# LRWR: Large-Scale Benchmark for Lip Reading in Russian language

###### Abstract

Lipreading, also known as visual speech recognition, aims to identify the
speech content from videos by analyzing the visual deformations of lips and
nearby areas. One of the significant obstacles for research in this field is
the lack of proper datasets for a wide variety of languages: so far, these
methods have been focused only on English or Chinese. In this paper, we
introduce a naturally distributed large-scale benchmark for lipreading in
Russian language, named LRWR, which contains 235 classes and 135 speakers. We
provide a detailed description of the dataset collection pipeline and dataset
statistics. We also present a comprehensive comparison of the current popular
lipreading methods on LRWR and conduct a detailed analysis of their
performance. The results demonstrate the differences between the benchmarked
languages and provide several promising directions for lipreading models
finetuning. Thanks to our findings, we also achieved new state-of-the-art
results on the LRW benchmark.

Index Terms: speech recognition, human-computer interaction, computational
paralinguistics

##  1 Introduction

In recent years, the machine learning community has made significant progress
in solving the problem of speech recognition from audio. Deep learning methods
for Automatic Speech Recognition are now showing impressive performance but
often struggle to understand the talk in noisy environments or the case of
multiple speakers. Adding lipreading methods can help in solving this problem
if there is a video with a speaker. Nowadays, when lot’s of people use voice
messages while walking or driving, news reports are filmed directly from the
place of events, such an expansion of speech recognition systems is becoming
more and more valuable. The word-level approach to solve the Lipreading task
is recognizing individual words from a previously known dictionary. Still,
even with this limitation, visual recognition is a challenging task for
computer vision. However, with the advent of deep learning, there has been
significant progress in this task.

Besides all the above progress, the primary catalyst for developing new
lipreading methods and architectures that could qualitatively solve the speech
recognition problem was the emergence of large datasets collected from non-
laboratory data. The first such dataset is LRW [1], proposed in 2016, which
consists of 500 classes and is based on BBC news footage. Another available
benchmark for lipreading in Chinese is the LRW-1000 [2], proposed in 2018,
which consists of 1000 classes and displays a wide variety of speech
conditions. Unfortunately, there was no such dataset for Russian language.

Table 1: Datasets statistics Datasets | number of classes | number of speakers | Resolution | Environment
---|---|---|---|---
OuluVS2: [3] | 10 | 53 | Fixed, with 6 different sizes | Lab.
LRW: [1] | 500 | >1000 |  Fixed ($256\times 256$) | TV
LRW-1000: [2] | 1000 | >2000 | Naturally distributed | TV
LRWR (ours) | 235 | 135 |  Fixed ($112\times 112$) | YouTube

To this end, we collect a naturally-distributed large-scale dataset for
lipreading in Russian language and provide a comprehensive comparison of the
current popular lipreading methods. The contributions in this paper are
summarized as follows.

Figure 1: Data collection pipeline

Our first contribution is a challenging and naturally distributed dataset for
lipreading in Russian language – LRWR. (section 3). There are 235 words from
135 speakers, with over 117500 samples in total. To the best of our knowledge,
it’s the first publicly available Russian lipreading dataset and Slavic-based
language dataset.

Our second contribution is benchmarking a various number of existing
architectures that demonstrate the best results on LRW and LRW-1000, such as
deformation Flow Based Two-Stream Network [4], D3D [2] and other architectures
[5, 6] on LRWR (section 4). Besides, we conduct a comprehensive quantitative
study and comparative analysis of the impact of different deep learning
techniques on the resulting lipreading methods performance (section 4.3). In
our analysis, we review different regularization and augmentation techniques,
as well as various methods for data sampling and learning rate scheduling
during model training.

The final point of our study is the proposal of a new architecture that
summarises our experiments’ findings and achieves the best results on our
dataset and the open LRW dataset, improving the previous state-of-the-art
result from 88.4% [6] to 89.1% accuracy. (section 4.4)

We hope that the open-sourced LRWR dataset and the proposed architecture will
serve as the foundation for developing lipreading techniques in Russian and
other languages of Commonwealth of Independent States.

##  2 Related work

Large-scale datasets have successively proven their fundamental importance in
many research fields. Word-level lipreading was no exception, and most recent
breakthroughs came with the emergence of such datasets as the LRW or the
LRW-1000. In this section, we observe the popular datasets and methods being
state-of-the-art or close to them based on results obtained on the datasets
for word-level lipreading.

###  2.1 Datasets

Several most popular world-level lipreading datasets are summarized in Table
1. All of them have a significant impact on the deep learning progress in
lipreading. In this part, we will give a short overview of these datasets
shown in the table.

OuluVS2 [3], released in 2015, consists of 10 phrases spoken by 53 speakers
with 9109 samples in total. This dataset was filmed in a laboratory
environment and contained various camera angles: frontal, profile, 30°, 45°,
60°. However, due to laboratory conditions, there are only a few variations
beyond the view positions, and the viewpoints are always identical, which
leads to poor robustness under real-life conditions for OuluVS2-trained
networks.

LRW: [1], the first large-scale dataset, released in 2016, contains 500
classes with more than 1000 recorded pronunciations per class. This dataset no
longer consists of videos with word-by-word pronunciation recorded in a
controlled lab environment but utilizes data from the BBC channel, which gives
various speakers, angles, and lighting conditions. All this together makes it
possible to use the LRW-trained network in the real world.

LRW-1000 [2], the largest dataset available for lipreading, released in 2018
and contains 1000 classes and more than 2000 individual speakers. The dataset
collection pipeline was optimized for real-world applications. Thus it aims at
covering a ”natural” variability over different speech modes and imaging
conditions.

Although there have been many lipreading datasets as listed above, there are
very few lipreading datasets for Slavic languages available up to now.
Therefore, we hope LRWR could help future ASR research in Russian.

###  2.2 Methods

Over the last decade, there was a lot of progress in lipreading methodology.
Early methods solved the Lipreading problem with expertly derived techniques
using mathematical image transformations such as Discrete Cosine Transform
(DCT) [7], Active Appearance Model (AAM) [8] and Local Binary Pattern (LBP)
[9].

With the advent of the LRW dataset, it became possible to train deep neural
networks and utilize them for real-world applications effectively. At first, a
lot of research was focused on 2D fully convolutional networks [10].
Nevertheless, thanks to hardware evolution, it soon became possible to use 3D
convolution over 2D convolutions [1] or recurrent neural networks [11] for
more efficient temporal information usage. This idea has grown into a type of
architecture where information about the local lip movement is extracted using
3D+2D convolutions backbone, while final temporal information is summarized
using recurrent layers at the end. This approach has been extremely
successful, and many state-of-the-art lipreading solutions used it as a meta-
architecture even nowadays [6].

In addition to the above methods, there have been many attempts to modify
architecture:

  * •

D3D architecture [2], where 3D+2D convolutions are replaced by 3D only.

  * •

Two-branch network [4], which takes into account the temporal component using
Deformaition-Flow, thus replacing the initial 3D convolutions in one branch.

Moreover, the researchers experimented with the replacement of recurrent
layers: temporal convolutional layers [5], and Transformer [12].

In this paper, we benchmark all of the above state-of-the-art methods on our
dataset and present a detailed analysis of the results, which could provide
directions for future research.

Table 2: LRWR benchmark Network | Initial weights | LRWR accuracy
---|---|---
D3D[2] | LRW pretrained | 54%
Feng20 [6] | LRW pretrained | 52%
Xiao20 [4] | LRW pretrained | 52%
Martinez20 [5] | LRW pretrained | 51%
D3D[2] | none | 49%
CSN50 [13] \+ GRU-backend + (2+1)D CNN | none | 49%
CSN50 [13] \+ GRU-backend | none | 49%
Martinez20 [5] | none | 48%
Xiao20 [4] | none | 47%
CSN34 [13] | none | 42%

##  3 Dataset

In this section, we describe the properties of the LRWR dataset: the pipeline
of raw data collection, its’ preprocessing logic, and the final statistics of
the resulting dataset.

###  3.1 Data collection

To make our benchmark suitable for a large variety of real-world use cases, we
made a few requirements during the data collection process:

  * •

First of all, we were carefully looking at the dataset diversity of speakers’
audio/video conditions – different rates of speech, facial expressions, and
the face’s visual appearance.

  * •

Secondly, we were interested in different lipreading conditions – a variety of
camera angles and lighting.

After an exploratory search, we found that YouTube videos could meet these
requirements. Downloaded videos were available in HD (1280 * 720) or full_HD
(1920 * 1080), 25 fps, with 192 Kbps sound quality, and a variety of Russian-
speaking podcasters and bloggers, resulting in a dataset of videos with a
large diversity of speakers in different environments. Moreover, we were able
to capture a wide range of topics in this case: history, art, travel, world
events, or just daily talks.

We also apply several extra preparations during the data collection pipeline.
Firstly, we filter the videos to select one with only one speaker present in
the frame. Then we prepare a vocabulary with 1500 most frequent words of
Russian-language blogging. The vocabulary was based on the first 50 hours of
video from the previous step, distilled into text using the speech-to-text
toolkit ”Kaldi” [14]. We did the preliminary selection of 1500 words, because
in Russian a word can have many forms, depending on the number, case, and
conjugation. Because of this, many words appear only once, and are not
suitable for inclusion in the dataset.

After these stages, we get a list of videos suitable for parsing, including
135 speakers and about 350 hours and the list of 1500 most frequently used
words.

###  3.2 Raw data preprocessing

The data preprocessing pipeline is shown in Fig 1. We process the downloaded
audio track using speech-to-text methods and find the frames with words from
our vocabulary.

As a speech-to-text baseline, we use the Kaldi toolkit [14], which outputs
text and the appropriate time frame in which this text was spoken. After that,
we cut this fragment from the video using FFmpeg111https://www.ffmpeg.org/.
Based on exploratory analysis of the data subset, we did not find any errors
in extracted word boundaries, so we didn’t have to implement video
synchronization to get more accurate time coordinates.

