# Fingerspelling PoseNet: Enhancing Fingerspelling Translation with Pose-Based
Transformer Models

Pooya Fayyazsanavi, Negar Nejatishahidin , and Jana Košecká
George Mason University
{pfayyazs, nnejatis<EMAIL_ADDRESS>
###### Abstract

We address the task of American Sign Language fingerspelling translation using
videos in the wild. We exploit advances in more accurate hand pose estimation
and propose a novel architecture that leverages the transformer based encoder-
decoder model enabling seamless contextual word translation. The translation
model is augmented by a novel loss term that accurately predicts the length of
the finger-spelled word, benefiting both training and inference. We also
propose a novel two-stage inference approach that re-ranks the hypotheses
using the language model capabilities of the decoder. Through extensive
experiments, we demonstrate that our proposed method outperforms the state-of-
the-art models on ChicagoFSWild and ChicagoFSWild+ achieving more than 10%
relative improvement in performance. Our findings highlight the effectiveness
of our approach and its potential to advance fingerspelling recognition in
sign language translation. Code is also available at
https://github.com/pooyafayyaz/Fingerspelling-PoseNet.

##  1 Introduction

American Sign Language (ASL) is a complex and expressive visual language, that
relies on hand gestures, facial expressions, and body movements to convey
meaning. It has its own unique grammar and syntax. In comparison to the
remarkable advancements achieved in Automatic Speech Recognition (ASR), sign
language recognition and translation are still in its early stages of
development. It encompasses diverse sub-tasks, including fingerspelling
translation, word-level recognition, and continuous translation. Sign language
translation faces challenges such as the availability of limited paired data
for training models and the complexity of extracting effective representation
from visual modality.

This paper focuses on fingerspelling translation, which involves accurately
detecting and interpreting the specific hand poses and movements used to spell
out individual letters. According to [27], fingerspelling accounts for
approximately 12-35% of communication in American Sign Language (ASL). This
functionality is crucial for recognizing proper nouns, technical terms, and
words that do not have dedicated signs.

Figure 1: Our overall architecture utilizes a sequence of hand poses as the
input. The encoder and decoder components work in conjunction to generate the
final output (”TALENT”).

There are some unique challenges in American fingerspelling translation. It
uses a single hand which involves relatively small and quick motions of the
hand and fingers, as opposed to the typically larger arm motions involved in
other ASL sub-tasks. Therefore, fingerspelling can be difficult to analyze
with standard approaches. Current fingerspelling methods [43, 44, 33, 28]
primarily rely on appearance-based techniques and often face limitations due
to high variability among signers, including differences in speed, hand
appearance, and other motion variations before and after signing. In contrast,
pose-based methods have the potential to be robust to these variations and
offer data efficiency while addressing privacy concerns.
In this work, we propose a novel pose-based approach using encoder-decoder
transformer model summarized in Figure 1. Transformers have demonstrated
remarkable success in various natural language processing tasks by effectively
capturing long-range dependencies and contextual information. Using
transformers in the domain of fingerspelling recognition, we were able to
exploit their language modeling capabilities and achieve significant
improvements on the ChicagoWild [42] and ChicagoWild+ [43] dataset. These
datasets consist of a diverse range of hand gestures corresponding to
individual letters, captured from multiple signers in various environments. In
summary, the contributions of the proposed approach are summarized as follows:

  * •

Transformer-based architecture that combines Connectionist Temporal
Classification (CTC) and language modeling for fingerspelling. The model
captures contextual information and enables effective language modeling and
seamless translation within a single framework.

  * •

Introducing a novel loss term for predicting the word length that enhances
translation accuracy and robustness, particularly in cases of missing letters.
This improvement benefits both training and inference.

  * •

Novel two-stage inference approach exploiting the learned language model for
re-ranking the hypotheses.

  * •

Our method surpasses existing SOTA models, achieving over 10% relative
improvement in finger spelling translation performance.

Figure 2: The figure illustrates our overall approach. Raw frames are
processed to extract keypoints, which are then cropped and normalized. These
normalized keypoints are passed through a fully connected layer for upscaling
and fed into a Transformers encoder. Additionally, a special token predicts
the letter size using normalized sine and cosine values. On the decoder side,
the letter sequence is augmented with BOS and EOS tokens which generate the
subsequent tokens in an autoregressive manner.

##  2 Related Work

Early works on sign language recognition from video focused on isolated signs
[30, 48, 26, 1, 20, 8, 15], where individual signs, words or letters, are
recognized in isolation. More recent advancements in the field have shifted
towards continuous sign language recognition [9, 40, 6, 5, 22], aiming to
parse and translate continuous signing sequences. More recently deep learning
techniques have been applied to fingerspelling recognition, leveraging the
power of convolutional neural networks (CNNs), recurrent neural networks
(RNNs) [38, 10], and Transformers [47]. The choice of representation plays a
crucial role in modeling sign language, as it directly impacts the performance
and robustness of recognition and translation systems. In RGB-based
approaches, a 2D/3D convolutional neural network backbone pre-trained on
datasets like ImageNet [7], DeepHand [45], or activity recognition datasets
[18] is commonly employed. For continuous translation, the backbone can be
pre-trained using word-level data, as demonstrated in [40, 34]. RGB-based
representations are often susceptible to lighting conditions, background
clutter, and high visual domain variation. These challenges impact the
accuracy of recognition and require a large amount of data for training from
scratch.

Skeleton-based representations use spatial positions of joints and body
landmarks of the signer. These models [16, 1, 24, 31] utilize off-the-shelf
pose estimation methods and then learn spatio-temporal features on the top of
these 2D or 3D keypoint coordinates. Authors in [31] proposed simple linear
layers to lift the 2D keypoints into the 3D instead of using 3D pose
estimation.

Previous works [14, 37, 36, 42, 43, 5, 6] have extensively employed LSTM and
RNN architectures for various sign language tasks and vary in the types of
input, model architectures, and fusion strategies when combining multiple
channels of information. These models excel in capturing sequential
dependencies and have been widely adopted for their ability to model temporal
information in video data. Following the advancements in Natural Language
Processing (NLP), transformer-based approaches have gained significant
attention in sign language processing [40, 1, 24, 33]. Transformers excel at
capturing long-range dependencies[25] and contextual information, making them
suitable for modeling the complex dynamics of sign language. The transformers-
based models either use features learned from the video frames [3, 8] or the
2D/3D pose estimates [1, 24]. The supervision can be both on the encoder or
decoder side. The decoder’s auto-regressive component is effective in modeling
the linguistic structure both in case of RNNs [2] and transformer
architectures [3]. in the case of word-level classification task, the decoder
of the transformer decodes the class query [1].

