††thanks: These authors contributed equally to this work.
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<EMAIL_ADDRESS>These authors contributed equally to this
work.
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# Precision Measurement of the Microwave Dielectric Loss of Sapphire
in the Quantum Regime with Parts-per-Billion Sensitivity

Alexander P. Read  Departments of Physics and Applied Physics, Yale
University, New Haven, Connecticut 06511, USA  Yale Quantum Institute, Yale
University, New Haven, Connecticut 06511, USA  Benjamin J. Chapman
Departments of Physics and Applied Physics, Yale University, New Haven,
Connecticut 06511, USA  Yale Quantum Institute, Yale University, New Haven,
Connecticut 06511, USA  Chan U Lei  Departments of Physics and Applied
Physics, Yale University, New Haven, Connecticut 06511, USA  Yale Quantum
Institute, Yale University, New Haven, Connecticut 06511, USA  Jacob C. Curtis
Departments of Physics and Applied Physics, Yale University, New Haven,
Connecticut 06511, USA  Yale Quantum Institute, Yale University, New Haven,
Connecticut 06511, USA  Suhas Ganjam  Departments of Physics and Applied
Physics, Yale University, New Haven, Connecticut 06511, USA  Yale Quantum
Institute, Yale University, New Haven, Connecticut 06511, USA  Lev Krayzman
Departments of Physics and Applied Physics, Yale University, New Haven,
Connecticut 06511, USA  Yale Quantum Institute, Yale University, New Haven,
Connecticut 06511, USA  Luigi Frunzio  Departments of Physics and Applied
Physics, Yale University, New Haven, Connecticut 06511, USA  Yale Quantum
Institute, Yale University, New Haven, Connecticut 06511, USA  Robert J.
Schoelkopf  Departments of Physics and Applied Physics, Yale University, New
Haven, Connecticut 06511, USA  Yale Quantum Institute, Yale University, New
Haven, Connecticut 06511, USA

###### Abstract

Dielectric loss is known to limit state-of-the-art superconducting qubit
lifetimes. Recent experiments imply upper bounds on bulk dielectric loss
tangents on the order of 100 parts-per-billion, but because these inferences
are drawn from fully fabricated devices with many loss channels, these
experiments do not definitely implicate or exonerate the dielectric. To
resolve this ambiguity, we devise a measurement method capable of separating
and resolving bulk dielectric loss with a sensitivity at the level of $5\times
10^{-9}$. The method, which we call the dielectric dipper, involves the in
situ insertion of a dielectric sample into a high-quality microwave cavity
mode. Smoothly varying the sample participation in the cavity mode enables a
differential measurement of the sample’s dielectric loss tangent. The
dielectric dipper can probe the low-power behavior of dielectrics at cryogenic
temperatures and does so without the need for any lithographic process,
enabling controlled comparisons of substrate materials and processing
techniques. We demonstrate the method with measurements of sapphire grown by
edge-defined film-fed growth (EFG) in comparison to high-grade sapphire grown
by the heat-exchanger method (HEMEX). For EFG sapphire we infer a bulk loss
tangent of $63(8)\times 10^{-9}$ and a substrate-air interface loss tangent of
$15(3)\times 10^{-4}$ (assuming a sample surface thickness of
$3~{}\text{nm}$). For a typical transmon, this bulk loss tangent would limit
device quality factors to $Q\lessapprox 20\times 10^{6}$, suggesting that bulk
loss is likely the dominant loss mechanism in the longest-lived transmons on
sapphire. We also demonstrate this method on HEMEX sapphire and bound its bulk
loss tangent to be less than $19(6)\times 10^{-9}$. As this bound is about 3
times smaller than the bulk loss tangent of EFG sapphire, use of HEMEX
sapphire as a substrate would lift the bulk dielectric coherence limit of a
typical transmon qubit to several milliseconds.

##  I Introduction

Superconducting circuits are a promising hardware platform for quantum
information science, prized for their union of rapid gates and long coherence
times. That success is partly due to a two-decade effort which has prolonged
the nanosecond-scale coherence of early superconducting qubits [1] by six-
orders of magnitude [2, 3, 4, 5].

