Example Feng et al. WS-MGI

# Extremely Weak Supervision Inversion of Multi-physical Properties

Shihang Feng1 Peng Jin1 Xitong Zhang1 Yinpeng Chen2 David Alumbaugh3 Michael
Commer3 and Youzuo Lin1
1Los Alamos National Laboratory, USA
2Microsoft Research, USA
3Lawrence Berkeley National Laboratory, USA

###### Abstract

Multi-physical inversion plays a critical role in geophysics. It has been
widely used to infer various physical properties (such as velocity and
conductivity). Among those inversion problems, some are explicitly governed by
partial differential equations (PDEs), while others are not. Without explicit
governing equations, conventional multi-physical inversion techniques will not
be feasible and data-driven inversion requires expensive full labels. To
overcome this issue, we develop a new data-driven multi-physics inversion
technique with extremely weak supervision. Our key finding is that the pseudo
labels can be constructed by learning the local relationship among geophysical
properties at very sparse well-logging locations. We explore a multi-physics
inversion problem from two distinct measurements (seismic and EM data) to
three geophysical properties (velocity, conductivity, and CO2 saturation). Our
results show that we are able to invert for properties without explicit
governing equations. Moreover, the label data on three geophysical properties
can be significantly reduced by 50 times (from 100 down to only 2 locations).

##  1 Introduction

Geophysical and fluid properties (such as velocity, conductivity and
$\mathrm{CO_{2}}$ saturation) provide structural and numerical information for
various geophysical applications, e.g. assessment of oil and gas reservoirs
and sequestration of $\mathrm{CO_{2}}$ (Lucia et al.,, 2003). These properties
are obtained from surface-based geophysical measurements including seismic
(Yilmaz,, 2001), electromagnetics (EM) (Zhdanov,, 2009), gravity (Li and
Oldenburg,, 1998), etc, by geophysical inversion.

These inversion problems have been studied separately along two directions:
physics-driven and data-driven. The physics-driven methods (Zhdanov et al.,,
2000; Virieux and Operto,, 2009; Feng and Schuster,, 2017, 2019; Chen et al.,,
2020) are applicable for seismic$\rightarrow$velocity and
EM$\rightarrow$conductivity by leveraging the known PDE, which is converted as
a forward modeling operator such that the input, velocity or conductivity
properties, is a function of seismic or EM data as output. Based on the
forward modeling, velocity and conductivity can be iteratively optimized. The
data-driven methods apply to the inversion problems by leveraging deep neural
networks to learn a correspondence from geophysical measurements to
geophysical properties (Araya-Polo et al.,, 2018; Wu and Lin,, 2019; Jin et
al.,, 2020; Feng et al.,, 2021). This type of work requires a large amount of
paired geophysical measurements and geophysical properties to train the
network.

Figure 1: Schematic illustration of our proposed method, which generate the
pseudo labels $\tilde{m}$ from the sparse samplings of the target property $m$
and the full labeling of the source property $v$.

Properties without explicit governing equations can be obtained with
supervised data-driven methods. However, the acquisition of the labeled data
is extremely expensive, only sparse labeled data can be acquired in the field
experiments. Sun et al., (2020) firstly present a joint inversion that
reconstructs salt geometry by combining seismic and electromagnetic data.
However, this method still relies on a large amount of labeled data, which is
impossible to be obtained in the real case.

In this work, we shift the data-driven inversion paradigm to jointly address
the following three inversion problems with extremely weak supervision. The
three inversion problems are as follows: (a) seismic$\rightarrow$velocity to
recover velocity models from seismic data, (b) EM$\rightarrow$conductivity to
recover conductivity models from EM data, and (c) seismic/EM$\rightarrow$CO2
to recover CO2 saturation models from seismic and EM data. In the first step,
a single-physics inversion is performed in an unsupervised way. In the second
step, we construct the pseudo labels by approximating the relationship between
the geophysical properties, which enable the inversion of the properties that
do not have explicit governing equations. The requirement of the multi-physics
labeled data is greatly reduced. We name our multi-physics method as Weakly
Supervised Multiple Geophysics Inversion (WS-MGI) and evaluate our methodology
on the Kimberlina reservoir data (Alumbaugh et al.,, 2021). These numerical
results demonstrate that WS-MGI can accurately reconstruct the subsurface
structures with sparsely labeled data.

##  2 Theory

Figure 2: (a) Velocity models given by UPFWI. (b) Conductivity dense
reconstruction and true conductivity model. (c) $\mathrm{CO_{2}}$ saturation
dense reconstruction and true $\mathrm{CO_{2}}$ saturation model.

###  2.1  Weakly Supervised Multiple Geophysics Inversion

Here we proposed a Weakly Supervised Multiple Geophysics Inversion (WS-MGI)
method to invert multi-physics properties (source property $v$ and target
property $m$) with sparse samplings. The samplings are well logs
$m(x=x_{k},z)$, where $x_{k}$ is the drilling location. The source property
$v$ is a property related to the geophysical measurements with PDEs while the
target property $m$ is the property without explicit governing equations. WS-
MGI is implemented in two stages.

Stage 1. Unsupervised Single Geophysical Inversion: Stage 1 is an unsupervised
inversion for the single geophysical property $v$, which had already been
proposed by Jin et al., (2022) as Unsupervised Physical-Informed Full Waveform
Inversion (UPFWI) for velocity models as $v$.

Stage 2. Pseudo Labels Building and Training: To build the pseudo labels in
Stage 2, we construct a simple regression model $\tilde{R}$ using support
vector regression (SVR) with Gaussian Kernel. The sparse sampling
$m(x=x_{k},z)$ and $v$ at its corresponding location $v(x=x_{k},z)$ are
discretized into $N$ training samples:

$\left(\left(\begin{array}[]{cc}v^{(i)}\\\
z^{(i)}\end{array}\right),m^{(i)}\right),$ (1)

where $i=1,2,...,N$. The model $\tilde{R}_{\theta}(\cdot)$ is trained by
minimizing

$\sum_{N}\left\\{\tilde{R}_{\theta}(v^{(i)},z^{(i)})-m^{(i)}\right\\},$ (2)

and then applied on $v(x,z)$ to build the dense reconstruction
$\tilde{R}_{\theta}(v(x,z),z)$. The dense reconstruction provides the global
information of $m$, but it is inaccurate due to the simplification of the
rock-physics model. To account for the inaccuracy, we add the well log data
$m(x=x_{k},z)$ as the sparse sampling. It is combined with the dense
reconstruction to composite the pseudo label:

$\tilde{m}(x,z)=\lambda_{1}\overbrace{\tilde{R}_{\theta}(v(x,z),z)}^{dense{\
}reconstruction}+\lambda_{2}\overbrace{\sum_{i}m(x,z)*\delta({x_{k})}}^{sparse{\
}sampling}$ (3)

where $\lambda_{1}$ and $\lambda_{2}$ are the weight for dense reconstruction
and sparse sampling. The dense reconstruction provides global but inaccurate
information while the sampling provides accurate but local information.

The end-to-end network takes the geophysical measurements, such as seismic and
EM data, as the inputs and generate geophysical properties $m_{pred}$. With
the pseudo labels $\tilde{m}$, the network $g$ can be trained with the loss
function $\mathcal{L}(m_{pred},\tilde{m})$ to approximate the inverse mapping
$f^{-1}(\cdot)$.

Figure 3:  Examples of profiles: the velocity profile provided by UPFWI, the
sparse samplings provided by well logs and the dense reconstructions provided
by SVR of conductivity and $\mathrm{CO_{2}}$ saturation models.

##  3 Numerical Tests

In this section, we apply this method to the Kimberlina reservoir dataset. The
original geophysical properties were developed under DOE’s National Risk
Assessment Program (NRAP) based on a potential $\mathrm{CO_{2}}$ storage site
in the Southern San Joaquin Basin of California (Alumbaugh et al.,, 2021). In
this data, there are 780 samples and each sample contains a set of seismic and
EM data as geophysical measurements, velocity, conductivity, and
$\mathrm{CO_{2}}$ saturation models as properties and two well log data that
provides CO2 saturation and conductivity. In our experiments, 750 samples are
set as training set and the rest are the validation set.

###  3.1 Kimberlina Data

The saturation, conductivity, and velocity models are with the size of
$59\times 100$. The grid is 60 $m$ in all dimensions. Two well logs are
located at 2 $km$ and 4 $km$. There are 5 seismic sources placed evenly on the
2D spatial grid over the surface with a shot interval of 1.2 $m$. Seismic data
are simulated using the finite-difference method (Moczo et al.,, 2007). Each
of them captures vibration signals as time-series data of length 1,001 with a
time spacing of 0.005 $s$. EM data are simulated by finite-difference method
(Commer and Newman,, 2008) with two sources location at $x=2.5$ $km$,
$z=3.025$ $km$ and $x=4.5$ $km$, $z=2.5$ $km$. There are 8 source frequencies
from 0.1 to 8.0 Hz and the data with each frequency has a real part and an
imaginary part. Both the seismic and EM data are collected by 100 receivers
uniformly distributed over the 2D earth surface with a receiver interval of 60
$m$.

