# Hate Speech Detection on Vietnamese Social Media Text using the
Bidirectional-LSTM Model

Hang Thi-Thuy Do, Huy Duc Huynh, Kiet Van Nguyen, Ngan Luu-Thuy Nguyen and Anh
Gia-Tuan Nguyen
University of Information Technology, VNU-HCM
{16520339<EMAIL_ADDRESS>{kietnv, ngannlt<EMAIL_ADDRESS>
###### Abstract

In this paper, we describe our system which participates in the shared task of
Hate Speech Detection on Social Networks of VLSP 2019 evaluation campaign. We
are provided with the pre-labeled dataset and an unlabeled dataset for social
media comments or posts. Our mission is to pre-process and build machine
learning models to classify comments/posts. In this report, we use
Bidirectional Long Short-Term Memory to build the model that can predict
labels for social media text according to Clean, Offensive, Hate. With this
system, we achieve comparative results with 71.43% on the public standard test
set of VLSP 2019.

###### Index Terms:

Bi-LSTM, Hate Speech Detection, Vietnamese, Social Media Text

##  I Introduction

In recent years, social networking has grown and become prevalent with every
people, it makes easy for people to interact and share with each other.
However, every problem has two sides. It also has some negative issues, hate
speech is a hot topic in the domain of social media. With the freedom of
speech on social networks and anonymity on the internet, some people are free
to comment on hate and insults. Hate speech can have an adverse effect on
human behavior as well as directly affect society. We don’t manually delete
each of those comments, which is time-consuming and boring. This spurs
research to build an automated system that detects hate speech and eliminates
them. With that system, we can detect and eliminate hate speech and thus
reduce their spread on social media. With Vietnamese, we can use methods to
apply specific extraction techniques manually and in combination with string
labeling algorithms such as Conditional Random Field (CRF)[1], Model Hidden
Markov (HMM)[2] or Entropy[3]. However, we have to choose the features
manually to bring the model with high accuracy. Deep Neural Network
architectures can handle the weaknesses of the above methods. In this report
we apply Bidirectional Long Short-Term Memory (Bi-LSTM) to build the model.
Also combined with the word embedding matrix to increase the accuracy of the
model.

The rest of the paper is organized as follows. In section 2, we presented the
related work. In section 3, we described our Bi-LSTM system. In sections 4 and
5, we presented the experimental process and results. Finally, section 6 gives
conclusions about the work.

##  II Related Work

Gao and Huang (2017)[4] used BiLSTMs with attention mechanism 372 to detect
hate speech. They illustrated that the Bi-directional LSTM model with
attention mechanism achieves the high performance. They hypothesize that this
is because hate indicator phrases are often concentrated in a small region of
a comment, which is especially the case for long comments. Davidson et al.
(2017)[5] train a model to differentiate among three classes: containing hate
speech, only offensive language, or neither.Jing Qian, Mai ElSherief,
Elizabeth Belding, William Yang Wang (2018) [6] worked on classifying a tweet
as racist, sexist or neither by multiple deep learning architectures. ABARUAH
at SemEval-2019 [7] presented the results obtained using bi-directional long
short-term memory (BiLSTM) with and without attention and Logistic Regression
(LR) models for multilingual detection of hate speech against immigrants and
women in Twitter. Animesh Koratana and Kevin Hu [8] use many machine learning
models to detect toxic words, in which the Bi-Lstm model got the highest
performance. Malmasi and Zampieri (2017)[9] made a similar study to compare
the performance of different features in detecting hate speech.

##  III Bi-LSTM model for Vietnamese Hate Speech Detection

As mentioned previously, we propose a framework based on the ensemble of Bi-
LSTM models to perform hate speech detection with the provided dataset.
Besides, we also implemented some more models to compare and find the optimal
model for the task.

###  III-A Long Short-Term Memory

LSTM takes words from an input sentence in a distributed word representation
format. LSTM’s network architecture includes memory cells and ports that allow
the storage or retrieval of information. These gates help the LSTM memory cell
to perform a write, read and reset operation. They enable the LSTM memory cell
to store and access information over a period of time.

###  III-B Bidirectional Long Short-Term Memory

One drawback of LSTM architecture[10] is that they are only considering the
previous context. However, the identification of a word depends not only on
the previous context but also on the subsequent context. Bidirectional LSTM
(Bi-LSTM)[11] was created to overcome this weakness. A Bi-LSTM architecture
usually contains two single LSTM networks used simultaneously and
independently to model input chains in two directions: forward LSTM and
backward LSTM.

Figure 1: Bi-LSTM architecture [12]

##  IV Pre-Processing

The pre-processing performed on the text includes the following:

  * •

The comments were converted to lowercase.

  * •

The URLs, mentions (@) and non-alphabetic characters are removed (number,
excess whitespace).

  * •

Several stopwords were removed from the comments. We don’t remove all stopword
because having a few stopwords affect the results.

  * •

A few characters that don’t affect the results are replaced by an empty
string.

  * •

Using Tokenizer to convert each comment into a sequence of integers.

##  V Experiments

###  V-A Dataset and Word Embeddings

VLSP Shared Task 2019: Hate Speech Detection on Social Networks: This dataset
includes 25431 items in csv format, the dataset was divided into two file,
training dataset with 20345 items and test dataset with 5086 items. Each data
line of training dataset is assigned 1 of 3 labels CLEAN, OFFENSIVE or HATE.
The test dataset is not assign label. The statistic summarization of the given
training dataset is described in Table I.

  * •

Hate speech (HATE) contains the abusive language, which often bears the
purpose of insulting individuals or groups, and can include hate speech,
derogatory and offensive language. An item is identified as hate speech if it
(1) targets individuals or groups on the basis of their characteristics; (2)
demonstrates a clear intention to incite harm, or to promote hatred; (3) may
or may not use offensive or profane words.

  * •

Offensive but not hate speech (OFFENSIVE) is an item (posts/comments) may
contain offensive words but it does not target individuals or groups on the
basis of their characteristics.

  * •

Neither offensive nor hate speech (CLEAN) is a normal item. It’s
conversations, expressing emotions normally. It does not contain offensive
language or hate speech.

In this paper, we use two different word embeddings to compare and find out
the best word embedding such as Word2Vec [17] and FastText [16]. We used pre-
trained vector with large dimensions to increase the accuracy of the model.
Through experiments we found FastText achieved better results.

TABLE I: THE STATISTIC OF VLSP 2019 HSDoSN TRAINING DATASET | CLEAN | OFFENSIVE | HATE | TOTAL
---|---|---|---|---
Frequency | 18614 | 1022 | 709 | 20345
Percentage | 91.49% | 5.02% | 3.49% | 100%

For this public dataset, we find that the dataset is an unbalanced dataset.
The CLEAN label has the highest rate with 91.49% and the HATE label is lowest
with 3.49%. Therefore, it is difficult and challenging to find a good model
for this task.

###  V-B Evaluation on each Model

For problems of this type, there are many models suitable to handle such as:
SVM, Bi-LTSM, LR, GRU, CNN and etc. To solve this problem, we implement four
different models (SVM, LR, Bi-LSTM, and GRU) to compare and find the most
suitable one. To evaluate the four models on this task, we divide the training
dataset into two parts training, testing rate of 80%, 20% respectively.

The details of our models are provided below.

1\. Support Vector Machine (SVM)

Support Vector Machines (SVMs) are a popular machine learning method for
classification, regression, and other learning tasks [13]. It is often used
for two-class classification problems. For this problem, it has three labels,
so we use the SVM to classify twice, two label at a time.Firstly, we classify
two label 0 and 1, we achieved accuracy, precision, recall, and F1-score rates
of 96.00%, 93.37%, 98.96%, and 96.08% respectively, on training dataset.
Second time, we classify two label 1 and 2, we achieved accuracy, precision,
recall, and F1-score rates of 84.34%, 87.38%, 78.86%, and 82.90% respectively.
We find that this model doesn’t classify well for two labels 1 and 2.
Moreover, when we check this model with the public-test, it brings the result
as not good as we expected with 63.87%.

