# Probing hundreds of individual quantum defects in polycrystalline and
amorphous alumina

Chih-Chiao Hung  Laboratory for Physical Sciences, 8050 Greenmead Drive,
College Park, Maryland 20740, USA  Quantum Materials Center, University of
Maryland, College Park, Maryland 20742, USA  Department of Physics, University
of Maryland, College Park, Maryland 20742, USA  Liuqi Yu  Laboratory for
Physical Sciences, 8050 Greenmead Drive, College Park, Maryland 20740, USA
Quantum Materials Center, University of Maryland, College Park, Maryland
20742, USA  Neda Foroozani  Laboratory for Physical Sciences, 8050 Greenmead
Drive, College Park, Maryland 20740, USA  Quantum Materials Center, University
of Maryland, College Park, Maryland 20742, USA  Stefan Fritz  Laboratory for
Electron Microscopy, Karlsruhe Institute of Technology, Karlsruhe, 76131,
Germany  Dagmar Gerthsen  Laboratory for Electron Microscopy, Karlsruhe
Institute of Technology, Karlsruhe, 76131, Germany  Kevin D. Osborn
Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, Maryland
20740, USA  Quantum Materials Center, University of Maryland, College Park,
Maryland 20742, USA  Joint Quantum Institute, University of Maryland, College
Park, MD 20742, USA

###### Abstract

Quantum two-level systems (TLSs) are present in the materials of qubits and
are considered defects because they limit qubit coherence. For superconducting
qubits, the quintessential Josephson junction barrier is made of amorphous
alumina, which hosts TLSs. However, TLSs are not understood generally – either
structurally or in atomic composition. In this study, we greatly extend the
quantitative data available on TLSs by reporting on the physical dipole moment
in two alumina types: polycrystalline $\mathrm{\mathrm{\gamma-
Al}_{2}\mathrm{O}_{3}}$ and amorphous $\mathrm{a-Al}\mathrm{O_{x}}$. To obtain
the dipole moments $p_{z}$, rather from the less-structural coupling parameter
g, we tune individual TLSs with an external electric field to extract the
$p_{z}$ of the TLSs in a cavity QED system. We find a clear difference in the
dipole moment distribution from the film types, indicating a difference in TLS
structures. A large sample of approximately 400 individual TLSs are analyzed
from the polycrystalline film type. Their dipoles along the growth direction
$p_{z}$ have a mean value of 2.6$\pm$0.3 Debye (D) (0.54$\pm$0.06
e$\mathring{A}$) and standard deviation $\sigma$ = 1.6$\pm$0.2 D
(0.33$\pm$0.03 e$\mathring{A}$). The material distribution fits well to a
single Gaussian function. Approximately 200 individual TLSs are analyzed from
amorphous films. Both the mean $p_{z}$ =4.6$\pm$0.5 D (0.96$\pm 0.1$
e$\mathring{A}$) and $\sigma$ =2.5$\pm$0.3 D (0.52$\pm 0.05$ e$\mathring{A}$)
are larger. Amorphous alumina also has very large $p_{z}$, > 8.6 D (1.8
e$\mathring{A}$), in contrast to polycrystalline which has none of this
moment. These large moments agree only with oxygen-based TLS models. Based on
data and the candidate models (delocalized O and hydrogen-based TLSs), we find
polycrystalline alumina has smaller ratio of O-based to H-based TLS than
amorphous alumina.

##  I Introduction

Long coherence times are essential for quantum information processing and this
implies high-quality Josephson junctions (JJs) in superconducting qubits [1,
2, 3, 4, 5, 6]. For over a dozen years, quantum tunneling two-level systems
(TLSs) have been known to be defects that cause loss and limit coherence of
qubits [7, 8]. In addition, TLSs create telegraphic noise and 1/f noise [9,
10, 11] in superconducting qubits [12, 13, 14], semiconducting qubits [15],
and astronomy photon detectors [16, 17]. There are several strategies to
improve the qubit coherence time such as material optimization [18, 19, 20,
21], surface treatments [21, 22, 23], and engineering of the qubit geometry to
decrease the participation of TLSs [24, 25, 26].

In the quintessential JJ, an amorphous alumina barrier is grown thermally on
the surface of aluminum [1, 2, 4]. Loss tangents of amorphous alumina in JJs
[18] and in grown films [27] are measured to be approximately $2\times
10^{-3}$, much higher than that in crystalline alumina from the sapphire
substrates used for qubit fabrication [28]. Accordingly, amorphous materials
are believed to have higher loss than crystalline ones due to additional
tunneling degrees of freedom (TLSs) in the former. HBN and other 2D materials
are being investigated for JJs [29, 30, 31], but an alternative method to
improve the JJ barrier uses annealed crystalline alumina [32, 33]. Crystalline
alumina studies show a decrease in both TLS density [32] and TLS-qubit
couplings, $g$, relative to amorphous alumina [33].

Recent TLS analysis techniques use dc-tuned electric [34, 35, 36, 37] or
strain field [38, 39, 40] for the observation of individual nanoscale defects.
According to the Standard Tunneling Model (STM) [41, 42], individual TLSs have
a dipole moment $p$, transition energy $E$ and tunneling energy $\Delta_{0}$.
Generally, the TLS is described as tunneling charge presumed to be an atom or
small group of atoms, though a recent study reports on a possibility of
trapped quasiparticles [43]. Their identification is a 50 years old mystery
[44].

TLSs have an ac-coupling to quantum systems,

$g=\frac{\Delta_{0}}{E}\,\frac{2\,p_{z}\mathrm{E}_{rms}}{\hbar},$ (1)

which is related to dipole moment component $p_{z}$ along zero-point
fluctuation of electric field $\mathrm{\mathbf{E}}_{rms}$. However, the most
common measurement of $g$ does not allow extraction of $p_{z}$ because
$\Delta_{0}$ is unknown or $\mathrm{E}_{rms}$ is not uniform [8, 32, 33]. On
the other hand, static dc-tuned measurements allow measurements of individual
$p_{z}$ [34, 35] and dynamically biased experiments extract averaged $p_{z}$
[45]. Such dynamical bias can induce Landau-Zener transitions, and recent work
shows that a resonator can even exhibit dynamical decoupling using these
transitions [46]. Recently, voltage bias gates are added above or below the
target area to tune TLSs [36, 37]. The extractions of dipoles are possible
from $g$ and extracted position, but the distribution is given as a function
of $g$ rather than $p_{z}$ [36]. Also, only small samples of $p_{z}$ were
directly extracted in the past: 13 in amorphous alumina [35] and 64 in silicon
nitride [34]. To the best of our knowledge, a comparison of $p_{z}$ in two
different materials has not been performed in a single study.

In this letter we study individual TLSs in both polycrystalline alumina
$\mathrm{\gamma-Al\mathrm{{}_{2}O}_{3}}$ and amorphous alumina
$\mathrm{a-Al}\mathrm{O_{x}}$. We follow the circuit schematic of Ref. [34],
and we now name it an Electrical-Bridge Quantum-Defect Sensor (EBQuDS). The
TLSs were analyzed in films with an approximate thickness of 20 nm. The TLSs
in $\mathrm{\gamma-Al\mathrm{{}_{2}O}_{3}}$ films are relatively stable, and
allow us to obtain a large distribution of 394 TLS dipole moments $p_{z}$. In
$\mathrm{a-Al}\mathrm{O_{x}}$ films, 189 TLS moments are extracted despite
higher TLS noise. Compared to the $\mathrm{\gamma-Al\mathrm{{}_{2}O}_{3}}$
film, larger averaged dipole and standard deviation $\sigma$ are extracted in
$\mathrm{a-Al}\mathrm{O_{x}}$ films and 10% of TLSs have larger dipole than
the maximum extracted in $\mathrm{\gamma-Al\mathrm{{}_{2}O}_{3}}$. Specific
TLS structures were proposed using available information from recent work on
density functional theory (DFT) analysis of TLSs in alumina, where both
hydrogen [47, 48] and oxygen [49, 50, 51] based TLS have been proposed as the
interstitial defects. A comparison of the dipole moments for both
$\mathrm{\gamma-Al\mathrm{{}_{2}O}_{3}}$ and $\mathrm{a-Al}\mathrm{O_{x}}$
film types allows possible TLS origin identification.

##  II Method

Figure 1: (a) Optical image of microwave resonator with an abbreviated wiring
schematic. The source voltage is filtered through an RC low pass filter and a
copper powder filter (CPF). The combination of resistance in the RC filter and
the 3dB attenuator gives $\mathrm{V_{bias}}$ = 9.56
$\times\mathrm{10^{-3}\times V_{source}}$. (b) $\left|S_{21}\right|$ of one
$\mathrm{\gamma-Al_{2}\mathrm{O}_{3}}$ resonator. It shows multiple dips
indicating TLSs strongly coupled to the resonator. The black curve is the
ensemble average $\left|S_{21,avg}\right|$, obtained by averaging $S_{21}$
from different bias voltages. The intrinsic quality $\mathrm{Q_{i}}$ is
$\mathrm{1\,/\,(1.47\times 10^{-3})\approx\,680}$ according to the red fitting
line .

Fabrication starts by $\mathit{in\>situ}$ growth of Al/alumina/Al trilayers on
a 3-inch Si substrate, where the alumina is the material hosting the TLSs. The
$\mathit{in\>situ}$ method prevents substantial hydrogen contamination, but
diffusion of hydrogen is also difficult to prevent in standard lithographic
processing [52]. Then, a first BCl3 etch forms a mesa into the top 2 layers,
defining 4 equal capacitors ($\mathrm{C_{1}-C_{4}}$). Next, a second BCl3 etch
forms the base-metal including a resonator inductor L and ground plane.
Finally, silicon nitride is deposited as a wiring dielectric, vias are etched
by SF6, and an Al wiring layer is defined to connect the inductor to the
capacitors. This creates the final resonator structure as shown in Fig. 1 (a).
Alumina in the dielectric layer is designed to have an approximate thickness
$d$ = 20 nm and a volume $\mathbb{V}=1.11\times 10^{-17}m^{3}$ in each
capacitor.

