# OmniAvatar: Geometry-Guided Controllable 3D Head Synthesis

Hongyi Xu1 Guoxian Song1 Zihang Jiang1,2 Jianfeng Zhang1,2 Yichun Shi1
Jing Liu1 Wanchun Ma1 Jiashi Feng1 Linjie Luo1
1ByteDance Inc 2National University of Singapore
{hongyixu, guoxian.song, zihang.jiang, jianfeng.zhang, yichun.shi,
jing.liu, wanchun.ma, jshfeng<EMAIL_ADDRESS>
###### Abstract

We present OmniAvatar, a novel geometry-guided 3D head synthesis model trained
from in-the-wild unstructured images that is capable of synthesizing diverse
identity-preserved 3D heads with compelling dynamic details under full
disentangled control over camera poses, facial expressions, head shapes,
articulated neck and jaw poses. To achieve such high level of disentangled
control, we first explicitly define a novel semantic signed distance function
(SDF) around a head geometry (FLAME) conditioned on the control parameters.
This semantic SDF allows us to build a differentiable volumetric
correspondence map from the observation space to a disentangled canonical
space from all the control parameters. We then leverage the 3D-aware GAN
framework (EG3D) to synthesize detailed shape and appearance of 3D full heads
in the canonical space, followed by a volume rendering step guided by the
volumetric correspondence map to output into the observation space. To ensure
the control accuracy on the synthesized head shapes and expressions, we
introduce a geometry prior loss to conform to head SDF and a control loss to
conform to the expression code. Further, we enhance the temporal realism with
dynamic details conditioned upon varying expressions and joint poses. Our
model can synthesize more preferable identity-preserved 3D heads with
compelling dynamic details compared to the state-of-the-art methods both
qualitatively and quantitatively. We also provide an ablation study to justify
many of our system design choices.

Figure 1: Our model can synthesize diverse identity-preserved 3D heads with
compelling dynamic details under full disentangled control over camera poses,
facial expressions, head shapes, articulated neck and jaw poses (left). Our
model can also reconstruct 3D heads from a single photo reference and enable
multi-view-consistent head reenactment (right).

##  1 Introduction

Photo-realistic face image synthesis, editing and animation attract
significant interests in computer vision and graphics, with a wide range of
important downstream applications in visual effects, digital avatars,
telepresence and many others. With the advent of Generative Adversarial
Networks (GANs) [15], remarkable progress has been achieved in face image
synthesis by StyleGAN [24, 25, 23] as well as in semantic and style editing
for face images [46, 54]. To manipulate and animate the expressions and poses
in face images, many methods attempted to leverage 3D parametric face models,
such as 3D Morphable Models (3DMMs) [6, 40], with StyleGAN-based synthesis
models [10, 53, 41]. However, all these methods operate on 2D convolutional
networks (CNNs) without explicitly enforcing the underlying 3D face structure.
Therefore they cannot strictly maintain the 3D consistency when synthesizing
faces under different poses and expressions.

Recently, a line of work has explored neural 3D representations by
unsupervised training of 3D-aware GANs from in-the-wild unstructured images
[8, 17, 37, 62, 7, 48, 61, 45, 11, 44, 67]. Among them, methods with
generative Neural Radiance Fields (NeRFs) [33] have demonstrated striking
quality and multi-view-consistent image synthesis [48, 7, 37, 17, 11]. The
progress is largely due to the integration of the power of StyleGAN in photo-
realistic image synthesis and NeRF representation in 3D scene modeling with
view-consistent volumetric rendering. Nevertheless, these methods lack precise
3D control over the generated faces beyond camera pose, as well as the quality
and consistency in control over other attributes, such as shape, expression,
neck and jaw pose, leave much to be desired.

In this work, we present _OmniAvatar_ , a novel geometry-guided 3D head
synthesis model trained from in-the-wild unstructured images. Our model can
synthesize a wide range of 3D human heads with full control over camera poses,
facial expressions, head shapes, articulated neck and jaw poses. To achieve
such high level of disentangled control for 3D human head synthesis, we devise
our model learning _in two stages_. We first define a novel _semantic signed
distance function_ (SDF) around a head geometry (i.e. FLAME [29]) conditioned
on its control parameters. This semantic SDF fully distills rich 3D geometric
prior knowledge from the statistical FLAME model and allows us to build a
differentiable _volumetric correspondence map_ from the _observation space_ to
a disentangled _canonical space_ from all the control parameters. In the
second training stage, we then leverage the state-of-the-art 3D GAN framework
(EG3D [7]) to synthesize realistic shape and appearance of 3D human heads in
the canonical space, including the modeling of hair and apparels. Following
that, a volume rendering step is guided by the volumetric correspondence map
to output the geometry and image in the observation space.

Figure 2: Overview of our training framework. Stage 1: Trained from
parameterized FLAME [29] mesh collections, a MLP-network $W$ maps a shape
$\bm{\mathbf{\alpha}}$, expression $\bm{\mathbf{\theta}}$ and articulated jaw
and neck pose $\bm{\mathbf{\theta}}$ into 3D point-to-point volumetric
correspondences from observation to canonical space, together with a signed
distance function of the corresponding FLAME head. Stage 2: Given a Gaussian
latent code $\bm{\mathbf{z}},$ our model generates a tri-plane represented 3D
feature space of a canonical head, disentangled with shape and expression
controls. The volume rendering is then guided by the volumetric correspondence
field to map the decoded neural radiance field from the canonical to
observation space. We condition the NeRF decoding with expression and joint
pose for modeling dynamic details. A super-resolution module synthesizes the
final high-resolution RGB image from the volume-rendered feature map. For
fine-grained shape and expression control, we apply the FLAME SDF as geometric
prior to the synthesized NeRF density, and self-supervise the image synthesis
to commit to the target expression $\bm{\mathbf{\beta}}$ and joint pose
$\bm{\mathbf{\theta}}$ by comparing the input code against the re-estimated
values $\hat{\bm{\mathbf{\beta}}},\hat{\bm{\mathbf{\theta}}}$ from synthesized
images.

To ensure the consistency of synthesized 3D head shape with controlling head
geometry, we introduce a _geometry prior loss_ to minimize the difference
between the synthesized neural density field and the FLAME head SDF in
observation space. Furthermore, to improve the control accuracy, we pre-train
an image encoder of the control parameters and formulate a _control loss_ to
ensure synthesized images matching the input control code upon encoding.
Another key aspect of synthesis realism is dynamic details such as wrinkles
and varying shading as subjects change expressions and poses. To synthesize
dynamic details, we propose to condition EG3D’s triplane feature decoding with
noised controlling expression.

Compare to state-of-the-art methods, our method achieves superior synthesized
image quality in terms of Frechet Inception Distance (FID) and Kernel
Inception Distance (KID). Our method can consistently preserve the identity of
synthesized subjects with compelling dynamic details while changing
expressions and poses, outperforming prior methods both quantitatively and
qualitatively.

The contributions of our work can be summarized as:

  * •

A novel geometry-guided 3D GAN framework for high-quality 3D head synthesis
with full control on camera poses, facial expressions, head shapes,
articulated neck and jaw poses.

  * •

A novel semantic SDF formulation that defines the volumetric correspondence
map from observation space to canonical space and allows full disentanglement
of control parameters in 3D GAN training.

  * •

A geometric prior loss and a control loss to ensure the head shape and
expression synthesis accuracy.

