# A Greedy Graph Search Algorithm Based on Changepoint Analysis for Automatic
QRS Complex Detection

Atiyeh Fotoohinasab  School of Informatics, Computing and Cyber Systems at
Northern Arizona University  Corresponding author: Atiyeh Fotoohinasab,
<EMAIL_ADDRESS>Toby Hocking  School of Informatics, Computing and Cyber
Systems at Northern Arizona University  Fatemeh Afghah  School of Informatics,
Computing and Cyber Systems at Northern Arizona University

###### Abstract

The electrocardiogram (ECG) signal is the most widely used non-invasive tool
for the investigation of cardiovascular diseases. Automatic delineation of ECG
fiducial points, in particular the R-peak, serves as the basis for ECG
processing and analysis. This study proposes a new method of ECG signal
analysis by introducing a new class of graphical models based on optimal
changepoint detection models, named the graph-constrained changepoint
detection (GCCD) model. The GCCD model treats fiducial points delineation in
the non-stationary ECG signal as a changepoint detection problem. The proposed
model exploits the sparsity of changepoints to detect abrupt changes within
the ECG signal; thereby, the R-peak detection task can be relaxed from any
preprocessing step. In this novel approach, prior biological knowledge about
the expected sequence of changes is incorporated into the model using the
constraint graph, which can be defined manually or automatically. First, we
define the constraint graph manually; then, we present a graph learning
algorithm that can search for an optimal graph in a greedy scheme. Finally, we
compare the manually defined graphs and learned graphs in terms of graph
structure and detection accuracy. We evaluate the performance of the algorithm
using the MIT-BIH Arrhythmia Database. The proposed model achieves an overall
sensitivity of 99.64%, positive predictivity of 99.71%, and detection error
rate of 0.19 for the manually defined constraint graph and overall sensitivity
of 99.76%, positive predictivity of 99.68%, and detection error rate of 0.55
for the automatic learning constraint graph.

###### Index Terms:

ECG segmentation, R-peak detection, Changepoint detection, Graph learning

##  I INTRODUCTION

The electrocardiogram (ECG) is a quasi-periodic biomedical signal that
provides information about cardiac muscle electrical activities. One cardiac
cycle in a typical ECG signal is delineated by arrangements of P, the QRS
complex, T waves, and PQ and ST segments. Correct R-peak detection is the
first and most critical step in almost all ECG analysis methods. The R-peak is
the highest and only positive peak within the QRS complex, reflecting the
ventricular depolarization of the heart’s electrical activity. Precise
detection of the R-peak location plays a critical role in obtaining the
morphology of the QRS complex and revealing the location of other ECG fiducial
points. Furthermore, R-peak localization serves as the basis for automated
determination of the heart rate, which is a significant criterion for heart
arrhythmia diagnoses such as premature atrial contraction, tachycardia, and
bradycardia. Many other diseases can also be diagnosed in a non-invasive way
using R-peak detection due to the relationship between heart rate variability
and several physiological systems (e.g., vasomotor, respiratory, central
nervous, and thermoregulatory).

Various approaches have been proposed in the literature for detecting R-peaks
in an ECG signal [1]. Typically, these methods consist of two main steps: pre-
processing and detection. In the pre-processing step, the algorithm attempts
to eliminate the noise and artifacts and to highlight the relevant sections of
the ECG [2, 3]. In the second step, various methods are used to locate R-peaks
based on the result of the pre-processing step, and then other waves are
detected by defining a set of heuristic rules [4]. However, these approaches
suffer from some critical drawbacks that limit their performance in practical
applications. First, in real-time data processing and ambulatory care
settings, where the collected data are highly noisy, preprocessing-based
algorithms are less effective. Second, these algorithms can fail to detect
R-peaks in some determinant morphological patterns resulting from certain
life-threatening heart arrhythmias due to the time-varying morphology of the
QRS complex. Incorrect detection of R-peaks can affect the correct
identification of subsequent waves.

The R-peak detection step can be generally accomplished either by implementing
a threshold-based technique or by employing an independent threshold
technique. The amplitude of the peak and time duration between two consecutive
R peaks (i.e., the RR interval) are typically used to determine a suitable
threshold [5]. A constant threshold is only efficient for detecting R-peaks
within records with normal morphological patterns. Therefore, recent studies
have employed adaptive thresholds, for which there is no need to determine the
threshold experimentally. In [6] and [7], the Hilbert transform with an
adaptive thresholding technique was utilized to detect R-peaks. Some
threshold-based techniques with other criteria have also been used to specify
the threshold. In [4], an adaptive threshold concerning the geometric angle
between two consecutive samples of the ECG signal was defined. The performance
of the threshold-based technique is highly dependent on the selection of
initial parameters; hence, it can lead to a significantly higher number of
false beats. Therefore, independent threshold techniques are more desirable
than the threshold-based technique.

Most of the state-of-the-art methods for R-peak detection are based on wavelet
transform [8, 9, 10], simple mathematical operations [6, 11, 12], hidden
Markov models, and machine learning. Wavelet transform is a suitable approach
for considering the non-stationary behavior of the ECG signal. However,
considering the various shapes of the QRS complex, it is difficult to select
the optimal mother wavelet or find the required threshold in the detection
step of the wavelet transform. Additionally, discrete wavelet transform fails
to provide reliable results in a short-recording duration. Mathematical
operation-based algorithms have a low computational cost, which is more
appropriate for real-time applications and large dataset analysis. However,
achieving high performance when the signal-to-noise ratio is high remains
challenging for these algorithms. Hidden Markov models are also widely used in
ECG segmentation because they are powerful tools for considering the temporal
dependency among the waveforms [13, 14, 15]. The majority of the studies on
machine learning-based methods have utilized sparse signal processing to
represent an approximation of the nonlinear ECG signal using sparsity
constraints [16, 17, 18, 19, 20, 21]. Some studies have also applied deep
learning techniques to detect the ECG waveforms considering its high
performance in various classification tasks [22, 23]. However, the caveat with
deep learning-based approaches is that they need large-scale datasets for the
training phase and often suffer from the imbalanced class problem [24, 25].

In this paper, we propose a new class of graphical models based on optimal
changepoint detection models, named the graph-constrained changepoint
detection (GCCD) model, to locate R-peaks in the ECG signal. A changepoint
detection model identifies abrupt changes in data when a property of the time
series changes. In the non-stationary ECG signal, ECG waves can also be
considered as abrupt up or down changes over time during the heart cycle. We
exploit the model introduced by Hocking et al. [26, 27], in which a graph-
based optimal changepoint detection model was used for detecting abrupt
changes in the genomics data. In their work, they propose a new class of
functional pruning algorithms with log-linear time complexity in the amount of
data, which is capable of handling the large datasets that are common to ECG
analysis.

Only a few studies in the literature have applied changepoint detection models
for cardiac analysis. Gold et al. [28] adopted a changepoint detection method
based on Bayesian inference to extract the onset of the QRS complex over a
small time window containing just one QRS complex. In [29], a changepoint
detection approach based on the Haar wavelet and Kolmogorov-Smirnov statistic
was applied to find normal and abnormal ECG segments within the assembled ECG
samples from different ECG datasets. Sinn et al. [30] analyzed heart rate
changes in ECG recordings by detecting abrupt changes in the ordinal pattern
distributions, which are used to represent the order structure of a time
series. Some studies have also applied changepoint detection models to
investigate sleep problems by analyzing heart rate variability in the ECG
signal during sleep [31, 32].

To the best of our knowledge, this is the first study in which changepoint
detection models have been proposed to detect ECG fiducial points in long
records of ECG signals. In this novel framework, prior biological knowledge
about the expected sequence of changes can be specified in a constraint graph.
Then, functional pruning dynamic programming algorithms can compute the
globally optimal model (mean, changes, and hidden states) in fast log-linear
time. We furthermore propose a new algorithm for learning the graph structure
using labeled ECG data. Therefore, the main contributions of this study are:

  * •

A new class of graphical models based on optimal changepoint detection models
to detect R-peak positions in the ECG signal. The proposed method does not
require any noise removal preprocessing step as it uses the sparsity of
changepoints to detect abrupt changes.

  * •

A new algorithm to learn the graph structure and parameters using labeled ECG
data. Thus, the model’s performance is no longer dependent on an expert to
encode prior knowledge into the constraint graph.

  * •

Comparison of the learned graphs with the manually constructed graphs in terms
of graph structure and detection accuracy. Results demonstrate that there can
exist different optimum graph structures for one subject, and the proposed
graph learning algorithm can find global optima depending on the initial graph
structure.

The rest of the paper is organized as follows. In the next section, we
describe the proposed model for R-peak detection in the ECG signal. We explain
the GCCD model in Section II-A and the constraint graph in Section II-B.
Section II-B also defines the manual graph and the proposed graph learning
algorithm. Section III provides a description of the dataset used in this
study and a discussion of the results as well as a comparison between the
performance of the manually defined graphs and learned graphs. Finally,
Section IV summarizes this research work and its contributions.