The two sample types are grown in separate chambers. The polycrystalline
sample has 20 nm thick $\gamma-\mathrm{Al_{2}O_{3}}$ film deposited by
electron-beam evaporation from 99.99% purity $\mathrm{Al_{2}O_{3}}$ pellets
with a base pressure of < $5\times 10^{-7}$ torr and alumina is sandwiched by
two 100 nm thick aluminum films. On the other hand, the
$\mathrm{a-Al}\mathrm{O_{x}}$ film is 14.7 nm thick with x = 1.3 $\pm$ 0.1 and
is grown by iterating eight rounds of 1 nm Al deposition followed by static
oxidation at 9.5 mbar of oxygen at 250 ∘ C as described in ref. [53]. Given
this oxidation condition, no long-range ordered (crystalline) structure was
detected from transmission electron microscopy and only 3% unoxidized aluminum
was found inside the $\mathrm{a-Al}\mathrm{O_{x}}$ layer. Transmission
electron microscopy shows that the thickness can vary from 10 to 20 nm in some
rare cases.

The resonator inductively couples to the transmission line so that a 2-port
microwave transmission measurement can be carried out. The applied voltage
from room temperature is filtered by an RC filter, 3dB attenuator and a copper
powder filter. It generates an dc biased field $\mathrm{E_{ex}}$ across each
capacitor. The maximum $\mathrm{E_{ex}}$ is $90\mathrm{\,kV/m}$ with which we
observe no refrigerator heating, thus no significant leakage current. Two
resonators were fabricated per chip with nominally the same capacitors, but
with different value inductors, giving resonance frequencies of approximately
$f_{0}=$ 5.0 GHz and 4.4 GHz. The resonators were measured at or below 60 mK.
A less than 1 probing photon number $\bar{n}$ is used for all the reported
data, to allow observation of TLSs near their ground state.

With a known external field $\mathbf{\mathrm{E_{ex}}}$, the asymmetry energy
$\Delta$ is shifted as
$\Delta^{\prime}\,=\,\Delta+2\,p_{z}\,\mathbf{\mathrm{E_{ex}}}$ [34].
Therefore, the resultant TLS energy is

$E\,=\,\sqrt{(\Delta+2\,p_{z}\,\mathbf{\mathrm{E_{ex}}})^{2}+\Delta_{0}^{2}}$
(2)

The resonator constitutes a circuit QED system with a Jaynes-Cummings model
modified for many TLSs. TLSs can be resolved individually by the resonator
when the cooperativity $g^{2}/\gamma_{\mathrm{TLS}}\,\kappa\,\geq 1$ [54],
where $\gamma_{\mathrm{TLS}}$ is the TLS decay rate for the strongly coupled
TLS, and $\kappa=\kappa_{e}+\kappa_{i}$ is the resonator decay rate from
external coupling and internal loss. We increase the $g$ and cooperativity by
reducing $\mathbb{V}$, since
$\mathrm{E}_{rms}=\sqrt{hf_{0}/8\epsilon_{r}\epsilon_{0}\mathbb{V}}$ in our
parallel-plate capacitor resonator [8].

A single transmission trace $\left|S_{21}\right|$ is shown in Fig. 1 (b) from
a $\mathrm{\gamma-Al_{2}\mathrm{O}_{3}}$ resonator. Within the bandwidth of
the resonator, a few fine resonance dips reveal the energies of individual
TLSs. However, the TLSs often only couple weakly to the resonator. Therefore,
we obtain an approximate intrinsic material loss tangent $\tan\delta_{0}$ by
using the averaged $S_{21}$ traces from different voltage biases, yielding an
ensemble-averaged $S_{21,avg}$. The |$S_{21,avg}$| component of the result is
shown as the solid black curve in Fig. 1 (b). A fit (dashed red curve) to
$S_{21,avg}$ yields $\tan\delta_{0}\,=\,1/Q_{i}\,=\,1/680\,=\,1.5\times
10^{-3}$ and the external (or coupling) quality factor is extracted as
$\mathrm{Q_{e}}\,=2\pi f_{0}/\kappa_{e}=\,590$. The same procedure performed
on $\mathrm{a-Al}\mathrm{O_{x}}$ gives an intrinsic loss tangent of
$\tan\delta_{0}\,=\,1/\overline{Q_{i}}\,=\,1/1020\,=\,9.8\times 10^{-4}$,
which is smaller than that of $\mathrm{\gamma-Al_{2}\mathrm{O}_{3}}$.
$\overline{Q_{i}}$ is the averaged $Q_{i}$ of $\mathrm{a-Al}\mathrm{O_{x}}$
and its $Q_{i}$ are distinct with every cooldown unlike
$\gamma-\mathrm{Al_{2}O_{3}}$ (Appendix C). Our loss tangents are similar to
alumina with different growth methods, where the loss is measured at
$\mathrm{tan\delta_{0}\,}=\,1.6\times 10^{-3}$ [8, 55], $7\times 10^{-4}$
[56], and $1.6\times 10^{-3}$ [57]. Below we discuss TLSs measured in both
alumina film types using two resonators for each type.

##  III Results and Discussion

Figure 2: Data on nanoscale thick
$\gamma-\mathrm{Al_{2}O_{3}}$(polycrystalline) film in cQED system (a) Color
scale plot of transmission $|S_{21}|$ vs. frequency $f$ and electric field
$\mathrm{E_{ex}}$. Data show a main resonance at 4.974 GHz. Several local
minima in $|S_{21}|$ reveal the energy of individual TLSs. (b) Several TLSs in
blue hyperbolas are fitted to the energy model (Eq. 2), where $p_{z}$ comes
from the curvature of the energy hyperbola. (c) The entire data set from this
film type yields 394 moments $p_{z}$. The red dashed line is a Gaussian
function multiplied by $p_{z}$ as fit to data. A Seaborn box is plotted, where
the red color line is the mean line and the right and left side of box
represent 25th and 75th percentage of dipole data. (d) Material TLS density of
$p_{z}$ after accounting for the experimental weighing factor. A red dashed
Gaussian line is shown with the corresponding fit parameters in (c). From the
fit we report the material mean $p_{z}$ of 2.6 D and standard deviation
$\sigma$ = 1.6 D. The black dashed line illustrates a possible material
density if we assume an isotropic TLS direction.

Fig. 2 (a) shows a TLS spectrum example, $\left|S_{21}\right|$, as a function
of frequency $f$ and the dc-field $\mathrm{E_{ex}}$ for $\mathrm{\gamma-
Al_{2}\mathrm{O}_{3}}$ measured on one of the two resonators during one cool-
down. TLS energies exhibit hyperbolic energies versus $\mathrm{E_{ex}}$, as
shown in Fig. 2 (a) in agreement with Eq. 2. Similar spectra have been
observed and analyzed in a previous study on silicon nitride [34]. One can
estimate the $\Delta_{0}$ from the minimum of the TLS energy
($hf_{m}$$=\Delta_{0}$), and $p_{z}$ from the hyperbola – a steeper curvature
gives a larger dipole moment. A optimized Monte Carlo fit is performed on each
TLS energy to extract $p_{z}$ of the specific TLS (see Appendix. D). Only well
defined TLS energy curves are selected for analysis. Example fits are plotted
as blue hyperbolas in Fig. 2 (b).

TLSs change their energies randomly during cool-downs from room temperature.
From four different cool-downs, we created different sets of TLSs in the two
resonators with one material type. According to the standard TLS distribution
[41], TLSs have log-uniform tunneling energies such that there are negligible
distribution changes for TLS tunneling energies that are only 0.6 GHz
different in our two resonator frequencies (see Appendix. E). We therefore
combine all the data from different runs in the two resonators to enlarge the
sampling number and improve the statistics: a total of 394 TLSs from two
resonators are analyzed to form the measured $p_{z}$ distribution $H(p_{z})$
with an average of 3.5 $\pm 0.4$ Debye (D) shown in Fig. 2 (c). The accuracy
of extraction is limited by the uniformity of thickness of alumina rather than
fitting process. A large amounts of individual TLSs allow relatively accurate
representation of the TLS moments in $\gamma-\mathrm{Al_{2}O_{3}}$.

Though the measured polycrystalline distribution $H(p_{z})$ has a mean value
of 3.5 D, this is not an intrinsic material property. At a given electric
field bias range $\Delta\mathrm{E}_{ex}$, TLSs with larger dipole moments have
a larger shift in asymmetry energy $\Delta^{\prime}$ relative to smaller
moments, and this leads to a higher probability of the former moments having
their energy minimum within the resonator bandwidth (see Appendix A and Ref
[34]). The intrinsic material TLS dipole distribution $D(p_{z})$ is related to
TLS material density $P_{0}\,=\,\intop D(p_{z})\,dp_{z}$ (in units of
$J^{-1}m^{-3}$), and can be calculated from
$D(p_{z})\,=\,\frac{1}{\mathbb{V}}\,\frac{H(p_{z})}{2\,p_{z}}\,\frac{1}{\,\Delta
E_{ex}}\,\frac{f_{0}}{\Delta f_{0}}$ , where $f_{0}$ is the resonator
frequency and $\Delta f_{0}$ is the frequency span of the $S_{21}$
measurement. The red dashed line in Fig. 2 (c) shows a fit using a maximum
likelihood estimation (MLE) method. The fitting function is a modified
Gaussian distribution, which is a Gaussian distribution multiplied by $p_{z}$.
The TLS material density $D(p_{z})$ is shown in Fig. 2 (d), where the red line
shows a Gaussian function matching the fit parameters in Fig. 2 (c).

