# PoseDet: Fast Multi-Person Pose Estimation Using Pose Embedding

Chenyu Tian1 Ran Yu1 Xinyuan Zhao2 Weihao Xia1 Haoqian Wang1&Yujiu Yang1
1Tsinghua University
2Northwestern University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Current methods of multi-person pose estimation typically treat the
localization and the association of body joints separately. It is convenient
but inefficient, leading to additional computation and a waste of time. This
paper, however, presents a novel framework PoseDet (Estimating Pose by
Detection) to localize and associate body joints simultaneously at higher
inference speed. Moreover, we propose the keypoint-aware pose embedding to
represent an object in terms of the locations of its keypoints. The proposed
pose embedding contains semantic and geometric information, allowing us to
access discriminative and informative features efficiently. It is utilized for
candidate classification and body joint localization in PoseDet, leading to
robust predictions of various poses. This simple framework achieves an
unprecedented speed and a competitive accuracy on the COCO benchmark compared
with state-of-the-art methods. Extensive experiments on the CrowdPose
benchmark show the robustness in the crowd scenes. Source code is available.

##  1 Introduction

Multi-person pose estimation is a fundamental yet challenging task in computer
vision, which plays a vital role in human understanding. It has broad
applications such as human-robot interaction, virtual reality, sports
analysis, _etc_., in which real-time performance is essential. The challenge
remains that estimating accurate poses of multi-person in the 2D image at a
real-time speed.

Figure 1: Keypoint-aware pose embedding

As an extension of single-person pose estimation, multi-person pose estimation
requires locating all joints and associating the joints with the same person.
Current approaches can be categorized as either top-down Sun et al. (2019);
Xiao et al. (2018); Chen et al. (2018); Huang et al. (2017); Papandreou et al.
(2017); Fang et al. (2017); Guo et al. (2019); Ruan et al. (2019); Jiang et
al. (2020) or bottom-up Newell et al. (2017); Cao et al. (2017, 2021); Cheng
et al. (2020); Pishchulin et al. (2016); Insafutdinov et al. (2016); Iqbal and
Gall (2016). Top-down approaches accomplish body joint association by a well-
trained person detector. They usually detect person instances by using a
person detector, then estimate the single-person pose within the detected box.
Bottom-up approaches instead detect all body joints in an image and then
associate them with individuals. Both of them carry out a two-stage strategy,
which remains an obstacle to real-time inference.

In this work, we propose a single-stage framework, PoseDet. It combines joint
localization and association using a simply designed network to eliminate
redundant operations. The central idea of PoseDet is to view multi-person pose
estimation as an object detection problem, _i.e._ , estimating poses by
detection. Object detection aims to regress two corner points of the bounding
box, while pose estimation can be regarded as the regression of a set of body
joints. By replacing corner points with key points of body joints, poses can
be estimated using an object detection system directly. However, only the
replacement of regression target decreases the performance due to the
complexity of regression space. Techniques including displacement refinement
and keypoint-based operations are adopted, making the detection model more
accessible to pose estimation.

Another challenge is the diversity of poses. To accurately regress the
keypoints of body joints, the representation of objects is supposed to be
informative and fine-grained. In modern detection systems, the object is
usually represented as axis-aligned bounding box Ren et al. (2015) or the
center point Tian et al. (2019). However, these bounding boxes contained
unexpected background areas that might lower the feature quality, and the
highly overlapped boxes could be incorrectly suppressed by non-maximum
suppression (NMS). Representation with the center point faces the problem that
center points of distinctive objects could be too close to distinguish. Both
representations provide rough information of the object and are not suitable
for pose estimation.

A more delicate representation is proposed by RepPoints Yang et al. (2019).
The object is represented using a set of sample points whose locations are
learned in an unsupervised way. Point-set Anchors Wei et al. (2020) utilize it
in pose estimation, where predefined anchors of poses are used as initial
locations of sample points. Inspired by Yang et al. (2019); Wei et al. (2020),
we propose the keypoint-aware pose embedding. As shown in Figure 1, we use the
body joints as keypoints to represent the person instance. In conjunction with
the deformable convolutional network (DCN) Dai et al. (2017), we generate the
pose embedding by aggregating the extracted feature vectors at locations of
keypoints. Pose embedding is informative as it contains both semantic
information of joint category and geometric information of the pose. The pose
embedding directly predicts the refined pose and the score of each instance.
Different from Point-set Anchors, the locations of keypoints are learnable and
supervised by GT keypoints, making them closer to target keypoints during
prediction, which allows PoseDet to be robust to various poses. Moreover,
PoseDet is an anchor-free detector, leading to lower complexity of the network
and a higher inference speed than those of Point-set Anchors. As a general and
flexible representation strategy, the proposed pose embedding can be easily
applied to understand other kinds of objects.

Due to the efficient pipeline and the adequate representation, the proposed
PoseDet achieves an unprecedented speed and a competitive accuracy for multi-
person pose estimation. We evaluate our method on COCO Lin et al. (2014)
benchmark. The inference speed of PoseDet is 2.5 times higher (10 FPS) than
Point-set Anchors with the same scale of the input image and backbone
HRNet-W48 Sun et al. (2019). By replacing the backbone with DLA-34 Yu et al.
(2018), the model runs faster at 42 FPS with the accuracy’s decline. The
effectiveness of keypoint-aware pose embedding is demonstrated on CrowdPose Li
et al. (2019) benchmark. PoseDet achieves comparable accuracy with the state-
of-the-art solutions and shows its robustness in the crowd scenes.

The contributions of our work are:

  * •

We reformulate the multi-person pose estimation as an object detection
problem. Our proposed PoseDet framework achieves an unprecedented speed and a
competitive accuracy compared with state-of-the-art approaches;

  * •

We propose the keypoint-aware pose embedding as a general and flexible
representation for person instances, making PoseDet being robust to real-world
crowd scenes with occlusion and various poses. It can be easily applied to
other instance-level recognition tasks like person re-identification;

  * •

The performance of PoseDet is further improved by displacement refinement and
keypoint-based operations.

##  2 Related Work

Due to the various applications in the real world, human pose estimation is an
active research topic for decades. In the early works, joint locations are
predicted based on hand-craft features, such as histogram of oriented gradient
Sun and Savarese (2011); Wang and Li (2013). Recently, state-of-the-art
performance is achieved using the Convolutional Neural Networks (CNNs). In
this work, based on CNNs, we focus on simplifying detection procedures and
improving object representation for multi-person pose estimation.

Real-time multi-person pose estimation. Multi-person pose estimation requires
dense prediction over spatial locations as well as detection of the person
instance. The complexity makes it hard to inference at real-time speed. The
key is to optimize processing procedures and eliminate redundant structures.
Cao et al. (2017) is the first real-time algorithm to tackle this problem.
They generate the ground-truth (GT) confidence maps to localize joints and
associate them via part affinity fields. Kocabas et al. (2018) predict the
heatmap of joints for joint localization and associate them with a person
detector and the pose residual network, where all subnets share the same
backbone. Nie et al. (2019) present SPM, a single-stage solution that predicts
hierarchical offsets of joints based on detected root points. The single-stage
strategy makes it faster than two-stage (_e.g._ , bottom-up or top-down)
strategies. We present a faster single-stage solution that combines joint
localization and association in the same pipeline.

Object representation in multi-person pose estimation. Object representation
serves as a guide to extract features for purposes such as object
classification and location refinement. He et al. (2017) use the bounding box
to represent an object. Features are extracted by the bounding box to predict
the heatmap of joints. Box-based representation faces the problem of including
the non-target area, resulting in low-quality feature and computation
overhead. Zhou et al. (2019) model an object as a single point and regress
joint offsets at each point. Center point-based representation is fast but
contains limited information for dense prediction of joints. Wei et al. (2020)
obtains features at a set of predefined anchors to acquire informative
features. For refinement of segmentation boundary, Kirillov et al. (2020)
regards an object as a continuous entity, which is encoded in feature maps and
can be accessed at any points by interpolation. We follow the point-set
representation and the concept of continuous entity, presenting pose embedding
that extracts keypoint-aware features.

##  3 Approach

Figure 2: An overview of PoseDet. We reformulate multi-person pose estimation
as an object detection problem. Regression and classification branch conduct
dense predictions over feature maps.

In this section, we first elaborate keypoint-aware pose embedding, a key
component of PoseDet in understanding objects effectively and efficiently.
Next, we present the multi-level anchor-free detector and show how to utilize
pose embedding in it to estimate 2D poses of persons in an image.

###  3.1 Keypoint-Aware Pose Embedding

Object representation is an important medium to recognize objects in visual
tasks (_e.g._ , classification and regression). As discussed in Section 1,
features extracted by the bounding box and center point are either redundant
or limited for pose estimation. In our system, objects are represented as
keypoints, and features at keypoints are extracted and aggregated for
classification and regression.

Given the preceding feature maps, the person instance can be regarded as a
continuous entity encoded in the feature maps. As shown in Figure 1, the
feature vectors at locations of body joints contain semantic and geometric
information of the pose.

In natural language processing, discrete text data are represented as
continuous vectors via word embedding to capture semantic and syntactic
features Paccanaro and Hinton (2001). Analogously, we extract and aggregate
the irregular feature vectors to represent the person instance. The
representation is mapped from a set of vectors to one vector, named pose
embedding. Formally, the process can be summarized as:

$f:\\{\psi_{i}\\}_{i=1}^{K}\rightarrow\varepsilon,$ (1)

where $\psi_{i}$ donates feature vector at point $i$ on the feature maps, $K$
is the number of joints. $\varepsilon$ is a vector, named pose embedding,
containing the information of $K$ joints.