  * •

A robust noised expression conditioning scheme to enable dynamic detail
synthesis.

##  2 Related Work

##### 3D-Aware Generative Image Synthesis.

Generative adversarial networks [15, 24, 25] gained popularity over the last
decade due to their remarkable ability in photo-realistic image synthesis.
Building on the success of 2D image-based GANs, recent works have extended the
capabilities to view-consistent image synthesis with unsupervised learning
from 2D single-view images. The key idea is to combine differential rendering
with 3D scene representations, such as meshes [50, 31], point clouds [1, 28],
voxels [57, 34, 35], and recently implicit neural representation [44, 8, 36,
7, 37, 17, 11, 48, 67]. We build our work on recent 3D GAN model by Chan et al
[7] that uses an efficient triplane-based NeRF generation, combined with 2D
CNN-based super-resolution. Even though 3D-aware GANs are able to control
camera viewpoints, they lack precise 3D control over the other attributes such
as shapes and expressions. In this work, we empower 3D-aware GANs with
disentangled precise control over shapes and expressions.

##### Controllable Face Image Synthesis.

Considerable work [10, 53, 9, 41, 52, 27, 14] has been devoted to incorporate
3D priors of statistical face models, such as 3D Morphable Models (3DMMs) [6,
40], in controllable face synthesis and animation. Among them, DiscoFaceGAN
[10] proposed imitative-contrastive learning to mimic the 3DMM rendering
process by the generative model. A similar strategy has also been adopted with
concurrent and follow-up works [53, 27, 14, 41]. However all of these
approaches suffer from 3D inconsistency due to the use of 2D CNNs as image
renderer. HeadNeRF [21] combines 3DMM with 3D NeRF representation, and is able
to synthesize 3D heads conditioned on 3DMM attributes. However, the training
relies on annotated multiview datasets whereas our approach learns the
disentangled 3D head synthesis with only single-view images. There has been
concurrent work [58, 49, 4, 64, 51] to ours in 3D-aware controllable face or
full-body GANs. Differently from these approaches, we use a full-head
parametric model FLAME [29], and fully exploit the spatial geometric prior
knowledge beyond the surface deformation and skinning. We have also achieved
fine-grained control with our novel losses and enhanced face animation with
rich dynamic details.

##### Controllable Neural Implicit Field of Face.

Neural implicit functions [59], have emerged as a powerful continuous and
differential representations of 3D scenes. Among them, Neural Radiance Field
[33, 3] has been widely adopted due to its superiority in modeling complex
scene details and synthesizing multiview images with inherited 3D consistency.
While initial proposals have focused on static scene modeling, recent work
have successfully demonstrated application of NeRF in modeling dynamic scenes
[42, 38, 60, 32, 55]. In particular, dynamic neural radiance fields of human
heads [39, 38, 13, 18, 56, 66, 22, 68] have enabled photo-realistic head
animation, often by conditioning with pose parameters or deforming the
radiance field with 3D morphable models. However, they do not leverage a
generative training paradigm and focus on scene-specific learning from video
sequences or multiview images. In contrast, our model learns generative and
controllable neural radiance fields from widely accessible single-view images.

##  3 Method

Our goal is to build a geometry-guided 3D head synthesis model with full
control of camera poses, face shapes and expressions, trained from in-the-wild
unstructured image collections.

###  3.1 Overview

##### Problem.

To achieve our goal, we leverage a 3D-aware generator (EG3D [7]) for photo-
realistic, multiview consistent image synthesis, while disentangle control of
head geometric attributes from image generation with a 3D statistical head
model (FLAME [29]). Specifically, given a random Gaussian-sampled latent code
$\bm{\mathbf{z}}$, a camera pose $\bm{\mathbf{c}}$ and a FLAME parameter
$\bm{\mathbf{p}}=(\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$
consisting of shape $\bm{\mathbf{\alpha}},$ expression $\bm{\mathbf{\beta}},$
jaw and neck pose $\bm{\mathbf{\theta}}$, the generator $G$ synthesizes a
photo-realistic human head image
$I_{RGB}(\bm{\mathbf{z}}|\bm{\mathbf{c}},\bm{\mathbf{p}})$ with corresponding
attributes as defined in $\bm{\mathbf{p}}.$

##### Framework.

As illustrated in our pipeline 2, our controllable 3D-aware GAN is trained in
two stages. From a large collection of 3D deformed FLAME meshes, we first pre-
train a deformable semantic SDF around the FLAME geometry that builds a
differential volumetric correspondence map from the observation to a
predefined canonical space (Section 3.2.1). In the second stage, guided by the
pre-trained volumetric mapping, we then deform the detailed 3D full heads
synthesized in the disentangled canonical space to the desired shapes and
expressions (Section 3.2.2). Fine-grained expression control is achieved by
supervising image synthesis such that expressions estimated from the generated
images is consistent with the input control(Section 3.2.3). Our approach
further enhances temporal realism with dynamic details, such as dimples and
wrinkles, synthesizing realistic shading and geometric variations as
expression changes (Section 3.2.4).

##### 3D-Aware GAN Background.

To ensure appearance consistency from different views, we choose EG3D [7] as
our backbone for 3D-aware image synthesis. The generator $G$ takes a random
latent code $\bm{\mathbf{z}}$ and conditioning camera label
$\hat{\bm{\mathbf{c}}}$, and maps to a manifold of triplane features
$\cal{M}(\bm{\mathbf{z}},\hat{\bm{\mathbf{c}}})$. For presentation clarity, we
absorb $\hat{\bm{\mathbf{c}}}$ to $\bm{\mathbf{z}}$ and simply denote triplane
as $\cal{M}(\bm{\mathbf{z}}).$ A low-resolution feature map
$I^{*}(\bm{\mathbf{z}}|\bm{\mathbf{c}})$ is then rendered from a desired
camera pose $\bm{\mathbf{c}}$ by sampling the triplane features and
integrating MLP-decoded neural radiance $(\sigma,\bm{\mathbf{f}})$ along
camera rays. A super-resolution module is followed to modulate the feature map
and synthesize the final RGB images at high resolution. We train $G$ and a
dual discriminator $D$ with adversarial training.

Figure 3: We synthesize different identities with the same shape and
expression code respectively in (a) and (b). We observe less shape variation
with our geometric prior loss (a) and more consistent expression with our
control loss (b). In (c), we show our expression-conditioned NeRF modeling
builds rich dynamic details as the subject varies expressions.

###  3.2 Controllable 3D-Aware Image Synthesis

To synthesize an image
$\bm{\mathbf{I}}(\bm{\mathbf{z}}|\bm{\mathbf{c}},\bm{\mathbf{p}})$ with
desired FLAME control $\bm{\mathbf{p}}$, we leverage the EG3D framework to
generate a triplane-based 3D volumetric feature space
$\cal{M}(\bm{\mathbf{z}})$ of a synthesized identity with canonical shape and
expression. Guided by our pretrained volumetric correspondence map, we deform
the synthesized feature volume into our target observation space, which is
further decoded and volume rendered into high-fidelity head appearance and
geometry with the target shape and expression. Our design explicitly
disentangles the underlying geometric variations of changing shape and
expression from canonical geometry and appearance synthesis. Following EG3D
[7], we associate each training image with a set of camera parameters
$\bm{\mathbf{c}}$ and control parameters $\bm{\mathbf{p}}$, which are obtained
from a nonlinear 2D landmarks-based optimization.