Despite the progress in both word-level recognition and continuous
translation, there remains a gap in the literature concerning the specific
task of fingerspelling translation. Fingerspelling translation in real-world
scenarios has been extensively explored in [43, 42, 33, 41]. These studies
collected videos from YouTube and Deaf social media platforms to capture
diverse fingerspelled words in natural contexts. In the work by [42] the
authors developed a hand detection method to locate the signing hand within
the video frames, followed by training a CNN-LSTM model for translation
purposes. The follow-up work [43] presented an end-to-end approach that
bypassed the explicit hand detection step and proposed an iterative attention
mechanism, leveraging a 2D-CNN to extract visual features from individual
frames, which were used as an input to RNN. To enhance representation
learning, [28] introduced a Siamese network architecture to distinguish
between similar and dissimilar hand shapes. These works primarily focus on
videos with exclusive fingerspelling content, which is a limitation in
realistic scenarios where the exact occurrence of fingerspelling is unknown.
In [39] the model first detects segment proposals in the video, and
subsequently performs recognition of these segments using the CTC [12] loss.
In [33], the authors employ a multi-stage training strategy to overcome the
need for labeled segmentation, leveraging additional cues such as mouthing. In
[11] authors explore the use of optical flow as additional input to the
Transformers Encoder. Meanwhile, in [29], translation is approached through
multi-modal fusion involving pose, optical flow, and CNN features. The work of
[17] employs an attention-based CNN approach for generating spatial features,
utilizing optical flow as a prior for LSTM modeling. Fingerspelling often
encounters the issue of distinct letters sharing highly similar hand-shapes,
leading to ambiguities. This issue is addressed by [21] by modifying the
transformers Encoder-Decoder to effectively discern these ambiguities in
visual representations.

In recent work [40], a new dataset combines continuous sign language and
fingerspelling, offering rich training data. Using pre-trained networks and
multi-modal transformers, the study reveals a BLEU-4 score decrease (7.74 to
6.33) in videos containing fingerspelling. This reveals a limitation in
existing models and emphasizes the potential for improvement in this area.

##  3 Approach

The aim of a finger-spelling translation system is to convert a collection of
video frames $I=\left\\{{I}_{1},{I}_{2},\ldots,{I}_{T}\right\\}$ into a letter
sequence, $W=\\{w_{1},w_{2},\ldots,w_{L}\\}$, thus translating the entire
video sequence. We have access to a set of $n$ pairs $\\{I,W\\}$ where $I$ is
a video and $w$ is the corresponding label. Our transformer-based model uses
the sequence of hand landmarks extracted from the video frames as input.
Additionally, our model incorporates a novel loss function designed to predict
the length of the word. The overall architecture is outlined in Figure 2. In
the following sections, each of these components will be described in detail.

###  3.1 Input Representation and Pre-processing

Pose Estimation. To estimate the human body pose from the video frames,
various off-the-shelf methods can be employed. While previous works mainly
used on OpenPose [4], this study utilizes the Google MediaPipe Holistic
framework [23]. In Section 4.5 we present the effect of different pose
estimation methods on the final translation task. MediaPipe provides 543 body
landmarks MediaPipe (33 body joints pose landmarks, 468 face landmarks, and 21
hand landmarks per hand), where the hand joints are specifically employed for
training the model. Each landmark comes with a $confidence$ value and $3$D
coordinates consisting of $x$, $y$, and $z$. In this work, only the $x$ and
$y$ coordinates are utilized for training purposes.

Signing Hand Detection. In American Sign Language (ASL), finger spelling is
performed using only one hand. Consequently, one of the initial steps in the
pre-processing stage involves identifying which hand does the fingerspelling.
Two techniques were employed to determine the hand involved in the process.
First, the finger joint positions obtained and used to analyze the movements
and gestures of each hand. The dominant hand typically exhibits more
variability(difference between consecutive frames) in joint movements.

$V=\sum_{t=1}^{T}\sum_{j=0}^{J}P^{t}_{j}-P^{t-1}_{j}\vspace{-5pt}$ (1)

Here, $P^{t}_{j}$ denotes the $j$-th hand joint at frame $t$, with $T$
representing the total number of frames and $J$ representing the count of hand
joints. The value $V$ is subsequently compared for the right and the left
hand, and the larger value is chosen to determine the dominant hand. To
further improve the accuracy of this heuristic, we leverage the consistency
observed in signers’ hand usage type across different videos. We check the
current predictions with the previous ones made by the same signer. This
approach takes advantage of the fact that signers tend to consistently use the
same hand for fingerspelling in all of their videos. By considering the past
patterns of hand usage for each signer, we can refine the predictions and
achieve more accurate results.

The estimated hand landmarks need to be normalized before the training.
Normalizing pose data ensures that different poses are represented
consistently across different individuals or scenarios like scale,
orientation, and position.

Hand Origin. To normalize all the $x$ and $y$ coordinates, we utilize the
wrist landmark origin of the hand coordinate system and normalize other joints
employing the following procedure:
$f_{\text{origin}}(x,y)=(x-x_{origin},y-y_{origin})$ where
$x_{origin},y_{origin}$ are the $x,y$ location of wrist landmark.

Mirror. In the case of signers utilizing the left hand, we employ a mirroring
technique to adjust the hand landmarks in the following manner:
$f_{\text{mirror}}(x)=-x+\max(X)$ Let $x$ represent the x-coordinate, and $X$
denote the array containing all the $x$ coordinates in one frame.

Scaling. In order to address the scaling issue, we uniformly resize the hand
bounding box to a dimension of $1\times 1$. This transformation guarantees
that all values are scaled within the range of 0 to 1, with the maximum value
set to 1 and the remaining values adjusted proportionally. By applying this
transformation, we ensure consistent scaling across all hand instances as
follows:
$f_{\text{scale}}(x,y)=\left(\frac{x}{\max(X)},\frac{y}{\max(Y)}\right)$ Where
the $X,Y$ represent the array that contains all the $x$ and $y$ locations in
one frame.

Lastly, all the hand joint coordinates are normalized by subtracting the mean
and dividing by the maximum absolute value. This process ensures that the
values are scaled in the range of $[-0.5,0.5]$ while being centered around
zero.

###  3.2 Model Architecture

Our approach utilizes transformer-based encoder-decoder architecture,
initially proposed in [47] and depicted in Figure 2. The input to our system
is a sequence of normalized body poses, each containing 21 keypoint
coordinates. The extraction of hand poses from the video involves applying the
procedure described in Section 3.1, utilizing MediaPipe and subsequent pre-
processing steps. The encoder takes in a tensor
$P=\left\\{p_{1},p_{2},\ldots,p_{T}\right\\}$ of size $T\times 21\times 2$,
which is then flattened to yield a tensor of size $T\times 42$. Subsequently,
a learnable positional encoding is added to the vector of poses. The sequence
then passes through the self-attention module and a feed-forward network
composed of two layers, to capture contextual information within the pose
sequence. The self-attention module has 8 attention heads in each of the 3
encoder layers.

Length Token. In the transformers encoder block we incorporate a learnable
parameter token and concatenate it with the vector of poses. This output token
is then mapped into a vector of size 2 using a fully connected layer in the
output. The role of this token is to predict the number of letters in the word
in sign language fingerspelling. We observed that existing models often
struggle with accurate prediction of certain letters, leading to performance
limitations. By introducing this token, we aim to improve the prediction of
missing letters. Furthermore, during the inference, we leverage this
prediction to enhance the accuracy and robustness of our model’s predictions.
To generate the ground truth data for this prediction, we transform the length
in:

$len=\left[\sin\left(2\pi*\left(\frac{L}{30}-0.5\right)\right),\cos\left(2\pi*\left(\frac{L}{30}-0.5\right)\right)\right]$
(2)

where $L$ represents the length of the word. Initially, we normalize the
length values, with $L=30$ being the longest word, transform them between
$[-\pi,\pi]$ and compute the sine and cosine of these normalized lengths. By
using sine and cosine representations the errors in length prediction are
mapped to points on a unit circle enabling more balanced treatment and making
the contribution of the errors less sensitive to the absolute scale of the
words.