Some of these gains have come from engineering matrix-elements [6] and
insensitivity to decoherence mechanisms like 1/f noise [7]. Other improvements
have been made by directly minimizing noise-spectral densities [8], for
example by improving the structure, materials, or fabrication process of the
device. For either approach, the first step in finding the next order-of-
magnitude improvement is to determine the dominant loss mechanism.

Identification of the dominant source of loss is complicated by the many loss
channels present in superconducting qubits [9]. These losses add together to
limit the quality factor $Q$ of the qubit.

$\displaystyle\frac{1}{Q}=\frac{1}{Q_{\text{conductor}}}+\frac{1}{Q_{\text{dielectric}}}+\frac{1}{Q_{\text{radiative}}}+...$
(1)

While loss $Q^{-1}$ is straightforward to measure, it is unclear how to best
improve qubit coherence without knowing which of these mechanisms is the
dominant source of loss. To overcome this challenge and distinguish between
various sources of loss, one can measure $Q^{-1}$ on either a suite of
devices, each designed to be more or less sensitive to particular loss
channels, or a single device with some in situ tunable experimental parameter,
which preferentially aggravates or alleviates one source of dissipation.

Studies on dielectric loss have principally employed the suite approach [10,
11, 12, 13, 14, 15, 16, 17, 18]. For transmons on sapphire with $Q\leq 4\times
10^{6}$, those studies measured $Q^{-1}$ that scaled proportionally with the
fraction of the qubit energy stored in surface dielectrics [12, 13]. This
directly implicated surface dielectric loss as the dominant loss mechanism in
these devices, motivating the use of fabrication processes compatible with
more aggressive surface cleaning. The result was some of the highest published
transmon lifetimes to-date [2, 3].

Advances in coherence make it necessary to reassess which loss channels limit
coherence. Quality factors of transmon qubits have now increased to $Q\approx
10^{7}$ [2, 3, 4]. Does bulk dielectric loss limit the coherence of these
devices?

A desirable technique to answer this question would uniquely distinguish bulk
loss from other sources of loss in the system, and do so with high resolution.
Specifically, it would have sensitivity to loss tangents much smaller than
those which would limit transmon coherence, which in current devices would be
a bulk loss tangent of $\text{tan}\delta_{\text{bulk}}\approx 10^{-7}$. It
would also complement previous methodologies [16, 17] by measuring both power-
dependent and power-independent losses without requiring lithography or other
processing. Such flexibility would allow measurements of materials for which
there is not yet an established process for depositing and patterning films
and Josephson junctions. Similarly, standard substrates could be studied mid-
process to determine the process stage at which losses manifest.

Here we introduce a suitable technique for this task, designed for the study
of dielectric loss at microwave frequencies, cryogenic temperatures, and low
powers, which allows dielectric loss to be measured independently from other
qubit loss-channels. The technique, which we call the dielectric dipper,
involves the in situ insertion of a dielectric sample into a high-quality
microwave cavity mode. Smooth variation of the sample participation enables a
differential measurement of the dielectric loss of the sample, with a
sensitivity of $5\times 10^{-9}$.

We use this technique to measure sapphire grown by edge-defined film-fed
growth (EFG), and determine its bulk and substrate-air interface loss tangents
to be $63(8)\times 10^{-9}$, and $15(3)\times 10^{-4}$, respectively [19]. We
also measure high-grade sapphire grown by the heat-exchanger method (HEMEX)
and bound the bulk loss tangent to below $19(6)\times 10^{-9}$. These
measurements suggest that current record-lifetime transmon qubits fabricated
on EFG sapphire are approaching the bulk dielectric loss limit, and that the
use of HEMEX sapphire as a substrate would suppress that loss, lifting the
transmon coherence limit from bulk dielectric loss to several milliseconds.