2\. Logistic Regression (LR)

Logistic regression is basically a supervised classification algorithm. In a
classification problem, the target variable(or output), can take only discrete
values for a given set of features(or inputs) [14]. We have applied it to this
problem as follows: Firstly, we use the TfidfVectorizer tool to convert text
into feature vectors that are used as input for the model. Then, we used the
Logistic Regression model to predict the classification results. When checking
it on training datasets, we achieved accuracy, precision, recall, and F1-score
rates of 94.17%, 88.87%, 55.54%, and 64.15% respectively. We also try
submitting this model’s result on the system, the result is worse we thought
with 51.15%

3\. Gated Recurrent Units (GRU)

The Recurrent Neural Network (RNN) handles the variable-length sequence by
having a recurrent hidden state whose activation at each time is dependent on
that of the previous time [15]. The GRU is a variant of RNN and it only has
two inputs. We have used it into this problem as follows: We have used it with
word embeddings Fasttext [16]. First, we use Tokenizer() for sequences because
GRU is good at processing long sequences. Then, we have applied this model to
the problem. We achieved accuracy, precision, recall, and F1-score rates of
94.61%, 67.12%, 59.66%, and 64.15% respectively, on the training dataset. When
we check this model with the public dataset, it brings the result quite good
with 65.01%

4\. Bidirectional Long Short-Term Memory (Bi-LSTM)

The LSTM is a famous variant of RNN. The Bidirectional Long Short Term Memory
can be trained using all available input information in the past and future of
a specific time frame. We have applied it with word embeddings Fasttext [16]
and baomoi.vn.model.txt [17]. As follows:

As well as the GRU, we also used GloVe Embedding for sequences and applied
this model for the problem. When using word embeddings baomoi.vn.model.txt
[17], we achieved accuracy, precision, recall, and F1-score rates of 93.26%,
90.74%, 50.30%, and 53.62% respectively, on the training dataset. The same
with word embeddings Fasttext [16], we achieved accuracy, precision, recall,
and F1-score rates of 95.67%, 85.61%, 67.36%, and 73.84% respectively, on the
training dataset. We find that when combining the Bi-LSTM with fasttext will
bring the result better. When we check it with the public dataset, it brings
the result good with 71.43%

###  V-C Experimental Results

After conducting experiments on many models, we obtained the following results
on public-test, shown in Table II.

TABLE II: The results table of models. Model | F1-Score
---|---
SVM | 63.87
LR | 51.15
GRU | 65.01
Bi-LSTM | 71.43

We achieved the best result with Bi-LSTM, ranking the 2nd of the scoreboard on
the public-test set shown in Table III. However, our result ranks the 6th of
the scoreboard on the private-test set.

TABLE III: The results table of the top 5 on public-test set Rank | Team | F1-score
---|---|---
1 | Try hard | 73.01
2 | HH_UIT | 71.43
3 | titanic | 70.74
4 | ABCD | 70.58
5 | TIN HUYNH | 70.57

##  VI Conclusion and Future Work

In this paper, we have presented our approach to address Vietnamese hate
speech detection task proposed at the VLSP Shared Task 2019. We develop the
system using Bidirectional Long Short Memory for classifying three different
labels in this task. We participate in this and evaluate the performance of
our system on this dataset. As a result, our result is 71.43% of F1-score,
ranking the 2nd of the scoreboard on the public-test set.

In the future work, we plan to address this problem in different ways to
enhance the performance of this task. We will investigate experiments both in
traditional machine learning and types of deep learning for this problem. In
addition, we also analyze experimental results on this task to choose the
efficient approach such as the hybrid approach which combines machine learning
and rule-based approaches to boost the result of detecting hate speech on
Vietnamese social media text.

## Acknowledgment

We would like to thank the VLSP Shared Task 2019 organizers for their really
hard work and providing the dataset of Vietnamese Hate Speech Detection on
social networks for our experiments.

## References

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  * [5] Thomas Davidson, Dana Warmsley, Michael Macy, and Ingmar Weber. 2017. Automated Hate Speech Detection and the Problem of Offensive Language. In Proceedings of the Eleventh International AAAI Conference on Web and Social Media (ICWSM 2017), pages 512–515, Montreal.
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$\displaystyle\left[M_{a},M_{b}\right]=\varepsilon_{ab}{}^{c}M_{c}\,,\quad\quad\left[M_{a},P_{b}\right]=\varepsilon_{ab}{}^{c}P_{c}\,,\quad\quad\left[P_{a},P_{b}\right]=\frac{1}{\ell^{2}}\varepsilon_{ab}{}^{c}M_{c}\,,$
(A.1)

where $M_{a}=\tfrac{1}{2}\varepsilon_{abc}M^{bc}$ are the dual Lorentz
generators, $P_{a}$ the translation generators and $\ell$ denotes the AdS
radius. Indices are raised and lowered with the $(-++)$ Minkowski metric in
$D=3$. To bring out the
${\mathfrak{sl}}(2,\mathbb{R})\oplus{\mathfrak{sl}}(2,\mathbb{R})$ form of
this algebra one defines the standard combinations

$\displaystyle
J_{a}=\frac{1}{2}\left(M_{a}+\ell\,P_{a}\right)\,,\quad\quad\tilde{J}_{a}=\frac{1}{2}\left(M_{a}-\ell\,P_{a}\right)\,,$
(A.2)

such that

$\displaystyle\left[J^{a},J^{b}\right]=\varepsilon_{ab}{}^{c}J_{c}\,,\quad\quad\left[\tilde{J}^{a},\tilde{J}^{b}\right]=\varepsilon_{ab}{}^{c}\tilde{J}_{c}\,,\quad\quad\left[J^{a},\tilde{J}^{b}\right]=0\,,$
(A.3)

making the two commuting ${\mathfrak{sl}}(2,\mathbb{R})$ manifest.

The AdS algebra (A.1) can also be written via a tensor product of
${\mathfrak{sl}}(2,\mathbb{R})$ with an associative abelian algebra
${\mathfrak{a}}_{\ell}$ on two generators $\mathcal{I}$ and $\mathcal{J}$
satisfying

$\displaystyle\mathcal{I}^{2}=\mathcal{I}\,,\quad\quad\mathcal{J}^{2}=\frac{1}{\ell^{2}}\mathcal{I}\,,\quad\quad\mathcal{I}\mathcal{J}=\mathcal{J}\mathcal{I}=\mathcal{J}\,.$
(A.4)

Since ${\mathfrak{a}}_{\ell}$ is abelian, one can turn
${\mathfrak{sl}}(2,\mathbb{R})\otimes{\mathfrak{a}}_{\ell}$ into a Lie algebra
and this Lie algebra is isomorphic to the AdS algebra.212121For $\ell=1$, the
algebra (A.4) is identical to the semi-group $S_{M}^{(1)}$ studied in [41]. As
the final step one defines

$\displaystyle{\mathfrak{A}}_{\ell}={\mathfrak{u}}(1,1)\otimes{\mathfrak{a}}_{\ell}$
(A.5)

as an associative algebra. Sending $\ell\to\infty$ for the flat space limit
and tensoring with the associative algebra ${\mathfrak{u}}(N)$ one obtains the
coloured Poincaré algebra in $D=3$ dimensions shown in (2.13).

##  Appendix B Free massive Poincaré particle

Here, we give a short exposition of the Lagrangian of a free massive particle
built on the standard Poincaré group using non-linear realisations, see e.g.
[57]. For easier comparison with the body of the paper we do everything in
$D=3$ space-time dimensions using the algebra (2.5) and write the algebra
${\mathfrak{so}}(2,1)\cong{\mathfrak{su}}(1,1)$ as $(2\times 2)$-matrices.

#### Massive particle

We begin with the construction from non-linear realisations. For a massive
particle in rest frame, the momentum eigenvalues take the form $p_{a}=(m,0,0)$
with $SO(2)\cong U(1)$ stabiliser in the Lorentz group generated by $M_{0}$.
The local subgroup $H$ of the non-linear realisation is thus $SO(2)$ and we
write group elements in the form

$\displaystyle g=e^{x^{a}P_{a}}b\,h\,,$ (B.1)

with $b=e^{v^{i}M_{i}}$ ($i=1,2$) the boost by the broken generators in the
Lorentz group. We choose the gauge $h=\mathbb{I}$ for the $G/H$ coset
representative.

The Poincaré algebra valued Maurer–Cartan form for this gauge-fixed coset
element is

$\displaystyle\Omega=g^{-1}dg=b^{-1}\left(dx^{a}P_{a}\right)b+b^{-1}db=\Omega_{(P)}^{a}P_{a}+\Omega^{a}_{(M)}M_{a}\,.$
(B.2)

The components of the Maurer–Cartan form are by construction invariant under
the global Poincaré group acting from the left on the group element (B.1).

If one wants to construct a Lagrangian that is also invariant under the right-
action of the unbroken local $SO(2)$ subgroup of the Lorentz group, one can
therefore choose any component of the Maurer–Cartan form that is invariant
under $M_{0}$. In the present case, there are two such candidates:
$\Omega_{(P)}^{0}$ and $\Omega_{(M)}^{0}$. The former is properly invariant
while the latter transforms as a gauge field into a total derivative and thus
is only quasi-invariant. This quasi-invariance is special to $D=3$ dimensions,
in general dimensions there is no such singlet from the Lorentz algebra
decomposed under spatial rotations.222222The form $\Omega_{(M)}^{0}$ is a
Wess–Zumino term in three space-time dimensions [70, 71, 72, 73]. We shall
therefore ignore $\Omega_{(M)}^{0}$ and take the Lagrangian to be given by the
pull-back of the component $\Omega_{(P)}^{0}$ to the world-line of the
particle:232323Note that $\Omega_{(P)}^{0}$ is nothing but the time column of
a Lorentz boost acting on a general momentum.