From the fit, we find that the polycrystalline material distribution
$D(p_{z})$ has a fit mean dipole moment of $\bar{p}_{z}$= 2.6 $\pm 0.3$ D
(=0.54$\pm 0.05$ e$\mathring{A}$) and $\sigma$ = 1.6 D (=0.33
e$\mathring{A}$). The computed TLS density $P_{0}$ is $1.0\pm 0.1\times
10^{44}\mathrm{(J^{-1}m^{-3})}$. This computed value of TLS density along with
the dipole moments agrees with the measured loss tangent. As a result we used
the material units in panel (d) for the distribution $D(p_{z})$.

For amorphous samples or random voids within polycrystals, we expect TLS
dipoles to be random in angle (isotropic). For a case of one single dipole
magnitude $p_{0}$ and uniform distribution in $\mathrm{cos}\theta$, where
$\theta$ is the angle of dipole to z-axis, results isotropy. Therefore,
$D(p_{z})$ is expected to be independent of $p_{z}$ until the maximum value
$p_{0}$. As a guide to the eye, an isotropic distribution (random angle) with
dipole moment $p_{0}$ = 4.5 D is shown as a black dashed line in Fig. 2 (d).
The positive slope in the observed distribution indicates that we have a
departure from isotropic distribution (isotropic TLSs give only non-positive
slope). Thus, data in Fig. 2 (d), shows that $\gamma-\mathrm{Al_{2}O_{3}}$
TLSs can be different than the standard model for TLSs (designated for
amorphous samples [41, 42]). The anisotropic angular distribution may be
caused by the polycrystalline film texture (crystallite orientation) which
influences the TLS orientation.

Figure 3: (a) Example of processed transmission data on amorphous
$\mathrm{a-Al}\mathrm{O_{x}}$ film resonator versus frequency $f$ and external
electric field $E_{ex}$ (see main text). The LC resonance is approximately
5.132 GHz (no longer visible after data processing). (b) The same data with
traces fit to TLS energy function. 7 fitted dipole moments are extracted at
values of 3.2 - 12.7 D. TLSs whose energy does not depend on bias voltage are
marked by arrows. Curved red dashed trace shows an anomalous TLS; it switches
between a hyperbola and a nearly constant energy of 5.116 GHz at $E_{ex}=-30$
kV/m and $E_{ex}=-18$ kV/m. (c) The measured distribution of 189 TLSs and a
Seaborn box. The green line represents the largest dipole measured in
$\gamma-\mathrm{Al_{2}O_{3}}$. (d) The probability of the material TLS dipole
distribution. Dashed lines in (c) and (d) are modified and regular Gaussian
functions and obtained by two methods. The red lines are acquired by
calculation, and he yellow lines are acquired by fitting. We report a mean
dipole = 4.6 $\pm 0.5$ D and $\sigma$ = 2.5 $\pm 0.3$ D. Analysis on missing
extracted dipoles was needed in the amorphous film, but it also gives a mean
value much larger than in the $\gamma-\mathrm{Al_{2}O_{3}}$ (see main text and
Appendix G).

Fig. 3 (a) shows transmission spectroscopy results for
$\mathrm{a-Al}\mathrm{O_{x}}$. However, the $\mathrm{a-Al}\mathrm{O_{x}}$
spectra are not as clear as in $\gamma-\mathrm{Al_{2}O_{3}}$ due to higher
noise in the spectra, despite using the same setup. To improve the TLS signal
contrast, the transmission ($S_{21}$) data is shown after processing, unlike
the polycrystalline film data shown earlier. For our first processing step we
chose to subtract $S_{21,avg}$ from $S_{21}$, and increase further contrast
using the formula $\,(S_{21}(dB)\,-\,S_{21,avg}(dB))\times|S_{21,avg}|$. As a
second and third processing step, we apply a low-pass filter in the frequency
direction and then take the derivative with respect to frequency. The final
result is plotted in arbitrary unit (AU) in Fig. 3 (a) and (b) and we added
hyperbolic fitting traces (blue lines) to the panel (b).

As we will show in detail below, TLSs within $\mathrm{a-Al}\mathrm{O_{x}}$ are
less stable than those in $\gamma-\mathrm{Al_{2}O_{3}}$ – the former TLSs show
sudden switchings in energy or even become invisible in time within the
resonator bandwidth, making it more difficult to identify individual TLSs.
This leads to a higher error in the Monte Carlo fit. We also observed energy
features that are almost independent of $E_{ex}$, as indicated by black
arrows. They are not expected because all coupled TLSs should be frequency
tunable in the device. At -8 kV/m, one hyperbola seems to change slope (as
indicated by the start of a blue dashed line), although instead this event may
represent a transition in observing two separate TLSs.

Surprisingly, one of them seems to be only partially described by hyperbola in
red-dashed curve. Increasing bias voltage at -30 kV/m, this TLS seems to
switch from a regular TLS state to an unknown state which has constant
transition energy under bias until -19 kV/m, and finally switch back to normal
TLS behavior. This indicates an unexpected state near its energy minima
$\Delta_{0}$, which we believe has not been identified previously.

Unlike the other sample, the amorphous films show most of the hyperbolas from
TLSs in the bias range of $-30$ to $30$ $\mathrm{kV/m}$ (more data in [58]).
Outside of this range TLSs tracks are seen, but they don’t trace out a smooth
hyperbola. Furthermore, we find that the most TLSs do not appear twice after
repeating the voltage scanning within the same cool-down. In a small fraction
of TLS hyperbola (< 3%) a TLS hyperbola has the same dipole and the minimum in
energy is within 1 MHz, such that it is regarded as the same TLS and
disregarded in distribution.

In $a-\mathrm{Al_{2}O_{3}}$, we identify and analyze a total of 189 TLSs using
multiple field sweeps and cool-downs according to the above procedure. The
measured distribution $H(p_{z})$ with counts and the probability of material
distribution $D(p_{z})$ are shown in Fig. 3 (c). The extracted $p_{z}$ shows a
broad range in value from 0.5 to 16 D with an average of 6.0 D and the
interquartile range (range from the 25th to 75th points) of 3.8 D. Because of
the large deviation, we cannot get a reasonable fitting to a Gaussian from the
MLE method as shown in yellow lines, which have mean value of 1.6 D and
standard deviation of 4.2 D. Instead, we calculate the material average dipole
moment $\bar{p}_{z}$ for $D(p_{z})$ from $H(p_{z})$, using $\bar{p}_{z}=\intop
p_{z}D(p_{z})dp_{z}/\intop D(p_{z})dp_{z}\,=\,$$\intop H(p_{z})dp_{z}/\intop
p_{z}^{-1}H(p_{z})dp_{z}$ , and the standard deviation in a similar way. From
this we find $\bar{p_{z}}\,=\,4.6\pm 0.5$ (= 0.96$\pm 0.1$ e$\mathring{A}$)
and $\sigma$ = 2.5$\pm$0.3 D (= 0.52$\pm 0.05$ e$\mathring{A}$). A Gaussian
curve with these parameters are plotted as red dashed line.

The calculated loss from the dipole distribution is $3.2\times 10^{-4}$ which
is smaller than $9.75\times 10^{-4}$ reported above. The missing TLS
extraction happens in two ways. On one hand, larger $p_{z}$ TLSs have a higher
possibility to interact with other TLSs and their frequencies are prone to
switch to other states or diffuse. On the other hand, small $p_{z}$ TLSs
require longer time to acquire TLS hyperbolas and the signal-to-noise ratio is
smaller due to smaller coupling to the resonator. As a result, the TLS
extraction does not include most of the TLS unlike the extraction in
$\gamma-\mathrm{Al_{2}O_{3}}$. We estimate the minimum mean moment of
amorphous alumina from missing TLSs. We make no claim about anisotropy in this
film since the missing TLSs may create one of the peaks in the distribution of
$p_{z}$ (Appendix G).

Figure 4: Time dependence of processed $|S_{21}|$ with a frequency range
$\geq$ 30 MHz. (a) $\gamma-\mathrm{Al_{2}O_{3}}$ spectroscopy versus time
shows the TLSs are relatively stable in frequency near the transmission
minimum for 10s of hours. (b) $\mathrm{a-AlO_{x}}$ spectroscopy versus time
shows relatively large TLS energy switching and drift.

To decipher the role of TLS-TLS interaction, we next conduct temporal
spectroscopes for the two different film types: $\gamma-\mathrm{Al_{2}O_{3}}$
and $\mathrm{a-Al}\mathrm{O_{x}}$. Fig. 4 shows processed $S_{21}$ traces
observed over many hours. This resonant TLS noise is believed to be caused by
interactions with thermally excited low-frequency TLSs [13, 59]. As shown in
Fig. 4 (a), TLSs biased at 0 V in $\gamma-\mathrm{Al_{2}O_{3}}$ films near the
resonance frequency are relatively stable – their energies drift by less than
2 MHz over tens of hours. On the contrary, TLSs in
$\mathrm{a-Al}\mathrm{O_{x}}$ behave similar to Ref. [13]. TLSs show irregular
drifts of more than 5 MHz, including multiple telegraphic switching events
(blue arrows) and abrupt TLS shifts (black arrow). Due to the larger dipole
moments observed in $\mathrm{a-Al}\mathrm{O_{x}}$, we expect a larger
interaction than that in $\gamma-\mathrm{Al_{2}O_{3}}$ (assuming that the low-
frequency thermally activated TLSs are similar to the high frequency ones).
The large unstable behaviors shown in the amorphous films occur in a few
hours, e.g., a 4 D hyperbola track in $\mathrm{a-Al}\mathrm{O_{x}}$ data took
about 5 hours to obtain.

