2}}{\sqrt{V}}\right),$
(10)

where $y^{i}$ is the class index of the $i$-th sample while $\bm{\mu}_{j}$,
$\bm{\mu}_{k}$, and $\bm{\mu}_{y^{i}}$ $\in\mathbb{R}^{V}$ are the center
vectors of the ${j}$-th, ${k}$-th, and $y^{i}$-th classes, respectively.
During the training phase, all center vectors are zero-initialized and
optimized using Equation III-D. Intuitively, the first term of
$\mathcal{L}_{D}$ encourages the feature vectors of one class to be close to
their corresponding center [center_loss] while the second term improves the
inter-class discrimination by pushing the cluster centers far away from each
other.

Combining Equation 1 and Equation III-D, we obtain the total loss for
training:

$\mathcal{L}=\mathcal{L}_{cls}+\gamma\mathcal{L}_{D},$ (11)

where $\gamma$ is the hyperparameter balancing between the two losses.

##  IV Introduction

Facial expression recognition (FER) plays an important role in understanding
people’s feelings and interactions between humans. Recently, automatic emotion
recognition has gained a lot of attention from the research community
[academia_industry] due to its applications in healthcare
[application_health_care], surveillance [application_surveillance], or human-
robot interaction [application_hci]. Most recent FER methods utilize deep
learning [cnn] and achieve better results than handcrafted features approaches
[hog, lbp]. The success of deep networks can be attributed to large-scale FER
datasets such as AffectNet [affectnet], EmotioNet [emotionet], and RAF-DB
[raf_db]. Some datasets describe emotion in terms of Action Units (AUs)
following the Facial Action Coding System [facs] or quantifying affection over
continuous scales such as valence and arousal [dim_model] while most of them
classify facial expressions into basic universal emotions [6emotions,
contempt] and the neutral state.

Unfortunately, large-scale FER datasets often suffer from the problem of label
uncertainty and annotation ambiguity [ipa2lt, ldl_alsg, dmue]. People with
different backgrounds might perceive and interpret facial expressions
differently, which can lead to inconsistent and uncertain labels [ipa2lt,
dmue]. In addition, real-life facial expressions usually manifest a mixture of
feelings [edl, ldl_alsg] rather than a single exaggerated emotion often found
in the lab-controlled setting. For example, Figure 3 shows that people may
have different opinions about the expressed emotion, particularly in ambiguous
images. We argue that distribution over emotion categories is better than a
single label because it takes all sentiment classes into account and can cover
various interpretations, thus mitigating the effect of ambiguity. However,
most current large-scale FER datasets only provide a single label for each
sample instead of a label distribution, which means we do not have a
comprehensive description for each facial expression. This can lead to a lack
of supervision during training and pose a big challenge for many FER systems.

Figure 3:  User study results by 50 volunteers on three random images from
RAF-DB dataset. The expression on the right image are more ambiguous, which
leads to high uncertainty in emotion labels. Labels at the bottom denote the
provided annotation from the dataset. (Su=Surprise, Fe=Fear, Di=Disgust,
Ha=Happy, Sa=Sad, An=Angry, Ne=Neutral)

To overcome annotation ambiguity in facial expression recognition, this paper
proposes a new uncertainty-aware label distribution learning method that
constructs emotion distributions for training samples. Specifically, for each
training instance, we leverage the valence-arousal information to identify a
set of neighbors and calculate their corresponding contributions using our
adaptive similarity mechanism. We then aggregate neighborhood information with
the provided single label, adjusted by its uncertainty factor, to generate the
target label distribution. Finally, we use the constructed distribution as
supervision signals to optimize the model via label distribution learning. We
also introduce a discriminative loss that reduces intra-class variations and
encourages inter-class differences to improve the model’s robustness against
ambiguous features. Note that the distribution construction only occurs during
training while the inference process remains intact. In summary, our
contributions are as follows:

  1. 1.

We propose a new label distribution learning method for FER under ambiguous
label conditions by exploiting the neighborhood information in the valence-
arousal space.

  2. 2.

Our uncertainty-aware label distribution construction can provide more
accurate and richer supervision for training deep FER networks, allowing them
to learn from ambiguous data effectively in an end-to-end manner.

  3. 3.

We perform extensive experiments under various synthetic and real-world
ambiguity settings and achieve state-of-the-art results on RAF-DB, AffectNet,
and SFEW datasets. Our source code and pretrained models will be released for
further study.

##  V Experiments

In this section, we first validate the effectiveness of our approach on
synthetic ambiguity caused by noisy label data. Next, we evaluate the
performance of our LDLVA in handling inconsistent labels caused by ambiguous
facial expressions. We then compare LDLVA with state-of-the-art methods to
demonstrate the robustness of our approach towards annotation ambiguity that
inherently exists in real-world data. Finally, we conduct ablation studies and
present qualitative results to investigate the effectiveness of each component
as well as the advantages of our method.

###  V-A Datasets

We perform experiments on three popular in-the-wild FER datasets: AffectNet
[affectnet], RAF-DB [raf_db] and SFEW [sfew]. They are created by collecting
data from the Internet and reflect real-life scenarios. AffectNet [affectnet]
has more than 400,000 facial images manually annotated with discrete emotions
and valence-arousal. Following previous work [ldl_alsg, ipa2lt, dacl_net], we
select approximately 280,000 and 3,500 images for training and testing, all of
which belong to six basic emotions (surprise, fear, disgust, happy, sad, and
angry) and neutral expression. RAF-DB [raf_db] is split into training and test
sets with more than 12,000 and 3,000 images, respectively. SFEW [sfew] has 879
training images and 406 testing images, all of which are extracted from movie
videos.

###  V-B Experimental Settings

By default, we use the pretrained ResNet-50 [resnet] as the CNN backbone. We
align the input image and perform on-the-fly augmentation during training by
randomly flipping the image horizontally and taking a random crop of size
224×224 after padding 16 pixels on each side. At test time, we use the central
crop of the image as input for the model. During training, for each instance,
we consider 8 nearest neighbors and initialize its uncertainty factor
$\lambda^{i}$ to zero. To optimize the discriminative loss (Equation III-D),
we follow the same settings as in [center_loss]. We train the network using
Adam optimizer [adam] with batch size 32 for 30 epochs with an initial
learning rate of $0.001$. The parameters $\delta$ in Equation 2 and $\gamma$
in Equation 11 are set to 0.5 and 0.1 based on validation results. Similar to
previous works [ldl_alsg, rul, scn], we use the overall accuracy as the metric
to evaluate the models.

###  V-C Experiments with Noisy Labels

Noise ratio | Method | Accuracy (%)
---|---|---
AffectNet | RAF-DB | SFEW
10% | Baseline | 60.14 $\pm$ 0.23 | 83.28 $\pm$ 0.45 | 45.98 $\pm$ 0.93
SCN [scn] | 61.57 $\pm$ 0.15 | 84.65 $\pm$ 0.32 | 49.51 $\pm$ 0.76
RUL [rul] | 62.89 $\pm$ 0.13 | 86.24 $\pm$ 0.22 | 47.82 $\pm$ 1.32
LDLVA (ours) | 64.37 $\pm$ 0.11 | 87.98 $\pm$ 0.10 | 53.33 $\pm$ 0.57
20% | Baseline | 58.37 $\pm$ 0.35 | 81.89 $\pm$ 0.61 | 41.25 $\pm$ 1.12
SCN [scn] | 60.83 $\pm$ 0.19 | 83.21 $\pm$ 0.49 | 46.26 $\pm$ 1.24
RUL [rul] | 61.74 $\pm$ 0.18 | 84.49 $\pm$ 0.24 | 44.78 $\pm$ 1.04
LDLVA (ours) | 63.89 $\pm$ 0.14 | 86.81 $\pm$ 0.12 | 51.53 $\pm$ 0.92
30% | Baseline | 56.94 $\pm$ 0.43 | 78.92 $\pm$ 0.59 | 38.51 $\pm$ 1.69
SCN [scn] | 58.80 $\pm$ 0.32 | 80.61 $\pm$ 0.54 | 43.28 $\pm$ 2.06
RUL [rul] | 60.77 $\pm$ 0.15 | 82.59 $\pm$ 0.42 | 41.79 $\pm$ 0.81
LDLVA (ours) | 62.57 $\pm$ 0.15 | 85.85 $\pm$ 0.09 | 50.3 $\pm$ 0.88

TABLE I: Accuracy with synthetic noise.