##  II Methodology

The proposed method treats ECG wave detection as a changepoint detection
problem for a non-stationary ECG signal. It extracts the R-peaks in the raw
ECG signal by representing the periodic non-stationary ECG signal as a
piecewise locally stationary time series with constant mean values (i.e., each
piece is the mean of one segment of datapoints). The model takes a raw ECG
signal and a constraint graph as inputs and computes the onset/offset and the
mean of desired segments (i.e., hidden states). Then, the center of each state
is associated with the location of a peak. The constraint graph allows the
incorporation of prior knowledge into the model and regularizes the model.
Figure 1 illustrates an overview of the proposed algorithm in the detection of
R-peak positions in the ECG signal. It is worth re-emphasizing that the model
takes the raw ECG signal as the input, without applying any preprocessing
step, as it leverages the sparsity of changepoints to denoise the signal and
to detect abrupt changes.

The constraint graph, which encodes the expected sequence of changes in the
ECG signal, can be defined manually by an expert or automatically from the
data. In the following sections, we describe the details of various parts of
the proposed model.

Figure 1: An overview of the GCCD model. The GCCD model takes a constraint
graph and a raw ECG signal as inputs and then detects segments corresponding
to the nodes of the constraint graph at the output.

###  II-A Graph-Constrained Changepoint Detection Model

ECG fiducial points detection can be defined as the problem of finding abrupt
changes over one cardiac cycle caused by changes in statistical
characteristics. From this point of view, a proper changepoint detection
algorithm can be employed to detect ECG waves in a fast and effective way. We
applied the optimal changepoint detection model introduced in [26] to localize
R-peak positions in the ECG signal. In this model, prior biological knowledge
about the expected sequence of changes is incorporated into the model as a
graph constraint. Then, a dynamic programming algorithm using functional
pruning computes the globally optimal model (mean, changes, and hidden states)
in fast log-linear $O(N\log N)$ time.

We assumed a directed graph $G=(V,E)$ as the constraint graph, where the
vertex set $V\in\\{1,\dots,|V|\\}$ represents the hidden states/segments (not
necessarily a waveform), and the edge set $E\in\\{1,\dots,|E|\\}$ represents
the expected changes between the states/segments. Each edge $e\in E$
incorporates the following associated prior knowledge about the expected
sequences of changes:

  * •

The source $\underline{v}_{e}\in V$ and target $\overline{v}_{e}\in V$ are
vertices/states for a changepoint $e$ from $\underline{v}_{e}$ to
$\overline{v}_{e}$.

  * •

A non-negative penalty constant $\lambda_{e}\in\mathbb{R}_{+}$ is the cost of
changepoint $e$.

  * •

A constraint function $g_{e}:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$
defines the possible mean values before and after each changepoint $e$. If
$m_{i}$ is the mean before the changepoint and $m_{i+1}$ is the mean after the
changepoint, then the constraint is $g_{e}(m_{i},m_{i+1})\leq 0$. These
functions can be used to constrain the direction (up or down) and/or the
magnitude of the change (greater/less than a certain amount).

Mathematically, given the input signal $Y=\\{y_{1},\dots,y_{n}\\}$ and the
directed graph $G=(V,E)$, the problem of finding changepoints $c$, segment
means $m$, and hidden states $s$ can be solved using the following
optimization problem:

$\displaystyle\operatorname*{minimize}_{\begin{subarray}{c}m\in\mathbb{R}^{N},\,s\in
V^{N}\\\ c\in\\{0,1,\dots,|E|\\}^{N-1}\\\ \end{subarray}}\ \
\sum_{i=1}^{N}\ell(m_{i},z_{i})+\sum_{i=1}^{N-1}\lambda_{c_{i}}$ (1) s. t.
$\displaystyle\ \text{no change: }c_{i}=0\Rightarrow
m_{i}=m_{i+1}\leavevmode\nobreak\ \&\leavevmode\nobreak\ s_{i}=s_{i+1}\,$ (2)
$\displaystyle\ \text{change: }c_{i}\neq 0\Rightarrow
g_{c_{i}}(m_{i},m_{i+1})\leq 0\leavevmode\nobreak\ \leavevmode\nobreak\ \&$
$\displaystyle(s_{i},s_{i+1})=(\underline{v}_{c_{i}},\overline{v}_{c_{i}}).$
(3)

The changepoints $c_{i}$ can be assigned to any of the pre-defined edges
($c_{i}\in\\{1,\dots,|E|\\}$). Consequently, $c_{i}=0$ indicates no change
with zero cost, $\lambda_{0}=0$. Function (1) consists of a data-fitting term
$\ell$ and a model complexity term $\lambda_{c_{i}}$ [33, 34]. $\ell$
represents the negative log-likelihood of each datapoint, and
$\lambda_{c_{i}}$ is a non-negative penalty on each changepoint. In other
words, $\lambda$ regularizes the number of predicted changepoints/segments by
the model so that a larger $\lambda$ reduces the number of changepoints by
estimating a more sparse changepoint vector. The constraint function $g_{e}$
also encodes the expected up/down change and the least amplitude gap between
the mean of two states. When there is no change $c_{i}=0$, Constraint (2)
forces the model to stay in the current state $s_{i}=s_{i+1}$ with no change
in mean $m_{i}=m_{i+1}$. However, when there is a change $c_{i}\neq 0$,
Constraint (3) imposes a change in the mean implied by the constraint function
$g_{c_{i}}(m_{i},m_{i+1})\leq 0$ as well as a change in the state
$(s_{i},s_{i+1})=(\underline{v}_{c_{i}},\overline{v}_{c_{i}})$. An open-source
implementation of the Generalized Functional Prunining Optimal Partitioning
(GFPOP) algorithm is available in C++ code inside an R package named GFPOP on
GitHub [35].

###  II-B Constraint Graph

The constraint graph $G=(V,E)$ in the optimization problem of Equation (1)
encodes prior biological knowledge about the expected sequences of changes
within one cardiac cycle. It can be designed manually by an expert or be
learned from the data by the model. The two following subsections detail both
the manual and learning-based designs.

####  II-B1 Manual Graph Definition

To manually define the constraint graph $G$, we took into account the possible
morphological categories for the ECG waves (i.e., P, QRS, and T waves) and the
overall morphological properties of the signal in each record [36]. An
expected hidden state/segment in the signal is characterized as a node in the
constraint graph, and the required conditions for transition between states
are encoded in the edges. The required conditions are determined based on the
expected minimum amplitude difference of two successive states and the
polarity of each transition (i.e., up/down).

The caveat with the manual definition of the constraint graph is that it can
be inefficient for ECG signal analysis considering the various morphological
patterns for each waveform. Furthermore, the model’s performance depends on
the expert knowledge encoded into the constraint graph. In the next
subsection, we explain the proposed graph learning algorithm for learning the
constraint graph using the R-peak labels provided by the gold standard.

####  II-B2 Constraint Graph Learning

To automate the R-peak detection task, we modified the previous model by
learning the constraint graph from the data (see the dashed part in Figure 1).
In this new framework, the proposed model takes the raw signal and an initial
graph structure as inputs and yields the desired outputs, including the
onset/offset and the mean of segments specified in the nodes of the learned
constraint graph [37]. Here, the model architecture is comprised of two
stages: training and detection. The training step tries to heuristically find
an optimum graph structure by which the label errors in the training set are
minimized (the block named “Graph Learning Algorithm” in Figure 1). The
detection step then extracts the R-peaks in the raw ECG record constrained to
the graph learned in the previous step (the block named “Changepoint Detection
Model” in Figure 1).

The novelty of this new structure lies in the training step, which is
comparable to the previous model in Section II-B1. The main idea of the
training step is to automatically discover the desired topology of the
constraint graph $G$ and the information about the edges from the data. As
described in Section II-A, each edge contains the following information: (1)
the expected up/down change in the segment means, (2) the least amplitude gap
between the means of two states, and (3) a non-negative penalty imposed by the
edge transition. Suppose that the initial graph for each record is denoted as
$G_{0}=(V_{0},E_{0})$, where $V_{0}$ and $E_{0}$ are the corresponding graph
node and edge sets, respectively. Each node in the $V_{0}$ set represents
initial hidden states in the model. Each edge in the $E_{0}$ set represents a
transition between two consecutive hidden states (i.e., a changepoint $e$ from
the source $\underline{v}_{e}$ to the target $\overline{v}_{e}$ in section
II-A) and also contains initial values for parameters of $t_{0}$, $g_{0}$, and
$\lambda_{0}$, which are the initial type, the initial gap between two states,
and the initial penalty, respectively. Figure 2a shows the simple initial
graph used for the optimization process. It should be noted that the initial
edge information was chosen based on the overall results obtained from the
manual definition of the constraint graph.