The pose embedding contains semantic and geometric information because both
the shape and joint categories are confirmed. It’s an informative
representation that can be used to estimate pose and classify positive or
negative candidates, but it’s hard to make full use of them by conventional
CNNs due to the restriction of the local receptive field and various poses.
Considering dense hypotheses over spatial locations of feature maps in modern
detection networks, which means dense pose embeddings are required. Following
Yang et al. (2019); Wei et al. (2020), we generate dense pose embeddings for
these hypotheses via one DCN layer simultaneously. The DCN layer takes offsets
as well as feature maps as input to generate pose embeddings. The offset is a
2K-dimensional vector for each hypothesis, which indicates the positions of K
keypoints based on the location of the hypothesis on the feature map. DCN
generates the pose embedding for each hypothesis according to different
offsets, which allows us to access features at arbitrary positions without
limitation of geometric structure efficiently.

The learning of offsets is crucial to the performance of DCN. Dai et al.
(2017) adopt an unsupervised way to drive the learning of offsets implicitly.
Yang et al. (2019) use weak supervision by minimizing the loss between the GT
bounding boxes and pseudo boxes generated by offsets. Wei et al. (2020) use
the most common poses as predefined anchors to generate fixed offsets. In this
work, a bypass is applied to generate the keypoint-aware offsets, which is
supervised by GT joints, driving the learning of offsets explicitly. This
allows us to generate meaningful feature representation stably. Moreover,
dynamic offsets provide more flexibility thanWei et al. (2020), making our
PoseDet robust to various poses.

Pose embedding is applied in PoseDet to estimate pose. As a general
representation strategy, we note that it’s easy to expand to other objects by
replacing the joint with other kinds of keypoint.

###  3.2 PoseDet

Recently, one-stage detectors show compatible accuracy to two-stage strategies
with higher inference speed. For fast pose estimation, we start from RepPoints
Yang et al. (2019), a one-stage anchor-free detector. By utilize pose
embedding in RepPoints, we propose PoseDet, a pose estimation machine that
combines human detection and poses estimation into the same pipeline.

The pipeline of our approach is illustrated in Figure 2. PoseDet comprises a
backbone for feature extraction, a bypass for offsets prediction, and two
branches for joint localization and candidate scoring.

The backbone is a feature pyramid network (FPN), producing a multi-scale
feature pyramid from a single resolution image. Objects are assigned to
different levels according to the scale, which partially solves the problem of
scale diversity of poses. PoseDet makes dense hypotheses over the spatial
location of multi-level feature maps, also known as candidates. We estimate
the pose of each candidate and score it.

Let $\mathbb{F}$ be the single-level feature maps of size $C\times H\times W$.
There are $H\times W$ candidates at that level, processed by the two branches
to estimate $H\times W$ possible poses. The bypass produces $H\times W$ sets
of offsets, where each set is a 2K-dimensional vector, indicating the offsets
from the location of a candidate to the body joints of a person instance.
Given the predicted offsets and feature maps extracted by FPN, the other two
branches implement task-specific pose embedding for displacement refinement
and candidate scoring, respectively. We utilize the above full convolutional
networks on each level of FPN, producing dense multi-level predictions, then
adopt NMS to eliminate redundant predictions.

In the following sections, we describe the details of regressing body joints
and scoring candidates and the intermediate supervision of FPN. Finally, we
introduce training and inference of PoseDet, including assignment strategies
and NMS.

####  3.2.1 Joint Localization

Multi-person poses in an image can be represented as:

$\mathbb{P}=\\{(x_{1}^{i},y_{1}^{i}),(x_{2}^{i},y_{2}^{i}),...,(x_{K}^{i},y_{K}^{i})\\}_{i=1}^{N},$
(2)

where N is the number of persons in images, and $(x_{j}^{i},y_{j}^{i})$
donates the coordinates of $j$ the joints of person $i$. To estimate the pose
of a person instance, we regress the displacements from the candidate’s point
to GT joints (each candidate is assigned with a GT pose). It’s hard to regress
the displacements directly due to the variety of poses. To tackle the problem,
following Nie et al. (2019), we implement a coarse-to-fine manner to
approximate a more accurate location estimation. The process can be formulated
as:

$(x_{j}^{i},y_{j}^{i})=(x,y)+(\delta_{x}^{c},\delta_{y}^{c})+(\delta_{x}^{f},\delta_{y}^{f}).$
(3)

Here, (x, y) is the coordinate of the candidate,
$(\delta_{x}^{c},\delta_{y}^{c})$ is coarse displacement, and
$(\delta_{x}^{f},\delta_{y}^{f})$ is refined displacement.

The offsets predicted by the bypass provide a coarse prediction of the pose,
which not only guides DCN to generate pose embedding but also gives the coarse
displacements. We estimate the refined displacements based on the coarse
prediction. The regression branch is composed of two $3\times 3$ conv layers,
and a deformable convolution layer is applied to produce pose embedding. The
refined displacements are regressed directly via $1\times 1$ conv layer, which
takes pose embedding as input. We get the final prediction by aggregating the
coarse prediction and the refinement (Equation 3).

Pose embedding encodes the bias of coarse prediction, leading to a more
precise estimation, solving the problem of regressing offsets on a large
scale, high-dimensional vector space.

####  3.2.2 Candidate Classification

Based on the coarse prediction, we evaluate the candidate by classifying
positive and negative predictions, where the predicted score indicates the
probability of positive prediction. We use Object Keypoint Similarity (OKS) as
a metric to evaluate the similarity between predicted pose and GT pose.
Predictions with OKS higher than 0.6 indicate positive, lower than 0.5
indicate negative. OKS is a collection of distances between every two joints
of the same category (_i.e._ , the similarity between two point sets). It
provides a point-wise evaluation for the evaluation of poses.

The classification branch after FPN is used to generate pose embedding for
candidate scoring, which is composed of two $3\times 3$ conv layers and a
deformable convolution layer. A following $1\times 1$ conv layer is applied to
estimate the score. The predicted scores and poses are used in NMS to suppress
redundant predictions.

####  3.2.3 Intermediate Supervision

The feature map of FPN is considered a mapping of the person instance, where
we extract the object representation to generate keypoint-aware pose
embeddings. To force the FPN to be sensitive to body joints, following Newell
et al. (2016); Wei et al. (2016), we utilize intermediate supervision with GT
joints. Specifically, we add a branch consisted of a $3\times 3$ conv layer
and a $1\times 1$ conv layer, which takes the multi-level feature maps of FPN
as input and generates multi-level heatmaps with the same resolutions. The
heatmaps indicate the probabilities of joints over spatial locations, where
the number of channels is equal to the number of joints. Naturally, for the
heatmaps of each level, we generate 2D Gaussian maps (with a standard
deviation of 2 pixels) centered on the GT joint location as targets. We
minimize the $l_{2}$ loss between predicted heatmaps and targets during
training.

We note that the intermediate supervision is a cost-free improvement because
this branch is not involved in testing. We demonstrate the benefit of it in
ablation study (Section 4.4).

Figure 3: Upper bounds of OKS-based NMS and IoU-based NMS on CrowdPose.

####  3.2.4 Training and Inference

In the training phase, we use different losses and assigning strategies in the
bypass, the classification branch, and the regression branch. The loss of
network can be summarized as:

$\begin{split}Loss=L_{2}(GTPose,\delta^{c})+L_{2}(GTPose-\delta^{c},\delta^{f})\\\
+FL(GT,scores)+L_{2}(GM,Heatmaps)\end{split}$ (4)

On the right side of the equation, the first item is loss of the bypass. The
bypass is responsible for providing a coarse prediction of each candidate. We
use GT joints as supervision. Each GT pose is allocated to a level in FPN
according to its size and assigned to a candidate at the same level closest to
the centroid of the GT pose. The centroid is calculated by mapping the spatial
locations of the joints to feature maps and averaging the coordinates. We
demonstrate the advantage of the centroid of joints compared with the centroid
of the bounding box in Section 4.4. Candidates assigned with targets are
trained with $l_{2}$ loss. The second item is the loss of the regression
branch, which is responsible for predicting more delicate poses based on the
coarse predictions. We compute OKS between the coarse prediction and every GT
pose, then assign the larges OKS to the candidate. Candidates with OKS higher
than 0.7 are trained with $l_{2}$ loss. The third item is the loss of the
classification branch. Both positive and negative candidates are trained with
focal loss Lin et al. (2017) (FL) to address the large class imbalance issue.
The last item is loss of intermediate supervision, which is $l_{2}$ loss
between Gaussian maps (GM) and predicted heatmaps.

For the inference, we predict each candidate’s pose and confidence score, then
implement OKS-based NMS Papandreou et al. (2017) to remove redundant
predictions.

Most of the modern detection systems use the IoU of bounding boxes as the
measurement to evaluate the overlap of predictions. When two predictions have
an overlap over the predefined threshold, the prediction with a lower
confidence score is suppressed. One problem is that the IoU-based NMS might
suppress positive samples that occlude heavily, which is not suitable for
multi-person pose estimation, especially for the crowd scenes.

In our system, the overlap is measured in terms of OKS. OKS is dependent on
distances between joints, leading to a more discriminate measurement of the
similarity. To examine the upper bound on persons with a high crowd index, we
use GT keypoints in the CrowdPose test set as input, performing OKS-based NMS
and IoU-based NMS on different thresholds, respectively. AP Hard is adopted as
the metric that contains the most crowd. As shown in Figure 3, the IoU-based
NMS wrongly suppress more positive samples than OKS-based NMS, showing that
the OKS is a better measurement to distinguish occluded persons.