The two main aspects of annotation ambiguity in FER are noisy labels and
uncertain visual features [scn]. In particular, it can be difficult for people
to accurately recognize the emotions on ambiguous facial images, which can
result in noisy and incorrect labels. Therefore, we conduct experiments to
study the robustness of our LDLVA on mislabelled data by adding synthetic
noise to AffectNet, RAF-DB, and SFEW datasets. More specifically, we randomly
flip the manual labels to one of the other categories. Three levels of noise
are studied in our experiment. We quantitatively evaluate our method and
compare with the baseline ResNet-50 [resnet] and recent noise-tolerant FER
methods including SCN [scn] and RUL [rul].

We perform each experiment three times and report the mean accuracy and
standard error in Table I. The results clearly show that our method
consistently outperforms other approaches in all cases. Particularly, our
model makes significant improvements over the baseline with an average
accuracy margin of 5.13%, 5.52%, and 9.81% on the AffectNet, RAF-DB, and SFEW
datasets, respectively. We also observe that the improvements are even more
apparent when the noise ratio increases, for example, the accuracy improvement
on RAF-DB is 4.7% with 10% noise and 6.93% with 30% noise. The consistent
results under various settings demonstrate the ability of our method to
effectively deal with noisy annotation, which is crucial in the robustness
against label ambiguity.

###  V-D Experiments with Inconsistent Labels

Method | Accuracy (%)
---|---
AffectNet | RAF-DB | SFEW | Average
AIR [air] | 54.23 | 67.37 | 49.88 | 57.16
NAL [nal] | 55.97 | 84.22 | 58.13 | 66.11
IPA2LT [ipa2lt] | 57.85 | 83.80 | 53.15 | 64.93
LDL-ALSG [ldl_alsg] | 58.29 | 85.33 | 55.87 | 66.50
LDLVA (ours) | 62.89 | 87.26 | 58.70 | 69.62

TABLE II: Accuracy with inconsistent labels.

Due to the ambiguous nature of facial expressions, different individuals can
assign different labels for the same image as illustrated in Figure 3. Since
the annotations for large-scale FER data are commonly obtained via crowd-
sourcing, this can create label inconsistency, especially between different
datasets. Therefore, to examine the effectiveness of the proposed methods in
dealing with this problem, we follow the cross-dataset protocol in previous
state-of-the-art methods [ldl_alsg, ipa2lt] and adopt the experimental
settings as proposed in [ldl_alsg] for a fair comparison. Specifically, the
model is trained using the joint training dataset from RAF-DB and AffectNet.
The resulting model is then tested on all three RAF-DB, AffectNet, and SFEW
datasets.

Table II reports the results of our experiments. Our method achieves the best
performance on all three datasets and the highest average accuracy. Notably,
LDLVA surpasses the current state-of-the-art LDL-ALSG [ldl_alsg] with an
improvement of 3.12% on average accuracy. Compared to our approach, LDL-ALSG
only uses the neighbors’ distributions to constrain the network prediction
without constructing a label distribution for the center instance. It also
lacks a mechanism to adaptively measure the contribution of each neighbor and
the uncertainty of the provided annotation. The favorable performance confirms
the advantages of our method over previous works and demonstrates the
generalization ability to data with label inconsistency, which is essential
for real-world FER applications.

###  V-E Experiments on Original Datasets

Method | Accuracy (%)
---|---
AffectNet | RAF-DB | SFEW
Island Loss [island_loss] | - | - | 52.52
IPFR [IPFR] | 57.40 | - | 55.10
EfficientFace [efficientface] | 63.70 | 88.36 | -
DACL [dacl_net] | 65.20 | 87.78 | -
MViT [mvit] | 64.57 | 88.62 | -
RAN [ran] | - | 86.90 | 56.4
SCN [scn] | - | 87.03 | -
DMUE [dmue] | - | 88.76 | 57.12
RUL [rul] | - | 88.98 | -
PSR [psr] | 63.37 | 88.98 | -
LDLVA (ours) | 66.23 | 90.51 | 59.90

TABLE III: Accuracy of different methods on original datasets.

We further perform experiments on the original AffectNet, RAF-DB, and SFEW to
evaluate the robustness of our method to the uncertainty and ambiguity that
unavoidably exists in real-world FER datasets. We compare the proposed LDLVA
with several state-of-the-art methods in Table III. By leveraging label
distribution learning on valence-arousal space, our model outperforms other
methods and achieves state-of-the-art performance on AffectNet, RAF-DB, and
SFEW. Although these datasets are considered to be “clean”, the results
suggest that they indeed suffer from uncertainty and ambiguity.

###  V-F Qualitative Analysis

Figure 4:  Comparison of the results from our survey and our model. More
results can be found in the supplementary materials.

Real-world Ambiguity. To understand more about real-world ambiguous
expressions, we conducted a user study in which we asked 50 participants to
choose the most clearly expressed emotion on random test images from RAF-DB
and AffectNet datasets. The numbers of votes per class are normalized to
obtain the emotion distribution. We compare our model’s predictions with the
survey results in Figure 4. We can see that these images are ambiguous as they
express a combination of different emotions, hence the participants do not
fully agree and have different opinions about the most prominent emotion on
the faces. It is further shown that LDLVA can give consistent results and
agree with the perception of humans to some degree, which suggests that our
model can effectively address the ambiguity problem in facial expressions.

Figure 5: The calibration scores of neighbor images with respect to the main
instance. The large image on the left is the main instance. The neighbor
images are shown at the top, and their corresponding scores are shown at the
bottom.

Adaptive Similarity. Figure 5 presents the normalized calibration scores of
different neighbors with respect to the center instance computed by our
adaptive similarity mechanism. It can be seen that some neighbors may look
visually similar to the central image but do not express the same emotion. By
giving low calibration values, our method can effectively suppress the
negative influence of these neighbors and lower their contribution, hence
resulting in a more robust and accurate estimation of the emotion
distribution.

Figure 6: Examples of the emotion distribution recovered by our method when
the dataset is contaminated with noisy labels. The label on top of each image
is the synthetic noisy label and the bottom denotes the human annotation.
Figure 7:  Visualization of uncertainty values of some examples from RAF-DB
dataset.

Constructed Label Distribution. In Figure 6, we visualize the emotion
distributions reconstructed by our method on mislabelled images. Despite the
incorrect annotations, our approach is able to construct plausible
distributions and discover the correct labels. It is also noteworthy that some
expressions manifest multiple emotions rather than only the single provided
category, which means the discovered distribution can provide more supervision
for training.

Uncertainty Factor. Figure 7 shows the estimated uncertainty factors of some
training images in the RAF-DB dataset and their original labels. The
uncertainty values decrease from top to bottom. Highly uncertain labels can be
caused by low-quality inputs (as shown in Angry and Surprise columns) or
ambiguous facial expressions. In contrast, when the emotions can be easily
recognized as those in the last row, the uncertainty factors are assigned low
values. This characteristic can guide the model to decide whether to put more
weights on the provided label or the neighborhood information. Therefore, the
model can be more robust against uncertainty and ambiguity.

###  V-G Ablation Study

Contribution of Each Component. In Table IV, we present the accuracy
corresponding to different combinations of our components: label distribution
(whether to construct $\bm{d}^{i}$ or not), adaptive similarity (whether to
compute calibration scores or directly use local similarity scores as
contribution degrees), uncertainty factor (whether to use separate
$\lambda^{i}$ for each instance or share a fixed value $\lambda$ for all
training samples), and discriminative loss (whether to incorporate
$\mathcal{L}_{D}$ in Equation 11 or not). By employing label distribution with
adaptive similarity (ii), we can significantly improve the accuracy of the
vanilla approach (i) by 1.89% on original RAF-DB and 3.77% on 30%-noise RAF-
DB. Further integrating uncertainty factor and discriminative loss
consistently boost the performance of the model, as shown in the results of
(iii) and (v), respectively. The results show the effectiveness of each
component as well as the advantages of their combination in our LDLVA method.

Setting |  | LD
---
| AS
---
| UF
---
| DL
---
| RAF-DB
---
(original)
| RAF-DB
---
(30% noise)
(i) | - | - | - | - | 87.06 | 78.92
(ii) | ✓ | ✓ | - | - | 88.95 | 82.69
(iii) | ✓ | ✓ | ✓ | - | 89.57 | 84.38
(iv) | ✓ | - | ✓ | ✓ | 89.31 | 83.56
(v) | ✓ | ✓ | ✓ | ✓ | 90.51 | 85.85

TABLE IV: Component analysis (LD: Label Distribution, AS: Adaptive Similarity,
UF: Uncertainty Factor, DL: Discriminative Loss)  Figure 8: Evaluation results
with different numbers of neighbors.