####  3.2.1 Semantic Signed Distance Function

For disentangled geometric modeling, we formulate an implicit semantic SDF
representation
$W(\bm{\mathbf{x}}|\bm{\mathbf{p}}=(\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}},\bm{\mathbf{\theta}}))=(s,\bar{\bm{\mathbf{x}}})$,
where $\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}}$ are the linear shape and
expression blendshape coefficients, and $\bm{\mathbf{\theta}}$ controls the
rotation of a 3-DoF jaw and neck joint. Specifically, given a spatial point
$\bm{\mathbf{x}}$ in observation space $\cal{O(\bm{\mathbf{p}})},$ $W$ returns
its 3D correspondence point $\bar{\bm{\mathbf{x}}}$ (i.e., semantics) in
canonical space $\cal{C(\bm{\mathbf{\bar{p}}})}$, with which we project and
query the triplane features $\cal{M}(\bm{\mathbf{z}}).$ Additionally it also
computes the closest signed distance $s(\bm{\mathbf{x}}|\bm{\mathbf{p}})$ to
the FLAME mesh surface $\bm{\mathbf{S}}(\bm{\mathbf{p}}).$ We illustrate the
function in Figure 4. We co-learn the highly-correlated volumetric
correspondence and SDF with the property that the signed distance is preserved
between canonical and observation correspondence points as
$s(\bar{\bm{\mathbf{x}}}|\bar{\bm{\mathbf{p}}})=s(\bm{\mathbf{x}}|\bm{\mathbf{p}})$.

We learn $W(\bm{\mathbf{x}}|\bm{\mathbf{p}})$ with a large corpus of 3D FLAME
meshes $\bm{\mathbf{S}}(\bm{\mathbf{p}})$ sampled from its parametric control
space. Similar to IGR [16] and imGHUM [2], we model our implicit field as an
MLP and optimize $W(\bm{\mathbf{x}}|\bm{\mathbf{p}})$ with losses,

$\displaystyle L_{iso}=\frac{1}{|N|}\sum_{\bm{\mathbf{x}}\in
N}(|s(\bm{\mathbf{x}}|\bm{\mathbf{p}})|+|\nabla
s_{\bm{\mathbf{x}}}(\bm{\mathbf{x}}|\bm{\mathbf{p}})-\bm{\mathbf{n}}(\bm{\mathbf{x}}|\bm{\mathbf{p}})|),$
(1) $\displaystyle L_{eik}=\frac{1}{|F|}\sum_{\bm{\mathbf{x}}\in F}\|\nabla
s_{\bm{\mathbf{x}}}(\bm{\mathbf{x}}|\bm{\mathbf{p}})-1\|_{2},$ (2)
$\displaystyle L_{sem}=\frac{1}{|N|}\sum_{\bm{\mathbf{x}}\in
N}(|\bar{\bm{\mathbf{x}}}(\bm{\mathbf{x}}|\bm{\mathbf{p}})-\bar{\bm{\mathbf{x}}}^{*}(\bm{\mathbf{x}}|\bar{\bm{\mathbf{p}}})|)$
(3)

where $N,F$ are a batch of on and off surface samples. For the surface
samples, the $L_{iso}$ encourages the signed distance values to be on the
zero-level-set and the SDF gradient to be equal to the given surface normals
$\bm{\mathbf{n}}$. The Eikonal loss $L_{eik}$ is derived from [16] where the
SDF is differentiable everywhere with gradient norm 1. The semantic loss
$L_{sem}$ supervises the mapping of surface samples $\bm{\mathbf{x}}\in F$ to
the ground-truth correspondence points $\bar{\bm{\mathbf{x}}}^{*}$ on the
canonical FLAME surface, where $\bar{\bm{\mathbf{x}}}^{*}$ and
$\bm{\mathbf{x}}$ share the same barycentric coordinates. The $L_{eik}$
provides a geometric regularization over the volumetric SDF, whereas the
$L_{iso}$ and $L_{sem}$ act as the boundary condition to the SDF and
volumetric correspondence field respectively.

Figure 4: Illustration to semantic SDF learning.

To guide the canonical correspondence learning for off-the-surface points, one
could try to minimize the signed distance difference between
$s(\bar{\bm{\mathbf{x}}}|\bar{\bm{\mathbf{p}}})$ and
$s(\bm{\mathbf{x}}|\bm{\mathbf{p}}).$ However, we note that the volume
correspondence between the observation and canonical space is still ill-
regularized, considering infinite number of points exist with the same signed
distance value. We therefore reformulate volumetric correspondence field as
$W(\bm{\mathbf{x}}|\bm{\mathbf{p}})=(\bar{s}(\bar{\bm{\mathbf{x}}}),\bar{\bm{\mathbf{x}}}),$
where the signed distance of an observation-space point is obtained by mapping
it into its canonical correspondence $\bar{\bm{\mathbf{x}}}$ and querying the
pre-computed canonical SDF $\bar{s}$. Thus we only learn a correspondence
field with which we can deform the canonical SDF to different FLAME
configurations, and then supervise the deformed SDFs with the FLAME surface
boundary, normals ($L_{iso}$) and Eikonal regularization ($L_{eik}$). As such,
even for off-the-surface samples, their canonical correspondences are well
regularized in space with the geometric properties of signed distance
functions via $L_{iso}$ and $L_{eik}$. In contrast to explicit mesh
deformation [4], our implicit volumetric correspondence is more accurate,
differentiable, smooth everywhere and semantically consistent with the
properties of SDF.

Figure 5: Qualitative comparisons in expression control. Under a similar
expression control, our method achieves the best perceptual quality with
better identity-preservation (compared to GAN-Control [47] and 3DFaceShop
[51]), more realistic dynamic details (compared to DiscoFaceGAN [10]) and
higher expression control accuracy (compared to AniFaceGAN [58], 3DFaceShop
[51], HeadNeRF [21]).

####  3.2.2 Canonical Generation with Geometric Prior

With our pretrained semantic SDF $W(\bm{\mathbf{x}}|\bm{\mathbf{p}})$ modeling
the shape and expression variation, we leverage the triplane for 3d-aware
generation of human heads with canonical shape and expression. In particular,
to generate a neural radiance feature $(\sigma,\bm{\mathbf{f}})$ for a point
$\bm{\mathbf{x}}$ in observation space $\cal{O}(\bm{\mathbf{p}})$, we use our
correspondence function $W$ to back warp $\bm{\mathbf{x}}$ into
$\bar{\bm{\mathbf{x}}}$, with which we project and sample the canonical
triplane features followed with a tiny MLP decoding.