A sketch of the proposed graph learning algorithm is summarised as Algorithm
1. The greedy graph search algorithm starts with the initial graph $G_{0}$ and
iteratively optimizes the graph structure and edge parameters to find a graph
that maximizes the accuracy regarding the provided labels. At the $t$-th
iteration, the function $Find\char 95\relax Graph\char 95\relax Candidates()$
finds the graph candidate set $G^{c}_{t}$ using the editing candidates for
each edge of the output graph from the previous iteration $G_{t-1}$. In this
study, the algorithm considers 11 editing candidates per edge to optimize the
graph topology and the three edge parameters. For example, in the iteration
$t$, if the parent graph (i.e., $G_{t-1}$) has two edges, the graph candidate
set $G^{c}_{t}$ will have no more than 22 members $|G^{c}_{t}|\leq 22$. These
editing candidates include three types of adding a node, two types of deleting
a node, one type of adding two nodes, changing the type of the abrupt change,
and increasing or decreasing the penalty and gap corresponding to an edge. We
believe all morphological patterns of the ECG waves can be constructed using
these editing candidates. Figure 2 illustrates the graph editing candidates
related to the edge $(V_{i},V_{j})$ with an up change.

###  II-C Computational Complexity

As can be seen in Algorithm 1, the time complexity of the GCCD algorithm is
theoretically proportional to the number of graph candidates at each iteration
(Line 10) and the number of required iterations to achieve an optimum graph
with minimum label errors (Line 5). There are three main aspects that
characterize the time complexity of the algorithm:

  * •

Given a record with $n$ data samples and a graph candidate $\hat{G}$ with $V$
vertices and $E$ edges, the time complexity to detect R-peaks (Lines 11–15) is
$S=O(En^{2})$ in the worst case (pathological simulated data) and $S=O(En\log
n)$ in the average case (typical in real data). Also note that since we
consider only graphs with a single circular path, $E=O(V)$, and the time
complexity is further reduced to $O(Vn\log n)$ (for average case/non-
pathological data).

  * •

Considering $C$ graph edit candidates in the iteration $t$, the time
complexity to compute all the models $\hat{G}\in\\{0,1,\dots,|G^{c}_{t}|=C\\}$
is $O(SC)$ (where $S$ is the time complexity of solving for optimal model
parameters given a single graph). It should be noted that the number of graph
candidates in the iteration $t$ depends on $G_{t-1}$, which is the graph from
the previous iteration (Line 10). The time complexity to compute the label
error given $L$ labels is $O(CL)$, which can be effectively ignored from the
overall time complexity as this task is fast.

  * •

Finally, iterating over $T$ iterations to obtain the graph with the minimum
label error (Line 5) causes the overall time complexity of the algorithm to be
$O(SCT)$, where $S$ is the time to solve for a single graph, and $C$ is the
number of edit candidates considered in each iteration.

Figure 2: (a:) The initial constraint graph structure with two nodes labeled
as $A$ and $R$, representing an alternative segment and the R-peak segment,
respectively, in a cycle. (b–g:) Some of the applied graph editing candidates
related to the edge $(V_{i},V_{j})$ with an up change. Algorithm 1 Greedy
Graph Learning

1:data, labels, initial graph structure $G_{0}$

2:$t\leftarrow 0$

3:$Best\\_Cost\leftarrow\inf$

4:$\hat{E}_{t}\leftarrow label\\_error(G_{t})$

5:while $\hat{E}_{t}<Best\\_Cost$ do

6: $Best\\_Cost\leftarrow\hat{E}_{t}$

7: $t\leftarrow t+1$

8: $\displaystyle G^{c}_{t}\leftarrow$ $\displaystyle
Find\\_Graph\\_Candidates(G_{t-1}),$ $\displaystyle Based\leavevmode\nobreak\
on\leavevmode\nobreak\ Figure\leavevmode\nobreak\
\ref{fig:Graph_Editing_Candidates}$

9: $\hat{E}_{t}\leftarrow Best\\_Cost$

10: for each $\hat{G}$ in $G^{c}_{t}$ do

11: $\hat{E}\leftarrow label\\_error(\hat{G})$

12: if $\hat{E}<\hat{E}_{t}$ then

13: $\hat{G}_{t}\leftarrow\hat{G}$

14: $\hat{E}_{t}\leftarrow\hat{E}$

15: end if

16: end for

17:end while

18:constraint graph $G_{t}$

##  III Experimental Studies

###  III-A Dataset

We applied the well-known MIT-BIH Arrhythmia (MIT-BIH-AR) database to evaluate
the GCCD model. This database contains 48 ECG recordings taken from 47
subjects. Each record’s duration is 30 min, and each recording is sampled at
360 Hz with a resolution of 200 samples over a 10 mV range [38, 39]. Each
recording consists of two ambulatory ECG channels from the modified lead II
(MLII) and one of the leads V1, V2, V4, or V5. In this study, all 48 records
with one MLII or V5 lead were used to evaluate the algorithm. The database has
been annotated with both RR intervals and heartbeat class information by two
or more expert cardiologists independently.

###  III-B Results and Discussion

This section presents a comprehensive discussion of the results obtained by
the proposed model and a detailed comparison between the manually defined
graphs and the learned graphs. We also provide some suggestions for the future
development of the GCCD model.

Figure 3 illustrates an example of the model’s performance with a manually
defined constraint graph in the R-peak detection task for a window of Record
230 of the MIT-BIH-AR dataset. However, as mentioned in Section II-B1, the
performance of the model using manually defined graphs depends on an expert
with prior knowledge. Furthermore, manual annotation by an expert is time
consuming and expensive. To address this issue, we proposed a new graph
learning algorithm that searches for a locally optimal constraint graph using
a greedy scheme on the labeled ECG data. Regarding the various morphological
patterns for the ECG signal, the proposed graph learning algorithm can relax
the model from the manual definition of the constraint graph for each record.

Figure 3: Demonstration of R-peak detection using the proposed model on Record
230 of the MIT-BIH-AR dataset. (Top:) The proposed model represents Record 230
as piece-wise locally stationary segments (blue lines). Extracted R-peak
positions are marked with a red “R.” (Bottom:) The graph structure for the
proposed model. The constraint graph has a vertex for each state including
state “R” for the R-wave. Below each edge $e$ we show the penalty
$\lambda_{e}$, which is either a constant $\lambda>0$ or $0$; above we show
the constants $\delta,\gamma$ in the constraint function
$g_{e}(m_{i},m_{i+1})=\delta(m_{i}-m_{i+1})+\gamma\leq 0$ , where $\delta=1$
for a non-decreasing change (shown with $\uparrow$), $\delta=-1$ for a non-
increasing change (shown with $\downarrow$), and $\gamma\geq 0$ is the minimum
magnitude of change. Figure 4: Demonstration of constraint graph optimization
using the proposed graph learning algorithm for Record 107 of the MIT-BIH-AR
dataset. (a–e, top:) Extracted R-peak positions given the learned constraint
graph in each learning iteration. The red and orange coverage bands show the
used labels in the training procedure, including training and validation sets.
The blue lines represent separate states at the model output over the raw ECG
signal (gray points). (a–e, bottom:) The learned constraint graph in each
learning iteration. For each edge, $\lambda$ is the penalty, $g$ is the gap
(the minimum magnitude of change), and the up/down arrow shows the type of
change. The part of the graph that is modified from the previous iteration is
shown in red. (f:) Testing results for the final learned constraint graph
using a new window of data. The pink coverage bands show the labels in the
testing set. Figure 5: Detection of R-peak positions for Record 219 of the
MIT-BIH-AR dataset using the proposed graph learning algorithm. (a–h, top:)
Extracted R-peaks given the constraint graph structure learned in each
iteration. The orange and red coverage bands present training and validation
labels, respectively. The blue line demonstrates locally stationary segments
at the model output, and the gray points also show the raw ECG signal at the
model input. (a–h, bottom:) The constraint graph learned in each iteration.
The red part of the constraint graph represents the selected graph editing
candidate in each iteration.

We adopted the intra-patient paradigm to train the constraint graph to address
the intra-patient variation in ECG morphologies. Thus, the training and
testing sets were generated by randomly splitting the intra-samples for each
record with an approximate ratio of $3:1$. We used a $k$-fold cross-validation
approach to evaluate the model performance with a $k$ size of $5$. More
specifically, we divided the intra-sample data into $k=5$ folds so that each
trial used four folds to train the model and one fold for validation.