.

###  3.2 Workflow

Stage 1: The velocity models are provided by UPFWI as in Fig. 2a. The
resolution of the UPFWI velocity models is lower than the true velocity models
due to the limitation of the frequency in full waveform inversion (Schuster,,
2017).

Stage 2: We use two well logs at $x=2$ $km$ and $x=4$ $km$ as the sparse
samplings ($\frac{1}{50}$$\times$full labels) and the UPFWI velocity models
(see Fig. 3) at the corresponding location in the training of SVR to predict
the dense reconstructions. The predicted dense reconstructions are shown in
Figs. 2b and 2c and their vertical profiles are shown in Fig. 3. We can see
the dense reconstruction are inaccurate, especially the reservoir area in the
conductivity model (see Green boxes in Fig. 2b) and the high saturation area
in $\mathrm{CO_{2}}$ saturation model (see Yellow boxes in Fig. 2c). Then we
combine the inaccurate dense reconstructions and the accurate sparse samplings
to construct pseudo labels with Eq. (3). The pseudo labels are fed into an
end-to-end network $g$ to learn the mapping from the seismic and EM data to
conductivity and CO2 saturation.

###  3.3 Main Results

The mean-square errors (MSE), mean-absolute errors (MAE), and structural
similarity (SSIM) are used for evaluating the conductivity and saturation. We
compare our methods with the supervised InversionNet method (Wu and Lin,,
2019; Zeng et al.,, 2022). There are 100 samples for the full label, we
gradually decrease the number of samples and evaluate the performance of the
methods when the sampling becomes more and more sparse. Fig. 4 compares the
results with the supervised InversionNet and our method on two scenarios:

Seismic+EM${\rightarrow}$$\mathbf{CO_{2}}$ Saturation: In this scenario,
seismic and EM data are set as the input measurement and the target property
$m$ is CO2 saturation. The ratio between the weight $\lambda_{1}$ and
$\lambda_{2}$ is set as 1. When the sampling number is less than 20
($\frac{1}{5}$$\times$full labels), the performance of the InversionNet
quickly degrades, the MAE becomes higher than 0.2, MSE increase to 0.03 and
SSIM decrease to 0.2. But our method always keeps MAE less than 0.05, MSE less
than 0.01, and SSIM higher than 0.6. Examples of the results are shown in Fig.
5. The saturation models given by InversionNet contain large amount of
artifacts in the background. The results with our WS-MGI method are consistent
with the ground truth. Moreover, the high saturation zone in the yellow box is
inverted clearly.

Seismic+EM${\rightarrow}$Conductivity: In this scenario, seismic and EM data
are the input measurement, and the target property $m$ is conductivity. The
ratio between the weight $\lambda_{1}$ and $\lambda_{2}$ is set as 10. The
relationship between the conductivity and EM data is governed by the PDE. When
the sampling number decreases, the performance of InversionNet decreases
slower than the saturation. However, our methods still have lower MAE, lower
MSE, and higher SSIM than those of the InversionNet for all the sampling
numbers. In Fig. 5, the thin layers in the blue boxes are reconstructed much
better in our result than the one obtained using InversionNet.

##  4 Conclusions

In this paper, we proposed Weakly Supervised Multiple Geophysics Inversion
(WS-MGI) that solve multi-physics inversion problem with sparse samplings.
With pseudo labels built from the sparse labeling of the properties, we can
train an end-to-end network that learns the mapping from the measurement to
the property. This network enables the inversion of geophysical properties
that only have an implicit relationship with the measurement. Moreover,
solving the multi-physics inversion in a weakly supervised way saves the
extremely high cost of the label collection, which is much more practical than
the previously existing supervised inversion methods.

Figure 4: Weakly Supervised Multiple Geophysics Inversion (ours) vs.
Supervised InversionNet (Wu and Lin,, 2019). Our method achieves better
performance e.g. lower Mean Absolute Error (MAE) and higher Structural
Similarity (SSIM).

.

Figure 5: Comparison of InversionNet and WS-MGI (ours) on inverted
$\mathrm{CO_{2}}$ saturation and conductivity models when sampling number
equal to 2.

##  5 ACKNOWLEDGMENTS

This work was funded by the Los Alamos National Laboratory (LANL) - Laboratory
Directed Research and Development program under project number 20210542MFR and
by the U.S. Department of Energy (DOE) Office of Fossil Energy’s Carbon
Storage Research Program via the Science-Informed Machine Learning to
Accelerate Real Time Decision Making for Carbon Storage (SMART-CS) Initiative.

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Current address: ]College of Natural Sciences, International Christian
University, Mitaka, Tokyo 180-8585, Japan

Current address: ]Komaba Institute for Science (KIS), The University of Tokyo,
Meguro-ku, Tokyo 153-8902, Japan

# Superconducting Acousto-optic Phase Modulator

Ayato Okada  Research Center for Advanced Science and Technology (RCAST), The
University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan  Rekishu
Yamazaki<EMAIL_ADDRESS>[  Research Center for Advanced Science and
Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo
153-8904, Japan  PRESTO, Japan Science and Technology Agency, Kawaguchi-shi,
Saitama 332-0012, Japan  Maria Fuwa  Research Center for Advanced Science and
Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo
153-8904, Japan  Atsushi Noguchi  [  Research Center for Advanced Science and
Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo
153-8904, Japan  PRESTO, Japan Science and Technology Agency, Kawaguchi-shi,
Saitama 332-0012, Japan  Yuya Yamaguchi  Network System Research Institute,
National Institute of Information and Communications Technology, 4-2-1,
Nukuikitamachi, Koganei, Tokyo 184-8795, Japan  Atsushi Kanno  Network System
Research Institute, National Institute of Information and Communications
Technology, 4-2-1, Nukuikitamachi, Koganei, Tokyo 184-8795, Japan  Naokatsu
Yamamoto  Network System Research Institute, National Institute of Information
and Communications Technology, 4-2-1, Nukuikitamachi, Koganei, Tokyo 184-8795,
Japan  Yuji Hishida  Advanced ICT Research Institute, National Institute of
Information and Communications Technology, 588-2 Iwaoka, Nishi-ku, Kobe, Hyogo
651-2492, Japan  Hirotaka Terai  Advanced ICT Research Institute, National
Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi-
ku, Kobe, Hyogo 651-2492, Japan  Yutaka Tabuchi  Research Center for Advanced
Science and Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-
ku, Tokyo 153-8904, Japan  Koji Usami  Research Center for Advanced Science
and Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku,
Tokyo 153-8904, Japan  Yasunobu Nakamura  Research Center for Advanced Science
and Technology (RCAST), The University of Tokyo, 4-6-1 Komaba, Meguro-ku,
Tokyo 153-8904, Japan  RIKEN Center for Quantum Computing (RQC), Wako, Saitama
351-0198, Japan

###### Abstract

We report the development of a superconducting acousto-optic phase modulator
fabricated on a lithium niobate substrate. A titanium-diffused optical
waveguide is placed in a surface acoustic wave resonator, where the electrodes
for mirrors and an interdigitated transducer are made of a superconducting
niobium titanium nitride thin film. The device performance is evaluated as a
substitute for the current electro-optic modulators, with the same fiber
coupling scheme and comparable device size. Operating the device at a
cryogenic temperature ($T=8$ K), we observe the length–half-wave-voltage
(length–$V_{\pi}$) product of 1.78 V$\cdot$cm. Numerical simulation is
conducted to reproduce and extrapolate the performance of the device. An
optical cavity with mirror coating on the input/output facets of the optical
waveguide is tested for further enhancement of the modulation efficiency. A
simple extension of the current device is estimated to achieve an efficient
modulation with $V_{\pi}=$ 0.27 V.

###### pacs:

77.65.Dq 78.35.+c 07.07.Mp 42.79.Jq

##  I Introduction

The advancement of optical communication technology using optical fibers is
the core of the modern ultra-high-speed networks. One of the optical devices
which play a central role in the optical communication is the optical
modulator that converts an electrical signal into an optical signal. A wide
range of electo-optic modulators (EOMs) including thin-film dielectrics
Stenger  _et al._ (2013); Wang  _et al._ (2018a), semiconductors Ogiso  _et
al._ (2017); Dong  _et al._ (2012), electro-optic polymer in slot waveguides
Palmer  _et al._ (2014), and electro-optic polymer in plasmonic waveguide Heni
_et al._ (2016) have been developed to further push its bandwidth as well as
the energy efficiency. For a conventional traveling-wave EOM, the half-wave
voltage ($V_{\pi}$), which is defined as the necessary voltage applied to
shift the optical phase by a half of the wavelength, is known as one of the
indices characterizing the performance of the optical modulator. The half-wave
voltage can be lowered in principle by extending the interaction length
between the light and the applied microwave signal. However, in practice, the
microwave signal is attenuated by the propagation loss of the electrodes in
the device chip, which limits the ultimate device length and the half-wave
voltage Doi  _et al._ (2006). Moreover, the electrical bandwidth of the
modulator is hindered by the profound electrical loss at higher frequencies,
making a trade-off between the efficiency and bandwidth. In general, a longer
device has a smaller half-wave voltage, while a shorter device has a wider
bandwidth. It is important to mitigate the electrical loss for a highly-
efficient wide-bandwidth device.