Number of Nearest Neighbors. We present the effect of the number of nearest
neighbors $K$ on the model performance in Figure 8. For original RAF-DB data,
higher values of $K$ give better results but also require more training time.
In particular, our training time with $K=8$ and $K=16$ on AffectNet is 12
hours and 20 hours, respectively. Under noisy conditions, the best result is
obtained with $K=8$ while larger or smaller $K$ can lead to slightly worse
performance. The reason is that using a large $K$ might include more corrupted
labels while using too few neighbors can limit the amount of exploitable
information.

##  VI Conclusion

This paper introduces a new label distribution learning method for facial
expression recognition by leveraging structure information in the valence-
arousal space to recover the intensities distributed over emotion categories.
We first employ the adaptive similarity to account for the errors caused by
pseudo valence-arousal and robustly measure the contribution degree of each
neighbor. Then, the target label distribution is constructed by incorporating
both the provided single label and the combination of neighbor distribution
guided by the uncertainty value. The constructed label distribution provides
rich information about the emotions, thus can effectively describe the
ambiguity degree of the facial image. Intensive experiments on popular
datasets demonstrate the effectiveness of our method over previous approaches
under inconsistency and uncertainty conditions in facial expression
recognition.


4pt 4pt

# Born Machines for Periodic and Open XY Quantum Spin Chains

Abigail McClain Gomez  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Susanne F. Yelin  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Khadijeh Najafi  Department
of Physics, Harvard University, Cambridge, Massachusetts 02138, USA  IBM
Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598 USA

###### Abstract

Quantum phase transitions are ubiquitous in quantum many body systems. The
quantum fluctuations that occur at very low temperatures are known to be
responsible for driving the system across different phases as a function of an
external control parameter. The XY Hamiltonian with a transverse field is a
basic model that manifests two distinct quantum phase transitions, including
spontaneous $Z_{2}$ symmetry breaking from an ordered to a disordered state.
While programmable quantum devices have shown great success in investigating
the various exotic quantum phases of matter, in parallel, the quest for
harnessing machine learning tools in learning quantum phases of matter is
ongoing. In this paper, we present a numerical study of the power of a
quantum-inspired generative model known as the Born machine in learning
quantum phases of matter. Data obtained from the system under open and
periodic boundary conditions is considered. Our results indicate that a Born
machine based on matrix product states can successfully capture the quantum
state across various phases of the XY Hamiltonian and close to a critical
point, despite the existence of long-range correlations. We further impose
boundary conditions on the Born machine and show that matching the boundary
condition of the Born machine and that of the training data improves
performance when limited data is available and a small bond dimension is
employed.

##  1 Introduction

Quantum many body systems have attracted the attention of physicists for many
years. These systems are far from being understood by any simple analytical
solution; their many body nature demands more elaborate techniques that
involve statistical approaches. Moreover, the competition between various
degrees of freedom (such as motion, charge, and spin) is the origin of many
exotic phases of matter, ranging from quantum magnets to superconducting and
topological phases. Recent developments in building coherent programmable
quantum devices have provided a compelling tool to probe the fundamental
properties of the quantum matter, and the quest for building and controlling
even larger devices is an ongoing process [1, 2]. However, in parallel with
the growth in system size, the number of measurements required for the full
reconstruction of quantum state scales exponentially, and the noisy nature of
the data becomes more pronounced.

Remarkably, machine learning (ML) tools have already shown great promise in
extracting effective information from large amounts of noisy data [3, 4]. One
category of machine learning applications that are inherently quantum is the
task of understanding or characterizing physical systems that exhibit many
body quantum effects, such as correlated electronic structure or phase
transitions from disorder to order in spin systems. Due to the complex
behavior of these systems, mean-field or semi-classical approaches are
insufficient, yet the naive approach of working in the entire quantum state
space would scale exponentially. Concretely, the exponential scaling of such
systems makes any scalable simulation on classical devices unfeasible. Quantum
simulations on quantum computers could lead to efficient simulations for many
body quantum systems or, more generally, any sparse quantum Hamiltonian.
However, it is not simple to extract meaningful knowledge of a physical
observable from a finite set of local measurements on a noisy processor, which
are prone to error and very expensive to query.

(a)

(b)

Figure 1: (a) Open Born machine architecture for a chain of $N$ atoms. Here,
$\\{i_{1},...i_{N}\\}$ denote the spin indices (which all have size 2 for a
two-level model of each atom) and $\\{D_{1},...D_{N-1}\\}$ denote the bond
dimensions (which all have size $D$ for our architectures). (b) Periodic Born
machine architecture for a chain of $N$ atoms. Note that there are $N$ bond
dimensions in this architecture.

While finding efficient machine learning-based algorithms for quantum state
tomography remains an important question, another intriguing possibility is
the direct characterization of quantum systems through the measurement of
many-body correlation functions. In particular, the behaviour of non-
equilibrium quantum systems undergoing a quantum phase transition (QPT) has
attracted much attention. Such transitions are driven by quantum fluctuations
at zero temperature and can be identified by the divergence of the correlation
length, an encompassing universal behaviour near the critical point [5, 6]. In
this paper, we address these questions by leveraging advanced ML algorithms
that enable quantum simulation. Specifically, we explore the task of learning
various phases of the one dimensional XY Hamiltonian with open and periodic
boundary conditions using the Born machine (BM). We further introduce the
Periodic Born machine (see Fig. 1) and show that the architecture of the Born
machine is sensitive to the boundary condition imposed on the training data.

Figure 2: Different regions in the phase diagram of the quantum XY chain
depicted for positive values of $\gamma$ and $h$. The model is critical for
$h=1$ as well as for $h<1,\gamma=0$, which respectively correspond to the
critical XY and XX chains. The model manifests a quantum phase transition from
disorder ($h>1$) to ordered phase ($h<1$). The point $(\gamma=1,h=1)$
corresponds to the critical point of the transverse field Ising model, and on
the line $\gamma^{2}+h^{2}=1$, the ground state can be factorized into a
product of single spin states.

We start with the Hamiltonian of the XY model in the transverse magnetic
field, described by:

$\displaystyle\
\mathbf{H}=-J\sum_{i=1}^{N-1}\Big{(}\frac{1+\gamma}{4}\sigma_{i}^{x}\sigma_{i+1}^{x}+\frac{1-\gamma}{4}\sigma_{i}^{y}\sigma_{i+1}^{y}\Big{)}-\frac{h}{2}\sum_{i=1}^{N}\sigma_{i}^{z}$
(1)

where $J$ is the spin-spin coupling, $\gamma$ is an asymmetric constant
capturing the relative strength of interaction along $x$ and $y$, and $h$ is
the external magnetic field coupled to spins along the $z$-axis. Despite the
fact that this Hamiltonian can be mapped to the free fermionic model via the
Jordan-Wigner transformation, it manifests a rich phase diagram (see Fig. 2)
[7]. The phase diagram involves two distinct quantum phase transitions;
namely, the critical XX line corresponding to $\gamma=0$ with $h\leq 1$
(belonging to $U_{1}$ symmetry) and the critical XY line corresponding to
$h=1$ (belonging to the $Z_{2}$ symmetry sector). The latter transition
occurring at $(|h|=J=1)$ is an Ising like transition, in that it exhibits
spontaneous breaking of $Z_{2}$ symmetry. This transition from a doubly
degenerate state in the ordered phase for $|h|<1$ to a single ground state
($|h|>1$) is similar to the classical 2D Ising phase transition, with the
magnetic field replacing temperature. The XY model is exactly solvable,
allowing one to attain analytical solutions for many interesting local and
non-local quantum observables and their corresponding time evolutions [8, 9,
10, 11]. These solutions can serve as valuable bench-marking tools for probing
the system close to a critical point [8]. In this work, rather than
calculating the exact solution, we used the density matrix renormalization
group (DMRG) for the simplicity of the numerical model. DMRG has shown great
success in evaluating many physical observables of interest for quantum spin
chains [12, 13].