##  4 Experiments

Method | Backbone | Input size | FPS | AP | $\text{AP}^{\text{50}}$ | $\text{AP}^{\text{75}}$ | $\text{AP}^{\text{M}}$ | $\text{AP}^{\text{L}}$
---|---|---|---|---|---|---|---|---
Ass.Embedding (2017) | Hourglass-4 | 512 | 4 | 56.6 | 81.8 | 61.8 | 49.8 | 67.0
CMU-Pose (2017) | 3CM-3PAF(102) | 368 | 10 | 61.8 | 84.9 | 67.5 | 57.1 | 68.2
PersonLab (2018) | ResNet-152 | 1401 | 2 | 66.5 | 88.0 | 72.6 | 62.4 | 72.3
Higher HRNet (2020) | HRNet-W48 | 640 | 4 | 65.7 | 86.3 | 72.3 | 60.8 | 72.6
Poi. Anchors (2020) | HRNet-W48 | 640 | 4 | 67.4 | 88.3 | 74.6 | 64.7 | 73.0
PoseDet (Ours) | HRNet-W48 | 640 | 10 | 66.2 | 87.5 | 73.2 | 62.3 | 73.2
PoseDet (Ours) | HRNet-W32 | 512 | 18 | 64.4 | 86.7 | 71.0 | 58.8 | 73.3
PoseDet (Ours) | DLA-34 | 512 | 42 | 59.2 | 84.3 | 64.6 | 52.7 | 68.8
Table 1: Comparison with state-of-the-art bottom-up approaches on COCO test-
dev, both keypoints AP and FPS are obtained without TTA. Input size indicates
the sort side of an input image.

###  4.1 Experimental Setting

Dataset. We evaluate PoseDet for accuracy and speed of multi-person pose
estimation on a widely adopt benchmark, COCO, and evaluate its robustness in
the crowd scenes on the CrowdPose benchmark.

COCO is a large dataset contains over 200K images, divided into
train/val/test-dev splits. Each person is labeled with 17 body joints. Models
are trained using train split (64k images). We report results on COCO test-dev
split (20K images) for comparisons to state-of-the-art methods. The official
Average Precision (AP), calculated in terms of OKS between the prediction and
GT pose, is adopted as a metric to evaluate the accuracy, and Frames per
Second (FPS) is adopted for the speed.

CrowdPose consists of 20,000 images, including about 80,000 persons. Each
person is labeled with 14 body joints. CrowdPose is divided into train,
validation, and test sets in the proportion of 5:1:4. The average intersection
over union(IoU) of human bounding boxes is 0.27 (0.06 in COCO). We evaluate
PoseDet on CrowdPose to validate the robustness in the crowd scenes. The test
set of CrowdPose is divided into three crowding levels, indicated as easy,
medium, and hard. Three metrics, $\text{AP}^{\text{E}}$,
$\text{AP}^{\text{M}}$, and $\text{AP}^{\text{H}}$ are adopted respectively,
where $\text{AP}^{\text{H}}$ contains the most crowd.

Implementation Details. We follow the conventional data augmentation
strategies and training schedules for multi-person pose estimation. The COCO
and CrowdPose dataset share the same training/testing settings for
convenience.

During model training, images are augmented with horizontally flipping and
resize factor in $[0.5,1.5]$, then cropped centered on a random person to
$512\times 512$ patches. The cropped patches are fed into the network as
training samples. All models are optimized with Adam Kingma and Ba (2015) over
8 NVIDIA V100 GPUs with a mini-batch of 48 patches (6 patches per GPU). The
initial learning rate is 0.0001, decreased at epoch 180 and 200 by a factor of
0.1. Training converges at epoch 210. We examine different backbones before
FPN, which is initialized with the weights pre-trained on ImageNet Deng et al.
(2009).

For evaluation, we conduct single-scale and multi-scale testing. For single-
scale testing, unless specified, the input images are resized to have a
shorter side of 640 and a longer side less than 1000, with an unchanged aspect
ratio. For multi-scale testing, the input image is resized to an image pyramid
with a sorter side of $\\{300,400,500,600,700,800\\}$ and augmented by
horizontal flip. NMS is conducted over the results of augmented images. The
inference speed is evaluated on GPU GTX 1080Ti and CPU Intel i9-7980XE. All
experiments are implemented based on mmdetection Chen et al. (2019).

###  4.2 Analysis of Accuracy and Speed

Both accuracy and speed are important in real-world applications of multi-
person pose estimation, while current pose estimation metrics are based on the
accuracy of keypoints. To provide a more complete comparison ,we evaluate
approaches on COCO test-dev and present keypoints AP and inference speed (FPS)
in Table 1. We compare the performances of bottom-up approaches without test-
time augmentation (TTA).

For multi-person pose estimation, models with the highest accuracy are
dominated by top-down approaches Sun et al. (2019); Cai et al. (2020). The
bounding boxes of persons are detected by a person detector and resized to a
fixed size, which avoids the scale diversity, leading to higher accuracy and
lower inference speed than bottom-up approaches. For example, HRNetSun et al.
(2019) achieves keypoints AP of 75.5 on COCO test-dev. They perform single-
person pose estimation on detected boxes (20 boxes per image roughly, each box
is resized to $384\times 288$), with an inference time of about 90 ms per box.
The performance of inference speed can not be analyzed due to the inference
time of the person detector is not reported. We only consider bottom-up
approaches in Table 1.

Moreover, TTA, including multi-scale testing and flip, is widely adopted in
multi-person pose estimation. With TTA, too big or too small targets can be
resized to a more proper size, making them easier to be detected. TTA can
improve the accuracy by additional detection but increases inference time and
computation overhead. TTA with a six-scale image pyramid and flip increase the
computation by about 12 times. We argue that discarding TTA is more practical
in application scenarios where real-time performance is required. Hence, we
provide a consistent comparison in Table 1, where both AP and FPS are obtained
without TTA.

Specifically, for CMU-Pose Cao et al. (2017) and Associative Embedding Newell
et al. (2017), we report the results cited from their papers or GitHub
repositories. For Point Anchors Wei et al. (2020) and Higher HRNet Cheng et
al. (2020), the results are based on our implementation, because either AP or
FPS without TTA is not reported by authors.

Given the results, we can see that PoseDet outperforms other methods in terms
of the tradeoff between accuracy and speed. The inference speed of PoseDet
with backbone HRNet-W48 is 2.5 times higher than Point Anchors and Higher
HRNet with the same backbone and scale of the input image. PoseDet with a
lighter backbone DLA-34 achieves a maximum speed (42 FPS) with lower accuracy
(59.2 AP). The main reason why backbone HRNet achieves higher accuracy than
DLA, especially refers to $\text{AP}^{\text{75}}$, is that HRNet extracts
high-resolution representation rich features, which are significant for the
pixel-level prediction of joints. To best our knowledge, PoseDet is the
fastest approach for multi-person pose estimation.

Without consideration of the inference speed, we also report results with TTA
in Table 2. The comparison indicates the ability of PoseDet to estimate
accurate poses. All reported numbers for comparison are cited from the papers.

Method | Backbone | AP | $\text{AP}^{\text{50}}$ | $\text{AP}^{\text{75}}$ | $\text{AP}^{\text{M}}$ | $\text{AP}^{\text{L}}$
---|---|---|---|---|---|---
PersonLab | ResNet-152 | 68.7 | 89.0 | 75.4 | 64.1 | 75.5
Ass.Embedding | Hourglass-4 | 65.5 | 86.8 | 72.3 | 60.6 | 72.6
SPM | Hourglass-8 | 66.9 | 88.5 | 72.9 | 62.6 | 73.1
Higher HRNet | HRNet-W48 | 70.5 | 89.3 | 77.2 | 66.6 | 75.8
Poi. Anchors | HRNet-W48 | 68.7 | 89.9 | 76.3 | 64.8 | 75.3
PoseDet (Ours) | HRNet-W48 | 67.8 | 89.3 | 75.0 | 63.3 | 75.5
PoseDet (Ours) | HRNet-W32 | 66.7 | 88.5 | 73.7 | 62.2 | 75.5

Table 2: Performance on COCO test-dev with TTA.

###  4.3 Robustness in the Crowd Scenes

Method | AP | $\text{AP}^{\text{50}}$ | $\text{AP}^{\text{75}}$ | $\text{AP}^{\text{E}}$ | $\text{AP}^{\text{M}}$ | $\text{AP}^{\text{H}}$
---|---|---|---|---|---|---
Mask-RCNN | 57.2 | 83.5 | 60.3 | 69.4 | 57.9 | 45.8
AlphaPose | 61.0 | 81.3 | 66.0 | 71.2 | 61.4 | 51.1
Simple Baseline | 60.8 | 81.4 | 65.7 | 71.4 | 61.2 | 51.2
SPPE | 66.0 | 84.2 | 71.5 | 75.5 | 66.3 | 57.4
Higher HRNet | 67.6 | 87.4 | 72.6 | 75.8 | 68.1 | 58.9
PoseDet (Ours) | 67.3 | 88.9 | 72.8 | 75.2 | 68.0 | 58.2

Table 3: Comparison with state-of-the-art methods on CrowdPose with TTA.

Another important issue in real-world applications is the robustness to
various poses, including occluded persons and various poses. Most of images in
popular datasets (_e.g._ , COCO Li et al. (2019) and MPII Andriluka et al.
(2014)) have no overlapped persons. We evaluate the robustness of PoseDet on
the CrowdPose benchmark, which has a larger crowd index and more persons in an
image than COCO. The inference speed of PoseDet is not affected by the number
of persons, which is the same on CrowdPose as COCO, so we focus on evaluating
the accuracy on CrowdPose. PoseDet with HRNet-W48 backbone is trained using
the train set and evaluated using the test set of CrowdPose with TTA. Table 3
summarizes the results compared with the state-of-the-art approaches.