On the decoder side, the model takes in the sequence of letters. We first
tokenize the letters, augment them with the beginning-of-sequence BOS and end-
of-sequence EOS tokens, and add the positional embeddings to the tokens
representing letters. The augmented and embedded sequence $W_{\text{word
}}=\left\\{w_{1},w_{2},\ldots,w_{L}\right\\}$, of length $L$, is then passed
through the decoder. The decoder employs a masked attention mechanism, where
each token can attend to only the preceding tokens, preventing the model from
accessing future information during training. This enables the decoder to
generate tokens autoregressively, attending only to the already generated
parts of the sequence. Following the masked attention step, the decoder
further utilizes self-attention mechanisms, allowing each token to attend to
all other tokens in the sequence capturing global dependencies and context.
The self-attention mechanism facilitates the decoder in generating the output
tokens one at a time, progressively constructing the final output sequence.
The decoder has 3 layers with 8 attention heads.

###  3.3 Loss Functions

In our fingerspelling translation task using a transformer encoder-decoder, we
employed three distinct loss functions that will be discussed in detail next.

CTC Loss. On the encoder side, where the input comprises a sequence of hand
poses without explicit alignments between the poses and the target sign
language letters. We use Connectionist Temporal Classification (CTC) loss
function. The CTC loss models all possible alignments between the hand shapes
and the sign language letters without requiring explicit alignment
supervision.

$\mathcal{L}_{\mathrm{CTC}}=-\log p(W\mid P)$ (3)

where $P$ is the vector of poses and $W$ is the target sequence of labels. In
more detail:

$\mathcal{L}_{CTC}=-\log\sum_{A\in\mathcal{A}_{P,W}}\prod_{t=1}^{T}p\left(c_{t}\mid
P\right)$ (4)

where, $A\in\mathcal{A}_{P,W}$ denotes the set of valid alignments
corresponding to the target sequence $W$, and $p\left(c_{t}\mid P\right)$
denotes the probability of corresponding letter at timestep $t$ of the input
sequence. The term $p\left(c_{t}\mid P\right)$ is the output of the encoder at
each timestep, where $c_{t}$ is the probability of the letter at the output of
the softmax layer.

MSE Loss. To further enhance the performance and learning capabilities of the
model, we introduced a learnable parameter to predict the length of the
letters during training. This additional parameter allowed the model to gain a
better understanding of the variations in letter sizes within sign language.
By training this parameter using a Mean Squared Error (MSE) loss function, the
model could improve its ability to accurately predict the length of the
letters. The length prediction could also be leveraged during the inference
stage, aiding in generating more accurate and visually consistent
translations.

$\mathcal{L}_{\mathrm{MSE}}=\frac{1}{N}\sum_{i=1}^{N}\frac{1}{2}\sum_{j=1}^{2}\left(\widehat{\operatorname{len}^{i}_{j}}-\operatorname{len}^{i}_{j}\right)^{2}$
(5)

In this equation, $[\widehat{len}^{i}_{1},\widehat{len}^{i}_{2}]$ represents
the predicted word length, which is a vector of size $2$ (sine and cosine)
(see Eq. 2). The ground truth length of $i$-th example is denoted as
$[{len}^{i}_{1},len^{i}_{2}]$ and $N$ represents the batch size.

Cross Entropy. On the decoder side, the task involved generating the sign
language letter translation one letter at a time, following an auto-regressive
approach. To optimize the decoder’s performance in this auto-regressive task,
we utilized a cross-entropy loss function. The cross-entropy loss encouraged
the model to produce more accurate and contextually appropriate letter
predictions.

$\mathcal{L}_{CE}=-\frac{1}{M}\sum_{i=1}^{M}y_{i}\cdot\log\left(\hat{y}_{i}\right)$
(6)

where $y_{i}$ is the ground truth label and $\hat{y_{i}}$ is the softmax
probability for the $i^{th}$ class, $M$ represents the total number of
classes.
The total loss is calculated as:

$\mathcal{L}=\lambda\mathcal{L}_{\text{CTC }}+\mathcal{L}_{\text{CE
}}+\mathcal{L}_{\text{MSE }}$ (7)

where $\lambda$ is utilized to regulate the relative contributions of loss
components.

###  3.4 Re-ranking Inference

Throughout the prediction process, the encoder employs a greedy decoding
strategy to generate the likelihood of each letter for every frame.

$\hat{W}=\underset{{\mathbf{W}}\in\bm{w}^{*}}{\operatorname{argmax}}\prod_{t=1}^{T}p_{ctc}\left(c_{t}\mid\varepsilon(P)\right)$
(8)

The beam search then refines the generated candidates by considering their
likelihood and selecting the most probable sequences, We refer to this set of
$k$ predictions as our hypotheses. In the proposed model, the contextualized
features $\varepsilon(P)$ are obtained from the encoder, where $P$ represents
the input pose sequence. The sequence length is denoted by $T$. At each
timestep $t$, $a_{t}$ represents the probability of characters. However,
utilizing beam search solely on the encoder side fails to capitalize on the
potential of a language model. The language model captures the probability
distribution of letters based on the generated letters up to a given point.

$p\left(w_{1},w_{2},\ldots,w_{L}\right)=\prod_{i=1}^{L}p\left(w_{i}\mid
w_{1},w_{2},\ldots,w_{i-1}\right)$ (9)

Therefore we employ autoregressive decoding on the decoder side. With this
approach, the model generates the output sequence token by token, taking into
account the previously generated tokens. This autoregressive process enables
the model to capture the context and dependencies within the sequence, leading
to coherent and contextually appropriate predictions.

$\hat{w_{t}}=\mathcal{D}\left(\hat{w}_{1:t-1},\varepsilon(P)\right)$ (10)

Using the input pose vector $P$, the encoder ($\varepsilon$) generates
contextualized tokens. The decoder starts with the SOS token and proceeds to
generate subsequent tokens until either the model generates the EOS token or
the maximum length is reached.

However, there are some drawbacks of using this method. Firstly, generating a
meaningful sequence, especially in the case of fingerspelling with limited
available data, necessitates a substantial amount of training data. Secondly,
prior research [35, 13, 46] has shown that the decoder is more sensitive to
target-side information rather than source-side information. Consequently,
even a minor mis-recognition can significantly degrade the overall predicted
performance. On the other hand, employing a separate language model, such as
[42], might overlook the rich contextual information encoded by the encoder
and focus solely on language aspects.

To address these limitations, we propose a hybrid approach that combines the
strengths of both methods. During the decoding process, we utilize the CTC
with beam decoding technique to generate a set of hypotheses. However, rather
than relying solely on an autoregressive method, we take these hypotheses as
input and employ a re-ranking strategy based on the generated probabilities on
the decoder. This integration allows us to benefit from the contextualized
encoder features while leveraging the hypothesis generation capability of the
CTC.