##  II Measurement technique

###  II.1 Overview

One way to improve the resolution of a dielectric loss measurement is to
heavily suppress the non-dielectric loss channels. This consideration suggests
the use of high-$Q$ microwave cavities as sensitive probes; the properties
that make them promising as quantum memories—relatively few physical loss
mechanisms and lifetimes from milliseconds to seconds [20, 21]—also make them
a well-suited platform for precision loss measurements.

To further increase sensitivity, one can distinguish dielectric loss from the
non-dielectric “background” by making a separate measurement of that
background, then subtracting it from the loss measured in the presence of the
dielectric under study [22]. If those measurements are made in separate
cooldowns, however, then comparison with the background may be corrupted by
cooldown-to-cooldown variation of cavity properties [23]. This problem can be
avoided by engineering an in situ tunable coupling between the cavity-mode and
the dielectric under study, which enables both measurements to be performed in
the same cooldown.

Figure 1: Measurement concept. (a) A $\lambda/4$ coaxial stub cavity is
machined from $4\text{N}$ aluminum, etched [24], and mounted to the base of a
dilution refrigerator. A dielectric sample is inserted into the cavity
waveguide to a position $z$, adjusted in situ by a piezoelectric positioner
(not pictured). (b) The fundamental frequency of the cavity is far below the
cutoff of the waveguide, making the participation $p_{\text{bulk}}$ of the
cavity mode in the dielectric sample change exponentially with $z$ (blue
line). As $p_{\text{bulk}}$ rises with insertion, the real part of the sample
permittivity causes the measured cavity frequency (purple circles) to drop, in
agreement with electromagnetic simulations (purple line). (c) Decay of the
cavity output power is measured by a ringdown experiment which can distinguish
between dephasing and decay (see App. A). The imaginary part of the sample
permittivity causes the energy decay rate of the cavity mode $1/\tau$ to
change when the dielectric is withdrawn (orange) or inserted (green). (d) (Top
view) In the withdrawn position, almost all of the measured loss comes from
either the metal-air dielectric interface (aluminum oxide), or from losses in
the conductor. (e). (Side view) As the sample is inserted, the loss channels
of the substrate under study (the substrate-air interface and bulk of the
dielectric) contribute to the measured loss.

We accomplish this using a coaxial stub cavity [25] with a high-quality
(internal quality factor $Q\approx 10^{8}$) $\lambda/4$ mode, and a
piezoelectric positioner that can insert a dielectric sample into the
waveguide of the stub cavity (Fig. 1a). The fundamental cavity mode is
designed to have a resonant frequency well below the cutoff of the waveguide.
This causes its electric field to be attenuated exponentially along that
waveguide, confining the cavity mode. Because of this attenuation, moving the
sample along the waveguide changes the ratio between the electric energy
stored in the volume of the dielectric sample $U_{\text{bulk}}$ and the total
energy stored in the cavity $U_{\text{tot}}$. This ratio is known as the bulk
participation $p_{\text{bulk}}$:

$\displaystyle
p_{\text{bulk}}\equiv\frac{U_{\text{bulk}}}{U_{\text{tot}}}\equiv\frac{\frac{1}{2}\int_{V_{\text{sub}}}\mathbf{D}\cdot\mathbf{E}~{}dV}{\frac{1}{2}\int_{V_{\text{tot}}}\mathbf{D}\cdot\mathbf{E}~{}dV}.$
(2)

This ratio $p_{\text{bulk}}$ can be tuned in situ (Fig. 1b), and with a large
on-off ratio that is set by the stroke of the piezoelectric positioner and the
attenuation length of the waveguide.

For materials with a permittivity different from vacuum, inserting the
dielectric shifts the frequency of the cavity. While the positioner does
feature a built-in resistive position indication, this frequency shift allows
frequency to serve as a proxy for sample position (see Fig. 1b), bypassing the
need for the position indication to determine system geometry. If the
permittivity is not purely real, insertion also alters the energy decay rate
of the cavity mode in a ringdown experiment, as shown in Fig. 1c (for details
of the ringdown measurement, see App. A). This in situ tunability enables a
differential measurement of sample loss to be performed in a single cooldown.