Higher HRNet Cheng et al. (2020) achieves the best accuracy on CrowdPose. With
the same backbone, PoseDet achieves higher inference speed and competitive
accuracy than Higher HRNet, showing the robustness of PoseDet. Further, The
accuracy of PoseDet suppress top-down methods (Mask-RCNN He et al. (2017),
AlphaPose Fang et al. (2017) and Simple Baseline Xiao et al. (2018)) by a
large margin, especially in the heavily occluded cases
($\text{AP}^{\text{H}}$). The main reason is that these top-down methods use a
person detector to separate person instances with rectangular boxes. As
discussed in 1, bounding boxes perform poorly for occluded cases.

The reason for robustness is that PoseDet adopts keypoint-aware pose
embedding. The offsets of pose embedding are learnable and supervised by
keypoints. As shown in Figure 4, PoseDet achieves a more accurate prediction
than Point Anchors Wei et al. (2020) in corner cases, where Point Anchors use
fixed anchors as offsets.

JR | AP | $\text{AP}^{\text{50}}$ | $\text{AP}^{\text{75}}$ | $\text{AP}^{\text{E}}$ | $\text{AP}^{\text{M}}$ | $\text{AP}^{\text{H}}$
---|---|---|---|---|---|---
| 51.7 | 85.2 | 54.9 | 57.8 | 52.1 | 45.5
✓ | 65.9 | 87.7 | 76.1 | 73.6 | 66.8 | 56.9

Table 4: Effect of joint refinement (JR) on CrowdPose with backbone HRNet-W48.

Centroid type | NMS type | IS | $\text{AP}^{\text{E}}$ | $\text{AP}^{\text{M}}$ | $\text{AP}^{\text{H}}$
---|---|---|---|---|---
Bounding box | IoU |  | 72.4 | 65.3 | 54.4
✓ | 73.3 | 65.9 | 55.2
OKS |  | 72.9 | 66.1 | 55.7
✓ | 73.6 | 66.7 | 56.6
Keypoints | IoU |  | 72.6 | 65.5 | 55.0
✓ | 73.3 | 66.1 | 55.7
OKS |  | 72.9 | 66.2 | 56.6
✓ | 73.6 | 66.8 | 56.9

Table 5: Effect of keypoint-based operations on CrowdPose with backbone
HRNet-W48. (IS: Intermediate supervision) Figure 4: Comparison of Point
Anchors and PoseDet. The upper row is predictions of Point Anchors, and the
lower row is results of PoseDet. PoseDet shows more robustness in corner cases
(_i.e._ , larger persons and rare poses). Figure 5: Predictions of PoseDet,
where the upper row is the coarse predictions and the lower row is the finer
predictions.

###  4.4 Ablation study

PoseDet is basically a detector. Displacement refinement and keypoint-based
operations are adopted to make it friendly for pose estimation. We perform
ablation experiments to analyze the effect of these strategies. All
experiments are conducted using PoseDet with backbone HRNet-W48. Models are
trained using the train set, tested on the test set of CrowdPose without TTA.

Effect of displacement refinement. Table 4 shows the results of using
displacement refinement (DR). PoseDet conducts DR via pose embedding based on
the coarse prediction. The metrics, $\text{AP}^{\text{50}}$ and
$\text{AP}^{\text{75}}$, measure predictions with OKS higher than 50 and 75
respectively. We can find out that PoseDet without DR already achieves strong
performance on the looser metric (85.2 $\text{AP}^{\text{50}}$). DR improve
the accurate prediction by a large margin (+21.2 $\text{AP}^{\text{75}}$). The
comparison demonstrates the effectiveness of pose embedding on encoding the
bias of the coarse prediction.

The qualitative analysis in Figure 5 comes to consistent results. In the upper
row, coarse predictions cover most keypoints. A high $\text{AP}^{\text{50}}$
is achieved with the coarse prediction. It also provides offsets for the DCN
layer to generate keypoint-aware pose embedding, where the offsets are closely
relative to keypoints. In the upper row, some keypoints that are not accurate
enough are refined by predictions of pose embeddings. Estimation of poses is
handled in a coarse-to-fine manner successfully.

Effect of keypoint-based operations. We compare PoseDet with different
strategies of target assignment and NMS, as well as intermediate supervision
in Table 5.

The centroid of the bounding box is widely used as a basis to assign targets
in object detection Qiu et al. (2020); Tian et al. (2019). Pose estimation
contains information on keypoints, which is more informative than the bounding
box. We make use of it by replacing the centroid of the bounding box with the
centroid of keypoints. We argue that the centroid of keypoints works better in
occluded cases. We compare the performance that assigning a GT pose to the
closest candidate according to the centroid of the bounding box or keypoints
during training. In the results, we can find out that the centroid of
keypoints improves the accuracy consistently.

As discussed in Section 3.2.4, OKS is a more discriminate measurement to
distinguish persons in the crowd than IoU. We conduct experiments with IoU NMS
and OKS NMS. OKS NMS improves the performance of PoseDet in the crowd scenes
(+1.4 $\text{AP}^{\text{H}}$ on average).

The keypoints are further used in intermediate supervision. We use GT
keypoints to generate Gaussian maps as supervision of FPN, forcing the FPN to
perceive body joints. The accuracy is improved consistently.

##  5 Conclusion and Future Works

In this paper, we present keypoint-aware pose embedding, which can be seen as
a general strategy to efficiently extract and aggregate informative feature
representation. We propose PoseDet, a simple yet efficient method for multi-
person pose estimation, in conjunction with the pose embedding. PoseDet
simplifies the procedures of estimating, predicting accurate poses at an
unprecedented speed. We conduct displacement refinement and keypoint-based
operations to improve the accuracy of PoseDet.

The future works include the applications to other instance-level recognition
tasks, _e.g._ , person re-identification, pose tracking, as well as the
investigation on selecting the number of keypoints adaptively.

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# Full counting statistics and large deviations in thermal 1D Bose gas

Maksims Arzamasovs  Department of Applied Physics, School of Science, Xi’an
Jiaotong University, Xi’an 710049, Shaanxi, China  Institute of Atomic Physics
and Spectroscopy, University of Latvia, Riga, LV-1586, Latvia  Dimitri M.
Gangardt  School of Physics and Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK

###### Abstract

We obtain the distribution of number of atoms in an interval (full counting
statistics) of Lieb-Liniger model of interacting bosons in one dimension. Our
results are valid in the weakly interacting regime in a parametrically large
window of temperatures and interval lengths. The obtained distribution
deviates strongly from a Gaussian away from the quasi-condensate regime, and,
for sufficiently short intervals, the probability of large number fluctuations
is strongly enhanced.

Introduction. In-situ measurements of particle number fluctuations in a one-
dimensional (1D) ultra cold Bose gas have been recently performed in
experiments with ultra cold 87Rb atoms on a chip Esteve  _et al._ (2006);
Armijo  _et al._ (2010); Jacqmin  _et al._ (2011). In these experiments
absorption images of a 1D gas of interacting bosons are divided into many
intervals of predetermined size $R$ of order of several microns and the number
of atoms in each pixel is inferred based on absorption intensity. The data
accumulated over several repetitions of such imaging was then used to extract
the second Esteve  _et al._ (2006) and third Armijo  _et al._ (2010) moments
of the obtained particle number distribution. This distribution, known as full
counting statistics (FCS) contains full information about many-particle
correlations. It is also an object which arises naturally in the experiments
Armijo  _et al._ (2010); Jacqmin  _et al._ (2011), so it is highly desirable
to have theoretical predictions for FCS.

Despite the fact that one-dimensional bosons with short range interactions are
amenable to description by the exactly solvable Lieb-Liniger model Lieb and
Liniger (1963), the theoretical treatment of this quantity is a formidable
task Korepin  _et al._ (1995); Bastianello  _et al._ (2018) as it involves
calculation of density correlations between an arbitrary number of different
spatial points Gaudin and Caux (2014); Korepin  _et al._ (1993). It was
suggested first in Ref. Armijo  _et al._ (2010) that one can use Yang-Yang
thermodynamics of Lieb-Liniger model Yang and Yang (1969) if the interval is
sufficiently large and can be viewed as a subsystem in contact with the
effective bath characterized by temperature $T$ and chemical potential $\mu$.
Then the moments of FCS can be obtained from an appropriate thermodynamic
relation involving mean density of particles $\bar{n}$ as a function of $\mu$
and $T$. This approach was later extended in Ref. Pietraszewicz and Deuar
(2017) to calculation of the fourth moment of FCS.

The results of these studies show that higher moments decay quickly with the
increasing of interval sizes and FCS becomes strongly peaked around mean
number of particles, $\bar{n}R$. This makes large deviations of particle
number from its mean value extremely improbable. In particular, the emptiness
formation probability, _i.e._ the probability to find a void of size $R$
considered in Ref. Korepin  _et al._ (1995); Abanov and Korepin (2002); Abanov
(2005) is exponentially small.

The situation is quite the opposite in the limit of microscopic intervals
$\bar{n}R\ll 1$. This limit was recently considered by Bastianello _et al._
Bastianello  _et al._ (2018) who obtained FCS using exact analytic Bethe
Ansatz calculations of local multiparticle correlations. An expected
consequence of these studies is that the most probable particle number is zero
and probability to find $N$ particles decays as $(\bar{n}R)^{N}$.