Comparable results on amorphous alumina exist. One study found $p_{z}$ in the
range of 2.3 -7.4 D, using a few analyzed TLSs [40]. Other field tuned
measurements in a-AlOx, studying the barrier of JJs, detected several moments
with $p_{z}$ = 1.0 - 2.9 D [35]. Measurements of the transition dipole moments
in a-AlOx of JJs indicate that $p_{z}$ $\leq$ 6.0 D [8] and $p_{z}$ $\leq$ 4.8
D [60] in two TLS dipoles measured. The existing data on amorphous alumina
TLSs seems consistent with our observations, though the growth methods are
slightly different.

Finally, we return to comment on the possible origins of TLSs in our alumina
from a comparison to recent TLS DFT [47, 48, 49, 50] and molecular dynamics
[51] simulations on aluminum oxide. Two models related to hydrogen(H) suggest
total dipole moment $p$ < 3.0 D [47, 48]. Holder et al. find that hydrogen
aluminum-vacancy TLSs $\mathrm{V_{Al}-H}$ in $\alpha-\mathrm{Al_{2}O_{3}}$
have $p$ = 3.0 D [47]. Separately, Gordon et al. simulated the interstitial
hydrogen in $\alpha-\mathrm{Al_{2}O_{3}}$ at various two oxygen(O) atoms
distances where $p$ = 2.2 – 2.7 D [48]. Besides H-based simulations, two
models of O-based TLSs suggest $p$ > 4.2 D. DuBois et al. studied models of
delocalized oxygen atoms with six neighboring aluminum atoms [49, 50]. They
found oxygen deficient $\mathrm{AlO_{x}}$ for x = 1.25 by varying distances
between O and Al atoms, with $p$ = 4.2 - 6.5 D for TLSs with tunneling energy
$\Delta_{0}/h$ = 4 GHz [49]. Additionally, Paz et al. find natural bi-stable
structures in amorphous alumina including only Al and O atoms and calculate an
average $p$ = 4.2 D from 7 TLSs [51]. Although there are difference between
theoretical models, they are consistent in that H-TLSs have smaller $p$ than
O-TLSs.

The $\gamma-\mathrm{Al_{2}O_{3}}$ $D(p_{z})$ has a single peak at
approximately $p_{z}$ = 2.6 D, which can be sourced from H-TLS or two
unresolved peaks of both H- and O-TLSs. Although small in statistics,
$\mathrm{a-Al}\mathrm{O_{x}}$ has a wider spread in $D(p_{z})$ and two
separate peaks. Furthermore, $\mathrm{a-Al}\mathrm{O_{x}}$ TLSs have 10 %
population with $p_{z}$ > 8.6 D, where $p_{z}$ = 8.6 D is the maximum in
$\gamma-\mathrm{Al_{2}O_{3}}$. Using a comparison between two alumina datasets
and the fact that O-based TLS is the larger dipole in DFT structures, we find
a higher ratio of O-TLSs to H-TLSs in a-AlOx than
$\gamma-\mathrm{Al_{2}O_{3}}$.

##  IV Conclusion

In summary, we have extracted the dipole moment $p_{z}$ of hundreds of
individual TLSs in nanoscale-thick films of (polycrystalline)
$\gamma-\mathrm{Al_{2}O_{3}}$ and (amorphous) $\mathrm{a-Al}\mathrm{O_{x}}$
alumina. We have used an Electrical-Bridge Quantum Defect Sensor (EBQuDS),
which we show is suitable to characterize number of TLSs as quantum defects.
Analysis of the measured histogram of $p_{z}$ reveals that polycrystalline
alumina fits well to a single Gaussian peak. From the material distribution
(algebraically related to the measured one), we obtain that the mean TLS
moment of the polycrystalline film is $p_{z}=$2.6$\pm 0.3$ D (= 0.54$\pm 0.05$
e$\mathring{A}$) and $\sigma$ = 1.6$\pm 0.2$ D (= 0.33$\pm 0.03$
e$\mathring{A}$). Furthermore, the material distribution disagrees with the
isotropic model commonly used in amorphous materials, indicative of a
preferred texture (orientation) of the polycrystalline grains which host TLSs.
On the other hand, we cannot conclude if amorphous alumina dipoles are
isotropic or not because of missing TLSs extraction.

The ability to extract an accurate mean $p_{z}$ puts constraints on its defect
type, and allows us to make first comparisons to new microscopic structures
used in DFT calculations. The polycrystalline data show one dominant peak, and
could be showing dominance of H-TLSs, or unresolved peaks of both H- and
O-TLSs in the distribution. We find that $\mathrm{a-Al}\mathrm{O_{x}}$ has a
larger mean $p_{z}$ = 4.6$\pm$0.5 D, which is consistent with previous
amorphous alumina results. In contrast to $\gamma-\mathrm{Al_{2}O_{3}}$, the
TLSs switch more rapidly, and our $p_{z}$ distribution in amorphous alumina
yields a larger standard deviation (= 2.5$\pm$0.3 D) and two peaks in contrast
to one. The moments above 8.6 D (10% of the distribution) are larger than any
TLS in polycrystalline alumina and agree only with calculations of delocalized
O atoms. Due to this and other amorphous distribution features, we find that
the ratio of O- to H-TLSs is higher in amorphous samples than in the
polycrystalline ones. Because of its relative simplicity in distribution,
alumina seems to be an important material for further JJ-barrier studies.

##  V Acknowledgements

We acknowledge Lukas Radke, Hannes Rotzinger (both from Physikalisches
Institut, Karlsruhe Institute of Technology, Germany) and Martin Weides (James
Watt School of Engineering, University of Glasgow, UK) for support of
discussions on amorphous alumina sample growth.

##  Appendix A Material Density and Loss tangent

In this section, we derive the relationship between material TLS density,
$D(p_{z}),$ and the measured histogram, $H(p_{z})$. In the Standard Tunneling
Model (STM) [41, 42], the authors assumes that the density of levels per unit
volume and energy, $n(\Delta,\Delta_{0})$, depends on tunneling energy
$\Delta_{0}$ but is uniform in $\Delta$ giving

$n(\Delta,\Delta_{0})\,d\Delta
d\Delta_{0}\,=\,\frac{P_{0}}{\Delta_{0}}\,d\Delta d\Delta_{0},$ (3)

where $P_{0}$ is a constant in unit of $1\,/(\,J\,m^{3}$). However, the model
only considers a single moment |$\overrightarrow{p}$| = $p$. In our
experiments, we notice the dipole moment in z axis is not uniform and we add
dipole direction and magnitude dependence. For a general case, we write

$n(\Delta,\,\Delta_{0},\overrightarrow{p})\,d\Delta
d\Delta_{0}d^{3}p\,=\,\frac{P^{\prime}_{0}}{\Delta_{0}}\,D(\overrightarrow{p})\,d\Delta
d\Delta_{0}d^{3}p,$ (4)

where $D(\overrightarrow{p})$ is a generalized material TLS distribution and
$P^{\prime}_{0}$ is a new constant. However, $D(\overrightarrow{p})$ depends
on 3 Cartesian coordinates, and we only have measurement access to one
component, $p_{z}$. The full investigation of $D(\overrightarrow{p})$ is
beyond the scope of this paper, but we assume that $D(\overrightarrow{p})$ is
separable in $p_{x}$, $p_{y}$, and $p_{z}$. Therefore, we consider the case

$n(\Delta,\,\Delta_{0},\,p_{z})d\Delta
d\Delta_{0}dp_{z}\,=\,\frac{D(p_{z})}{\Delta_{0}}\,\,d\Delta
d\Delta_{0}dp_{z},$ (5)

where $\intop D(p_{z})\,dp_{z}\,=\,P_{0}$ and $D(p_{z})$ is the material
density mentioned in the main text. In TLS spectroscopy, $V_{bias}$ is
controlled such that $p_{z}$, $\Delta_{0}$, and $\Delta|_{V_{bias}=0}$ can be
extracted for individual TLSs and
$\Delta\,=\,\Delta|_{V_{bias}=0}\,+\,2\,p_{z}V_{bias}/l_{0}$, where $l_{0}$ is
the thickness of dielectric. Next, we change variables to include $V_{bias}$
through Jacobian transformation giving

$d\Delta
dp_{z}\,=\,dp_{z}dV_{bias}|\begin{array}[]{cc}\frac{\partial\Delta}{\partial
p_{z}}&\frac{\partial\Delta}{\partial V_{bias}}\\\ \frac{\partial
p_{z}}{\partial p_{z}}&\frac{\partial p_{z}}{\partial
V_{bias}}\end{array}|\,=(\frac{2\,p_{z}}{l_{0}})\,dp_{z}dV_{bias}.$ (6)

$N_{tot}$ is the total number of observed TLSs from measurement histogram,
$H(p_{zi})$,

$N_{tot}\,=\,\sum_{i}\,H(p_{zi})\,\Delta p_{zi},$ (7)

where $p_{zi}$ is the center value and $\Delta p_{zi}$ is the bin width of the
i-th bin. $N_{tot}$ can also be written related to Eq. 5 as

$N_{tot}=\mathbb{V}\,\int n(\Delta,\,\Delta_{0},\,p_{z})d\Delta
d\Delta_{0}dp_{z}.$ (8)

Substituting Eq. 6 into the above equation, we have

$N_{tot}\,=\mathbb{V}\,\int_{p_{min}}^{p_{max}}\,D(p_{z})\cdot(\frac{2\,p_{z}}{l_{0}})\,dp_{z}\,\int_{V_{1}}^{V_{2}}dV_{bias}\,\int\frac{d\Delta_{0}}{\Delta_{0}}.$
(9)

Similarly, we consider the i-th bin of $H(p_{zi})$ and the number of TLSs in
this bin

$N_{i}=H(p_{zi})\,\Delta p_{zi}$ (10)

In the case of small enough $\Delta p_{zi}$ and $\Delta_{0}$, we obtain

$H(p_{zi})\,\Delta
p_{zi}=\mathbb{V}\,(\frac{2\,p_{zi}}{l_{0}})\,D(p_{zi})\,\Delta p_{zi}\,\Delta
V_{bias}\,\frac{\Delta f_{0}}{f_{0}}$ (11)

or

$D(p_{z})\,=\frac{1}{\mathbb{V}}\,\frac{H(p_{z})}{2\,p_{z}}\,\frac{l_{0}}{\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}},$ (12)

where $\Delta f_{0}$ is the measurement frequency span. Thus, we prove that
$D(p_{z})$ is not proportional to measured histogram $H(p_{z})$, but
$D(p_{z})\propto\frac{H(p_{z})}{p_{z}}$.

