# Reliable Conflictive Multi-View Learning

Cai Xu, Jiajun Si, Ziyu Guan, Wei Zhao, Yue Wu, Xiyue Gao Corresponding author

###### Abstract

Multi-view learning aims to combine multiple features to achieve more
comprehensive descriptions of data. Most previous works assume that multiple
views are strictly aligned. However, real-world multi-view data may contain
low-quality conflictive instances, which show conflictive information in
different views. Previous methods for this problem mainly focus on eliminating
the conflictive data instances by removing them or replacing conflictive
views. Nevertheless, real-world applications usually require making decisions
for conflictive instances rather than only eliminating them. To solve this, we
point out a new Reliable Conflictive Multi-view Learning (RCML) problem, which
requires the model to provide decision results and attached reliabilities for
conflictive multi-view data. We develop an Evidential Conflictive Multi-view
Learning (ECML) method for this problem. ECML first learns view-specific
evidence, which could be termed as the amount of support to each category
collected from data. Then, we can construct view-specific opinions consisting
of decision results and reliability. In the multi-view fusion stage, we
propose a conflictive opinion aggregation strategy and theoretically prove
this strategy can exactly model the relation of multi-view common and view-
specific reliabilities. Experiments performed on 6 datasets verify the
effectiveness of ECML. The code is released at
https://github.com/jiajunsi/RCML.

## Introduction

Artificial intelligence systems usually perceive and understand the world from
multi-view data. For example, automated vehicle systems sense their
surroundings through multiple sensors (e.g., camera, lidar, radar);
recommender systems capture users’ preferences from their multi-view generated
content such as textual review and visual review. Integrating the consistent
and complementary information of multiple views could obtain a more
comprehensive description of data instances, which boosts various tasks such
as clustering (Xu et al. 2019; Huang et al. 2022; Wen et al. 2022; Ektefaie et
al. 2023), retrieval (Mostafazadeh et al. 2017; Qin et al. 2022) and
recommendation (Fan et al. 2023; Tan et al. 2022).

Figure 1: Visualization of the conflictive multi-view data: the text view is
related to food “sauce”; however, the other views show conflictive
information, i.e., food “burger”.

Most of the previous studies on multi-view learning (Liang, Zadeh, and Morency
2022; Zhao et al. 2023; Zhang et al. 2023a) always assume that data of
different views are strictly aligned. For example, different views
consistently belong to the ground-truth category in a classification task.
However, in real-world settings, this assumption cannot always be guaranteed.
Fig. 1 visualizes a case of users’ multi-view generated content: the text and
image views show conflictive food categories. As a result, this conflictive
information in different views makes most multi-view learning methods
inevitably degenerate or even fail.

The prevalent solutions to this problem mainly aim to eliminate the
conflictive data instance. Pioneer works regard the conflictive data as
outliers (Marcos Alvarez et al. 2013; Hou et al. 2022). They usually consist
of 3 steps: 1) measuring the consistency of views; 2) identifying outliers as
instances with significant inconsistencies across views; 3) removing outliers
to construct a clean dataset. Recently, some multi-view learning methods
(Huang et al. 2020; Zhang et al. 2023d) dedicate to learning alignment
relations for the original data and constructing new data instances
accordingly. For example, the text view in Fig. 1 would be replaced with this
view of another aligned instance. Therefore, the conflict in the original
instances would be solved.

Nevertheless, real-world applications usually require making decisions for
conflictive instances rather than only eliminating them. For instance,
recommender systems need to predict users’ preferences from their conflictive
multi-view review. Considering the decision of a conflictive instance might be
unreliable, we need the model can answer “should the decision be reliable?”.
Therefore, we point out a new problem in this work, Reliable Conflictive
Multi-view Learning (RCML) problem, which requires the model to provide the
decision results and the attached reliabilities for conflictive multi-view
data.

Figure 2:  Illustration of ECML. View-specific DNNs collect evidence, which
could be termed as the amount of support to each category. Then we form view-
specific opinions consisting of belief masses of all categories and
uncertainty (inverted to reliability). Finally, we integrate opinions by
conflictive opinion aggregation. The uncertainty of the aggregated opinion
might increase if view-specific opinions are conflictive.

In this paper, we propose an Evidential Conflictive Multi-view Learning (ECML)
method for the RCML problem. As shown in Fig. 2, we first construct the view-
specific evidential Deep Neural Networks (DNNs) to learn view-specific
evidence, which could be termed as the amount of support to each category
collected from data. Then the view-specific distributions of the class
probabilities are modeled by Dirichlet distribution, parameterized with view-
specific evidence. From the distributions, we can construct opinions
consisting of belief mass vector and decision reliability. Specifically, we
calculate the conflictive degree according to the projected distance and
conjunctive certainty among views. In the multi-view fusion stage, we propose
a conflictive opinion aggregation strategy and establish a simple and
effective average pooling fusion layer accordingly. We theoretically prove the
final reliability would be less than view-specific reliabilities for
conflictive instances.

The first contribution is recognizing the importance of explicitly providing
decision results and associated reliabilities when dealing with conflicting
multi-view data. Another contribution is that we develop a conflictive opinion
aggregation strategy and theoretically prove it can exactly model the relation
of multi-view common and view-specific reliabilities. Finally, we empirically
compare ECML with state-of-the-art multi-view learning baselines on 6 publicly
available datasets. Experiment results show that ECML outperforms baseline
methods on accuracy, reliability and robustness.

## Related Work

We briefly review related work about conflictive multi-view learning and
uncertainty-aware deep learning.

Conflictive Multi-View Learning: The prevalent conflictive multi-view learning
methods are mainly dedicated to eliminating the conflictive data instance. One
line is based on multi-view outlier detection, which is developed to detect
outliers with abnormal behavior in the multi-view context. There are two main
categories for classifying these methods: cluster-based (Huang et al. 2023;
Zhang et al. 2023b) and self-representation-based (Wang et al. 2019; Wen et
al. 2023b). Cluster-based methods employ separate clustering in each view and
generate affinity vectors for each instance accordingly (Marcos Alvarez et al.
2013; Zhao et al. 2018). Outliers are subsequently identified by comparing the
affinity vectors across multiple views. Self-representation-based methods
identify outliers by recognizing that they are difficult to represent using
normal views (Hou et al. 2022).

Another line is based on partially view-aligned multi-view learning (Wen et
al. 2023c; Zhang et al. 2021). Earlier work (Lampert and Krömer 2010)
introduces weakly-paired maximum covariance analysis to overcome the
limitations of unaligned data. Recently, Huang et al. (Huang et al. 2020)
employ the differentiable agent of the Hungarian algorithm to establish
alignment relationships for unaligned data. Along this line, researchers
propose noise-robust contrastive learning (Yang et al. 2021), the self-focused
mechanism (Zhang et al. 2023d), et al., to compute the alignment matrix.
However, these methods aim to eliminate conflictive instances, while real-
world applications usually require making decisions for them. Therefore, we
propose to make reliable decisions for conflictive instances.

Uncertainty-aware Deep Learning: Deep neural networks have achieved remarkable
success in various tasks, but often fail to capture the uncertainty of their
predictions, especially for low-quality data (Wen et al. 2023a). Uncertainty
can be categorized into aleatoric uncertainty (related to data uncertainty)
and epistemic uncertainty (associated with model uncertainty). Deep learning
for estimating uncertainty (Gawlikowski et al. 2023) can be classified into:
single deterministic method (Sensoy, Kaplan, and Kandemir 2018), bayesian
neural networks (Gal and Ghahramani 2016), ensemble methods (Lakshminarayanan,
Pritzel, and Blundell 2017) and test-time augmentation methods (Lyzhov et al.
2020). Specifically, the representative single deterministic method,
Evidential Deep Learning (EDL) (Sensoy, Kaplan, and Kandemir 2018) calculates
the category-specific evidence according to a single DNN.