In spite of the control disentanglement, there is no explicit loss that
constrains the triplane-generated neural radiance density to conform to the
shape and expression as defined in $\bm{\mathbf{p}}.$ Therefore, we guide the
generation of neural radiance density field by minimizing its difference to
the underlying FLAME head geometry represented with SDF, as

$\displaystyle L_{prior}=\frac{1}{|R|}\sum_{\bm{\mathbf{x}}\in
R}e^{-\gamma|s(\bm{\mathbf{x}}|p)|}|\sigma(\bm{\mathbf{x}}|\bm{\mathbf{z}},\bm{\mathbf{p}})-\sigma^{*}(\bm{\mathbf{x}}|\bm{\mathbf{p}})|,$
(4)
$\displaystyle\sigma^{*}(\bm{\mathbf{x}}|\bm{\mathbf{p}})=\frac{1}{\kappa}\cdot\textrm{Sigmoid}(\frac{-s(\bm{\mathbf{x}}|\bm{\mathbf{p}})}{\kappa})$
(5)

where $R$ is the stratified ray samples for volume rendering and $\kappa$ is a
learnable scalar controlling the density tightness around the SDF boundary.
Following [63, 37], we convert SDF value $s(\bm{\mathbf{x}}|\bm{\mathbf{p}})$
to proxy 3D density $\sigma^{*}(\bm{\mathbf{x}}|\bm{\mathbf{p}})$ assuming
non-hollow surfaces. We decay the weights for our geometric prior loss
$L_{prior}$ as the point moving away from the SDF boundary, allowing higher
degrees of freedom in generation of residual geometries, such as hair and
glasses. The geometric prior loss effectively guides the 3D head geometry
learning but should not be overpowered which otherwise might lead to loss of
geometric details.

####  3.2.3 Fine-Grained Expression Control

Our geometric prior loss $L_{prior}$ provides local 3D point-wise guidance,
and is able to well regularize the shape generation and achieve coarse-level
expression control. However, for delicate expressions, such as eye blinks,
$L_{prior}$ provides little supervision as the geometric variation is subtle.
Moreover, for regions with complex correspondences, such as around the lips,
it is challenging to guide the formation of correct expressions globally, just
with point-wise geometric losses. To improve the control granularity, we
propose an image-level supervision loss that requires a synthesized image
$I_{RGB}(\bm{\mathbf{z}}|\bm{\mathbf{c}},\bm{\mathbf{p}})$ matching the target
expression as defined in the input $\bm{\mathbf{p}}$. Using our training
images with estimated control labels $\bm{\mathbf{p}},$ we first pretrain an
image encoder
$E(I_{RGB})=(\tilde{\bm{\mathbf{\beta}}},\tilde{\bm{\mathbf{\theta}}})$ that
regresses the expression coefficients $\tilde{\bm{\mathbf{\beta}}}$ and joint
poses $\tilde{\bm{\mathbf{\theta}}}.$ During our 3D GAN training, we then
apply our image-level control supervision as

$\displaystyle
L_{enc}=|\tilde{\bm{\mathbf{\beta}}}-\bm{\mathbf{\beta}}|+|\tilde{\bm{\mathbf{\theta}}}-\bm{\mathbf{\theta}}|+|\bm{\mathbf{S}}(\bm{\mathbf{\alpha}},\tilde{\bm{\mathbf{\beta}}},\tilde{\bm{\mathbf{\theta}}})-\bm{\mathbf{S}}(\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})|+$
$\displaystyle+|\bm{\mathbf{J}}\bm{\mathbf{S}}(\bm{\mathbf{\alpha}},\tilde{\bm{\mathbf{\beta}}},\tilde{\bm{\mathbf{\theta}}})-\bm{\mathbf{J}}\bm{\mathbf{S}}(\bm{\mathbf{\alpha}},\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})|,$
(6)

where $\bm{\mathbf{S}}$, $\bm{\mathbf{J}}$ are the FLAME mesh vertices and 3D
landmarks regressor respectively. While being straightforward for the first 2
terms, the last two terms in $L_{enc}$ penalize deviation of 3D vertex
coordinates and surface landmarks after mesh decoding. We note that we do not
supervise shape $\bm{\mathbf{\alpha}}$ in $L_{enc}$ since our geometric prior
loss $L_{prior}$ suffices in shape control already and also due to the
ambiguity of shape scaling estimated from monocular images.

####  3.2.4 Dynamic Details Modeling

To this end, we have achieved controllable static image synthesis. However,
one should observe the variation of shading and geometric details in a dynamic
head motion, such as the appearance of dimples and eye wrinkles in smiling. We
consider the appearance of dynamic details is highly correlated with the
driving expression $(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$. To model such
temporal effects, one could try to condition the triplane generation on
expression as
$\cal{M}(\bm{\mathbf{z}},\bm{\mathbf{\beta}},\bm{\mathbf{\theta}}).$ However,
such design results in entanglement of expression control with the generative
latent code $\bm{\mathbf{z}},$ inducing identity or appearance changes when
varying expressions. It is also hard to synthesize images out of training
distribution of expressions $(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$. We
therefore leave our triplane generation disentangled but when decoding neural
feature field $(\sigma,\bm{\mathbf{f}})$ from sampled triplane features, we
additionally condition the MLP-decoder on $\bm{\mathbf{\beta}}$ and
$\bm{\mathbf{\theta}}.$ Specifically, we have

$\displaystyle(\sigma(\bm{\mathbf{x}}),\bm{\mathbf{f}}(\bm{\mathbf{x}})|\bm{\mathbf{z}},\bm{\mathbf{p}})=\Phi(\cal{M}(\bar{\bm{\mathbf{x}}}(\bm{\mathbf{x}}|\bm{\mathbf{p}})|\bm{\mathbf{z}}),\phi(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})),$
(7)

where both $\Phi$ and $\phi$ are tiny MLPs and $\phi$ regresses an expression-
dependent feature vector from $(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$
after positional encoding. For better extrapolation to novel expressions and
jaw poses, we add Gaussian noise to the conditioning parameters to prevent MLP
overfitting.

##  4 Experiments

Model |  FID $\downarrow$ |  KID $\downarrow$ |  DSα$\uparrow$ |  DSβ,θ$\uparrow$ |  DSc $\uparrow$ |  IS$\uparrow$
---|---|---|---|---|---|---
DiscoFaceGAN-SR [10] | 42.6 | 0.04 | - | 27.2 | 0.11 | 0.59
GAN-Control [47] | 9.8 | 0.005 | - | 43.7 | 0.28 | 0.62
HeadNeRF [21] | 163 | 0.168 | 1.33 | 8.9 | 0.25 | 0.678
AniFaceGAN-SR [58] | 30.5 | 0.024 | 2.0 | 11.1 | 0.04 | 0.73
3DFaceShop [51] | 24.8 | 0.018 | 0.45 | 64.2 | 0.029 | 0.594
Ours | 5.7 | 0.0016 | 3.1 | 71.9 | 0.35 | 0.80
(a) Baseline comparisons.

Model |  Ours wo $L_{enc}$ |  Ours wo $L_{prior}$ | Ours
---|---|---|---
FID $\downarrow$ | 6.0 | 6.1 | 5.7
KID $\downarrow$ | 0.00172 | 0.00174 | 0.0016
ASD (cm) $\downarrow$ | 0.124 | 0.135 | 0.124
$var({\bm{\mathbf{\alpha}}})\downarrow$ | 0.000525 | 0.000577 | 0.000515
AED (cm) $\downarrow$ | 0.1233 | 0.0813 | 0.0797
$var({\bm{\mathbf{\beta}},\bm{\mathbf{\theta}}})\downarrow$ | 0.004707 | 0.00406 | 0.00404
(b) Ablation study.