As a measure to overcome this limitation, superconducting electrodes and the
operation at low temperature can be employed to mitigate the ohmic loss.
Several earlier studies have been conducted for the exploitation of a
high-$T_{c}$ superconducting film Yoshida  _et al._ (1993); Rozan  _et al._
(1999). However, optical modulators using superconductors have not yet been
put to a commercial use. The number of optical channels for the mobile
connections are still increasing due to the rapid progress of the internet-of-
things (IoT) technology, and it becomes more and more important to be able to
reduce the energy consumption in electrical and optical communication devices.
As in the case of single-flux-quantum logic circuits for computation Holmes
_et al._ (2013), it could be an effective solution to use superconducting
circuits even with the cooling overhead.

The use of acoustic waves, more precisely the photoelastic effect, for the
optical modulation has long been overlooked, and there has been tremendous
progress recently. The acoustic waves typically have much lower velocity than
that of the electromagnetic waves, and their wavelength at the GHz frequency
range approximately matches that of the optical waves. The short wavelength
enables tight confinement of the acoustic mode, resulting in a significant
modulation with small device size. Also, the piezo-acoustic devices have a
long history with matured technologies and well-studied materials Datta
(1986); Campbell (1998). Various useful designs of the piezoelectric
transducers, acoustic mirrors, and acoustic resonators are readily available.
Many acoustic materials are also known to have a high internal quality factor
(Q-factor), reaching above a billion Galliou  _et al._ (2013). These well-
developed tools and knowledge in piezo-acoustic devices should accelerate the
development of novel acousto-optical devices.

Integrated acousto-optical devices on lithium niobate (LN) are of particular
importance, and various developments, including the acousto-optic frequency
shifter Shao  _et al._ (2020); Yu and Sun (2021), and the acousto-optic
modulators Cai  _et al._ (2019); Sarabalis  _et al._ (2020) are reported. Many
of these devices also utilize the thin-film LN on various insulators to
further confine the acoustic mode. Other recent activities on the integrated
acousto-optical devices are compiled in Ref. Zhu  _et al._ (2021)

Additionally, the development of coherent conversion between microwave and
optical photons are recently on call, stimulated by the rapid progress in the
field of quantum computing Arute  _et al._ (2019) and its extension to the
quantum network Awschalom  _et al._ (2021). A superconducting resonator
coupled to an electro-optic microresonator is theoretically investigated
Javerzac-Galy  _et al._ (2016), and experimentally demonstrated Witmer  _et
al._ (2017); Fan  _et al._ (2018); McKenna  _et al._ (2020); Holzgrafe  _et
al._ (2020). Highly efficient acousto-optical devices at room temperature Shao
_et al._ (2019) and in cryogenic environment with superconducting electrodes
Jiang  _et al._ (2020) are reported.

In this paper, we develop an acousto-optic phase modulator consisting of
superconducting electrodes and a titanium-diffused (Ti-diffused) optical
waveguide on a LN substrate to reduce the half-wave voltage of the modulator.
The optical waveguide is placed in a surface-acoustic-wave (SAW) resonator and
the photoelastic effect induced by the SAW is employed as a modulating
mechanism, along with the electro-optic (Pockels) effect utilized in the
conventional EOM. By using the SAW whose energy is localized on the substrate
surface, it is possible to efficiently modulate the optical signal guided in
the waveguide. The Ti-diffused waveguide also provides an opportunity for
excellent connectivity with fiber optics. The length–half-wave-voltage
(length–Vπ) product obtained is more than 10-fold smaller than that of the
commercially available traveling-wave EOM, with the device size smaller by
about fifty times. We also develop a facet-coated optical waveguide cavity
with the finesse of $\mathcal{F}=43$, which could additionally reduces the
half-wave voltage approximately by $2\mathcal{F}/\pi$.

Figure 1: (a) Schematic of the superconducting acousto-optic phase modulator
architecture. A surface-acoustic-wave (SAW) resonator is fabricated on a Y-cut
lithium niobate (LN) substrate with the SAW propagation direction in the
$Z$-direction. Both the interdigitated-transducer (IDT) and mirror electrodes
(blue) are made of a superconducting niobium titanium nitride (NbTiN) film of
50-nm thick. A titanium-diffused (Ti-diffused) waveguide (orange) is
fabricated on the same substrate with the optical propagation in the
$X$-direction. End surfaces of the substrate on both ends of the waveguide are
polished to optical quality. Additional mirror coating (red) can be applied to
form a waveguide optical cavity. The zoomed section shows a schematic of the
cross section of the sample across the optical waveguide. The optical
waveguide (OWG) is designed to be placed at the anti-node of the
$u_{z}$-component of the SAW standing waves in the SAW resonator. (b)
Simulated optical intensity distribution in the Ti-diffused waveguide for the
transverse-magnetic (TM) mode. The substrate surface is located at
$y=0~{}\mu$m. We estimate the mode diameter of $2w_{0}=6.7~{}\mu$m and
$9.7~{}\mu$m for the $y$\- and $z$-directions, respectively. (c) Optical
micrograph of a sample chip. Light-colored part on the chip is the NbTiN film.
The optical waveguide is located on the vertical gap between the Bragg mirrors
of the SAW resonator.

##  II Superconducting acousto-optic phase modulator: device description

A schematic of the superconducting acousto-optic phase modulator is shown in
Fig. 1(a). In order to realize highly efficient optical modulation by elastic
waves, a system composed of a SAW resonator and an optical waveguide, is
fabricated on a Y-cut LN substrate. LN has a wide transparent window in the
optical domain along with large piezoelectric and photoelastic coefficients.
The SAW resonator is fabricated with the SAW propagation direction in the
$Z$-axis of the LN crystal. We choose the device coordinate system ($x$, $y$,
and $z$) to coincide with the crystal axes ($X$, $Y$, and $Z$) from here on.
Both the interdigitated transducer (IDT) and Bragg-mirror electrodes are made
of superconducting niobium titanium nitride (NbTiN) film. The NbTiN film has a
superconducting transition at the temperature of around 12 K for the thickness
of 50 nm, which is higher than the operating temperature of our refrigerator,
typically running at 8 K (Montana Instruments). The Ti-diffused waveguide is
also fabricated on the same substrate with the optical propagation direction
along the $X$-axis of the LN crystal. The Rayleigh-type SAW used in the
experiment has the displacement field components denoted as $u_{i}$ $(i=y,z)$
and the electric potential $\phi$. The optical waveguide is patterned in
between the SAW mirrors and designed to sit at the anti-node of the
$u_{z}$-component of the SAW standing wave.

When a radio-frequency (RF) signal, tuned to the resonance frequency of the
SAW resonator, is applied to the IDT, standing waves of the SAW are excited in
the SAW resonator. An optical signal is input from one of the end facets of
the waveguide. The deformation and strain due to the SAW cause the modulation
of the refractive index of the optical waveguide, resulting in the phase
modulation of the input optical signal by the SAW frequency. The modulation
can be enhanced by the optical cavity, which effectively increases the
interaction length between the SAW and optical input signal.

##  III Experiment

###  III.1 Device fabrication and sample preparation

First, an optical waveguide is fabricated on a LN substrate using the Ti
diffusion method. The waveguide is nearly a single mode for both the
transverse-electric (TE) and transverse-magnetic (TM) polarizations at the
laser wavelength of $\lambda_{\mathrm{opt}}=1064$ nm. The optical field
distribution of the TM mode, obtained by numerical simulation (OptiBPM), is
shown in Fig. 1(b). We obtain the fundamental mode $1/e^{2}$ diameter (defined
at $1/e^{2}$ of the maximum power density) of $2w_{0}=6.7~{}\mu$m and
$9.7~{}\mu$m, for the $y$\- and $z$-directions, respectively.

Next, the SAW resonator is fabricated with a 50-nm-thick superconducting NbTiN
thin film. The resonator is designed for the SAW wavelength
$\lambda_{\mathrm{SAW}}=40$ $\mu$m, corresponding to the resonance frequency
of approximately 87 MHz. The SAW wavelength is chosen so that the half-wave of
the SAW is large enough compared to the optical mode diameter in the
waveguide. After the SAW resonator fabrication, the substrate is diced and the
end facets of the optical waveguide are polished to optical quality. Figure
1(c) shows an optical micrograph of a typical device.