Recently, researchers extend EDL to the multi-view learning area. The
pioneering work, Trusted Multi-View Classification (TMC) (Han et al. 2021)
involves Dempster’s combination rule, which assigns small weights to highly
uncertain views. Following this line, multiple opinion aggregation methods
(Jung et al. 2022; Liu et al. 2022, 2023; Zhang et al. 2023c) are proposed.
However, an important characteristic of them is “After integrating another
opinion into the original opinion, the obtained uncertainty mass will be
reduced” (Han et al. 2023). We argue that this since when incorporating an
unreliable or conflicting opinion, the uncertainty should increase. To solve
this, we propose a conflictive opinion aggregation strategy and theoretically
prove the uncertainty would increase for conflictive instances.

## The Method

In this section, we first define the RCML problem, then present ECML in
detail, together with the theoretical prove discussion, and analyses.

Figure 3: Notations for conflictive multi-view data. The two categories are
marked as yellow and green respectively. Conflictive instances contain noise
and unalignment views: noise views are marked as blue and do not belong to any
ground-truth categories; unaligned views show different categories from other
views.

### Problem Definition

In the RCML setting, suppose we are given a dataset with $V$ views, $\bar{N}$
normal instances and $\tilde{N}$ conflictive instances as shown in Fig. 3. We
use $\mathbf{x}_{n}^{v}\in\mathbb{R}^{D_{v}}(v=1,...,V)$ to denote the feature
vector for the $v$-th view of the $n$-th instance $(n=1,...,N)$, where $D_{v}$
is the dimensionality of the $v$-th view. The one-hot vector
$\mathbf{y}_{n}\in\left\\{0,1\right\\}^{K}$ denotes the ground-truth label of
the $n$-th instance, where $K$ is the total of all categories. The training
tuples
$\\{\left\\{\mathbf{x}_{n}^{v}\right\\}_{v=1}^{V},\mathbf{y}_{n}\\}_{n=1}^{\bar{N}_{train}}$
contain $\bar{N}_{train}$ normal instances. The other
$\bar{N}-\bar{N}_{train}$ normal instances and $\tilde{N}$ conflictive
instances form the test set. The goal of RCML is to accurately predict
$\mathbf{y}_{n}$ for the test instances and provide the attached prediction
uncertainties ${{u_{n}\in[0,1]}}$111The “ground truth” uncertainties are
usually not available. We manually construct conflictive instances and expect
large prediction uncertainties. We elaborate the construction approach in the
experiment section. to measure the decision reliability ($1-u_{n}$).

### Evidential Conflictive Multi-view Learning

As shown in Fig. 2, the overall architecture consists of view-specific
evidential learning and evidential multi-view fusion stages. In the first
stage, we learn view-specific evidence by evidential DNNs, which could be
termed as the amount of support to each category collected from data. Then the
view-specific distributions of the class probabilities are modeled by
Dirichlet distribution, parameterized with view-specific evidence. From the
distributions, we can construct opinions consisting of belief mass vector and
decision reliability. Specifically, we calculate the conflictive degree
according to the projected distance and conjunctive certainty among views. By
minimizing the conflictive degree, we force the view-specific DNNs to well
capture multi-view common information in the training stage. This would reduce
the decision conflict caused by view-specific DNNs, i.e., normal instances
mistake for conflictive instance since view-specific models make wrong
decisions. In the second stage, we propose a conflictive opinion aggregation
strategy and establish a simple and effective average pooling fusion layer
accordingly. Details will be elaborated as below.

#### View-specific Evidencial Deep Learning.

Most existing deep multi-view learning methods commonly rely on employing a
softmax layer atop deep DNNs for classification purposes. However, these
softmax-based DNNs face limitations in accurately estimating predictive
uncertainty. This is due to the fact that the softmax score essentially
provides a single-point estimation of a predictive distribution, leading to
over-confident outputs even in cases of false predictions.

To solve this, we employ EDL (Sensoy, Kaplan, and Kandemir 2018) in the view-
specific learning stage. EDL was developed to address the above limitation by
introducing the evidence framework of subjective logic (SL) (Jøsang 2016). In
this context, evidence refers to the metrics collected from the input to
support the classification process. We collect evidence,
$\\{\boldsymbol{\mathit{e}}_{n}^{v}\\}$ by view-specific evidential DNNs
$\\{f^{v}(\cdot)\\}_{v=1}^{V}$.

For $K$ classification problems, a multinomial opinion over a specific view of
an instance ($\mathbf{x}_{n}^{v}$)222In this section, we omit the super- and
sub-scripts for clarify. can be represented as an ordered triplet
$\mathbf{\mathit{w}}=(\boldsymbol{\mathit{b}},\mathit{u},\boldsymbol{\mathit{a}})$,
where the belief mass
$\boldsymbol{\mathit{b}}=(\mathit{b}_{1},...,\mathit{b}_{k})^{\top}$ assigns
belief masses to possible values of the instance based on the evidence support
for each value. The uncertainty mass $\mathit{u}$ captures the degree of
ambiguity or vacuity in the evidence, while the base rate distribution
$\boldsymbol{\mathit{a}}=(\mathit{a}_{1},...,\mathit{a}_{k})^{\top}$
represents the prior probability distribution over each class $k$. Subjective
logic dictates that both $\boldsymbol{\mathit{b}}$ and $\mathit{u}$ must be
non-negative and their sum should equal one:

$\displaystyle\sum_{k=1}^{K}\mathit{b}_{k}+\mathit{u}=1,\forall
k\in\left[1,...,K\right],$ (1)

where $\mathit{b}_{k}\geq 0$ and $\mathit{u}\geq 0$. The projected probability
distribution of multinomial opinions is given by:

$\displaystyle\mathit{P}_{\mathit{k}}=\mathit{b}_{\mathit{k}}+\mathit{a}_{\mathit{k}}\mathit{u},\forall
k\in\left[1,...,K\right].$ (2)

Normally, the prior probabilities are manually set according to prior
knowledge. For example, a common approach is to set the prior probabilities
equal for each category, denoted as $\mathit{a}_{\mathit{k}}=1/K$. This
implies all categories have similar data instance numbers.

The Dirichlet Probability Density Function (PDF) is used for forming category
distribution. It can model second-order uncertainty, while the probability
values in the softmax layer only capture first-order uncertainty. The
probability density function of the Dirichlet distribution is given by:

$\displaystyle\mathit{D}(\mathbf{p}|\boldsymbol{\alpha})=\left\\{\begin{matrix}\frac{1}{\mathit{B}(\boldsymbol{\alpha})}{\textstyle\prod_{k=1}^{K}\mathit{p}_{k}^{\mathit{\alpha}_{k}-1}},&for\
\mathbf{p}\in\mathcal{S}_{K},\\\ 0,&otherwise,\end{matrix}\right.$ (3)

where $\boldsymbol{\mathit{p}}=(\mathit{p}_{1},...,\mathit{p}_{k})^{\top}$ is
the probability that the instance is assigned to $k$-th class,
$\boldsymbol{\mathit{\alpha}}=(\mathit{\alpha}_{1},...,\mathit{\alpha}_{k})^{\top}$
represents the Dirichlet parameters, $\mathcal{S}_{K}$ is the $K$-dimensional
unit simplex, defined as:

$\displaystyle\mathcal{S}_{K}=\left\\{\mathbf{p}|\sum_{k=1}^{K}\mathit{p}_{k}=1\
and\ 0\leq\mathit{p}_{1},...,\mathit{p}_{k}\leq 1\right\\},$ (4)

and $\mathit{B}(\boldsymbol{\alpha})$ is the $K$-dimensional multinomial beta
function.

The Dirichlet PDF naturally reflects a random sampling of statistical events,
which is the basis for the aleatory interpretation of opinions as statistical
measures of likelihood. Then, uncertainty mass can be well expressed in the
form of Dirichlet PDFs. Uncertainty mass in the Dirichlet model reflects the
vacuity of evidence. Interpreting uncertainty mass as vacuity of evidence
reflects the property that “the fewer observations the more uncertainty mass”.