Table 1:  (a) Our method outperforms the prior 2D and 3D controllable image
synthesis methods in both image quality and control disentanglement. (b) Our
method demonstrates the best control accuracy over shape and expression with
geometric prior $L_{prior}$ and self-supervised reconstruction loss $L_{enc}.$

##### Training and Dataset.

Our training is devised into two stages. In the pretraining stage, we build
our semantic SDF $W$ using a four 192-dimensional MLP with a collection of
150K FLAME meshes by Gaussian sampling of the control parameters
$\bm{\mathbf{p}}.$ For each mesh, we sample 4K surface samples $N$ with
surface normals $\bm{\mathbf{n}}$ and ground-truth correspondence point
$\bar{\bm{\mathbf{x}}}^{*}$, and 4K off surface samples $F$ distributed
uniformly inside the box-bounded volume. Our canonical mesh has a neutral
shape $\bm{\mathbf{\alpha}}=0$, expression $\bm{\mathbf{\beta}}=0$ and neck
pose, but with an opening jaw at 10 degrees. We close the face connectivity
between the lips for a watertight geometry and also as a proxy geometry for
the mouth cavity modeling. We open the jaw at the canonical space such that
the spatial geometry and appearance inside the mouth can be modeled with
distinguishable triplane features. In the second training stage, we freeze the
weights of $W$ for best efficiency and train our model on the FFHQ [24], a
human face dataset with $70,000$ in-the-wild images. For each image, we
estimate the camera pose $\bm{\mathbf{c}}$ and FLAME parameters
$\bm{\mathbf{p}}$ with a nonlinear optimization in fitting 2D landmarks,
assuming zero root and neck rotation and camera located $1.0$m away from the
world origin. We refer to the supplementary materials for more details. We
rebalance the FFHQ dataset by duplicating images with large head poses and jaw
opening expressions. We note that we also use pairs of images and
$(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$ for fine-tuning of a light-weight
image encoder $E,$ using a ResNet50 [19] backbone followed with a single-layer
MLP.

###  4.1 Qualitative Comparisons

##### Controlled Portrait Synthesis.

Our framework enables high-fidelity head synthesis with disentangled control
over camera pose, shape and expression, as shown in Figure 1. We compare our
method with prior controllable portrait synthesis works include DiscoFaceGAN
[10], GAN-Control [47], HeadNeRF [21], AniFaceGAN [58] and 3DFaceShop [51].
The first two methods integrates 3DMM controls into 2D face synthesis.
HeadNeRF [21] is a 3D parametric NeRF-based head model built from both indoor
multiview image capture datasets and in-the-wild monocular image collections.
AniFaceGAN [58] and 3DFaceShop [51] are concurrent 3D-aware controllable GANs
in face synthesis.

We qualitatively compare with prior work in Figure 5 for expression control.
GAN-Control shows high-quality expression manipulation but with noticeable
identity variation with differences in hair, head shape contour and
background, whereas DiscoFaceGAN better maintains identity but with lower
perceptual quality, and lacks dynamic details as expression varies. The
expression control space of 3DFaceShop [51] is sensitive, suffering from
appearance changes even with minor expression variation. AniFaceGAN
demonstrates visually-consistent expression editing but with limited
resolution and image quality, e.g., with blurry artifacts in hair and teeth.
HeadNeRF ensures decent consistency in image generation by rendering the
conditional NeRF but lacks fine details with very limited perceptual quality.
In contrast, our approach produces the most compelling images with consistent
appearance and dynamic details under expression changes. Moreover,
controllable neck pose is a key factor towards realistic video avatar in
applications like talking head synthesis. As shown in Figure. 1, our method
achieves explicit control of neck and jaw poses which are seldom explored in
prior work. Please refer to supplementary material for qualitative comparison
in view consistency and shape editing.

##### Dynamic details.

Our method presents highly-consistent image synthesis in control of shape,
expression and camera poses, but also depicts temporal realism in portrait
animation, credited to the modeling of expression-dependent dynamic details.
As shown in the last row of Figure 3, the appearance of wrinkles around the
mouth and eyes when transiting from a neural expression to smiling largely
enhances the animation realism. In comparison, without explicitly modeling the
dynamic details, the wrinkles are embedded in the appearance and do not vary
with expressions.

##### Expressive Head Synthesis with Extrapolated Controls.

AniFaceGAN [58] and 3DFaceShop [51] depict dynamic details as well since their
generation of the neural fields is conditioned on the input expression latent
code. However, their designs result in shape and expression entanglement with
the appearance generation, as reflected in the identity changes as shown in
Figure. 5. By embedding the controls in a Gaussian latent space with such as
Variation AutoEncoder (VAE), their approaches sacrifice expressiveness. The
synthesized image quality is also highly correlated to the distribution of
training images with target expressions.

In contrast, our tri-plane generation is explicitly disentangled from shape
and expression control. Moreover, the volume deformation is independently
learnt from the deformed FLAME mesh collections which offer abundant 3D
geometric knowledge with largely augmented control space. Therefore we are
much less dependent on the distribution of the training images and support
better extrapolation to unseen novel expressions. In Figure. 1, we show high-
quality synthesized head with extreme jaw and neck articulated movements which
do not exist in our training images. Our expression control is also more
expressive, supporting subtle expressions like eye blinks (Figure. 1 3).

###  4.2 Quantitative Comparisons

##### Image Quality

We measure the image quality with Frechet Inception Distance (FID) [20] and
Kernel Inception Distance [5] between 50K randomly synthesized images and 50K
randomly sampled real images at the resolution of $512\times 512$. Since
DiscoFaceGAN [10] and AniFaceGAN [58] only synthesize images at $256\times
256$ resolution, we utilize a state-of-the-art super-resolution model, SwinIR
[30] to upsample into $512\times 512$ for a fair comparison. As shown in
Table. 1(a), our method is superior in both FID and KID than all prior work,
demonstrating the most compelling image quality. We note that the original
EG3D has a slightly lower FID at $4.8$ which is expected since we introduce
controllability with additional loss regularization.

##### Disentanglement

To evaluate the disentangled controllability of our model over shape,
expression and camera pose, we measure the disentanglement score [10] of
synthesized images as DSα, DSβ,θ and DSc respectively. DS measures the
stability of other factors when a single attribute is modified in image
synthesis. We employ DECA [12] for estimation of FLAME parameters from
generated images and calculate the variance of the estimated parameters
$(\bm{\mathbf{\alpha}},\\{\bm{\mathbf{\beta}},\bm{\mathbf{\theta}}\\},\bm{\mathbf{c}}).$
Specifically the DSi is calculated as

$DS_{i}=\prod_{j\neq
i}\frac{var(i)}{var(j)},\,\,i,j\in\\{\bm{\mathbf{\alpha}},\\{\bm{\mathbf{\beta}},\bm{\mathbf{\theta}}\\},\bm{\mathbf{c}}\\}.$
(8)

Higher value of DS indicates better disentanglement. Additionally, we evaluate
the identity similarity between pairs of synthesized images with random camera
poses and expressions by calculating the cosine similarity of the face
embeddings with a pre-trained face recognition module [26]. Our approach
demonstrates the best disentanglement numerically over all prior work as
indicated in Table. 1(a).