Figures 4 and 5 show representative examples of the R-peak detection task
performed by the model integrated with the graph learning algorithm for two
records from the MIT-BIH-AR database. These figures illustrate how the
proposed graph learning algorithm iteratively edits the graph structure to
yield a model with maximum accuracy in detecting R-peaks. We initialized the
constraint graphs using the graph structure in Figure 2a with the initial
values of $g_{0}=100$ and $\lambda_{0}=5\times 10^{5}$ for Record 107 and
$g_{0}=100$ and $\lambda_{0}=10^{5}$ for Record 219. It should be noted that
the initial edge information was assigned based on the overall results derived
from the manually defined graphs in all experiments. However, graph candidates
2f and 2g can adjust the parameters $g$ and $\lambda$ for the optimum values.
For these two examples, we chose the initial edge information so that all the
training steps could be completely displayed. Label errors are omitted from
Figures 5a–5c to reduce clutter in the figures. The red part of the graph
structure in each iteration presents the chosen editing candidate in the
current iteration over the graph in the previous iteration. More
interestingly, Figure 5 demonstrates the model’s capability to detect R-peaks
in the presence of a baseline wandering artifact, which is a typical artifact
in the ECG signal. Baseline wandering can change the shape of the QRS complex
and thereby causes incorrect detection of the R-peak. The performance of the
Pan and Tompkins [11] algorithm, algorithms derived from the ECG signal slope
[40], and methods based on wavelet transform are highly dependent on the
removal of this artifact. Figure 6 shows the test result for this record over
two different time windows of data. Figure 7 illustrates the training progress
for these two records, where the Y-axis shows the sum of false negative and
false positive error rates. Indeed, the training progress curve reflects the
number of label errors produced by the model in each iteration given the
provided labels for the training and validation sets. It is worth mentioning
that the proposed graph learning algorithm avoids possible overfitting issues
as it tries to extract the morphology of the ECG signal that contains multiple
various morphological patterns.

Figure 6: Test result for Record 219 for two different windows of time. The
pink coverage bands represent the labels in the testing set, and the blue
lines demonstrate model output.

(a)

(b)

Figure 7: Training progress for (a) Record 107 and (b) Record 219 of the MIT-
BIH-AR dataset. Figure 8: Comparison of the training progress initialized with
two simple and complex initial graph structures for the record 106 in the MIT-
BIH-AR dataset. Figure 9: Comparison of the trained models for Record 106 of
the MIT-BIH-AR dataset inititalized with (a) a simple graph structure and (b)
a complex graph structure.

The proposed graph learning algorithm employs a greedy search scheme to select
the best performing graph in terms of detection accuracy (see Section II-B2).
Therefore, the performance of the algorithm depends heavily on the initial
graph structure and will likely lead to local optima. Figure 8 compares the
training progress for Record 106 of the MIT-BIH-AR database initialized with
the two simple (see Fig. 2a) and complex graph structures (i.e., a graph with
eight nodes representing the morphology of a normal ECG signal). Figure 9 also
presents a comparison of the final selected graphs and their performances for
a window of this record. As these figures show, the model initialized with the
complex graph structure can achieve higher accuracy (i.e., a lower number of
label errors) in a lower number of iterations than the model initialized with
the simple graph structure.

The investigation of the experimental results shows that the greedy graph
search algorithm can achieve optimal performance for the model trained with
the manually defined graphs, although its performance is affected by the
initial graph. We noticed that for most of the records from the MIT-BIH-AR
database, the learned graphs could reach the performance of the manually
defined graphs but with different graph structures. This means that the GCCD
model can obtain global optima using various initialization structures, which
will likely lead to different final graph structures. Figure 10 compares the
constraint graph structures defined manually vs. those learned automatically
using the initial graph structure in Figure 2a for Record 100 of the MIT-BIH-
AR dataset. As shown in this figure, the manually defined graph and the
learned graph both achieved the optimal performance but with different graph
structures. We also noticed that for some records from the MIT-BIH-AR
database, the graph learning algorithm chose the same structure as the
manually defined graph structure. Figure 11 shows the model performance using
the graph learning algorithm for Record 232 from the MIT-BIH-AR dataset, for
which the manually defined constraint graph and the learned graph had the same
structures.

Figure 10: Comparison of the constraint graph structures (a) defined manually
and (b) learned using the proposed graph learning algorithm for Record 100 of
the MIT-BIH-AR dataset. Figure 11: The model’s performance for Record 230 of
the MIT-BIH-AR dataset. (a) The training progress, (b) extracted R-peaks for a
window of this record, and (c) the constraint graph structure.

Different metrics were adopted to evaluate the performance of the proposed
model with both the manual and learning-based graph designs. These metrics
included sensitivity ($Sen$), positive predictivity rate ($PPR$), and
detection error rate ($DER$), which are calculated by:

$\displaystyle\textit{Sen}(\%)=\frac{TP}{TP+FN}\times{100}$ (4)
$\displaystyle\textit{PPR}(\%)=\frac{TP}{TP+FP}\times{100}$ (5)
$\displaystyle\textit{DER}(\%)=\frac{FN+FP}{TP+FN}\times{100}$ (6)

where $TP$ is true positives, $FP$ is false positives, $FN$ is false
negatives, and $TN$ is true negatives. Table I presents the performance of the
proposed model regarding both the manually defined and learning graphs against
other state-of-the-art methods for R-peak detection (QRS complex). As shown in
the table, the proposed algorithm achieved Sen = %99.76, PPR = %99.68, and DER
= 0.55 for the manual definition of the constraint graph and Sen = %99.64, PPR
= %99.71, and DER = 0.19 for the learning constraint graph using the MIT-BIH-
AR database. Note that the model constrained to the manually defined graphs
outperformed the model combined with the graph learning algorithm because in
the latter, the model’s performance was largely dependent on the initial graph
structure.

Table I: Comparison of the performance of several R-peak detection methods
using the MIT-BIH-AR database

Method | Sen (%) | PPR (%) | DER (%)
---|---|---|---
Park et al. [8] | 99.93 | 99.91 | 0.163
Farashi [41] | 99.75 | 99.85 | 0.40
Sharma and Sunkaria [3] | 99.50 | 99.56 | 0.93
Castells-Rufas and Carrabina [2] | 99.43 | 99.67 | 0.88
GCCD Model with Manual Definition of the Constraint Graph | 99.76 | 99.68 | 0.55
GCCD Model with Learning of the Constraint Graph | 99.64 | 99.71 | 0.19

ECG recordings in the MIT-BIH-AR database were chosen to challenge the R-peak
detection task because they represent a wide variety of QRS morphologies with
real-world variability. Our proposed model yielded outstanding results when
detecting R-peaks in these tricky records. Records 103, 104, 105, 108, 111,
112, 116, 200, 201, 203, 205, 208, 210, 217, 219, 222, and 228 are comprised
of abrupt changes in ECG morphology, and they are severely affected by noise
and artifacts. Figure 5 shows the capability of the model to detect R-peaks in
the presence of baseline wandering noise. We re-emphasize that these
comparable results were obtained without applying any preprocessing
operations, as opposed to other methods in the literature. Records 108, 113,
117, 201, 202, 203, 213, 219, 222, 223, 231, and 232 contain many peaks with
unusual amplitudes. Small-amplitude R-peaks or high-amplitude P- and T-peaks
embedded in high-amplitude QRS complexes can lead to high FN and FP errors in
the R-peak detection task. As a representative example, Figure 4 illustrates
the efficiency of the GCCD model in R-peak detection for Record 117, which
contains many beats with high-amplitude T-peaks.

The experimental results obtained using the proposed model justify changepoint
detection models as a potential approach to extract ECG fiducial points. In
this study, we demonstrated the capability of the GCCD model in locating
R-peaks within various morphological patterns of ECG. The proposed greedy
graph search algorithm can potentially detect ECG waves other than the R wave
(i.e., P, Q, S, and T waves) by considering corresponding prior knowledge of
the graph editing candidates. We noticed that in Records 114, 200, 203, 207,
and 210, the Sen and PPR values were less than 99%. These records contain
multiple different morphological patterns, including negative QRS complexes,
and Records 200 and 203 have several QRS complexes with ventricular
arrhythmias. The constraint graph for these records involves learning a graph
with more than one optimum graph path. Learning a multi-path constraint graph
is also required to detect all ECG waves due to the various morphological
patterns of each wave incorporated into the graph. The other point that should
be considered here is that the GCCD model estimates the ECG signal using a
Gaussian function. A modified model with a multi-Gaussian fitting method can
drastically improve the ECG-related changepoint detection task.

Future work should focus on developing the proposed model with a multi-
Gaussian fitting and a multi-path graph learning algorithm. Incorporating
these modifications into the proposed model could provide a promising platform
for evolving new graph-based tools to detect and classify heart arrhythmias. A
multi-path graph learning algorithm could reveal the morphology of the ECG
signal (time duration, amplitude, and direction of each wave) in each cardiac
cycle. Subsequently, new graph-based features could be extracted from the
constraint graph path for an ECG cycle to classify heartbeats.