Having received a video, we find the speaker’s face and highlight the face’s
landmarks with the FaceAlignment network [15]. Then we normalize the video
using landmarks, making the line between lips corners horizontal. Finally, we
crop the lip’s region during the entire video with an algorithm inspired by
the paper [2] using the following equation:

$w=min(3d_{mn},max(1.5d_{mn},1.05x_{r}-0.95x_{l}))$ (1)

where $w$ is a crop width and height, $d_{mn}$ is a distance between nose and
center of lips and $x_{l}$, $x_{r}$ – x coordinate of lip corners. To filter
the inaccuracies of the FaceAligment network, we employ a binary classifier
based on ResNet-18 [16]. Moreover, to filter blurry footage or unuseful camera
angles, we utilize the IoU tracker [17] to check the face’s sharp movement in
the frame and filter such cases out.

The resulting dataset was extremely unbalanced: some classes had more than
2,000 words, most of them – less than 200. Moreover, some classes of words
differed in the pronunciation of one letter. For instance, the dataset
contains words pronounced kotoriy and kotoria as different classes. Our
initial experiments demonstrate that too many classes that differ in only one
phoneme lead to a significant deterioration in quality. In such a case, we
leave only one class not to disrupt the neural networks learning process.
After such a strict selection, there were 235 classes with more than 500 words
were selected.

In the final step, we balance classes, leaving 500 words in each class. The
resulting dataset was used for further benchmarking of the networks.

###  3.3 Dataset Statistics

The LRWR contains 235 words, 135 speakers with over 117500 samples. Each class
represents by 450 train samples and 50 test samples. The dataset has a broad
coverage of different ages, genders, and pronunciation habits of the speakers.
Lighting conditions range from natural daylight to artificial. The face
rotation angle varies from 0°to 20°. By these factors, the dataset presents a
challenging task for current lipreading methods. Moreover, for the speaker’s
privacy presented in the data, we make publicly available222The dataset is
available on request to authors upon submitting a license agreement. only
cropped speakers’ lip area as a final benchmark dataset as shown in Fig 1.

Table 3: Training tips analysis on LRWR Network | LRW weights | Label Smoothing | CutOut | MixUp | CosineScheduler | upsampling | gru size | LRWR accuracy
---|---|---|---|---|---|---|---|---
D3D[2] | ✓ | ✓ | × | ✓ | ✓ | $450\rightarrow 750$ | 2048 | 61%
D3D[2] | ✓ | ✓ | × | ✓ | ✓ | $450\rightarrow 750$ | 512 | 59%
D3D[2] | ✓ | ✓ | × | ✓ | ✓ | × | 512 | 55%
D3D[2] | ✓ | ✓ | (1,32) | × | × | × | 512 | 54%
D3D[2] | ✓ | ✓ | (32,8) | × | × | × | 512 | 54%
D3D[2] | ✓ | ✓ | (8,8) | × | × | × | 512 | 54%
D3D[2] | ✓ | ✓ | × | × | × | × | 512 | 54%
D3D[2] | ✓ | × | × | × | × | × | 512 | 54%
D3D[2] | ✓ | ✓ | × | ✓ | ✓ | $450\rightarrow 950$ | 512 | 52%
D3D[2] | × | × | × | × | × | × | 512 | 49%

##  4 Experiments

In this section, we evaluate popular lipreading networks on our dataset,
perform the analysis of training tricks effectiveness and summarise our
finding into a new architecture proposal that achieved state-of-the-art
accuracy on LRWR and LRW.

###  4.1 Experimental settings

For the benchmarking, the final version of the LRWR dataset with balanced
classes was taken. Additionally, we randomly flip all the frames in the video
horizontally and randomly crop area (88*88) through all video, which is common
practice for lipreading [2, 4, 5]. Furthermore, we explore neural networks
training from random and pretrain weights initialization to estimate
finetuning capabilities.

###  4.2 Methods

To provide a comprehensive benchmark, we review recent papers on lipreading,
most common baselines, and the best-performing approaches in this task. We
highlight three of them:

  * •

3D+2D frontend with recurrent layers backend approach, represented by the
article ”Learn an effective lipreading model without pains” [6] which
currently has the highest score on the LRW and the LRW-1000.

  * •

Modifications to the previous method with recurrent layers replaced by the
temporal convolution layers [5], Using the second branch, where 3D
convolutions are replaced by the Deformation-Flow-Network [4].

  * •

3D-only frontend architecture, presented by the D3D [2].

Moreover, we adapted a method from the field of action recognition – the
channel-separated convolutional networks [13]. The adaptation includes usage
of recurrent layers as a backend, following the idea presented in article
[11]. As another approach for backend architecture, we replace 3D convolution
with (2+1)D-convolution [18]. The final results of methods comparison are
shown in the Table 2.

###  4.3 Analysis

After selecting the most successful candidate - D3D, we investigated the
impact of various regularization and augmentation techniques or learning
process modification on the final quality. We review the following techniques:
Label Smoothing [19], MixUp [20], CutOut [21]. Moreover, we investigate the
impact of the hidden size of the recurrent layers and Cosine scheduler usage
[22] during training. In addition, we examine upsampling. Since not all parsed
data entered the LRWR, as mentioned in chapter 3.3, we add data that did not
get into the LRWR due to class balancing. If there was not enough unused data,
then the samples were copied. The results of the experiments are presented in
Table 3.

Experiments have shown that the combination of LabelSmoothing, MixUp, and
CosineScheduler with increased RNN hidden size positively affects the
resulting performance. We also found a general fact for LRWR: the quality of
any network increases if we upsample each class up to 750 samples, while
further upsampling leads to performance degradation. Moreover, pretrained
weights initialization leads to better results in all our experiments that
provide promising directions for lipreading models finetuning in future
research. Finally, while D3D demonstrates the greatest performance on the LRWR
dataset (Table 2), it has the lowest accuracy on the LRW (Table 4), which
could demonstrate the differences between the benchmarked languages

###  4.4 LRW improvement – AttentionLipreadingNet

Using our findings from the LRWR benchmark, we would like to introduce a new
architecture named AttentionLipreadingNet (ALN) and describe its training
process, leading to the new state-of-the-art results on LRW dataset.

AttentionLipreadingNet. Analyzing the differences between published (Table 4)
and benchmarked (Table 2) results, we would like to propose a neural
architecture that will work effectively on both datasets. Despite the fact
that on LRWR dataset D3D[2] shows the best results, this network demonstrate
lower quality on the LRW compared to networks using a combination of 3D and 2D
convolutions. Therefore, to test our developments on the LRW, we need to
develop new network, and apply all the techniques that led to the accuracy
improvement on the LRWR.

The new network was inspired by the paper [6], which architecture shows the
best results on the LRW and LRW-1000. The frontend of ALN consists of 3D
convolutions followed by 2D ResNeSt-based convolutions with split-attention
[23], and a backend consisting of recurrent layers.

Train Loop. For each video submitted for training, a region of size 88 * 88
was randomly selected, and only a crop of this region was used for training.
After that, we use horizontal flip augmentation with 50% probability, followed
by MixUp, LabelSmoothing, and DropBlock augmentations. We used Adam Optimizer
with an initial learning rate of 1e-4, combined with CosineScheduler and a
batch size of 32. LogSoftmax averaged over the length of the video was used as
a loss.

We trained the network on the LRW dataset for 60 epochs on the NVidia V100
GPU. As a result, we got an accuracy of 89.1%, which is superior to the
previous best result (Table 4).

Table 4: Results on LRW and LRWR. All LRWR networks were pretrained on LRW

.  Method LRW Accuracy LRWR Accuracy ALN (ours) 89.1% 61% Feng20 [6] 88.4% 52%
Martinez20 [5] 85.3% 51% Xiao20 [4] 84.1% 52% D3D [2] 78.0% 54%

After achieving this result on the LRW dataset, we train the ALN network on
the LRWR using the same training settings and LRW pretrain weights, resulting
in the 61.1% accuracy. We explain such a significant disparity (61% and 89%
accuracy) in the final quality on LRW and LRWR due to the lip region quality
difference: 112 (LRWR) versus 256 (LRW) (Table 1), and more complex and varied
shooting conditions. Another possible reason for the difference in the
resulting quality may be the phonetic difference of the languages themselves
and its impact on the lipreading methods’ performance, which we see as another
possible direction for future research.

##  5 Conclusion

In this paper, we introduce the first dataset for the lipreading benchmark in
Russian language – LRWR, containing 235 classes with 117500 samples. We
evaluate current lipreading methods and investigate the impact of different
techniques on the final performance. As a result of our benchmark, we
developed a network that accomplishes the best score on both LRWR and LRW
datasets, improving the accuracy on the last one from 88.4% to 89.1% achieving
new state-of-the-art results.

##  6 Acknowledgements

The authors are grateful to Konstantin Osminin and Pavel Kalaidin for useful
discussions and valuable comments.