We calculate the Dirichlet distribution parameters $\boldsymbol{\alpha}$ by
$\boldsymbol{\alpha}=\boldsymbol{\mathit{e}}+\textbf{1}$ to guarantee the
parameters are larger than one, and hence the Dirichlet distribution is non-
sparse. A mapping between the multinomial opinion and Dirichlet distribution
can be given by:

$\displaystyle\mathit{b_{k}}=\frac{\mathit{e}_{k}}{S}=\frac{\mathit{\alpha_{k}}-1}{S},\mathit{u}=\frac{K}{S},$
(5)

where
$S={\textstyle\sum_{k=1}^{K}}\left(\mathit{e}_{k}+1\right)={\textstyle\sum_{k=1}^{K}}\mathit{\alpha}_{k}$
is the Dirichlet strength,
$\boldsymbol{\mathit{e}}=(\mathit{e}_{1},...,\mathit{e}_{K})^{\top}$. It is
important to note that the level of uncertainty is inversely proportional to
the amount of total evidence available. In the absence of any evidence, the
belief for each view is 0, resulting in maximum uncertainty, i.e., 1.
Specifically, the class probability $\mathit{p}_{k}$ could be computed as
$\mathit{p}_{k}=\mathit{\alpha}_{k}/{S}$.

Through the view-specific evidential learning stage, we obtain the view-
specific opinion and the corresponding category distribution.

#### Evidential Multi-view Fusion via Conflictive Opinion Aggregation.

In this subsection, we focus on multi-view fusion according to view-specific
opinions. The noise views of the conflictive multi-view data would show high
uncertainty. We would diminish their impact in the fusion stage. The unaligned
views of the conflictive multi-view data would provide highly conflicting
opinions with low uncertainty, which may indicate that one or more views are
unreliable. In this case, we are hard to judge which view is high-quality. In
fact, the uncertainty of multi-view learning results should not decrease with
the increase of the number of views, but should be related to the quality of
the perspectives to be fused, especially when the learning results of two
views conflict. To solve this, we propose a new conflictive opinion
aggregation method.

Definition 1 Conflictive Opinion Aggregation. Let
$\boldsymbol{\mathit{w}}^{\mathit{A}}=(\boldsymbol{\mathit{b}}^{\mathit{A}},\mathit{u}^{\mathit{A}},\boldsymbol{\mathit{a}}^{\mathit{A}})$
and
$\boldsymbol{\mathit{w}}^{\mathit{B}}=(\boldsymbol{\mathit{b}}^{\mathit{B}},\mathit{u}^{\mathit{B}},\boldsymbol{\mathit{a}}^{\mathit{B}})$
be the opinions of view $\mathit{A}$ and $\mathit{B}$ over the same instance,
respectively. The conflictive aggregated opinion
$\boldsymbol{\mathit{w}}^{A\underline{\Diamond}B}$ is calculated in the
following manner:

$\displaystyle\boldsymbol{\mathit{w}}^{A\underline{\Diamond}B}=\boldsymbol{\mathit{w}}^{\mathit{A}}\underline{\Diamond}\boldsymbol{\mathit{w}}^{\mathit{B}}=(\boldsymbol{\mathit{b}}^{A\underline{\Diamond}B},\mathit{u}^{A\underline{\Diamond}B},\boldsymbol{\mathit{a}}^{A\underline{\Diamond}B}),$
(6)
$\displaystyle{\mathit{b}}_{k}^{A\underline{\Diamond}B}=\frac{{\mathit{b}}_{k}^{A}\mathit{u^{B}}+{\mathit{b}}_{k}^{B}\mathit{u^{A}}}{\mathit{u^{A}}+\mathit{u^{B}}},$
(7)
$\displaystyle\mathbf{\mathit{u}}^{A\underline{\Diamond}B}=\frac{2\mathit{u^{A}}\mathit{u^{B}}}{\mathit{u^{A}}+\mathit{u^{B}}},\boldsymbol{\mathit{a}}_{k}^{A\underline{\Diamond}B}=\frac{\boldsymbol{\mathit{a}}_{k}^{A}+\boldsymbol{\mathit{a}}_{k}^{B}}{2}.$
(8)

The opinion $\boldsymbol{\mathit{w}}^{A\underline{\Diamond}B}$ represents the
combination of the dependent opinions of $A$ and $B$. This combination is
achieved by mapping the belief opinions to evidence opinions using a bijective
mapping between multinomial opinions and the Dirichlet distribution.
Essentially, the combination rule ensures that the quality of the new opinion
is proportional to the combined one. In other words, when a highly uncertain
opinion is combined, the uncertainty of the new opinion is larger than the
original opinion. The averaging belief fusion can be computed simply by
averaging the evidence. A more detailed explanation is shown in Proposition 1.

Following Definition 1, we can fusion the final joint opinions
$\boldsymbol{\mathit{w}}$ from different views with the following rule:

$\displaystyle\boldsymbol{\mathit{w}}=\boldsymbol{\mathit{w}}^{\mathit{1}}\underline{\Diamond}\boldsymbol{\mathit{w}}^{\mathit{2}}\underline{\Diamond}...\underline{\Diamond}\boldsymbol{\mathit{w}}^{\mathit{V}}.$
(9)

According to the above fusion rules, we can get the final multi-view joint
opinion, and thus get the final probability of each class and the overall
uncertainty.

We also aim to: 1) ensure the consistency of the model in different views
during the training stage (using normal instances); 2) get an intuitive sense
of the level of conflict. Therefore, we introduce a measure named the
conflictive degree in Definition 2, which is established according to opinion
entropy.

Definition 2 Conflictive Degree. Given two opinions $\boldsymbol{w}^{A}$ and
$\boldsymbol{w}^{B}$ over an instance, the conflictive degree between
$\boldsymbol{w}^{A}$ and $\boldsymbol{w}^{B}$ is defined as:

$\displaystyle
c(\boldsymbol{w}^{A},\boldsymbol{w}^{B})=c_{p}(\boldsymbol{w}^{A},\boldsymbol{w}^{B})\cdot
c_{c}(\boldsymbol{w}^{A},\boldsymbol{w}^{B}),$ (10)

where $c_{p}(\boldsymbol{w}^{A},\boldsymbol{w}^{B})$ is the projected distance
between $\boldsymbol{w}^{A}$ and $\boldsymbol{w}^{B}$,
$c_{c}(\boldsymbol{w}^{A},\boldsymbol{w}^{B})$ is the conjunctive certainty
between $\boldsymbol{w}^{A}$ and $\boldsymbol{w}^{B}$, which can be formulated
as follows:

$\displaystyle
c_{p}(\boldsymbol{w}^{A},\boldsymbol{w}^{B})=\frac{\sum_{k=1}^{K}\left|p^{A}_{k}-p^{B}_{k}\right|}{2},$
(11) $\displaystyle
c_{c}(\boldsymbol{w}^{A},\boldsymbol{w}^{B})=(1-u^{A})(1-u^{B}).$ (12)

Intuitively, this metric ensures two things: (1) The scenario where $c=0$
arises when the same projected probability distributions are observed,
indicating non-conflicting opinions; (2) $c=1$ arises when absolute opinions
are present but with different projected probabilities. Specifically, when
$c_{c}=0$, it indicates that the vacuous condition is present in one or both
opinions. On the other hand, when $c_{c}=1$, it signifies that the opinions
are considered credible, meaning they have zero uncertainty mass.

#### Loss Function.

In this subsection, we will introduce the training DNN to obtain the multi-
view joint opinion. Traditional DNN can be easily converted into evidential
DNN with minimal modifications, as demonstrated in (Sensoy, Kaplan, and
Kandemir 2018). This transformation primarily involves replacing the softmax
layer with an activation layer (e.g., ReLU) and considering the non-negative
output of this layer as evidence. By doing so, we can obtain the parameters of
the Dirichlet distribution.