##  IV Conclusion

The accurate delineation of R-peaks in the ECG signal plays a crucial role in
most automated ECG analysis tools. This paper proposed a novel graphical model
based on changepoint detection techniques for detecting R-peaks within a non-
stationary ECG signal. The proposed model was highly successful at detecting
R-peaks in noisy ECG data without applying any preprocessing steps. To our
knowledge, this is the first time that a changepoint detection model has been
applied for ECG fiducial points detection. In this new framework, prior
biological knowledge about the expected sequences of changes was incorporated
into the model using a graph. We defined the constraint graph manually and
automatically using a proposed greedy graph search algorithm. Using the
proposed graph learning algorithm, the initial graph structure can develop
into a structure containing edge parameters with maximum detection accuracy
for a record. The experimental results provided in this paper demonstrate that
the GCCD model can be a promising approach for detecting ECG waves and
developing new graph-based tools for further ECG analysis. The proposed
graphical model approach can be advanced by learning a multi-path constraint
graph and fitting a multi-Gaussian curve model to the ECG signal, which should
be considered in future studies.

## ACKNOWLEDGMENT

This material is based on work supported by the National Science Foundation
under Grant Number 1657260. Research reported in this publication was
supported by the National Institute on Minority Health and Health Disparities
of the National Institutes of Health under Award Number U54MD012388.

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# Ultrastable vacuum-gap Fabry-Pérot cavities operated in air

Yifan Liu  1,2,* Naijun Jin  3 Dahyeon Lee  1,2 Charles McLemore  1,2 Takuma
Nakamura  1,2 Megan Kelleher  1,2 Haotian Cheng  3 Susan Schima  2 Nazanin
Hoghooghi  2 Scott Diddams  1,2,4 Peter Rakich  3 and Franklyn
Quinlan2,4,${\dagger}$ 1Department of Physics, University of Colorado Boulder,
Boulder, CO 80309, USA
2Time and Frequency Division, National Institute of Standards and Technology,
Boulder, CO 80305, USA
3Department of Applied Physics, Yale University, New Haven, CT 06520, USA
4Electrical, Computer & Energy Engineering, University of Colorado Boulder,
Boulder, CO 80309, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We demonstrate a vacuum-gap ultrastable optical reference cavity that does not
require a vacuum enclosure. Our simple method of optical contact bonding in a
vacuum environment allows for cavity operation in air while maintaining vacuum
between the cavity mirrors. Vacuum is maintained long term, with no observed
degradation in cavity stability for over 1 year after bonding. For a 1550 nm
laser stabilized to a 9.7 mL in-vacuum bonded cavity, the measured Allan
deviation is $2.4\times 10^{-14}$ at 1 s and its phase noise is thermal-noise-
limited from 0.1 Hz to 10 kHz, reaching about -105 dBc/Hz at 10 kHz offset
frequency. This represents the highest stability of any oscillator operated
without a vacuum enclosure. Furthermore, we demonstrate a 0.5 mL in-vacuum
bonded cavity created using microfabricated mirrors and cavity dicing, with
phase noise reaching -95 dBc/Hz at 10 kHz offset frequency. By relieving the
need for high-vacuum enclosures, we greatly enhance the portability and
utility of low noise, compact cavity-stabilized lasers, with applications
ranging from environmental sensing to mobile optical clocks to ultralow noise
microwave generation.

##  1 Introduction

Ultrastable laser systems play a pivotal role across various technological and
scientific domains, including precision timekeeping[1, 2], precision
spectroscopy[3], photonic microwave generation[4, 5, 6], and gravitational
wave detection[7]. The conventional approach to constructing an ultrastable
laser system involves locking a laser to a highly stable vacuum-gap Fabry-
Pérot (FP) cavity[8]. These rigid FP cavities, typically ranging in length
from several centimeters to half a meter, can achieve remarkable stability
through careful design and environmental control. Amazingly, proper cavity
mounting and isolation results in vacuum-gap FP cavities that reach the
stability limit determined by stochastic volumetric fluctuations in the cavity
mirrors and high reflection coatings, resulting in room-temperature length
fluctuations below $10^{-16}$ m, equivalent to only a fraction of a proton
charge radius[9]. By combining decimeter-long cavities with cryogenic
operation, length instability below $10^{-17}$ m has been achieved,
corresponding to a laser fractional frequency instability of only $4\times
10^{-17}$ [10].

Figure 1: Compact ultrastable laser applications. Compact and field-deployable
ultrastable laser systems can be utlized in various applications, such as
photonic-based radar systems [11, 12], satellite ranging and
interferometry[13], mobile optical atomic clocks[14], and environmental
sensing[15, 16].

Robust and field-deployable ultrastable laser systems are attractive for out-
of-the-lab applications that can benefit from the low phase noise and high
stability of ultrastable lasers, such as satellite ranging and
interferometry[13], photonic-based radar systems [11, 12], mobile optical
atomic clocks[14], and environmental sensing[15, 16] (Fig. 1). Traditional
vacuum-gap FP cavity systems, despite their superior performance, have
significant drawbacks for such out-of-lab applications, mainly due to their
large size and weight, as well as the need to maintain ultrahigh vacuum
through constant active pumping. Therefore, there have been efforts to
miniaturize these laser systems, broadly along two distinct paths: the
development of solid-state dielectric resonators and the miniaturization of
vacuum-gap FP cavities. Stable dielectric resonators include fiber delay
lines[17, 18, 19], integrated spiral resonators[20, 21, 22], bulk fused silica
FP[23, 24, 25], whispering-gallery-mode (WGM) resonators[26, 27], and
stimulated Brillouin scattering (SBS) in ring resonators[28]. Dielectric
resonators, particularly on-chip resonators, can have the advantage of small
size and manufacturability with lithographic techniques. But with performance
limited by thermorefractive noise and the large thermal expansion coefficient
typical of fiber, crystalline, and on-chip waveguide materials, the fractional
instability of such resonators is typically above $1\times 10^{-13}$. Indeed,
it is useful to note that the highest stability dielectric resonators are
housed in vacuum enclosures with exquisite temperature control.

Alternatively, the size of traditional vacuum-gap FP cavities can be reduced
to make them compact and portable [29, 30, 31]. For example, a vacuum-gap FP
with a volume of only 8 mL has been used to demonstrate fractional frequency
instability of only $7\times 10^{-15}$ at 1 second [31]. While such high
performance has been demonstrated with greatly reduced cavity size, the
requirement for a vacuum enclosure and vacuum pump is not eliminated, and
represents a barrier to the realization of compact and portable field-
deployable systems.

In this work, we demonstrate laser stabilization with $10^{-14}$ level
fractional frequency instability with a sub-10 mL volume cavity while
operating without a vacuum enclosure. The cavity performance represents the
highest stability ever achieved across optical, microwave or radio frequency
domains without vacuum operation. To accomplish this, we have devised a simple
and straightforward method to bond the cavity in vacuum, essentially turning
the cavity itself into a vacuum cell. With this bonding, and with the cavity
surrounded by atmospheric pressure, the optical phase noise is at the mirror
coating noise limit across 5 decades of offset frequencies, from 0.1 Hz to 10
kHz, reaching a phase noise level below -100 dBc/Hz at 10 kHz offset. The
fractional frequency stability likewise reaches the thermal noise limit, and
is only $2.4\times 10^{-14}$ at 1 second. For one of our in-vacuum bonded
cavities that has been operated for over 1 year, we see no degradation in
either phase noise or frequency stability. Furthermore, we combine the in-
vacuum bonding method with micro-fabricated mirrors to demonstrate a 0.5 mL-
volume miniature cube cavity. This cavity is diced from an array of miniature
cavities, and achieves phase noise of -95 dBc/Hz at 10 kHz offset and
competitive Allan deviation performance. Thus these in-vacuum bonded cavities
enable a significant simplification of the vacuum-gap FP-based ultra-stable
laser system, removing the bulky vacuum enclosure and active vacuum pumping
requirements, without degrading the performance. With unprecedented frequency
stability and phase-noise properties in a compact setup, this work points
towards a truly portable and field-deployable ultrastable optical reference.

##  2 Vacuum requirement analysis and measurement

The vacuum requirement for ultrastable FP cavity systems arises from the
desire to eliminate cavity length fluctuations coming from variations of the
refractive index in the optical beam path, such that the cavity can operate at
the thermal noise limit determined by the cavity mirror substrates and
coatings. Vacuum in ultrastable FP cavity systems is typically held with an
ion pump, since this pump type has no moving parts that could couple
vibrations to the cavity. However, ion pumps best operate in the high and
ultrahigh vacuum regimes, that is, at a pressure levels below $10^{-6}$ hPa.
As we show in this section, this is far below what is required for the cavity
to operate at the thermal noise limit.