Ideally, all the additional loss observed after insertion of the sample could
be attributed to the sample. However, as insertion of the sample perturbs the
field distribution of the cavity mode, “background” loss from the cavity also
changes. This effect can be modeled with the participation formalism, in which
the loss of an electromagnetic mode is the weighted-sum of material loss-
factors $q_{j}^{-1}$:

$\displaystyle Q^{-1}=\sum_{j}p_{j}q_{j}^{-1},$ (3)

where the weights $p_{j}\equiv U_{j}/U_{\text{tot}}$ given to each loss factor
are known as the participations [24] (for details on the participations
$p_{j}$, see App. B). Implicit in this formalism are the assumptions that
mechanisms for loss are linear and uniform within each material region. For
comparison with other studies [26, 12], we assume that surface dielectric
interfaces have a thickness of 3 nm (see App. B). We note that the
$q_{j}^{-1}$ have natural physical interpretations: when $j$ indexes a
dielectric loss channel, $q_{j}^{-1}$ is a loss tangent
$\text{tan}\delta_{j}$; and when $j$ indexes a conductor loss channel,
$q_{j}^{-1}$ is the quotient of surface resistance and surface reactance:
$q_{\text{cond}}^{-1}={R_{\text{s}}}/({\omega\mu_{0}\lambda_{\text{L}}})$,
where $\lambda_{\text{L}}$ is the London penetration depth [27].

We consider a model with losses that originate in the cavity and losses that
come from the dielectric sample. Fig. 1d indicates the two material regions of
the cavity responsible for “background” losses: conductor loss in the
superconducting walls of the cavity, and dielectric loss on the metal-air (MA)
interface of those walls. While some energy does propagate along the waveguide
and radiate out of the cavity, the waveguide length has been chosen to
suppress this effect such that $Q^{-1}_{\text{radiative}}<10^{-10}$, as
verified by electromagnetics simulations. This is small compared to the sample
losses intended to be resolved in this study, so we neglect radiative loss in
our model.

Fig. 1e indicates the two regions associated with loss from the dielectric
under study: the substrate-air (SA) interface [19] and the bulk of the
dielectric. For a given dielectric sample, these two sources of dielectric
loss depend sensitively on the sample position $z$. However, their ratio is
predominantly set by the surface-to-volume ratio of the sample, which in the
case of a thin sample, is largely insensitive to changes in $z$. This
proportionality prevents the separate extraction of these individual loss
rates from a measurement of a single sample. For this reason, we combine these
two substrate loss channels into a single effective loss channel:

$\displaystyle Q^{-1}_{\text{sub}}\equiv
p_{\text{bulk}}q^{-1}_{\text{bulk}}+p_{\text{SA}}q^{-1}_{\text{SA}}.$ (4)

From this composite loss channel we define an effective loss tangent of the
sample: $q_{\text{sub}}^{-1}\equiv Q^{-1}_{\text{sub}}/p_{\text{bulk}}$.

Having chosen the loss channels under consideration, a set of measurements
$Q^{-1}_{i}$, taken at a series of $N$ different positions $z_{i}$, can be
analyzed using a system of $N$ equations:

$\displaystyle\left(\begin{array}[]{c}Q_{1}^{-1}\\\ Q_{2}^{-1}\\\ \vdots\\\
Q_{N}^{-1}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}\tilde{p}_{\text{cond,1}}&p_{\text{MA,1}}&p_{\text{{bulk,1}}}\\\
\tilde{p}_{\text{{cond,2}}}&p_{\text{{MA,2}}}&p_{\text{{bulk,2}}}\\\
\vdots&\vdots&\vdots\\\
\tilde{p}_{\text{cond},N}&p_{\text{MA},N}&p_{\text{bulk},N}\\\
\end{array}\right)\left(\begin{array}[]{c}\tilde{q}_{\text{cond}}^{-1}\\\
q_{\text{MA}}^{-1}\\\ q_{\text{sub}}^{-1}\\\ \end{array}\right)$ (16)