The rest of the paper is organized as follows. In Section 2, we first
introduce the quantum inspired generative model known as the Born Machine,
based on an important category of quantum many body states called matrix
product states (MPS) [14, 15]. We further introduce the Periodic Born machine
and demonstrate its ability to learn the underlying boundary condition of
quantum training data. In Section 3, we present various data from different
points of the phase diagram of the XY quantum spin chain obtained using DMRG
simulations with ITensor [16]. We then examine a successful example of
learning a ground state of the XY Hamiltonian with the Born machine.
Subsequently, we present our results for different boundary conditions with a
discussion of how the loss function and quantum fidelity of the model is
affected. In Section 4, we conclude the paper with a brief discussion of our
results, an investigation into the conditions that cause the Born machine to
fail, and an explanation of how these obstacles can be overcome by
generalizing the algorithm to a complex-valued basis-enhanced Born machine.

##  2 Open and periodic Born machines

A generative model has two distinct objectives: 1) to learn some joint
probability distribution from training data, and 2) to subsequently generate
new data from the trained model. The Born machine is a kind of generative
machine learning model with a quantum-inspired framework. Rather than directly
modeling the joint probability distribution, the Born machine models the
quantum wave function $\Psi$, or more generally, the probability amplitude
distribution [14, 15]. To differentiate the actual quantum state $\Psi$ and
the estimated quantum state modeled by the Born machine, we denote our trained
model with a hat: $\hat{\Psi}$. One can arrive at the probability distribution
of the model through the probabilistic interpretation of quantum mechanics,
also known as Born’s rule:

$\displaystyle\hat{\mathbbm{P}}(\mathbf{v})=\frac{\lvert\hat{\Psi}(\mathbf{v})\rvert^{2}}{\sum_{\mathbf{v}\in\mathcal{V}}{\lvert\hat{\Psi}(\mathbf{v})\rvert^{2}}}$
(2)

Here, $\mathcal{V}$ refers to the entire Hilbert space. Training occurs
through the minimization of the negative log likelihood (NLL), defined below:

$\displaystyle\mathcal{L}=-\frac{1}{\lvert\mathcal{T}\rvert}\sum_{\mathbf{v}\in\mathcal{T}}\ln\hat{\mathbbm{P}}(\mathbf{v})$
(3)

In the above expression, $\hat{\mathbbm{P}}(\mathbf{v})$ is the probability
distribution of our model, defined in Eq. (2), and $\mathcal{T}$ represents
the set of training data. Minimization of the NLL will maximize the model’s
probability of configurations $\mathbf{v}$ that appear in the training data.
If the probability distribution $\hat{\mathbbm{P}}(\mathbf{v})$ perfectly
captures the probability distribution of the training data
$\mathbbm{P}(\mathbf{v})$, then the NLL will approach the Shannon entropy of
the training data $\mathcal{T}$:
$S=-\sum_{\mathbf{v}\in\mathcal{T}}{\mathbbm{P}}(\mathbf{v})\ln({\mathbbm{P}}(\mathbf{v}))$.

The matrix product state (MPS) is a natural and efficient way of classically
representing the quantum wave function of a one-dimensional chain of atoms:

$\displaystyle\ {\hat{\Psi}}$ $\displaystyle=$
$\displaystyle\sum\limits_{D_{1}=1}^{D}\cdots\sum\limits_{D_{N-1}=1}^{D}{M_{1,i_{1}}^{D_{1}}\,M_{2,i_{2}}^{D_{1},D_{2}}\,\cdots\,M_{N,i_{N}}^{D_{N-1}}}$
(4) $\displaystyle=$
$\displaystyle\sum\limits_{\mathbf{v}=1}^{2^{N}}\hat{\Psi}(\mathbf{v})\lvert\mathbf{v}\rangle$
(5)

The tensors listed in Eq. (4) can be contracted along each interior bond
dimension $D$ to produce a single rank-N tensor, which can be cumbersomely
expressed with $2^{N}$ parameters $\hat{\Psi}(\mathbf{v})$ in the
computational basis – see Eq. (5). Here, $\mathbf{v}$ again represents a
single configuration in the computational basis
($\lvert\mathbf{v}\rangle=\lvert\lambda_{1},\lambda_{2},\cdots,\lambda_{N}\rangle,\lambda_{j}\in\\{0,1\\})$.

It is known that the expressive power of matrix product states arises from the
amount of entanglement encoded in their bond dimensions, which depends on the
particular architecture. Taking advantage of this fact, we use an MPS as the
ansatz for the model with one tensor for each atomic site. As shown in Fig.
1(a), our selected MPS architecture involves an array of rank-3 tensors
connected to each other via a virtual bond dimension of size $D$, while the
dangling bonds indicated by $i$ are spin indices. The parameters trained by
the Born machine are simply the elements of each tensor in the MPS. For
simplicity, we use the same bond dimension $D$ between each tensor and select
$D$ prior to learning (although in theory each bond dimension could be
adaptively learned as well).

We employ two subtly distinct MPS architectures. The first involves an open
boundary condition and is referred to as the Open Born machine. In this MPS
architecture, the first and last tensors have only two indices: one spin index
and one bond dimension linking the tensor to the rest of the chain. The second
MPS architecture involves a periodic boundary condition and is aptly named the
Periodic Born machine. In this case, each tensor has three indices: one spin
index and two bond dimensions linking the tensor to its neighbors on either
side. A bond dimension is added between the first and last tensor to complete
the ring.

##  3 Learning a one dimensional XY quantum spin chain

A necessary ingredient to any machine learning algorithm – quantum or
classical – is data. As a proof of concept, in this work we assembled training
data by sampling from DMRG simulations (see Fig. 3). DMRG has proven to be an
effective tool for accurately calculating ground state properties of quantum
spin chains, including quantum correlations [6]. A variety of points across
the phase diagram were selected for study: the Ising critical point
($\gamma=1,h=1$), a point from the ordered phase ($\gamma=1.5,h=0.5$), a point
from the disordered phase ($\gamma=2,h=2$), and a point from the oscillatory
region ($\gamma=0.5,h=0.5$). DMRG was used to calculate the ground state at
these points of interest for chain lengths of $N=$ 13, 16, and 19 atoms. Both
open and periodic boundary conditions were considered for comparison (OBC and
PBC, respectively – note that the first sum in Eq. (1) runs from $i=1$ to $N$
for the case of PBC). The MPS output by DMRG can be efficiently sampled in the
computational basis according to the probability distribution of the quantum
state, effectively simulating data that might be acquired through measurement.

Once the training data is produced, it can be fed into either Born machine to
train an MPS model of the state. We used PyTorch’s Adam optimizer to train the
model with 200 samples per batch and 20 epochs [17]. The final MPS can be
sampled efficiently, generating new data that can be compared to the original
training data [14, 18]. Overlapping probability distributions of the training
data and data sampled from the trained model (Fig. 4(a)) provide visual
confirmation that the model can produce data that mimics state measurement.
Note that in this figure, data from the $x$\- and $y$-bases are included,
although only $z$-basis data was used for training. There is high overlap in
each basis, indicating that the full quantum state has been learned. It is not
generally true that a single-basis Born machine can learn the full quantum
state. It has been shown that the MPS based Born machine utilizing gradient
based schemes for optimization fails to learn the parity data obtained from
GHZ state, which is considered to be nonlocal quantum data [18]. We will
further discuss this in Section 4.

Figure 3: General procedure for generating training data for learning and
generating new results from the trained model.

(a)

(b)

Figure 4: (a) Overlaid probability distributions of training data and data
generated from the trained MPS at the critical Ising point, for a chain of
$N=13$ atoms with open boundary conditions. The Open Born machine was used for
training with $\lvert\mathcal{T}\rvert=30000$ and bond dimension $D=4$. (b)
Top to bottom: The NLL loss function calculated during training compared with
the Shannon entropy of the training data, indicated by $S(\mathcal{T})$. The
infidelity $(1-F)$ between the model and the quantum state during training.
Only data in the computational basis (that is, the Z configuration) is
included in $\mathcal{T}$, indicating the high training power of the Born
machine for this model. Correlation function $G(r)$ of the MPS after training,
plotted over the correlation function of the DMRG simulations. The correlation
length $\xi$ of the model during training plotted alongside the “target”
correlation length of the DMRG results to highlight the power of Born machine
in capturing the long range ordering at the quantum critical point.