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needsurl

# Spectrally multimode integrated SU(1,1) interferometer

A. Ferreri  Department of Physics, Paderborn University, Warburger Strasse
100, D-33098 Paderborn, Germany  M. Santandrea  Department of Physics,
Paderborn University, Warburger Strasse 100, D-33098 Paderborn, Germany  M.
Stefszky  Department of Physics, Paderborn University, Warburger Strasse 100,
D-33098 Paderborn, Germany  K. H. Luo  Department of Physics, Paderborn
University, Warburger Strasse 100, D-33098 Paderborn, Germany  H. Herrmann
Department of Physics, Paderborn University, Warburger Strasse 100, D-33098
Paderborn, Germany  C. Silberhorn  Department of Physics, Paderborn
University, Warburger Strasse 100, D-33098 Paderborn, Germany  P. R. Sharapova
Department of Physics, Paderborn University, Warburger Strasse 100, D-33098
Paderborn, Germany

###### Abstract

Nonlinear SU(1,1) interferometers are fruitful and promising tools for
spectral engineering and precise measurements with phase sensitivity below the
classical bound. Such interferometers have been successfully realized in bulk
and fiber-based configurations. However, rapidly developing integrated
technologies provide higher efficiencies, smaller footprints, and pave the way
to quantum-enhanced on-chip interferometry. In this work, we theoretically
realised an integrated architecture of the multimode SU(1,1) interferometer
which can be applied to various integrated platforms. The presented
interferometer includes a polarization converter between two photon sources
and utilizes a continuous-wave (CW) pump. Based on the potassium titanyl
phosphate (KTP) platform, we show that this configuration results in almost
perfect destructive interference at the output and supersensitivity regions
below the classical limit. In addition, we discuss the fundamental difference
between single-mode and highly multimode SU(1,1) interferometers in the
properties of phase sensitivity and its limits. Finally, we explore how to
improve the phase sensitivity by filtering the output radiation and using
different seeding states in different modes with various detection strategies.

##  1 Introduction

One of the main tasks in Quantum Metrology research is to improve both methods
and techniques to estimate the phase sensitivity of interferometers [1]. It is
well known that optical phase sensing is generally limited by the noise due to
both photons statistics and the quantum nature of light [2, 3]. The first
bound, commonly called shot noise limit (SNL), describes the noise performance
of an ideal classical field and is limited by classical correlations. The SNL
is proportional to $1/\sqrt{N}$, where N is the number of photons in the
interferometer [4, 5]. The second bound condition, known as Heisenberg limit
(HL), stems from the quantum uncertainty principle and is proportional to
$1/N$ [6].

The shot noise limit can be overcome by using quantum states in a typical
Mach-Zehnder interferometer. For example, one can insert a Fock state or the
more exotic NOON state [7] into input channels of such an interferometer.
However, the preparation of such exotic states is a challenging task. The
nonlinear SU(1,1) interferometers can beat the SNL even without using quantum
states as inputs [8]. This type of interferometer consists of two nonlinear
processes [9] (sometimes one in a truncated version [10, 11]), such as four-
wave-mixing (FWM) or parametric down-conversion (PDC).

The phase sensitivity of this class of interferometers has been investigated
for spectrally single-mode sources [12, 13, 14], including gain-unbalanced
[15, 16] and fully quantum three-mode [17] configurations, as well as
configurations with different input states, such as coherent light [18],
squeezed states [19], a mixture of coherent and squeezing states [20, 21, 22]
and a mixture of thermal and squeezed states [23]. These studies have shown
that it is generally possible to overcome the shot noise limit and even reach
the Heisenberg limit. Moreover, SU(1,1) interferometers can provide different
benefits with respect to other interferometers, not only in terms of high-
precision measurements, but also because of the possibility to perform a joint
measurement of multiple observables [24] and due to the advantages of
robustness against external losses, namely losses due to an inefficient
detection system [25, 26, 27, 28].

Nevertheless, the single-mode picture, commonly used to analyse SU(1,1)
interferometers, is merely an approximation of what can realistically be
obtained in experiments. Indeed, PDC sources generate photon pairs with finite
spectral and spatial bandwidths and generally in more than a single mode.
Therefore, the complete description of SU(1,1) interferometers should account
for the presence of such spatial [29, 30, 31] and spectral modes [32, 33, 34]
in order to correctly engineer the system [35] and accordingly maximize the
phase sensitivity [36].

The theoretical analysis of SU(1,1) interferometers has been limited so far to
free space bulk [37] or fiber-based systems [38, 39]. However, it is of
particular interest to extend such analysis to other integrated systems [40,
41, 42], which can provide higher efficiencies, smaller footprints and pave
the way to quantum-enhanced on-chip interferometry. Such integrated systems
have been already utilized for instance in boson sampling [43] and other
elements of quantum computing, such as quantum walk [44] and Hong-Ou-Mandel
interference [45].

In this work, we present a theoretical description of the multimode integrated
SU(1,1) interferometer. The presented architecture can be realized on various
integrated platforms. As an example, we investigate the spectral properties of
the monolithic SU(1,1) interferometer based on the potassium titanyl phosphate
(KTP) platform and demonstrate conditions for obtaining the phase sensitivity
below the shot noise limit.

The paper is organized as follows: in the second section, a brief introduction
of the theoretical model and the phase sensitivity calculation strategy are
derived and discussed. In the third section, we outline two different sample
designs: the conventional bulk architecture implemented on the integrated
platform and a new developed integrated design. We demonstrate that this new
design allows the phase sensitivity to overcome the SNL and generate
supersensitivity bands. In the fourth section, we show how phase sensitivity
can be improved by using a frequency filter. In the last section, we extend
the analysis by considering non-vacuum input states along with different
detection strategies, i.e. direct detection and homodyne detection.

##  2 Theoretical model

The Hamiltonian describing the whole interference process inside the SU(1,1)
interferometer can be represented via the joint spectral amplitude (JSA)
$F(\omega_{s},\omega_{i})$ [32]:

$\hat{H}=i\hbar\Gamma\int
d\omega_{s}d\omega_{i}F(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}\hat{a}_{i}^{\dagger}+h.c.,$
(1)

where $\Gamma$ is the coupling constant containing the second order
susceptibility $\chi^{(2)}$, the pump intensity $I$ and the length of PDC
sections; $\omega_{s}$ and $\omega_{i}$ are the frequencies of the signal and
idler photons, respectively. Here we neglect the effects of time ordering.
However, such an approximation is well satisfied for the SU (1,1)
interferometer due to the effective narrowing of the spectrum because of the
nonlinear interference [33]. The joint spectral amplitude depends on both the
spectral properties of the pump laser and the geometry of the system, and it
is generally expressed as:

$\begin{split}F(\omega_{s},\omega_{i})=C\,\alpha(\omega_{s},\omega_{i})f(\omega_{s},\omega_{i}),\end{split}$
(2)

where $C$ is the normalization constant. The function
$\alpha(\omega_{s},\omega_{i})$ depends on the spectral properties of the pump
and has a Gaussian profile:

$\alpha(\omega_{s},\omega_{i})=e^{-\frac{(\omega_{s}+\omega_{i}-\omega_{p})^{2}\tau^{2}}{2}},$
(3)

where $\tau$ is the pulse duration of the pump laser, $\omega_{p}$ is the pump
frequency.

The function $f(\omega_{s},\omega_{i})$ is the phase matching function. In a
general form, for both integrated circuits with two PDC sections of equal
length $L$ investigated in this work, the function $f(\omega_{s},\omega_{i})$
is given by needsurlklyshko1993ramseyneedsurlklyshko1994parametric[46, 47, 48,
49]:

$f(\omega_{s},\omega_{i})=\frac{1}{2L}\int_{0}^{2L+l}dz\,g(z)\,e^{i\int_{0}^{z}d\xi\Delta
k(\xi)},$ (4)

where $g(z)$ is the spatial profile of the second-order nonlinear
susceptibility [50], $l$ is the distance between two PDC sections, $\Delta
k(\xi)$ is the phase matching profile, corresponding to the type-II PDC
process.

By solving the Heisenberg equations for the Hamiltonian in Eq.(1), one can
calculate the mean number of signal photons $\langle N\rangle$ and its
variance $\langle\Delta^{2}N\rangle$ for a given input state of the
interferometer, see Appendix. Consequently, the phase sensitivity can be
calculated as:

$\lvert\Delta\phi\rvert=\bigg{\lvert}\frac{\Delta N}{\partial\langle
N\rangle/\partial\phi}\bigg{\rvert},$ (5)

where $\phi$ is the phase shift undergone by photons inside the
interferometer. Interferometers are characterized by the supersensitivity
regions if the phase sensitivity overcomes the shot noise limit:

$\Delta\phi_{SNL}=\frac{1}{\sqrt{\langle N_{in}\rangle}},$ (6)

where $N_{in}$ is the number of interfering photons inside the interferometer.
Starting from the next section we will use the phase sensitivity normalized in
respect to the shot noise limit.

##  3 Interferometer optimization

In contrast to bulk optics, where temporal dynamics of the PDC photons can
often be neglected, the long interaction lengths in integrated SU(1,1) require
a careful design that takes into account the temporal walk-off and stretching
of pulses in the material due to group velocity dispersion.

###  3.1 The conventional bulk architecture implemented on the integrated
platform

The conventional bulk optics design of the SU(1,1) interferometer, transferred
to a fully integrated platform, is shown in Fig.2.

Figure 1: The joint spectral intensity $|F(\omega_{s},\omega_{i})|^{2}$
depending on the phase implemented by the phase modulator: (a) $\phi=0$, (b)
$\phi=\pi/2$ and (c) $\phi=\pi$. The following parameters are chosen: the CW
laser, the KTP (potassium titanyl phosphate) platform, the pump wavelength
$\lambda_{p}=766nm$, the crystal length $L=8mm$, the distance between the two
poling sections $l=10mm$, the period of poling $\Lambda=126\mu m$.

In this scheme, a pump laser interacts with the first crystal creating a pair
of signal-idler photons. Along the propagation path, the phase of one or more
of the fields can be modulated via an active element. Henceforth, we consider
only a phase modulation of the e-polarized idler field, which can be achieved
e.g. via electrooptic modulation, that changes the refractive index profile
$n(\omega_{i})$. For this reason, in the modulator region we will consider a
new idler wavevector $k^{\prime}_{i}$, such that $\delta
k_{e}=k_{e}-k^{\prime}_{e}$ is the variation of the idler wavevector induced
by the phase modulator. Afterwards, the three fields interact in the second
PDC source and finally the generated photons are detected.

Figure 2: A schematic model of the collinear integrated SU(1,1)
interferometer. An incoming pump laser interacts with a periodic poled PDC
section, generating a signal-idler photon pairs. In the propagation path,
idler photon undergoes a phase modulation (PM). Afterwards, both the generated
photons and the pump beam interact in the second PDC section. Finally, the
signal photon is detected. Different colours for signal and idler photons are
used in order to distinguish vertical and horizontal polarizations.