In fact, large deviations of number of particles become appreciable already
for intervals still containing typically a large number of particles,
$\bar{n}R\gg 1$, but shorter than a certain correlation length scale. For such
_mesoscopic_ intervals the central limit theorem does not hold and FCS
deviates strongly from the thermodynamic Gaussian distribution expected for a
collection of many independent intervals.

In this Letter we study FCS on intervals of arbitrary length and provide an
elegant and simple method for its calculation based on the exact mapping of
one-dimensional field theory onto a quantum mechanical problem, introduced in
Ref. Castin  _et al._ (2000). In the limit of short intervals the form of FCS
is shown to change continuously from a Gaussian to an exponential one as
temperature is increased, see Fig. 1. The limit of long intervals is
represented in Fig. 2 and our FCS agrees with the results of previous studies.
We also trace FCS as function of the interval length in Fig. 3. These results
show enhanced large deviations of the number of particles for mesoscopic
intervals where fluctuations play major role.

Figure 1: Normalized FCS $p(\nu,R)$ defined in Eq. (4) in the limit of short
intervals. Dimensionless temperature is $T/c\bar{n}=\xi/\ell_{\varphi}=0.1$,
0.25, 0.5, 1.0, 5.0.  Figure 2: Normalized FCS $p(\nu,R)$ defined in Eq. (4)
in the limit of long intervals. Dimensionless temperatures are the same as in
Fig. 1 and $R/\ell_{\varphi}=5$. Inset: reduced width $w(x)$ defined in Eq.
(22) characterizing fluctuations of number of particles.  Figure 3: Normalized
FCS $p(\nu,R)$ defined in Eq. (4) for intermediate intervals and high
temperature regime $T/c\bar{n}=\xi/\ell_{\varphi}\gg 1$. Dashed line
represents exponential distribution, Eq. (10).

Full counting statistics. The main quantity studied in this Letter is so-
called full counting statistics (FCS) defined as the probability $P_{N}(R)$ to
find exactly $N$ particles in an interval of length $R$. We define it via the
generating function,

$\displaystyle\chi(\lambda,R)$ $\displaystyle=\sum_{N=0}^{\infty}e^{-\lambda
N}P_{N}(R)=\left\langle e^{-\lambda\hat{N}_{R}}\right\rangle\,.$ (1)

Here $\hat{N}_{R}=\int_{0}^{R}\hat{\psi}^{\dagger}\hat{\psi}\,\mathrm{d}x$ is
the operator of number of particles in the interval. The statistical average
in Eq. (1) is performed in the equilibrium state of uniform 1D Bose gas with
contact interactions. For normal-ordered operators it is given by the
imaginary-time functional integral

$\displaystyle\left\langle:\\!F[\hat{\psi}^{\dagger},\hat{\psi}]\\!:\right\rangle=\frac{1}{Z}\int\mathcal{D}\bar{\psi}\psi\,F[\bar{\psi},\psi]\,e^{-S[\bar{\psi},\psi]}\,,$
(2)

where configurations of the complex-valued fields
$\bar{\psi}(x,\tau),\psi(x,\tau)$ are weighted by the action

$\displaystyle
S=\int_{0}^{1/T}\\!\mathrm{d}\tau\\!\int\\!\mathrm{d}x\left(\bar{\psi}\partial_{\tau}\psi+\frac{1}{2m}|\partial_{x}\psi|^{2}-\mu|\psi|^{2}+\frac{g}{2}|\psi|^{4}\right)\,,$
(3)

with $m$ being the atomic mass and $g$ the strength of 1D contact interaction.
The inverse thermodynamic partition function $Z$ ensures normalization and the
units are chosen such that $\hbar=1$, $k_{B}=1$. We are using grand canonical
formalism, but use the average density
$\bar{n}=\left\langle\bar{\psi}(x,\tau)\psi(x,\tau)\right\rangle$ as a control
parameter and adjust chemical potential $\mu$ accordingly.

For intervals containing a large number of particles, $\bar{n}R\gg 1$, it is
convenient to define the distribution

$\displaystyle
p(\nu,R)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\mathrm{d}k}{2\pi\mathrm{i}}e^{k\nu}\chi(k/\bar{n}R,R)=\bar{n}RP_{\nu\bar{n}R}(R)$
(4)

of the fraction of particles, $\nu=N/\bar{n}R$ treating it as a continuous
variable. We represent the generating function, Eq. (1), by a functional
integral as it were normal-ordered. It follows from the relation $\langle
e^{-\lambda\hat{N}_{R}}\rangle=\langle:\\!e^{(e^{-\lambda}-1)\hat{N}_{R}}\\!:\rangle$
that the errors introduced by this procedure are of order $1/\bar{n}R$ and we
neglect them.

Here we consider regime where the mean inter-particle separation $1/\bar{n}$
is the smallest of the characteristic length scales. The other two length
scales, in addition to the interval length, $R$, are the healing length
$\xi=1/\sqrt{mg\bar{n}}$ and the phase coherence length
$\ell_{\varphi}=\bar{n}/mT$. Both are much longer than $1/\bar{n}$ in the
regime of weak interactions $mg/\bar{n}\ll 1$ and degenerate bosons
$T<\bar{n}^{2}/m$. This leaves only two independent dimensionless parameters
which can be chosen to be $R/\ell_{\varphi}$ and $\xi/\ell_{\varphi}$. The
latter equals $T/c\bar{n}$, where $c=\sqrt{g\bar{n}/m}$ is the sound velocity
at zero temperature. We study FCS as function of these two parameters.

Classical field theory and effective quantum mechanics. The main obstacle in
calculating FCS is the non linearity of the action (3) which is responsible
for correlations between the particles and which makes the exact calculation
of FCS extremely difficult if at all possible. In the hydrodynamic approach of
Ref. Abanov and Korepin (2002); Abanov (2005) this difficulty was overcome by
expanding the action (3) near configurations of the fields contributing the
most to FCS. This method is limited to sufficiently low temperatures,
$\xi/\ell_{\varphi}\ll 1$, and sufficiently large intervals $R/\xi\gg 1$ where
the contribution of quantum and thermal fluctuations are small. Here we use an
alternative classical field method of Ref. Castin  _et al._ (2000) which
accounts properly for thermal fluctuations of arbitrary magnitude, but not the
quantum ones. The latter can be safely neglected under condition of
sufficiently high temperature, $T\gg g\bar{n}$, equivalent to
$\xi/\ell_{\varphi}\gg 1/\bar{n}\xi$. This condition and the condition of
quantum degeneracy $\xi/\ell_{\varphi}\ll\bar{n}\xi$ define a parametrically
wide range of temperatures where classical field method provides reliable
results for macroscopic intervals of _any length_ thus extending the validity
domain of the hydrodynamic approach of Refs. Abanov and Korepin (2002); Abanov
(2005).

Neglecting quantum fluctuations amounts to retaining only $\tau$-independent
configurations of fields in Eq. (3), leading to a $1+0$ dimensional field
theory described by the action

$\displaystyle S\simeq
S_{\mathrm{cl}}=\frac{1}{T}\int\mathrm{d}x\left(\frac{1}{2m}|\partial_{x}\psi|^{2}-\mu|\psi|^{2}+\frac{g}{2}|\psi|^{4}\right)\,.$
(5)

This action can be reformulated as an effective quantum mechanical problem if
we treat the rescaled spatial coordinate $\bar{n}x$ as an effective imaginary
time. The components of the complex field
$\psi=\sqrt{\bar{n}}re^{\mathrm{i}\theta}$ are parametrized by dimensionless
polar coordinates $(r,\theta)$ of a fictitious quantum particle moving in a
plane with the rotationally symmetric Hamiltonian,

$\displaystyle
H_{0}=-\frac{1}{2M}\nabla^{2}-\frac{\mu}{T}r^{2}+\frac{g\bar{n}}{2T}r^{4}$ (6)

with effective mass $M=\bar{n}\ell_{\varphi}$. Due to the infinite extension
of the integration in Eq. (5) the effective particle is in the ground state
$|0\rangle$ of the Hamiltonian (6) for $\bar{n}x=\pm\infty$.

The shape of the potential in Eq. (6) experienced by the effective quantum
particle is controlled by the value of $\mu/T$ obtained from the condition
$\langle 0|r^{2}|0\rangle=1$. It was shown in Ref. Castin  _et al._ (2000)
that for low temperatures where $\xi/\ell_{\varphi}\ll 1$, the chemical
potential is positive, $\mu/T>0$, and the potential experienced by the
effective particle has a characteristic “Mexican hat” shape, with the
effective particle localized near the valley $r\simeq 1$. This temperature
range corresponds to the quasi-condensate regime Kheruntsyan  _et al._ (2003);
Sykes  _et al._ (2008). For high temperatures, $\xi/\ell_{\varphi}\gg 1$,
corresponding to quantum degenerate regime of Refs.Kheruntsyan  _et al._
(2003); Sykes  _et al._ (2008), $\mu/T<0$ and the effective particle explores
vicinity of the minimum at $r=0$, where the potential is almost harmonic.

In the language of effective quantum mechanics the generating function (1) has
the following meaning. The ground state $|0\rangle$ is evolved for imaginary
time $\bar{n}R$ by the modified Hamiltonian $H_{\lambda}=H_{0}+\lambda r^{2}$
resulting in the modified state $e^{-\bar{n}RH_{\lambda}}|0\rangle$. The
generating function $\chi(\lambda,R)$ is then given by the normalized overlap

$\displaystyle\chi(\lambda,R)$ $\displaystyle=\langle
0|e^{-\bar{n}R(H_{\lambda}-E_{0})}|0\rangle\,,$ (7)

where $E_{0}$ is the ground state energy of $H_{0}$.

