[affiliation=1,2]GasserElbanna [affiliation=2,3]ZohrehMostaani
[affiliation=2]MathewMagimai.-Doss

# Predicting Heart Activity from Speech using Data-driven and Knowledge-based
features

###### Abstract

Accurately predicting heart activity and other biological signals is crucial
for diagnosis and monitoring. Given that speech is an outcome of multiple
physiological systems, a significant body of work studied the acoustic
correlates of heart activity. Recently, self-supervised models have excelled
in speech-related tasks compared to traditional acoustic methods. However, the
robustness of data-driven representations in predicting heart activity
remained unexplored. In this study, we demonstrate that self-supervised speech
models outperform acoustic features in predicting heart activity parameters.
We also emphasize the impact of individual variability on model
generalizability. These findings underscore the value of data-driven
representations in such tasks and the need for more speech-based physiological
data to mitigate speaker-related challenges.

###### keywords:

Speech processing, Biological signals, ECG, Heart Rate, Self-supervised
learning

##  1 Introduction

In the contemporary era of telemedicine, harnessing ubiquitous biological
signals such as human speech has been of great importance. Thanks to the
advances in machine learning, a significant body of research has emerged,
exploring paralinguistic analysis in speech. This multi-faceted discipline
encompasses tasks such as detecting various stress indicators [1, 2],
predicting emotional states [3], and modeling physiological parameters such as
heart activity [4], respiration patterns [5], and skin conductance responses
[6].

There has been an effort to examine the association between speech signals and
heart activity. Orlikoff et al. [7] in one of the very early studies showed
that cardio vascular system can influence the vocal fundamental frequency (F0)
indicating that the absolute F0 perturbation (jitter) during a sustained
phonation could vary between 0.5% to 20%. In other studies, heart activity was
studied in relation with speech in different emotional states. Williams et al.
[8] demonstrated that the emotion variation might cause an increase in blood
pressure (BP), heart rate (HR), sub-glottal pressure, and the depth of
respiratory movements. James et al. [9] suggested a strong correlation between
speech, emotion, and heart rate using spectral features from speech. Smith et
al. [10] showed that HR increases when the person is speaking compared to when
they are silent and this increase is greater when they are frustrated. In
another study [11], the HR was predicted using linear models in different
emotional states. Jati et al. [12] predicted physiological signals from speech
during stressful conversations.

However, the influence of inter- and intra-individual variability on the
efficacy of speech-based models remains relatively unexplored. For example in
[6, 13, 10, 14], it has been shown that acoustic features, in particular
spectral features, can be used as good predictors of heart activity parameters
like beats per minute (BPM) in the context of regression and classification
tasks. Nevertheless, the speaker-dependent data split used in these efforts
might be confounding the reported performance since both training and testing
sets have included data samples belonging to the same speakers. Furthermore,
an interesting study conducted by [4] has addressed the speaker-dependency in
their experimental setup showing the superior performance achieved using
speaker-dependent split over leave-one-speaker-out (LOSO) approach. However,
the level of variability between and within speakers has not been examined.

Part of the challenge in properly addressing these confounding parameters has
been manifested in the lack of large speech corpora that includes recorded
biological signals, leading to overfitting models on speakers' samples. One
way to alleviate this limitation is to pre-train models on large scale corpora
and evaluate the pre-trained models on downstream paralinguistic tasks with
limited data samples. One major breakthrough in this domain is introducing
self-supervised models (SSMs). These are models trained to optimize a certain
task objective without using labels. Consequently, several benchmarks have
evaluated the predictive power of these data-driven representations on
paralinguistic tasks [3, 15] as well as speech [16] and audio tasks in general
[17]. Additionally, SSMs outperformed acoustic and knowledge-based features on
multiple audio-based tasks.

Figure 1: Training pipeline for predicting BPM and HRV values from knowledge-
based and data-driven speech representations.

In this study, we evaluate, for the first time, the ability of SSMs to be used
for predicting heart activity parameters such as BPM and heart rate
variability (HRV) and assess its predictive power compared to acoustic
features. Importantly, we highlight the impact of both inter- and intra-
individual variability on performance and generalizability. We further examine
the optimal context window duration of speech for this task. Lastly, we
pinpoint the salient features for this task and compare it to previous work in
literature. Collectively, these analyses highlight the importance of
collecting data systematically to provide better understanding of the
underlying physiological variability across individuals.

The rest of the paper is organized as follows. We introduce our proposed
method and the database used in Section 2. The results has been given in
Section 3. We conclude the paper in Section 4.