The spatial profile $g(z)$ of the second-order nonlinear susceptibility of the
sample under consideration can be written as:

$g(z)=\begin{cases}square_{\Lambda}(z)&0<z<L\vee L+l<z<2L+l\\\
1&L<z<l+L\end{cases},$ (7)

where $square_{\Lambda}(z)$ is the periodic square function that oscillates
between -1 and +1 with periodicity $\Lambda$ and describes periodic poling.
Taking into account the first term of the Fourier expansion of Eq.(7) in the
poling region, the JSA of the sample, as calculated by equations (2), (4), is
given by:

$\begin{split}F(\omega_{s},\omega_{i})=\frac{C}{2L}\alpha(\omega_{s},\omega_{i})\bigg{(}\int_{0}^{L}dz\,e^{i\frac{2\pi
z}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta
k(\xi)}+\int_{L}^{L+l}dz\,e^{i\int_{0}^{z}d\xi\Delta
k(\xi)}+\int_{L+l}^{2L+l}dz\,e^{i\frac{2\pi
z}{\Lambda}}e^{i\int_{0}^{z}d\xi\Delta k(\xi)}\bigg{)}\\\
=C\alpha(\omega_{s},\omega_{i})\operatorname{sinc}\left[\frac{\Delta\beta
L}{2}\right]\times\cos\left(\frac{2\Delta\beta L+\Delta\beta
l+\Delta\beta^{\prime}l}{4}\right)e^{i(\Delta\beta L+\Delta\beta
l/4+\Delta\beta^{\prime}l/4)},\end{split}$ (8)

where
$\Delta\beta=k_{o}(\omega_{p})-k_{o}(\omega_{s})-k_{e}(\omega_{i})+2\pi/\Lambda$
is the phase mismatch, compensated by the poling period $\Lambda$,
$\Delta\beta^{\prime}=k_{o}(\omega_{p})-k_{o}(\omega_{s})-k^{\prime}_{e}(\omega_{i})+2\pi/\Lambda$
is the phase mismatch in the modulator region, while indices "o" and "e"
denote the "ordinary" and "extraordinary" polarization, respectively. The
first integral in Eq.(8) contains the phase matching function describing the
first PDC section, whereas the third integral describes both the propagation
of all beams inside the interferometer and the phase matching in the second
PDC section. Note, that terms corresponding to the linear propagation in the
waveguide are also characterized by the mismatch $\Delta\beta$. This is a
consequence of the fact that, in our consideration, the poling profile of the
second PDC section is in phase with the poling profile of the first PDC
section. This does not apply to two independent gratings. The second integral
corresponds to the generation of PDC light between the two poling regions, the
contribution of this integral is neglected, since the absence of the grid
makes the integrand a fast oscillating function resulting in a negligible
magnitude of the integral.

The variation of $n(\omega_{i})$ can be experimentally realized by applying an
external voltage. In theory, since the phase solely affects the idler photon,
we can write

$\frac{\Delta\beta^{\prime}l}{2}=\frac{\Delta\beta
l}{2}+\frac{k_{e}(\omega_{i})l-k_{e}^{\prime}(\omega_{i})l}{2}=\frac{\Delta\beta
l}{2}+\frac{\delta k_{e}(\omega_{i})l}{2}.$ (9)

The average phase imparted by the phase modulator to the idler field can be
then defined as $\phi=\delta k_{e}(\omega_{p}/2)l/2$, and Eq.(9) can be
rewritten as:

$\frac{\Delta\beta^{\prime}l}{2}=\frac{\Delta\beta l}{2}+\frac{\delta
k_{e}(\omega_{i})\phi}{\delta k_{e}(\omega_{p}/2)}.$ (10)

Although the mathematical model presented so far does not depend on the
waveguide material explicitly, here onwards we will focus on an integrated
SU(1,1) interferometer based on the KTP platform. Fig.1 shows how the applied
phase shift drastically modifies the shape of the joint spectral intensity
(JSI) $|F(\omega_{s},\omega_{i})|^{2}$ and leads to the modulation of both
signal and idler spectra. It is important to notice that, in such
configuration, for any values of phase $\phi$ a non-vanishing mean number of
photons exists: due to different polarizations and, as a consequence,
different refractive index profiles (and group velocities) of signal and idler
photons, there is no $\phi$ values that can lead to perfect destructive
interference in the JSA. Since destructive interference is required for the
observation of supersensitivity [1], the proposed design based on the
conventional bulk SU(1,1) architecture is inadequate for photon sensing
applications in integrated optics.

Nevertheless, this problem can be overcome as shown in the next section.

###  3.2  The new integrated architecture

To obtain almost ideal destructive interference at the output of an integrated
SU (1,1) interferometer, the compensation for the time delay between signal
and idler photons, arising due to the difference in their group velocities, is
strictly required. The integrated SU(1,1) interferometer which takes into
account such a compensation is depicted in Fig. 3. This interferometer has the
same layout as the one previously presented, with the addition of a
polarisation converter (PC) at the centre of the device. The PC switches
polarizations of the signal and the idler photons in order to compensate for
the arising time delay.

The JSA characterizing such interferometer can be obtained by following the
same procedure as before, but taking into account the change in the phase
matching after PC:

$\begin{split}F(\omega_{s},\omega_{i})=\\\
=\frac{C}{2}\alpha(\omega_{s},\omega_{i})\bigg{[}\operatorname{sinc}\bigg{[}\frac{\Delta\beta
L}{2}\bigg{]}e^{i\Delta\beta L/2}+\operatorname{sinc}\bigg{[}\frac{\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5muL}{2}\bigg{]}\\\ \times\exp\bigg{\\{}i\bigg{(}\frac{\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5muL}{2}+\Delta\beta L+\frac{(\Delta\beta^{\prime}+\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5mu)l}{2}\bigg{)}\bigg{\\}}\bigg{]},\end{split}$ (11)

where $\mkern 1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5mu=k_{o}(\omega_{p})-k_{e}(\omega_{s})-k_{o}(\omega_{i})+2\pi/\Lambda$ is
the phase matching after the polarization switch. Note that the polarization
of the signal and idler fields in $\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern 1.5mu$ is different
compared to $\Delta\beta$.

Figure 3: A schematic model of the integrated SU(1,1) interferometer with the
time delay compensation. An incoming pump laser interacts with a periodic
poled PDC section, where signal-idler photons pairs are generated. The idler
photon undergoes a phase modulation (PM). Afterwards, a polarization converter
(PC) located in the middle of the device switches polarizations of signal and
idler photons. Finally, all beams interact in the second PDC section. In the
end, the signal photon is detected. Different colours are used in order to
distinguish vertical (red) and horizontal (blue) polarizations. Figure 4:  The
KTP platform. (a) The pump spectral bandwidth $\alpha(\omega_{s},\omega_{i})$,
(b) the phase matching function $\lvert f(\omega_{s},\omega_{i})\rvert$ and
(c) the joint spectral intensity $\lvert F(\omega_{s},\omega_{i})\rvert^{2}$
in the pulsed regime when $\phi\simeq\pi$. The pulse duration is
$\tau=0.35ps$, $L=8mm$, $\lambda_{p}=766nm$, $l=10mm$, $\Lambda=126\mu m$. By
increasing the pulse duration, $\alpha(\omega_{s},\omega_{i})$ gets narrower
with respect to the $\omega_{s}=\omega_{p}-\omega_{i}$ diagonal, this leads to
zero overlap between $\alpha(\omega_{s},\omega_{i})$ and
$f(\omega_{s},\omega_{i})$ and purely destructive interference.

To avoid additional compensation for the time delay between signal/idler and
pump photons and, at the same time, to reduce the uncompensated second-order
effects, we use a CW laser. Indeed, the stretching of the pump pulse is
inversely proportional to the square of the pulse duration. Therefore, in the
CW pump case, this stretching can be neglected. The high degree of correlation
between the signal and the idler photons determined by the employment of the
CW laser allows us to assume $\omega_{i}=\omega_{p}-\omega_{s}$. Substituting
this expression into the Taylor expansion of $\Delta\beta$ in the proximity of
$\omega_{p}/2$ and considering terms up to the first order inclusive, we have:

$\begin{split}\Delta\beta\approx\underbrace{k_{o}(\omega_{p})-k_{o}(\omega_{p}/2)-k_{e}(\omega_{p}/2)+\frac{2\pi}{\Lambda}}_{=0}\\\
-\frac{\partial
k_{o}}{\partial\omega}\mid_{\frac{\omega_{p}}{2}}(\omega_{s}-\omega_{p}/2)-\frac{\partial
k_{e}}{\partial\omega}\mid_{\frac{\omega_{p}}{2}}(\omega_{p}/2-\omega_{s}).\end{split}$
(12)

Since the first line in Eq.(12) vanishes due to the definition of periodic
poling, the phase matching is simply:

$\Delta\beta\approx-\frac{\Omega}{v_{o}(\omega_{p}/2)}+\frac{\Omega}{v_{e}(\omega_{p}/2)},$
(13)

where the frequency detuning $\Omega=\omega_{s}-\omega_{p}/2$ and the group
velocity $v=(\partial k/\partial\omega)^{-1}$ are introduced. By applying the
same strategy to $\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern 1.5mu$ we finally
obtain:

$\mkern 1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5mu\approx-\frac{\Omega}{v_{e}(\omega_{p}/2)}+\frac{\Omega}{v_{o}(\omega_{p}/2)},$
(14)

which means that $\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5mu\simeq-\Delta\beta$.