Next, we derive the loss tangent, $\mathrm{tan}\delta_{0}$, from TLS
histogram. Following Ref. [17], loss due to TLSs can be described as

$\mathrm{tan}\delta\,=\,\int\frac{p_{z}^{2}}{\varepsilon}\,\frac{-\frac{1}{T_{2}}\,\mathrm{tanh}(\frac{E_{TLS}}{2k_{B}T})}{(T_{2}^{-2}+\Omega^{2}\frac{T_{1}}{T_{2}})+(\frac{E_{TLS}}{\hbar}-2\pi
f_{0})^{2}}d^{3}n$

$=\,\int
D(p_{z})\frac{p_{z}^{2}}{\varepsilon}\,\frac{-\frac{1}{T_{2}}\,\mathrm{tanh}(\frac{E_{TLS}}{2k_{B}T})\,d\Delta\frac{d\Delta_{0}}{\Delta_{0}}}{(T_{2}^{-2}+\Omega^{2}\frac{T_{1}}{T_{2}})+(\frac{E_{TLS}}{\hbar}-2\pi
f_{0})^{2}}dp_{z},$ (13)

where $\varepsilon$ is the permittivity constant, $\Omega$ is the Rabi
frequency, and $T_{1}$ ($T_{2}$) is TLS relaxation (decoherence) time. The
derivation of a similar integral has been presented in reference [17]. In the
case when $\Omega^{2}T_{1}T_{2}$ is much smaller than 1,

$\mathrm{tan}\delta\,=\,\frac{\pi}{\varepsilon}\int D(p_{z})p_{z}^{2}dp_{z}.$
(14)

Next, we estimate the loss tangent of $\gamma-\mathrm{Al_{2}O_{3}}$. By
replacing $D(p_{z})$ from Eq. 12, we get

$\mathrm{tan}\delta=\frac{\pi}{\varepsilon}\,\frac{1}{\mathbb{V}}\,\int\frac{H(p_{z})}{2\,p_{z}}\,\frac{l_{0}}{\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}}\,p_{z}^{2}\,dp_{z}$

$=\,\frac{\pi}{2\,\varepsilon}\,\frac{l_{0}}{\mathbb{V\,}\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}}\,\int H(p_{z})\,p_{z}\,dp_{z}$

$\approx\,\frac{\pi}{2\,\varepsilon}\,\frac{l_{0}}{\mathbb{V\,}\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}}\,\sum H(p_{z})p_{z}\Delta p_{z}$

$=\frac{\pi}{2\,\varepsilon}\,\frac{l_{0}}{\mathbb{V\,}\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}}\,\sum_{i}^{394}p_{zi}\,=\,1.4(1)\times
10^{-3}.$ (15)

The function is factor of $\pi$ larger than the equation (S6) in Ref. [34].
Notice that the loss tangent from the TLS histogram is similar to the bulk
resonator loss tangent, $\mathrm{tan}\delta_{0}\,=\,1.5\times 10^{-3}$,
reported in main text. Last, we estimate the material constant in the same
film

$P_{0}\,=\,\int D(p_{z})dp_{z}$

$\approx\,\sum_{i}\frac{1}{\mathbb{V}}\,\frac{1}{p_{zi}}\,\frac{l_{0}}{2\,\Delta
V_{bias}}\,\frac{f_{0}}{\Delta f_{0}}\,=\,1.0(1)\times 10^{44}(J^{-1}m^{-3}).$
(16)

In contrast to $\gamma-\mathrm{Al_{2}O_{3}}$, the loss tangent of amorphous
alumina is 3.2(3) $\times 10^{-4}$ which is few times smaller than the
measured amorphous alumina loss tangent (=9.8(9)$\times 10^{-4}$). It is
expected because we do not fit those TLSs with uncompleted hyperbola curves.

##  Appendix B Internal Quality Factor Fitting And Bias Filtering

In this section, we discuss the effect of bias line filtering and TLS noise on
resonator data fittings. Filtering noise in the bias line is essential to
study the individual TLSs in both film types. We performed a control
experiment with additional bias-line noise. We start from a setup where the
bias line has only a low-pass copper powder filter and a 12GHz K&L filter.
Fig. 5 (a) shows measurements of $\gamma-\mathrm{Al_{2}O_{3}}$ TLS
spectroscopy with low frequency noise and Fig. 5 (b) shows one |$S_{21}$| at
fixed bias. There is no observation of any individual TLS. Surprisingly, the
fit gives an internal quality factor, $\mathrm{Q_{i,noise}}$ = 1600, which is
higher than $\mathrm{Q_{i}}$ = 680 reported in the main text. It is believed
that without proper noise filtering, the bias voltage noise strongly affects
the visibility of the TLSs. The change of TLS frequency due to voltage noise
is

$\delta\omega_{TLS}\,=\,\frac{\Delta}{\hbar
E_{TLS}}\delta\Delta\,=\,\frac{\Delta}{\hbar E_{TLS}}\times 2p_{z}\delta
V_{ex}/l_{0}.$ (17)

In the case of multiple TLSs coupled to a resonator, the transmission rate
[61]

$S_{21}(\omega)\,=\,1-\frac{\kappa_{c}/2}{\frac{\kappa_{c}+\gamma_{c}}{2}+i\,(\omega-\omega_{c})+\sum\frac{g_{i}^{2}}{\gamma_{i}/2\,+\,i\,(\omega-\omega_{i})}},$
(18)

where $\kappa_{c}\,$is the cavity’s decay rate to the transmission line,
$\gamma_{c}$ is the cavity decay rate to the environment when no TLSs exist,
$\omega_{c}$ is the cavity resonance, $g_{i}$ is the coupling strength of each
TLS, $\gamma_{i}$ is the decoherence rate, and $\omega_{i}$ is the resonant
frequency of the i-th TLS. Inspired by Ref. [62], we assume that every TLS
experiences a Gaussian voltage noise and we rewrite transmission rate as

$S_{21}\,=\,1-\frac{\kappa_{c}}{2}\,\dotsintop\frac{\prod_{i}\sqrt{\frac{1}{2\pi\sigma_{i}}}\,exp[\frac{-\delta\omega_{i}^{2}}{2\,\sigma_{i}}]\,d\delta\omega_{i}...d\delta\omega_{n}}{\frac{\kappa_{c}+\gamma_{c}}{2}+i\Delta_{c}+\sum\frac{g_{i}^{2}}{\frac{\gamma_{i}}{2}+i(\Delta_{i}+\delta\omega_{i})}},$
(19)

where $\Delta_{c}=\omega-\omega_{c}$ and $\Delta_{i}=\omega-\omega_{i}$yy The
standard deviation, $\sigma_{i}$, of the i-th TLS depends on $\delta V_{ex}$
and $p_{z}$ such that the sensitivity to voltage noise is described as an
averaged effect. The enhancement of $\sigma_{i}$ exterminates the effect of
the i-th TLS. A simulation of different $\sigma_{i}$ is shown in Fig. 6 (a).
Two TLSs are coupled to the resonator. The upper TLS in the plot is not
affected by voltage noise and the lower TLS in the plot has $g$, $\gamma_{0}$,
and a random frequency shift with deviation, $\sigma$. Simulations show that
the lower TLS has smaller and smaller effect on the resonator while voltage
noise (or $\sigma$) increases. The phenomena of the voltage noise is
qualitatively similar to the increasing decoherence $\gamma$ of TLS as shown
in Fig. 6 (b).

Figure 5: (a) Spectroscopy of DC bias sweep on $\gamma-\mathrm{Al_{2}O_{3}}$
with voltage noise. (b) One example of $|S_{21}|$ from (a) without proper
filtering. Voltage noise obscures TLSs within the spectrum and $Q_{i}=1600$.
Figure 6: Simulation of Eq. 19 with resonator parameters
$\kappa_{c},\,\gamma_{c}$ = 4 MHz, $f_{c}$ = 4 GHz, and two TLSs having $g$ =
0.7 MHz, $\gamma_{0}$ = 0.5 MHz. Only one of them is affected by voltage noise
(lower one). (a) The solid blue line shows resonator coupled to two TLSs and
both have no voltage noise. The colored dashed lines show the enhancement of
$\sigma$ from 0.1 $\times\,g$ to 10 $\times\,g$ on the lower TLS. (b)
Simulation of the lower TLS under enhanced decoherence rate. The color lines
indicate the behavior of the lower TLS are similar to enhancement of $\sigma$
in (a).