###  4.3 Ablation Studies

We ablate the efficacy of the individual component by removing it from our
full pipeline. As shown in Figure. 3, we show the loss of control accuracy
over shape and expression respectively when removing the geometric prior loss
$L_{prior}$ and control loss $L_{enc}.$ Conditioning the neural radiance field
on expression is also critical to the modeling of dynamic details. Numerically
we validate the efficacy of $L_{prior}$ and $L_{enc}$ with Average Shape
Distance (ASD) and Average Expression Distance (AED). From the prior
distribution, we randomly sample $500$ shapes and expressions, with which we
synthesize images with $10$ different identities. We then reconstruct FLAME
parameters from the synthesized images and compare against the input control.
We compute 3D per-vertex $L_{1}$ distance of FLAME meshes for ASD while we use
3D landmarks $L_{1}$ distance for AED calculation. Additionally we compute the
variance of estimated shapes and expressions within each control group, with
lower value indicating more precise control. The efficacy of $L_{prior}$ and
$L_{enc}$ is well evidenced in Table. 1(b), with even slight image quality
improvements.

###  4.4 Applications

##### Talking Head Video Generation.

In Figure. 1, we showcase talking head video generation in controllable views
driven with animation sequences of FLAME. Thanks to the high control accuracy,
our method is able to synthesize various talking head videos with the same
head movements and expressions performed by different identities. Our method
is expressive in depicting both large articulated neck and jaw movements and
subtle facial expressions like eye blinks, with rich dynamic details. Shape
manipulation is easily achievable as well by modifying the shape parameters.
Please refer to our supplementary materials for more results in high
resolution.

##### Portrait Image Manipulation and Animation.

As illustrated in Figure. 1, our model also supports 3D-aware face reenactment
of a single-view portrait to a video sequence. To achieve that, we perform an
optimization in the latent Z+ space [24] to find the corresponding latent
embedding, with FLAME parameter and camera pose estimated from the input
portrait. With a frozen generator, the optimization is performed by measuring
the similarity between generated image and real image using the $L_{2}$ loss
and LPIPS loss [65]. For better reconstruction quality, we alter the
parameters of the tri-plane synthesis module with a fixed optimized latent
code [43]. After that one can explicitly manipulate the portrait with a
preserved identity and in a different camera pose and expression. With the
expression codes $(\bm{\mathbf{\beta}},\bm{\mathbf{\theta}})$ reconstructed
from a video sequence, we are also able to reenact the portrait to the video
motion.

##### Societal Impact.

Our work focuses on improving the controllability of 3D-aware GANs in
technical aspects and is not specifically designed for any malicious uses.
This being said, we do see that the method could be potentially extended into
controversial applications such as generating fake videos. Therefore, we
believe that the synthesized images and videos should present themselves as
synthetic.

##  5 Conclusion

In this work, we introduce OmniAvatar, a novel 3D-aware generative model for
synthesis of controllable high-fidelity human head. Our model achieves
disentangled semantic control by factoring the generative process into
3D-aware canonical head synthesis and implicit volume deformation to target
shapes and expressions. By learning a deformable signed distance function with
canonical volumetric correspondence, we instill the geometric prior knowledge
of a 3D statistical head model into the generation of neural radiance fields,
enabling superior generative capability to novel shapes and expressions. Our
approach demonstrates expressive and compelling talking head generation and
portrait image animation with fine-grained control accuracy and temporal
realism. We believe the proposed method presents an interesting direction for
3D avatar creation and animation, which sheds light on many potential
downstream tasks.

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1]Faculty of Applied Sciences, University POLITEHNICA of Bucharest, 060042
Bucharest, Romania, email<EMAIL_ADDRESS>2]Extreme Light
Infrastructure – Nuclear Physics, 077125 Măgurele-Bucharest, Romania, email:
<EMAIL_ADDRESS>3]Physics Doctoral School, Bucharest University,
077125 Măgurele-Bucharest, Romania 4]Department of Physics, Faculty of Applied
Sciences, University POLITEHNICA of Bucharest, 060042 Bucharest, Romania,
email<EMAIL_ADDRESS>5]Centre for Research in Fundamental
Sciences Applied in Engineering, University POLITEHNICA of Bucharest, 060042
Bucharest, Romania 6]Horia Hulubei National Institute of Physics and Nuclear
Engineering, 077125 Bucharest–Măgurele, Romania, email:
<EMAIL_ADDRESS>
# Construction and characterization of a Sagnac-based entangled-photon source

Vasile-Laurenţiu Dosan  Andrei Nazîru  Mona Mihăilescu 111Corresponding author
Radu Ionicioiu  [  [  [  [  [  [

###### Abstract

Entangled photons are crucial for the development of quantum technologies and
especially important in quantum communications. Hence it is paramount to have
a reliable, high-fidelity source of entangled photons. Here we describe the
construction and characterization of a polarization-entangled photon source.
We generate maximally-entangled Bell states using a type-II PPKTP nonlinear
crystal inside a Sagnac interferometer. We characterize the source in terms of
brightness, visibility, Bell-CHSH test and we perform quantum-state tomography
of the density matrix. Our source violates Bell-CHSH inequality $|S|\leq 2$ by
$n=22$ standard deviations. With visibilities up to ${\cal V}=98.9\%$ and
fidelity ${\cal F}=97\%$, our source is highly competitive, with state-of-the-
art performance. To the best of our knowledge, this is the first source of
entangled-photons designed and build in Romania and as such, represents an
important step for the development of quantum technologies in our country. We
envisage that our results will stimulate the progress of quantum technologies
in Romania and will educate the future generation of quantum engineers.

###### keywords:

entanglement, polarization-entangled photon source, Sagnac configuration,
quantum technologies

##  1 Introduction

The new generation of quantum technologies use counter-intuitive aspects of
quantum mechanics like superposition and entanglement [1, 2]. Entanglement is
a key resource for quantum technologies and has several applications in
quantum communications, quantum computing and quantum imaging. As such, a
robust source of entangled photons is essential for any further development of
the field.

Recently, entanglement generation techniques have been progressing
considerably. A widely-used method to generate polarization-entangled photon
pairs is spontaneous parametric down-conversion (SPDC) in nonlinear crystals
[3]. In bulk optics, one of the most stable sources of polarization-entangled
photons use a Sagnac-interferometer setup. Sagnac-based quantum sources are
widespread due to their phase-stability, versatility and the use of a single-
cristal configuration [4, 5]. Nevertheless, this scheme requires polarization
optics for different wavelengths (pump and down-converted photons) which makes
the alignment process difficult.

Due to its stability, an entanglement source based on the Sagnac configuration
has been installed on the first quantum-communication satellite (Micius) [6]
and used to demonstrate satellite-to-ground quantum key distribution [7],
entanglement distribution over 1200 km [8] and ground-to-satellite quantum
teleportation [9].

In this article we describe the experimental realization of a Sagnac-based
entanglement source and we characterize its output state. Our polarization-
entangled photon source can be tuned to generate one of two maximally-
entangled Bell states:

$|\Psi^{\pm}\rangle_{AB}=\frac{1}{\sqrt{2}}(|H_{A}V_{B}\rangle\pm|V_{A}H_{B}\rangle)$
(1)

where $H(V)$ labels horizontal (vertical) polarization and $A$ and $B$ the
photons.

The structure of the article is the following. In Section 2 we discuss the
architecture of the entanglement source and its main components. We briefly
describe the alignment procedure, which is essential for obtaining a high-
fidelity entangled state. In Section 3 we characterize the source and the
output state in terms of several parameters (spectral brightness, visibility,
state fidelity, purity, Bell parameter etc) and we perform the full quantum
state tomography of the density matrix. Finally, we conclude in Section 4.