By substituting this last equality in Eq.(11), we get:

$\begin{split}F(\omega_{s},\omega_{i})\simeq
C\,\delta(\omega_{p}-\omega_{s}-\omega_{i})\operatorname{sinc}\left[\frac{\Delta\beta
L}{2}\right]\cos\left[\frac{\delta k_{e}(\omega_{i})\phi}{2\delta
k_{e}(\omega_{p}/2)}\right]\exp\left\\{\frac{i}{2}\left(\Delta\beta
L+\frac{\delta k_{e}(\omega_{i})\phi}{2\delta
k_{e}(\omega_{p}/2)}\right)\right\\},\end{split}$ (15)

where $\delta(\omega_{p}-\omega_{s}-\omega_{i})$ is the Dirac delta function
describing the CW pump. Numerically, the CW regime is achieved by setting the
pulse duration $\tau$ in Eq.(11) so long that the pump can cover entirely the
whole length of the interferometer; this condition is mathematically expressed
as $c\tau\gg 2L+l$.

Since the relation $\frac{\delta k_{e}(\omega_{i})}{\delta
k_{e}(\omega_{p}/2)}\simeq 1$ in the vicinity of $\omega_{p}/2$, it can be
clearly seen that Eq.(15) results in the JSA of a single PDC process modulated
by $\cos(\phi/2)$. Due to this modulation it is possible to achieve almost
perfect destructive interference in a large bandwidth around $\omega_{p}/2$ in
the JSA using specific values of $\phi$ (which is in stark contrast to the
non-compensated configuration, whose JSA profiles are presented in Fig.1).

This regime, where the JSA is drastically attenuated, can be achieved with
relative simplicity thanks to the spectral feature of our device based on the
KTP (potassium titanyl phosphate) platform. In this framework, the natural
orientation of the phase matching function $f(\omega_{s},\omega_{i})$ in KTP,
whose slope is typically positive in the ’$\omega_{s}-\omega_{i}$’ diagram,
together with the presence of a PC make sure that the phase matching function
of the SU(1,1) interferometer lies along the $\omega_{s}=\omega_{i}$ diagonal
(Fig. 4b); moreover, the additional phase induces a modulation leading to a
destructive interference at $\phi=\pi$, in which the phase matching function
vanishes along the antidiagonal $\omega_{p}=\omega_{s}+\omega_{i}$.

Although the presence of the phase modulator and the PC changes the structure
of the phase matching function and inhibits the output light intensity, the
pulsed regime with a broad pump spectrum $\alpha(\omega_{s},\omega_{i})$, see
Fig.4a, leads to the generation of a notable number of photons and a non-
vanishing JSA with two high peaks, as demonstrated in Fig.4c. However, such
residual light can be further inhibited by decreasing the pump spectral
bandwidth (using a CW laser), in order to dramatically reduce the overlapping
region between $\alpha(\omega_{s},\omega_{i})$ and $f(\omega_{s},\omega_{i})$
and minimize the number of generated photons. In this context, the use of the
CW pump laser is an important and necessary requirement in order to achieve a
notable reduction of the mean number of photons.

The interference profile, namely the integral number of photons depending on
the phase, is presented in Fig.5 for two different types of pump lasers: the
pulsed and the CW laser. A comparison of the two curves was performed by
making use of different pump powers in order to achieve a similar number of
output photons. One can clearly observe that the use of the CW laser
drastically increases the visibility of the modulation and allows the photon
number to drop close to zero.

Figure 5: The comparison between the number of photons in the case of pulsed
laser (the pulse duration $\tau=0.35ps$, red line) and CW regime. In order to
have a situation with the resulting number of photons are nearly identical at
$\phi=0$, different pump intensities are used for the two pumping regimes. The
following parameters are chosen: $L=8mm$, $\lambda_{p}=766nm$, $l=10mm$,
$\Lambda=126\mu m$.

The behaviour of the phase sensitivity, defined in Eq.(5) and normalized to
the shot noise limit Eq.(6), is presented in Fig.6 for different gains, where
we introduce the gain parameter $\gamma=G(0)\sqrt{\lambda_{1}(0)}$, see
Appendix. In this framework, the variation of $\gamma$ is determined by
changing the coupling constant $\Gamma$. This allows to span a range of mean
photon number between $\langle N(\phi=0)\rangle\approx 0.12$ for $\gamma\simeq
0.04$, and $\langle N(\phi=0)\rangle\approx 10^{9}$ for $\gamma\simeq 10.0$.

Figure 6: The phase sensitivity normalized to SNL versus phase at different
gains. The SNL is shown by the black line. The following parameters are chosen
and fixed for all further calculations: the CW laser, $L=8mm$,
$\lambda_{p}=766nm$, $l=10mm$, $\Lambda=126\mu m$. Figure 7: The minimum
values of the normalized phase sensitivity presented in Fig.6 versus gain
$\gamma$. The higher the gain, the faster the phase sensitivity grows. The SNL
is plotted in black.

The central peak at $\phi=\pi$ in Fig.6 is a feature of the multimode
structure of the considered interferometer and stems from the non-perfect
interference process: due to the fact that the relation $\mkern
1.5mu\overline{\mkern-1.5mu\Delta\beta\mkern-1.5mu}\mkern
1.5mu\simeq-\Delta\beta$ holds only for the first order of the Taylor
expansion around the central frequency $\omega_{p}/2$, and not strictly for
large frequency detuning $\lvert\Omega\rvert>0$. As a result, the mean number
of photons and its variance are not entirely zero for this value of $\phi$,
whereas the derivative $\partial\langle N\rangle/\partial\phi$ is exactly
zero. This leads to a degradation in phase sensitivity with increasing gain
and reveals one of the deepest differences between phase sensitivity analysis
for single-mode plane-wave interferometers, typically presented in the
literature [12, 18, 19, 20], and the multimode scenario considered here, where
the broad spectral distribution of the generated light is taken into account.
Although the proposed scheme improves the visibility of interference pattern
significantly, a perfect interference can never occur because of the complex
material dispersion of the integrated device.

Figure 8: The normalized JSI at $\phi=0$: (a) without filter, (b) with the
filter (bandwidth 5.71$\cdot 10^{12}$ rad/s).

Fig.7 shows the minimum values of the normalized phase sensitivity for each
gain. It is clearly seen that the phase sensitivity gets worse with enhancing
gain or pump power. Explicit calculations of the mean photon number and its
variance can be found in the Appendix, where it is shown that the main trend
of the normalized phase sensitivity with increasing gain is roughly defined
as:

$\frac{\Delta\phi}{\Delta\phi_{SNL}}\approx\frac{\sinh[\gamma/2]}{\gamma}.$
(16)

This trend can be observed in Fig.7 and reflects a progressive worsening of
$\Delta\phi$ as the gain increases. However, the phase sensitivity can be
improved by using a filtering of the output radiation or a seed, both
strategies will be investigated below.

##  4 Filtering

In this section, we demonstrate how to improve the phase sensitivity of the SU
(1,1) interferometer by filtering a specific frequency range in the output
radiation.

Since filters select specific spectral bandwidths, working in the Schmidt
basis is no longer meaningful as the Schmidt operators include the integration
over all frequencies. Hence, it is more convenient to calculate the relevant
quantities by using the output annihilation and creation operators expressed
in the plane wave basis, introduced in [29] and directly presented in
Appendix, see Eq.(28). Using the output plane wave operators Eq.(28), the
number of signal photons in a certain spectral range $2\delta$ around the
central frequency $\omega_{p}/2$ is given by:

$\begin{split}\langle
N_{s}(\phi)\rangle=\int_{\omega_{p}/2-\delta}^{\omega_{p}/2+\delta}d\omega_{s}\langle
a_{s}^{\dagger
out}(\omega_{s})a_{s}^{out}(\omega_{s})\rangle=\sum_{k}\sinh^{2}\gamma_{k}\int_{\omega_{p}/2-\delta}^{\omega_{p}/2+\delta}d\omega_{s}\lvert
u_{k}(\omega_{s},\phi)\rvert^{2},\end{split}$ (17)

where $\delta$ determines the filter bandwidth and
$\gamma_{k}=G(\phi)\sqrt{\lambda_{k}(\phi)}$, see Appendix. Similarly, we can
calculate the variance of the number of photons:

$\begin{split}\langle\Delta^{2}N_{s}(\phi)\rangle=\langle
N_{s}(\phi)\rangle+\sum_{kk^{\prime}}\left\lvert\int_{\omega_{p}/2-\delta}^{\omega_{p}/2+\delta}d\omega_{s}u_{k}^{*}(\omega_{s},\phi)u_{k^{\prime}}(\omega_{s},\phi)\sinh\gamma_{k}\sinh\gamma_{k^{\prime}}\right\rvert^{2},\end{split}$
(18)

where the first term corresponds to Eq.(17). Using both the variance in
Eq.(18) and the derivative of Eq.(17), the phase sensitivity in a filtering
case can be calculated according to Eq.(5).

It is important to specify that considered filters should also be taken into
account when calculating the SNL via Eq.(6). In this way, the number of
photons in the SNL corresponds to the number of signal photons generated by a
single PDC section and subject to the band-pass filter, these photons then
enter the phase modulator:

$\begin{split}\langle
N_{in}\rangle=\sum_{k}\sinh^{2}\left[G_{1}\sqrt{\eta_{k}}\right]\int_{\omega_{p}/2-\delta}^{\omega_{p}/2+\delta}d\omega_{s}\lvert\bar{u}_{k}(\omega_{s})\rvert^{2},\end{split}$
(19)

where $\bar{u}_{k}(\omega_{s})$ is the orthonormal set of signal Schmidt modes
corresponding to a single periodic poled waveguide, both $G_{1}$ and
$\eta_{k}$ are defined in the Appendix.

The filter we used selects a range with bandwidth $2\delta=5.71\cdot
10^{12}rad/s$ around the central frequency, see Fig.(8). This filter cuts the
main part of the JSA with the central peak, excluding the second order peaks
of the sinc function in Eq.(15), and allows us to investigate the influence of
the side-lobes of the JSA on the phase sensitivity.