##  2 Materials & Methods

###  2.1 Dataset

In this work, we use Ulm-TSST database presented in the MuSe challenges [18,
19]. This corpus included 69 German-speaking subjects (49 Females and 20
Males) performing free speech tasks following the Trier Social Stress Test
(TSST) protocol [20]. Multiple physiological signals such as ECG, BPM,
respiration (RESP), and EDA were captured during the speech tasks at a
sampling rate of 1 kHz. This dataset was originally used to study emotions
(i.e., arousal and valence) of people in stressful dispositions. However, in
this paper, we are only interested in the recorded heart activity (ECG).

###  2.2 Data Preprocessing

The utterances were originally acquired with 6 channels. We select the channel
that featured the highest loudness (the first channel). Then, we re-sample all
mono-channel utterances to 16 kHz and standardize them. Subsequently, we chunk
the data into clips of varying window sizes ranging from 3 to 5 seconds with a
hop size of 500 ms. The ECG signals are preprocessed using NeuroKit package
111https://neuropsychology.github.io/NeuroKit/functions/ecg.html. BPM is
computed from the ECG signal using the same package. Then, for each audio
clip, we extract the corresponding ECG signal and compute HRV. Lastly, we
compute the average value for BPM and HRV for each clip to have one value per
audio sample/clip.

Figure 2: Performance of different speech features in both speaker conditions.
The reported distributions show the evaluation across multiple regressors and
window sizes as well as the performance for predicting both targets (i.e. BPM
and HRV).

###  2.3 Speech Feature Extraction

We compared the performance of knowledge-based features against speech
representations generated from a pre-trained self-supervised model.

Knowledge-based: We use openSMILE [21], an open source toolkit for extracting
low-level descriptors from audio utterances. We extract two commonly-used
feature sets from openSMILE, namely eGeMAPS (88 features) [22] and ComParE
(6373 features) [23].

Data-driven: We use Hybrid BYOL-S model [2], a self-supervised model derived
from Bootstrap Your Own Latent (BYOL-A) learning framework [24]. BYOL-A learns
general-purpose audio representations from juxtaposing two augmented views of
a single input utterance. The two augmented views are fed to two networks, an
online and a target network. The task objective involves the online network
predicting the generated representations from the target network. The new
variant of BYOL-A (i.e. Hybrid BYOL-S) is a speech-specific derivation that
trains the online network to predict data-driven representations from the
target network as well as knowledge-based representations (ComParE features)
simultaneously yielding robust speech embeddings of size 2048. We selected
Hybrid BYOL-S due to its superior performance on multiple audio downstream
tasks [25] as part of the HEAR 2021 challenge [17].

###  2.4 Evaluation Pipeline

We evaluate the performance of the three feature sets to predict BPM and HRV.
To examine the generalizability of the methods to various speakers, we create
two data splits, one speaker-independent split where we used 70% of the
speakers (N=48) for training and hold out 30% (N=21) for testing. The second
split is speaker-dependent in which we keep 70% of the clips for each speaker
in the training set and the remaining clips per speaker are evaluated.
Additionally, we train speaker-specific regression models for each speaker
where the model is only trained on data samples from one speaker and these
samples are split into 70% and 30% for training and testing, respectively. For
each experiment, multiple regressors are used such as Ridge regression, Random
Forrest (RF), Gradient boosting tree (GBT), and multi-layer perceptron (MLP).
We run a grid-search for hyperparameters that are model-specific. In case of
speaker-independent, we run a 5-fold group shuffle split where speakers in the
training set were further divided into train and validation sets (70% and 30%,
respectively). Thus, ensuring that the validation set includes unseen speakers
during training with cross validation. On the other hand, when running
speaker-dependent experiments, we use a 5-fold time series split where the
training samples for all speakers are split into train and validation (70% and
30%, respectively) while considering the temporal dependency between the data
samples. Lastly, for speaker-specific, we train regression models on data
samples from a single speaker using 5-fold time series split as well. After
training the regression models, we report performance on a held out test set
using coefficient of determination ($R^{2}$) score as well as Pearson's
correlation coefficient, as illustrated in Figure 1.

##  3 Results & Analysis

Figure 2 illustrates the performance of candidate speech representations on
predicting BPM and HRV under two different speaker conditions for unseen test
sets; speaker-dependent and speaker-independent protocols. We report the
$R^{2}$ and Pearson's correlation between the predicted and the ground truth.
The plotted distributions highlight the performance across different
regressors, targets (i.e., BPM and HRV), and window sizes (i.e., 3, 4, and 5
seconds). We show that Hybrid BYOL-S outperforms knowledge-based features in a
speaker-dependent setting. Whereas, all representations perform equally poorly
in the speaker-independent setting. This result highlight the limitations of
generalizability of speech features for this downstream task.