Furthermore, we incorporate the predicted length to improve the ranking
process. This aspect proves particularly beneficial, as one of the limitations
is the potential omission of certain letters. By integrating the predicted
length, our model generates more consistent predictions and improves the
overall performance.

$\small\hat{W}=\underset{W\in w^{*}}{\operatorname{argmax}}\log
p_{\mathrm{ctc}}(W\mid P)+\beta\log
p_{\operatorname{lm}}(W\mid\varepsilon(P))-\gamma E_{L}$ (11)

Where,

$\small E_{L}=\left|\hat{L}-L_{Y}\right|$ (12)

In our approach, the decoder, denoted as $p_{\text{lm}}(W\mid\varepsilon(P))$,
takes as input the generated hypotheses from the beam search. $\hat{L}$ is the
predicted Length token and $L_{Y}$ is length of the hypotheses generated on
the encoder.

Decoding Strategy | Letter Accuracy%
---|---
Encoder only
Encoder Only(CTC) Greedy | $57.3$
Encoder Only(CTC) + Beam | $58.5$
Encoder Only(CTC) + LSTM [42] | $59.8$
Encoder-Decoder
Encoder-Decoder(only CE) | $54.6$
Encoder-Decoder(CTC + CE) | $56.3$
Ours | $66.3$
Table 1: Comparison of Training and Decoding Strategies for FingerSpelling
translation. For training, we can incorporate CTC loss, CE, or both. During
inference, decoding includes auto-regressive on the decoder or beam search
decoding on the encoder side.

##  4 Experiments

We report the results of our approach on ChicagoFSWild[42], and
ChicagoFSWild+[43] datasets. We also provide information regarding the
training schema, inference ranking, used datasets, and our ablation study.

###  4.1 Training

We implement our model using PyTorch [32] framework. The Adam optimizer [19]
is employed to train our network with $\beta_{1}=0.9,\beta_{2}=0.999$. The
network is trained on one NVIDIA GeForce GPU for $20$ epochs on both
ChicagoFSWild and ChicagoFSWild+ datasets, with a batch size of $1$. In
addition, we set hyper-parameters in Eq. 7 as $\lambda=5$. All the
hyperparameters are determined using the validation set.

###  4.2 Dataset

The Chicago Fingerspelling Dataset[42] is a collection of videos that feature
individuals performing American Sign Language (ASL) fingerspelling. This
dataset was created “in the wild”, using videos collected from websites.
ChicagoFSWild includes $7304$ ASL sequences by 160 signers, while
ChicagoFSWild+[43] contains $55,232$ sequences by 260 signers. The datasets
offer video-level annotations but lack individual frame-level segmentation.

###  4.3 Inference

The inference stage plays a crucial role in generating accurate predictions.
In this section, we present three main stages employed during the inference
stage, namely CTC with beam search, autoregressive decoding, and our re-
ranking inference. Following the prior works [41, 43, 42, 21] we evaluate the
performance based on the metrics of letter accuracy $\text{ ErrorRate
}=\frac{(S+D+I)}{N}$, where $S,D,I$ are the number of substitutions,
deletions, and insertions in the alignments, and N is the number of letters.

CTC with Beam Search. First, the CTC with beam search technique is commonly
used to generate multiple hypotheses or candidate sequences. Table 1 presents
the results for non-autoregressive decoding using the CTC approach. Our
experiments conducted in two scenarios. In the first scenario, we performed
greedy decoding, selecting the most probable character at each time step.
Secondly, in order to enhance the prediction quality, we incorporate beam
search with a beam width of $5$ to consider multiple hypotheses, as
demonstrated in Table 1.

CTC with Language Model. In this experiment, we leverage the language model
trained specifically for finger spelling, as introduced in [42]. This
dedicated language model is employed to refine the generated hypotheses,
leading to improved results, as demonstrated in Table 1. The language model
consists of an LSTM trained separately on the training set of labels.

Autoregressive Decoding. Another approach is to employ autoregressive decoding
on the decoder side. With this approach, we solely rely on the decoder to
generate the output sequence. The utilization of the CTC loss during training
leads to improved results during inference, as demonstrated in Table 1.
However, as shown in Table 1, the performance of the models in this scenario
still lags behind that of the non-autoregressive counterparts. This outcome
was expected, as explained in Section 3.4.

Our Method. In our approach, we aim to leverage the strengths of language
models while giving importance to the contextualized features from the encoder
as described in Section 3.4. The values of $\beta$ and $\gamma$ are assigned
as $0.4$ and $1.2$, respectively in Eq. 11. As shown in Table 1, the model can
outperform all other inference strategies.

Model | FSWild [42] | FSWild+ [43]
---|---|---
Resnet Whole Frame | $22.3\%$ | $24.7\%$
Hand Det.+ CNN + RNN [42] | $41.9\%$ | $41.2\%$
Iterative Attention + LM [43] | $45.1\%$ | $46.7\%$
Weakly Supervised [28] | $48\%$ | -
Fine-Grained Attention [11] | $48.36\%$ | -
TDC-SL [29] | $50\%$ | -
Attention(optical flow+Res) [17] | $57.84\%$ | -
FSS-Net [41] | $52.5\%$ | $64.4\%$
CtoML [21] | $54.9\%$ | -
Ours(Enc-Dec Transformers) | $66.3\%$ | $71.1\%$
Table 2: Comparing different models on the test set of the ChicagoWild [42]
and the ChicagoWild+ [43] datasets, we evaluate the performance using the
metric of Letter Accuracy ($\%\uparrow$).

###  4.4 Result

In this section, we present the results of our experiments and evaluations
conducted to assess the performance of our proposed method. We aim to provide
an analysis and interpretation of the outcomes obtained, showcasing the
advancements and contributions made toward the problem. We adopted the
train/val/test split introduced in the original paper[42]. The results, as
shown in Table 2, are compared with various models on both datasets,
demonstrate that our model surpasses all other models by a significant margin.
Our study also investigates the impact of various factors, including model
architectures, different inference techniques, and hyperparameters to
establish a robust and reliable framework for tackling the challenges at hand.
All the ablations are using the Chicago Wild [42] dataset.

Figure 3: Qualitative results on ChicagoFSWild[42]. Only a subset of frames is presented here. | Letter Accuracy
---|---
OpenPose[4] | 55.7
3D MediaPipe [23] | 64.1
Ours without Length Token | 64.1
Ours | 66.3
Table 3: Ablation Analysis of Various Components. Our method is compared with
alternative pose estimation approaches. Additionally, we examine the impact of
utilizing 3D coordinates instead of 2D, along with assessing the influence of
the Length Token.

###  4.5 Ablation Study

In this section, we present a series of ablation studies to evaluate the
contribution and effectiveness of various components in our proposed method.
Specifically, we investigate the impact of different factors and variations,
including the selection of the pose method, diverse decoding formulations used
during inference, and the influence of length tokens.