##  Appendix C Fitting of the averaged $S_{21}$ of $\mathrm{a-AlO_{x}}$

Due to the low Q (<250) in $\mathrm{a-AlO_{x}}$ resonator, the resonator’s
transmission data is easily perturbed by frequency-dependent background. For
example, there will be (1) spurious modes: existing in the input-output
cables, the sample box or on the chip and (2) electrical components:
amplifiers, circulators, or attenuation. One obvious result is that the off-
resonant part of IQ plot deviates from the circle fitting curve where
|$S_{21}$| is not constant. Thus, we modified the fitting function to better
extract resonator’s parameters. We consider the background as multiple low-Q
modes, which have no direct coupling with each others and our resonator and
the their transmission rate

$S_{21,lowQ}=\prod_{k=1}^{N}(1-\frac{Q_{k}/Q_{e,k}}{1-2\,i\,Q_{k}\,(\omega-\omega_{k})/\omega_{k}}),$
(20)

where $Q_{k},Q_{e,k}$ and $\omega_{k}$ are total quality factor, external
quality factor and resonant frequency, respectively. In general, in a small
range around resonant frequency $\omega_{0}$, the background can be simplified
to $(1+(a+ib)(\omega-\omega_{0}))$, where a and b is a constant. In the case
that the $Q$ of the nearest mode to the measured resonator is in the order of
10 and $Q_{e}/Q_{i}\lesssim 1$ and each mode separates few hundreds of MHz,
$S_{21,lowQ}$ can be expanded around $\omega_{0}$ to the first order of
$\omega$. As a result, the final fitting function is

$S_{21}=C(1+(a+ib)(\omega-\omega_{0}))e^{i\theta}(1-\frac{Q/Q_{e}e^{i\phi}}{1+2i\frac{Q}{\omega_{0}}(\omega-\omega_{0})}).$
(21)

Fig. 7 shows two averaged $S_{21}$, which are normalized to 1, from two
different cooldowns: (a) and (b) are from the dataset of Fig. 12 ; (c) and (d)
are from the dataset shown in supplementary [58]. To our surprise, the $Q_{i}$
of $\mathrm{a-AlO_{x}}$ sample are distinct in different cooldowns and the
fits vary from 710 to 1510. We do not understand this variance in amorphous
sample and no significant $Q_{i}$ changes for our
$\gamma-\mathrm{Al_{2}O_{3}}$ sample in different cooldowns.

The new averaged of $Q_{i}$ is 1020 for amorphous alumina which is 1.5 times
higher than $\gamma-\mathrm{Al_{2}O_{3}}$. However, the loss tangent is lower
in amorphous alumina can probably because of the below two reasons. First,
amorphous alumina can have smaller density of TLSs with higher average dipole
moment and lower loss tangent in the end. Second, amorphous alumina has higher
voltage noise which eventually reduces the coherent time of TLS and loss
tangent. Due to the smaller film thickness and a larger portion of large
dipole TLS in $\mathrm{a-AlO_{x}}$, TLSs in $\mathrm{a-AlO_{x}}$ suffer from
larger voltage noise than those in $\gamma-\mathrm{Al_{2}O_{3}}$. One example
of extreme cases of high voltage noise is shown in Fig. S1, where we see no
TLS and $Q_{i}=1600$ is larger than $Q_{i}=680$ reported in the main text.

Figure 7: The normalized $S_{21,avg}$ of TLS spectrum of $\mathrm{a-AlO_{x}}$,
which is similar to the black line in Fig. 1 (b). Unlike Fig 1 (b) in the main
text, we add an frequency dependent on I and Q for a better fit. The red line
represents the fitting range and the black line is the extension of fit to the
measurement frequency range. Panel (a) and (b): the fitting result shows
$Q_{i}$ = 710 and the dataset is from Fig. 12 below. Panel (c) and (d): the
dataset is shown in the supplementary. The fitting result shows $Q_{i}$ =
1030.

##  Appendix D Fitting of hyperbolas in DC sweep plot

This section is describing the procedure of fitting TLS dipole for Fig. 2 and
3 in the main text. We search the local minimum (hybridized state) of each
$\left|S_{21}\right|$ from Fig. 2 (a) and apply a Gaussian filter on the 2D
plot of local minimum (see Fig. 8 (a) for example). We obtain the initial
guess values of $\Delta_{0}$, $V_{bias}$, and $p_{z}$ for all possible TLS
candidates. Due to the complexity of TLS spectrum, TLSs are fit separately in
the order of their $\Delta_{0}$ from the largest to the smallest. The first
candidate TLS with the largest $\Delta_{0}$ is fitted by Monte Carlo method
and the result is shown in Fig 8 (b). We assign those local minimum to first
TLS and subtract them from Fig. 8 (a) to get Fig. 8 (c). For next TLS, we use
the local minimum plot from Fig. 8 (c) and the next fitting will not be
affected by the previous TLS result. We repeat the process until all TLSs are
fit and discard those TLSs not crossing their minimum energies.

Figure 8: (a) Plot of extracted minimum energies in Fig 2 of main text. (b)
The final result of first dipole fit. (c) Subtract the first dipole fitting
result from (a).

##  Appendix E $\gamma-\mathrm{Al_{2}O_{3}}$ TLS histograms in two different
resonators

Fig. 9 and Table. 1 show the dipole moment measured in two
$\gamma-\mathrm{Al_{2}O_{3}}$ resonators. The mean $p_{z}$ are 3.5(1) and
3.6(1), for Res1 with $f_{0}$ = 4.35 GHz and Res2 with $f_{0}$ = 4.95 GHz,
respectively. Fig 10 shows the dipoles measured in 4 different cooldowns vs
either $\Delta_{0}$ or the external field $E_{ex,min}$, where TLSs meet their
minimum energies.

Amorphous materials rearrange their atom positions when their temperature is
above glass transition temperature ($T_{g}$). The thermal cycle to room
temperature ($\leq T_{g}$) is likely not enough. There are chances that TLSs
are measured twice in two cooldowns. However, we found the $p_{z}$ of
different cooldowns scatter randomly in the plots (see Fig. 10).
Theoretically, TLSs do not change their $\Delta_{0}$ without reaching $T_{g}$.
However, we cannot tell if TLSs’ $\Delta_{0}$ and $p_{z}$ have changed
sufficiently to be considered as new TLSs. But from Fig. 10 (b) and (d), most
TLSs ($\gtrsim$ 85%) have different $\Delta_{0}$ and $p_{z}$ in different
cooldowns. Whether the thermal cycle is constructing a new set of TLSs is
beyond the scope of this paper.

Figure 9: Histograms of dipole moments from two different $\mathrm{\mathrm{\gamma-Al}_{2}\mathrm{O}_{3}}$ resonators.  | mean $p_{z}$ (D) | standard deviation (D)
---|---|---
Res1 (4.35 GHz) | 3.55 | 1.47
Res2 (4.95 GHz) | 3.64 | 1.3
Table 1: Mean and standard deviation of extracted TLS $p_{z}$ for each
$\gamma-\mathrm{Al_{2}O_{3}}$ resonator. Figure 10: Dipole moments $p_{z}$ vs
$\Delta_{0}$ or the field of minimum TLS energy $E_{ex,min}$ in four
cooldowns. Panel (a) and (b) is from the resonator of frequency $\approx 4.35$
GHz. Panel (c)and (d) is from the resonator of frequency $\approx 4.97$GHz.
Markers with different color represent four different cooldowns.

##  Appendix F Fitting of TLS density by maximum likelihood estimation

Here, we show the procedure of applying Fisher’s maximum likelihood estimation
(MLE) on our statistics. For simplicity, we choose truncated normal
distribution as our target function $f(p_{z};\,\mu,\,\sigma)$ to fit our
material density $D(p_{z})$, and

$f(p_{z};\,\mu,\,\sigma)\,=\,C(\mu,\sigma)\,\frac{1}{\sigma\sqrt{2\pi}}exp(-\frac{(p_{z}\,-\,\mu)^{2}}{2\sigma^{2}})$
(22)

$\mathrm{for}\,p_{z}\in[0,\infty)$, where $\mu$ is the mean value, $\sigma$ is
the standard deviation. The normalized constant

$C(\mu,\sigma)\,=\,\frac{2}{1-\mathrm{erf}(-\mu/\sigma\sqrt{2})}$ (23)

depends only on $\mu$ and $\sigma$, where erf(x) is error function defined as

$\mathrm{erf}(x)\,=\,\frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{exp}(-t^{2})dt.$
(24)

The likelihood function

$L\,=\,\prod_{i}f(p_{zi};\,\mu,\,\sigma).$ (25)

The necessary conditions for the occurrence of a maximum (or a minimum) are

$\frac{\partial ln(L)}{\partial\mu}\,=\,0,\,\frac{\partial
ln(L)}{\partial\sigma}\,=\,0.$ (26)

As mentioned above in Sec. S-I, $D(p_{z})$ is not a directly measurement
result, but $H(p_{z})$ is. Since there is a weighing factor $p_{z}$
transferring $D(p_{z})$ to $H(p_{z})$ , we have

$H(p_{z};\,\mu,\,\sigma)\,=\,N_{tot}\,C_{1}(\mu,\sigma)p_{z}\,\mathrm{exp}(-\frac{(p_{z}\,-\,\mu)^{2}}{2\sigma^{2}}).$
(27)

A new normalization constant is

$C_{1}(\mu,\sigma)\,=\,\sigma^{2}\mathrm{exp}(-\frac{\mu^{2}}{2\sigma^{2}})+\,\mu\,\sigma\,\sqrt{\frac{\pi}{2}}\,(\,1\,-\mathrm{erf}(-\frac{\mu}{\sigma\sqrt{2}})),$
(28)

and $N_{tot}$ is the total observed TLS.