Moreover, we study the effect of context window duration on performance.
Figure 3 shows that increasing the window size improves performance in all
feature candidates. This figure reports the Pearson's correlation on a
speaker-dependent test set using GBT (best-performing regressor) for both
targets. Importantly, we observe significant improvement between 3 sec window
and 4 sec window while between 4 sec and 5 sec duration performance is
relatively overlapping.

Figure 3: Performance of different speech features with varying context window
duration. The reported distributions show the evaluation using speaker-
dependent and GBT regression model for predicting both targets (i.e., BPM and
HRV). Figure 4: Predictions from GBT model using Hybrid BYOL-S features with 5
sec window size. Predictions are shown for speakers 52, 13, and 46,
respectively. Figure 5: Performance of Hybrid BYOL-S features for speaker-
specific protocol across all window sizes (3, 4, 5 seconds) of audio and a GBT
regressor model. The figure shows a random sample of 20 speakers for both
targets BPM and HRV

Given the discrepancy between both speaker-related protocols, we further train
regression models on data samples from a single speaker resulting in building
a model per speaker (i.e., speaker-specific protocol). Figure 5 demonstrates
the high variability in performance across a randomly selected sample of 20
speakers. In this figure, we report the Pearson's correlation evaluated on
Hybrid BYOL-S to predict BPM and HRV across different window sizes.

We further plot the predictions from three speakers; the speaker exhibiting
the lowest correlation (i.e., speaker 52), the speaker with the highest
positive correlation (i.e., speaker 46) and the speaker with the highest
negative correlation (i.e., speaker 13), as shown in Figure 4. This figure
provides insights regarding the model's ability to capture the variability
within a single speaker.

Furthermore, we opted to pinpoint the salient acoustic features for predicting
heart activity. Thus, we perform feature importance on the trained GBT
regression model using eGeMAPS and ComParE features. Figure 6 shows the top 10
features for each feature set that contribute to predicting both BPM and HRV.
The feature importance is computed from the best estimator and ranked
accordingly.

Figure 6: Feature importance showing top 10 acoustic features for both targets
BPM and HRV extracted from both openSMILE feature sets eGeMAPS and ComParE.

##  4 Discussion & Conclusions

In this paper, we studied the robustness of acoustic and data-driven features
to predict heart activity parameters. We showed that Hybrid BYOL-S, a self-
supervised model, outperformed acoustic features on this downstream task. This
finding aligns with previous work showing the feasibility of using speech
representations as predictors for heart activity [6, 14]. Additionally, we
highlight the significance of leveraging data-driven, in particular SSMs,
representational spaces for better predictive power.

Nevertheless, we noticed that all features performed equally poorly in the
speaker-independent protocol as shown in Figure 2. This result exposes the
limitations of these features and their sensitivity to inter-individual
variability confirming the work done in [4]. Importantly, this dataset was
originally collected to study speech as a biomarker for stress. Accordingly,
one might hypothesize that the high variability in heart activity might have
been a consequence of variability in stress response across individuals.
Hence, the inability of models to generalize across speakers. Thus, it is
crucial for future work to consider emotional states and cognitive aspects as
confounding variables for predicting heart activity from speech. We further
observe varying correlation scores when training models on a single speaker
data as in Figure 5. The poor performance in some speakers indicates that even
intra-individual variability plays a role as well in hindering the model to
properly predict unseen data samples from the same speaker which has not been
previously discussed. Moreover, we demonstrated that increasing the context
window duration more than 3 sec provides significantly better predictions.

Lastly, we ran feature importance as shown in Figure 6 and found that spectral
features and loudness were the most important acoustic features for this
downstream task which line up with the previous findings in literature [7, 13,
6]. These results together provide insights on the robustness and limitations
of acoustic and data-driven features in studying heart activity as well as
shed light on the importance of adding the mental state as an essential
dimension to study the relationship between speech and physiological signals
such as heart activity.

##  5 Acknowledgments

This work was partially funded by the Swiss National Science Foundation
through the project Towards Integrated processing of Physiological and Speech
signals (TIPS), grant no. 200021_188754. The first author carried out this
work as an intern at the Idiap Research Institute, Martigny, Switzerland.