First, we estimate the residual gas pressure that allows us to reach our
cavity thermal noise limit using a model developed for LIGO[32]. In this
model, noise caused by refractive index fluctuations is derived from a
microscopic picture that considers the effects of individual molecules
entering the optical beam path. As shown in more detail in the supplement,
assuming a constant beam radius $w_{0}$ and ideal gas condition, the resulting
single-sideband optical phase noise on an optical carrier at frequency $\nu$
at offset frequencies below 100 kHz is well approximated by

$\mathscr{L}(f)\approx\sqrt{\frac{m}{2}}\frac{8\pi^{2}\alpha^{2}}{Lw_{0}}\frac{P}{(k_{B}T)^{3/2}}\frac{\nu^{2}}{f^{2}},$
(1)

where $\alpha$ is the polarizability of the gas molecules, $m$ is the mass of
the individual gas molecule, $L$ is the cavity length, $P$ is the residual gas
pressure, $k_{B}$ is the Boltzmann constant, $T$ is the system temperature,
and $f$ is the offset frequency.

A key assumption in the derivation of Eq. 1 is that the residual gas molecules
pass through the beam without collisions with other molecules. This means that
the mean free path of the molecules should be much larger than the cross-
sectional diameter of the beam, which puts an upper bound on the pressure for
this assumption to be valid. For our compact optical reference cavities, the
average beam radius is around 0.2 mm, such that the applicable pressure range
is below about 0.1 hPa (assuming the gas is predominately nitrogen molecules
maintained at 300 K). If the cavity length is 6.35 mm and the optical carrier
frequency is 1550 nm, Eq. 1 suggests
$\mathscr{L}(1~{}\text{Hz})\approx-44~{}\text{dBc/Hz}$ and
$\mathscr{L}(1~{}\text{kHz})\approx-104~{}\text{dBc/Hz}$ at a pressure level
of 0.1 hPa, which are well below the thermal noise limit such a cavity can
achieve [33]. Thus, it should not be necessary to have a high-vacuum
environment for the cavity to perform at its thermal noise limit. In fact, the
required vacuum level is expected to be quite moderate.

Figure 2: Vacuum requirement measurement. (a) Experimental setup. A 1550 nm
continuous wave (CW) laser was stabilized via the Pound-Drever-Hall (PDH)
method to the 6.35 mm long reference cavity, which was mounted in a pressure-
variable vacuum chamber. A portion of the light after the acoustic-optic
modulator (AOM) was taken to beat with a stable reference for phase noise
measurements at different pressure levels. PZT: piezoelectric transducer. EOM:
electro-optic modulator. DBM: double-balanced mixer. PD: photodetector. UHV:
ultra-high-vacuum. (b) Selected phase noise traces at different pressure
levels plotted with the estimated thermal noise limit of the cavity and the
residual gas noise for 0.1 hPa of nitrogen molecules. After 1 kHz offset, the
measurement is limited by the signal-to-noise ratio (SNR) of the heterodyne
beat. (c) The change of phase noise levels at 100 Hz offset frequency and 1
kHz offset frequency as pressure increases. The uncertainty in the pressure
reading is estimated to be about 30% for the pressures below 100 hPa and 50%
for the pressures above 100 hPa.

To verify this conclusion, we experimentally measured the phase noise of a
1550 nm laser locked to a 6.35 mm-long cavity operating at different pressure
levels. The experimental setup is shown in Fig. 2a. A 1550 nm commercial fiber
laser is locked to the cavity via the Pound-Drever-Hall (PDH) method, with
fast feedback control achieved through an acousto-optic modulator (AOM) and
slow feedback control through the laser’s piezoelectric tuning port. The
cavity has a vent hole and is mounted in a vacuum chamber placed inside an
acoustic isolation box on a vibration isolation stage. Connected to the vacuum
chamber are a vacuum gauge and an ultra-high-vacuum (UHV) valve. First, we
pump the vacuum chamber pressure down to $10^{-7}$ hPa using a turbo pump,
then close the UHV valve and disconnect the pump. Without active pumping, the
vacuum chamber does not hold at $10^{-7}$ hPa and the pressure will slowly
rise. A heterodyne beat is taken between the cavity stabilized light and an
optical frequency comb (OFC) fully stabilized to the local oscillator of a Yb
atomic clock[34]. While the pressure inside the chamber rises, phase noise
measurements are performed on the heterodyne beat at different pressure levels
indicated by the vacuum gauge. The experimental results are shown in Fig. 2b
and Fig. 2c. The phase noise was limited by the cavity thermal noise limit at
low pressures and only starts to deviate from this limit after reaching 0.30
hPa. (After 1 kHz offset, the measurement is limited by the signal-to-noise
ratio of the heterodyne beat and noise from the optical frequency comb.) This
measurement confirms that only a very moderate level of vacuum (< 0.3 hPa) is
needed for the cavity to be thermal-noise-limited out to an offset frequency
of at least 1 kHz, and that continuous vacuum pumping may not be necessary for
best phase noise performance.

##  3 In-vacuum bonding and cavity performance

Demonstrations of vacuum-tight glass-to-glass optical contact bonds date back
at least 60 years[35]. However, we are unaware of any prior investigation that
shows that the achievable vacuum seal from an optical contact bond can support
thermal noise-limited, ultrastable cavities, or that the required vacuum level
can be held long-term. In this section, we describe experiments where we bond
cavities in vacuum, then lock lasers to the cavities without a vacuum
enclosure to demonstrate thermal noise-limited phase noise performance.
Moreover, with repeated low phase noise measurements in the months since the
initial bonding, we establish the longevity of the in-vacuum bond.

Figure 3: In-vacuum bonding method. (a) The mirror substrates and the vent-
hole-less spacer are aligned by a vacuum compatible holding structure, which
holds the top mirror substrate about 2 mm above the spacer. The holding
structure is then placed inside a high-vacuum (HV) bellows valve, which is
connected to a turbo pump. Rotating the handwheel of the valve presses the top
mirror down and bonds the cavity. (b) Photograph of an in-vacuum bonded
cavity. (c) Photograph of the holding structure containing the mirror
substrates and spacer.

We have bonded several cavities to date using standard off-the-shelf
components. Here we describe the essential elements of the bonding technique
and results from one of our cavities. More details on the bonding and results
from our other cavities may be found in the supplement. The mirror substrates
and cavity spacer are all made with ultra-low-expansion (ULE) glass. They have
a cylindrical shape with a diameter of 25.4 mm and a length of 6.35 mm. The
spacer has a 10 mm diameter center bore hole but, importantly, does not
contain a vent hole. One of the mirror substrates has a 1 m radius of
curvature (ROC), whereas the other mirror is flat. Both mirrors have a highly
reflective (>99.999%) dielectric coating, and the coating diameter is less
than that of the bore hole. The contacting surfaces on the mirrors and spacer
are superpolished to the level typical of optical contact bonding, with an
average surface roughness of a few angstroms. The in-vacuum bonding setup is
illustrated in Fig. 3a. After cleaning, the substrates were placed into a
specially designed, vacuum-compatible holding structure that easily aligns the
mirrors and spacer concentrically. One mirror substrate was placed at the
bottom of the holding structure with the spacer resting directly on top of it.
The top mirror substrate is held by friction about 2 mm above the top surface
of the spacer. The holding structure containing all three pieces is then
placed in a vacuum bellows valve, which is connected to a turbo pump. A vacuum
gauge is included between the bellows valve and the turbo pump to monitor the
pressure. The pressure quickly falls below $10^{-5}$ hPa, after which we
rotate the hand wheel of the valve to lower the inner top surface such that it
presses the top mirror substrate into contact with the spacer. While
maintaining a compressive force on the bonded cavity, we then shut down and
remove the turbo pump. The bonded cavity, surrounded by atmosphere pressure,
is then held under the continuous compression force inside the vacuum valve
for a few days before being removed for measurements. A picture of an in-
vacuum bonded cavity is shown in Fig. 3b, and a picture of the holding
structure with mirror substrates is shown in Fig. 3c.

Figure 4: Measurements on the in-vacuum bonded cavity. (a) Simplified diagram
illustrating the measurement setup. A 1550 nm fiber laser was stabilized to
the in-vacuum bonded cavity via the method of PDH. Part of the stabilized
light was taken for phase noise and Allan deviation measurements. The in-
vacuum bonded cavity was mounted in an air-tight metal box that is actively
maintained at 30∘C with acoustic and vibration isolation. (b) Photograph of
the in-vacuum bonded cavity inside the V-groove holder made with Macor. The
cavity rests on two Viton strips. (c) Phase noise of the laser stabilized to
the in-vacuum bonded cavity, measured at different times after the cavity was
bonded, together with the free-running laser noise and the cavity’s estimated
thermal noise limit. (d) Allan deviation of the laser stabilized to the in-
vacuum bonded cavity, measured at different times after the cavity was bonded,
together with the cavity’s estimated thermal noise limit.

A 1550 nm laser was locked to the cavity using the PDH technique as shown in
Fig. 4a, with locking electro-optics similar to what is shown in Fig. 2a. The
cavity finesse is 591,000, corresponding to a full-width at half-maximum
linewidth of 40 kHz. To test the phase noise and frequency stability of the
cavity, we mount the cavity into a Macor V-groove holder (Fig. 4b) enclosed by
an air-tight metal enclosure actively maintained at 30∘C. Without vacuum
surrounding the cavity, we find an air-tight enclosure critical to keeping the
cavity’s temperature and outer-air pressure constant, yielding the best
results. Air pressure sensitivity of the cavity is discussed further in the
supplement. Since we are primarily concerned with assessing the low noise
abilities of in-vacuum bonded cavities, the cavity geometry is not optimized
for minimum acceleration sensitivity and was placed within a passive acoustic
isolation box with active vibration cancellation.