Figure 9: The normalized phase sensitivity with and without a filter for
different gains: (a) $\gamma=1.3$, (b) $\gamma=2.5$ and (c) $\gamma=5.0$. (d)
The zoom of the supersensitivity region for $\gamma=2.5$ . The solid red line
shows the case without a filter, the dashed blue line presents the filter
case.

The normalized phase sensitivity in the filtered case at different gains is
presented in Fig.9. It can be seen that in the region far from $\phi=\pi$, the
filter provides similar results to the no-filter case. However, around
$\phi=\pi$, filtering gives a significant improvement in the phase sensitivity
both in depth and in the expansion of the supersensitivity region, highlighted
in Fig.9d. This improvement indicates the negative influence of the side
lobes. Indeed, removing the side lobes does not dramatically change the final
number of photons with respect to the case without a filter, nevertheless, it
reduces the influence of the residual photons stemming from the non-perfect
compensation in JSA for large frequency detuning $\lvert\Omega\rvert>0$ at
$\phi\simeq\pi$.

Figure 10: The minimum values of the normalized phase sensitivity versus gain
$\gamma$ with and without the use of the filter. The SNL is represented by a
solid black line.

The best phase sensitivity achieved with and without the filter at varying
gains is shown in Fig.10. This picture demonstrates that the action of the
filter guarantees better phase sensitivity with respect to the no-filter case,
but at the same time confirms the tendency of worsening the phase sensitivity
with gain as in the case without a filter.

##  5 Seeded SU(1,1)

The description of the SU(1,1) interferometer in terms of broadband spectral
modes reveals the possibility to prepare the seed either at the selected
frequency or at the selected spectral mode of the interferometer (or their
superposition). In this section, we analyze the seeding of the interferometer
by a single photon in the first Schmidt mode. Other cases are analysed in the
Appenix B. The seed is performed only for the signal mode of the
interferometer (the vacuum input for the idler mode), which can be done due to
the distinguishability of photons because of their different polarizations.
Since the Schmidt modes of the SU(1,1) interferometer depend on the internal
phase, we suppose to utilise the radiation generated by the second
interferometer to prepare a seed in the selected phase-dependent Schmidt mode.

###  5.1 Direct detection

When seeding, the detection strategy is important for the phase sensitivity.
In this subsection we consider direct detection, as was done in the previous
section without seeding. Therefore, the protocol as well as the set of
operators we need in order to estimate the phase sensitivity in Eq.(5) are the
same as before, but the initial states are different.

####  5.1.1 Single photon in the first Schmidt mode

Having a single photon in the first Schmidt mode, the input state of the
interferometer is given by:

$|\psi\rangle_{in}=|1\rangle_{A_{1}}|0\rangle_{A_{n\neq 1}}|0\rangle_{B_{n}}.$
(20)

This state shows that only the first Schmidt mode of the signal photon is
seeded, while the vacuum stands for other input signal Schmidt modes and all
idler modes. Direct calculations show that the output number of signal photons
in this configuration is:

$\langle N_{s}\rangle=\sum_{k}\sinh^{2}\gamma_{k}+1+\sinh^{2}\gamma_{1}.$ (21)

The first term in Eq.(21) is the vacuum term obtained without a seed and
already known in Eq.(29). The second and the third terms appear because of the
seed: the unity term describes the single photon, whereas the last term is the
surplus of photons in the signal mode generated by the seed. The variance of
the number of photons is given by:

$\langle\Delta^{2}N_{s}\rangle=\sum_{k}\sinh^{2}\gamma_{k}\cosh^{2}\gamma_{k}+\sinh^{2}\gamma_{1}\cosh^{2}\gamma_{1}.$
(22)

The normalized phase sensitivity as a function of phase for different gains is
plotted in Fig.11. Fig.12 presents the best phase sensitivity for each gain.
Overall, this figure shows a similar behaviour with respect to the case with a
vacuum seed, except for small gains when the number of PDC photons is less
than one: for this region, the phase sensitivity becomes drastically worse.

This is due to the fact that the shot noise limit in the case of the single-
photon-seed

$\begin{split}\Delta\phi_{SNL}=\frac{1}{\sqrt{1+\sinh^{2}[C_{1}\sqrt{\eta_{1}}]+\sum_{k}\sinh^{2}[C_{1}\sqrt{\eta_{k}}]}}\end{split}$
(23)

tends to unity for small gains, or in other words, only the seeded photon
contributes to SNL; while, the non-normalized phase sensitivity is still
proportional to $1/\Gamma$.

Figure 11: The normalized phase sensitivity on the phase for different gains.
The first Schmidt mode of the signal photon is seeded with a single photon.
The SNL is represented by a solid black line.  Figure 12: The phase
sensitivity trend in Fig.11 at the point of minimum with respect to the gain
(red circles) compared to the no-seeding case (black squares). The SNL is
plotted in black. The insert shows the zoom of the phase sensitivity in the
seeded case for small gain values.

For gain values greater than unity, the second and third terms dominate in the
SNL, and the phase sensitivity grows exponentially, similarly to the case
without a seed, see Fig.(12). In other words, the single-photon seed leads to
an imbalance in the signal-idler photon correlations for low gains, which
destroys the phase sensitivity in this region. The analysis of other single-
mode seeding cases and detection stratagies shows that such inefficiency of a
single-mode seed is an intrinsic property of a multimode interferometer that
fundamentally distinguishes it from a single-mode system.

##  6 Conclusion

In this work, we presented a detailed theoretical description of a fully
integrated multimode SU(1,1) interferometer. We demonstrated that material
dispersion complicates the interferometer setup and special integrated designs
are required. The presented interferometer includes a polarization converter
located between the two photon sources, which allows us to compensate for the
time delay arising between signal and idler photons. The use of the CW laser
allows us to avoid additional time compensation for the pump photons and, at
the same time, to reduce the influence of uncompensated second-order effects,
that results in the supersensitivity regions below the shot noise limit. The
spectral properties of such interferometer were investigated and appropriate
conditions for the best phase sensitivity below the shot noise limit were
found and analysed. Interestingly, the presented configuration leads to a
situation where the multimode system realized with using a CW laser
outperforms the single-Schmidt-mode system realized with a pulsed laser.

In addition, we investigated the features of the phase sensitivity with an
increase in the gain of the process and show the degradation of the phase
sensitivity with increasing gain due to the multimode structure of generated
light.

Finally, we discuss the impact of filtering and seeding on the phase
sensitivity. We show that using filtering one can remove the uncompensated
radiation and improve the phase sensitivity both in depth and in the expansion
of the supersensitivity regions. However, the investigated seeds in the signal
mode do not improve the phase sensitivity and even completely destroy the
supersensitivity regions in the case of a bright single-mode seed. The reason
is the destruction of signal-idler photon correlations, which are strongly
required for the observation of supersensitivity in the highly multimode
regime.

This work stresses the fundamental difference between the single-plane-wave-
mode and strongly multimode interferometers in the behaviour of the phase
sensitivity. Taking into account the spectral features of photon sources is an
important point for more pragmatic descriptions of this type of interferometer
and future experimental implementations and investigations. Moreover, this
work demonstrates that the introduction of spectral modes in the phase
sensitivity analysis could open new frontiers in quantum metrology: it was
shown how an accurate spectral engineering in terms of mode scaling and
frequency filtering determines deep changes in final results; furthermore, a
large spectrum of seeding strategies can be explored. Finally, the impact of
internal losses on the performance of the multimode integrated nonlinear
interferometer is an important open question that will be a scope for future
research.

##  7 ACKNOWLEDGMENTS

We acknowledge the financial support of the Deutsche Forschungsgemeinschaft
(DFG) via TRR 142/2, project C02. P. R. Sh. thanks the state of North Rhine-
Westphalia for support by the Landesprogramm für geschlechtergerechte
Hochschulen. We also thank the $\mathrm{PC}^{2}$ (Paderborn Center for
Parallel Computing) for providing computing time.

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##  Appendix A Schmidt modes calculation

The PDC process is a well-known source of broadband squeezed light. In order
to make the squeezing and broadband behaviour mathematically explicit, we
firstly need to introduce the Schmidt operators, namely a special set of
annihilation and creation operators equipped with a spectral distribution. For
doing this, we should use the Schmidt decomposition of the JSA presented in
Eq.(2),

$F(\omega_{s},\omega_{i},\phi)=\sum_{k}\sqrt{\lambda_{k}(\phi)}u_{k}(\omega_{s},\phi)v_{k}(\omega_{i},\phi),$
(24)

where we made explicit the parametric dependence of the JSA on the phase
$\phi$ experienced by the idler photon. In this framework, $\lambda_{k}(\phi)$
are the Schmidt eigenvalues and $u_{k}(\omega_{s},\phi)$ and
$v_{k}(\omega_{i},\phi)$ are an orthonormal set of spectral Schmidt modes for
signal and idler photons, respectively. At this point it is convenient to
introduce the input Schmidt operators as following:

$\begin{split}(A_{k}^{\dagger})^{in}=\int
d\omega_{s}u_{k}(\omega_{s},\phi)a^{\dagger}(\omega_{s}),\\\
(B_{k}^{\dagger})^{in}=\int
d\omega_{i}v_{k}(\omega_{i},\phi)a^{\dagger}(\omega_{i}),\end{split}$ (25)

which describe the signal and idler photons characterized by the spectra
$\lvert u_{k}(\omega_{s},\phi)\rvert^{2}$ and $\lvert
v_{k}(\omega_{i},\phi)\rvert^{2}$, respectively. These operators fulfill the
typical bosonic commutation rules:

$[A_{k},A^{\dagger}_{j}]=\delta_{kj},\,[A_{k},B^{\dagger}_{j}]=0.$ (26)

Following the strategy detailed in [29], we can define the Hamiltonian in
terms of Schmidt operators and solve the Heisenberg equations in order to find
both the output Schmidt operators:

$\begin{split}A_{k}^{out}=A_{k}^{in}\cosh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]+\\\
(B_{k}^{\dagger})^{in}\sinh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right],\\\
B_{k}^{out}=B_{k}^{in}\cosh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]+\\\
(A_{k}^{\dagger})^{in}\sinh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right],\end{split}$
(27)

and the output plane wave operators:

$\begin{split}a^{out}(\omega_{s})=a^{in}(\omega_{s})+\sum_{k}u_{k}(\omega_{s},\phi)\\\
+\bigg{[}A_{k}^{in}\left(\cosh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]-1\right)\\\
+(B_{k}^{\dagger})^{in}\sinh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]\bigg{]},\\\
\\\ b^{out}(\omega_{i})=b^{in}(\omega_{i})+\sum_{k}v_{k}(\omega_{i},\phi)\\\
+\bigg{[}B_{k}^{in}\left(\cosh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]-1\right)\\\
+(A_{k}^{\dagger})^{in}\sinh\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]\bigg{]},\end{split}$
(28)

where $G(\phi)=\int C(\phi)\Gamma dt$, $C(\phi)$ is the normalization constant
depending on the phase, see Eq. 2, the coupling constant $\Gamma$ is
proportional to the pump power.