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  *[]:
  *[GWs]: Gravitational Waves
  *[GR]: General Relativity
  *[BBH]: Binary Black Hole
  *[BNS]: Binary Neutron Star
  *[BH-NS]: Black Hole-Neutron Star
  *[GW]: Gravitational-Wave
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  *[BCJ]: Bern-Carrasco-Johansson
  *[SQED]: Scalar Quantum Electrodynamics
  *[QED]: Quantum Electrodynamics
  *[NLO]: Next to Leading Order
  *[LO]: Leading Order
  *[irreps.]: irreducible representations
  *[SSC]: Spin Supplementary Condition
  *[QCD]: Quantum Chromodynamics
  *[KK]: Kaluza-Klein
  *[DoF]: Degrees of Freedom
  *[CoM]: Center of Mass
  *[LHS]: Left Hand Side
  *[SYM]: Super Yang-Mills
  *[RHS]: Right Hand Side
  *[HCL]: Holomorphic Classical Limit
  *[GME]: Gravitational Memory Effect
  *[PW]: Plane Wave


††thanks: Present address: Laser Centre Vrije Universiteit Amsterdam, The
Netherlands

# Frequency Comparison of Two High-Accuracy Al+ Optical Clocks

C. W. Chou<EMAIL_ADDRESS>D. B. Hume  J. C. J. Koelemeij  D. J. Wineland
T. Rosenband  Time and Frequency Division, National Institute of Standards and
Technology, Boulder, Colorado 80305

###### Abstract

We have constructed an optical clock with a fractional frequency inaccuracy of
$8.6\times 10^{-18}$, based on quantum logic spectroscopy of an Al+ ion. A
simultaneously trapped Mg+ ion serves to sympathetically laser-cool the Al+
ion and detect its quantum state. The frequency of the 1S0$\leftrightarrow$3P0
clock transition is compared to that of a previously constructed Al+ optical
clock with a statistical measurement uncertainty of $7.0\times 10^{-18}$. The
two clocks exhibit a relative stability of $2.8\times 10^{-15}\tau^{-1/2}$,
and a fractional frequency difference of $-1.8\times 10^{-17}$, consistent
with the accuracy limit of the older clock.

Optical clocks based on petahertz ($10^{15}$ Hz) transitions in isolated atoms
have demonstrated significant improvements over the current cesium primary-
frequency-standards at 9.2 GHz. They also shed light on fundamental physics,
such as the possible variation of physical constants. While the merits of
laser-cooled ion optical frequency standards were known HGD1982monoion ;
DJW1987LaserCoolingLimits , further developments were required to permit their
use. Sub-hertz linewidth lasers BCY1999subhertz  enabled single ions to be
probed with sufficient resolution for high-stability clock operation, and
control of external-field perturbations allowed such clocks to operate with an
inaccuracy below $10^{-16}$ HgCs2006 ; TR2008AlHg . For comparison, cesium
standards that realize the SI second have reached an inaccuracy of $3\times
10^{-16}$ 2008ParkerEFTF , and an optical lattice clock based on Sr atoms has
been reported Campbell2008  with an inaccuracy of $1.5\times 10^{-16}$. Here
we describe an Al+ ion clock with an inaccuracy of $8.6\times 10^{-18}$.

The 1S0$\leftrightarrow$3P0 transition in Al+ at 1.121 PHz is of interest due
to its low sensitivity to electromagnetic perturbations and its narrow natural
linewidth of 8 mHz Dehmelt1992 ; TR2007Al3P0observed . Al+ has the smallest
sensitivity to blackbody radiation TR2006BBRshift ; Mitroy2009BBR  among
atomic species currently under consideration for clocks, thus relaxing the
requirement on ambient temperature control. However, the absence of an
accessible allowed optical transition prevents the internal state of Al+ ions
from being detected by conventional methods, and the ion cannot currently be
directly laser-cooled. Quantum logic spectroscopy (QLS) POS2005BeAl  overcomes
these difficulties by trapping a “logic ion” that can be directly laser-cooled
together with the Al+ clock ion. The coupled motion of the two ions allows for
sympathetic cooling, as well as the transfer of the clock ion’s quantum state
to the logic ion, where the state can be measured.

The clock described here shares features with our previously-constructed Al-Be
clock TR2007Al3P0observed ; TR2008AlHg , but also includes many changes,
making a comparison of the two clocks a valuable test of systematic errors. In
the new clock, the 9Be+ logic ion has been replaced by 25Mg+, whose mass
closely matches that of 27Al+. Laser-cooling inefficiencies due to mass-
mismatch are thus suppressed. The Al-Mg ion trap is a linear Paul trap built
from tool-machined all-metal electrodes (Fig. 1). This construction differs
from the Al-Be trap that was built from laser-machined and gold-coated alumina
electrodes Rowe2002 , and it exhibits reduced RF-micromotion-inducing electric
fields.