To further quantify how well the trained MPS captures the quantum state, we
examined the loss function during training as well as the quantum fidelity
between our model and the quantum state of the system
$F=\lvert\langle\Psi|\hat{\Psi}\rangle\rvert^{2}$, which can be calculated
exactly by taking the inner product of the trained MPS model and the MPS
output of DMRG simulation. This would be a daunting task if measurement data
was our only resource, as full state tomography would be required to obtain
the quantum state. We include the quantum fidelity here in order to emphasize
the power of the Born machine. In addition to these metrics, we investigated
how well the Born machine is able to capture an observable – in particular,
the correlation function $G(r)$. $G(r)$ is known to provide insightful
information about the underlying phase of matter. It is defined as:

$G(r)=\sum_{i}\frac{\langle\sigma^{z}_{i}\sigma^{z}_{i+r}\rangle-\langle\sigma^{z}_{i}\rangle\langle\sigma^{z}_{i+r}\rangle}{N_{p}}$
(6)

Here, $\sigma_{z}^{i}$ denotes the Pauli $z$ operator acting on the $i^{th}$
site, and the normalization factor $N_{p}$ is the number of distinct pairs of
atoms in the chain separated by $r$. One expects that the correlation function
will decay exponentially with distance, e.g: $G(r)=A\,e^{-r/\xi}$, where $\xi$
represents the correlation length. $\xi$ is known to provide information about
the underlying quantum phase; for example, due to the long range ordering
close to a critical point, the correlation function exhibits a power law decay
and the corresponding correlation length is expected to be larger.

(a)

(b)

(c)

Figure 5: (a) Loss and fidelity during training of the Ising critical point
for a chain of 13 atoms with $\lvert\mathcal{T}\rvert=10000$ and bond
dimension $D=2$. (b) Same as (a), with bond dimension increased to $D=4$. (c)
Same as (a), with chain increased to 19 atoms and training data increased to
$\lvert\mathcal{T}\rvert=30000$. The loss and fidelity data presented here has
been smoothed for clarity of comparison.

We first examine the power of the Born machine in learning the quantum
critical point. We feed data from the ground state of the transverse field
Ising quantum spin chain with length $N=13$ at the critical point
($\gamma=1,h=1$), simulated by DMRG. It has been shown that due to the long
range ordering at a critical point, both the Boltzmann machine and Born
Machine based on projected entangled pair state (PEPS) have failed to learn
the 2-dimensional Ising model [19, 20]. To our surprise, the MPS-based Born
Machine was able to successfully capture the state at the critical point. In
Fig. 4(b) we present the training power of the MPS-based Born machine for a
variety of quantities, including the correlation function and correlation
length. The loss approaches the Shannon entropy of the training data,
indicating that the Born machine was able to fully learn and memorize the
data. This is perhaps more evident from the fidelity, which approaches $>99\%$
at the end of the training period. Furthermore, the local Pauli operators
($\sigma_{x}$ being the order parameter) were also successfully learned as the
generated data has good overlap with the original data. We further demonstrate
the power of the Born machine in learning the correlation function. The bottom
left panels of Fig. 4(b) indicate that the MPS Born machine is able to capture
the long range correlations at the quantum critical point. Additional results
for different points across the phase diagram are provided in Fig. 6.

(a)

(b)

Figure 6: (a) Top two panels: overlaid probability distributions of training
data and data generated from the trained MPS at the point $\gamma=1.5,h=0.5$
from the ordered phase, for a chain of $N=13$ atoms with open boundary
conditions. The Open Born machine was used for training with
$\lvert\mathcal{T}\rvert=30000$ and bond dimension $D=4$. The infidelity
$(1-F)$ between the model and the quantum state during training. Correlation
function $G(r)$ of the MPS after training, plotted over the correlation
function of the DMRG simulations. (b) Top two panels: overlaid probability
distributions of training data and data generated from the trained MPS at the
point $\gamma=2,h=2$ from the disordered phase, for a chain of $N=13$ atoms
with open boundary conditions. The Open Born machine was used for training
with $\lvert\mathcal{T}\rvert=30000$ and bond dimension $D=4$. The infidelity
$(1-F)$ between the model and the quantum state during training. Correlation
function $G(r)$ of the MPS after training, plotted over the correlation
function of the DMRG simulations.

While it is more natural to consider an open boundary condition, a periodic
boundary condition allows translational invariance in the physical system,
leading to an advantage in the mathematical description and a solution for
quantum spin chains. Consequently, the boundary condition imposes different
behaviour in physical quantities as well. In order to better address the
learnability of the boundary condition and uncover any advantage in using one
Born machine architecture over the other, here we present all combinations of
boundary conditions between training data and Born machine architecture.

(a)

(b)

Figure 7: (a) Overlaid probability distributions of training data and data sampled from the trained MPS at the point $h=0.5,\gamma=0.5$ for a chain of 13 atoms with open boundary conditions. The Open Born machine was used for training with $\lvert\mathcal{T}\rvert=30000$ and bond dimension $D=4$. (b) Top to bottom: Loss of the model during training with the Shannon entropy of the training data $S(T)$. Fidelity between the model and the quantum state during training. Correlation function $G(r)$ of the model after training, plotted over the correlation function of the DMRG results. The correlation length $\xi$ of the model during training, plotted alongside the “target” correlation length of the DMRG results. Model | Data | $D$ | $\Omega$ | $F$
---|---|---|---|---
Open | Open | 2 | 96 | 0.9888
3 | 210 | 0.9924
4 | 368 | 0.9936
6 | 816 | 0.9944
8 | 1440 | 0.9944
10 | 2240 | 0.9946
Open | Periodic | 2 | 96 | 0.8958
3 | 210 | 0.9491
4 | 368 | 0.9808
6 | 816 | 0.9909
8 | 1440 | 0.9908
10 | 2240 | 0.9910
Periodic | Open | 2 | 104 | 0.9922
3 | 234 | 0.9928
4 | 416 | 0.9935
6 | 936 | 0.9949
8 | 1664 | 0.9944
10 | 2600 | 0.9935
Periodic | Periodic | 2 | 104 | 0.9882
3 | 234 | 0.9948
4 | 416 | 0.9961
6 | 936 | 0.9964
8 | 1664 | 0.9966
10 | 2600 | 0.9964
Table 1: Fidelity after training the Ising critical point for a chain of 13
atoms with $|\mathcal{T}|=10000$. Various combinations of model architecture,
data boundary condition, and bond dimension are provided. Bond dimension and
model architecture fix the total number of model parameters, $\Omega$.

Fig. 5 provides a few examples of learning the Ising critical point for
different bond dimensions $D$ and atom chain lengths $N$. Our computations
indicate that in general, if enough training data is supplied and a large
enough bond dimension is selected, each Born machine architecture is
successful in learning from either OBC or PBC data. Perhaps unsurprisingly, in
Fig. 5(a), for a spin chain of $N=13$ and with only 10000 samples, we see that
the Periodic Born machine performs considerably better than the Open Born
machine when learning from PBC data. One might attribute this result to the
larger expressibility of the Periodic Born machine due to its increased number
of model parameters. However, this explanation seems insufficient when one
considers that the infidelity was decreased by an order of magnitude with the
addition of only eight ($=2i(D^{2}-D)$) parameters. It is the strategic
location of these parameters in the architecture as a whole, greatly expanding
the connectivity and entanglement-encoding potential of the MPS, that
contributes to the success of the Periodic Born machine. This is further
explored in Table 1, which lists the final fidelities for different
combinations of model architecture and data boundary condition for a variety
of bond dimensions $D$. The exact number of model parameters, $\Omega$, is
provided for clarity. Notice the success of the Periodic Born machine in
learning from PBC data; the quantum fidelity reaches $>99\%$ for $D=3$
($\Omega=234$). In order for the Open Born machine to reach this level of
quantum fidelity, a bond dimension of at least 6 is required ($\Omega=816$).
Thus, one can attribute the success of the Periodic Born machine to the
architecture of the model – the physical placement of the model parameters –
rather than their total number. In all situations, the fidelity reaches
$>99\%$ well before $D=10$ with little improvement beyond that threshold.

In Fig. 5(b), the bond dimension was increased from $2$ to $4$. The Open Born
machine was more successful in learning PBC data with more parameters in the
model, though it was still out-performed by the Periodic Born machine.
Finally, in Fig. 5(c), we consider the scenario where there is limited data
available (relative to the size of the Hilbert space, as $30000\ll 2^{N}$ for
$N=19$) and a small bond dimension has been selected for the model. In this
situation, interestingly, the advantage of matching the boundary conditions of
the model to those of the data is apparent: for both OBC and PBC data, the
opposite model is unable to minimize the loss function and learn the state,
while the “matching” model can learn the state with fidelities $>90\%$ –
despite the few number of parameters and small size of the training data.