The output operators can be used to calculate different quantities, for
instance, the output integral mean number of signal photons is given by:

$\begin{split}\langle
N_{s}(\phi)\rangle=\sum_{k}\sinh^{2}\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right].\end{split}$
(29)

Since we only consider the signal photon observables, the index "s" will be
omitted below along the text.

Similarly, the variance of the integral number of signal photons is given by:

$\langle\Delta^{2}N\rangle=\frac{1}{4}\sum_{k}\sinh^{2}\left[2G(\phi)\sqrt{\lambda_{k}(\phi)}\right].$
(30)

The derivative of the mean number of photons $\langle N\rangle$ respect to the
phase gives

$\begin{split}\frac{d\langle
N\rangle}{d\phi}=\sum_{k}\sinh\left[2G(\phi)\sqrt{\lambda_{k}(\phi)}\right]\\\
\times\left(\frac{dG(\phi)}{d\phi}\sqrt{\lambda_{k}(\phi)}+\frac{G(\phi)}{2\sqrt{\lambda_{k}(\phi)}}\frac{d\lambda_{k}(\phi)}{d\phi}\right).\end{split}$
(31)

In all cases analysed in this work, the JSA never vanishes totally due to the
multimode structure of the light. This means that both $G(\phi)$ and the
eigenvalues $\lambda_{k}(\phi)$ cannot reach zero, as a result, both the mean
photon number and its variance are not fully suppressed. Therefore, by the
definition of phase sensitivity in Eq.5, the non vanishing variance causes the
presence of a peak in the phase sensitivity plot.

It is fundamental to assert that the plotted phase sensitivity is normalized
with respect to the SNL, which depends on the number of signal photons inside
the interferometer:

$\langle N_{in}\rangle=\sum_{k}\sinh^{2}\left[G_{1}\sqrt{\eta_{k}}\right],$
(32)

where $\eta_{k}$ are the Schmidt eigenvalues corresponding to the single
periodically poled waveguide. $G_{1}=\int C_{1}(\Gamma/2)dt$, where it is
taken into account that the coupling constant of a single PDC section is twice
smaller compared to the double-PDC-section configuration; $C_{1}$ is the
normalization constant corresponding to the JSA of a single PDC section.

In order to give an analytical justification of the trend of the normalized
phase sensitivity in Fig. 7, it is more practical to split our argumentation
in two parts, distinguishing a low gain regime, in which
$G(0)\sqrt{\lambda_{1}(0)}\ll 1$, from a high gain regime, in which
$G(0)\sqrt{\lambda_{1}(0)}\gg 1$.

In the former case, an estimation of the photon number in Eq.(29) and in
Eq.(32) gives

$\langle N(\phi)\rangle\simeq
G^{2}(\phi)\sum_{k}\lambda_{k}(\phi)=G^{2}(\phi),$ (33)

and

$\langle N_{in}\rangle\simeq G_{1}^{2}\sum_{k}\eta_{k}=G_{1}^{2},$ (34)

respectively, whereas using the Taylor expansion of the variance Eq.(30) we
obtain

$\langle\Delta^{2}N\rangle\simeq
G^{2}(\phi)\sum_{k}\lambda_{k}(\phi)=G^{2}(\phi).$ (35)

Since the SU(1,1) interferometer, which we are dealing with, has the JSA at
$\phi=0$ almost identical to the JSA of a single waveguide, we can assume that
$C_{1}\approx C(0)$ and $\lambda_{k}(0)\approx\eta_{k}$. Therefore, by
calculating the derivative of the photon number in Eq.(33) with respect to the
phase, it turns out that the normalized phase sensitivity is described by:

$\frac{\langle\Delta\phi\rangle}{\langle\Delta\phi_{SNL}\rangle}\approx\left\lvert\frac{G(0)}{4}\left(\frac{\partial
G(\phi)}{\partial\phi}\right)^{-1}\right\rvert\approx\frac{1}{2\sin(\phi/2)},$
(36)

where the last step stems from the assumption that in the low intensity
regime, according to the form of JSA Eq.(15), the gain is a modulated function
of the form $G(\phi)=G(0)\lvert\cos(\phi/2)\rvert$ (this expression holds at
$0<\phi<\pi$). It is interesting to notice that the normalized phase
sensitivity in the low gain regime does not depend on the gain anymore, which
can be seen in Fig. 7.

In the high gain regime, due to the redistribution of the eigenvalues [29],
the contribution of the first mode to the signal and variance will be
dominated, and we can temporarily underestimate the number of photons of both
equations (29) and (32) by taking into account the contribution stemming from
the first mode, obtaining:

$\langle
N_{s}(\phi)\rangle\approx\sinh^{2}\left[G(\phi)\sqrt{\lambda_{1}(\phi)}\right],$
(37)

and

$\langle N_{in}\rangle\approx\sinh^{2}\left[G_{1}\sqrt{\eta_{1}}\right],$ (38)

respectively. We underestimate the variance consequently:

$\langle\Delta^{2}N\rangle\approx\frac{1}{4}\sinh^{2}\left[2G(\phi)\sqrt{\lambda_{1}(\phi)}\right].$
(39)

By assuming again $C_{1}\approx C(0)$ and $\lambda_{k}(0)\approx\eta_{k}$, the
gain of the process can also be expressed as
$\gamma=G(0)\sqrt{\lambda_{1}(0)}\simeq 2G_{1}\sqrt{\eta_{1}}$. Moreover, the
shape of the JSA Eq.(15) modulated by $\cos(\phi/2)$ allows us to write
$G(\phi)\sqrt{\lambda_{1}(\phi)}\approx\gamma\lvert\cos(\phi/2)\rvert$.
Finally, by collecting all assumptions written above, we can estimate the
normalized phase sensitivity:

$\frac{\Delta\phi}{\Delta\phi_{SNL}}\approx\frac{\sinh[\gamma/2]}{\gamma\sin(\phi/2)}.$
(40)

If the angle $\phi$ is close to $\pi$ (the point of almost perfect destructive
interference), the eigenvalues distribution is very close to the flat
distribution. In this case, the underestimation with only the first Schmidt
mode is not correct, but one can use the overestimation, suggesting that all
Schmidt modes have the same weights. It means that the integral number of
photons is equivalent to

$\langle N(\phi)\rangle\approx
K\sinh^{2}\left[G(\phi)\sqrt{\lambda_{1}(\phi)}\right],$ (41)

where we assume that the all $K$ Schmidt modes have the same contribution,
$K=1/(\sum_{k}\Lambda_{k}^{2})$ is the effective number of modes (the Schmidt
number) and

$\Lambda_{k}=\frac{\sinh^{2}\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]}{\sum_{k}\sinh^{2}\left[G(\phi)\sqrt{\lambda_{k}(\phi)}\right]}$
(42)

is a set of Schmidt coefficients taking into account the redistribution of the
weights in the high gain regime [32]. The variance would change consequently:

$\langle\Delta^{2}N\rangle=\frac{K}{4}\sinh^{2}\left[2G(\phi)\sqrt{\lambda_{1}(\phi)}\right].$
(43)

By modifying the mean photon number in the SNL similarly to Eq.(41), along
with all assumptions made above, the overestimated normalized phase
sensitivity would be identical to Eq.(40), which in turn, with $\phi=\pi$
equals to Eq.(16).

##  Appendix B Further seeding configurations

In this appendix, we analyse different types of seeding as well as different
detection strategies to find the optimal conditions for the best phase
sensitivity. In all cases presented below, the seed consists of a coherent
state, moreover, seeding is performed only for the signal mode of the
interferometer (the vacuum input for the idler mode), which can be done due to
the distinguishability of photons because of their polarizations. The seed of
coherent states in specific modes can be carried out experimentally by
injecting a laser whose spectral features are shaped to one of the specific
modes of the interferometer. Furthermore, since the Schmidt modes of the
interferometer depend on the internal phase, we can suppose to tune the
spectral profile of the laser accordingly.