Short intervals. We first consider the case of a short interval $R$. In this
limit the imaginary time evolution of the ground state in Eq. (7) is obtained
by a multiplication of the rotationally symmetric ground state wave function
$\langle r|0\rangle=\Phi_{0}(r)$ by an exponential factor $e^{-\bar{n}R\lambda
r^{2}}$ so that

$\displaystyle\chi(\lambda,R)=2\pi\int r\mathrm{d}r\,e^{-\bar{n}R\lambda
r^{2}}|\Phi_{0}(r)|^{2}\,.$ (8)

The corresponding probability distribution

$\displaystyle p(\nu,R)$ $\displaystyle=\int\mathrm{d}k\int
r\mathrm{d}r\,e^{\mathrm{i}k(\nu-r^{2})}|\Phi_{0}(r)|^{2}$
$\displaystyle=2\pi\int
r\mathrm{d}r\,\delta\left(\nu-r^{2}\right)|\Phi_{0}(r)|^{2}=\pi|\Phi_{0}(\sqrt{\nu})|^{2}\,$
(9)

is independent of the interval length $R$ and is proportional to the ground
state probability density of the effective 2D quantum mechanical problem.

For high temperature, $\xi/\ell_{\varphi}\gg 1$, the ground state
$\Phi_{0}(r)$ is that of a two-dimensional harmonic oscillator, which is
simply $\Phi_{0}(r)=e^{-r^{2}/2}/\sqrt{\pi}$, as $1/M\omega_{0}=1$. Using Eq.
(Full counting statistics and large deviations in thermal 1D Bose gas) we see
immediately that FCS is exponential,

$\displaystyle p(\nu,R)=e^{-\nu}\,.$ (10)

The low temperature limit, $\xi/\ell_{\varphi}\ll 1$, corresponds to the
quasi-condensate regime. Expanding the Mexican hat shaped potential near the
minimum at $r=1$ we get an effective one-dimensional harmonic oscillator,

$\displaystyle MV(1+\delta
r)\simeq\frac{1}{2}\left(\frac{\ell_{\varphi}}{\xi}\right)^{2}\left(-1+4\delta
r^{2}\right),$ (11)

with the temperature independent frequency $\omega=2/\bar{n}\xi$. The
corresponding ground state wave-function

$\displaystyle\Phi_{0}(\delta
r)=\left(\frac{M\omega}{4\pi^{3}}\right)^{\frac{1}{4}}e^{-\frac{M\omega}{2}\delta
r^{2}}$ (12)

yields the approximate Gaussian distribution

$\displaystyle
p(\nu,R)=\sqrt{\frac{\ell_{\varphi}}{2\pi\xi}}e^{-\frac{1}{2}\frac{\ell_{\varphi}}{\xi}\left(\nu-1\right)^{2}}\,.$
(13)

The quadratic approximation (11) fails for large deviations $\nu-1\sim 1$ and
the corresponding quantum mechanical problem has to be solved numerically. We
find numerically the ground state of the Hamiltonian (6) and plot the
corresponding distributions in Fig. 1 for several values of
$\xi/\ell_{\varphi}=T/c\bar{n}$. The plots show how the exponential
distribution in Eq. (10) transforms into the Gaussian distribution of Eq. (13)
with decreasing temperature.

Long intervals. For sufficiently long intervals $R$ the evolution operator in
Eq. (7) becomes a projector

$\displaystyle e^{-\bar{n}R(H_{\lambda}-E_{0})}\simeq|\lambda\rangle
e^{-\bar{n}R\delta E_{\lambda}}\langle\lambda|\,,$ (14)

onto the ground state $|\lambda\rangle$ of the modified Hamiltonian,
$H_{\lambda}|\lambda\rangle=(E_{0}+\delta E_{\lambda})|\lambda\rangle$. The
precise criterion separating long intervals from short ones is thus
$\bar{n}R\,\delta E_{\lambda}\gg 1$. We rewrite this condition by extracting
the kinetic energy scale and defining $\Delta(s,\xi/\ell_{\varphi})=M\delta
E_{\lambda}$, where $s=\lambda M$. Using the fact that
$M=\bar{n}\ell_{\varphi}$ we see that the long interval condition becomes

$\displaystyle\frac{R}{\ell_{\varphi}}\Delta(s,\xi/\ell_{\varphi})\gg 1\,.$
(15)

Provided this condition is satisfied, the generating function (7) has the
following form

$\displaystyle\chi(\lambda,R)=A(\lambda
M,\xi/\ell_{\varphi})e^{-\frac{R}{\ell_{\varphi}}\Delta(\lambda
M,\xi/\ell_{\varphi})}\,,$ (16)

where the amplitude $A=|\langle\lambda|0\rangle|^{2}$ is independent of the
interval length. The same condition (15) allows to find the distribution (4)
by the saddle point method,

$\displaystyle p(\nu,R)$
$\displaystyle=\frac{R}{\ell_{\varphi}}\int\frac{\mathrm{d}s}{2\pi\mathrm{i}}\,A(s,\xi/\ell_{\varphi})e^{\frac{R}{\ell_{\varphi}}\left(s\nu-\Delta\left(s,\xi/\ell_{\varphi}\right)\right)}$
$\displaystyle\simeq
D(\nu,\xi/\ell_{\varphi})e^{\frac{R}{\ell_{\varphi}}\Gamma(\nu,\xi/\ell_{\varphi})}$
(17)

where Legendre transform
$\Gamma(\nu,\xi/\ell_{\varphi})=s\nu-\Delta(s,\xi/\ell_{\varphi})$ and the
prefactor

$\displaystyle
D(\nu,\xi/\ell_{\varphi})=\sqrt{\frac{R}{2\pi\ell_{\varphi}}}\frac{A(s,\xi/\ell_{\varphi})}{\sqrt{\left|\partial^{2}_{s}\Delta(s,\xi/\ell_{\varphi})\right|}}\,.$
(18)

are calculated at the saddle point obtained from the condition
$\nu=\partial_{s}\Delta(s,\xi/\ell_{\varphi})$.

In the high-temperature limit, $\xi/\ell_{\varphi}\gg 1$, the rescaled ground
state energy shift and the overlap become independent of temperature
$\Delta(s,0)=\sqrt{1+2s}-1$,
$A(s,0)=4\sqrt{1+2s}\left(1+\sqrt{1+2s}\right)^{-2}$ and we obtain

$\displaystyle p(\nu,R)=\sqrt{\frac{R}{2\pi\ell_{\varphi}}}\
\frac{4e^{\frac{R}{\ell_{\varphi}}\left(1-\frac{1}{2}\left(\nu+\frac{1}{\nu}\right)\right)}}{\sqrt{\nu}\left(1+\nu\right)^{2}}\,.$
(19)

For small deviations $|\nu-1|\ll 1$ this expression for FCS becomes a Gaussian
with variance $\overline{\delta\nu^{2}}=\ell_{\varphi}/R$.

In the low temperature regime, $\xi/\ell_{\varphi}\ll 1$, to the lowest order,
the rescaled ground state energy shift is a quadratic function
$\Delta(s,\xi/\ell_{\varphi})\simeq s-(\xi/\ell_{\varphi})^{2}s^{2}/2$ and
$A(s,\xi/\ell_{\varphi})\simeq 1$, so by performing Gaussian integration we
get

$\displaystyle
p(\nu,R)=\sqrt{\frac{R\ell_{\varphi}}{2\pi\xi^{2}}}\exp\left[-\frac{R\ell_{\varphi}}{2\xi^{2}}(\nu-1)^{2}\right]$
(20)

in full agreement with the hydrodynamic result of Refs. Abanov (2005); Abanov
and Korepin (2002). The variance is
$\overline{\delta\nu^{2}}=\xi^{2}/\ell_{\varphi}R$. In Fig. 2 the results for
$p(\nu,R)$ based on numerical calculations of Eq. (Full counting statistics
and large deviations in thermal 1D Bose gas) are shown for intermediate values
of dimensionless temperature $T/c\bar{n}=\xi/\ell_{\varphi}$.

Intermediate intervals. In the limiting cases of high and low temperature the
probability distribution, can be obtained for an interval of arbitrary length
$R$. The method is based on exact evolution of harmonic oscillator wave
functions under time-dependent variation of frequency and external force
Perelomov and Zeldovich (1998) as explained in Supplemental Material Sup . For
high temperatures, $\xi/\ell_{\varphi}\gg 1$ , FCS is shown in Fig. 3. It
interpolates between Eq. (10), and Eq. (19) and has distinctive non-Gaussian
shape.

For low temperatures, $\xi/\ell_{\varphi}\ll 1$, the distribution remains very
close to a Gaussian with variance depending on the interval length,

$\displaystyle p(\nu,R)=\sqrt{\frac{1}{2\pi
C(R/\xi)}\frac{\ell_{\varphi}}{\xi}}e^{-\frac{1}{2C(R/\xi)}\frac{\ell_{\varphi}}{\xi}(\nu-1)^{2}}\,.$
(21)

The crossover function $C(x)=(2x-1+e^{-2x})/2x^{2}$ behaves as $C(x)\simeq 1$
for $x\ll 1$ and $C(x)\simeq 1/x$ for $x\gg 1$ and interpolates between Eqs.
(13) and (20). This result could have otherwise been obtained using
hydrodynamical approach of Ref.Abanov (2005) with gradient terms included in
the action.