For instance $\\{\mathbf{x}_{n}^{v}\\}_{v=1}^{V}$,
$\mathbf{e}^{v}_{n}=f^{v}(\mathbf{x}^{v}_{n})$ represent the evidence vector
predicted by the network for the classification.
$\boldsymbol{\alpha}^{v}_{n}=\mathbf{e}^{v}_{n}+\mathbf{1}$ is the parameters
of the corresponding Dirichlet distribution. In the case of conventional
neural network-based classifiers, the cross-entropy loss is typically
employed. However, we need to adapt the cross-entropy loss to account for the
evidence-based approach:

$\displaystyle\mathit{L_{ace}}(\boldsymbol{\alpha}_{n})$
$\displaystyle=\int\left[\sum_{j=1}^{K}-y_{nj}\log{p_{nj}}\right]\frac{\textstyle\prod_{j=1}^{K}p_{nj}^{{\alpha}_{nj}-1}}{B\left(\boldsymbol{\alpha}_{n}\right)}d\mathbf{p}_{n}$
$\displaystyle=\sum_{j=1}^{K}y_{nj}(\psi(S_{n})-\psi({\alpha}_{nj})),$ (13)

where $\psi(\cdot)$ is the digamma function.

The above loss function does not guarantee that the evidence generated by the
incorrect labels is lower. To address this issue, we can introduce an
additional term in the loss function, namely the Kullback-Leibler (KL)
divergence:

$\displaystyle L_{KL}(\boldsymbol{\alpha}_{n})$
$\displaystyle=KL\left[D(\boldsymbol{p}_{n}|\tilde{\boldsymbol{\alpha}}_{n})\parallel
D(\boldsymbol{p}_{n}|\mathbf{1})\right]$ (14)
$\displaystyle=\log{(\frac{\Gamma({\textstyle\sum_{k=1}^{K}}\tilde{\alpha}_{nk})}{\Gamma(K){\textstyle\prod_{k=1}^{K}\Gamma(\tilde{\alpha}_{nk})}})}$
$\displaystyle+\sum_{k=1}^{K}(\tilde{\alpha}_{nk}-1)\left[\psi(\tilde{\alpha}_{nk})-\psi(\sum_{j=1}^{K}\tilde{\alpha}_{nj})\right],$

where $D(\mathbf{p}_{n}|\mathbf{1})$ is the uniform Dirichlet distribution,
$\tilde{\boldsymbol{\alpha}}_{n}=\mathbf{y}_{n}+(\mathbf{1}-\mathbf{y}_{n})\odot\boldsymbol{\alpha}_{n}$
is the Dirichlet parameters after removal of the non-misleading evidence from
predicted parameters $\boldsymbol{\alpha}_{n}$ for the $n$-th instance, and
$\Gamma(\cdot)$ is the gamma function.

Therefore, given the Dirichlet distribution with parameter $\alpha_{n}$ for
the $n$-th instance, the loss is:

$\displaystyle
L_{acc}(\boldsymbol{\alpha}_{n})=L_{ace}(\boldsymbol{\alpha}_{n})\ +\
\lambda_{t}L_{KL}(\boldsymbol{\alpha}_{n}),$ (15)

where $\lambda_{t}=min(1.0,t/T)\in\left[0,1\right]$ is the annealing
coefficient, $t$ is the index of the current training epoch, and $T$ is the
annealing step. By gradually increasing the influence of KL divergence in
loss, premature convergence of misclassified instances to uniform distribution
can be avoided.

In order to ensure the consistency of results between different opinions
during training, minimizing the degree of conflict between opinions was
adopted. The consistency loss for the instance
$\\{\mathbf{x}_{n}^{v}\\}_{v=1}^{V}$ is calculated as:

$\displaystyle L_{con}=\frac{1}{V-1}\sum_{p=1}^{V}\left(\sum_{q\neq
p}^{V}c(\boldsymbol{w}^{p}_{n},\boldsymbol{w}^{q}_{n})\right).$ (16)

To sum up, the overall loss function for a specific instance
$\\{\mathbf{x}_{n}^{v}\\}_{v=1}^{V}$ can be calculated as:

$\displaystyle
L=L_{acc}(\boldsymbol{\alpha}_{n})+\beta\sum_{v=1}^{V}L_{acc}(\boldsymbol{\alpha}_{n}^{v})+\gamma
L_{con}.$ (17)

The model optimization is elaborated in Algorithm 1 (Technical Appendix).

### Discussion and Analyses

In this subsection, we theoretically analyze the advantages of ECML,
especially the conflictive opinion aggregation for the conflictive multi-view
data. The following propositions provide the theoretical analysis to support
the conclusions. The proof is shown in the Technical Appendix.

Proposition 1 The conflictive opinion aggregation
$\boldsymbol{\mathit{w}}^{A\underline{\Diamond}B}=\boldsymbol{\mathit{w}}^{\mathit{A}}\underline{\Diamond}\boldsymbol{\mathit{w}}^{\mathit{B}}$
is equivalent to averaging the view-specific evidences
$\boldsymbol{\mathit{e}}^{A\underline{\Diamond}B}=\frac{1}{2}(\boldsymbol{\mathit{e}}^{\mathit{A}}+\boldsymbol{\mathit{e}}^{\mathit{B}})$.

Based on this proposition, in the multi-view fusion stage, we establish a
simple and effective average pooling fusion layer, $g(\cdot)$, to realize
conflictive opinion aggregation.

Proposition 2 For the conflictive opinion aggregation, after aggregating a new
opinion into the original opinion, if the uncertain mass of the new opinion is
smaller than the original uncertain mass, the uncertain mass of the aggregated
opinion would be smaller than the original one; conversely, it would be
larger.

An important characteristic of most existing trust multi-view learning methods
(Han et al. 2021; Jung et al. 2022; Liu et al. 2022; Xu et al. 2022; Liu et
al. 2023; Zhang et al. 2023c) is “After integrating another opinion into the
original opinion, the obtained uncertainty mass will be reduced.” We argue
that this is unreasonable since: 1) when integrating a reliable perspective,
the fusion process should ideally reduce the overall uncertainty; 2) when
incorporating an unreliable or conflicting perspective, the fusion should
increase the uncertainty. Furthermore, existing methods often overlook the
possibility of conflicts between opinions gathered from different views. These
conflicts may arise due to misaligned data or variations in the model’s
performance across different views.

To illustrate this issue, let’s consider the scenario of two observers, A and
B, observing colored balls drawn from a box. The observers could be seem as
sensors to collect multi-view data. Different kinds of observers (normal or
color blind) can produce conflictive multi-view data. The balls can be one of
four colors: black, white, red, or green. Observer B is color blind,
specifically having difficulty distinguishing between red and green balls
while being able to differentiate between other color combinations. On the
other hand, Observer A has perfect color vision and can usually identify the
correct color when a ball is selected. Consequently, when a red ball is
chosen, observer A typically identifies it as red, while observer B may
perceive it as green. This disagreement between A and B leads to conflicting
opinions regarding the same object.

Assuming that it is initially unknown whether one of the observers is color
blind, their opinions are considered equally reliable. However, the existing
fusion methods would erroneously reduce the uncertainty after combining the
opinions of both observers. But in this case, we should treat the information
from each perspective equally, given the possibility of conflicting opinions,
and not automatically reduce the uncertainty. In summary, existing fusion
methods should take into account the potential conflicts between opinions
collected from different views and treat each perspective’s information
equally, rather than assuming a reduction in uncertainty based solely on the
fusion process.

## Experiments

In this section, we evaluate ECML on 6 real-world multi-view datasets.
Furthermore, we also analyze the conflictive degree and uncertainty on
conflictive multi-view data.