Selection of Pose Estimator. We begin by analyzing the effect of the Pose
Estimator on the overall performance. We evaluate the impact of different pose
methods on translation accuracy. We employed OpenPose[4] and MediaPipe[23] as
the pose extractor methods. When comparing MediaPipe[23] Holistic to
OpenPose[4], notable differences arise in their approach to predicting body
keypoints. MediaPipe first predicts the body keypoints and subsequently
employs separate models for hand and face keypoints on cropped patches. In
contrast, OpenPose predicts all keypoints together from the input image. A
distinguishing feature of MediaPipe is its consideration of the consistency
between predictions across subsequent frames. This approach promotes smoother
and more consistent predictions, reducing the likelihood of detection failures
or missed keypoints. Also, MediaPipe directly predicts the keypoints in 3D,
offering a more direct estimation. On the other hand, OpenPose relies on
triangulation techniques to infer the 3D pose from the detected keypoints. The
results of these evaluations are presented in the first row of Table 3. These
findings highlight the potential for enhancing the accuracy of the methods by
further advancements in pose estimation. Pose models demonstrate greater
robustness in handling variations compared to RGB-based methods. Furthermore,
pose-based approaches exhibit improved data efficiency during training and
also can be advantageous in scenarios where data privacy is a concern.

3D vs 2D. We further investigate the impact of utilizing 3D coordinates
instead of 2D from the Mediapipe[23] Holistic approach. The results, presented
in the second row of Table 3, indicate a degradation in performance. This
suggests that the 3D coordinates may not be reliable and can introduce
significant noise to the model.

Length Token. To assess the impact of the length token in our approach, we
conducted experiments where we removed it from the training and decoding
process. The third row of Table 3 shows the results of the method with/without
the Length Token.

| Deletions | Substitutions | Insertions
---|---|---|---
Error Count | 768 | 488 | 231
Error Rate | 17.37 | 11.04 | 5.22
Table 4: Error Counts and Rates in three Scenarios: Deletions, Substitutions,
and Insertions.

Figure 4: This figure shows the distribution of hand pose availability in the
dataset. The x-axis represents the percentage of missing hand poses in each
video, while the y-axis indicates the percentage of videos in the dataset
falling into each category.

###  4.6 Limitations and Failure Cases

In this section, we discuss the method’s failure cases and limitations. The
primary errors involve deletions, followed by insertions as shown in Table 4.
Furthermore, regarding substitutions, the top-5 letter pairs that exhibit the
highest confusion rates are
$(\mathrm{e}\rightarrow\mathrm{o}),(\mathrm{i}\rightarrow\mathrm{y}),(\mathrm{r}\rightarrow\mathrm{u}),(\mathrm{a}\rightarrow\mathrm{o}),(\mathrm{i}\rightarrow\mathrm{j})$.
An additional limitation concerns the performance of the pose models employed.
Some video frames contain low-quality and frequently blurry images due to fast
movements. Figure 4 shows the distribution of the missing hand joints
estimated using openpose[4] method. Given that our model relies solely on pose
keypoints, instances of failure in the pose model directly lead to the overall
failure of our approach. Figure 3 showcases some of the video frames alongside
our model’s output, displaying both accurate translations and other errors.

###  4.7 Conclusion

In conclusion, we have presented a novel approach that combines transformer
architecture with hand pose models for fingerspelling translation. Our
proposed method leverages the language modeling capabilities of transformers
while effectively capturing the temporal dynamics of hand poses. Through
extensive experiments on the ChicagoWild and ChicagoWild+ datasets, we have
demonstrated significant improvements in accuracy and translation performance
compared to state-of-the-art models.

Acknowledgments: This work was supported by the 2023 Amazon Research Awards
Program.

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[figure]style=plain,subcapbesideposition=top ††thanks: These authors
contributed equally to this work. ††thanks: These authors contributed equally
to this work. ††thanks: Author to whom correspondence should be addressed:
<EMAIL_ADDRESS>
# Demonstration and modelling of time-bin entangled photons from a quantum dot
in a nanowire

Philipp Aumann  Institute for Theoretical Physics, University of Innsbruck,
Innsbruck, Austria  Maximilian Prilmüller  Institute for Experimental Physics,
University of Innsbruck, Innsbruck, Austria  Florian Kappe  Institute for
Experimental Physics, University of Innsbruck, Innsbruck, Austria  Laurin
Ostermann  Institute for Theoretical Physics, University of Innsbruck,
Innsbruck, Austria  Dan Dalacu  National Research Council of Canada, Ottawa,
Canada  Philip J. Poole  National Research Council of Canada, Ottawa, Canada
Helmut Ritsch  Institute for Theoretical Physics, University of Innsbruck,
Innsbruck, Austria  Wolfgang Lechner  Institute for Theoretical Physics,
University of Innsbruck, Innsbruck, Austria  Parity Quantum Computing GmbH,
Innsbruck, Austria  Gregor Weihs  Institute for Experimental Physics,
University of Innsbruck, Innsbruck, Austria

###### Abstract

Resonant excitation of the biexciton state in an InAsP quantum dot by a phase-
coherent pair of picosecond pulses allows preparing time-bin entangled pairs
of photons via the biexciton-exciton cascade. We show that this scheme can be
implemented for a dot embedded in an InP nanowire. The underlying physical
mechanisms can be represented and quantitatively analyzed by an effective
three-level open system master equation. Simulation parameters including decay
and intensity dependent dephasing rates are extracted from experimental data,
which in turn let us predict the resulting entanglement and optimal operating
conditions.

In a future quantum world long-distance quantum communication will allow users
to communicate in perfect privacy and it will connect quantum computers for
distributed and blind computation tasks. Quantum repeaters [1] will be
necessary in order to establish the required long-distance entanglement and
for building even the simplest quantum repeaters we will need reliable, high-
rate and high-fidelity sources of entangled photon pairs besides quantum
memories and local quantum processing. The emitted photon pairs must propagate
with low loss and low decoherence in order to cover as much a distance as
possible. While the propagation loss in optical fibers is limited by intrinsic
material properties, the decoherence can be minimized by choosing a suitable
quantum information encoding [2]. Time-bin entanglement [3, 4] has emerged as
the optimal encoding for optical fiber quantum communication, because it is
immune to residual fiber birefringence as well as thermal and mechanical
fluctuations up to very high frequencies.

Figure 1: Scheme of the Time-Bin Entanglement Setup. Three phase-stable
interferometers facilitate the generation and projection of time-bin entangled
states. The delay of the pump interferometer, $\Delta t$ is chosen to be much
longer than the coherence time of the emitted photons to rule out photonic
first-order interference. The phases of the three interferometers
$\varphi_{\mathrm{P}}$, $\varphi_{b}$ and $\varphi_{x}$ are controlled via
phase plates. Each pump pulse excites the system with very low probability in
order to ensure that on average maximally one photon pair is created. The
interference of these two time bins can be observed when looking at the
coincident photon detections between outputs of the different analysis
interferometers. Inset: Quantum dot as a three level system (without dark
states). Green arrows indicate the direct population of the biexciton state
($\ket{b}$) via a virtual level (gray line). The single photon transition is
detuned from the exciton state ($\ket{x}$). Relaxation into the groundstate
($\ket{g}$) results in the successive emission of two photons at different
wavelengths.