$\displaystyle L\,dt=m\left[\Omega_{(P)}^{0}\right]^{*}\,.$ (B.3)

We note that the component $\Omega_{(P)}^{0}$ can also be obtained by pairing
the space of translations, generated by the $P_{a}$, with its dual. In other
words, using as the dual basis $\mathcal{P}^{a}$ with pairing

$\displaystyle\langle P_{a},\mathcal{P}^{b}\rangle=\delta_{a}^{b}\,,$ (B.4)

one has

$\displaystyle\Omega_{(P)}^{0}=\langle\Omega_{(P)},\mathcal{P}^{0}\rangle=\langle
b^{-1}\left(dx^{a}P_{a}\right)b,\mathcal{P}^{0}\rangle=\langle\mathrm{Ad}_{b^{-1}}\left(dx^{a}P_{a}\right),\mathcal{P}^{0}\rangle=\langle\left(dx^{a}P_{a}\right),\mathrm{Ad}_{b^{-1}}^{*}\mathcal{P}^{0}\rangle\,,$
(B.5)

where we have used $\mathrm{Ad}_{b^{-1}}$ to denote the ‘adjoint’ action of
the Lorentz group element $b^{-1}$ on its vector representation when viewing
the Poincaré group as a semi-direct product. Similarly,
$\mathrm{Ad}^{*}_{b^{-1}}$ is the co-adjoint action on the dual space.

Writing $x^{a}P_{a}=\mathbb{X}$ we therefore have the equivalent form of the
Lagrangian (B.3)

$\displaystyle L=\langle\mathbb{P},\dot{\mathbb{X}}\rangle\,,$ (B.6)

where $\mathbb{P}$ is any element in the Lorentz group orbit of
$m\mathcal{P}^{0}$. Here, it is important that the orbit of $\mathcal{P}^{0}$
can be written as the group coset $SO(2,1)/SO(2)$ by the orbit stabiliser
theorem. This orbit can be parametrised by some parameters $v^{i}$ that then
appear algebraically in the Lagrangian: the boost parameters. The derivative
in $\dot{\mathbb{X}}$ denotes the derivative with respect to the parameter of
the world-line and so we have explicitly carried out the pull-back.

Other orbits of the action of the Lorentz group on the space of momenta can
also be considered and the form (B.6) is universal in all cases. If the orbit
is of the form $SO(2,1)/H$ for some subgroup $H$ then the Lagrangian
corresponds to the non-linear realisation of the Poincaré group with local
subgroup $H$.

This can also be treated as the $N=1$ case of the matrix analysis in the main
body of the paper. We take as reference momentum

$\displaystyle\widehat{\mathbb{P}}=m\mathcal{P}^{0}=m\begin{pmatrix}\text{i}\,&0\\\
0&-\text{i}\,\end{pmatrix}\,.$ (B.7)

The generalised boost (4.33b) becomes

$\displaystyle b=\exp\left[\frac{1}{2}\begin{pmatrix}0&v\\\
\bar{v}&0\end{pmatrix}\right]=\begin{pmatrix}\cosh\frac{|v|}{2}&v\frac{\sinh\frac{|v|}{2}}{|v|}\\\
\bar{v}\frac{\sinh\frac{|v|}{2}}{|v|}&\cosh\frac{|v|}{2}\end{pmatrix}$ (B.8)

with $v\in\mathbb{C}$. The orbit of the reference momentum is

$\displaystyle
b\,\widehat{\mathbb{P}}\,b^{-1}=m\text{i}\,\begin{pmatrix}\cosh|v|&-v\frac{\sinh|v|}{|v|}\\\
\bar{v}\frac{\sinh|v|}{|v|}&-\cosh|v|\end{pmatrix}=\begin{pmatrix}\text{i}\,\sqrt{|p|^{2}+m^{2}}&p\\\
\bar{p}&-\text{i}\,\sqrt{|p|^{2}+m^{2}}\end{pmatrix}\,,$ (B.9)

where we have introduced $p=-m\text{i}\,\frac{\sinh|v|}{|v|}v$ as a change of
coordinates for the algebraically appearing boost parameters.

The velocity vector is the matrix

$\displaystyle\dot{\mathbb{X}}=\begin{pmatrix}\text{i}\dot{x}^{0}&\dot{z}\\\
\dot{\bar{z}}&-\text{i}\dot{x}^{0}\end{pmatrix}$ (B.10)

where $z=x^{1}+\text{i}x^{2}$. With this the Lagrangian

$\displaystyle L$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left(\dot{\mathbb{X}}\,b\,\widehat{\mathbb{P}}\,b^{-1}\right)=m\left[\dot{x}^{0}\cosh|v|+\mathrm{Im}(\dot{z}\bar{v})\frac{\sinh|v|}{|v|}\right]$
$\displaystyle=-\dot{x}^{0}\sqrt{|p|^{2}+m^{2}}+\mathrm{Re}(\dot{z}\bar{p})\,.$
(B.11)

The equation of motion obtained by varying with respect to the algebraic
variable $p$ is

$\displaystyle\frac{\dot{x}^{0}}{\sqrt{|p|^{2}+m^{2}}}\bar{p}=\dot{\bar{z}}\,.$
(B.12)

Solving this for $p$ and substituting back into the Lagrangian leads to the
expected

$\displaystyle L=-m\sqrt{\left(\dot{x}^{0}\right)^{2}-|\dot{z}|^{2}}\,.$
(B.13)

Going back to the action (B) we can also perform another analysis that is
closer to the main body of the text. For this we vary the Lagrangian (B) with
respect to $x^{0}$. This leads to

$\displaystyle\sqrt{|p|^{2}+m^{2}}=p^{0}\,,$ (B.14)

where $p^{0}$ is a constant of integration while the spatial $p$ is still a
dynamical variable. As $p^{0}$ sits at the right place in the momentum matrix
we deduce that it transforms as the zero component of the contravariant
momentum $p^{a}$ would. Enforcing the square of the above relation with a
Lagrange multiplier $e$ leads to the Lagrangian

$\displaystyle L=\dot{x}^{a}p_{a}+e\left(p^{a}p_{a}+m^{2}\right)\,.$ (B.15)

This Lagrangian has manifest Poincaré invariance. In this Lagrangian, all
components of $p_{a}$ are algebraic variables and we can integrate them out to
obtain

$\displaystyle
L=-\frac{1}{4e}\dot{x}^{a}\dot{x}_{a}+em^{2}=-m\sqrt{-\dot{x}^{a}\dot{x}_{a}}\,,$
(B.16)

where in the second step we have also integrated out $e$ (with a choice of
square root). This Lagrangian agrees with (B.13) above.

#### Massless case

Let us also consider the case of a massless particle in this formulation. The
representative momentum is now given by

$\displaystyle\widehat{\mathbb{P}}=\frac{1}{2}\begin{pmatrix}\text{i}\,E&E\\\
E&-\text{i}\,E\end{pmatrix}\,,\hskip
28.45274pt\mathrm{tr}\left(\widehat{\mathbb{P}}\,\widehat{\mathbb{P}}\right)=0\,,$
(B.17)

and it is stabilised by the one-parameter group generated by
$M_{0}+M_{1}=\begin{pmatrix}\text{i}\,&1\\\ 1&-\text{i}\,\end{pmatrix}$.

The boost can still be chosen as (B.8) and an arbitrary element of the orbit
of the reference momentum is

$\displaystyle
b\,\widehat{\mathbb{P}}\,b^{-1}=\begin{pmatrix}\text{i}\,\omega&e^{\text{i}\,\theta}\omega\\\
e^{-\text{i}\,\theta}\omega&-\text{i}\,\omega\end{pmatrix}$ (B.18)

in a convenient parametrisation. The Lagrangian becomes

$\displaystyle L$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left(\dot{\mathbb{X}}\,b\,\widehat{\mathbb{P}}\,b^{-1}\right)=-\omega\left(\dot{x}^{0}-\mathrm{Re}\left(e^{-i\theta}\dot{\bar{z}}\right)\right)$
(B.19)

The equation of motion when varying $\theta$ is

$\displaystyle e^{2\text{i}\,\theta}=\frac{\dot{z}}{\dot{\bar{z}}}\,,$ (B.20)

leading to

$\displaystyle L$
$\displaystyle=-\omega\left(\dot{x}^{0}-\sqrt{|\dot{z}|^{2}}\right)\,.$ (B.21)

This enforces correctly the constraint $\dot{x}^{a}\eta_{ab}\dot{x}^{b}=0$,
but is not quite the usual form for a massless Poincaré particle.