Here $\tilde{p}_{\text{cond},i}\equiv p_{\text{cond},i}/\omega_{i}$, and
$\tilde{q}_{\text{cond}}^{-1}\equiv q_{\text{cond}}^{-1}\omega_{i}$. This
accounts for the frequency dependence of $q_{\text{cond}}$. We refer to the
matrix in Eq. 16 as the participation matrix $\mathbf{P}$ [24, 12] and
determine it with electromagnetic simulations (see App. B). The material loss
factors $q_{j}^{-1}$ are extracted from Eq. (16) using bounded (see App. E.3)
least-squares regression.

To decompose $q_{\text{sub}}^{-1}$ into its components $q_{\text{bulk}}^{-1}$
and $q_{\text{SA}}^{-1}$, a second measurement may be made on a sample of the
same material but with different thickness. We demonstrate this process with
EFG sapphire in Sec. III.

When analyzing various samples measured with the same cavity, we concatenate
multiple instances of Eq. (16), providing each sample its own substrate loss
factor and associated matrix column. Because this joint analysis permits only
a single value of $q_{\text{MA}}^{-1}$ and $q_{\text{cond}}^{-1}$, the number
of free parameters in the model is reduced by four (compared to fitting three
samples individually), ensuring consistency in the extracted substrate loss
factors. Joint analysis is valid only if the cavity’s bare losses do not
change between cooldowns but this can be verified by measuring the cavity with
each sample in the withdrawn position.

###  II.2 Sensitivity

When solving for the inferred material loss factors $q_{j}^{-1}$, the
structure of $\mathbf{P}$ determines how errors in the measured $Q_{i}^{-1}$
propagate into loss factor errors $\sigma_{q_{j}^{-1}}$. If the propagated
fractional error $\sigma_{q_{j}^{-1}}/q_{j}^{-1}$ is greater than 100%, then
the loss factor is said to be unresolved as it is below the sensitivity of the
measurement. One option for predicting $\sigma_{q_{j}^{-1}}$ is to use the
condition number [28] of $\mathbf{P}$ to infer a bound on the severity of this
error amplification. Such a perspective is useful in experimental design, and
has been used to improve the conditioning of matrix problems similar to Eq.
(16) [16, 17].

Another tool for predicting the error and sensitivity of an experiment is the
covariance matrix. Given some participation matrix $\mathbf{P}$ and an
approximate expectation for the measurement errors $\sigma_{Q_{i}^{-1}}$, the
covariance matrix $\mathbf{C}$ allows the individual $\sigma_{q_{j}^{-1}}$ to
be calculated (and not just bounded) from its diagonal elements
$C_{jj}=\sigma_{q_{j}^{-1}}^{2}$. For the case of linear least-squares,

$\displaystyle\mathbf{C}\equiv(\mathbf{\tilde{P}}^{\text{T}}\mathbf{\tilde{P}})^{-1},$
(17)

where $\tilde{P_{ij}}\equiv P_{ij}/\sigma_{Q_{i}^{-1}}$ [29]. The specificity
of this information makes the covariance matrix a powerful tool for
experimental design. We use it to predict the sensitivity of the dielectric
dipper.