##  4 Discussion and future directions

The transverse field XY Hamiltonian is a simple model manifesting two distinct
quantum phase transitions. The exact mathematical solution of this model has
provided a powerful tool to investigate a wide variety of observables, making
the XY model very valuable in understanding many aspects of quantum phase
transitions and in benchmarking. In this paper, we investigated the power of
the MPS-based Born machine in learning across the quantum phase transition
from disorder to order (see Fig. 6), as well as at critical point, (see Fig.
4). Surprisingly, our simple Born machine was able to learn the quantum states
across different phases and even at critical point despite the long range
ordering. We also investigated the learnabilty of the periodic and open Born
machines when different boundary conditions are imposed on the training data.
Our numerical results indicate that when given enough sampling data and a
large enough bond dimension, the Periodic Born machine has a considerably
better performance in learning from training data with a periodic boundary
condition, and comparable performance in learning from training data with an
open boundary condition. This can partially be attributed to the extra
parameters provided by the extra bond dimension, but is largely due to the
enhanced connectivity that the strategically placed extra bond dimension
provides. However, for limited amounts of training data and a small bond
dimension, the Born machine becomes sensitive to the original boundary imposed
on the quantum data, and the matching Born machine performs better than the
opposite architecture, as shown in Fig. 5(c).

Finally, although many points on the phase diagram can be learned to a high
degree of accuracy by the Born machine (as evidenced by Fig. 4), this is not
true of every point. In particular, the Born machine is less successful in
learning points from the oscillatory region of the phase diagram, where the
correlation function shows oscillatory behavior. This region is highlighted in
blue in Fig. 2. See Fig. 7 for an example of learning the point
$h=0.5,\gamma=0.5$. The $z$-basis probability distribution associated with
this point is spread across most of the Hilbert space, resulting in a large
value for the Shannon entropy. This is an obstacle to learning – the loss does
not reach the global minimum, and although some aspects of the state can be
captured by the model (such as the correlation function and correlation
length), the full quantum state has not been learned faithfully. It is
possible to learn such states by incorporating training data from a different
basis with a lower Shannon entropy (the $x$-basis, for this particular phase
of the system) and introducing complex parameters to increase the
expressibility of the model, but this discussion will be left for a future
study.

## Acknowledgments and disclosure of funding

A.M.G acknowledges support from the NSF through the Graduate Research
Fellowships Program, as well as support through the Theodore H. Ashford
Fellowships in the Sciences. S.F.Y would like to acknowledge funding by the
NSF and DOE.

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# Slow light with electromagnetically induced transparency in cylindrical
waveguide

Agus Muhamad Hatta<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, Jazan University, Jazan, Saudi Arabia  Department of Engineering
Physics, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia  Ali A.
Kamli  Department of Physics, Faculty of Science, Jazan University, Jazan,
Saudi Arabia  Ola A. Al-Hagan  Department of Physics, Faculty of Science, King
Khalid University, Saudi Arabia  Sergey A. Moiseev  Kazan Quantum Center,
Kazan National Research Technical University, Kazan, Russia

###### Abstract

Slow light with electromagnetically induced transparency (EIT) in the core of
cylindrical waveguide (CW) for an optical fiber system containing three-level
atoms is investigated. The CW modes are treated in the weakly guiding
approximation which renders the analysis into manageable form. The
transparency window and permittivity profile of the waveguide due to the
strong pump field in the EIT scheme is calculated. For a specific permittivity
profile of the waveguide due to EIT, the propagation constant of the weak
signal field and spatial shape of fundamental guided mode are calculated by
solving the vector wave equation using the finite difference method. It is
found that the transparency window and slow light field can be controlled via
the CW parameters. The reduced group velocity of slow light in this
configuration is useful for many technological applications such as optical
memories, effective control of single photon fields, optical buffer and delay
line.

###### pacs:

##  I Introduction

Electromagnetically induced transparency (EIT) is a quantum interference
phenomenon in which the absorption of a medium can be highly reduced within a
frequency band of transparency window pt-1997 . In EIT system, a three level
atomic system is considered where there are two fields, namely a strong
control pump field and a weak signal field, interact with the medium. The pump
coherent field changes the refractive index of the medium through which the
weak signal field propagates and this leads to low absorption of the signal
field. In addition, within the transparency window, a dispersion of the medium
varies rapidly and it induces significantly reduced group velocity which leads
to slow light n-1999 . Slow light in a medium can be utilized for optical
signal processing such as optical buffer, data storage, delay lines opn-2006 .
Due to the capability to change the susceptibility or refractive index, the
EIT scheme has been also explored for several applications such as an all-
optical tunable mirror, a wavelength filter, a wavelength monitor, an all-
optical switch, a quantum surface plasmon resonance system, and etc. pier-2008
; apa-2012 .

In terms of the medium’s field propagation, as an advancement to the bulk
medium pt-1997 , EIT in cavity and waveguides have been investigated ol-1998 ;
pra-2013 . EIT in cavity can yield very strong pulling of combined cavity-atom
resonance frequency towards the EIT resonance frequency ol-1998 . EIT in
planar waveguide has been investigated as jmse-2011 . In prl-2008 , EIT medium
was proposed at an interface between two materials for an all-optical control
of surface polaritons. EIT inside the core of cylindrical waveguide with metal
or metamaterial cladding has been considered for possible plasmonic devices
applications pra-2013 . These authors considered slow light for TM surface
modes that occur in cylindrical waveguides (with metal or metamaterial
cladding), with reduced loss for the metamaterial cladding.

In an optical fiber system, the slow light of the fundamental guided mode can
be realized using several schemes such as: a birefringent optical fiber and
other standard telecom components prl-2004 , stimulated Brillouin scattering
oe-2005-1 , and Raman assisted fiber optical parametric amplifier oe-2005-2 .
Using EIT based slow light in the optical fibers as well as in the waveguides
promises to be very useful for deterministic control of the light fields and
as a basic building block for fiber-optic communication and convenient tool
for the light-atoms interactions enhancement. In this paper, we focus on the
effect of cylindrical waveguide (CW) on the properties of EIT and the slow
light due to spatial inhomogeneous pump coherent field and transverse spatial
confinement of the resonant three-level atomic system in the optical fiber.
Unlike the work in pra-2013 , our work here is concerned with EIT and slow
light when both the pump and signal fields are in the fundamental guided mode
of CW with dielectric core containing resonant three-level atoms and
dielectric cladding rather than surface modes and metal or metamaterial
claddings. For our analysis we adopt the weakly guiding approximation kk-2001
where the difference in the refractive indices of core and caldding is assumed
very small. This is true in many practical fiber optical applications where
the refractive index mismatch is very small. This approximation is useful and
reduces the mathematical analysis to more manageable forms. Our focus in this
work is on the optimal conditions for the transverse light field confinement
in the system under consideration to realize enhanced light-atom interaction
that could be easily integrated with conventional optical fiber network. After
this introduction we present the model and basic formulation and equations.
Results and discussions are given in the following section and the paper
concludes with summary of main results.

(a)

(b)

(c)

Figure 1:  (a) A schematic of the three-level $\Lambda$ atom, (b) a schematic
ofcylindrical waveguide (CW) where the core embedded with the three level
$\Lambda$ atom, (c) Pump and signal fields are injected to the optical fiber
system

##  II Formulations and Equations

A three level atomic system in the $\Lambda$ configuration is considered as
shown in Fig. 1(a). A weak signal electric field ${\bf{E}}_{\text{s}}$
interacts with the atoms on the $|\text{g}\rangle$ $\leftrightarrow$
$|\text{e}\rangle$ transition, and a strong pump coherent electric field
${\bf{E}}_{\text{p}}$ drives the atomic transition $|\text{s}\rangle$
$\leftrightarrow$ $|\text{e}\rangle$ with Rabi frequency $\Omega_{\text{p}}$.
The three-level $\Lambda$ atom system is embedded in the dielectric core of CW
with a surrounding dielectric cladding as depicted in Fig. 1(b). The two
fields of pump and signal are injected via an optical fiber system where the
fields are coupled into an input singlemode fiber (SMF) as shown in Fig. 1(c).
The input SMF is aligned to the CW and again is directed along CW axis to the
output SMF to be used for any further signal processing applications in the
optical fiber system.