Let us also go the alternative route for the Lagrangian (B.19), meaning that
we vary with respect to $x^{0}$ first. This forces the energy of the particle
to be constant

$\displaystyle\omega=p^{0}$ (B.22)

for some constant $p^{0}$. We enforce the square of this relation as a
constraint

$\displaystyle
L=p_{0}\dot{x}^{0}+p_{i}\dot{x}^{i}+e(\omega^{2}-p_{0}^{2})=p_{a}\dot{x}^{a}+ep_{a}p^{a}\,,$
(B.23)

where we have used that $\omega e^{\text{i}\,\theta}=p^{1}+\text{i}\,p^{2}$
represents the spatial momentum. Integrating out the covariant $p^{a}$ leads
to expected action

$\displaystyle L=-\frac{1}{4e}\dot{x}^{a}\dot{x}_{a}\,.$ (B.24)

Both Lagrangians (B.21) and (B.24) can be obtained from the original (B.19)
(modulo the fact that for (B.24) a square for anti-particles was introduced).
Let us study their equivalence in general. The equations of motion implied by
(B.21) are

$\displaystyle\dot{x}^{0}-|\dot{\bf x}|$ $\displaystyle=0\,,$ (B.25a)
$\displaystyle\frac{d}{d\tau}\omega$ $\displaystyle=0\,,$ (B.25b)
$\displaystyle\frac{d}{d\tau}\left(\frac{\dot{\bf x}}{|\dot{\bf x}|}\right)$
$\displaystyle=0\,,$ (B.25c)

while from (B.24) we get

$\displaystyle\dot{x}^{a}\dot{x}_{a}$ $\displaystyle=0\,,$ (B.26a)
$\displaystyle\frac{d}{d\tau}\left(\frac{\dot{x}^{a}}{e}\right)$
$\displaystyle=0\,.$ (B.26b)

We first observe that (B.25a) and (B.26a) are equivalent, up to the choice of
square root. Let us set

$\displaystyle\omega=\frac{\dot{x}^{0}}{e}\quad\Leftrightarrow\quad
e=\frac{\dot{x}^{0}}{\omega}\underset{\eqref{eq1}}{=}\frac{|\dot{\bf
x}|}{\omega}$ (B.27)

and this is conserved consistently by (B.25b) and the first component of
(B.26b). The spatial components of (B.26b) are

$\displaystyle\frac{d}{d\tau}\left(\frac{\dot{\bf
x}}{e}\right)=\frac{d}{d\tau}\left(\omega\frac{\dot{\bf x}}{|\dot{\bf
x}|}\right)=\omega\frac{d}{d\tau}\left(\frac{\dot{\bf x}}{|\dot{\bf
x}|}\right)=0$ (B.28)

consistently with (B.25c) when using (B.25b).

##  Appendix C General remarks on particle actions

In this appendix, we collect some general remarks on the different ways of
writing particle actions based on symmetries.

We go back to the action of nonlinear realisation or the coadjoint orbit,

$S=\int dt\,\langle\varphi,g^{-1}\frac{d}{dt}g\rangle$ (C.1)

where $g\in G$ and $\varphi\in\mathfrak{g}^{*}$. The coadjoint action of $a\in
G$ is defined as

$\langle{\rm Ad}^{*}_{a}\,\varphi,T\rangle=\langle\varphi,{\rm
Ad}_{a^{-1}}\,T\rangle\,,\qquad\forall\,T\in\mathfrak{g}\,.$ (C.2)

For a given $\varphi$, let us denote the stabiliser subgroup by
$G_{\varphi}=\\{a\in G\,|\,{\rm Ad}^{*}_{a}\,\varphi=\varphi\\}$ and its Lie
algebra by $\mathfrak{g}_{\varphi}=\\{T\in\mathfrak{g}\,|\,{\rm
Ad}^{*}_{T}\,\varphi=0\\}$ . The dimension of the coadjoint orbit is simply,

${\rm dim}\,{\cal O}_{\varphi}={\rm
dim}\,\mathfrak{g}/\mathfrak{g}_{\varphi}={\rm dim}\,{\mathfrak{g}}-{\rm
dim}\,{\mathfrak{g}_{\varphi}}\,.$ (C.3)

As an example, consider $G$ as the $D$-dimensional Poincaré group
$ISO(D{-}1,1)$ and $\varphi=m\,{\cal P}^{0}$ where ${\cal P}^{0}$ is the dual
basis generator corresponding to $P_{0}$. Then,242424Note that this contrasts
with  $\mathfrak{poin}_{P_{0}}={\rm
Span}\\{M_{ij},P_{0},P_{i}\\}\,,\qquad\mathfrak{poin}/\mathfrak{poin}_{P_{0}}\sim{\rm
Span}\\{M_{0i}\\}\,.$

$\mathfrak{poin}_{m\,{\cal P}^{0}}={\rm
Span}\\{M_{ij},P_{0}\\}\,,\qquad\mathfrak{poin}/\mathfrak{poin}_{m\,P_{0}^{*}}\sim{\rm
Span}\\{M_{0i},P_{i}\\}\,.$ (C.4)

Hence, we find the orbit has $2\times(D-1)$ dimensions, i.e. it is a phase
space for $(D-1)$-dimensional mechanical system so can describe the motion of
a massive particle in $D$-dimensional space-time. If we want to see this more
explicitly, we can consider an element of $g\in G$ such that

$g=a\,b\,,\qquad b\in G_{\varphi}\,,$ (C.5)

then,

$\langle\varphi,g^{-1}dg\rangle=\langle\varphi,a^{-1}\,da\rangle+\langle\varphi,b^{-1}\,db\rangle\,.$
(C.6)

The second term is closed and can only contribute when $G_{\varphi}$ has a
non-trivial topology. At the classical level, we can ignore this term.
Therefore, we see that the action depends only on $G/G_{\varphi}\simeq{\cal
O}_{\varphi}$. In order to get an explicit expression of the Lagrangian for a
given $G$ and $\varphi$, it is sufficient to take an appropriate
parametrisation of $G/G_{\varphi}\simeq{\cal O}_{\varphi}$. For instance, in
the Poincaré case with $\varphi=m\,{\cal P}^{0}$, we can use

$g=\exp(x^{a}\,P_{a})\,\exp(v^{i}\,M_{0i})\,R\,,$ (C.7)

where $R$ is a rotation element in $SO(D{-}1)$, and we can drop it as it
contributes to the action only as a boundary term. We can also remove $e^{\
x^{0}\,P_{0}}$ using

$\exp(v^{i}\,M_{0i})\,e^{\tau\,P_{0}}=\exp\left(\tau\left(\cosh
v\,P_{0}+\tfrac{\sinh v}{v}\,v^{i}\,P_{i}\right)\right)\exp(v^{i}\,M_{0i})\,.$
(C.8)

With a choice of $\tau\,\cosh v=x^{0}$ , we find

$g=\exp\left(\tilde{x}^{i}\,P_{i}\right)\exp(v^{i}\,M_{0i})\,\exp\left(-\tfrac{x^{0}}{\cosh
v}\,P_{0}\right)R\,,$ (C.9)

with

$\tilde{x}^{i}=x^{i}-x^{0}\,\tfrac{\tanh v}{v}\,v^{i}\,.$ (C.10)

Discarding the last two factors in (C.9) since they belong to the stabiliser,
we obtain the action,

$\displaystyle S$ $\displaystyle\\!=\\!$ $\displaystyle m\int dt\,\langle{\cal
P}^{0},\exp(-v^{i}\,M_{0i})\,\dot{\tilde{x}}^{i}\,P_{i}\,\exp(v^{i}\,M_{0i})\rangle$
(C.11) $\displaystyle\\!=\\!$ $\displaystyle-m\int dt\,\frac{\sinh
v}{v}\,v_{i}\,\dot{\tilde{x}}^{i}\,.$

If we redefine $v^{i}$ as

$\tilde{p}_{i}=-m\,\frac{\sinh v}{v}\,v_{i}\,,$ (C.12)

we end up with an action without Hamiltonian,

$S=\int dt\,\tilde{p}_{i}\,\dot{\tilde{x}}^{i}\,.$ (C.13)

If we had not done the reordering of factors in $g$, we would have obtained

$S=m\int dt\left(\cosh v\,\dot{x}^{0}-\frac{\sinh
v}{v}\,v_{i}\,\dot{x}^{i}\right).$ (C.14)

One can also derive the above form of the action by plugging (C.10) into
(C.11) and performing an integration by part. Since the solution of (C.11) is
simply,

$\tilde{x}^{i}=\text{const},\qquad v^{i}=\text{const}\,,$ (C.15)

the relation (C.10) provides the general solution of the ordinary relativistic
massive particle system (C.14).

The fact that we can bring the action (C.14) to the form (C.13) is not
surprising because the starting point of the action was the pullback of the
Liouville one-form on the orbit. Locally, we can always choose Darboux
coordinates, then the action will take the form of (C.13). From this
perspective, the dynamics of the system is inherited from the coordinate
choice for the orbit, or equivalently the group element $g$. In order to make
certain symmetries of an orbit manifest it is useful to choose appropriate
coordinates on the orbit. Moreover, global properties of the orbit have to be
taken into account.