Before performing RF and optical measurements in the cryogenic environment,
the device is glued to a sample holder, as shown in the inset picture of Fig.
2(a), and fixed to the cold finger of the refrigerator. The free-space laser
light is coupled into and out of the optical waveguide through collimating
lenses mounted on the sample holder. More details on the sample fabrication
and preparation are compiled in Appendix A.

Figure 2: (a) Experimental setup: As shown in the inset picture, the device
chip is fixed on a sample holder with a SMP connector and mounted inside a
refrigerator. For the RF spectroscopy, the sample is directly connected to
port 1 of the network analyzer for the reflection measurement. For the optical
spectroscopy, an RF signal from port 1 of the network analyzer is mixed with
another signal from a function generator to drive either the SAW-based
modulator or a commercial EOM. The optical signal is input to the modulator
and the output signal is combined with an optical local oscillator before the
measurement with a photodiode. The inset frequency diagram indicates the
output signal and LO frequencies arriving at the photodiode. The photodiode
signal is sent to port 2 of the network analyzer to determine the phase
modulation imparted by the modulators. (b) Reflection spectrum of the SAW
resonator obtained from RF spectroscopy. The blue dots are data and the red
solid line is a Lorentzian fit to the data, from which we obtain the total
mechanical loss rate of the SAW resonator $\Gamma/2\pi=26.4$ kHz and the SAW-
resonator quality factor of 3,300. (c) Sideband signal ($S_{21}$) obtained in
the heterodyne measurement. The blue solid and orange dashed lines correspond
to the heterodyne signal obtained for the SAW device and the commercial EOM,
respectively. While the EOM shows almost no frequency dependence, the SAW
device shows clear signal enhancement at the SAW resonance frequency.
Difference between the two signals at the SAW resonance frequency is 11.8 dB.

###  III.2 RF spectroscopy of the SAW resonator

RF reflection of the SAW resonator is measured using a network analyzer. The
sample is directly connected to port 1 of the network analyzer for the
reflection measurement. The loss due to the ohmic resistance of the NbTiN
electrodes is large at room temperature, and the signal from the SAW resonator
could not be found. As the sample temperature was lowered, a dip in the RF
reflection spectrum appeared around 11 K, as expected from the superconducting
transition of NbTiN. At temperature below 11 K, the shape of the reflection
spectrum remains almost unchanged. The RF reflection spectrum of the SAW
resonator measured at 7.8 K is shown in Fig. 2(b). From the fitting, we obtain
the resonance frequency $\Omega/2\pi=87.6$ MHz and the Q-factor of 3,300, with
the internal dissipation rate $\Gamma_{\mathrm{in}}/2\pi=23.9$ kHz, and
external coupling rate $\Gamma_{\mathrm{ex}}/2\pi=2.5$ kHz.

###  III.3 Optical spectroscopy and measurement of $V_{\pi}$

The SAW excitation by the RF signal input results in the phase modulation of
the optical signal in the waveguide. We first evaluate the magnitude of the
phase modulation by a heterodyne measurement Okada  _et al._ (2018). In order
to quantify the depth of the phase modulation by the SAW, we perform the same
heterodyne measurement on a reference EOM (Photline NIR-MPX-LN-10) and compare
the sideband signal strength. The half-wave voltage of the reference EOM is
pre-determined by measuring the sideband and carrier amplitudes using a
scanning Fabry-Pérot resonator.

The optical setup used for the measurement is shown in Fig. 2(a). The input
optical signal polarization is in the $y$-direction to excite the fundamental
TM mode of the optical waveguide. The light emitted from the optical waveguide
is coupled to a single-mode fiber, mixed with a local-oscillator (LO) light by
a beam combiner, and detected by a photodiode. The LO light is prepared by
partially splitting the incident light (frequency $\omega_{0}$) to the optical
waveguide and shifting the frequency with an acousto-optic modulator by
$\omega_{\mathrm{AOM}}$. As shown in the inset frequency diagram of Fig. 2(a),
the first blue sideband generated by the phase modulation creates a beat
signal with the LO beam at frequency
$\omega_{\mathrm{AOM}}-\omega_{\mathrm{RF}}$. The signal is appropriately
filtered and measured with the network analyzer.

The blue solid and orange dashed lines in Fig. 2(c) show the results of the
sideband generation for the SAW modulator and the commercial EOM,
respectively, while sweeping the excitation RF frequency with a network
analyzer. For the SAW modulator, the sideband amplitudes are maximized at the
SAW resonance frequency, while the EOM showing a flat spectrum within the
frequency range of the data. Both measurements are taken with the same RF
power. At the SAW resonance, the sideband power difference of 11.8 dB is
measured. The half-wave voltage of the EOM is $V_{\pi}=4.8$ V at the driving
frequency of 87 MHz. Since the sideband power is inversely proportional to the
square of the half-wave voltage, the half-wave voltage of the SAW modulator
for the single-pass is determined as $V_{\pi}=18.7$ V. Due to a large
difference in the device length, the product of the interaction length and
half-wave voltage, which is one of the figure of merit for optical modulators,
can be calculated for a better comparison. For the reference EOM, whose
pigtail to pigtail packaging size is 100 mm, we estimate a device length of 50
mm, which is a typical length for these devices. We calculate length–$V_{\pi}$
product of 24 V$\cdot$cm. For the SAW modulator, the interaction length of the
SAW and light is assumed to be equal to the Bragg-mirror width $W=$ 950
$\mu$m, which gives the length–$V_{\pi}$ product of 1.78 V$\cdot$cm. The small
length–$V_{\pi}$ product shows the efficient modulation delivered by the SAW
modulator, achieved by the high Q-factor of the SAW resonator with
superconducting electrodes. We discuss further enhancement of the modulation
efficiency with a use of an optical cavity later.

Figure 3: Calculated displacement and electric potential in the SAW resonator.
The color scale represents the displacement and the electric potential of the
SAW when resonantly driven by an external RF signal generator with an
amplitude of 1 V. (a) Longitudinal displacement $u_{z}$, (b) transverse
displacement $u_{y}$ and (c) electric potential $\phi$. The green dashed lines
represent the mode spread (defined at $1/e^{2}$ of the maximum power density)
of the light propagating in the Ti-diffused optical waveguide.

##  IV Theoretical analysis

In order to quantitatively understand the observed phase modulation, we
examine the spatial distribution of the SAW-induced phase modulation imparted
on the light propagating in the optical waveguide.

The refractive index variation due to the photoelastic and electro-optic
effects in terms of the electric field $E_{k}=\frac{\partial\phi}{\partial
x_{k}}$ and strain tensor $S_{kl}=\frac{1}{2}\left(\frac{\partial
u_{k}}{\partial x_{l}}+\frac{\partial u_{l}}{\partial x_{k}}\right)$ reads,

$\Delta\\!\left(\frac{1}{n^{2}}\right)_{ij}=r_{ijk}E_{k}+p_{ijkl}S_{kl},$ (1)

where $r_{ijk}$ is the electro-optic tensor at zero strain (clamped) and
$p_{ijkl}$ the photoelastic tensor at constant electric field Jazbinšek and
Zgonik (2002). Here, we use the coordinate indices (${i,j,k,l}={1,2,3}$)
instead of (${x,y,z}$). For the case where the TM mode ($y$-polarization) is
chosen for the input and output optical modes, the refractive index variation
in terms of the SAW displacement $u_{i}$ and electric potential $\phi$ is,

$\delta n_{y}=\frac{n_{y}^{3}}{2}\left[r_{22k}\frac{\partial\phi}{\partial
x_{k}}-\frac{p_{22kl}}{2}\left(\frac{\partial u_{k}}{\partial
x_{l}}+\frac{\partial u_{l}}{\partial x_{k}}\right)\right],$ (2)

where $n_{y}$ is the unperturbed refractive index for the $y$-polarization.

Table 1: Material constants of LiNbO3 used for the determination of SAW-induced refractive index Jazbinšek and Zgonik (2002).  Constant | Value
---|---
$n_{y}$ | 2.23
$p_{2222}$ | $-0.0261$
$p_{2233}$ | 0.0832
$p_{2223}$ | 0.1335
$r_{222}$ |  $3.40\times 10^{-12}$ m/V
$r_{223}$ |  $9.10\times 10^{-12}$ m/V

The contributions of the terms proportional to $u_{i}$ and $\phi$, have
different relative phases. Numerical results of the distributions of
longitudinal displacement $u_{z}$($=u_{3}$), transverse displacement
$u_{y}$($=u_{2}$), and electric potential $\phi$ in the vicinity of the
optical waveguide are shown in Fig. 3. The dashed lines in Fig. 3 indicate the
spread of the optical mode propagating in the optical waveguide. From the
variation of the refractive index $\delta n_{y}$ calculated using Eq. 2 for a
given RF input power, the half-wave voltage $V_{\pi}$ is calculated from the
following formula,

$\pi=\delta n_{y}(V_{\pi})\,k_{\mathrm{opt}}W,$ (3)

where $k_{\mathrm{opt}}$ is the optical wavenumber. Material constants used in
the calculation are listed in Table 1 Jazbinšek and Zgonik (2002).