### Experimental Setup

#### Datasets.

HandWritten333https://archive.ics.uci.edu/ml/datasets/Multiple+Features
comprises 2000 instances of handwritten numerals ranging from ’0’ to ’9’, with
200 patterns per class. It is represented using six feature sets.
CUB444http://www.vision.caltech.edu/visipedia/CUB-200.htmlconsists of 11788
instances associated with text descriptions of 200 different categories of
birds. In this study, we focus on the first 10 categories and extract image
features using GoogleNet and corresponding text features using doc2vec.
HMDB555https://serre-lab.clps.brown.edu/resource/hmdb-a-large-human-motion-
database is a large-scale human action recognition dataset containing 6718
instances from 51 action categories. We extract the HOG and MBH features as
multiple views for this dataset.
Scene15666https://doi.org/10.6084/m9.figshare.7007177.v1 includes 4485 images
from 15 indoor and outdoor scene categories. We extract three types of
features GIST, PHOG, and LBP. Caltech101777http://www.vision.caltech.edu/Image
Datasets/Caltech101 comprises 8677 images from 101 classes. We select the
first 10 categories and extract two deep features (views) using DECAF and
VGG19 models. PIE888http://www.cs.cmu.edu/afs/cs/project/PIE/MultiPie/Multi-
Pie/Home.html contains 680 instances belonging to 68 classes. We extract
intensity, LBP, and Gabor as 3 views. Table 1 summarizes a summary of the
datasets.

Dataset | Size | $\mathit{K}$ | Dimensionality
---|---|---|---
HandWritten | 2000 | 10 | 240/76/216/47/64/6
CUB | 11788 | 10 | 1024/300
HMDB | 6718 | 51 | 1000/1000
Scene15 | 4485 | 15 | 20/59/40
Caltech101 | 8677 | 101 | 4096/4096
PIE | 680 | 68 | 484/256/279
Table 1:  Dataset summary. Data | DCCAE | CPM-Nets | DUA-Nets | TMC | TMDL-OA | Ours | $\bigtriangleup\%$
---|---|---|---|---|---|---|---
HandWritten | 95.45$\pm$0.35 | 94.55$\pm$1.36 | 98.10$\pm$0.32 | 98.51$\pm$0.13 | 99.25$\pm$0.45 | $\mathbf{99.40\pm 0.00}$ | 0.15
CUB | 85.39$\pm$1.36 | 89.32$\pm$0.38 | 80.13$\pm$1.67 | 90.57$\pm$2.96 | 95.43$\pm$0.20 | $\mathbf{98.50\pm 2.75}$ | 3.21
HMDB | 49.12$\pm$1.07 | 63.32$\pm$0.43 | 62.73$\pm$0.23 | 65.17$\pm$2.42 | 88.20$\pm$0.58 | $\mathbf{90.84\pm 1.86}$ | 2.99
Scene15 | 55.03$\pm$0.34 | 67.29$\pm$1.01 | 68.23$\pm$0.11 | 67.71$\pm$0.30 | 75.57$\pm$0.02 | $\mathbf{76.19\pm 0.12}$ | 0.82
Caltech101 | 89.56$\pm$0.41 | 90.35$\pm$2.12 | 93.43$\pm$0.34 | 92.80$\pm$0.50 | 94.63$\pm$0.04 | $\mathbf{95.36\pm 0.38}$ | 0.77
PIE | 81.96$\pm$1.04 | 88.53$\pm$1.23 | 90.56$\pm$0.47 | 91.85$\pm$0.23 | 92.33$\pm$0.36 | $\mathbf{94.71\pm 0.02}$ | 2.57
Table 2:  Accuracy (%) on normal test sets. The best and the second best results are highlighted by boldface and underlined respectively. $\bigtriangleup\%$ denotes the performance improvement of ECML over the best baseline. Data | DCCAE | CPM-Nets | DUA-Nets | TMC | TMDL-OA | Ours | $\bigtriangleup\%$
---|---|---|---|---|---|---|---
HandWritten | 82.85$\pm$0.38 | 83.34$\pm$1.07 | 87.16$\pm$0.34 | 92.76$\pm$0.15 | 93.05$\pm$0.05 | $\mathbf{94.40\pm 0.05}$ | 1.45
CUB | 63.57$\pm$1.28 | 68.82$\pm$0.17 | 60.53$\pm$1.17 | 73.37$\pm$2.16 | 74.43$\pm$0.26 | $\mathbf{76.50\pm 1.15}$ | 2.78
HMDB | 29.62$\pm$1.79 | 42.62$\pm$1.43 | 43.53$\pm$0.28 | 47.17$\pm$0.15 | 67.62$\pm$0.28 | $\mathbf{70.84\pm 1.19}$ | 4.76
Scene15 | 25.97$\pm$2.86 | 29.63$\pm$1.12 | 26.18$\pm$1.31 | 42.27$\pm$1.61 | 48.42$\pm$1.02 | $\mathbf{56.97\pm 0.52}$ | 17.66
Caltech101 | 60.90$\pm$2.32 | 66.54$\pm$2.89 | 75.19$\pm$2.34 | 90.16$\pm$2.50 | 90.63$\pm$2.05 | $\mathbf{92.36\pm 1.48}$ | 1.91
PIE | 26.89$\pm$1.10 | 53.19$\pm$1.17 | 56.45$\pm$1.75 | 61.65$\pm$1.03 | 68.16$\pm$0.34 | $\mathbf{84.00\pm 0.14}$ | 23.24
Table 3:  Accuracy (%) on conflictive test sets.

#### Compared Methods.

The baselines based on feature fusion include: (1) DCCAE (Deep Canonically
Correlated AutoEncoders) (Wang et al. 2015) is the classical method, which
employs autoencoders to seek a common representation. (2) CPM-Nets (Cross
Partial Multi-view Networks) (Zhang et al. 2022) is a SOTA multi-view feature
fusion method, which focuses on learning a versatile representation to handle
complex correlations among different views. (3) DUA-Nets (Dynamic Uncertainty-
Aware Networks) (Geng et al. 2021) is an uncertainty-aware method, which
utilizes reversal networks to integrate intrinsic information from different
views into a unified representation. The baselines based on decision fusion
include: (1) TMC (Trusted Multi-view Classification) (Han et al. 2021) is the
pioneer uncertainty-aware method, which addresses the uncertainty estimation
problem and produces reliable classification results. (2) TMDL-OA (Trusted
Multi-View Deep Learning with Opinion Aggregation) (Liu et al. 2022) is a SOTA
multi-view decision fusion method, which is also based on the evidential DNN
and proposes a consistency measure loss to achieve trustworthy learning
results.

To create a test set with conflictive instances, we perform the following
transformations: (1) For noise views, we introduce Gaussian noise with varying
levels of standard deviations $\sigma$ to a partial percentage of the test
instances. (2) For unaligned views, we select a portion of the instances and
modify the information of a random view, causing the label corresponding to
that view to be misaligned with the true label of the instance. We conduct 10
runs for each method and report the mean values and standard deviations.

### Experiment Results

#### Performance Comparison.

Tables 2 and 3 show the classification performance on normal and conflictive
test sets, respectively. We obtain that: (1) Even on normal test sets, ECML
outperforms all the other baselines. For instance, on the HMDB dataset, ECML
achieves an accuracy improvement of approximately $2.64\%$ compared to the
second-best (TMDL-OA) model. The reason would be attributed to the
incorporation of consistency loss, which enhances the model’s learning
capability, as validated by the ablation study. (2) When evaluating on
conflictive test sets, the accuracy of all the compared methods notably
decreases. Nonetheless, thanks to the conflictive opinion aggregation, ECML
exhibits an awareness of view-specific conflicts, leading to impressive
results across all datasets. This highlights the effectiveness of ECML for
both normal and conflictive multi-view data.

#### Conflictive Degree Visualization

Fig. 4 shows the conflictive degree on the HandWritten dataset with six views.
The left and right parts show the conflictive degree of normal and conflictive
instances, respectively. To create conflicts, we modify the content from the
first view, causing unaligned content from the other views. The results
clearly demonstrate that ECML can effectively capture and quantify conflictive
degrees between views. This finding further validates the reliability of ECML.

Figure 4: Conflictive degree visualization.

#### Uncertainty Estimation

To further evaluate the estimated uncertainty, we visualize the distribution
of normal and conflictive test sets on the CUB dataset. To construct
conflictive test sets, we introduce Gaussian noise with standard deviation
$\sigma=0.1,1,5,10$ to $50\%$ of the test instances. The experimental results
are presented in Fig. 5. The results reveal that, when the noise intensity is
low ($\sigma=0.1$), the distribution curve of the conflictive instances
closely aligns with that of the normal instances. However, as the noise
intensity increases, the uncertainty of the conflictive instances also
increases. This finding indicates that the estimated uncertainty is correlated
with the quality of the instances, thereby validating the capability of our
method in uncertainty estimation.