So far all sources of time-bin entanglement have been probabilistic, even the
ones that used single quantum dots[5, 6]. Most work on quantum dots as
entanglement sources has concentrated on maximizing polarization entanglement,
for which elaborate growth and tuning techniques have been developed [7].
Polarization entanglement can be converted probabilistically to time-bin
entanglement [6] or by using ultra high-speed optical modulators, which,
however, are always very lossy and thus do not allow a near-deterministic
source. Therefore we consider the direct creation of single time-bin entangled
photon pairs from semiconductor quantum dots an important goal. The only known
way to achieve this involves using at least three energy levels in the quantum
dot, one of which must be metastable [8]. While research into deterministic
time-bin entanglement from quantum dots is on the way in our laboratory, in
this letter, as an intermediate step, we present the realization of
probabilistic time-bin entanglement from a quantum dot in an optimized
photonic structure.

[] []

Figure 2: Rabi Oscillations of a Quantum Dot Embedded in a Nanowire and Fit of
the Emission Probabilities to the Photon Counts (a) The solid lines correspond
to emission collected at the exciton wavelength, while the dashed lines
correspond to biexciton emission, respectively. The horizontal axis represents
the average laser power scaled such that the first maximum of the observed
oscillations occurs at $\pi$. Pink: $85\text{\,}\mathrm{ps}$ FWHM linearly
polarized pump. Brown: $20\text{\,}\mathrm{ps}$ FWHM linearly polarized pump.
Orange: $20\text{\,}\mathrm{ps}$ FWHM circular polarized pump, scaled to the
biexciton excitation angle of the brown curve. (b) We fitted the emission
probabilities predicted by the theoretical model to biexciton and exciton
emission counts for a pulse length of $85$ ps FWHM. The resulting parameter
values can be found in section S3 of the supplemental material. The dashed
line indicates the position of the $\pi/15$ power that has been used for the
time-bin measurement. The measurement error is estimated by the square root of
the number of the counts resulting in error bars smaller than the symbols.

In the past two decades a lot of work has gone into improving the out-coupling
efficiencies of photons from quantum dots [9] e.g. via the implementation of
circular Bragg gratings [10], enhancing emission into a collectable mode.
Alternatively, realizing quantum dots embedded in tapered nanowires turned out
to be a promising platform for coherent photon emission [11, 12, 13, 14]. The
tapered part of the nanowire acts as an antenna that matches the impedance of
the nanowire waveguide mode to the vacuum and thus achieves efficient out-
coupling [15].

In the following, we report the generation of time-bin entangled photon pairs
in indium arsenide phosphide (InAsP) quantum dots embedded in indium phosphide
(InP) nanowires via a resonant two-photon excitation [16, 17] (see figure 1).
Furthermore, we present an extension of our theoretical model from previous
work [18] that includes the density matrix of the time-bin entangled photons,
which allows suggesting optimal parameter values.

Experiment – The core of our setup is constituted by a quantum dot embedded in
a nanowire. Our samples were manufactured utilizing a selective-area vapor-
liquid-solid epitaxy which produced InAsP quantum dots embedded in defect-free
wurtzite InP nanowires [19]. A single electron-hole pair trapped in the
quantum dot is referred to as an exciton ($\ket{x}$), while the confinement of
two pairs is called a biexciton ($\ket{b}$). A recombination of a single pair
leads to the emission of a photon at a characteristic wavelength, as depicted
in the inset of figure 1. The biexciton-exciton photon cascade is used in
order to operate the quantum dot as a source of correlated photon pairs. The
emission spectrum of our quantum dot can be found in figure S1 in the
supplementary material.

Figure 3: Procedure for Simulating the Photonic Density Matrix from the
Quantum Dot Dynamics. After setting our model parameters to fit the
experimental Rabi data, we simulate the dynamics of the quantum dot’s density
matrix, obtaining the photon coincidence counts via calculating the two-photon
detection probabilities and thus reconstruct the photonic density matrix by
means of state tomography. More details on the procedure to estimate the
density matrix of the photons $\rho_{\text{photons}}$ from our theoretical
model are given in section S2 of the supplementary material.

The basic principle of the time-bin encoding scheme relies on the ability to
create a coherent superposition of two well-defined excitation processes. Its
simplest realisation relies on addressing the emitter with two pump pulses of
very low excitation probability each, and postselecting on biexciton - exciton
photon pair detection events. The two excitation pulses are created in an
unbalanced Mach-Zehnder interferometer and denoted by e (early) and l (late).
The phase between the two pulses $\Delta_{\phi}$ can be modified via a phase
plate and determines the phase of the entangled state. Denoting biexciton and
exciton photons by $b$ and $x$ respectively, the created state can be written
as:

$\displaystyle\ket{\Phi}$
$\displaystyle=\frac{1}{\sqrt{2}}\Bigl{(}\ket{e}_{b}\ket{e}_{x}+e^{i\Delta_{\phi}}\ket{l}_{b}\ket{l}_{x}\Bigr{)}$
$\displaystyle=:\frac{1}{\sqrt{2}}\left(\ket{ee}+e^{i\Delta_{\phi}}\ket{ll}\right).$
(1)

Using another two unbalanced Mach-Zehnder interferometers that are phase
stable with respect to the pump interferometer we carry out projective
measurements on the created entangled state. In order to perform quantum state
tomography, we analyze the result of $16$ different pairs of phase settings
and use a maximum likelihood approach. [20, 21] For collecting the $16$
different projections necessary for the quantum state tomography we employ
four different phase settings in the analysis interferometers each and detect
photons at each of the four output ports. We collect time tags of the detected
photons for $3600\text{\,}\mathrm{s}$ per phase setting and identify
coincident photon pairs by imposing a coincidence window of
$400\text{\,}\mathrm{p}\mathrm{s}$. The integration time was chosen such that
it would yield sufficient statistics for the maximum likelihood reconstruction
method [22].

For the generation of biexciton-exciton photon pairs, we employ resonant
pulsed two-photon excitation from $\ket{g}$ to $\ket{b}$ (see inset in figure
1). In order to calibrate and characterize the system, we observe Rabi
oscillations by measuring the photon counts as a function of the average laser
power as shown in figure 2. We see that it is critical to identify an
appropriate polarization as well as a sensible pulse duration. Choosing a
circular pump polarization violates optical selection rules and leads to
incoherent excitations rather than to a two-photon coherent coupling of the
ground and biexciton state. By comparing the oscillations resulting from a
linearly polarized pump and pulse lengths of $25$ ps and $85$ ps, we find a
significantly stronger coherence for the longer pulse. The similar slopes at
low excitation power of the biexciton and exciton emission probabilities for a
linearly polarized pump indicate the superior pair production efficiency of
this excitation scheme.

For the creation of time-bin entangled photons we thus use the optimized pulse
duration of $85$ ps [18] resulting in a substantial increase of the excitation
coherence and we determine the energy of a $\pi/15$-pulse to be adequate,
yielding an excitation probability of about $7.5$ $\%$ per pulse which reduces
the probability of emitting at both time bins to below $0.6$ $\%$. Our
theoretical model (see below) underpins the feasibility of the chosen
parameters and provides the basis for even further improvements in future
work.