##  Appendix D From the coloured Hamiltonian to the Lagrangian action

In the uncoloured case, we can obtain a quadratic action by integrating out
the momenta $p_{a}$. We can try the same for the coloured particle (4.26)
where the equation for momenta becomes matrix equations

$\dot{\mathbb{X}}+\mathbb{P}\,\mathbb{L}+\mathbb{L}\,\mathbb{P}=0\,.$ (D.1)

In order to solve this equation, we assume the invertibility of $\mathbb{L}$
and deform the equation by introducing $\epsilon$ as

$\mathbb{P}=-\mathbb{L}^{-1}\dot{\mathbb{X}}-\epsilon\,\mathbb{L}^{-1}\,\mathbb{P}\,\mathbb{L}\,.$
(D.2)

Eventually, we will take $\epsilon\to 1$ limit, but in the intermediate step
we will keep $\epsilon$ which plays the role of expansion parameter and
regulator. The solution of the above as a power seires in $\epsilon$ is

$\mathbb{P}=-\sum_{n=0}^{\infty}(-\epsilon)^{n}\,\mathbb{L}^{-n-1}\dot{\mathbb{X}}\,\mathbb{L}^{n}\,.$
(D.3)

The Lagrangian (4.26) becomes

$\displaystyle 2N\,L=$
$\displaystyle-\sum_{n=0}^{\infty}(-\epsilon)^{n}\,\mathrm{Tr}(\mathbb{L}^{-n-1}\dot{\mathbb{X}}\,\mathbb{L}^{n}\,\dot{\mathbb{X}})$
$\displaystyle+\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-\epsilon)^{m+n}\,\mathrm{Tr}(\mathbb{L}^{-n+m}\dot{\mathbb{X}}\,\mathbb{L}^{n-m-1}\,\dot{\mathbb{X}})+m^{2}\,\mathrm{Tr}(\mathbb{L})\,.$
(D.4)

We can perform a resummation for the double sum in the second line as

$\displaystyle\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(-\epsilon)^{m+n}\,\mathrm{Tr}(\mathbb{L}^{-n+m}\dot{\mathbb{X}}\,\mathbb{L}^{n-m-1}\,\dot{\mathbb{X}})=\frac{1}{1+\epsilon}\sum_{n=0}^{\infty}(-\epsilon)^{n}\,\mathrm{Tr}(\mathbb{L}^{-n-1}\dot{\mathbb{X}}\,\mathbb{L}^{n}\,\dot{\mathbb{X}})\,.$
(D.5)

Above, we split the double sum into two parts, $m\geq n$ and $n>m$, and each
parts gives the prefactors $\frac{1}{1-\epsilon^{2}}$ and
$\frac{-\epsilon}{1-\epsilon^{2}}$ respectively, hence it is important to keep
$|\epsilon|<1$ for the convergence of these terms. Note however that the final
result (the sum of two prefactors) is regular in $\epsilon$ . Using the above
solution, we find the quadratic action as

$2N\,L=-\frac{\epsilon}{1+\epsilon}\,\sum_{n=0}^{\infty}(-\epsilon)^{n}\,\mathrm{Tr}(\mathbb{L}^{-n-1}\dot{\mathbb{X}}\,\mathbb{L}^{n}\,\dot{\mathbb{X}})+m^{2}\,\mathrm{Tr}(\mathbb{L})\,,$
(D.6)

As mentioned above, the above series do not exhibit any apparent divergence in
the $\epsilon\to 1$ limit, but we do not know whether the series converges.

For a more rigorous treatment, let us assume that the matrix $\mathbb{L}$ can
be diagonalised as

$\mathbb{L}=\mathbb{S}\,{\rm
diag}(\lambda_{1},\ldots,\lambda_{2N})\,\mathbb{S}^{-1}\,,\qquad|\lambda_{1}|>|\lambda_{2}|>\cdots>|\lambda_{2N}|>0\,.$
(D.7)

Then $\mathbb{P}$ can be solved as

$(\mathbb{S}^{-1}\,\mathbb{P}\,\mathbb{S})_{ij}=-\frac{1}{\lambda_{i}+\lambda_{j}}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ij}\,,$
(D.8)

and the Lagrangian reads

$2N\,L=-\frac{1}{2}\sum_{i,j=1}^{2N}\frac{1}{\lambda_{i}+\lambda_{j}}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ij}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ji}+m^{2}\,\sum_{i=1}^{2N}\lambda_{i}\,.$
(D.9)

Since the factor $(\lambda_{i}+\lambda_{j})^{-1}$ prevents us to re-express
the above as a trace, we can try to expand it so that each term in the
expansion could be written as a trace. However, for the convergence of the
expansion we need to distinguish cases $i>j$, $i<j$ and $i=j$, and we can
obtain at best

$2N\,L=-\sum_{n=0}^{\infty}(-1)^{n}\,\sum_{i>j}\lambda_{i}^{-n-1}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ij}\,\lambda_{j}^{n}(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ji}-\frac{1}{2}\,\sum_{i}\,\lambda_{i}^{-1}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ii}^{2}+m^{2}\,\sum_{i=1}^{2N}\lambda_{i}\,.$
(D.10)

If there were no restriction $i>j$, we could express the first series as

$-\sum_{n=0}^{\infty}(-1)^{n}\,\sum_{i,j}\lambda_{i}^{-n-1}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ij}\,\lambda_{j}^{n}(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ji}=-\sum_{n=0}^{\infty}(-1)^{n}\,\mathrm{Tr}(\mathbb{L}^{-n-1}\dot{\mathbb{X}}\,\mathbb{L}^{n}\,\dot{\mathbb{X}})\,,$
(D.11)

but because of this restriction we cannot. Since the restricted sum
$\sum_{i>j}$ can be understood as “one half” of the unrestricted sum
$\sum_{i,j}$, one may have an intuitive understanding of the expression (D.5).

Now, let us integrate out the Lagrange multiplier. We can do it again in the
diagonalised case. The equation for the eigenvalue $\lambda_{i}$ is

$\sum_{j=1}^{2N}\frac{1}{(\lambda_{i}+\lambda_{j})^{2}}\,A_{ij}+m^{2}=0\,,\qquad
i=1,\ldots,2N\,,$ (D.12)

where we defined

$A_{ij}=A_{ji}=(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ij}\,(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{ji}\,.$
(D.13)

For $N=1$, we find

$\frac{1}{4\,\lambda_{1}^{2}}\,A_{11}+\frac{1}{(\lambda_{1}+\lambda_{2})^{2}}\,A_{12}=-m^{2}\,,\qquad\frac{1}{4\,\lambda_{2}^{2}}\,A_{22}+\frac{1}{(\lambda_{1}+\lambda_{2})^{2}}\,A_{12}=-m^{2}\,.$
(D.14)

The equations can be re-expressed as a quartic one so they have four different
solutions. Since the Lagrangian is proportional to the trace
$\mathrm{Tr}(\mathbb{L})=\lambda_{1}+\lambda_{2}$, we can focus on it and find

$\displaystyle-4\,m^{2}(\lambda_{1}+\lambda_{2})^{2}$
$\displaystyle=4\,A_{12}+(\sqrt{A_{11}}\pm\sqrt{A_{22}})^{2}$
$\displaystyle=2(A_{11}+2\,A_{12}+A_{22})-(\sqrt{A_{11}}\mp\sqrt{A_{22}})^{2}$
$\displaystyle=2\,\mathrm{Tr}(\dot{\mathbb{X}}^{2})-\left((\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{11}\mp(\mathbb{S}^{-1}\,\dot{\mathbb{X}}\,\mathbb{S})_{22}\right)^{2}\,.$
(D.15)

When the relative sign in the last line is postive then it can be also
expressed as a trace (which is zero in this case), and we find

$-4\,m^{2}(\lambda_{1}+\lambda_{2})^{2}=2\,\mathrm{Tr}(\dot{\mathbb{X}}^{2})-(\mathrm{Tr}(\dot{\mathbb{X}}))^{2}=2\,\mathrm{Tr}(\dot{\mathbb{X}}^{2})\,.$
(D.16)

For $N=2$, we find a set of equations equivalent to a single order-eight one,
and hence would contain eight solutions.

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# Rotation of the noise ellipse for squeezed vacuum light generated via four-
wave-mixing

Neil V. Corzo  Joint Quantum Institute, National Institute of Standards and
Technology and the University of Maryland, Gaithersburg, Maryland 20899, USA.
Center for Photonic Communication and Computing, EECS Department, Northwestern
University, 2145 Sheridan Road, Evanston, Illinois 60208-3118, USA.  Quentin
Glorieux<EMAIL_ADDRESS>Joint Quantum Institute, National
Institute of Standards and Technology and the University of Maryland,
Gaithersburg, Maryland 20899, USA.  Group of Applied Physics, University of
Geneva, Chemin de Pinchat 22, CH-1211 Geneva, Switzerland  Alberto M. Marino
Homer L. Dodge Department of Physics and Astronomy, The University of
Oklahoma, 440 W. Brooks St., Norman, Oklahoma 73019, USA.  Jeremy B. Clark
Joint Quantum Institute, National Institute of Standards and Technology and
the University of Maryland, Gaithersburg, Maryland 20899, USA.  Paul D. Lett
<EMAIL_ADDRESS>Joint Quantum Institute, National Institute of Standards
and Technology and the University of Maryland, Gaithersburg, Maryland 20899,
USA.

###### Abstract

We report the generation of a squeezed vacuum state of light whose noise
ellipse rotates as a function of the detection frequency. The squeezed state
is generated via a four-wave mixing process in a vapor of 85Rb. We observe
that rotation varies with experimental parameters such as pump power and laser
detunings. We use a theoretical model based on the Heisenberg-Langevin
formalism to describe this effect. Our model can be used to investigate the
parameter space and to tailor the ellipse rotation in order to obtain an
optimum squeezing angle, for example, for coupling to an interferometer whose
optimal noise quadrature varies with frequency.