(a) $\sigma=0.1$

(b) $\sigma=1$

(c) $\sigma=5$

(d) $\sigma=10$

Figure 5:  Density of uncertainty.

## Conclusion

In this paper, we proposed an Evidential Conflictive Multi-view Learning
(ECML) method for the RCML problem. ECML tries to form view-specific opinions
consisting of belief mass vector and decision reliability. It further
aggregates conflictive opinions by a simple and effective average pooling
layer. We theoretically proved it can exactly model the relation of multi-view
common and view-specific reliabilities. Furthermore, we also extended our
method by minimizing the degree of conflict between opinions to guarantee the
consistency of results between different opinions. Experimental results on six
real-world datasets confirmed the effectiveness of ECML.

## Acknowledgments

This research was supported by the National Natural Science Foundation of
China (Grant Nos. 62133012, 61936006, 62103314, 62302370, 62073255, 62303366),
the Key Research and Development Program of Shanxi (Program No.
2020ZDLGY04-07), Innovation Capability Support Program of Shanxi (Program No.
2021TD-05), Natural Science Basic Research Program of Shaanxi under Grant
No.2023-JC-QN-0648, the Open Project of Anhui Provincial Key Laboratory of
Multimodal Cognitive Computation, Anhui University, No. MMC202105.

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[1]S. P. Dinesh

[1]Light and Matter Physics, Raman Research Institute, Sadashivanagar,
Bangalore, 560080, Karnataka, India

# Dynamic Fabry-Pérot cavity stabilization technique for atom-cavity
experiments

<EMAIL_ADDRESS>V. R. Thakar  V. I. Gokul  Arun Bahuleyan  S. A. Rangwala
*

###### Abstract

We present a stabilization technique developed to lock and dynamically tune
the resonant frequency of a moderate finesse Fabry-Pérot (FP) cavity used in
precision atom-cavity quantum electrodynamics (QED) experiments. Most
experimental setups with active stabilization either operate at one fixed
resonant frequency or use transfer cavities to achieve the ability to tune the
resonant frequency of the cavity. In this work, we present a simple and cost-
effective solution to actively stabilize an optical cavity while achieving a
dynamic tuning range of over 100 MHz with a precision under 1 MHz. Our unique
scheme uses a reference laser locked to an electro-optic modulator (EOM)
shifted saturation absorption spectroscopy (SAS) signal. The cavity is locked
to the PDH error signal obtained from the dip in the reflected intensity of
this reference laser. Our setup provides the feature to efficiently tune the
resonant frequency of the cavity by only changing the EOM drive without
unlocking and re-locking either the reference laser or the cavity. We present
measurements of precision control of the resonant cavity frequency and vacuum
Rabi splitting (VRS) to quantify the stability achieved and hence show that
this technique is suitable for a variety of cavity QED experiments.

###### keywords:

Fabry-Pérot cavity, Cavity stabilization, Cavity QED.

##  1 Introduction

Cavities find applications in quantum electrodynamics (QED) [1, 2, 3, 4],
precision spectroscopy [5, 6, 7, 8] all the way to gravitational wave
detection [9, 10], to give a few diverse examples of the versatility of these
systems. Several experiments study a multitude of physical phenomena
originating from a combined system containing an optical cavity and an intra-
cavity single atom or an ensemble of atoms [11, 12, 13, 14, 15, 16, 17]. In
these experiments, the optical cavity can be tuned to a desired frequency
close to the atomic transition, and an ensemble of atoms can be cooled and
confined to have spatial overlap with the mode defined by the geometric
properties of the cavity, thus coupling the atoms to the cavity. In our
experimental setup, we study the effects of interactions between ultra-cold
dilute gas of atoms in a magneto-optical trap (MOT) and a resonant moderate
finesse Fabry-Perot (FP) cavity [18, 19, 20, 21] in which a weak probe beam
along the cavity axis is typically scanned over several tens of megahertz
within a few milliseconds. For effective measurements at low signal levels,
the experiment needs multiple repetitions. To achieve this, the resonant
condition of the cavity needs to be held constant with respect to the cavity
mode coupled atoms.

To perform these measurements efficiently, it is necessary to either actively
or passively stabilize the FP cavity to compensate for any drifts in the
length of the cavity due to temperature fluctuations, acoustic noise, etc.
Passive stabilization is achieved by using ultralow thermal expansion
materials and vibration-resistant mounting designs to build the cavity [22,
23, 24, 25, 26, 27, 28] and active stabilization techniques use feedback loops
to maintain the resonant frequency of the cavity [29, 30, 31, 32, 33]. The
existing active stabilization techniques use transfer cavities to shift the
lock point and hence achieve the dynamic tuning property. In this paper, we
propose a Pound-Drever-Hall (PDH) [34, 35, 36] based active stabilization
scheme of the cavity that uses only two lasers. The FP cavity is locked to a
reference laser which is in turn locked to an electro-optic modulator (EOM)
shifted, saturation absorption spectroscopy (SAS) signal. A feature of our
setup allows us to tune the resonant frequency of the cavity over a range of
over a hundred MHz with a sensitivity of 1MHz by introducing a shift in the
reference laser frequency without unlocking either the cavity or the reference
laser. Thus, a locking technique that can be dynamically tuned over a wide
frequency range can be implemented in experiments where we want to study the
dependence of light scattered by the coupled atom-cavity system as a function
of the detuning between the atomic transition and cavity frequency. Our main
objective in presenting this work is to show how one can develop a relatively
low investment and robust lock setup that bypasses using the transfer cavity
to achieve the dynamic tuning ability. In this paper, we describe, the general
experimental setup, the technique implemented to tune and stabilize the
cavity, the configuration of the reference laser, the setup used to generate
the necessary error signal, and the sensitivity, effectiveness, and tuning
range of the implemented technique.

Figure 1: Schematic of the experiment. Six laser cooling beams (light brown)
create the MOT between the cavity mirrors. The other element shown are annular
piezoelectric transducer (PZT), The description of the operation is in the
text.

##  2 Experimental setup

The six laser cooling beams and a gradient magnetic field form the 85Rb MOT
that confines atoms at the center of the FP cavity formed by two high
reflectivity mirrors, as shown in Fig . 3. The Fabry-Perot cavity consists of
two identical spherical mirrors of diameter ${12.5}$ mm, radius of curvature
${\approx}$ ${50}$ mm, separation of ${\approx}$ ${45}$ mm, a beam waist of
${\approx}$ ${78}$ ${\mu}$m and has a linewidth of nearly ${\approx}$ ${8.8}$
MHz at 780nm. The input mirror is attached to an annular piezoelectric
transducer (PZT). The 780 nm resonant probe beam (red) and the 767 nm cavity
lock beam (blue) are coupled into the cavity so that they are both
simultaneously resonant with the cavity mode. On the input side, a quarter-
wave plate (QWP) and half-wave plate (HWP) combination in conjunction with two
polarization beam splitting cubes (PBS) are used to separate the back
reflected beams from the input mirror of the cavity, and the 767 light is
captured by the photodiode (PD) to measure the light intensity dip in the
reflected light when the cavity is resonant with the 767 nm as shown in 3.
This allows the creation of a feedback signal to the PZT and locks the cavity
length as discussed further in section B.