###  B.1 Direct detection: coherent state in the first Schmidt mode

In this configuration, we utilise an intense coherent state $|\alpha\rangle$
in the first Schmidt mode of the signal photon, the input state in this case
is given by:

$|\psi\rangle_{in}=|\alpha\rangle_{A_{1}}|0\rangle_{A_{n\neq
1}}|0\rangle_{B_{n}}.$ (44)

To simplify calculations, we consider a coherent state with a real $\alpha$
and high number of photons $\lvert\alpha\rvert^{2}=10^{6}$. Such seeding
changes the SNL accordingly:

$\begin{split}\Delta\phi_{SNL}=\\\
\frac{1}{\sqrt{\alpha^{2}\cosh^{2}[C_{1}\sqrt{\eta_{1}}]+\sum_{k}\sinh^{2}[C_{1}\sqrt{\eta_{k}}]}}.\end{split}$
(45)

Analogous to the previous case, direct calculations give the following
expressions for the mean photon number:

$\langle
N_{s}\rangle=\sum_{k}\sinh^{2}\gamma_{k}+|\alpha|^{2}+|\alpha|^{2}\sinh^{2}\gamma_{1},$
(46)

and for the photon number variance:

$\begin{split}\langle\Delta^{2}N_{s}\rangle=\sum_{k}\sinh^{2}\gamma_{k}\cosh^{2}\gamma_{k}\\\
+|\alpha|^{2}\cosh^{2}\gamma_{1}\cosh 2\gamma_{1}.\end{split}$ (47)

However, as it is shown in Fig.13, for any values of the gain this
configuration does not present supersensitivity regions. Analytically, this
can be demonstrated by checking the normalized phase sensitivity in the
asymptotic limit when $\alpha^{2}\gg\sum_{k}\sinh^{2}\gamma_{k}$, namely

$\frac{\Delta^{2}\phi}{\Delta^{2}\phi_{SNL}}\approx(1+\coth^{2}\gamma_{1})\cosh^{2}\left[G_{1}\sqrt{\eta_{1}}\right]\left\lvert\frac{\partial\gamma_{1}}{\partial\phi}\right\rvert^{-2}.$
(48)

Due to the similarity between the JSA in Eq.(15) at $\phi=0$ and the JSA of a
single waveguide, we can assume $\lambda_{1}(0)\simeq\eta_{1}$ and $G(0)\simeq
2G_{1}$; hence the gain $\gamma$ can be expressed as $\gamma\simeq
2G_{1}\sqrt{\eta_{1}}$. Furthermore, we can also assume
$\gamma_{1}\approx\gamma\lvert\cos(\phi/2)\rvert$, therefore in the range
$\\{0-\pi\\}$ we have

$\frac{\Delta^{2}\phi}{\Delta^{2}\phi_{SNL}}\approx\frac{4\left(1+\coth^{2}[\gamma\cos(\phi/2)]\right)\cosh^{2}[\gamma/2]}{\gamma^{2}\sin^{2}(\phi/2)}.$
(49)

By plotting this curve, or alternatively by calculating the minimum point with
respect to $\phi$, it is clear that this function cannot even reach the shot
noise limit.

Figure 13: The normalized phase sensitivity on the phase for different gains.
The first Schmidt mode of the signal photon is seeded with an intense coherent
state having $\lvert\alpha\rvert^{2}=10^{6}$ photons. The SNL is represented
by a solid black line.

###  B.2 Homodyne detection

In this subsection we discuss seeded interferometers with the homodyne
detection strategy. In the case of homodyne detection, we must properly modify
the definition of the phase sensitivity using the homodyne operator
$\hat{H}_{d}$ (not the photon number operator as before) [19] to calculate
observables. According to this modification, the phase sensitivity is defined
as:

$\Delta\phi=\bigg{\lvert}\frac{\langle\Delta\hat{H}_{d}\rangle}{\partial\langle\hat{H}_{d}\rangle/\partial\phi}\bigg{\rvert}.$
(50)

The homodyne operator depends on the local oscillator properties, which is why
it is necessary to ensure that the output light of the SU(1,1) interferometer
and the local oscillator are spectrally matched. To ensure this, both the
output radiation and the local oscillator are spectrally filtered. The
spectral characteristic of the filters will be modelled depending on the
different type of seeding used. In particular, in case 1 below, we will
consider a spectral filter selecting only the first Schmidt mode, while in
case 2 we will consider a passband filter around the central frequency of
generated squeezed light. The SNL calculation takes these modifications into
account as well: the mean number of photons in the SNL corresponds to the
number of signal photons generated by a single periodically poled waveguide,
averaged with respect to the input seeding state, and then filtered to match
the spectral properties of the output squeezed light.

####  B.2.1 Coherent state in the first Schmidt mode

The first case we consider is an intense coherent seed in the first Schmidt
mode with the same initial state as in Eq.(44). At the end of the
interferometer we filter the output radiation of the signal photon, selecting
the first Schmidt mode. A local oscillator is fixed in the same frequency
range, corresponding to the first Schmidt mode. In this case, the homodyne
operator is defined as:

$\hat{H}_{d}=\lvert\beta_{lo}\rvert\left(e^{i\theta_{a}}\hat{A}_{1}^{out}+e^{-i\theta_{a}}\left[\hat{A}_{1}^{out}\right]^{\dagger}\right),$
(51)

where $\lvert\beta_{lo}\rvert$ and $\theta_{a}$ are the amplitude and the
phase of the local oscillator (described by the coherent state
$|\lvert\beta_{lo}\rvert e^{i\theta_{a}}\rangle_{A_{1}}$). Averaging the
homodyne operator with respect to the input state we obtain:

$\langle\hat{H}_{d}\rangle=2\alpha\lvert\beta_{lo}\rvert\cos\theta_{a}\cosh\gamma_{1}.$
(52)

The variance of the homodyne operator is given by:

$\langle\Delta^{2}\hat{H}_{d}\rangle=\lvert\beta_{lo}\rvert^{2}\cosh{2\gamma_{1}},$
(53)

whereas the expression for the shot noise limit is:

$\Delta\phi_{SNL}=\frac{1}{\sqrt{\alpha^{2}\cosh^{2}[C_{1}\sqrt{\eta_{1}}]+\sinh^{2}[C_{1}\sqrt{\eta_{1}}]}}.$
(54)

The optimum phase sensitivity is reached when $\theta_{a}=0$. According to our
analysis, this seeding case does not present any supersensitivity behaviour,
and the phase sensitivity curves, very similar to Fig.13 and therefore not
shown, lies entirely above the SNL. This result, obtained through the homodyne
detection, together with the previous outcomes achieved via direct detection,
suggests that a strong coherent seed in the spectral mode does not provide any
improvement in the supersensitivity and therefore other seeding strategies are
recommended.

####  B.2.2 Coherent state in the plane-wave mode

In this section we consider an intense coherent seed in the plane-wave mode
$\omega_{p}/2$. The output radiation is filtered in the same frequency. The
input state in this case is given by:

$|\psi\rangle_{in}=\int
d\omega_{s}\delta(\omega_{s}-\omega_{p}/2)\rvert\alpha\rangle_{\omega_{s}}|0\rangle_{\omega_{i}},$
(55)

where we again assume a seeding state containing one million photons. In order
to prevent any time dependence in the output, the local oscillator is also
defined in a plane-wave mode at the frequency $\omega_{p}/2$. The homodyne
operator therefore takes the following expression:

$\begin{split}\hat{H}_{d}=\\\
\lvert\beta_{lo}\rvert\left(e^{i\theta_{a}}\hat{a}^{out}\left(\frac{\omega_{p}}{2}\right)+e^{-i\theta_{a}}\left[\hat{a}^{out}\left(\frac{\omega_{p}}{2}\right)\right]^{\dagger}\right),\end{split}$
(56)

where the output plane wave operators $\hat{a}^{out}$ and
$\left[\hat{a}^{out}\right]^{\dagger}$ are defined in the Appendix.

The average value of the homodyne operator gives:

$\begin{split}\langle\hat{H}_{d}\rangle=2\alpha\lvert\beta_{lo}\rvert\cos\theta_{a}\\\
\times\left(1+\sum_{k}\left\lvert
u_{k}\left(\frac{\omega_{p}}{2},\phi\right)\right\rvert^{2}(\cosh\gamma_{k}-1)\right),\end{split}$
(57)

while the variance of the homodyne operator takes the form:

$\begin{split}\langle\Delta^{2}\hat{H}_{d}\rangle=\\\
\lvert\beta_{lo}\rvert^{2}\left(1+2\sum_{k}\left\lvert
u_{k}\left(\frac{\omega_{p}}{2},\phi\right)\right\rvert^{2}\sinh^{2}\gamma_{k}\right),\end{split}$
(58)

where the integrals in the frequency domain have already been performed.

Figure 14: The normalized phase sensitivity in relation to the phase. The
plane wave mode with the frequency $\omega_{p}/2$ is seeded by the intense
coherent light having $\lvert\alpha\rvert^{2}=10^{6}$ photons. The SNL is
represented by a solid black line. Figure 15: The minimum values of the
normalized phase sensitivity presented in Fig. 14 versus gain (red circles)
compared to the no-seeding case (black squares). The SNL is plotted in black.

The calculation of the shot noise limit is carried out via Eq.(6), taking into
account the number of photons inside the interferometer after the frequency
integration:

$\begin{split}N_{in}=\alpha^{2}+\sum_{k}\left\lvert\bar{u}_{k}\left(\frac{\omega_{p}}{2}\right)\right\rvert^{2}\sinh^{2}[G_{1}\sqrt{\eta_{k}}]\\\
+\alpha^{2}\left[\sum_{k}\left\lvert\bar{u}_{k}\left(\frac{\omega_{p}}{2}\right)\right\rvert^{2}(\cosh[G_{1}\sqrt{\eta_{k}}]-1)\right]^{2}\\\
+2\alpha^{2}\sum_{k}\left\lvert\bar{u}_{k}\left(\frac{\omega_{p}}{2}\right)\right\rvert^{2}(\cosh[G_{1}\sqrt{\eta_{k}}]-1).\end{split}$
(59)

According to these expressions, the phase sensitivity was calculated,
normalized with respect to the SNL, optimized for $\theta_{a}=0$, and finally
plotted in Fig.14. The minimum values of the phase sensitivity for different
gains are presented in Fig.15. This figure demonstrates that the phase
sensitivity with this seeding choice also does not have any supersensitivity
regions. The reason is that a strong seeding in the signal mode alone creates
an imbalance between the number of signal and idler photons and destroys the
signal-idler photon correlations required for the observation of the
supersensitivity in the highly multimode regime.