Accurate prediction of an extracted loss factor error $\sigma_{q_{j}^{-1}}$ by
solving Eq. 17 requires two things: the participation matrix $\mathbf{P}$
(obtained from simulations) and an anticipated absolute measurement error,
$\sigma_{Q_{i}}^{-1}$. The error in the measurement of a line-width often
scales with the line-width, so we assume a fractional measurement error
$\sigma_{Q_{i}^{-1}}/Q_{i}^{-1}$. The conversion an anticipated fractional
error into an anticipated absolute error requires an accurate prediction of
$Q_{i}^{-1}$, which can be made using participations from simulation, and
assumed values for $q_{j}^{-1}$ as inputs to eq. (3). In this experiment,
reasonable values for $q_{\text{cond}}^{-1}$ and $q_{\text{MA}}^{-1}$ can be
chosen during the design phase from measurements of a cavity made of the same
materials as the planned design (or the exact cavity to be used if it is
available). However, during the design phase, the substrate material loss
factors $q_{\text{bulk}}^{-1}$ and $q_{\text{SA}}^{-1}$ are unknown. We
therefore calculate $\sigma_{q_{\text{sub}}^{-1}}$ as a function of these
unknown substrate loss factors. The result of this error analysis is plotted
in color in Fig. 2 in terms of the 95% confidence interval
$\text{C.I.}=2\sigma_{q_{\text{sub}}^{-1}}$. Also seen in Fig. 2 are white
contours which indicate the result of converting these absolute errors into
fractional errors, $2\sigma_{q_{\text{sub}}^{-1}}/q_{\text{sub}}^{-1}$. These
provide a visualization of which loss tangent combinations would be resolvable
by this measurement with 95% confidence.

Figure 2: Simulated sensitivity. Estimated $q_{\text{sub}}^{-1}$ 95%
confidence interval (C.I.), plotted (in color) as a function of hypothetical
dielectric material quality factors $q_{\text{SA}}^{-1}$ and
$q_{\text{bulk}}^{-1}$. White curves indicate the $3\%$, $10\%$ and $100\%$
fractional error contours. We assume fractional measurement error
$\sigma_{Q_{i}^{-1}}/Q_{i}^{-1}=1\%$ (limited by our ability to separate
coupling loss from total loss), cavity material quality factors
$q_{\text{MA}}^{-1}=3\times 10^{-2}$, and $q_{\text{cond}}^{-1}=2\times
10^{-5}$ and $N=30$ measurement positions. Participations are as simulated for
the 460 $\upmu$m thick EFG sapphire sample.

When the losses from the sample are sufficiently small, the fractional error
exceeds one and the substrate material quality factor $q_{\text{sub}}^{-1}$
can no longer be distinguished from zero. The $100\%$ fractional error contour
indicates the boundary of this region. The calculated error near the boundary
of this region, $\sigma_{q_{\text{sub}}^{-1}}<5$ parts-per-billion (ppb),
indicates the predicted sensitivity of the experiment. The sensitivity of our
experiment is predominantly limited by our ability to know the composition of
the cavity losses, and in turn our ability to distinguish substrate losses
from losses in the metal-air interface of the cavity (for details, see App.
C). Thus, even though multiple participations change with sample insertion to
some degree, we can still measure dielectric losses with 5 ppb sensitivity.

##  III Sapphire Measurements

Figure 3: Power dependence of loss. Measured internal loss $Q^{-1}$ with the
sample withdrawn (orange) and inserted (green) as a function of the average
initial cavity photon number $n$ in the ringdown measurement, with (a) $100$
$\upmu$m thick EFG sapphire, (b) $460$ $\upmu$m thick EFG sapphire, and (c)
$440$ $\upmu$m thick HEMEX sapphire. The curves are fits of the TLS model in
Eq. (18) and the shaded region reflects propagation of the standard error of
the fit parameters $q_{j}^{-1}$ to $Q^{-1}$. For the inserted measurements of
EFG sapphire, fits are only to measurements with $n\lesssim 2\times 10^{10}$,
as above this photon number the loss deviates from the TLS model. The origin
of this deviation is unclear but we conjecture it to be related to heating of
the dielectric sample.

To demonstrate the utility of the dielectric dipper, we use it to determine
whether bulk dielectric loss plays a role in the lifetimes of state-of-the-art
superconducting qubits on sapphire [2, 3]. To do this, we measure samples from
wafers of sapphire cut from sheets grown by the same edge-defined film-fed
growth method (EFG) [30] as the substrates used in those studies. Samples of
two different thicknesses (and the same length and width) were measured in
order to localize substrate loss to either the bulk or the substrate-air
interface using Eq. (4). Motivated by previous studies [31, 13] on low-loss
sapphire substrates grown with a different process, we also measure a third
sapphire sample which is cut from a boule grown by the heat-exchanger method
(HEM) [30]. After production, this wafer was screened by the manufacturer for
low optical absorption and wavefront aberration, qualifying for the highest
grade: HEMEX [32]. For all three samples, we perform power sweeps with the
sample inserted to ensure that following measurements are made in the low
power regime. Withdrawn power sweeps are also performed to infer bounds on the
background losses of the cavity. We then perform a position sweep at low
excitation power to extract the substrate loss tangents.