The contribution of the electro-optic effect, driven by the piezoelectrically-
induced electric field, and that of the photoelastic effect from the strain
are found to be comparable. We calculate $V_{\pi}=17.3$ V from the numerical
simulation, slightly lower than the observed value. In the same simulation, we
also found that a misplacement of the optical waveguide by $-0.75$ $\mu$m in
the $z$-direction results in a diminished modulation with $V_{\pi}=$ 18.7 V.
We estimate the misalignment between the optical waveguide and the SAW
resonator during the fabrication to be at most $1~{}\mu$m, and the numerical
simulation is consistent with our measurement.

As shown in Fig. 3, $u_{z}$ has a mirror symmetric distribution around $z=0$,
while $u_{y}$ and $\phi$ show asymmetric distributions. Different strain
components ($S_{kl}$) and electric fields ($E_{k}$) derived from the
displacement and electric potential also have either symmetric or asymmetric
distribution around $z=0$. Among them, only the ones with symmetric
distribution contributes to the phase modulation. We find $S_{23}$ and $E_{3}$
to be the main contributors for the photoelastic and electro-optic phase
modulations, respectively. We also find that the center of the optical mode is
approximately 4 $\mu$m below the surface, while the largest variation of the
refractive index is found nearly at the surface. Possible improvement may be
found by lifting the optical mode location towards the surface by constructing
a ridge-structured waveguide.

Figure 4: Optical waveguide cavity transmission spectra for the TE and TM
modes shown in blue and red lines, respectively. The detuning refers to the
first peak of the TE mode. The amplitude of the transmission spectra is
normalized with the maximum peak height of the TM mode. The TM mode signal is
offset by unity. The input laser frequency is swept more than the free-
spectral-range of the cavity. The cavity finesse $\mathcal{F}=43$ and the
linewidth $\kappa/2\pi=284$ MHz are obtained for the TM mode. The quality
factor of the cavity is $Q=1.0\times 10^{6}$. We observe nearly the same
performance for the two input optical polarizations. The higher spatial modes
are observed in the TE mode presumably due to a smaller mode area in the
optical waveguide.

##  V Optical waveguide cavity

In order to further improve the performance of the modulator, one can apply
mirror coating on both end facets of the optical waveguide to form an optical
cavity. The ratio of the output optical power in the first sideband for a
device with the optical cavity ($P_{\mathrm{SB,cavity}}$) to that for a device
with only the waveguide ($P_{\mathrm{SB,waveguide}}$) can be derived as,

$\frac{P_{\mathrm{SB,cavity}}}{P_{\mathrm{SB,waveguide}}}=\left(\frac{2\mathcal{F}}{\pi}\right)^{2}\frac{\kappa_{\mathrm{ex}}^{2}}{\Omega^{2}+(\kappa/2)^{2}},$
(4)

where $\kappa_{\mathrm{ex}}$, $\kappa$, and $\mathcal{F}$ are the external
coupling rate, total dissipation rate, and finesse of the optical cavity,
respectively. When the SAW frequency $\Omega$ is less than the linewidth of
the cavity $\kappa$ $(\Omega<\kappa)$, the power enhancement of approximately
$\left(\frac{2\mathcal{F}}{\pi}\right)^{2}$, equivalently $V_{\pi}$ reduction
of $2\mathcal{F}/\pi$, can be achieved at the critical coupling condition
($\kappa_{\mathrm{ex}}=\kappa/2$). Contrary, when the SAW frequency is larger
than $\kappa$, the enhancement drops as the density of states for the
scattered photon diminishes rapidly outside the cavity linewidth.

To show this possibility, we apply dielectric multi-layer film coating with
reflectivity $R=95~{}\%$ on both end facets of the optical waveguide chip used
for the current device. Transmission spectra of the optical waveguide cavity
for the TE and TM modes are shown in Fig. 4. We obtain the free-spectral
ranges of 12.6 and 12.1 GHz for the TE and TM modes, respectively, which is
consistent with the birefringence of LN. For the TM mode, we observe the
finesse of $\mathcal{F}=43$ ($Q=1.0\times 10^{6}$), with the corresponding
total cavity dissipation rate $\kappa/2\pi=284$ MHz, and find a negligibly
small difference of the finesse between the two input laser polarizations. The
cavity finesse did not seem to change from the room temperature to the
cryogenic temperature during the cool-down, either. The propagation loss of
the optical waveguide is estimated to be less than $0.2$ dB/cm. The worst case
would limit the maximum finesse of $\mathcal{F}\approx 150$ ($Q=1.4\times
10^{6}$), for a 5-mm-long cavity at the critical coupling. The effect of the
optical cavity can be understood as a simple elongation of the interaction
length by approximately $2\mathcal{F/\pi}$, as long as the SAW frequency
$\Omega$ is much smaller than the cavity linewidth.

##  VI Discussion

The modulation efficiency $V_{\pi}=$ 18.7 V for the current device is
partially limited by the short SAW cavity width (950 $\mu$m) along the optical
waveguide. By extending the length of the SAW cavity to 50 mm, which is about
a typical length in commercial EOMs, we expect to obtain $V_{\pi}=$ 2.58 V for
a single-pass device. Additional optical cavity implementation with the
finesse of $\mathcal{F}=15$ ($Q=1.4\times 10^{6}$), corresponding to the
channel propagation loss of 0.2 dB/cm, would lead to $V_{\pi}=$ 0.27 V. For
the future mobile transport networks, a narrow-band and high-efficient optical
modulator is one of the key components for the radio-over-fiber (RoF) links
Murata (2020). In those links, narrow-band radio signals are transmitted
through optical fibers by using electro-optic (EO) and opto-electric (OE)
conversion without frequency mixing in electrical domain. The resonance-type
narrow-band modulator with high efficiency, presented here, could potentially
be applied to such future communication networks.

While we routinely observe the Q-factor of an acoustic resonator exceeding
$10^{5}$ with aluminum electrodes on a bulk LN in a cryogenic environment, the
Q-factor of the current device is limited to 3,300 even with the use of NbTiN
superconducting electrodes. We presume the degradation of the Q-factor
originates from the reactive ion etching of NbTiN electrodes on LN with CF4
plasma during the fabrication. We expect the Q-factor to reach the same level
as other acoustic resonators made with alternative fabrication techniques,
enhancing the modulation efficiency. The high Q-factor acoustic resonator in a
cryogenic environment may also pave the way towards using these devices in the
quantum regime.

There are a number of reports, similarly improving the efficiency of the
optical modulation. Among different types of EOMs, devices made of thin-film
LN has attracted much attention recently. Development of various EOMs
including LN ridge-structured waveguides on silicon Stenger  _et al._ (2019);
Wang  _et al._ (2018b), and Ti-diffused waveguides Yamaguchi  _et al._ (2019)
are reported with estimated $V_{\pi}=$ 2.5, 1.4, and 1.2 V, respectively. With
an optical resonator, an estimated $V_{\pi}=$ 11 V is also reported Wang  _et
al._ (2018a). All of these measurements were conducted at room temperature.
One of the drawbacks of the current device may be a relatively large optical
mode width of the Ti-diffused waveguide, limiting the modulation efficiency.
The large extent of the optical mode could be an obstacle for increasing the
acoustic frequency. Incorporating this thin-film LN technology, at the expense
of the fiber connectivity, may be the key to improve the performance of the
current device further.

In another research area, where the focus is on coherent wavelength conversion
between optical and microwave photons, tremendous efforts have been made to
utilize both electro-optic and photoelastic effects along with superconducting
circuits to efficiently convert the photons while mitigating the loss. SAW
resonators coupled with an optical racetrack resonator Holzgrafe  _et al._
(2020), optical and microwave hybrid-ring resonator Fan  _et al._ (2018), and
photonic-phononic crystal resonator Jiang  _et al._ (2020), have been reported
with estimated $V_{\pi}=$ 0.088, 0.032, and 0.024 V, respectively. Many of
these works are at the frontier of efficient optical modulation. However, the
use of low-temperature superconducting electrodes requiring a temperature
below 4 K and the intricacy of fabrication could be overwhelming and may not
be the near-future solution for the replacement of the commercial EOMs.