Variance of the particle number. For a macroscopic interval, Eq. (15), the
above results suggest the following scaling form for the variance of the
number of particles,

$\displaystyle\frac{\overline{\delta
N^{2}}}{\bar{n}R}=\bar{n}R\overline{\delta\nu^{2}}=\bar{n}\xi\,w(\xi/\ell_{\varphi})\,,$
(22)

where the universal function has the limiting behavior $w(x)=x$ for $x\ll 1$
and $w(x)=1/x$ for $x\gg 1$. For intermediate values of $x$ the numerical
results for $w(x)$ are shown in inset in Fig. 2 and confirm the non-monotonic
dependence of the particle number variance on temperature anticipated from the
limiting behaviors of $w(x)$. The right hand side of Eq. (22) is greater than
1 in the whole range of validity of our approach,
$1/\bar{n}\xi<\xi/\ell_{\varphi}<\bar{n}\xi$, and thus the fluctuations of
particle number are super-Poissonian in agreement with findings of Ref.
Jacqmin  _et al._ (2011).

Higher moments of FCS can also be obtained from the knowledge of generating
function $\chi(\lambda,R)$ and we calculate the third and the fourth moments
in Supplemental Material. They are in full agreement with the results of
previous studies Refs.Armijo  _et al._ (2010); Pietraszewicz and Deuar (2017).

Concluding remarks. The departure of FCS from Poisson distribution expected
for a classical ideal gas Sup  is a direct consequence of quantum statistics
and is closely related to bosonic bunching. At high enough temperatures we
found another manifestation of these quantum effects which lead to an enhanced
probability to find large (on the scale of mean inter-particle separation)
regions of depleted number of particles. For lower temperatures the inter-
particle interactions tend to suppresses such large density deviations from
its mean value. Our findings are relevant for temperatures, interactions and
interval lengths used in current experiments and can provide a novel way to
characterize the temperature and interaction strength due to the strong
dependence of FCS on these parameters.

Acknowledgments. MA would like to thank Prof. Bo Liu for encouragement. DMG is
grateful to IBS PCS, Daejeon, South Korea, for hospitality.

## References

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  * Armijo  _et al._ (2010) J. Armijo, T. Jacqmin, K. V. Kheruntsyan, and I. Bouchoule, Phys. Rev. Lett. 105, 230402 (2010).
  * Jacqmin  _et al._ (2011) T. Jacqmin, J. Armijo, T. Berrada, K. V. Kheruntsyan, and I. Bouchoule, Phys. Rev. Lett. 106, 230405 (2011).
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  * Gaudin and Caux (2014) M. Gaudin and J.-S. Caux,  _The Bethe Wavefunction_ (Cambridge University Press, Cambridge, England, 2014).
  * Korepin  _et al._ (1993) V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,  _Quantum Inverse Scattering Method and Correlation Functions_ (Cambridge University Press, Cambridge, England, 1993).
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  * Abanov and Korepin (2002) A. G. Abanov and V. E. Korepin, Nucl. Phys. B 647, 565 (2002).
  * Abanov (2005) A. G. Abanov, arXiv:cond-mat/0504307 (2005).
  * Castin  _et al._ (2000) Y. Castin, R. Dum, E. Mandonnet, A. Minguzzi, and I. Carusotto, J. Mod. Opt. 47, 2671 (2000).
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  * Sykes  _et al._ (2008) A. G. Sykes, D. M. Gangardt, M. J. Davis, K. Viering, M. G. Raizen, and K. V. Kheruntsyan, Phys. Rev. Lett. 100, 160406 (2008).
  * Perelomov and Zeldovich (1998) A. M. Perelomov and Y. B. Zeldovich,  _Quantum Mechanics - Selected Topics_ (World Scientific Publishing, 1998).
  * (17) See Supplemental Material at [URL will be inserted by publisher] for details of calculation.

## Supplementary material for
“Full counting statistics and large deviations in thermal 1D Bose gas”

##  I Full counting statistics of non-degenerate ideal Bose gas

Full counting statistics for classical (Boltzmann) non-interacting gas (e.g.
see the review of stochastic methods by Chandrasekhar Chandrasekhar (1943))
follows from combinatorial arguments and is given by the binomial distribution

$P_{N}(R)=\frac{N_{\mathrm{tot}}!}{N!(N_{\mathrm{tot}}-N)!}\left(\frac{R}{L}\right)^{N}\left(1-\frac{R}{L}\right)^{N_{\mathrm{tot}}-N}$
(S1)

where $N_{\mathrm{tot}}$ is the total number of particles confined to the
system of size $L$ and held at temperature $T$. In thermodynamic limit,
$N_{\mathrm{tot}}\rightarrow\infty$, $L\rightarrow\infty$, and
$\bar{n}=const$, the distribution Eq. (S1) converges to Poisson distribution,

$P_{N}(R)\rightarrow\frac{\left(\bar{n}R\right)^{N}e^{-\bar{n}R}}{N!}.$ (S2)

In the grand canonical case the distribution is still given by Eq. (S2), where
the equation of state of classical gas has to be specified,

$\bar{n}\lambda=e^{\mu/T},$

where $\lambda$ is the thermal (de Broglie) wavelength,

$\lambda=\sqrt{\frac{2\pi}{mT}}.$ (S3)

and $\mu$ is chemical potential. This results in the average and variance

$\overline{N}=\overline{\delta
N^{2}}=\overline{N^{2}}-{\overline{N}}^{2}=\bar{n}R.$ (S4)

For $\bar{n}R>1$ the distribution Eq. (S2) has a maximum at non-zero $N$ and
the distribution looks almost Gaussian: going from $N$ to $\nu=N/\bar{n}R$ and
treating it as continuous quantity, obtain

$p(\nu,R)\sim\exp(-\bar{n}R+\nu\bar{n}R-\nu\bar{n}R\ln(\nu)).$ (S5)

The maximum of the above expression occurs when $\nu=1$ and for $\nu\simeq 1$
obtain

$p(\nu,R)=\sqrt{\frac{\bar{n}R}{2\pi}}e^{-\frac{\bar{n}R}{2}(\nu-1)^{2}}.$
(S6)

In real systems the above results are relevant in the regime $T>T_{d}$, where
$T_{d}=\bar{n}^{2}/m$ is the temperature of quantum degeneracy, regime beyond
the reach of classical field approximation and therefore not considered in the
main text.

Extending the above analytic result to degenerate non-interacting Bosons is
non-trivial because of correlations caused by quantum statistics. This
corresponds to the regime $c\bar{n}<T<T_{d}$. The crossover from degenerate
quantum (corresponding to what is called high temperature regime in this
Letter) to classical (corresponding to gas of classical particles discussed
above) regimes is manifested by considering the variance of particle number
(also see Ref. Sykes  _et al._ (2008)). The variance is given by

$\overline{\delta
N^{2}}=\overline{N}+\int_{0}^{R}\int_{0}^{R}g(x-y)g(y-x)\,\mathrm{d}x\mathrm{d}y,$

where
$g(x-y)=\left\langle\psi^{\dagger}\left(x\right)\psi\left(y\right)\right\rangle$
is one-particle correlation function

$g(x-y)=\frac{1}{\sqrt{\pi}\lambda}\int_{-\infty}^{\infty}\frac{\exp\left(2\mathrm{i}\sqrt{\pi}w\left(y-x\right)/\lambda\right)}{z^{-1}e^{w^{2}}-1}\,\mathrm{d}w$

where $z=e^{\mu/T}$ stands for fugacity. For $T>T_{d}$, $z\ll 1$ and the
Bosonic occupation numbers can be replaced by Boltzmann factors,
$1/\left(z^{-1}e^{w^{2}}-1\right)\approx ze^{-w^{2}}$. This leads to the
following expression for variance,

$\overline{N^{2}}\approx\overline{N}+\overline{N}\sqrt{\frac{T_{d}}{T}}\left\\{\frac{e^{-x^{2}}-1+\sqrt{\pi}x\,\mathrm{erf}\left(x\right)}{x}\right\\},$

where $x=\overline{N}\sqrt{T/T_{d}}$ and the expression in curly brackets
takes values between $0$ and $\sqrt{\pi}$. In other words, for $T>T_{d}$ the
variance is essentially Poissonian for all interval sizes.

On the other hand, for $T<T_{d}$, $z\approx 1$ and the bosonic occupation
number can be approximated by $1/\left(z^{-1}e^{w^{2}}-1\right)\approx
1/\left(1-z+w^{2}\right)$ which leads to

$\overline{\delta
N^{2}}\approx\overline{N}+\overline{N}^{2}\left\\{\frac{2x-1+e^{-2x}}{2x^{2}}\right\\},$

where $x=\overline{N}/\bar{n}l_{\varphi}=R/\ell_{\varphi}$ and the expression
in curly brackets takes values ranging from $0$ (for $x\gg 1$) to $1$ (for
$x=0$). Thus in the quantum degenerate regime and for short intervals the full
counting statistics is manifestly non-Poissonian - standard deviation becomes
comparable to the average number of particles - which remains true in other
temperature regimes as well.