###  III.1 Power-dependence of loss

Due to the saturability of dielectric loss, measurements which aim to be
representative of qubit performance must be taken in this same low-power
regime where superconducting qubits operate. To identify this regime, we begin
with the sample completely inserted, and measure the cavity’s impulse response
as a function of the excitation power in the ringdown experiment. We
parameterize cavity excitation by the average number of photons $n$ created by
the impulse (see App. A.4). The green circles in Fig. 3 show the results for
the 100 $\upmu$m EFG sample (a), the 460 $\upmu$m EFG sample (b), and the 440
$\upmu$m HEMEX sample (c). In each case, the total internal loss of the cavity
asymptotically approaches some high-power value $Q^{-1}_{\text{hp}}$, and some
low-power value $Q^{-1}_{\text{lp}}=Q^{-1}_{\text{hp}}+Q^{-1}_{\text{sat}}$,
with a smooth transition between these two regimes. This power-dependence can
be captured by a two-level-system (TLS) model for dielectric loss [33] which
we fit to the data:

$\displaystyle
Q^{-1}=Q^{-1}_{\text{hp}}+\frac{Q^{-1}_{\text{sat}}}{\sqrt{1+\left(\frac{n}{n_{\text{c}}}\right)^{\alpha}}}.$
(18)

Here the critical photon number $n_{\text{c}}$ sets the scale for saturation
and $\alpha$ sets the width of this transition. Table 1 shows the extracted
fit parameters for these sample-inserted power sweeps.

Sample |  $n_{\text{c}}$ (millions) | $\alpha$
---|---|---
$100$ $\upmu$m EFG sapphire | $600(300)$ | $0.42(5)$
$460$ $\upmu$m EFG sapphire | $2000(1800)$ | $0.29(4)$
$440$ $\upmu$m HEMEX sapphire | $300(30)$ | $0.40(1)$
Table 1: Fit results for the inserted power sweeps in Fig. (3) using the TLS
model in Eq. (18).

The inserted power sweeps show that cavity population $n=10^{5}$ is
sufficiently below $n_{\text{c}}$, such that measured loss is only weakly
dependent on power and closely approximates the low-power behavior
$Q^{-1}_{\text{lp}}$, as is appropriate for predicting their performance as
superconducting qubit substrates. We therefore fix the excitation photon
number for the remaining measurements to $n=10^{5}$ (for details, see App.
E.1). This results in a peak electric field in the sample of about $4\times
10^{-8}$ V/m. Note that the extracted values of $n_{\text{c}}$ are a factor of
$10^{6}$ greater than those found in coplanar waveguide resonators [34, 35],
reflecting the much larger mode volume of a three-dimensional cavity [36].

To measure the power dependence of the cavity background alone and bound the
low-power values of $q_{\text{cond}}$ and $q_{\text{MA}}$, we repeat the power
sweep with the sample completely withdrawn (orange circles in Fig. 3).
Previous studies on conductor losses in aluminum cavities have found no power
dependence up to $10^{10}$ photons [20]. If we assume that this trend
continues up to $10^{14}~{}\text{photons}$, the measured power dependence may
be attributed entirely to saturation of dielectric loss from the metal-air
interface of the cavity. This measurement therefore provides two useful
constraints on the cavity loss factors $q_{\text{cond}}^{-1}$ and
$q_{\text{MA}}^{-1}$: first, the conductor loss factor is bounded from above
by the measurement with the smallest loss $\text{min}(Q^{-1})$ (typically a
high-power measurement, ideally described by $Q_{\text{hp}}^{-1}$); and
second, the metal-air loss factor is bounded from below by the change in the
cavity loss $\Delta Q^{-1}$ (ideally described by $Q_{\text{sat}}^{-1}$). More
precisely,