###### pacs:

42.50.Dv, 42.50.Lc, 42.50.Nn

##  I Introduction

The generation of non-classical states of light has been a topic of interest
since the first experimental demonstration of squeezed light by Slusher et al.
using four-wave mixing (4WM) in sodium vapor Slusher  _et al._ (1985). More
recently interest in 4WM techniques has seen a resurgence due to the potential
for the generation of very high multi-spatial modes Boyer  _et al._ (2008)
quantum correlations (two-mode twin beams) in the so-called phase-insensitive
configuration McCormick  _et al._ (2007); Glorieux  _et al._ (2010a); Liu  _et
al._ (2011); Qin  _et al._ (2012); Glorieux  _et al._ (2012). 4WM in the
phase-sensitive configuration can also produce single-mode vacuum squeezed
states Corzo  _et al._ (2011) that might be used for improving sensitivity in
precision interferometry.

Gravitational wave detection using optical interferometers with long arms is a
promising but highly challenging technique Schnabel  _et al._ (2010). Since a
pioneering experiment with modest sensitivity Forward (1978), the performance
of interferometers has improved dramatically Abadie et al (2011); Vahlbruch
_et al._ (2010, 2008). In these experiments, gravitational waves would be
detected as a phase or path difference between the two arms of the
interferometer and the sensitivity is ultimately limited by the quantum noise.
The standard quantum limit (SQL) has been introduced for this class of
interferometers Braginsky and Khalili (1996), and it is equivalent to the
detection limit for the arm length difference:

$\Delta_{SNL}=\sqrt{\frac{\hbar\tau}{m}},$ (1)

where $\tau$ is the measurement time and $m$ is the mass of the mirrors. In
1980, Caves emphasized that the SQL in interferometers is a property of the
interferometer itself and it is a consequence of the vacuum noise entering
from the unused port Caves (1980). Subsequently, he proposed to inject
squeezed light into the unused port to improve the sensitivity of these
interferometers Caves , which has been experimentally demonstrated Grangier
_et al._ (1987); Xiao  _et al._ (1987) and is currently used on a daily basis
in the LIGO project Abadie et al (2011). This technique, however, cannot
overcome the SQL at all frequencies because the reduction in photon noise at
low frequencies is overwhelmed by an increase in the random motion of the
mirrors known as radiation pressure fluctuations. In an extension of Caves
idea, Unruh demonstrated that the SQL can be violated across a large band of
frequencies if the appropriate quadrature component of the mode entering the
unused port is squeezed Meystre and Scully (1983). The key feature of the
Unruh proposal is that the squeezed vacuum angle (equivalent to the direction
of the noise ellipse) be frequency dependent. Since the dominant source of
noise at low frequencies is the radiation pressure noise, and at high
frequencies is the photon shot noise, this determines that the choice of the
noise ellipse direction where the squeezing is present should rotate between
the phase quadrature at low frequencies and the amplitude quadrature at high
frequencies. Several papers have discussed this proposal Jaekel and Reynaud
(1990); Pace  _et al._ (1993) and experimental demonstrations have been
achieved Kimble  _et al._ (2001). Different techniques have been proposed to
implement a frequency dependent noise ellipse Khalili  _et al._ (2009);
Corbitt  _et al._ (2004); Mikhailov  _et al._ (2006) Most of these techniques
are based on pre-filtering cavities as initially proposed in Kimble  _et al._
(2001), and they can all introduce losses and mode-matching problems. More
recently, Horrom et al. have demonstrated an example of angle rotation of the
quantum noise quadratures using the frequency-dependent absorption of the EIT
windowHorrom  _et al._ (2013).

In this paper we discuss a different technique that allows for the rotation of
the noise ellipse of vacuum squeezed light using four-wave mixing (4WM) in a
hot atomic vapor. In this technique, we use a double-$\Lambda$ system Glorieux
_et al._ (2011) in a phase-sensitive configuration to generate squeezed vacuum
light, described in detail in Corzo  _et al._ (2011). We demonstrate up to
$-4.0$ dB of squeezing bellow the shot noise limit (SNL) and we show that the
squeezing spectrum can be modified to maximize squeezing at various analysis
frequencies. We propose an explanation for this effect, based on the fact that
two frequency sidebands (positive and negative sidebands) contribute
differently to the measured noise, and we verify the consistency of this
description with our experimental data. We report an experimental observation
of the noise ellipse rotation over a frequency bandwidth of several MHz. We
suggest that the rotation of the noise ellipse in this system is due to a
frequency dependent phase shift induced by the non-linear susceptibility of
the atomic medium. We adapt the theoretical model of Glorieux  _et al._
(2010b) to our system in order to provide a quantitative estimation of the the
phase shift. For the experimental parameters used in this paper we observe a
noise ellipse rotation of $\pi/5$ over 10 MHz frequency bandwidth. This
combination of experimental results and theoretical model might allow one to
tailor the noise ellipse rotation to follow the optimal squeezing angle
specific to a given interferometer.

##  II Experimental setup

Figure 1:  (Color online.) a) Ball-and-stick representation for bright beams
and vacuum states. b) Energy levels of the D1 line in 85Rb and the optical
frequencies arranged in the double-$\Lambda$ system. c) Experimental setup to
generate squeezed light and measure the squeezing level. AOM: acousto-optic
modulator, TA: tapered amplifier, SA: radio-frequency spectrum analyzer, BS:
non-polarizing beamsplitter, PBS: Polarizing beamsplitter, BB: beam block. d)
Typical vacuum squeezing noise measurement at 1 MHz analysis frequency
(red/gray) compared to the shot noise limit (black).

A common representation of the fluctuations of a quantum state as a phase
space diagram is known as the ”ball-and-stick” picture Bachor and C Ralph
(2004). In this representation, the stick corresponds to the mean value of the
field and the ball corresponds to the area of the fluctuations around the mean
value. For a vacuum state, the mean value is zero and therefore the stick
disappears and the ball is centered around the origin (see Fig. 1 a). The
standard quantum limit in this representation is described by a circle with a
diameter normalized to 1. A squeezed vacuum state is therefore an ellipse with
one direction for which the axis is smaller than 1, referred to as the noise
ellipse direction (see Fig. 1(a)). In an ideal case, the area of the
fluctuation ellipse is preserved, otherwise the area is increased. In the
following, we study the compression factor of this ellipse (referred to as the
squeezing value) and the angle of this ellipse generated by the four-wave
mixing process in 85Rb.

The experimental setup is similar to the one presented in Corzo  _et al._
(2011) and is shown in Fig. 1(c). A Ti:sapphire laser at frequency $\nu_{p}$
is tuned approximately $0.8$ GHz to the blue of the 85Rb
5S${}_{1/2},F=3\rightarrow$ 5P1/2 transition (red of the 85Rb
5S${}_{1/2},F=2\rightarrow$ 5P1/2 transition), as shown in Fig. 1(b). The one-
photon detuning is referred to as $\Delta$. Two pump frequencies are generated
by frequency-shifting the light of the Ti:sapphire laser using two double-
passed 1.52 GHz acousto-optic modulators (AOMs). One pump beam (frequency
$\nu_{1}$) is downshifted and the other upshifted (frequency $\nu_{2}$) by the
same amount, around $3$ GHz, that is related to the ground state hyperfine
splitting $\nu_{HF}=3036$ MHZ in 85Rb. The beams out of the AOMs (0.5 mW each)
are used to seed two tapered amplifiers and generate two strong ($\sim$500 mW
each) pump beams. The pump beams are spatially-filtered with polarization-
maintaining fibers and we obtained maximum powers of 230 mW and 190 mW for the
pump beams at frequencies $\nu_{1}$ and $\nu_{2}$, respectively. The two pumps
have the same linear polarization which are orthogonal to the polarization of
the probe. These pump beams are directed into a 12.5 mm long vapor cell filled
with isotopically pure 85Rb heated to 90 ∘C and with an angle of 0.4 degrees
between the beams, as shown in Fig. 1(b). The pumps are focused at the center
of the cell with waists ($1/e^{2}$ power radius) of 600 $\mu$m. After the cell
the pump light is filtered out using a Glan-Taylor polarizer and the output
squeezed vacuum beam is sent to a balanced homodyne detection system and the
noise is then measured using a RF spectrum analyzer.

The local oscillator (LO) for homodyne detection, at frequency $\nu_{LO}$, is
taken directly from the Ti:sapphire laser. The LO carrier frequency selects
the frequency of the generated photons that are being analyzed. We summarize
in Fig. 2 (a), the various frequencies involved. The two photon detuning
$\delta$ is defined as $\delta=\nu_{LO}-(\nu_{1}+\nu_{HF})$ and is also
equivalent to $\nu_{LO}-\nu_{p}$ for the measurements presented here. Fig.
1(d) shows a typical measurement of $-4.0$ dB of vacuum quadrature squeezing
at $\Delta=0.8$ GHz and $\delta=4$ MHz for an analysis frequency $\omega_{a}=$
1 MHz.