To measure its phase noise, we split the cavity-stabilized light into two
channels and compared against two independent ultrastable optical references:
an optical frequency comb stabilized to the local oscillator of a Yb optical
atomic clock, and a 1550 nm laser stabilized to a 10 cm long vacuum-gap
cavity. The two heterodyne beat notes with the two references are digitally
sampled and cross-correlated to reveal the noise of our cavity-stabilized
laser [29]. We have performed periodic measurements of the laser phase noise,
allowing us to assess the longevity of the vacuum-tight bonding.
Representative data are shown in Fig. 4c along with the cavity’s thermal noise
limit given by Brownian noise in the coatings. At both 6 months and our most
recent measurement at 11 months after bonding, the laser phase noise
corresponds to the thermal noise limit of the cavity across nearly 5 decades
of offset frequencies, from 0.1 Hz to nearly 10 kHz. At 2 weeks after bonding
the noise below 10 Hz offset was slightly elevated, likely due to laser
intensity noise coupling to the cavity length [36]. Laser intensity noise was
improved before the 6- and 11-month measurements were made. As shown in the
supplement, our other bonded cavities have been tested with less acoustic
isolation and temperature control, and over 1 year since bonding, yet all
achieve phase noise limited by the cavity thermal noise, with phase noise
$\leq-100$ dBc/Hz at 10 kHz offset.

The laser’s fractional frequency stability was also recorded, and is shown in
Fig. 4d. For measurements at 6 months and 11 months after bonding, the Allan
deviation (ADEV) reaches the thermal noise limit at $1.6\times 10^{-14}$ for
timescales less than 0.3 second. At 1 second, the ADEV is below $3\times
10^{-14}$, and for longer timescales corresponds to a cavity drift rate in the
range of 3-5 Hz/s. While the drift rate does vary, it has remained less than
or equal to 5 Hz/s. The long-term drift is slightly larger than is typical of
all ULE vacuum-gap cavities that are held in thermally shielded vacuum
enclosures. Sources of drift are still under investigation. However, we note
that without vacuum surrounding the cavity the temperature isolation is
reduced, and that our spacer was not acid etched after machining, which is
known to improve the dimensional stability of ULE glass[37, 38]. Acid etch is
common practice for ultrastable cavity spacers and will be incorporated in
future cavity builds.

##  4 In-vacuum bonded cavity with lithographically fabricated mirrors

The successful creation of ultrastable cavities with in-vacuum bonding creates
the opportunity to construct extremely compact cavity-stabilized laser
systems. Towards this end, we combined the in-vacuum bonding technique with
microfabricated mirrors to create an ultrastable resonator whose volume is
only 0.5 mL.

Figure 5: Measurements on the in-vacuum bonded cavity with lithographically
fabricated mirror. (a) Photograph of the cavity under test that was diced out
of an array of cavities. (b) Photograph of the cavity inside the air-tight
metal enclosure. (c) Phase noise measurement on a 1550 nm laser stabilized to
the cavity. Blue: free running laser noise. Red: phase noise at 9 months after
bonding. Black: estimated thermal noise limit of the cavity. (d) The
corresponding measured Allan deviation.

We fabricated an array of nine high-finesse, 35-cm-ROC mirrors on a 2-inch-
diameter, 2-mm-thick ULE substrate, using the technique described in [39]. The
mirror substrate was matched to a 2-inch-diameter, 3-mm-thick ULE spacer with
nine bore holes, and a high-finesse 2-inch-diameter, 2-mm-thick ULE flat. The
mirror substrates and spacer were bonded in vacuum, similarly to the procedure
described above. After bonding, the multi-cavity structure was annealed and
diced into separate individual cavities. More details on the creation of these
cavities will appear in a future publication. One of these miniature cavities
was then selected for phase noise and frequency stability testing. The
measured cavity finesse is 400,000. The cavity was mounted in a purpose-built,
air-tight enclosure made of aluminum as shown in Fig. 5b, and the whole cavity
system is placed within a passive acoustic isolation box on a vibration
isolation stage without active temperature stabilization. The experimental
results are shown in Fig. 5c and Fig. 5d. The measured phase noise is thermal-
noise-limited from about 10 Hz to about 300 Hz, reaching about -95 dBc/Hz at
10 kHz offset frequency. The Allan deviation is around $7.5\times 10^{-14}$ at
0.01 s and $6.5\times 10^{-13}$ at 1 s with a linear frequency drift of around
27 Hz/s. The higher drift rate is assumed to be primarily due to the lack of
active temperature control, and we anticipate a much lower frequency drift
rate and close-to-carrier phase noise in a well-controlled temperature
environment. Additionally, intensity noise on the laser may be contributing to
an increase in the phase noise for offset frequencies below 10 Hz, but can be
readily reduced with a laser intensity servo. Despite this, the low phase
noise can immediately be utilized in applications where short-term phase
stability is needed. For example, through the use of optical frequency
division [4], this extremely compact cavity can support 10 GHz microwave
generation with phase noise below -180 dBc/Hz at 10 kHz offset. With this
bond-and-dice approach, we foresee a path towards mass production of such
miniature ultrastable optical reference cavities.

##  5 Discussion and Conclusion

Figure 6: Performance comparison of this work with various state-of-the-art
compact optical resonators and ultrastable laser systems. (a) Comparison of
fractional frequency instability expressed as Allan deviation. (b) Comparison
of laser frequency noise (scaled to 1550 nm). On-chip Si3N4 4 m coil[22],
Fused silica microrod WGM[26], Fiber SBS[28], Fused silica 25 mm bulk FP[25],
100 m fiber delay line[18], MgF2 WGM[40], vacuum-gap 10 mm FP[31], 5 km fiber
delay line[19], vacuum-gap 25 mm FP[29]. Dashed lines represent the presence
of vacuum environment.

The context by which we may evaluate the performance of our in-vacuum-bonded
compact optical reference cavities is shown in Fig. 6. Vacuum-gap FP cavities
are now approaching the form factor associated with on-chip and whispering
gallery mode resonators, but with superior phase noise and frequency stability
performance. With the optical mode primarily propagating in vacuum, the
thermorefractive noise that dominates solid-state dielectric resonators is
largely eliminated. Furthermore, the use of ULE can provide orders of
magnitude lower thermal expansion coefficient than crystalline, optical fiber,
or on-chip SiN resonators, leading to a commensurately lower frequency drift.
We note that the lowest noise dielectric resonators require operation in high
vacuum and multilayer thermal shielding to achieve their high stability
results [27, 19].

Compared to larger vacuum-gap FP resonators, the thermal noise-limited phase
noise of our cavities is higher due to the shorter optical cavity length.
However, there are many applications for which this higher phase noise floor
does not pose a limitation, while a smaller form-factor more easily enables
out-of-the-lab use. As noted above, a compact optical reference cavity used in
conjunction with an optical frequency comb can support microwave generation
with phase noise much lower than conventional electronic sources[4], and
common-mode rejection of laser noise in many fiber sensing applications
relaxes the phase noise requirements at small offset frequencies[41].
Importantly, the higher thermal noise of compact cavities is less of an
obstacle at higher offset frequencies – our demonstrated optical phase noise
of -105 dBc/Hz at 10 kHz offset is one of the lowest at this offset frequency
of which we are aware for a cavity of any size. However, it is also worth
noting that, since the phase noise is limited by Brownian noise in the
coatings, a longer cavity should result in improved noise performance.

In-vacuum bonded cavities can be further engineered for improved robustness
and integration in compact laser systems. While our cavity demonstrations are
of simple cylindrical and cubic geometries, in-vacuum bonding is amenable to
other designs optimized for low acceleration sensitivity and rigid holding
[29, 30, 42, 43, 44, 45]. This will be critical for use beyond the laboratory.
Furthermore, recent developments in integrating small FP cavities with
waveguides on chip [46], when used in conjunction with low noise on-chip
lasers [47], should enable a low noise, fully integrated chip-based reference-
cavity system.

In summary, we have developed an in-vacuum bonding technique to provide
vacuum-gap ultrastable optical reference cavities that operate surrounded by
air. Despite the lack of a vacuum enclosure, the cavities perform at the
thermal noise limit determined by Brownian noise of the coatings. By combining
in-vacuum bonding with microfabricated mirrors, we created an ultrastable
reference cavity whose volume is only 0.5 mL. We anticipate this advance in
ultrastable laser technology will accelerate a variety of out-of-the-lab
applications, including mobile optical atomic clocks, photonic radar, and
sensing.