##  2 Experimental setup

In this section we describe in detail the configuration of the entangled-
photons source. Briefly, it consists of a laser (405 nm) who pumps
bidirectionally a nonlinear PPKTP crystal situated in a Sagnac interferometer,
Fig. 1. The source generates polarization-entangled photon pairs by SPDC from
405 nm to 810 nm.

Specifically, the source is based on collinear type-II quasi phase matching
(QPM) on a 15 mm-long PPKTP crystal (Raicol) with a grating period of
$10\,\mu$m. The source is pumped with a free-space continuous-wave (CW) laser
diode at $\lambda_{p}=$ 405 nm (Ondax compact laser module). The photons
emerging from the laser diode are vertically-polarized. A half-wave plate
(HWP) is set at $22.5^{\circ}$ to prepare the superposition
$|+\rangle=\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$.

Figure 1: Sagnac-based source of polarization-entangled photons. The pump
laser (down-converted photons) are in blue (red). HWP: half-wave plate; DW-
HWP: dual-wavelength HWP; QWP: quarter-wave plate; M: mirror; I: iris; FL:
focusing lens; DM: dichroic mirror; PBS: polarization beam splitter; LPF:
linear polarizer filter; BPF: bandpass filter; C: fiber coupler; PS1,2:
polarization stages (QWP, LPF); SMF: single-mode fiber; APD: avalanche photo
diode.

The pump beam is focused in the middle of the crystal by a convex lens (FL,
$f=75$ cm). The beam passes a dichroic mirror (DM1) which transmits 405 nm and
reflects 810 nm. The $H$ component is transmitted by the polarization beam
splitter (PBS) and enters the Sagnac loop in counter-clockwise direction,
while the $V$ component is reflected by the PBS and follows a clockwise path.

Inside the loop (in the clockwise arm) we place a dual-wavelength half-wave
plate (DW-HWP) for 405 nm and 810 nm, which is set to $45^{\circ}$, such that
$H$\- and $V$-polarizations are swapped, $H\rightarrow V$, $V\rightarrow H$.
This ensures that the crystal is pumped (at 405 nm) from both directions by
$H$-polarized photons. This is crucial, since SPDC occurs in the crystal only
for $H$-polarized pump photons. At the exit, both $H$ and $V$ pump photons
(405 nm) will exit the loop and will be transmitted back by the dichroic
mirror (DM). The PPKTP crystal is kept inside a Peltier oven at a controlled
temperature (between 20∘ and 60∘C) required for QPM. Inside the crystal SPDC
occurs and a part of $H$-polarized pump photons are down-converted into pairs
of 810 nm with orthogonal polarization.

The 810 nm photons travel back to the PBS. After the PBS, the down-converted
photons exit the loop via the two outputs. The first output consists of a $HV$
pair that are reflected by a dichroic mirror (DM2) to the polarization stage
(PS2). The second output emits $VH$ pairs which are also reflected by another
dichroic mirror DM1 to the polarization stage PS1. In other words, in the
Sagnac interferometer the two paths are coherently superposed and, if the pump
is in the $|+\rangle$ state, both paths will have the same amplitude.

After the two polarization stages PS1/PS2, the photons are coupled into
single-mode fibers (SMFs) using opto-couplers (OCs). A 810 $\pm$ 10 nm
bandpass filter (BPF) is placed before each OC. The BPF are used to minimize
the noise and to filter better the 405 nm photons. The FC/PC SMFs are
connected to a quED quTools module which contains two Si-based avalanche photo
diodes (Si-APDs, quantum efficiency $\eta=30\%$) and a coincidence module; the
coincidence window width is set to 30 ns.

The main operating parameters of the source are shown in Table 1.

Parameters | min | typ | max
---|---|---|---
Laser diode input current | 50 mA | 80 mA | 95 mA
Laser diode stabilization time | 0.5 hrs | 2 hrs | –
Pump wavelength | 404 nm | 405 nm | 406 nm
Output (down-converted) wavelength | – | 810 nm | –
Oven temperature | 20 ∘C | 33.2 ∘C | 60 ∘C
Single count rate | 1 kHz | 30 kHz | 50 kHz
Dark count rate | – | 500 Hz | –
Detection efficiency | – | 30 % | –
Coincidence window length | – | 30 ns | –
Table 1: Operating parameters of the quantum source, with min, max and typical
values.

Building and aligning the source is non-trivial and several steps are required
in order to obtain a highly-entangled output state. We now briefly describe
the alignment procedure.

The construction of the setup took several months because the alignment
process was difficult. From our experience, a few key features of the
alignment procedure are to: (i) visualize fringes at 405 nm before inserting
the PPKTP crystal inside the loop; (ii) ensure the proper degrees of freedom
for each optical component; (iii) keep the crystal at the right temperature in
order to generate the target quantum state; (iv) position precisely the PPKTP
crystal in the middle of the interferometer.

The setup misaligns easily between measurements performed in different days.
In this case, a few hints are: the distances between the optical components
should be as small as possible; a constant temperature and humidity should be
maintained in the lab; the setup should be on a vibration-free optical table;
the lab windows should be covered to minimize the dark counts. In order to re-
align the setup, we have to do the following steps: improve the coupling;
remove the LPFs and achieve a constant coincidence rate for each HWP1 position
using the PBS and M2,3,4; find a proper crystal temperature for good
entanglement visibilities.

##  3 Source characterization

The next important step after aligning the quantum source is to characterize
its output in terms of brightness, entanglement, state fidelity, purity etc.
These parameters will be described in the present section. To begin with, the
quality of the alignment is directly reflected in the characteristics of the
output quantum state.

From the coincidence-to-single-photon coupling ratio
($\eta_{C}=\frac{S_{C}}{\sqrt{S_{i}S_{s}}}\approx 9.87\,\%$, where $S_{C}$,
$S_{i}$, $S_{s}$ are the coincidence, idler and signal rates, respectively),
the detection efficiency ($\eta_{D}\approx 30\,\%$) and the single-mode
bandwidth of the down-converted photons ($\Delta\lambda=\frac{5.52\cdot
10^{-3}}{L}=0.4\,\mathrm{nm}$, where $L=15\,\mathrm{mm}$ is the crystal
length), we estimated the spectral brightness, $S\approx 62500\,(\rm mW\cdot
s\cdot nm)^{-1}$ [11, 10]. This gives the number of entangled photon pairs per
second, per mW of pump power and per nm of wavelength bandwidth. By
integrating over the wavelength bandwidth, we obtain the source brightness
$B=25000\,(\rm mW\cdot s)^{-1}$, i.e., the number of entangled photon pairs
per second, per mW of pump power.

One of the most important parameters is the entanglement of the output state.
The easiest way to test the entanglement is to measure the correlation curves
in two mutually-unbiased bases. This can be done by keeping a linear polarizer
filter at a fixed angle ($\theta_{1}$) while changing the orientation
($\theta_{2}$) of the other one. For the $H/V$ basis $\theta_{1}=0^{\circ}$
and for $+/-$ basis $\theta_{1}=45^{\circ}$. The entanglement visibility is
given by:

${\cal V}=\frac{C_{\rm max}-C_{\rm min}}{C_{\rm max}+C_{\rm min}}$ (2)

where $C_{\rm min/max}$ is the minimum/maximum coincidence rate determined
from the correlation curves (Fig. 2). From our experimental data, the
visibilities in the two bases are ${\cal V}_{H/V}=(98.9\pm 0.4)\,\%$ and
${\cal V}_{+/-}=(93.7\pm 1.1)\,\%$.