Figure 1: Setup for comparing the frequencies of the two Al+ clocks. The 4th
harmonic of a fiber laser is locked to the Al+ 1S0$\leftrightarrow$3P0 clock
transition in the Al-Mg apparatus with a fixed offset frequency (applied to
AOM1). Another laser beam derived from the same laser probes the clock
transition in the Al-Be apparatus, where the laser frequency is locked to the
clock transition in a separate digital feedback loop that controls AOM2. The
record of frequencies applied to AOM2 represents the difference in clock
frequencies. BS: beam splitter; AOM: acousto-optic modulator; x2: frequency
doubler.

QLS with a Mg+ logic ion proceeds in the same way as with Be+, but the ground-
state-cooling process CM1995GScooling  has been modified to require only two
lasers rather than three TwoLaserGSC . This cools the out-of-phase axial
motional mode to an average quantum number of $\bar{n}<0.05$. It also enables
quantum-non-demolition transfer (QNDT) of the Al+-clock-ion state to Mg+ with
approximately 80 % fidelity in a single QNDT repetition and over 99 % fidelity
after typically five QNDT repetitions DBH2007detection .

The trap utilized in the new Al+ clock has blade-shaped gold-coated beryllium-
copper electrodes (Fig. 1) whose edges are approximately 400 $\mu$m from the
ions. The 25Mg+ logic ion ($I=5/2$) is manipulated with light of 279.5 nm
wavelength from two independent lasers. A frequency-doubled dye laser
resonantly drives $|^{2}S_{1/2},F\in\\{2,3\\}\rangle$ $\rightarrow$
${}^{2}P_{3/2}$ cycling and repumping transitions, while a frequency-
quadrupled fiber laser is detuned by 40-60 GHz to drive Raman transitions
between the 25Mg+ hyperfine qubit states. Both ions are created via multi-
photon-ionization of neutral atoms from ovens. A 396 nm diode laser produces
27Al+ Hurst1979PI , while a frequency-doubled dye laser at 285 nm creates
25Mg+ Drewsen2000PI . Typically the clock transition is probed with 150 ms
duration $\pi$-pulses with a duty cycle of approximately 65 %. The remaining
35 % is occupied with state preparation and state detection functions as well
as interleaved experiments that allow real-time measurement and reduction of
micromotion.

Table 1: Systematic effects that shift the clock from its ideal unperturbed frequency. Shifts and uncertainties given are in fractional frequency units ($\Delta\nu/\nu$). See text for discussion. Effect | Shift | Uncertainty
---|---|---
| ($10^{-18}$) | ($10^{-18}$)
Excess micromotion | -9 | 6
Secular motion | -16.3 | 5
Blackbody radiation shift | -9 | 3
Cooling laser Stark shift | -3.6 | 1.5
Quad. Zeeman shift | -1079.9 | 0.7
Linear Doppler shift | 0 | 0.3
Clock laser Stark shift | 0 | 0.2
Background gas collisions | 0 | 0.5
AOM freq. error | 0 | 0.2
Total | -1117.8 | 8.6

Systematic shifts of the Al-Mg clock are listed in Table 1. Two types of
residual motion cause time-dilation shifts: micromotion near the trap drive
frequency of $\nu_{RF}=59$ MHz, and harmonic-oscillator (secular) motion at
the ion’s normal mode frequencies (Table 2). In both cases the clock frequency
shifts by $\frac{\Delta\nu}{\nu}=-\frac{\langle
v^{2}\rangle}{2c^{2}}\left(1+(\frac{f}{\mathrm{400MHz}})^{2}\right)$, where we
add to the relativistic time-dilation $\langle v^{2}\rangle/(2c^{2})$ a
frequency-dependent term that corresponds to the Stark shift from the motion-
inducing electric fields. Here $\langle v^{2}\rangle$ is the mean-squared
Al+-ion velocity and $f$ is the frequency of motion. For the highest motional
frequencies the Stark shift correction is 2 %.