The 780 nm probe light is transmitted through the appropriately tuned locked
cavity for probing the atoms. The transmitted light through the cavity is
passed through a premium 780 nm band pass filter (BPF) (Thorlabs-FBH780-10)
and detected by the CCD camera and the photomultiplier tube (PMT). The spatial
overlap of the cavity mode and the MOT density profile is measured to obtain
the number of atoms coupled to the cavity. In our experiment, prominent
collective strong coupling effects such as vacuum Rabi splitting (VRS) [37,
38, 4] can be detected when a few thousands of atoms are coupled to the
cavity. A ${780}$ nm laser referenced to a Rb saturated absorption
spectroscopy (SAS) signal is used to probe VRS. This probe laser is incident
along the axis of the cavity. As the atoms of interest are 85Rb, whose 5S1/2
\- 5P3/2(D2) transition wavelength is at ${780}$nm a separate wavelength has
to be used to lock the cavity. This laser has to be far off-resonance from
85Rb D2 transitions to avoid any interaction with the cavity-coupled atoms.
The coatings of the cavity restricted the choice of the reference laser to
within ${780\pm}$100 nm. Keeping these conditions in mind, a ${767}$ nm laser
is chosen as the reference laser to lock the cavity. A ${40}$ dB premium
bandpass filter is placed in the cavity transmission collection setup to
ensure that the contribution of the reference laser in the measured
transmitted signal is well below the average noise level. The ${767}$ nm laser
needs to be locked to a potassium SAS signal since the drift in its frequency
changes the resonant frequency of the cavity and impacts the precision of the
experiment This SAS locked frequency is noted using a wavemeter.

###  2.1 Scheme of the lock

Initially, the ${780}$ nm probe laser is locked to an independent rubidium SAS
setup while the FP cavity and the ${767}$ nm reference laser are kept
unlocked. A CCD camera is used to observe the spatial profile of the
transmitted light from the cavity. The unlocked cavity is tuned to transmit
TEM00 mode of ${780}$ nm probe laser by adjusting the voltage of the cavity
PZT. As the cavity frequency is scanned about this TEM00 mode over a range of
${40}$ MHz, the characteristic Lorentzian peak is observed in the cavity
transmission signal. Any major drifts or fluctuations in the cavity length are
countered by manually adjusting the cavity PZT voltage in order to ensure that
this transmission peak, corresponding to the resonance condition for TEM00
mode of probe laser, would be seen in each scan cycle. While maintaining this
transmission peak within the scan range, the frequency of the ${767}$ nm
reference laser is changed by gradually varying the laser controller offset.
This is done until a dip in the reflected intensity of the reference laser and
the corresponding PDH error signal is observed to coincide with the
transmission peak of the probe on an oscilloscope. Thus, a condition is
experimentally achieved where the cavity can be stabilized using the PDH error
signal produced by the ${767}$ nm reference laser while being simultaneously
nearly resonant to the TEM00 mode of the ${780}$ nm probe. At this point, the
cavity length is approximately a common multiple of both the wavelengths
corresponding to the probe and the reference laser. Let, ${L}$ be the cavity
length, ${c}$ the speed of light, ${\nu}$${p}$ and ${n}$${p}$ respectively, be
the frequency and the longitudinal eigen mode number corresponding to the
TEM00 mode of the probe laser sustained in the cavity, ${\nu}$${r}$ and
${n}$${r}$ be the frequency and longitudinal eigen mode number of the mode of
reference laser. Then while keeping ${\nu}$${p}$ and ${n}$${p}$ constant,
${\nu}$${r}$ and in turn, ${n}$${r}$ are varied until the condition ${L}$
${\approx}$ ${c}$ ${\cdot}$np/${\nu}$p ${\approx}$ c${\cdot}$nr/${\nu}$${r}$
is satisfied. The reference laser frequency corresponding to this condition is
noted on the wavemeter. This frequency is different from the SAS locked value.
This preliminary measurement allows an appropriate choice combination of EOM
and acoustic-optic modulator (AOM) required to introduce the necessary shift
in the reference laser frequency from its SAS locked value. The reference
laser is stabilized at the approximately required frequency and the FP cavity
is in turn locked to the corresponding PDH error signal. Finally, the
reference laser frequency is fine-tuned, without unlocking either the cavity
or the laser itself, by adjusting only the EOM drive to maximize the
transmitted intensity of the probe, thus achieving the desired condition, L =
${c}$ ${\cdot}$n${p}$/${\nu}$${p}$ = ${c}$ ${\cdot}$nr/${\nu}$r.

Figure 2: Setup of 767 nm reference laser. Schematic representation of the
optical circuit of the reference laser. Labels: M mirror, L lens, K potassium,
HWP half-wave plate, QWP quarter wave plate, PBS1, PBS2, PBS3, PBS4 are
polarizing beam splitters, AOM acoustic-optic modulator, EOM1 fiber electro-
optic modulator, EOM2 free space electro-optic modulator. The output of DL
${100}$ is passed through a combination of AOM and fiber-based EOM to obtain
the desired shift in the frequency. Before the output is taken to the
experimental table, it is passed through a free-space EOM to create sidebands
for PDH locking.

###  2.2 Configuration of the reference laser

A Toptica DL-100 extended cavity diode laser operated at ${767}$ nm,
stabilized to potassium SAS is used as the reference to lock the cavity, see
Fig. 2 for the schematic of the experimental setup of the reference laser. The
${767}$ nm reference laser is transmitted through a fiber-based EOM to achieve
the relative frequency shift between the probe and reference resonances,
modulo the cavity-free spectral range (FSR). In an earlier attempt, the EOM
output was fed to a potassium SAS setup to obtain three SAS signals. One for
the central frequency and one for each sideband generated by the EOM. In this
scheme, the ${767}$ nm laser is directly referenced to the SAS signal
corresponding to the appropriate sideband.

To avoid this issue in the present scheme the reference laser output from the
Toptica DL-${100}$ is divided into two arms. The first arm is passed through a
double pass AOM setup driven at ${200}$ MHz to generate a ${400}$ MHz shift in
the reference laser frequency. The second arm is passed through the EOM to
achieve the remainder of the shift. At this drive frequency, sufficient power
can be transferred to the sideband, and a high-quality error signal for the
sideband SAS can be obtained to lock the reference laser. After the double
pass AOM setup, the first arm is further subdivided into two parts, one of
which is fed to a wavemeter to monitor the reference laser frequency, and the
other is passed through a free space Qubig EOM and coupled into a polarization
maintaining single mode optical fiber to transfer it to the experimental
table. When the cavity is tuned in resonance with the reference laser, the
intensity of reference laser light reflected from the input mirror of the
cavity dips.

The free space EOM is driven by the internal oscillator of the PDH module used
to lock the cavity to generate the sidebands required to obtain the error
signal from this reflection dip. On stabilizing the ${767}$ nm reference laser
to the potassium SAS signal, the wavemeter readings showed a standard
deviation of ${\approx}$ ${100}$ kHz, which is about the least count of the
instrument. Since the cavity is made to follow the frequency of the reference
laser, the length of the cavity is stabilized at a comparable frequency
linewidth.

Figure 3: Schematic of the experimental setup implemented to reference the
cavity. The stabilized reference laser is transferred to the experimental
table through an optical fiber. Labels: QWP quarter waveplate, HWP half
waveplate, PBS polarizing beam splitter, ML mode matching lens, HWP half-wave
plate, QWP quarter waveplate, BPF premium bandpass filter, M mirror, CCD
charge-coupled device, PD photodiode and PMT photomultiplier tube, RFO rf
oscillator, LPF low pass filter, PBS1, PBS2 are polarizing beam splitters.

###  2.3 Cavity referencing setup

The probe beam and the modulated reference laser are mixed using a half-wave
plate and PBS2 as shown in Fig. 3. The output of the PBS is aligned along the
axis of the cavity. A ${150}$ mm focal length convex lens is used to match the
spatial mode profile of the two lasers with that defined by the FP cavity.
This lens ensured that the two lasers are maximally coupled to the cavity. A
quarter-wave plate is placed close to the window of the vacuum chamber. This
quarter-wave plate introduced the shift in polarization required to separate
out and measure the reflected intensity of the reference laser on a
photodiode. On resonance, the cavity transmits the reference laser and a
corresponding dip can be observed in the photodiode signal. The reflective dip
is the order of ${1\%}$ dip in the reflected intensity.

Figure 4: PDH error signal to lock the cavity. Inset is the oscilloscope trace
of a locked signal. Cavity is scanned across the resonance of the ${767}$ nm
reference laser to obtain the standard PDH error signal (red curve) from the
dip in the reflected intensity. The inset graph (blue curve) shows the
oscilloscope trace of a locked signal.