While the current device shows moderate modulation efficiency, it requires
much simpler fabrication. The facet mirror coating shown here can also be
implemented to various devices. The well-developed fiber-based butting
technique for efficient coupling into a Ti-diffused waveguide is available
with or without the waveguide facet mirror coating to form an optical cavity.
The thin-film LN technology often suffers from the fiber connectivity due to
the tight confinement of the optical mode and the size mismatch with standard
single-mode fibers. While the mode conversion is possible, it still remains a
challenge to reach near unity efficiency. These advantages make the current
device attractive for a mass production and as a replacement of the commercial
EOMs.

##  VII Summary

In summary, we developed a superconducting acousto-optic phase modulator.
Operated at the cryogenic temperature of 8 K, the device showed a
length–$V_{\pi}$ product of 1.78 V$\cdot$cm, which is more than 10-fold
improvement compared to commercially available electro-optic modulators.
Implementing an optical cavity in the current device structure is shown to
further improve the overall performance.

###### Acknowledgements.

The authors acknowledge K. Kusuyama and K. Nittoh for the help in sample
fabrication. This work was partly supported by JSPS KAKENHI (Grant No.
26220601), NICT NEXT-project, JST PRESTO (Grant No. JPMJPR1429, JPMJPR1667),
and JST ERATO (Grant No. JPMJER1601).

##  Appendix A Fabrication and sample preparation

An optical waveguide is fabricated using the titanitum diffusion method, in
which we sputter a Ti strip with a width of 3 $\mu$m and a thickness of 90 nm
on a LN substrate and diffuse the titanium at the temperature of 960∘C for 21
hours.

Next, a 50-nm-thick superconducting NbTiN thin film is sputtered on the entire
surface of the substrate on which the optical waveguide is fabricated. The SAW
resonator is patterned by electron beam lithography and dry-etched with CF4.
The resonator is designed for the SAW wavelength $\lambda_{\mathrm{SAW}}=40$
$\mu$m. The SAW wavelength is chosen, so that the half-wavelength of the SAW
is large enough compared to the optical mode diameter in the waveguide. Using
the SAW velocity of 3,488 m/s in LN, the corresponding resonance frequency is
approximately 87 MHz. The interdigitated transducer and Bragg mirrors are 950
$\mu$m wide and have four (two pairs) and two-hundred electrodes,
respectively. The distance between the mirrors, measured between the closest
electrodes, is 380 $\mu$m. The angular misalignment between the optical
waveguide and the SAW resonator is approximately within 0.06 degrees, which
may shift the location of the SAW standing wave by 1 $\mu$m with respect to
the optical waveguide.

After the SAW resonator fabrication, the substrate is diced with a sintered
diamond blade and the end facets of the optical waveguide are polished to
optical quality. The length of the optical waveguide after polishing is 5.5
mm. The waveguide is nearly a single mode for both the TE and TM polarizations
at the laser wavelength of $\lambda_{\mathrm{opt}}=1064$ nm.

The optical-quality polishing and mirror coating of the end facets of the
optical waveguide are conducted externally (LAMBDA PRECISION CORPORATION). The
multi-layer mirror-coating consists of SiO2, Al2O3, and Nb2O5 layers with a
total thickness of 1830 nm.

Before performing RF and optical measurements in the cryogenic environment,
the device is fixed to the measurement jig made of copper. The sample is glued
on the jig with varnish next to an external printed circuit board with a SMP
connector soldered on it. The RF input port for the IDT is wire-bonded to the
external printed circuit board. Next, the jig is fixed to the cold finger of
the refrigerator, and an alignment laser is introduced by butt-ending a bare
fiber to the output of the optical waveguide. An input collimator lens mounted
on the side of the sample is adjusted to collimate the alignment beam out of
the optical waveguide. We then introduce a free-space laser to couple the beam
into the waveguide through the input collimator and optimize the coupling by
aligning the input laser to the alignment laser. Once the input laser is
optimized, the alignment laser is removed, and an output collimator lens is
placed and adjusted. After the optical alignment is completed, the sample is
covered with a radiation shield, and the refrigerator is closed for cool-down.
The free-space laser light is coupled into and out of the waveguide via the
optical windows of the refrigerator throughout the experiment.

##  Appendix B Estimation of the SAW amplitude

In order to calculate the SAW amplitude in the SAW resonator, we use the
relationship between the input RF power and the phonon number in the cavity.
First, the zero-point fluctuation amplitude of the SAW is defined as,

$U_{\mathrm{zpf}}=\sqrt{\frac{\hbar}{2\rho V_{\mathrm{mode}}\Omega}},$ (S1)

where $\rho$, $V_{\mathrm{mode}}$, $\Omega$ are the mass density of the SAW
substrate, the mode volume of the SAW resonator, and the SAW frequency,
respectively. The mode volume is defined as $V_{\mathrm{mode}}=\lambda LW$,
where $\lambda$ is the SAW wavelength. $L$ and $W$ are the length and width of
the SAW resonator, respectively Schuetz  _et al._ (2015).

The amplitude of the SAW, $U_{0}$, is related to the number of phonons in the
SAW resonator, $N$, by

$U_{0}^{2}=NU_{\mathrm{zpf}}^{2}=\frac{N\hbar}{2\rho
V_{\mathrm{mode}}\Omega}.$ (S2)

The number of phonons in the single-port SAW resonator for the input power $P$
is given as,

$N=\frac{\Gamma_{\mathrm{ex}}}{\Delta^{2}+(\Gamma^{2}/4)}\frac{P}{\hbar\Omega},$
(S3)

where $\Delta=\omega_{\mathrm{RF}}-\Omega$ is the detuning between the RF
input frequency $\omega_{\mathrm{RF}}$ and the SAW resonance frequency
$\Omega$. $\Gamma_{\mathrm{ex}}$ and $\Gamma$ are the external coupling and
total loss rates of the SAW resonator, respectively. At the SAW resonance
frequency ($\Delta=0$),

$N=\frac{4\Gamma_{\mathrm{ex}}}{\Gamma^{2}}\frac{P}{\hbar\Omega},$ (S4)

and the SAW amplitude $U_{0}$ can be rewritten in a form,

$U_{0}^{2}=\frac{4\Gamma_{\mathrm{ex}}}{\Gamma^{2}}U_{\mathrm{zpf}}^{2}\frac{P}{\hbar\Omega}.$
(S5)

In the simulation, we use $U_{0}$ determined from the input power $P$ as the
amplitude of the SAW.

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[a,b]Pietro Butti

# Testing (asymptotic) scaling in Yang-Mills theories in the large-$N_{c}$
limit

Antonio González-Arroyo

###### Abstract

TEK reduction is a well-established technique that allows single-site
simulations of Yang-Mills theory in the large-$N_{c}$ limit by exploiting
volume reduction induced by twisted boundary conditions. We performed
simulations for $SU(841)$ for several gauge couplings and applied standard
Wilson flow techniques combined with a tree-level improvement methodology to
set the lattice scale. The wide range of gauge couplings covered by our
simulations allows us to explore a region in the coupling space where our data
exhibits asymptotic scaling and perturbation theory could be used to analyze
the behaviour of the $\beta$-function. In this talk, I will review the
methodology used and go through the main results we obtained, including a
determination of the $\Lambda$-parameter of Yang-Mills theory at large-$N_{c}$
in $\overline{\text{MS}}$-scheme.

IFT-UAM/CSIC-23-58

##  1 Introduction

In a generic bare-coupling definition scheme labelled by $s$, the
Renormalization Group (RG) equation for the bare ’t Hooft coupling
$\lambda_{s}=g^{2}N_{c}$ dependence on the cutoff scale $a$ reads

$\beta_{s}(\lambda_{s})=-\derivative{\lambda_{s}}{\log a^{2}}\,.$ (1)

It is well-known that the $\beta$-function has the following perturbative
expansion around $\lambda_{s}\sim 0$

$\beta(\lambda_{s})\sim-
b_{0}\lambda_{s}^{2}-b_{1}\lambda_{s}^{3}-b_{2}^{(s)}\lambda_{s}^{4}+\order{\lambda_{s}^{5}}\,,$
(2)

where $b_{0}$ and $b_{1}$ are known to be universal (scheme independent) and
in a pure Yang-Mills theory they amount to $b_{0}=\frac{11}{3(4\pi)^{2}}$ and
$b_{1}=\frac{34}{3(4\pi)^{4}}$, while higher-order coefficients are known to
be dependent on the scheme chosen. Using a fully perturbative $\beta$-function
truncated at $\order{\lambda^{3}}$, upon integration, Eq. (1) gives