Table 2: Motional modes of the Al-Mg ion pair. For each of the six normal modes, the oscillation frequency and zero-point motional amplitude for each ion is listed. The trap axis corresponds to $\hat{z}$, and $\hat{x},\hat{y}$ are two orthogonal radial directions whose orientation is determined by the trap geometry. Also shown are the calculated ($\bar{n}_{C}$) and measured ($\bar{n}_{M}$) Doppler-cooled average motional quantum numbers, and the time-dilation (TD) per motional quantum as well as the total TD per mode. f | [MHz] | 6.53 | 5.66 | 5.20 | 4.64 | 3.41 | 3.00
---|---|---|---|---|---|---|---
25Mg+ | [nm] | 4.9$\hat{y}$ | 2.9$\hat{y}$ | 4.6$\hat{z}$ | 5.5$\hat{x}$ | -4.2$\hat{x}$ | 5.6$\hat{z}$
27Al+ | [nm] | 2.6$\hat{y}$ | -5.0$\hat{y}$ | -4.1$\hat{z}$ | 3.5$\hat{x}$ | 6.2$\hat{x}$ | 5.8$\hat{z}$
$\bar{n}_{M}$ |  | 2.9 | 4.5 | 3.4 | 6.3 | 10.0 | 7.0
$\bar{n}_{C}$ |  | 3.3 | 3.8 | 3.4 | 5.9 | 8.0 | 5.9
TD/quantum | $[10^{-18}]$ | 0.226 | 0.731 | 0.197 | 0.290 | 0.771 | 0.133
Total TD | $[10^{-18}]$ | 0.77 | 3.66 | 0.77 | 1.97 | 8.10 | 1.00

Excess micromotion (EMM) refers to the rapid ion motion at $\nu_{RF}$
Berkeland1998MM . It is caused by electric fields that force the ion away from
the RF-minimum of the ion trap, or phase shifts between trap electrodes that
cause the RF-fields to be non-zero at the pseudo-potential minimum. We measure
the amplitude of this motion at $\nu_{RF}$ by observing the motional-sideband
strength of the Al+ 1S0 $\rightarrow$ 3P1 transition in three orthogonal
directions. For small amplitude of motion, the time dilation shift is
$\Delta\nu/\nu=-|\vec{\eta}\nu_{RF}/\nu_{L}|^{2}=-2.8\times
10^{-15}|\vec{\eta}|^{2}$, where $\nu_{L}=1.12$ PHz is the probe laser
frequency, and $\vec{\eta}=(\eta_{1},\eta_{2},\eta_{3})$ is the measured EMM
Lamb-Dicke parameter (the ratio of sideband and carrier Rabi rates) for the
three orthogonal directions. Typical values for $\eta_{1,2,3}$ are 0.01 to
0.04. We note that this EMM measurement method detects slow electric field
fluctuations such as those caused by the migration of photo-electrons, as well
as faster fluctuations caused by periodic line noise (50 to 60 Hz).
Fluctuations that are shorter than the laser probe period of 0.05 to 0.1 ms
will not be detected.

In a perfect linear Paul trap EMM along the trap axis does not occur, but
imperfections in the ion-trap geometry can lead to axial EMM . For the Al-Mg
trap there exists a sharp minimum of axial EMM at one spatial location. The
Al+ ion is held at this point, but random background-gas collisions cause the
Al-Mg ion pair to spontaneously re-order, which causes the Al+ ion to move by
3 $\mu$m every 200 s, on average. The Al+ ion is maintained at the position of
minimal micromotion by adjusting the electrode voltages every 10 s to force
the ion-pair into the desired order. When the ions are in the wrong order
(about 5 % of the time), the Al+ ion experiences excess axial micromotion and
a corresponding clock shift of $\Delta\nu/\nu=-2.7\times 10^{-17}$. This
additional shift is included in Table 1.

During each Al+ clock interrogation pulse, the Mg+ ion is simultaneously
Doppler cooled by a laser that is tuned 21 MHz below the
$|^{2}S_{1/2},F=3,m=-3\rangle$ $\rightarrow$ $|^{2}P_{3/2},F=4,m=-4\rangle$
cycling transition. The amplitude of secular motion (corresponding to the
motional temperature) is extracted from the ratio of amplitudes for the red-
and blue-sidebands of a Raman transition for all six normal modes
CM1995GScooling . Measured values are shown in Table 2, together with values
calculated from laser-cooling theory for a single ion Itano1982Cooling . The
single-Mg+-ion Doppler cooling limit is also valid when applied to each of the
six normal modes of the Al-Mg ion pair. The measured and calculated motional
quantum numbers agree within the measurement uncertainty, and we consider the
stated 30 % uncertainty for this shift to be a conservative limit.