##  II Imaginary-time evolution of harmonic oscillator under sudden change of
frequency and external force.

For high temperatures the modification $H_{0}\to H_{\lambda}(r)$ amounts to a
sudden change of the oscillator frequency
$\omega_{0}\to\omega_{1}=\omega_{0}\sqrt{1+2s}$, where
$s=\lambda\bar{n}\ell_{\varphi}$ and $\omega_{0}=1/\bar{n}\ell_{\varphi}$. We
define dimensionless imaginary time $t_{R}=\bar{n}R$ and calculate the
evolution of the ground state, $\Phi_{\lambda}(r;R)=\langle
r|{e^{-t_{R}H_{\lambda}}}|0\rangle$ by adopting the methods of Ref. Perelomov
and Zeldovich (1998) to the imaginary time evolution, with the result

$\displaystyle\Phi_{\lambda}(r;R)=\frac{1}{\sqrt{\pi}b}e^{-\frac{r^{2}}{2}\left(\frac{1}{b^{2}}+\frac{1}{\omega_{0}}\frac{\dot{b}}{b}\right)-\omega_{0}\int_{0}^{t_{R}}\frac{\mathrm{d}\tau}{b^{2}(\tau)}}\,.$
(S7)

Here the scaling factor $b=b(t_{R})$ is obtained from the solution of the
second order differential equation

$\displaystyle-\ddot{b}+\omega_{1}^{2}b=\frac{\omega_{0}^{2}}{b^{3}}\,,\qquad
b(0)=1,\qquad\dot{b}(0)=0\,.$ (S8)

Solving Eq. (S8) we obtain

$\displaystyle
b(\tau)=\sqrt{1+\frac{\omega_{1}^{2}-\omega_{0}^{2}}{\omega_{1}^{2}}\sinh^{2}\omega_{1}\tau}\,,$
(S9)

and

$\displaystyle\omega_{0}\int_{0}^{t_{R}}\frac{\mathrm{d}\tau}{b^{2}(\tau)}=\frac{1}{2}\log\frac{1+\frac{\omega_{0}}{\omega_{1}}\tanh\omega_{1}t_{R}}{1-\frac{\omega_{0}}{\omega_{1}}\tanh\omega_{1}t_{R}}\,.$
(S10)

Substituting these results into Eq. (S7) for the evolved ground state and
calculating its overlap with the initial state yields the generating function

$\displaystyle\chi(\lambda,R)$
$\displaystyle=\frac{2e^{-\omega_{0}\int_{0}^{t_{R}}\frac{\mathrm{d}\tau}{b^{2}(\tau)}}}{b\,e^{-\omega_{0}t_{R}}}\int_{0}^{\infty}\\!r\mathrm{d}r\,e^{-\frac{r^{2}}{2}\left(\frac{1+b^{2}}{b^{2}}+\frac{1}{\omega_{0}}\frac{\dot{b}}{b}\right)}$
$\displaystyle=\frac{2e^{\omega_{0}t_{R}}e^{-\omega_{0}\int_{0}^{t_{R}}\frac{\mathrm{d}\tau}{b^{2}(\tau)}}}{b+\frac{1}{b}+\frac{\dot{b}}{\omega_{0}}}$
$\displaystyle=\frac{e^{\omega_{0}t_{R}}\left(\cosh\omega_{1}t_{R}-\frac{\omega_{0}}{\omega_{1}}\sinh\omega_{1}t_{R}\right)}{1+\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}\omega_{1}}\sinh\omega_{1}t_{R}\left(\cosh\omega_{1}t_{R}+\frac{\omega_{0}}{\omega_{1}}\sinh\omega_{1}t_{R}\right)}$
$\displaystyle=\frac{e^{R/\ell_{\varphi}}}{\cosh\sqrt{1+2s}\frac{R}{\ell_{\varphi}}+(1+s)\frac{\sinh\sqrt{1+2s}\frac{R}{\ell_{\varphi}}}{\sqrt{1+2s}}}\,.$
(S11)

valid in the high-temperature regime for an interval of arbitrary length $R$.
The latter has to be compared to the microscopic length $\ell_{\varphi}$. In
the limit $R\ll\ell_{\varphi}$ we recover the generating function

$\displaystyle\chi(\lambda,R)\simeq\frac{1}{1+\frac{\omega_{1}^{2}-\omega_{0}^{2}}{2\omega_{0}}t_{R}}=\frac{1}{1+\lambda\bar{n}R}\,,$
(S12)

_i.e._ $\chi(k/\bar{n}R,R)=1/(1+k)$ is Laplace transform of the exponential
probability density, Eq. (10). For long intervals, $R\gg\ell_{\varphi}$ we
have

$\displaystyle\chi(\lambda,R)$ $\displaystyle\simeq$
$\displaystyle=\frac{4\sqrt{1+2s}}{\left(1+\sqrt{1+2s}\right)^{2}}e^{-\left(\sqrt{1+2s}-1\right)\frac{R}{\ell_{\varphi}}}\
$ (S13)

leading to Eq. (19).

In the low temperature regime, $\ell_{\varphi}/\xi\gg 1$ one uses the
expansion (11) together with with the approximation

$\displaystyle\lambda r^{2}=\lambda(1+\delta
r)^{2}\simeq\lambda+2\lambda\delta r$ (S14)

mapping the problem on a 2D harmonic oscillator under influence of a time
dependent force $f(\tau)=-2\lambda$ acting for $0<\tau<t_{R}$. Again, the
evolution of the ground state can be found adapting methods of Ref. Perelomov
and Zeldovich (1998) to imaginary time and is given by

$\displaystyle\Phi_{\lambda}(1+\delta r;R)=e^{-F(\delta r;t_{R})-\lambda
t_{R}}\Psi_{0}\left(\delta r-\eta\right)\,,$ (S15)

where $\Psi_{0}(\delta
r)=(\ell_{\varphi}/2\pi^{3}\xi)^{1/4}e^{-(\ell_{\varphi}/\xi)\delta r^{2}}$
and $\eta=\eta(\tau)$ is the solution of the classical equation of motion

$\displaystyle\ddot{\eta}-\omega^{2}\eta=-\frac{f}{M}=\,$ (S16)

with initial conditions $\eta(0)=\dot{\eta}(0)=0$ and

$\displaystyle F(\delta r;t_{R})$ $\displaystyle=M\dot{\eta}\left(\delta
r-\eta\right)+\int_{0}^{t_{R}}\\!\mathrm{d}\tau\left(\frac{M\dot{\eta}^{2}}{2}+\frac{M\omega^{2}\eta^{2}}{2}-f\eta\right)$
$\displaystyle=M\dot{\eta}\left(\delta
r-\eta/2\right)-\frac{1}{2}\int_{0}^{t_{R}}\\!\mathrm{d}\tau\,f\eta\,,$ (S17)

where we have used equation of motion, Eq. (S16) to simplify the integral.
Solving Eq. (S16) we get

$\displaystyle\eta(\tau)=\frac{f}{M\omega^{2}}\left(1-\cosh\omega\tau\right)\,.$
(S18)

Substituting it into Eq. (S15) and truncating at the second order in $\lambda$
in the exponent we get

$\displaystyle\chi(\lambda,R)\simeq\exp\left[-\lambda\bar{n}R+\frac{C(R/\xi)\xi}{2\ell_{\varphi}}(\lambda\bar{n}R)^{2}\right]\,,$
(S19)

where the crossover function

$\displaystyle
C(x)=\frac{1}{x^{2}}\left(x-\frac{1}{2}\left(1-e^{-2x}\right)\right)\,.$ (S20)

Performing the inverse Laplace transform we get Eq. (21) of the main text.

##  III Third and fourth moments.

Figure S1: The second cumulant, $\kappa_{2}=\overline{\delta N^{2}}$, of the
large-R distribution as a function of $\overline{N}$ at fixed $R$ and $T$.
Dependence on dimensionless parameters $TR/g$ and $mTR^{2}$ is shown for
generality.  Figure S2: The third cumulant, $\kappa_{3}=\overline{\delta
N^{3}}$, of the large-R distribution as a function of $\overline{N}$ at fixed
$R$ and $T$. Dependence on dimensionless parameters $TR/g$ and $mTR^{2}$ is
shown for generality.  Figure S3: The fourth cumulant, $\kappa_{4}$, of the
large-R distribution as a function of $\overline{N}$ at fixed $R$ and $T$.
Dependence on dimensionless parameters $TR/g$ and $mTR^{2}$ is shown for
generality.

Similarly to expressing variance as a function of $T/c\bar{n}$, Eq. (22),
second, third, and fourth cumulants can be plotted as functions of
$\overline{N}$ at fixed $R$ and $T$ to be compared with the existing results
obtained using Yang-Yang thermodynamics in Ref. Armijo  _et al._ (2010).
Defining $\overline{\delta N^{3}}=\overline{\left(N-\overline{N}\right)^{3}}$
and $\overline{\delta N^{4}}=\overline{\left(N-\overline{N}\right)^{4}}$ we
present second, third and fourth cumulants, $\kappa_{2}=\overline{\delta
N^{2}}$, $\kappa_{3}=\overline{\delta N^{3}}$ and $\kappa_{4}=\overline{\delta
N^{4}}-3\left(\overline{\delta N^{2}}\right)^{2}$, as functions of
$\overline{N}$ in Figs. S1, S2 and S3. Notice that cumulants are expressed in
terms of universal functions which should be scaled appropriately depending on
the values of dimensionless parameters
$TR/g=\left(\bar{n}\xi\right)^{2}R/\ell_{\varphi}$ and
$mTR^{2}=\bar{n}R^{2}/\ell_{\varphi}$. Shapes shown on Figs. S1 and S2 are in
full agreement to Fig. 2(b) of Armijo  _et al._ (2010).

## References

  * Chandrasekhar (1943) S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
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  * Perelomov and Zeldovich (1998) A. M. Perelomov and Y. B. Zeldovich,  _Quantum Mechanics - Selected Topics_ (World Scientific Publishing, 1998).
  * Armijo  _et al._ (2010) J. Armijo, T. Jacqmin, K. V. Kheruntsyan, and I. Bouchoule, Phys. Rev. Lett. 105, 230402 (2010).