Figure 2: Correlations curves for $+/-$ basis (left) and $H/V$ basis (right).

Another way to quantify the entanglement is to test the violation of Bell-CHSH
inequality. As shown by Bell and others, any local hidden-variable theory
satisfies [12, 13, 14]:

$|S|\leq 2$ (3)

where the Bell-CHSH parameter $S$ is defined as

$S=E(\alpha,\beta)-E(\alpha,\beta^{\prime})+E(\alpha^{\prime},\beta)+E(\alpha^{\prime},\beta^{\prime})$
(4)

Here $\alpha,\alpha^{\prime},\beta,\beta^{\prime}$ are different polarizer
angles and $E(\alpha,\beta)$ is the expectation value of joined polarization
measurements (of the two photons) along directions $\alpha$ and $\beta$:

$E(\alpha,\beta)=\frac{C(\alpha,\beta)+C(\alpha_{\perp},\beta_{\perp})-C(\alpha,\beta_{\perp})-C(\alpha_{\perp},\beta)}{C(\alpha,\beta)+C(\alpha_{\perp},\beta_{\perp})+C(\alpha,\beta_{\perp})+C(\alpha_{\perp},\beta)}$
(5)

where $C(\alpha,\beta)$ is the coincidence rate for polarizer angles $\alpha$
and $\beta$; $\alpha_{\perp}=\alpha+90^{\circ}$ and
$\beta_{\perp}=\beta+90^{\circ}$.

However, in quantum mechanics the inequality (3) is violated for certain
entangled states. The maximum violation is obtained for maximally-entangled
(Bell) states and for polarizer angles
$(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=(0,22.5^{\circ},45^{\circ},67.5^{\circ})$.
In this case, quantum mechanics gives

$|S|_{QM}=2\sqrt{2}\approx 2.828$ (6)

Thus, to test Bell-CHSH inequality we need to measure polarization
correlations for joint measurements of the two-photons state at different
polarizer angles $\alpha,\beta$.

We have measured the 16 coincidence rates for the previous angles
$(\alpha,\beta,\alpha^{\prime},\beta^{\prime})=(0,22.5^{\circ},45^{\circ},67.5^{\circ})$
for our source. We have obtained

$|S|=2.684\pm 0.03$ (7)

Thus our source violates the Bell-CHSH inequality by more than $n_{\Delta}=22$
standard deviations, with $n_{\Delta}=(\overline{|S|}-2)/\Delta S$,
$\overline{|S|}=2.684,\Delta S=0.03$. Consequently, there is reliable and
strong evidence that our source generates highly-entangled states violating
Bell-CHSH inequality (3).

Figure 3: The reconstructed (maximum-likelihood) density matrix, with the real
part (left) and imaginary part (right).

To fully characterize the output state we performed quantum state tomography.
This procedure reconstructs the full density matrix of a state from the set of
coincidence measurements for different polarization measurements [15]. This
procedure uses: (i) coincidence counts for measurements in different bases
($HH,HV$ etc), and (ii) a numerical optimization technique known as maximum-
likelihood (ML) [16].

Using our experimental data for coincidence counts in different bases, we have
reconstructed the maximum-likelihood density operator $\rho_{ML}$ using an
open-source (online) tool from Kwiat group (Illinois University) [17]. Our
reconstructed maximum-likelihood density operator $\rho_{ML}$ is given below,
see Fig. 3.

$\small\rho_{ML}=10^{-3}\cdot\begin{bmatrix}4.24&-6.17-9.42\rm
i&-0.275+19.6\rm i&1.26+3.78\rm i\\\ -6.17+9.42\rm i&486&-487-032\rm
i&0.754-0.574\rm i\\\ -0.275+19.6\rm i&-487-32\rm i&505&7.02-7.84\rm i\\\
1.26+3.78\rm i&0.754+0.574\rm i&7.02-7.84\rm i&4.02\end{bmatrix}$ (8)

For a mixed state the amount of entanglement is quantified by the concurrence
$C$ [18]; $C=0$ for separable states and $C=1$ for maximally-entangled states.
For our source we have:

$C=0.951$ (9)

The source is aligned to generate the $|\Psi^{-}\rangle$ state. The fidelity
$\cal F$ measures how close is the output state from the target state:

${\cal
F}(\rho_{ML},\rho_{|\Psi^{-}\rangle})=\mathrm{tr}^{2}\left.\sqrt{\rho_{ML}^{1/2}\
\rho_{|\Psi^{-}\rangle}\ \rho_{ML}^{1/2}}\right.=0.9743$ (10)

This shows that our output state is very close to the desired maximally-
entangled state $|\Psi^{-}\rangle$. Another relevant parameter is the purity:

$P={\rm tr}(\rho_{ML}^{2})=0.953$ (11)

The purity varies between $\frac{1}{4}$ (for the totally mixed state) to 1
(for pure states).

The main output parameters of our source are summarised in Table 2.

Parameter | value
---|---
Brightness | 2.5$\times 10^{4}$ $\rm(mW\cdot s)^{-1}$
Spectral brightness | 6.25$\times 10^{4}$ $\rm(mW\cdot s\cdot nm)^{-1}$
${\cal V}_{H/V}$ visibility | $98.9\pm 0.4\,\%$
${\cal V}_{+/-}$ visibility | $93.7\pm 1.1\,\%$
Bell-CHSH parameter | $|S|=2.684\pm 0.03$
Concurrence | $C=0.951$
Fidelity | ${\cal F}=0.9743$
Purity | $P=0.953$
Table 2: The output parameters of our entangled-photons source.

##  4 Discussion and conclusions

Year | ref. | visibility ($\%$) | fidelity
---|---|---|---
2022 | this work | 98.9 | 0.97
2022 | [19] | 98.2 | 0.96
2021 | [16] | 99.9 | 0.97
2021 | [20] | - | 0.99
2018 | [21] | 96.0 | 0.95
2018 | [22] | 98 | 0.87
Table 3: Performance of our source compared to state-of-the-art.

In this article we have described the experimental realization of a
polarization-entangled photon source. Maximally-entangled states are generated
through SPDC in a type-II nonlinear crystal inside a Sagnac interferometer.
The quality of the alignment is directly reflected by the parameters of the
output state. We have characterized the quantum source in terms of several
parameters: brightness, visibility, Bell-CHSH test, concurrence and we have
performed quantum state tomography of the density matrix.

A comparison between our source and state-of-the-art is given in Table 3. As
we can see, our source is highly competitive, and in some cases equals or
exceeds existing experiments, in terms of visibility and fidelity.

To the best of our knowledge, this is the first source of entangled-photons
designed and build in Romania. As such, it represents an important step for
the development of quantum technologies in our country. In the future, our
source can be used to demonstrate quantum teleportation and entanglement-based
quantum key distribution. We envisage that our results will stimulate the
progress of quantum technologies in Romania and will educate the future
generation of quantum engineers.

Acknowledgements. The authors acknowledge support from a grant of the Romanian
Ministry of Research and Innovation, PCCDI-UEFISCDI, project number PN-
III-P1-1.2-PCCDI-2017-0338/79PCCDI/2018 (QUTECH-RO), within PNCDI III. R.I.
acknowledges support from PN 19060101/2019-2022.

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