In the optical fiber and CW, the core has radius $a$ and permittivity
$\epsilon_{1}$, and is surrounded by a dielectric cladding of permittivity
$\epsilon_{2}$. The electric field in the optical fiber and CW is expressed in
cylindrical coordinates as kk-2001

$\displaystyle{\widetilde{{\bf{E}}}}(r,\phi,z,t)={\bf{E}}(r,\phi)e^{j(\omega
t-\beta z)}$ (1)

where $\omega$ is an angular frequency of the field and $\beta$ is the
propagation constant to be determined by solving a dispersion relation. In
this paper, we are concerned with CW fundamental mode. In the weakly guiding
approximation where the difference in the refractive indices for the core and
cladding is very small the fundamental mode is designated HE11 mode or LP01
kk-2001 . The dispersion relation for HE11 mode (with the undoped core i.e. in
the absence of the three level atoms) is determined by solving the expression
kk-2001

$\displaystyle\frac{J_{0}(u)}{uJ_{1}(u)}=\frac{K_{0}(\text{w})}{\text{w}K_{1}(\text{w})}$
(2)

where $J_{i}(u)$, $K_{i}(\text{w})$ are i${}^{\text{th}}$-order of first kind
of Bessel functions and first kind of modified Bessel functions, respectively
kk-2001 . The parameters $u^{2}$ and $\text{w}^{2}$ are defined as

$\displaystyle u^{2}=a^{2}(k_{0}^{2}\epsilon_{1}-\beta^{2})$ (3)
$\displaystyle\text{w}^{2}=a^{2}(\beta-k_{0}^{2}\epsilon_{2})$ (4)

and related by $u^{2}+\text{w}^{2}=V^{2}$, where
$V=k_{0}a\sqrt{\epsilon_{1}-\epsilon_{2}}$ is a normalized frequency,
$k_{0}=\omega/c$ is the free space wavenumber with $c$ is the speed of light
in vacuum, and $\omega$ is the field frequency. The electric field is most
conveniently expressed in terms of Cartesian coordinates. The field profile of
the HE11 mode in the transverse component is assumed to be linearly polarised
and can be expressed by kk-2001

$\displaystyle{E_{x}(r)}$
$\displaystyle=\begin{cases}\frac{E_{0}}{J_{0}(u)}J_{0}\left(\frac{ur}{a}\right),&\text{
$r\leqslant a$}\\\
\frac{E_{0}}{K_{0}(\text{w})}K_{0}\left(\frac{\text{w}r}{a}\right),&\text{
$r>a$}\\\ \end{cases}$ (5)

where
$E_{0}=\frac{\text{w}}{V}\frac{J_{0}(u)}{J_{1}(u)}\left(\frac{2z_{0}}{\pi
a^{2}\epsilon_{2}^{1/2}}\right)^{1/2}$ is an amplitude coefficient and $z_{0}$
is a vacuum impedance.

Figure 2:  (Colour online) Dispersion relation of cylindrical or fiber
waveguide as a normalized frequency varied with the normalized effective
index.

(a)

(b)

Figure 3:  (Colour online) EIT transparency window for (a) variation of the
amplitude of pump field of $\Omega_{0}$ with a core radius $a$ = 4.5
$\lambda_{p}$, (b) variation of core radii with a fixed $\Omega_{0}=10\times
10^{4}$ $\text{s}^{-1}$

Equation (2) is solved numerically for different waveguide structure
parameters and frequency. The dispersion relation can be plotted as the
normalized frequency $V$ versus a normalized effective index
$b=\frac{(\beta/k_{o})^{2}-\epsilon_{2}}{\epsilon_{1}-\epsilon_{2}}$. Figure 2
shows the dispersion relation for the fundamental guided mode of the
cylindrical waveguide in the weakly guiding approximation. It should be noted
that for the fundamental mode, the waveguide structure parameters and
frequency should satisfy the condition such that the normalized frequency is
below a cutoff $J_{0}(V)=0$ or $V=2.4048$ kk-2001 . In this paper, the fiber
and cylindrical waveguide with undoped core are designed to propagate the
fundamental mode only.

So far we have been discussing the undoped weveguide structure. Now, we dope
the waveguide core with three level-atoms in the $\Lambda$ configuration which
will induce EIT effects. The presence of three-level atoms inside the core
will lead to modifications in the core permittivity and is expected to lead to
interesting features which we like to explore next. A procedure to model EIT
in the waveguide has been presented in refs. jmse-2011 ; pra-2013 , and it is
adapted in this formulation. The susceptibility of the undoped core is
$\chi_{0}=\epsilon_{1}/\epsilon_{0}-1$. The presence of the pumped three-level
$\Lambda$ atoms under EIT in the core changes the susceptibility into
$\chi_{0}+\chi(\omega,r)$. The susceptibility $\chi(\omega,r)$ can be written
as sp-2007

$\displaystyle\chi(\omega,r)=\frac{2a_{0}}{n_{\text{d}}\omega}\frac{i\gamma_{\text{eg}}}{\gamma_{\text{eg}}-i\Delta+|\Omega_{\text{p}}(r)|^{2}(\gamma_{\text{sg}}-i\Delta_{\text{R}})^{-1}},$
(6)

where $n_{\text{d}}=\sqrt{\epsilon_{1}\mu_{1}}$ is the refractive index of the
core with $\mu_{1}$ the permeability of the core, $\gamma_{ij}$ is the atomic
decay rate from level $i$ to level $j$, and

$\displaystyle
a_{0}=\frac{3\pi}{n_{\text{d}}^{2}\omega^{2}}\rho_{\text{a}}\approx\frac{3\pi}{n_{\text{d}}^{2}\omega_{\text{eg}}^{2}}\rho_{\text{a}}$
(7)

with $\rho_{\text{a}}$ the number density of three-level atoms, and
$\omega_{\text{ij}}$ is the transition frequency for the $|\text{i}\rangle$
$\leftrightarrow$ $|\text{j}\rangle$ transition. The detunings are
$\Delta=\omega_{\text{eg}}-\omega_{s}$,
$\Delta_{\text{R}}=\Delta-\omega_{\text{es}}+\omega_{\text{p}}$. The signal
field frequency is $\omega_{\text{s}}$ and pump field has frequency
$\omega_{\text{p}}$ and is set on resonance with the $|\text{s}\rangle$
$\leftrightarrow$ $|\text{e}\rangle$ transition such that
$\Delta_{\text{R}}=\Delta$. The Rabi frequency of the pump is

$\displaystyle\Omega_{\text{p}}(r)=\frac{1}{\hbar}{\bf
d}\cdot{\bf{E}}_{\text{p}}(r),$ (8)

where ${\bf d}$ is the dipole moment of the $|\text{s}\rangle$
$\leftrightarrow$ $|\text{e}\rangle$ transition, and ${\bf{E}}_{\text{p}}(r)$
is the pump field given by expression (5). For a given pump field in the core,
the Rabi frequency of the pump field becomes

$\displaystyle\Omega_{\text{p}}(r)=\Omega_{0}J_{0}\left(\frac{u_{p}r}{a}\right),$
(9)

where $\Omega_{0}=\frac{E_{0}d}{J_{0}(u_{\text{p}})\hbar}$ is the amplitude of
the pump field with ${d=\bf d\cdot\hat{r}}$ lies along the field direction,
$u_{\text{p}}^{2}=a^{2}\left(k_{\text{es}}^{2}\epsilon_{1}-\beta^{2}\right)$,
and $k_{\text{es}}=\omega_{\text{es}}/c$.

The change of core susceptibility due to three level atoms will modify the
dispersion and absorption coefficient $(\alpha)$ of the doped core in a manner
that depends on the pump field, and can be calculated from the real and
imaginary parts of the $\chi(\omega,r)$, respectively. The permittivity of the
waveguide with three-level $\Lambda$ atoms embedded in the core becomes

$\displaystyle\epsilon_{\text{eff}}\left(\omega,r\right)$
$\displaystyle=\begin{cases}\epsilon_{0}\left[1+\chi_{0}+\chi\left(\omega,r\right)\right],&\text{
$r<a$}\\\ \epsilon_{2},&\text{ $r\geqslant a$.}\\\ \end{cases}$ (10)

The weak signal field ${\bf{E}}_{\text{s}}$ propagates through the waveguide
with an appropriate permittivity profile of the cylindrical waveguide due to
the EIT. Since the permittivity profile at the core has non-step index
profile, the eqs. (2) and (5) can not be used to solve the propagation
constant of the signal field. Therefore, a vector wave equation should be
solved to obtain the propagation constant. It is convenient to write the
transverse component of the signal field and the permittivity profile in the
Cartesian coordinate as ${\bf{E}}_{\text{s}}(x,y)$ and
$\epsilon_{\text{eff}}(\omega,x,y)$, respectively; and the vector wave
equation can be written as

$\displaystyle\bigtriangledown^{2}{\bf{E}}_{\text{s}}+\bigtriangledown\left(\frac{1}{\epsilon_{\text{eff}}(\omega,x,y)}\bigtriangledown\left(\epsilon_{\text{eff}}(\omega,x,y)\cdot{\bf{E}}_{\text{s}}\right)\right)+\epsilon_{\text{eff}}(\omega,x,y)k_{0}^{2}{\bf{E}}_{s}=\beta^{2}{\bf{E}}_{\text{s}}.$
(11)

A finite difference method (FDM) can be used to solve (11) and obtain the
propagation constant $\beta$ and the corresponding field profile of the signal
field oe-2002 . With the propagation constant determined from equation (11)
for a frequency range, the group velocity can be calculated using

$\displaystyle
v_{\text{g}}=\left(\frac{\text{d}\beta}{\text{d}\omega}\right)^{-1}.$ (12)

It is clear that the weak signal propagation constant $\beta$ and hence group
velocity $v_{\text{g}}$ depend on the new core susceptibility due the
interaction of the three level atoms with signal and pump fields. The wave
equation above will be solved numerically using the FDM and results will be
presented in the next section.