Funding This work is funded by DARPA and NIST.

Acknowledgments We thank Andrew Ludlow and the NIST Yb optical clock team for
ultrastable reference light, and Matthew Hummon and Tobias Bothwell for
helpful comments on the manuscript. Product names are given for scientific
purposes only and do not represent an endorsement by NIST.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

Supplemental document See Supplement 1 for supporting content.

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#  $\gamma$-Chiral is same as Chiral

Shrinit Singh
Department of Mathematics, Shiv Nadar Institution of Eminence, India-201314
(e-mail<EMAIL_ADDRESS>
###### Abstract

A word $w$ in a free group is called chiral if there exists a group $G$ such
that image of word map corresponding to word $w$ is not closed with respect to
inverse. Similarly a group $G$ is said to be chiral if there exists a word $w$
in free group such that $w$ exhibits chirality on the group $G$. Gordeev et
al. [2] extended the concept of chirality to introduce $\gamma$-chirality in
both cases. We show that the notion of $\gamma$-chirality is equivalent to
chirality.

Key Words: Free group, Word map, Chiral word
AMS(2020):20F10

## Introduction

Let $F_{d}$ be a free group with a basis $x_{1},\ldots,x_{d}$. By a word $w\in
F_{d}$, we mean a reduced word in $F_{d}.$ Each word $w\in F_{d}$ defines a
map from $G^{d}$ to $G$ which is an evaluation map. For example, if
$w=x_{1}x_{3}^{2}\in F_{3},$ then $w:G^{3}\to G$ defined as
$w((g_{1},g_{2},g_{3}))=g_{1}g_{3}^{2}.$ We denote the image of the word map
$w(G^{d})$ as $G_{w}.$

Let $G$ be a group. An anti-automorphism $\delta$ of the group $G$ is a
bijective map from $G$ to $G$ such that $\delta(ab)=\delta(b)\delta(a)$ for
all $a,b\in G$. Let $AA(G)$ be the set all anti-automorphisms of $G$ and
$A(G)$ be the set of all automorphisms of $G.$ Then it is easy to see that
there is a one to one correspondence between $A(G)$ and $AA(G).$ In
particular, if $f\in A(G),$ then $g:G\to G$ defined as $g(x)=f(x^{-1})$
belongs to $AA(G)$ and vice versa.

In [2] Gordeev et al. generalized the concept of chirality to
$\gamma$-chirality in following way.

######  Definition 1 ([2]).

  1. 1.

A word $w\in F_{d}$ is a $\gamma$-chiral, where $\gamma\in AA(F_{d})$, if
there exists a group $G$ such that $G_{w}\neq G_{\gamma(w)}.$ In this case,
chirality of word $w$ is a special case of $\gamma$-chirality for
$\gamma(u)=u^{-1}$ for all $u\in F_{d}.$

  2. 2.

Let $G$ be a group, and let $\gamma\in AA(G)$. For every word $w\in F_{n}$,
define $w_{\gamma}:G^{(n)}\rightarrow G$ by
$w_{\gamma}(g_{1},\cdots,g_{n})=\gamma(w(g_{1},\cdots,g_{n}))$. Then $G$ is
said to be $\gamma$-chiral if there exists a word $w\in F_{d}$ for some $d$
such that the images of $w$ and $w_{\gamma}$ are different. In this case also,
chirality of group $G$ is a special case of $\gamma$-chirality for
$\gamma(g)=g^{-1}$ for all $g\in G$.

In this work, our goal is to establish that a word or group is chiral if and
only if it is $\gamma$-chiral for an anti-automorphism $\gamma$ if and only if
it is $\gamma$-chiral for all anti-automorphism $\gamma$.

## Main Result

First we will prove a key lemma.

######  Lemma 2.

Let $F_{d}$ and $G$ be a free group with basis $x_{1},x_{2},\ldots,x_{d}$ and
a group respectively. Let $\theta$ and $\zeta$ be automorphisms of $F_{d}$ and
$G$ respectively. Then for a word $w\in F_{d}$, we have
$G_{w}=G_{\theta(w)}=\zeta(G_{w}).$

###### Proof.

Let $w(g_{1},g_{2},\ldots,g_{d})=g\in G_{w}$ for
$g,g_{1},g_{2},\ldots,g_{d}\in G.$ It is easy to see that

$\theta(w(\theta^{-1}(x_{1}),\theta^{-1}(x_{2}),\ldots,\theta^{-1}(x_{d}))=w.$

Hence

$\theta(w(\theta^{-1}(x_{1})(g_{1},\ldots,g_{d}),\theta^{-1}(x_{2})(g_{1},\ldots,g_{d}),\ldots,\theta^{-1}(x_{d})(g_{1},\ldots,g_{d})))=w(g_{1},\ldots,g_{d}).$

Therefore $G_{w}=G_{\theta(w)}$.

We also have

$\zeta(w(\zeta^{-1}(g_{1}),\zeta^{-1}(g_{2}),\ldots,\zeta^{-1}(g_{d}))=g.$

Hence $\zeta(G_{w})\subseteq G_{w}.$ Similarly $\zeta^{-1}(G_{w})\subseteq
G_{w}.$ Applying $\zeta$ will give $\zeta(G_{w})=G_{w}.$ Hence
$G_{w}=G_{\theta(w)}=\zeta(G_{w}).$ ∎

######  Theorem 3.

Let $w\in F_{d},$ for some $d\in{\mathbb{N}}$ and $\gamma\in AA(F_{d}).$ Then
$w$ is chiral if only if $w$ is $\gamma$-chiral.

###### Proof.

First we start with a small observation. If $\gamma\in AA(F_{d}),$ then there
exists $\theta\in A(F_{d})$ such that $\theta(w^{-1})=\gamma(w).$ But we have
$G_{\theta(w)}=G_{w}$ by lemma 2. Hence, for every group $G,$
$G_{\gamma(w)}=G_{w^{-1}}.$

Suppose $w\in F_{d}$ is chiral. Then there is a group $G$ such that $G_{w}\neq
G_{w^{-1}}.$ Consequently $G_{w}\neq G_{\gamma(w)}$ for every $\gamma\in
AA(F_{d}).$ Hence $w$ is $\gamma$-chiral for every $\gamma\in AA(G).$

If the word is $\gamma$-chiral, then $G_{w}\neq G_{\gamma(w)}.$ But
$G_{\gamma(w)}=G_{w^{-1}}$. This implies $G_{w}\neq G_{w^{-1}}$ proving that
$w$ is chiral. ∎

Now we prove the equivalence for groups. We follow the same steps as in the
above proof. Suppose $G$ is a group and $\gamma$ is an anti-automorphism of
the group $G$. Then there exists an automorphism $\zeta$ of $G$ such that
$\zeta(g^{-1})=\gamma(g)$ for all $g\in G$. Images of word maps are preserved
by automorphism by lemma 2, so we have $G_{w}=\zeta{(G_{w}})$.

Assume $G$ is $\gamma$-chiral, then there exists a word $w$ such that
$G_{w}\neq\gamma(G_{w}).$ Thus
$\gamma(G_{w})=\zeta(G_{w})^{-1}=\zeta(G_{w^{-1}})=G_{w^{-1}}$. This implies
$G_{w}\neq G_{w^{-1}}$ proving $G$ is chiral. We can reverse the process to
prove the converse. Hence we proved the following.

######  Theorem 4.

Let $G$ be a group and let ${\gamma}\in AA(G).$ Then $G$ is chiral if and only
if $G$ is ${\gamma}$-chiral.

######  Definition 5.

A finite group $G$ with an anti-automorphism $\gamma$ is said to be weakly
$\gamma$-chiral if there exists $g\in G$ and $w\in F_{d}$ such that the
cardinalities of fibres of $g$ with respect to word map $w$ and $w_{\gamma}$
are distinct. We say the pair is $(G,w)$ is weakly $\gamma$-chiral.

######  Remark 6.

It is worth noting that the property of weak $\gamma$-chirality is not
dependent on the specific anti-automorphism $\gamma$ chosen.

Gordeev et al. [2] further posed an intriguing question whether there exists a
group $G$ along with an anti-automorphism $\gamma$ of $G$ such that $G$
exhibits both $\gamma$-achirality and weakly $\gamma$-chirality. Notably, this
question has been addressed by William Cocke [1] for anti-automorphism defined
by $\gamma(g)=g^{-1}$, but it is essential to emphasize that for all anti-
automorphisms $\gamma$ considered, the property of weak $\gamma$-chirality
remains invariant.

## References

  * [1] William Cocke.  Idle Words:Word Maps on Finite Groups.  Ph.D thesis, 2019.
  * [2] Nikolai Leonidovich Gordeev, Boris Èmmanuilovich Kunyavskiĭ, and Eugene Bīrisovich Plotkin.  Geometry of word equations in simple algebraic groups over special fields.  Russian Mathematical Surveys, 73(5):753, 2018.