The Mg+ Doppler cooling laser beam maintains the Al-Mg ion pair at a constant
motional temperature during the clock interrogation pulse, but because it also
overlaps the Al+ ion, it causes an AC Stark shift by coupling off-resonantly
to allowed transitions that connect to the ground (3s${}^{2})$1S0 and excited
(3s3p)3P0 clock states. Following the evaluation of the blackbody radiation
shift TR2006BBRshift , we estimate the differential clock polarizability at
$279.5$ nm as $\Delta\nu/\nu=(-3.5\pm 0.6)\times 10^{-17}S$, where
($S=I/I_{S}$) is the saturation parameter for 25Mg+ ($I_{S}\approx 2470$
W/m2). The intensity $I$ of the Mg+ Doppler cooling laser is estimated from
the rate at which this laser repumps the $|^{2}S_{1/2},F=2,m=-2\rangle$ dark
hyperfine ground state. The ion fluorescence photo-multiplier counts $F(t)$
collected in a duration $t$ due to repumping of the dark state may be written
as $F(t)=b(t+\tau(e^{-t/\tau}-1))$ where $\tau=(0.217/S)$ ms and $b$ is the
bright-state counting rate. We extract $S$ by fitting $F(t)$ to the observed
ion fluorescence. Typically, we measure $\tau=2.1\pm 0.8$ ms, and find
$\Delta\nu/\nu=(-3.6\pm 1.5)\times 10^{-18}$.

Another Stark shift is caused by thermal blackbody radiation. The temperature
of the Al-Mg ion trap is measured with two platinum sensors at opposite ends
of the trap structure which are expected to be at temperature extremes. Heat
is removed primarily through thermal conduction at one end of the trap where
we measure a temperature of 35 ∘C. At the higher thermal resistance trap-end
the temperature is 40 ∘C, and the thermal radiation field impinging upon the
ion is bounded by this maximum temperature and the laboratory room temperature
of 22 ∘C: $T_{ion}=(31\pm 9)$ ∘C.

During operation, the Al+ clock servo alternates between probing of the
$|^{1}S_{0},m_{F}=5/2\rangle$$\leftrightarrow$$|^{3}P_{0},m_{F}=5/2\rangle$
and
$|^{1}S_{0},m_{F}=-5/2\rangle$$\leftrightarrow$$|^{3}P_{0},m_{F}=-5/2\rangle$
transitions every 5 s, and the apparatus synthesizes an average of these two
frequencies to eliminate first-order Zeeman shifts Madej1998 ;
TR2007Al3P0observed . Each transition’s resonance is probed several times at
the high- and low-frequency half-maximum points to derive a frequency-
correction signal. The frequency-difference between the transitions is
proportional to the mean magnetic field $\langle B\rangle$, which allows an
accurate estimate of the quadratic Zeeman shift due to the quasi-static
quantization field of typically $\langle B\rangle=0.1$ mT. However, the
quadratic Zeeman shift is proportional to $\langle B^{2}\rangle=\langle
B\rangle^{2}+B_{AC}^{2}$, where $B_{AC}^{2}$ is the variance of the magnetic
field about its mean. The dominant sources of varying magnetic fields are
currents at $\nu_{RF}$ in conductors near the ion. We vary the trap RF drive
power $P$ and measure the frequency of the hyperfine clock transition in 25Mg+
$|F=3,m_{F}=0\rangle\rightarrow|F=2,m_{F}=0\rangle$ near 1.789 GHz, which has
a strong quadratic dependence on the magnetic field, to find
$B_{AC}^{2}=1.45\times 10^{-12}(P/\textrm{W})$ T2. For Al+ clock operation
$P=15$ W, and $B_{AC}^{2}=2.17\times 10^{-11}\textrm{T}^{2}$, which alters the
quadratic Zeeman shift by $\Delta\nu/\nu=(-1.4\pm 0.3)\times 10^{-18}.$

Other potential systematic shifts are listed in Table 1. When stabilizing the
clock laser to the ion we probe the clock transition alternately with counter-
propagating laser beams to observe and cancel potential first-order Doppler
shifts TR2008AlHg . For the clock comparison described below, we observe a
differential shift for the two probe directions of $(1.2\pm 0.7)\times
10^{-17}$ but this effect is suppressed by taking the average. The suppression
factor is limited because the atomic line-shape and hence the servo gains
differ slightly for the two probe directions. During clock operation the mean
fractional gain imbalance was 1.5 %, thereby reducing the possible first order
Doppler shift to $3\times 10^{-19}$.