$-\log{a\Lambda_{s}}=\frac{1}{2b_{0}\lambda_{s}(a)}+\frac{b_{1}}{2b_{0}^{2}}\log(b_{0}\lambda_{s}(a))+\frac{c_{1}^{(s)}}{2b_{0}}\lambda_{s}(a)+\order{\lambda_{s}^{2}}\,,$
(3)

where the coefficient of the linear term is given by
$c_{1}^{(s)}=\frac{b_{2}^{(s)}}{b_{0}}-\frac{b_{1}^{2}}{b_{0}^{2}}$, and is
scheme-dependent, as well as the $\Lambda$-parameter itself. On the lattice,
it is natural to use the coupling parameter in the lattice action as a natural
coupling scheme. For Wilson action we label it as
$\lambda_{w}\equiv\nicefrac{{1}}{{b}}=2N^{2}/\beta$. This scheme is known to
have large higher-order perturbative corrections, since the ratio between the
Wilson and $\overline{\text{MS}}$ scales is large [1, 2]
$\frac{\Lambda_{\overline{\text{MS}}}}{\Lambda_{w}}=38.853$. This induces
large violations of the scale dependence of the coupling with respect to the
truncated perturbative prediction, which we refer to as asymptotic scaling.
This showed up from the earliest studies, and several authors proposed
adopting a different definition of the coupling which is better behaved. These
are called improved couplings. The goal of this work is to test how well the
asymptotic predictions hold for the case of Yang-Mills theory in the
large-$N_{c}$ limit. Our results are based upon an extensive analysis. Here we
present some preliminary results. The full analysis will follow in a future
publication [3].

Our strategy is to make use of the volume reduction property at large $N_{c}$
which allows to obtain results about standard $SU(N_{c})$ gauge theory by
simulations on a single site lattice with twisted boundary conditions [4, 5].
After a change of variables, the resulting Wilson action of the reduced model
(TEK model) becomes

$S_{\text{TEK}}=bN_{c}\sum_{\mu\neq\nu}\tr\quantity[1-z_{\mu\nu}U_{\mu}U_{\nu}U_{\mu}^{\dagger}U_{\nu}^{\dagger}]$
(4)

where $U_{\mu}$ are $SU(N_{c})$ matrices, $b=1/\lambda_{w}$ is the inverse of
’t Hooft coupling $\lambda=g^{2}N_{c}$, while $z_{\mu\nu}$ is a complex phase
encoding the twist and given by $e^{i\frac{2\pi k}{N_{c}}}$ for $\nu>\mu$ ($k$
is an integer coprime with $\sqrt{N_{c}}$). This particular choice of the
twist factor is called symmetric twist.

The vacuum configurations for the TEK action (4) are called twist-eaters and
they are given by $U_{\mu}=\Gamma_{\mu}$ which are the solutions of the twist
equation:

$\Gamma_{\mu}\Gamma_{\nu}=z_{\nu\mu}\Gamma_{\nu}\Gamma_{\mu}$ (5)

The equivalence of the TEK model and the infinite volume large $N_{c}$ gauge
theory has been proven both perturbatively and non-perturbatively.
Furthermore, the proof shows that finite $N_{c}$ corrections take the form of
finite volume corrections in a lattice whose effective size is
$V=(\sqrt{N_{c}})^{4}$ [4, 5]. Thus our choice of $N_{c}=841$ implies a
lattice of size $29^{4}$ allowing us to study the approach to the continuum
limit and matching the expected behaviour of the infinite volume large $N_{c}$
gauge theory.

##  2 The Wilson flow scale(s)

To set the scale we use an observable which allows a simple determination with
great precision. This is the gradient flow scale [6]. This is based upon
evolving the gauge field with the equation
$\partial_{t}A_{\mu}(x,t)=D_{\nu}G_{\mu\nu}(x,t)$, $t$ being the flow time. As
observable we take the flowed energy density
$E(t)=\expectationvalue{\tr\quantity(G_{\mu\nu}(x,t)G_{\mu\nu}(x,t))/2}$. On
the single-site twisted lattice the evolution takes place with respect to the
dimensionless lattice flow time $T=\frac{t}{a^{2}}$, with the corresponding
observable being

$E=-\frac{1}{128}\sum_{\mu\neq\nu}\tr[z_{\nu\mu}(U_{\nu}U_{\mu}U_{\nu}^{\dagger}U_{\mu}^{\dagger}+U_{\mu}U_{\nu}^{\dagger}U_{\mu}^{\dagger}U_{\nu}+U_{\nu}^{\dagger}U_{\mu}^{\dagger}U_{\nu}U_{\mu}+U_{\mu}^{\dagger}U_{\nu}U_{\mu}U_{\nu}^{\dagger})-\text{h.c.}]^{2}\,,$
(6)

in terms of which we define the dimensionless observable
$\Phi(T)=\frac{\expectationvalue{T^{2}E(T)}}{N_{c}}$ normalized to the number
of colours in order to have a finite large $N_{c}$ limit.

Figure 1: Left-hand side: Flowed dimensionless energy density for pure Yang-
Mills for different numbers of colours at $b=0.37$. Right-hand side: Same flow
curves with the norm correction applied. The scaling windows in Eq. (9) are
represented with a vertical stripe of the same colour as the flow curve they
are associated with. They all start at $T=1.25$ but only the ones with the
smallest $N_{c}$ are depicted so as not to make the colours overlap.

This observable allows to define a scale as follows

$\evaluated{\Phi(T)}_{T=T_{1}}\equiv\frac{1}{N_{c}}\evaluated{\expectationvalue{T^{2}E(T)}}_{T=T_{1}}=0.05.$
(7)

The choice of $0.05$ is taken as a compromise that minimizes finite volume
effects (finite $N_{c}$) while still being very slightly affected by lattice
artefacts. To exemplify this, we display in the left side of Fig. 1 the flow
curves for several values of $N_{c}$ for pure Yang-Mills reduced model
($N_{f}=0$) at $b=0.37$, where the two reference scales $0.1$ and $0.05$ are
depicted with corresponding horizontal grey lines. As it is visible from the
plot, although the scale $T_{1}$ is less affected by the $N_{c}$-dependence of
the flow, the systematic error induced is still sizeable for the smaller
values of $N_{c}$. The situation can be strongly ameliorated by the method
employed in Ref. [7] to be briefly explained below.

Our approach is to consider a new lattice observable to replace $\Phi(T)$,
which nonetheless coincides with its infinite $N_{c}$ and vanishing lattice
spacing limit. It is clear that this quantity is as good an observable as the
previous one, but is expected to have smaller finite $N_{c}$ and lattice
corrections. This is given by

$\hat{\Phi}(T)=\frac{3}{128\pi^{2}\hat{\mathcal{N}}_{N}(T)}\Phi(T,N)$ (8)

where the prefactor $\hat{\mathcal{N}}_{N}(T)$ is a function of
$\sqrt{\nicefrac{{8T}}{{N_{c}}}})$ such that
$\hat{\mathcal{N}}_{\infty}(0)=\frac{3}{128\pi^{2}}$. It is determined in such
a way as to eliminate the $N_{c}$ dependence at the lowest order of
perturbation theory. This was analytically calculated in Ref. [8]. We refer to
our modified method as norm correction and we expect it to simultaneously
reduce finite-$N_{c}$ and lattice artefacts in an effective window in $T$. The
lower limit is set to eliminate any interference from lattice artefacts, while
the upper limit is determined by ensuring that the smearing radius remains
reasonably smaller than a fraction of the overall size of the effective
lattice, where residual finite-volume effects are negligible. We typically
refer to this range in $T$ as the scaling window, which can be expressed as

$\quantity[1.25,\gamma^{2}\frac{N_{c}}{8}]\text{ with }\gamma=0.22$ (9)

After the application of the norm correction, we define $T_{1}$ in the same
way but in terms of $\hat{\Phi}(T)$, and we extract the scale
$T_{1}=t_{1}/a^{2}$ by interpolation. The right-hand side of Fig. 1 shows the
effect. We observe that, apart from the case of $N_{c}=169$, all the scales
$T_{1}$, interpolated from $\hat{\Phi}$, coincide within errors, signalling
that the norm-correction can effectively correct the finite-$N_{c}$/finite-
size effects. The norm-correction method was used in Ref. [7] to determine the
scale from data of $N_{c}=289$ and $361$. In our case, since $N_{c}=841$ is
much larger we can safely claim that the corresponding scales amount to the
corresponding one at infinite $N_{c}$. A detailed discussion about the norm
correction and its applications to the case of this and other theories can be
found also in[9].

We have measured the scale $T_{1}$ for a large number of values of $b$ for the
TEK model. The obtained values of $1/\sqrt{8T_{1}}$ are listed in the second
column of Tab. 1.