We also apply a gamma distribution function to fit the data, where

$f(x,\alpha,\beta)\,=\,\frac{x^{\alpha}\,e^{-\beta\,x}}{\Gamma(\alpha)\beta^{\alpha}},$
(29)

and $\Gamma(\alpha)$ is a gamma function. The fitting result gives $\alpha$ =
5.15 and $\beta$ = 1.72 and mean = 2.99 D. However, we expect there is a
distribution > 0 when dipole equals 0 so that gamma distribution is not
suitable.

MLE method applied to the amorphous alumina measured histogram does not give
the main peak feature adequately in the material distribution $D(p_{z})$.
Thus, we do not use MLE to report averaged dipole in the amorphous data, but
rather a calculated mean and standard deviation. The calculation method is
described in the main text.

Figure 11: Gamma distribution MLE analysis on $\gamma-\mathrm{Al_{2}O_{3}}$
data added to Fig. 2 (c) and (d) in the main text. Panel (a): measured
distribution (b) Material distribution. The black dashed lines use the target
function of Eq. S28 to fit the distribution. The red dashed lines are Gaussian
function as the main text. The Gaussian function shows a better agreement to
the data than Gamma distribution.

##  Appendix G Estimation of averaged dipole in $a-\mathrm{AlO_{x}}$ under TLS
frequency noise

Figure 12: One TLS spectroscopy of $a-\mathrm{AlO_{x}}$ TLS. (a) original data
and (b) TLS fitting results and their dipole $p_{z}$ (D). (c) TLSs with
incomplete hyperbolas in red and their potential $p_{z}$ (D) Figure 13:
Computed average dipole for different conditions of plausible missing TLSs in
the amorphous film. We assume 10th (2.8 Debye) or 20th (3.5 Debye) percentile
in original data as small $p_{z}$ TLS in square or in star marks respectively.
Small $p_{z}$ TLSs in the distribution are multiplied by the multiplication
factor M to simulate possible missing TLSs. The original data is when M = 1.
The black dashed line represents the average dipole in
$\gamma-\mathrm{Al_{2}O_{3}}$. We take M=5 as the most possible case for our
data, implying $p_{z,avg}$ > 3.3 D in amorphous alumina film.

Because of the unstable TLS during voltage bias, and resonator noise level, we
are prone to measure the larger dipole moments than the smaller ones
especially in amorphous sample. Those TLSs without crossing their minimum are
not counted in statistics due to the high uncertainty. We show few examples in
Fig. 12 (c), where preliminary manual fits are shown in dashed line. A closer
look, one can see some other potential small dipole moment TLSs at the edges
of the figure.

Here we estimate a case when we miss M-1 out of M small dipole moment by
multiplying the small $p_{z}$ TLS distribution. First, we define the small
$p_{z}$ TLSs as the 10th or 20th percentile, which equals 2.8 and 3.5 Debye
respectively. The dipole smaller than the above value are multiplied M times
in counts for a new distribution. The resulting new average dipole $p_{z}$ and
ratio of small $p_{z}$ TLS number to the total number are plotted in Fig. 13.
Suppose an extreme case with 8 missing TLS and 2 fitted TLS with $p_{z}$ < 3.5
D, it means 4 out of 5 are missed. This implies multiple factor M of 5 and the
new average is at least 3.3 Debye. This is still larger than $\bar{p_{z}}$ =
2.6 Debye in $\gamma-\mathrm{Al_{2}O_{3}}$. Now, we turn to the isotropy of
dipole orientation. There are missing TLS extractions, especially of those
small $p_{z}$ TLS. The portion of small $p_{z}$ TLS could be underestimated.
As a result, if we add those small $p_{z}$ back, the material distribution
$D(p_{z})$ could be monotonic decreasing as expected in the standard model.
More studies are needed to understand the small dipole TLSs in amorphous
phase.