(a)

(b)

Figure 4:  (Colour online) (a) The dispersion for various core radii at a
fixed $\Omega_{0}=10\times 10^{4}$ $\text{s}^{-1}$, (b) the permittivity
profile for $\Delta/\gamma_{eg}=-0.005,0,005$, at $\Omega_{0}=10\times 10^{4}$
$\text{s}^{-1}$ and the core radius $a=3\lambda_{p}$

##  III Results and Discussions

For the three-level atoms embedded in the core, we select a material system of
Pr3+ ions embedded in bulk Y2SiO5 crystal and under cryogenic conditions as in
ref pra-2013 . To calculate the absorption coefficient, dispersion, and
permittivity profile, we use some parameters as follows: the core and cladding
parameters are $\epsilon_{1}=2.1037\epsilon_{0}$,
$\epsilon_{2}=2.0872\epsilon_{0}$, the core radius $a$ which is comparable to
$\lambda_{\text{p}}=2\pi c/\omega_{\text{p}}=0.336$ $\mu\text{m}$. For the
three-level atoms embedded in the core, we set the number of density
$\rho_{\text{a}}=1.26\times 10^{15}$ $\text{cm}^{-3}$. The decay rates are
$\gamma_{\text{eg}}=1\times 10^{5}$ s-1, $\gamma_{\text{sg}}=5\times 10^{2}$
s-1. The pump and signal frequencies are
$\omega_{\text{p}}=\omega_{\text{es}}$ and
$\omega_{\text{s}}=\omega_{\text{eg}}$, respectively.

Figure 5:  (Colour online) Normalized signal field profiles without- and with-
EIT.

(a)

(b)

Figure 6:  (Colour online) (a) The dispersion relation of signal field for the
core radius $a=3\lambda_{p}$, $\Omega_{0}=1\times 10^{4}$ $\text{s}^{-1},$ and
$1\times 10^{5}$ $\text{s}^{-1}$ , (b) the group velocity as a function of
$\Omega_{0}$ for three core radii $a=1.5\lambda_{p},3\lambda_{p},$ and
$4.5\lambda_{p}.$

For a specific Rabi frequency of the pump field as in equation (9), the
absorption spectrum $\alpha/\alpha_{0}$ can be obtained by calculating the
imaginary part of (6). Figure 3 shows the EIT transparency window at the
centre of the core embedded with the three level $\Lambda$ atom as a function
of the normalized detuning. In Fig. 3(a), the interaction with the pump field
results in a splitting of the absorption spectrum into two peaks, and the line
centre of the medium becomes transparent to the resonant signal field. For a
fixed core radius $a=4.5\lambda_{\text{p}}$, the transparency window can be
broadened by increasing the amplitude of pump field $\Omega_{0}$ as well known
in EIT. The transparency window can also be increased by reducing the core
size as depicted in Fig. 3(b), for the core radii
$a=4.5\lambda_{\text{p}},3\lambda_{\text{p}},$ and $1.5\lambda_{\text{p}},$
and a fixed $\Omega_{0}=10\times 10^{4}$ $\text{s}^{-1}$.

Figure 4(a) shows the dispersion given by the real part of $\chi(\omega,r)$
for several core radii at fixed $\Omega_{0}$ and the same parameters as in
Fig. 3(b). One can see the dispersion yields a steep and approximately linear
slope at the transparency window near the resonance frequency $\Delta=0.$ The
dispersion depends on the core radius while increasing the core radius
increases the slope gradient. Figure 4(b) shows the permittivity profile for
several normalized detunings $\Delta/\gamma_{\text{eg}}=-0.005,0,0.005$ at
$\Omega_{0}=10\times 10^{4}$ $\text{s}^{-1}$ and the core radius
$a=3\lambda_{\text{p}}$. For the normalized detuning
$\Delta/\gamma_{\text{eg}}=0$, the core permittivity has a step function
profile , whereas for non-zero detunings $\Delta/\gamma_{\text{eg}}\neq 0$,
the core permittivity shows a graded profile.

When the signal field propagates through a given permittivity profile as in
Fig. 4(b), the field profile and propagation constant of the signal field can
be evaluated by solving equation (11) using the FDM. Figure 5 shows the
normalized signal field profiles without and with EIT. One can see that for
the fundamental mode, the signal field profile without EIT, calculated using
the weakly guiding approximation as given by equation (5) and the field
obtained from more exact FDM are in a good agreement. For the EIT condition at
$\Delta/\gamma_{\text{eg}}=0.005$, the permittivity in the core becomes higher
and the graded profile is as shown previously in Fig. 4(b), and therefore the
signal field is more confined inside the core compared to the condition
without-EIT.

Figure 6(a) shows the calculated propagation constant within a range of
normalized detuning for the core radius $a=3\lambda_{\text{p}}$,
$\Omega_{0}=1\times 10^{4}$ $\text{s}^{-1},$ and $1\times 10^{5}$
$\text{s}^{-1}$. The propagation constant varies rapidly near the resonance,
with steeper slope for lower Rabi frequency $\Omega_{0}$. Once the dispersion
profile obtained, the group velocity can be computed using equation (12) and
it is depicted in Fig. 6(b). The group velocity is displayed as a function of
$\Omega_{0}$ for three core radii
$a=1.5\lambda_{\text{p}},3\lambda_{\text{p}},$ and $4.5\lambda_{\text{p}}$.
The group velocity can be controlled and slowed down by reducing the Rabi
frequency $\Omega_{0}$. However, for a smaller $\Omega_{0}$, the absorption
coefficient is higher at the transparency window as shown in Fig. 3(a). The
group velocity also depends on the core radius, it decreases as the core
radius gets larger.

Thus, the slow light of signal field in cylindrical waveguide can be
controlled by adjusting the amplitude of pump field of $\Omega_{0}$ which is
determined from the solutions of CW fundamental guided mode. It also depends
on the core radius of the cylindrical waveguide. The light can be almost
stopped for a choice of lower value of Rabi frequency $\Omega_{0}$, and it
leads to potential applications such as optical memories, optical buffer, and
optical delay line. Meanwhile, the core radius has also important role to
control the slow light and the bandwidth of the transparency window.

##  IV Conclusions

We have investigated the EIT inside the core of CW in the optical fiber system
in the weakly guiding approximation where the core and cladding indices
difference is very small. This approximation turns out to practical and
convenient tools for the anaylisis of EIT inside complex CW system. The
results agree very well with more elaborate finite difference calculations for
the lowest fundamental mode considered here. Higher order modes can also be
treated but are not considered in this work. For EIT inside CW it is found
that the transparency window can be broadened by increasing the amplitude of
pump field of fundamental mode and decreasing the core radius of the CW
containing three level atomic system. For a given permittivity profile due to
the EIT, the propagation constant of the signal field can be calculated by
solving the vector wave equation using the finite difference method. The group
velocity of the signal field can be reduced by increasing the core radius of
the CW, and by decreasing the Rabi frequency parameter as EIT demands. This
study is useful for slow light development in the optical fiber system. The
analyzed scheme can be easily integrated with conventional optical fiber
systems at the conditions of enhanced light-atom interaction due to transverse
light field confinement. It can be also useful for realization of enhancement
of light atom interaction which is essential for optical quantum repeaters and
other quantum optical communications.

S.A.M. acknowledges the Russian scientific fund through Grant No. 14-12-01333
for financial support.

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