The PDH internal oscillator drove the free space EOM to modulate the laser
frequency at ${20}$ MHz to produce sidebands in the reflection dip. Since the
cavity linewidth at ${767}$ nm is over ${10}$ MHz, the two sidebands can not
be well resolved from the central feature in the dip. The shape of the error
signal can be adjusted to produce a steep slope near resonance by carefully
choosing the phase. At low oscillator power, the error signal strength
increased with an increase in the amplitude of the PDH internal oscillator.
However, as the sidebands are not well separated, a further increase in the
amplitude of the internal oscillator would distort the signal. Despite the
small amplitude and limited resolution of the dip, a good error signal is
generated by optimizing the amplitude and the phase of the PDH internal
oscillator, see Fig. 4. The optimized error signal is fed to a proportional,
integration and derivative (PID) module which controls the cavity PZT voltage,
thus stabilizing the cavity length. Fig. 4 shows the optimized PDH error
signal and the PID signal upon locking the cavity. The ${780}$ nm bandpass
filter in the collection setup ensured that of both the frequencies
transmitted by the cavity, only the contribution from ${780}$ nm probe is
measured by the photo-multiplier tube (PMT).

Figure 5: Transmitted intensity of probe versus reference laser detuning. The
plot shows the intensity of transmitted ${780}$ nm probe light from the cavity
versus wavemeter reading reference laser frequency. In this measurement, the
cavity is locked to the reference laser, and the transmission of the probe is
measured for various detunings of the reference laser for approximately two
seconds. Each black dot represents the mean and the error bar is the standard
deviation of the transmitted intensity at a given detuning. The value of the
error bar is observed to increase near zero detuning. This is because of the
Lorentzian nature of the transmission of light through an FP cavity. The
reference laser frequency is changed in steps and the intensity of the
transmitted probe light is measured.

##  3 Sensitivity and tuning ability of the lock

Following the extensive setup, an experiment is conducted to measure the
intensity of the probe transmitted through the empty cavity as a function of
reference laser frequency. The probe laser is locked to F=${3}$ to F′=${3}$ D2
line of 85Rb transition. The reference laser is locked to the appropriate
sideband SAS signal and the cavity is locked to the error signal of the
reflection dip. The drive frequency of the fiber-based EOM (Eospace) is
changed in steps of ${1}$ MHz by monitoring the wavemeter reading of the
reference laser frequency. This effectively changes the resonant frequency of
the cavity in steps of ${1}$ MHz. It must be noted that since
$\Delta\nu_{p}=\Delta\nu_{r}\cdot n_{p}/n_{r}$, where $n_{p}/n_{r}\neq 1$, a
change of ${1}$ MHz in ${767}$ nm reference laser frequency will not translate
to ${1}$ MHz change in the resonant frequency of the ${780}$ nm probe. The
intensity of the TEM00 mode of the ${780}$ nm probe transmitted by the cavity
is measured at each step for 10 seconds. The graph in Fig. 5 shows the average
intensity and the standard deviation versus the resonant frequency of the
cavity. The ability to distinguish between the average intensity in steps of
${1}$ MHz and the flexibility to tune the cavity over a range ${100}$ MHz
without unlocking and relocking any of the components demonstrates the
precision, sensitivity, and tunability of the cavity lock.

To demonstrate the utility of the cavity stabilization mechanism in cavity QED
experiments, we present a comparison of symmetry and precision in measuring
VRS in our atom-cavity experiment before and after the implementation of the
locking scheme. In the collective strong coupling regime, the split between
the two normal modes of VRS is given by $g_{0}\sqrt{N_{c}}$, where $g_{0}$ is
the atom-cavity interaction strength for a single atom and $N_{c}$ is the
effective number of atoms coupled to the cavity.

We observe a double peak structure characteristic of VRS on probing at F=${3}$
to F′=${3}$ of 85Rb D2 by using a weak light incident along the axis of the
cavity [18]. The F=${3}$ to F′=${4}$ of 85Rb D2 laser is used for cooling
transition for our MOT and needs to be turned off to ensure that the maximum
number of atoms are prepared at the ground state of the probing transition.
The MOT isotropically expands with a time constant of the order of a few
milliseconds. Therefore, the probe is scanned within a millisecond, demanding
a minimum scan rate of the order of ${1}$ kHz. This demands a higher bandwidth
PMT pre-amplifier. In the higher bandwidth setting, the pre-amplifier
generates higher noise, thus reducing the signal-to-noise ratio (SNR) in the
transmission. To improve the SNR, the transmitted signal has to be averaged
over several cycles. Before implementing the cavity stabilizing mechanism,
small drifts, and fluctuations in the cavity length during the several
consecutive cycles, would make the ${2}$ peaks in the VRS asymmetric as seen
in Fig. 6a, 6b, and 6c. Once the cavity is stabilized using the reference
laser, an average of over ${100}$ cycles can be successfully performed and we
get a symmetric VRS as shown in Fig. 6d, which can be used to estimate the
number of atoms coupled to the cavity with much higher reliability.

Figure 6: VRS of a locked versus an unlocked cavity The plot shows the
averaged transmitted intensity of the probe versus detuning of the probe light
from F=${3}$-${3}$′ transition of 85Rb under collective strong coupling
condition. The two peak structure VRS is obtained by scanning the probe laser
around the atomic transition. Fig (a), Fig (b), and Fig (c) are plotted by
taking the averages of the data from $0-10$, $45-55$, and $90-100$
respectively. The symmetric VRS peak shown in Fig (d) is obtained when there
is an active stabilization on the length of the cavity. In the unlocked case,
during the measurement, the resonant frequency of the optical cavity changes
due to the drift and fluctuations in the length, as a result, the expected
symmetry in the double-peak structured VRS is lost.

Figure 7: The plot (a) shows the drift of the unlocked cavity with time and
(b) is the measured VRS split of an unlocked cavity versus a locked cavity. We
allowed the cavity to drift and the probe laser is continuously scanned across
the resonant transition to obtain the VRS split at various times, each dot is
data of the VRS split averaged over 10 such scans. In the unlocked case, the
fluctuations in the length of the cavity cause changes in resonant frequency
and this introduces an unequal detuning between the atom and the cavity during
the measurement cycle. In Fig (7a) we show the drift of the left and right
peaks of the double peak structured VRS, the solid red plot shows the
progression of the left peak without the lock, and the solid blue is the left
peak with the cavity lock, the dotted red and blue shows the progression of
the right peak without and with the cavity lock respectively. It can be
observed that measured drifts in frequency for blue lines do not change with
time. In Fig (7b) we show the VRS split observed for unlocked (blue plot) and
locked (red plot). A consistent VRS split with a standard deviation of $0.83$
MHz, when there is an active lock on the cavity is in place. The red dots are
experimental data, The black solid line is the mean, and the light red band
shows the standard deviation.

It can be clearly observed that the split in the VRS changes with time in the
absence of the cavity lock due to cavity drifts, while it remains unchanged
with the cavity lock in place as shown in Fig. 7. This demonstrates the
efficiency of our FP cavity stabilization technique.

##  4 Conclusion

In this paper, we have discussed in detail the experimental setup and the
techniques used to stabilize a Fabry-Perot cavity. We have demonstrated the
ability to dynamically tune the resonant cavity frequency over a range of
${100}$ MHz with under ${1}$ MHz precision. The techniques present here are
suitable for a variety of cavity QED experiments. They are especially
applicable in studying the interaction between a detuned cavity and ultra-cold
dilute gas of atoms [20]. Such a lock strategy can be made compact and
portable for future quantum electrodynamics experiments.

## Acknowledgments

The authors would like to thank the Department of Science and Technology and
Ministry of Electronics and Information Technology (MeitY), Government of
India, under a Centre for Excellence in Quantum Technologies grant with Ref.
No. 4(7)/2020-ITEA. The authors would also like to acknowledge the inputs from
Dr. Arijit Sharma, Mohamed Ibrahim, and Meena M S.

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