Figure 4: Reconstructed Density Matrix of the Emitted Photons. Left: Real and
imaginary part of the reconstructed density matrix from the experiment
$\rho_{\mathrm{photons}}^{\mathrm{exp}}$. Right: Real and imaginary part of
the simulated density matrix $\rho_{\mathrm{photons}}^{\mathrm{sim}}$. The
agreement between the experimentally obtained and the simulated density matrix
is calculated by means of equation 12 yielding a value of around
$F_{\rho}\approx 0.96$.

Theoretical Model – We propose a quantum optical model in order to identify a
suitable set of parameter values to enhance the quality of entanglement of the
created photon pairs. This allows us to estimate the reconstructed density
matrix as shown in figure 4. Extending our earlier work [18], where we used a
model for the internal dynamics of the quantum dot, we include a procedure for
obtaining the photons’ density matrix from said quantum dot dynamics. Our
strategy is outlined in figure 3. The conceptual procedure introduced here is
not restricted to this particular experimental setup and thus can be seen as a
more fundamental framework for a general setting of time-bin entangled photons
from a quantum dot.

The pulsed laser driving field couples the ground state to the biexciton via
the ground state to exciton and exciton to biexciton transition dipoles. The
Hamiltonian in the effective interaction picture reads (for a derivation see
section S4 of the supplementary material)

$\displaystyle H=$
$\displaystyle(\Delta_{x}-\Delta_{b})\ket{x}\bra{x}-2\Delta_{b}\ket{b}\bra{b}$
(2)
$\displaystyle+\Omega(t)\Bigl{(}\ket{g}\bra{x}+\ket{x}\bra{b}+\mathrm{h.c.}\Bigr{)}.$

Here, $\Delta_{x}$ denotes the detuning from the exciton level to the laser
frequency, while $\Delta_{b}$ is the detuning from the biexciton level to the
two-photon transition, $\Omega(t)$ is the Rabi frequency featuring a Gaussian
time profile,

$\Omega(t)=\Omega_{0}\exp\left(-\frac{4\ln(2)(t-t_{0})^{2}}{\tau^{2}}\right),$
(3)

with amplitude $\Omega_{0}$, pulse duration (FWHM) $\tau$ and time offset
$t_{0}$.

To simulate the dynamics we solve the master equation in Lindblad form
numerically, i.e.

$\dot{\rho}=i\left[\rho,H\right]+\frac{1}{2}\sum_{j=1}^{6}\left(2R_{j}\rho
R_{j}^{\dagger}-R_{j}^{\dagger}R_{j}\rho-\rho R_{j}^{\dagger}R_{j}\right).$
(4)

where $\rho=\rho_{\mathrm{dot}}(t)$ is the quantum dot density matrix. We
consider six dissipative channels associated with six different Lindblad
operators $R_{j}$, where

$\displaystyle R_{1}=$ $\displaystyle\sqrt{\gamma_{x}}\ \ket{g}\bra{x},$ (5)
$\displaystyle R_{2}=$ $\displaystyle\sqrt{\gamma_{b}}\ \ket{x}\bra{b},$ (6)

describes the radiative decay of the biexciton and exciton levels with rates
$\gamma_{b}$ and $\gamma_{x}$, respectively, while

$\displaystyle R_{3}=$
$\displaystyle\sqrt{\gamma^{\text{const}}_{xg}+\gamma_{xg}}\
(\ket{x}\bra{x}-\ket{g}\bra{g})$ (7) $\displaystyle R_{4}=$
$\displaystyle\sqrt{\gamma^{\text{const}}_{bx}+\gamma_{bx}}\
(\ket{b}\bra{b}-\ket{x}\bra{x})$ (8)

introduce dephasing. The rates
${\gamma_{bx}=\gamma^{I_{0}}_{bx}\Bigl{(}\tfrac{\Omega(t)}{\Omega_{S}}\Bigr{)}^{n}}$
and
${\gamma_{xg}=\gamma^{I_{0}}_{xg}\Bigl{(}\tfrac{\Omega(t)}{\Omega_{S}}\Bigr{)}^{n}}$
are comprised of their amplitudes $\gamma^{I_{0}}_{bx}$ and
$\gamma^{I_{0}}_{xg}$ as well as the scaled time-dependent Rabi frequency to
the $n$-th power. Throughout this work we consider $n=2$. This laser intensity
dependent dephasing can be explained by phonons coupling to the quantum dot
[23]. The scaling factor ${\Omega_{S}=1}$ THz accounts for the correct
numerical values and leads to a unitless expression for the Rabi frequency. A
minor role is attributed to the rates of constant dephasing
$\gamma^{const}_{xg}$ and $\gamma^{const}_{bx}$ by the fit in figure 2.

In order to account for the decrease of photon counts for higher laser power
as depicted in figure 2, we introduce dark states modelling a laser power
dependent loss mechanism, as states outside the three-level approximation
become more prominent for higher laser powers. Moreover, this additional dark
state loss counteracts the increased exciton population via a single photon
transition that appears at higher laser intensities based on the broadening of
the spectral linewidth due to the laser dependent dephasing. For bookkeeping
purposes, we introduce two dark states $\ket{d_{x}}$ and $\ket{d_{b}}$, which
are populated by laser dependent exciton and biexciton decay, whereas in
general one dark state would suffice to constitute the same effect. The
corresponding Lindblad operators are given by

$\displaystyle R_{5}=$ $\displaystyle\sqrt{\gamma_{xd}}\ \ket{d_{x}}\bra{x},$
(9) $\displaystyle R_{6}=$ $\displaystyle\sqrt{\gamma_{bd}}\
\ket{d_{b}}\bra{b},$ (10)

with laser intensity dependent decay rates
${\gamma_{xd}=\gamma^{I_{0}}_{xd}\Bigl{(}\tfrac{\Omega(t)}{\Omega_{S}}\Bigr{)}^{n}}$
and
${\gamma_{bd}=\gamma^{I_{0}}_{bd}\Bigl{(}\tfrac{\Omega(t)}{\Omega_{S}}\Bigr{)}^{n}}$,
decay amplitudes $\gamma^{I_{0}}_{xd}$ and $\gamma^{I_{0}}_{bd}$, as well as
the same power $n$ as the dephasing mechanism.

Exemplary dynamics of the quantum dot when driven by a laser pulse are
depicted in figure S2 and numerical values for the system parameters can be
found in table S2 and S3 of the supplementary material.

[] []

Figure 5: Fidelity with the Bell State and Normalized Number of Photon Counts.
(a) Fidelity $F_{\ket{\Phi^{+}}}$ (see equation 14) between the theoretically
predicted density matrix $\rho_{\mathrm{photons}}$ and $\ket{\Phi^{+}}$ for
various values of Rabi frequency amplitude $\Omega_{0}$ and the pulse duration
$\tau$. The red dot marks the parameters chosen for figure 4, i.e. an
excitation angle of $\pi/15$ ($\Omega_{0}\approx 0.05$) and a pulse duration
of $85$ ps FWHM for the measurement of time-bin entangled photons. The dashed
lines indicate areas of constant pulse energy proportional to
$\Omega_{0}^{2}\tau$. (b) Normalized number of total counts predicted by the
simulated projective measurements. The blue line in the main plot follows a
constant count number of $0.32$. Additionally, the inset depicts the fidelity
along this line when going from small to large $\Omega_{0}$.