We have looked for Stark shifts due to the clock pulse itself by raising the
intensity of the probe beam in one clock, and comparing the frequency to that
produced by the other Al+ clock. With an increase of 40 dB in the clock probe
intensity, we observe no statistically significant frequency difference with a
fractional uncertainty of $2\times 10^{-15}$, constraining any effect that
scales linearly with this intensity to $2\times 10^{-19}$. We observe a rate
of background gas collisions similar to the Al-Be clock and assign a $5\times
10^{-19}$ uncertainty to this potential shift TR2008AlHg . Thermally induced
frequency errors from beam switching AOMs (Fig. 1) have been evaluated
previously TR2008AlHg , and this uncertainty is reduced to $2\times 10^{-19}$
by operating the AOMs at less than 1 mW.

Figure 2: Clock stability. Fractional frequency uncertainty vs. averaging
period ($\tau$) for a comparison between the two Al+ clocks (10700 s
duration). Overlapping Allan deviation and N-sample standard deviation are
shown Riley2008Stability . For each comparison measurement the coefficient of
the $\tau^{-1/2}$ asymptote is estimated and used to derive the measurement’s
statistical uncertainty. The $2.8\times 10^{-15}\tau^{-1/2}$ asymptote is
reached for averaging periods that are longer than the servo time constant of
10 s.

We directly compared the Al-Mg clock with the Al-Be clock to perform
independent tests that could reveal unaccounted-for clock shifts. The Al-Be
clock was evaluated with an accuracy of $2.3\times 10^{-17}$ TR2008AlHg , and
this evaluation remained valid during the two-clock comparison. The two clocks
were compared in 56 separate measurements, each of duration $1000-11000$ s. As
shown in Fig. 1, the frequency of the 1S0$\leftrightarrow$3P0 probe laser was
actively steered to the Al+ ion in the Al-Mg apparatus with a servo time-
constant of about 10 s. A portion of this laser light simultaneously probed
the Al+ ion in the Al-Be clock, where it was servoed to the clock transition
in a separate digital feedback loop. The frequency produced by this feedback
loop represents the frequency difference between the two clocks, and was
recorded and analyzed for stability (Fig. 2). Average frequencies of the
individual measurements corrected for known shifts are shown in Fig. 3, where
the overall weighted mean is $(\nu_{AlMg}-\nu_{AlBe})/\nu=$ $(-1.8\pm
0.7)\times 10^{-17}$. This value is consistent with the 1-$\sigma$ error of
$2.5\times 10^{-17}$ that is calculated by adding in quadrature the
inaccuracies of the two clocks and the statistical uncertainty. The
reduced-$\chi^{2}$ for this data set is $1.02$, indicating that the error bars
derived from the estimated stability correctly capture the scatter of the
data. For this data set the total in-loop servo error for the Al-Mg clock was
$\Delta\nu/\nu=6\times 10^{-19}$, and for a well-designed servo loop this
error declines faster than the statistical error. It is therefore not included
as a systematic shift in Table 1.

Figure 3: Measurements of the fractional frequency difference between the two
Al+ clocks (blue points). Error bars represent the statistical uncertainty
(see Fig. 2). The horizontal line shows the weighted mean of $-1.8\times
10^{-17}$ with an overall statistical uncertainty of $\pm$$7.0\times 10^{-18}$
(shaded band).

In summary, we have built an Al+ ion clock with a fractional frequency
inaccuracy of $8.6\times 10^{-18}$. Its frequency is compared to that of a
previously constructed Al+ clock, and the measured fractional frequency
difference of $(-1.8\pm 0.7)\times 10^{-17}$ is consistent with the inaccuracy
of the previous clock. Significantly, the statistical uncertainty in the
frequency comparison of $7.0\times 10^{-18}$ is smaller than the inaccuracy of
either clock, and the average measurement stability was $2.8\times
10^{-15}\tau^{-1/2}$ (total measurement duration: 164 967 s). This result may
be compared to other direct same-species atomic clock comparisons, where two
Cs microwave clocks have reached an agreement of $(4\pm 3)\times 10^{-16}$
Vian2005Cs , and two Yb+ single ion clocks showed agreement of $(3.8\pm
6.1)\times 10^{-16}$ Schneider2005 . Future frequency ratio measurements of
the Al+ clock and the NIST Hg+ optical clock would enable improved constraints
on present-era changes in the fine-structure constant TR2008AlHg .

We thank J. C. Bergquist for use of his stable Fabry-Perot cavities, T. M.
Fortier for optical frequency measurements, W. M. Itano for atomic structure
calculations, and S. Bickman and D. Leibrandt for helpful comments on the
manuscript. This work was supported by ONR. J.C.J.K. acknowledges support from
the Netherlands Organisation for Scientific Research (NWO). Contribution of
NIST, not subject to U.S. copyright.

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