## References

  * Nakamura  _et al._ [1999] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, Coherent control of macroscopic quantum states in a single-cooper-pair box, nature 398, 786 (1999).
  * Wallraff  _et al._ [2004] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics, Nature 431, 162 (2004).
  * Manucharyan  _et al._ [2009] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, Fluxonium: Single cooper-pair circuit free of charge offsets, Science 326, 113 (2009).
  * Devoret and Schoelkopf [2013] M. H. Devoret and R. J. Schoelkopf, Superconducting circuits for quantum information: an outlook, Science 339, 1169 (2013).
  * Yan  _et al._ [2016] F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, D. Rosenberg, G. Samach, S. Weber, _et al._ , The flux qubit revisited to enhance coherence and reproducibility, Nature communications 7, 1 (2016).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Simmonds  _et al._ [2004] R. W. Simmonds, K. Lang, D. A. Hite, S. Nam, D. P. Pappas, and J. M. Martinis, Decoherence in josephson phase qubits from junction resonators, Physical Review Letters 93, 077003 (2004).
  * Martinis  _et al._ [2005] J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, _et al._ , Decoherence in josephson qubits from dielectric loss, Physical review letters 95, 210503 (2005).
  * Burnett  _et al._ [2014] J. Burnett, L. Faoro, I. Wisby, V. Gurtovoi, A. Chernykh, G. Mikhailov, V. Tulin, R. Shaikhaidarov, V. Antonov, P. Meeson, _et al._ , Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator, Nature communications 5, 1 (2014).
  * Meißner  _et al._ [2018] S. M. Meißner, A. Seiler, J. Lisenfeld, A. V. Ustinov, and G. Weiss, Probing individual tunneling fluctuators with coherently controlled tunneling systems, Physical Review B 97, 180505 (2018).
  * Burnett  _et al._ [2019] J. J. Burnett, A. Bengtsson, M. Scigliuzzo, D. Niepce, M. Kudra, P. Delsing, and J. Bylander, Decoherence benchmarking of superconducting qubits, npj Quantum Information 5, 1 (2019).
  * Müller  _et al._ [2015] C. Müller, J. Lisenfeld, A. Shnirman, and S. Poletto, Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits, Physical Review B 92, 035442 (2015).
  * Klimov  _et al._ [2018] P. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends, K. Arya, B. Chiaro, Y. Chen, _et al._ , Fluctuations of energy-relaxation times in superconducting qubits, Physical review letters 121, 090502 (2018).
  * Schlör  _et al._ [2019] S. Schlör, J. Lisenfeld, C. Müller, A. Bilmes, A. Schneider, D. P. Pappas, A. V. Ustinov, and M. Weides, Correlating decoherence in transmon qubits: Low frequency noise by single fluctuators, Physical review letters 123, 190502 (2019).
  * Connors  _et al._ [2019] E. J. Connors, J. Nelson, H. Qiao, L. F. Edge, and J. M. Nichol, Low-frequency charge noise in si/sige quantum dots, Physical Review B 100, 165305 (2019).
  * Noroozian  _et al._ [2009] O. Noroozian, J. Gao, J. Zmuidzinas, H. G. LeDuc, and B. A. Mazin, Two-level system noise reduction for microwave kinetic inductance detectors, in  _AIP Conference Proceedings_ , Vol. 1185 (American Institute of Physics, 2009) pp. 148–151.
  * Gao [2008] J. Gao,  _The physics of superconducting microwave resonators_ , Ph.D. thesis, California Institute of Technology (2008).
  * O’Connell  _et al._ [2008] A. D. O’Connell, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, C. McKenney, M. Neeley, H. Wang, E. M. Weig, _et al._ , Microwave dielectric loss at single photon energies and millikelvin temperatures, Applied Physics Letters 92, 112903 (2008).
  * Lock  _et al._ [2019] E. H. Lock, P. Xu, T. Kohler, L. Camacho, J. Prestigiacomo, Y. J. Rosen, and K. D. Osborn, Using surface engineering to modulate superconducting coplanar microwave resonator performance, IEEE Transactions on Applied Superconductivity 29, 1 (2019).
  * Chang  _et al._ [2013] J. B. Chang, M. R. Vissers, A. D. Córcoles, M. Sandberg, J. Gao, D. W. Abraham, J. M. Chow, J. M. Gambetta, M. Beth Rothwell, G. A. Keefe, _et al._ , Improved superconducting qubit coherence using titanium nitride, Applied Physics Letters 103, 012602 (2013).
  * Goetz  _et al._ [2016] J. Goetz, F. Deppe, M. Haeberlein, F. Wulschner, C. W. Zollitsch, S. Meier, M. Fischer, P. Eder, E. Xie, K. G. Fedorov, _et al._ , Loss mechanisms in superconducting thin film microwave resonators, Journal of Applied Physics 119, 015304 (2016).
  * Quintana  _et al._ [2014] C. Quintana, A. Megrant, Z. Chen, A. Dunsworth, B. Chiaro, R. Barends, B. Campbell, Y. Chen, I.-C. Hoi, E. Jeffrey, _et al._ , Characterization and reduction of microfabrication-induced decoherence in superconducting quantum circuits, Applied Physics Letters 105, 062601 (2014).
  * Grünhaupt  _et al._ [2017] L. Grünhaupt, U. von Lüpke, D. Gusenkova, S. T. Skacel, N. Maleeva, S. Schlör, A. Bilmes, H. Rotzinger, A. V. Ustinov, M. Weides, _et al._ , An argon ion beam milling process for native alox layers enabling coherent superconducting contacts, Applied Physics Letters 111, 072601 (2017).
  * Paik  _et al._ [2011] H. Paik, D. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. Sears, B. Johnson, M. Reagor, L. Frunzio, L. Glazman, _et al._ , Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit qed architecture, Physical Review Letters 107, 240501 (2011).
  * Wang  _et al._ [2015] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. Devoret, and R. J. Schoelkopf, Surface participation and dielectric loss in superconducting qubits, Applied Physics Letters 107, 162601 (2015).
  * Dial  _et al._ [2016] O. Dial, D. T. McClure, S. Poletto, G. Keefe, M. B. Rothwell, J. M. Gambetta, D. W. Abraham, J. M. Chow, and M. Steffen, Bulk and surface loss in superconducting transmon qubits, Superconductor Science and Technology 29, 044001 (2016).
  * Deng [2015] C. Deng,  _Fast control and decoherence in superconducting quantum circuits_ , Ph.D. thesis (2015).
  * Creedon  _et al._ [2011] D. L. Creedon, Y. Reshitnyk, W. Farr, J. M. Martinis, T. L. Duty, and M. E. Tobar, High q-factor sapphire whispering gallery mode microwave resonator at single photon energies and millikelvin temperatures, Applied Physics Letters 98, 222903 (2011).
  * Lee  _et al._ [2019] K.-H. Lee, S. Chakram, S. E. Kim, F. Mujid, A. Ray, H. Gao, C. Park, Y. Zhong, D. A. Muller, D. I. Schuster, _et al._ , Two-dimensional material tunnel barrier for josephson junctions and superconducting qubits, Nano letters 19, 8287 (2019).
  * Wang  _et al._ [2018] J. I.-J. Wang, D. Rodan-Legrain, L. Bretheau, D. L. Campbell, B. Kannan, D. Kim, M. Kjaergaard, P. Krantz, G. O. Samach, F. Yan, _et al._ , Quantum coherent control of a hybrid superconducting circuit made with graphene-based van der waals heterostructures, arXiv preprint arXiv:1809.05215 (2018).
  * Shim and Tahan [2014] Y.-P. Shim and C. Tahan, Bottom-up superconducting and josephson junction devices inside a group-iv semiconductor, Nature communications 5, 1 (2014).
  * Oh  _et al._ [2006] S. Oh, K. Cicak, J. S. Kline, M. A. Sillanpää, K. D. Osborn, J. D. Whittaker, R. W. Simmonds, and D. P. Pappas, Elimination of two level fluctuators in superconducting quantum bits by an epitaxial tunnel barrier, Physical Review B 74, 100502 (2006).
  * Weides  _et al._ [2011] M. P. Weides, J. S. Kline, M. R. Vissers, M. O. Sandberg, D. S. Wisbey, B. R. Johnson, T. A. Ohki, and D. P. Pappas, Coherence in a transmon qubit with epitaxial tunnel junctions, Applied Physics Letters 99, 262502 (2011).
  * Sarabi  _et al._ [2016] B. Sarabi, A. N. Ramanayaka, A. L. Burin, F. C. Wellstood, and K. D. Osborn, Projected dipole moments of individual two-level defects extracted using circuit quantum electrodynamics, Physical review letters 116, 167002 (2016).
  * Bilmes  _et al._ [2021] A. Bilmes, S. Volosheniuk, J. D. Brehm, A. V. Ustinov, and J. Lisenfeld, Quantum sensors for microscopic tunneling systems, npj Quantum Information 7, 1 (2021).
  * Bilmes  _et al._ [2020] A. Bilmes, A. Megrant, P. Klimov, G. Weiss, J. M. Martinis, A. V. Ustinov, and J. Lisenfeld, Resolving the positions of defects in superconducting quantum bits, Scientific reports 10, 1 (2020).
  * de Graaf  _et al._ [2021] S. de Graaf, S. Mahashabde, S. Kubatkin, A. Y. Tzalenchuk, and A. Danilov, Quantifying dynamics and interactions of individual spurious low-energy fluctuators in superconducting circuits, Physical Review B 103, 174103 (2021).
  * Lisenfeld  _et al._ [2010] J. Lisenfeld, C. Müller, J. H. Cole, P. Bushev, A. Lukashenko, A. Shnirman, and A. V. Ustinov, Measuring the temperature dependence of individual two-level systems by direct coherent control, Physical review letters 105, 230504 (2010).
  * Grabovskij  _et al._ [2012] G. J. Grabovskij, T. Peichl, J. Lisenfeld, G. Weiss, and A. V. Ustinov, Strain tuning of individual atomic tunneling systems detected by a superconducting qubit, Science 338, 232 (2012).
  * Brehm  _et al._ [2017] J. D. Brehm, A. Bilmes, G. Weiss, A. V. Ustinov, and J. Lisenfeld, Transmission-line resonators for the study of individual two-level tunneling systems, Applied Physics Letters 111, 112601 (2017).
  * Anderson  _et al._ [1972] P. W. Anderson, B. Halperin, and C. M. Varma, Anomalous low-temperature thermal properties of glasses and spin glasses, Philosophical Magazine 25, 1 (1972).
  * Phillips [1972] W. Phillips, Tunneling states in amorphous solids, Journal of Low Temperature Physics 7, 351 (1972).
  * de Graaf  _et al._ [2020] S. de Graaf, L. Faoro, L. Ioffe, S. Mahashabde, J. Burnett, T. Lindström, S. Kubatkin, A. Danilov, and A. Y. Tzalenchuk, Two-level systems in superconducting quantum devices due to trapped quasiparticles, Science Advances 6, eabc5055 (2020).
  * Yu and Carruzzo [2021] C. C. Yu and H. M. Carruzzo, Two-level systems and the tunneling model: A critical view, arXiv preprint arXiv:2101.02787 (2021).
  * Khalil  _et al._ [2014] M. Khalil, S. Gladchenko, M. Stoutimore, F. Wellstood, A. Burin, and K. Osborn, Landau-zener population control and dipole measurement of a two-level-system bath, Physical Review B 90, 100201 (2014).
  * Matityahu  _et al._ [2019] S. Matityahu, H. Schmidt, A. Bilmes, A. Shnirman, G. Weiss, A. V. Ustinov, M. Schechter, and J. Lisenfeld, Dynamical decoupling of quantum two-level systems by coherent multiple landau–zener transitions, npj Quantum Information 5, 1 (2019).
  * Holder  _et al._ [2013] A. M. Holder, K. D. Osborn, C. Lobb, and C. B. Musgrave, Bulk and surface tunneling hydrogen defects in alumina, Physical review letters 111, 065901 (2013).
  * Gordon  _et al._ [2014] L. Gordon, H. Abu-Farsakh, A. Janotti, and C. G. Van de Walle, Hydrogen bonds in al 2 o 3 as dissipative two-level systems in superconducting qubits, Scientific reports 4, 1 (2014).
  * DuBois  _et al._ [2013] T. C. DuBois, M. C. Per, S. P. Russo, and J. H. Cole, Delocalized oxygen as the origin of two-level defects in josephson junctions, Physical review letters 110, 077002 (2013).
  * DuBois  _et al._ [2015] T. C. DuBois, S. P. Russo, and J. H. Cole, Atomic delocalization as a microscopic origin of two-level defects in josephson junctions, New Journal of Physics 17, 023017 (2015).
  * Paz  _et al._ [2014] A. P. Paz, I. V. Lebedeva, I. V. Tokatly, and A. Rubio, Identification of structural motifs as tunneling two-level systems in amorphous alumina at low temperatures, Physical Review B 90, 224202 (2014).
  * Van de Walle and Neugebauer [2003] C. G. Van de Walle and J. Neugebauer, Universal alignment of hydrogen levels in semiconductors, insulators and solutions, Nature 423, 626 (2003).
  * Fritz  _et al._ [2019] S. Fritz, L. Radtke, R. Schneider, M. Luysberg, M. Weides, and D. Gerthsen, Structural and nanochemical properties of al o x layers in al/al o x/al-layer systems for josephson junctions, Physical Review Materials 3, 114805 (2019).
  * Sarabi  _et al._ [2015] B. Sarabi, A. N. Ramanayaka, A. L. Burin, F. C. Wellstood, and K. D. Osborn, Cavity quantum electrodynamics using a near-resonance two-level system: Emergence of the glauber state, Applied Physics Letters 106, 172601 (2015).
  * Stoutimore  _et al._ [2012] M. Stoutimore, M. Khalil, C. Lobb, and K. Osborn, A josephson junction defect spectrometer for measuring two-level systems, Applied Physics Letters 101, 062602 (2012).
  * Khalil  _et al._ [2013] M. Khalil, M. Stoutimore, S. Gladchenko, A. Holder, C. Musgrave, A. Kozen, G. Rubloff, Y. Liu, R. Gordon, J. Yum, _et al._ , Evidence for hydrogen two-level systems in atomic layer deposition oxides, Applied Physics Letters 103, 162601 (2013).
  * Pappas  _et al._ [2011] D. P. Pappas, M. R. Vissers, D. S. Wisbey, J. S. Kline, and J. Gao, Two level system loss in superconducting microwave resonators, IEEE Transactions on Applied Superconductivity 21, 871 (2011).
  * [58] _See Supplemental Material at [URL will be inserted by publisher] for [give brief description of material]_.
  * Faoro and Ioffe [2012] L. Faoro and L. B. Ioffe, Internal loss of superconducting resonators induced by interacting two-level systems, Physical review letters 109, 157005 (2012).
  * Shalibo  _et al._ [2010] Y. Shalibo, Y. Rofe, D. Shwa, F. Zeides, M. Neeley, J. M. Martinis, and N. Katz, Lifetime and coherence of two-level defects in a josephson junction, Phys. Rev. Lett. 105, 177001 (2010).
  * Sarabi [2014] B. Sarabi,  _Cavity quantum electrodynamics of nanoscale two-level systems_ , Ph.D. thesis (2014).
  * Brock  _et al._ [2020] B. Brock, M. Blencowe, and A. Rimberg, Frequency fluctuations in tunable and nonlinear microwave cavities, Physical Review Applied 14, 054026 (2020).