In order to obtain the density matrix of the emitted photons from the quantum
dot dynamics, we mimic the measurement of the photon coincidence counts in the
experiment: first, we calibrate our model by fitting the emission
probabilities

$\displaystyle
P_{i}=\gamma_{i}\int\bra{i}\rho_{\mathrm{dot}}(t)\ket{i}\mathrm{d}t$ (11)

for $i\in\\{x,b\\}$ to the biexciton and exciton Rabi data (see figure 2).
This comprehensive and numerical demanding fitting loop is outlined in section
S2 of the supplemental material. Then, the density matrix of the quantum dot
as a function of time $\rho_{\mathrm{dot}}(t)$ is the result of numerically
solving the master equation. Relating the density matrix of the photons
$\rho_{\mathrm{photons}}$ to $\rho_{\mathrm{dot}}(t)$ is achieved by
calculating the resulting photon coincidence counts, where we derive analytic
expressions for the detection probabilities of all $16$ projective
measurements (see S2 in the supplement) and subsequently use this estimate as
an input for the conventional state tomography. This procedure is depicted
schematically in figure 3. The density matrix resulting from this approach is
shown in figure 4 and compared to the density matrix obtained in the
experiment. In order to quantitatively compare the experimental results to our
simulation we employ to following definition of the fidelity for two mixed
states [24]:

$\displaystyle F_{\rho}=\mathrm{tr}\
\sqrt{\sqrt{\rho_{\mathrm{photons}}^{\mathrm{exp}}}\
\rho_{\mathrm{photons}}^{\mathrm{sim}}\
\sqrt{\rho_{\mathrm{photons}}^{\mathrm{exp}}}}\approx 0.96.$ (12)

The density matrix from theory and experiment have the same structural
appeareance, as they show similar values at the prominent matrix elements.
While the remaining entries of the simulated density matrix appear to be
rather flat, we observe additional small fluctuations of these entries for the
density matrix from the experiment.

Entanglement Quality of the Photons – Ultimately, our goal is to achieve two-
photon emission in a perfect Bell state,

$\displaystyle\ket{\Phi^{+}}=\frac{1}{\sqrt{2}}\Bigl{(}\ket{ee}+\ket{ll}\Bigr{)}.$
(13)

Therefore, we identify suitable values for the laser intensity
$I\propto\Omega^{2}$ and its pulse duration $\tau$ in our simulation, which
can assist in maximizing the fidelity,

$\displaystyle
F_{\ket{\Phi^{+}}}=\sqrt{|\bra{\Phi^{+}}\rho_{\mathrm{photons}}\ket{\Phi^{+}}|},$
(14)

to a Bell state in the experiment. Figure 5 shows a scan of the fidelity
$F_{\ket{\Phi^{+}}}$ over the corresponding parameter space spanned by
$\Omega_{0}$ and $\tau$. Here, we study the influence of the parameter
$\Omega_{0}$ instead of the intensity $I$ as this parameter is more natural to
the theoretical model. Once the model is calibrated to the experimental data,
$\Omega_{0}$ can be converted to the average laser power.

Similar to the Rabi oscillations in figure 2, we observe an oscillatory
pattern, which becomes less and less pronounced towards regions of higher
energy (upper right corner). This can mostly be attributed to the intensity-
dependent dephasing. For lower energies (lower left corner) the pattern
roughly follows areas of constant energy, indicated by the yellow dashed
lines. The red dot indicates the values chosen in the measurements that yield
the reconstructed time-bin encoded photonic density matrix in figure 4. We
show simulated density matrices for the same pulse length but different
average laser power in figure S4 of the supplemental material, where we
observe an increase of the diagonal entries of the density matrix towards
regions of lower fidelity which means that the photonic state is becoming more
classical in low fidelity regions for this pulse length. Reaching the regime
of maximal fidelity has to be deferred to a future experimental setup, where
our theoretical model can prove even more useful in fine-tuning the
experimental parameters.

For a source of entangled photons it is desirable to not only achieve a high
fidelity, but also to yield sufficient output. Figure 5 depicts the normalized
number of total expected counts of all simulated projective measurements (see
supplementary material S2). Again, we observe an oscillatory behaviour where
we find some degree of anti-correlation between the pattern of the counts and
the fidelity, i.e. dark areas with less output correspond to a relatively high
fidelity, whereas bright areas are connected to a smaller fidelity. Yet, these
two patterns are not perfectly anti-correlated, as we find slightly varying
fidelity for contours of constant counts. For some applications, a minimum
amount of photons is required. Consequently, one might be interested in the
optimal fidelity for a given photon count rate. For instance, we observe the
fidelity along a contour of constant counts in the inset of figure 5. For this
particular contour, we find the highest fidelity for long pulses with a
relatively low intensity. In cases where the rate of output photons is not an
issue, our study suggests that the optimal parameter regime is that of low
pulse energy (lower left corner).

Conclusions – In this work we have shown the coherent coupling of ground to
biexciton state of a InAsP quantum dot embedded in an InP nanowire via an
optimised two-photon resonant excitation scheme. We have used this method to
generate time-bin entangled photons, yielding a fidelity of
$F_{\ket{\Phi^{+}}}\approx 0.90$ (see equation 14) with respect to the
maximally entangled $\ket{\Phi^{+}}$ Bell state.

Additionally, we have presented a quantum optical model for simulating the
dynamics of the quantum dot. By making use of the experimental reconstruction
method, we have introduced a scheme for predicting the density matrix of the
emitted photons based on the simulation of the dynamics of the quantum dot.
The results of the model have been compared to the outcome of the experiment.
With this, we are able to identify optimal parameter regimes in order to
further increase the fidelity of the photons’ density matrix to a Bell state
and to provide a more general toolbox for the study of time-bin entangled
photons from a quantum dot.

## Acknowledgements

We want to thank Doris Reiter and her group for fruitful discussions. P.A. and
W.L. are supported by the Austrian Science Fund (FWF) through a START grant
under Project No. Y1067-N27 and the SFB BeyondC Project No. F7108, the Hauser-
Raspe foundation and the European Union’s Horizon 2020 research and innovation
program under grant agreement No. 817482. M.P. F.K. and G.W. acknowledge
partial support by the Austrian Science Fund (FWF) through projects W1259 (DK-
ALM), I4380 (AEQuDot), and F7114 (SFB BeyondC). This material is based upon
work supported by the Defense Advanced Research Projects Agency (DARPA) under
Contract No. HR001120C0068. Any opinions, findings and conclusions or
recommendations expressed in this material are those of the author(s) and do
not necessarily reflect the views of DARPA.

## Supplementary Material

The supplementary material shows the quantum dot’s emission spectrum and
details on the experimental methods. It features exemplary dynamics of the
quantum dot upon excitation by a laser pulse and provides an in-depth
mathematical assessment of the reconstruction of the photons’ density matrix
from the quamtum dot’s density matrix. Furthermore, it contains a summary of
the chosen values for the simulation parameters including the fit of the decay
rates and a derivation of the Hamiltonian in equation 2.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Conflict of Interest

The authors have no conflicts to disclose.

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