As in Corzo  _et al._ (2011), the homodyne detection system can be locked to
measure the quadrature with the minimum noise level of the squeezed vacuum
beams. The locking system McKenzie  _et al._ (2005) works as follows: the
phase of the squeezed beam is modulated at a frequency of 5 kHz, which
translates into a modulation of the noise power signal. This signal is then
detected at an analysis frequency of 1 MHz (zero span) and demodulated with
the use of a lock-in amplifier. As a result we obtain an error signal that has
a zero crossing at the minimum and maximum noise levels, which allows us to
lock the homodyne detection system and measure the squeezing and
anti–squeezing levels of the output squeezed beam without phase fluctuations.
The main result of this paper is the comparison between measurements with and
without a locking system for the phase of the LO.

Figure 2:  (Color online.) a) Diagram to summarize the various frequencies
involved. $\nu_{1}$ and $\nu_{2}$ are the carrier frequencies of pumps 1 and
2, respectively. $\nu_{LO}$ is the carrier frequency of the local oscillator
and $\nu_{p}=\nu_{1}+\nu_{HF}$. The two-photon detuning
$\delta=\nu_{LO}-\nu_{p}$. The analysis frequency is denoted $\omega_{a}$, and
the upper and lower sidebands are at $\nu_{LO}\pm\omega_{a}$. The grey area
represents the gain region of the process (4WM bandwidth). b) Measured noise
level of the generated beam as a function of the two-photon detuning $\delta$
at an analysis frequency of $\omega_{a}=1$ MHz. Black squares corresponds to
the choice of phase that maximizes the noise and red circles corresponds to
the phase that minimizes the noise. The state is squeezed for a noise level
less than zero. Black line: shot noise level.  Figure 3:  (Color online.)
Measured squeezing as a function of frequency for different values of the two-
photon detuning $\delta$. (a) $\delta=0$ MHz (b) $\delta=-4$ MHz (c)
$\delta=-12$ MHz (d) $\delta=-20$ MHz. Red traces (light gray): Measured noise
power when the phase of the LO is scanned. Blue traces (dark gray): Measured
noise power when the phase of the LO is locked (at $1$ MHz). Green (gray)
solid line: envelopes of the detected noise minima, calculated by applying a
low pass filter to the red traces. Black line at 0 dB: shot noise level.
$\Delta=0.8$ GHz, the cell temperature $\rm T_{cell}=90^{\circ}C$, and the
power in pump 1 is 230 mW and in pump 2 is 160 mW.

##  III Squeezing bandwidth

In order to study the noise level on the squeezed beam as a function of the
two-photon detuning $\delta$, we show in Fig. 2(b) the measured noise (minimum
and maximum noise) as a function of $\delta$. We observe the largest squeezing
around $\delta=-2$ MHz and the largest excess noise around $\delta=+18$ MHz.

Similarly, we can measure the noise power for a fixed quadrature phase and
two-photon detuning $\delta$ and report it as function of the analysis
frequency. This measurement is known as the squeezing spectrum. In this
section, we clarify the relationship between these two measurements and verify
the consistency of the two sets of data. In Fig. 3, we present the measured
squeezing spectrum for different values of the two-photon detuning. As
described earlier, two measurements of the noise power are reported for each
value of $\delta$, with (blue/dark gray trace) and without (red/light gray
trace) locking the phase of the LO. In order to quantify the behavior of the
noise power as a function of the analysis frequency, we introduce the shot
noise limit frequency (SNLF) defined as the analysis frequency where the
minimum of the detected noise (red/light gray traces) is equal to the shot
noise limit. To evaluate this quantity, we calculate the envelope of the
detected noise minima, by applying a low pass filter and we plot the resulting
filtered curves in Fig. 3. These envelopes represent the maximum squeezing
obtainable independent of the choice of phase for the LO. We then report in
Table 1 the frequencies at which these envelope curves cross the standard
quantum limit.

Two-photon detuning | SNLF
---|---
0 MHz | 4.9 MHz
$-$4 MHz | 9.1 MHz
$-$12 MHz | 17.7 MHz
$-$20 MHz | 24.8 MHz
Table 1: Behavior of the measured noise power as a function of the two-photon
detuning $\delta$. SNLF (shot noise limit frequency) is the analysis frequency
at which the squeezing is lost (envelopes of red traces in Fig.3).

We note that the values reported on Table 1 show that $4.8$ MHz$<$
SNLF$+\delta$ $<5.7$ MHz, and suggest that SNLF$+\delta$ is a constant within
experimental uncertainties. This point will be addressed in detail at the end
of this section.

When we measure the noise properties of the generated probe beam we beat it
with a LO and the frequency of the LO defines the two-photon detuning
$\delta$. The 4WM process, in the phase sensitive configuration used here,
spontaneously generates photons in the spatial mode of the probe beam over a
range of frequencies, and the frequency spread is defined as the bandwidth of
the process. The spatial mode is defined by the mode-matching conditions. In
consequence, the LO beats with multiple independent frequencies spontaneously
generated in the probe mode, and each of these contributes incoherently to the
measured noise spectrum. The contribution of the upper and lower sidebands
must be taken into account separately to describe the noise power at a given
analysis frequency $\omega_{a}$. More precisely, if we fix the two-photon
detuning of the LO ($\delta$), we can write the noise power ($N$) at
$\omega_{a}$ as the sum of the contribution from the two sidebands (see Fig. 2
(a)) for a noise power measured at $\omega_{a}=0$ with an apparent two photon
detuning $\delta+\omega_{a}$ and $\delta-\omega_{a}$. The noise power can be
expressed as followsper :

$N(\omega_{a},\delta)=\frac{N(0,\delta+\omega_{a})+N(0,\delta-\omega_{a})}{2}.$
(2)

The noise power at $\omega_{a}=0$ Hz is a quantity virtually impossible to
measure experimentally. Nonetheless we can extrapolate its value using the
data reported in Fig. 2, measured at $\omega_{a}=1$ MHz. Indeed, the quantity
plotted as the red dots in Fig. 2 is $N(\omega_{a}=1~{}\text{MHz},\delta)$.
Using Eq. (2), we know that :

$N(\omega_{a}=1~{}\text{MHz},\delta)=\frac{N(0,\delta+1~{}\text{MHz})+N(0,\delta-1~{}\text{MHz})}{2}.$
(3)

Estimating the noise power at $\omega_{a}=0$ Hz is then equivalent to solving
the following equation:

$f(\delta)=g(\delta+1)+g(\delta-1),$ (4)

where we want to obtain the function $g(\delta)$ in terms of the function
$f(\delta)$. $f(\delta)$ corresponds to $N(\omega_{a}=1~{}\text{MHz},\delta)$,
and $g(\delta)$ corresponds to $N(\omega_{a}=0,\delta)$. To solve this
equation we make use of the Fourier transform and find the solution:

$\displaystyle
g(\delta)=\frac{1}{2\pi}\int^{\infty}_{-\infty}\frac{\tilde{f}(t)}{2\cos(2\pi
i\delta t)}e^{2\pi i\delta t}dt,$ (5)

where $\tilde{f}(t)$ is the Fourier transform of the function $f(\delta)$. We
can then use the data in Fig. 2 to calculate the noise level at an analysis
frequency of 0 Hz as function of $\delta$ (see Fig. 4).

Figure 4:  (Color online.) Noise level at an analysis frequency of 0 MHz as a
function of the two-photon detuning $\delta$. The red circles correspond to
the measured noise power as a function of $\delta$ at 1 MHz. Blue solid line:
calculated noise at an analysis frequency of 0 MHz as a function of $\delta$.
Black solid line at 0 dB: shot noise level.

With the estimated noise power at $\omega_{a}=0$ Hz, it becomes
straightforward to calculate the expected squeezing spectrum using Eq. (2). In
Fig. 5, we plot these calculated traces together with the envelope of the
maximum obtainable squeezing (envelope of the oscillating red/light gray
traces in Fig. 3). The qualitative agreement between these two data sets
demonstrates the validity of our approach using the incoherent sum of the two
sidebands. The values of SNLF are also well predicted using the simulated
noise traces. Moreover, we can explain why SNLF$+\delta$ is a constant within
the experimental uncertainties. If one of the sidebands exhibits a much larger
excess noise in comparison to the other one, the noise power can be
approximated as coming only from the noisiest sideband. As we can see in Fig.
4, a sideband at a $\delta$ higher than 1 MHz will exhibit excess noise
compared to the SNL, while on the other side ($\delta$ lower than -1 MHz) the
noise power in the sideband is below the SNL and its contribution to the total
noise power can therefore be neglected. Therefore the squeezing bandwidth
measurements (noise level as a function of $\omega_{a}$, as in Fig. 5) for
values of $\delta\leq 0$ can simply be seen as a translation of the frequency
axis by the value of $-\delta$.

##  IV Heisenberg Langevin theoretical model

In this section, we address the question of the discrepancy in Fig. 3 between
the envelope of the red (light grey) traces representing the minimum
obtainable noise independent of the choice of the LO phase indicated along the
bottom axis (referred to as $N_{min}$) and the blue (dark gray) traces
representing the noise for a choice of LO phase that minimizes the noise at
$\omega=1$ MHz (referred to as $N_{1}$). As seen in the previous section, a
model that sums the contribution of the two frequency sidebands in order to
calculate the spectrum of the squeezing can qualitatively explain the global
behavior of $N_{min}$.