$b$ | $\frac{a}{\sqrt{8t_{1}}}$ | $a\sqrt{\sigma}$
---|---|---
$0.355$ | $0.37689(62)$ | $0.2410(30)$
$0.360$ | $0.31754(47)$ | $0.2058(25)$
$0.365$ | $0.2716(10)$ | $0.1784(17)$
$0.370$ | $0.23582(69)$ | $0.1573(19)$
$0.375$ | $0.20605(69)$ | $0.1361(17)$
$0.380$ | $0.17734(86)$ | $0.1191(17)$
$0.385$ | $0.1584(15)$ | $0.1049(11)$
Table 1: Lattice scale in units of the Wilson flow improved scale and of the
string tension (taken from [10]).

We can confront the extracted scales with a previous, less precise,
determination in units of the string tension $\sigma$ given in Ref. [10]. The
numbers are reported in the third column of Tab. 1. In Fig. 2, we depict each
value of the dimensionless ratio $R=\sqrt{\sigma}{\sqrt{8t_{1}}}$ as a
function of $(a/\sqrt{8t_{1}})^{2}$.

Figure 2: Ratio of the string tension on the lattice and the Wilson flow scale
for each value of the gauge coupling $b$. The horizontal band corresponds to
the fit to a constant, while the linear fit gives a value of $0.6742(83)$ for
the extrapolated value.

Exact scaling would imply the ratio to be constant. Such a fit has a
$\chi^{2}/\\#\mathrm{dof}=1.6$, mainly spoiled by the value at the 2 coarsest
point. Nevertheless, we notice that a linear fit in $(a/\sqrt{8t_{1}})^{2}$
gives an $\chi^{2}/\\#\text{dof}=0.24$ and might suggest slight scaling
violations. As a final value, we will consider $0.674(8)(18)$ where the
systematic error is the dispersion between the two previously obtained values.
This comparison with the string tension is a remarkable confirmation of the
validity of scaling for a wide range of ’t Hooft couplings with only small
violations in the case of the coarsest ensembles.

##  3 Asymptotic scaling and $\Lambda_{\overline{\text{MS}}}$

In this section, we will analyze if the scales we determined follow the
asymptotic predictions reviewed in the introduction. For the large $N_{c}$
theory a similar analysis was performed earlier [11, 10]. We employ three
different improved couplings: $\lambda_{I}$ (from [11]), $\lambda_{E}$ (from
[12, 13]) and $\lambda_{E^{\prime}}$, defined in Tab. 2.

symbol | definition | $\frac{\Lambda_{w}}{\Lambda_{s}}$ | $c_{1}$
---|---|---|---
$\lambda_{w}$ | $\frac{1}{b}$ | $1$ | $-0.00438798235$
$\lambda_{I}$ | $\frac{\lambda_{w}}{P(\lambda_{w})}$ | $0.0677656398$ | $-0.00172791$
$\lambda_{E}$ | $8(1-P(\lambda_{w}))$ | $0.4148791463$ | $-0.0005030$
$\lambda_{E^{\prime}}$ | $-8\log P(\lambda_{w})$ | $0.1080025976$ | $-0.00042426$
Table 2: Possible definitions for the gauge coupling on the lattice.

Having at our disposal such a large range of values for the ’t Hooft couplings
and given the small errors of our scale determination we are certainly making
stringent tests to our data. A global analysis can be done by fitting Eq. (3)
to our data for the case of the $E^{\prime}$ scheme only leaving free the
value of $\Lambda_{E^{\prime}}$. The result is displayed in Fig. 3 where the
data points and the continuous best-fit curve are shown. The overall
impression is very good since the curve follows quite nicely the evolution of
the data. However, given the small errors, the $\chi^{2}$ of the fit is not
good. Indeed, for the range of scales covered, certain non-perturbative
scaling violations are to be expected.

Figure 3: Logartihm of the lattice spacing in units of the Wilson flow scale
$\sqrt{8t_{1}}$ listed in 1 as a function of the improved coupling
$\lambda_{E^{\prime}}$ defined in Tab. 2. The green line is the result of a
fit to Eq. (3).

We aim at a determination of the $\Lambda$ parameters which are consistent
with the predictions of perturbation theory. It is customary to choose as
preferred scale the one coming from the $\overline{\text{MS}}$ scheme in the
continuum: $\Lambda_{\overline{\text{MS}}}$. The ratio of the
$\Lambda$-parameters in any scale to $\Lambda_{\overline{\text{MS}}}$ can be
determined by passing through the Wilson scheme as follows:

$\frac{\Lambda_{\overline{\text{MS}}}}{\Lambda_{s}}\equiv\frac{\Lambda_{\overline{\text{MS}}}}{\Lambda_{w}}\frac{\Lambda_{w}}{\Lambda_{s}}\,,$
(10)

where the separate factors can easily calculated in perturbation theory and
are reported in Tab. 2. Now using Eq. (3) truncated to order $\lambda_{s}$ and
fixing $\Lambda_{s}$ in terms of $\Lambda_{\overline{\text{MS}}}$ we can
determine this quantity in units of $\sqrt{8t_{1}}$ from each of our values of
$b$ and from all of the improved couplings. The result is displayed in Fig. 4.
Obviously, a perfect asymptotic scaling would mean that all the values
obtained were the same. However, it is natural to expect deviations coming
from higher order perturbative and non-perturbative corrections to Eq. (3).
These corrections should nonetheless vanish as we approach the continuum limit
($a=0$). This seems to be the case for our data since the larger difference
obtained when comparing $\lambda_{I}$ with $\lambda_{E}$ approaches zero as we
move towards smaller values of the lattice spacing. To give an estimate we
perform a linear plus quadratic extrapolation in $a$ for each of the improved
couplings separately displayed by the coloured bands in Fig. 4. The
extrapolated values of $\Lambda_{\overline{\text{MS}}}$ in units of
$\sqrt{8t_{1}}$ are $0.3582(47)$, $0.3620(46)$ and $0.3652(47)$ for
$\lambda_{I}$, $\lambda_{E}$ and $\lambda_{E^{\prime}}$, respectively. Notice
that the three values are consistent within errors. The $\chi^{2}$ of the fits
is not good but this is due entirely to the result at $b=0.38$ which is seen
to deviate considerably from the others. We attribute this phenomenon to a bad
estimate of the error based on the large autocorrelation times of the data at
this value. A similar phenomenon has been observed and reported in similar
studies of the flow in the literature [14, 15]. Our final results having good
$\chi^{2}$ have been obtained by excluding the value at $b=0.38$, but the
extrapolated values do not change significantly if we include it. A more
detailed analysis of this behaviour will be covered in a future publication
[3].

To quote a final estimate of the $\Lambda$-parameter for the Yang-Mills theory
at large-$N_{c}$ in the $\overline{\text{MS}}$ scheme, we give the mean values
between the previous results and assign the dispersion as a systematic error.

$\Lambda_{\overline{\text{MS}}}=0.3618(47)(29)\,\quantity(\sqrt{8t_{1}})^{-1}=0.5366(94)(120)\,\sqrt{\sigma}\,,$
(11)

Notice that the use of the flow scale allows one to obtain an estimate at the
level of 1-2 per cent. In the previous formula, we also report the final value
in units of $\sqrt{\sigma}$ using the dimensionless ratio $R$ that we found in
the previous section. For the systematic error, we have added in quadrature
the systematic uncertainty on $\Lambda$ and the one on $R$. Although this
other estimate implies a loss in precision, it has the advantage of allowing a
comparison with previous determinations. These are $0.503(2)(40)\sqrt{\sigma}$
in [11] and with the result $0.525(2)$ in the TEK model from [10] and also
with the number of $0.5093(15)(250)$ given in [16] and $0.542(16)(48)$ from
[17]. To collect and compare visually all these results we plot all the values
in Fig. 5 in a FLAG-style plot.

Figure 4: (Logarithm of the) $\Lambda$-parameter in the $\overline{\text{MS}}$
scheme in units of $\sqrt{8t_{1}}$ versus the lattice spacing in the same
units using three different improved couplings in Tab. 2. We also depict the
corresponding quadratic continuum extrapolations. In the fit, we excluded the
point corresponding to $b=0.38$ (whose points are depicted with an empty
marker), obtaining $\chi^{2}/\\#\text{dof}=0.49,0.61,0.62$ for $\lambda_{I}$,
$\lambda_{E}$ and $\lambda_{E^{\prime}}$, respectively. Figure 5: Values of
$\Lambda_{\overline{\text{MS}}}$ present in the literature compared. The red
band represents a weighted average.

##  4 Conclusions

We used volume reduction to study large $N_{c}$ Yang-Mills theory by
simulating the TEK reduced model. This study focuses on testing predictions of
asymptotic scaling for the evolution of the scale of the theory. We performed
simulations for $SU(841)$ and for 8 values of the coupling and obtained a
precise determination of the scale. Our study shows consistent results using
different improved couplings, having small corrections and allowing a precise
determination of the Lambda parameter of the theory in the
$\overline{\text{MS}}$ scheme. Our work is part of a more extensive study to
appear in a future publication [3].

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