Figure 5:  (Color online.) Noise power as a function of the analysis frequency
for $\delta=0$ MHz (red), $-4$ MHz (orange), $-12$ MHz (blue), $-20$ MHz
(green). Dashed lines (regular, double, condensed and large) are the envelopes
corresponding to the minimum of noise traces shown in Fig. 3 for respectively
$\delta=0$ MHz, $-4$ MHz, $-12$ MHz, $-20$ MHz. Solid lines are obtained
numerically by summing the contributions of the two sidebands using the noise
level at $\omega_{a}=0$ plotted in Fig. 4.

In order to understand the behavior of $N_{1}$, we introduce a new parameter:
the phase of the LO. As explained in the experimental section, when we lock
the phase of the LO, the reference signal for the lock is derived from the
measured and modulated noise at $\omega=1$ MHz. We denote the phase that
minimizes the noise for $\omega=1$ MHz as $\phi_{1}$. We also introduce the
phase dependency to the noise power and $N$ becomes a function of three
parameters : $\omega_{a}$, $\delta$, and $\phi$. We can use Eq. (2) to express
$N_{min}$:

$\displaystyle N_{min}(\omega_{a},\delta)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\min_{\phi}\left[N(0,\delta-\omega_{a},\phi)\right.$
(6) $\displaystyle+\left.N(0,\delta+\omega_{a},\phi+\Delta\phi)\right],$

and $N_{1}$:

$\displaystyle N_{1}(\omega_{a},\delta)$ $\displaystyle=$
$\displaystyle\frac{\left[N(0,\delta-\omega_{a},\phi_{1})+N(0,\delta+\omega_{a},\phi_{1}+\Delta\phi)\right]}{2}.\quad$
(7)

In these equations, $\Delta\phi$ corresponds to an arbitrary phase shift
between the two sideband contributions. This parameter is the main difference
with the model discussed in the previous section (where $\Delta\phi$ was
assumed to be 0). The physical origin of this phase shift will be discussed
later.

We have shown in the previous section (see Fig. 4) that we can compute
$N(0,\delta-\omega_{a},\phi_{m})$ for $\phi_{m}$ that minimizes $N$.
Similarly, it is straightforward to compute this quantity for $\phi_{M}$ that
maximizes $N$. We can then calculate the expected value of
$N(0,\delta-\omega_{a},\phi)$ for any value of $\phi$ :

$\displaystyle N(0,\delta-\omega_{a},\phi)$ $\displaystyle=$ $\displaystyle
N_{-}~{}\text{Sin}(\phi)+N_{+},$ (8)

where

$\displaystyle
N_{-}=\frac{N(0,\delta-\omega_{a},\phi_{M})-N(0,\delta-\omega_{a},\phi_{m})}{2},$
$\displaystyle
N_{+}=\frac{N(0,\delta-\omega_{a},\phi_{M})+N(0,\delta-\omega_{a},\phi_{m})}{2}.$
(9)

$\phi$ is chosen such that for $\phi=0$, we have
$N(0,\delta-\omega_{a},0)=N(0,\delta-\omega_{a},\phi_{M})$ and for $\phi=\pi$,
we have $N(0,\delta-\omega_{a},\pi)=N(0,\delta-\omega_{a},\phi_{m})$. Using
these equations, we can compute $N_{min}(\omega_{a},\delta)$ by minimizing the
expression of Eq. (6) over $\phi$ and we can calculate
$N_{1}(\omega_{a},\delta)$. The results of these simulations are plotted in
Fig. 6 for $\delta=-4$ MHz and $\delta=-20$ MHz, with $\Delta\phi=0.2\pi$. The
difference between $N_{min}$ and $N_{1}$, can then be explained by the
presence of a phase shift between the two sidebands that contribute to the
noise power spectrum. The phase that minimizes the noise for $\omega_{a}=1$
MHz, is not necessarily the phase that minimize the noise at higher
frequencies. This effect can be interpreted as a rotation of the noise ellipse
in phase space. In the last section of this paper we propose a way to estimate
the phase shift theoretically.

Figure 6:  (Color online.) Noise power as a function of the analysis
frequency. Red traces (light gray) in a) and c) are identical to the red
traces in Fig. 3 and correspond to the measured noise power when the phase of
the LO is scanned. The black solid lines in a) and c) are
$N_{min}(\omega_{a},\delta)$. The blue traces (dark gray) in b) and d) are
identical to the blue traces in Fig. 3 and correspond to the measured noise
power when the phase of the LO is locked for lowest noise at $1$ MHz. The
black solid lines in b) and d) are $N_{1}(\omega_{a},\delta)$. In a) and b)
$\delta=-4$ MHz, and in c) and d) $\delta=-20$ MHz. For the numerical
simulations, we used $\Delta\phi=0.2\pi$.

The model that we use is based on a microscopic model for a double-$\Lambda$
system, similar to the work of Glorieux  _et al._ (2010b). Using the
Heisenberg-Langevin formalism, we write the evolution equations for the atomic
system and solve them analytically in the steady-state regime. Using the
slowly varying envelope approximation we can write the evolution of the probe
field amplitude along the $z$ direction as:

$\displaystyle\frac{\text{d}}{\text{d}z}\begin{bmatrix}\alpha\\\
\alpha^{*}\end{bmatrix}=\begin{bmatrix}A&B\\\
B^{*}&A^{*}\end{bmatrix}\begin{bmatrix}\alpha\\\ \alpha^{*}\end{bmatrix},$
(10)

where $\alpha$ is the probe field amplitude and $A$ and $B$ are coefficients
independent of $z$ that describe the steady state of the system. $A$ and $B$
are calculated following a procedure similar to the derivation of Eq. (11) in
Glorieux  _et al._ (2010b) and they depend on the pumps power, $\Delta$,
$\delta$. After introducing a term for the second pump field in Eq. (2) of
Glorieux  _et al._ (2010b), we wrote the propagation equations similar to Eq.
(6) and Eq. (7) of Glorieux  _et al._ (2010b) for $\alpha$ and $\alpha^{*}$.
Finally, we can write $\alpha=\alpha_{0}\text{e}^{\text{i}\phi_{\alpha}}$,
where $\alpha_{0}$ and $\phi_{\alpha}$ are real quantities. The evolution
equation for the phase $\phi_{\alpha}$ can then be written as:

$\displaystyle\frac{\text{d}}{\text{d}z}\phi_{\alpha}=\text{Im}[A]+\text{Im}[B]\cos(2\phi_{\alpha})-\text{Re}[B]\sin(2\phi_{\alpha}).\hskip
8.5359pt$ (11)

We solve this equation and plot the value of $\phi_{\alpha}$ as function of
the two-photon detuning in Fig. 7. The numerical parameters used for the
simulation are given in the figure caption and are directly calculated from
the experimental values. For this configuration we see that the phase shift is
negligible for sidebands located bellow 0 MHz or above 20 MHz. The value of
the maximum observed phase shift (approximately 0.3$\pi$) is in agreement with
the model we have developed in this paper. The physical origin of this phase
shift lies in the susceptibility of the atomic medium, and that it depends on
the two-photon detuning. It is well known that in a three-level system in the
$\Lambda$ configuration, the gain line around the Raman resonance, derived
from the real part of the susceptibility, is coupled with a fast variation of
the index of the medium derived from the imaginary part of the susceptibility.
The system we consider in this paper, presents the same qualitative behavior
and we have observed experimentally the gain line appears around
$\delta\approx 15$ MHz. Similarly, our theoretical model suggests a variation
of the index (and in consequence a phase shift) that appears around
$\delta\approx 10$ MHz, which is in qualitative agreement with the
experimental results.

Figure 7:  (Color online.) Phase shift as function of $\delta$. Numerical
simulation parameters are : $\Delta=0.8$ GHz, pump power: 200 mW, optical
depth: 1000.

##  V Conclusion

In this paper, we have presented the squeezing spectrum of the squeezed vacuum
light generated via four-wave mixing in a double-$\Lambda$ system. We observe
a rotation of the noise ellipse as function of different experimental
parameters such as the detuning and pump power. This rotation of the squeezing
quadrature is of interest in applications such as gravitational wave detection
because it could allow one to tailor the squeezed quadrature to match the
dominant measurement noise at different frequencies, thus improving the noise
performance across the spectrum. Traditional techniques for generation of
frequency-dependent squeezing rotation are based on the use of pre-filtering
cavities. Here we have shown that 4WM in a hot atomic vapor can provide this
effect without the use of cavities. An interesting feature of our system is
the fact that it is spatially multimode and it has been shown that this
property can be useful in sub-shot noise measurement techniques Treps  _et
al._ (2002). We use a theoretical model to describe the frequency dependent
squeezing that we have observed and we suggest that this model can be used to
investigate the parameter space (e.g. one-photon detuning, pump power, two-
photon detuning) and to tailor the ellipse rotation in order to obtain the
optimum squeezing angle rotation.

###### Acknowledgements.

This work is supported by the AFSOR and the Physics Frontier Center of the
Joint Quantum Institute. QG is supported by the Marie Curie IOF FP7 Program -
(Multimem - 300632).

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