$\displaystyle q_{\text{cond}}^{-1}$ $\displaystyle\leq$
$\displaystyle\frac{\text{min}(Q^{-1})}{p_{\text{cond}}},$ (19) $\displaystyle
q_{\text{MA}}^{-1}$ $\displaystyle\geq$ $\displaystyle\frac{\Delta
Q^{-1}}{p_{\text{MA}}}.$

Table 6 in App. E.3 lists the bounds obtained in this way.

Figure 4: Measurements of the bulk and surface loss tangents. (a) Measured
internal loss $Q^{-1}$ plotted as a function of bulk participation
$p_{\text{bulk}}$, for $100$ $\upmu$m thick EFG sapphire, $460$ $\upmu$m thick
EFG sapphire, and $440$ $\upmu$m thick HEMEX sapphire. The excitation pulse
for these measurements create approximately $10^{5}$ photons (low-power
regime), and the mixing chamber temperature is 20 mK (low-temperature regime).
Solid lines are fits to the three-loss model in Eq. (16), consistent with
bounds provided by the power sweeps (see App. E.3). The shaded region shows
propagation of the 95% confidence interval of $q^{-1}_{\text{sub}}$, and the
dashed line is the estimated cavity background, comprised of conductor and MA
dielectric losses. (b) The effective substrate loss factor
$q_{\text{sub}}^{-1}$ is a weighted average of substrate bulk and surface loss
factors. By Eq. (4), a measurement of $q_{\text{sub}}^{-1}$ defines a line in
the two-dimensional parameter space of $q_{\text{bulk}}^{-1}$ and
$q_{\text{SA}}^{-1}$. The intersection of two such lines, each defined by
measurements on the same material with different form factors, identifies the
bulk and SA loss factors of that material. The single measurement of HEMEX
sapphire does not separately resolve bulk and SA losses, but bounds both the
bulk and SA loss factors.

###  III.2 In situ insertion of the sample

With the cavity loss factors constrained and the edge of the low-power regime
identified, we proceed with the position sweeps described in Sec. II. The
results are shown in Fig. 4a. For each sample in this study, internal loss
$Q^{-1}$ of the cavity mode is plotted against the substrate bulk
participation $p_{\text{bulk}}$. When the substrate is withdrawn from the
cavity, $p_{\text{bulk}}$ approaches zero, and the measured loss is that of
the empty cavity. As the sample is inserted, $p_{\text{bulk}}$ rises, and the
cavity mode inherits loss from the sample. All samples are inserted until
$z\approx 30$ $\upmu$m.

Examining the position sweeps for the two EFG samples, one can see that the
total change in $Q^{-1}$ is larger for the thicker sample. As the samples have
almost identical surface areas, this indicates that surface loss does not
account for all of the added loss. To check that intuition and distinguish the
bulk and surface contributions, we first extract the substrate loss tangent
$q_{\text{sub}}^{-1}$ for each sample by solving Eq. (16) (constrained by Eq.
(19)). This separates substrate loss from the “background” cavity losses. As
$q_{\text{sub}}^{-1}$ is a weighted sum of the bulk and surface loss tangents
of the sample material per Eq. (4), a measurement of $q_{\text{sub}}^{-1}$
constrains those loss tangents, in the space of all possible ordered pairs
$(q_{\text{bulk}}^{-1},q_{\text{SA}}^{-1})$, to lie along a line (see Fig.
4b). A single constraining line has several useful geometric interpretations:
its $x$ and $y$ intercepts correspond to upper bounds on the bulk and SA
interface loss tangents; its distance from the origin is proportional to the
measured value of $q_{\text{sub}}^{-1}$; and its slope is determined by the
ratio of surface and bulk participations $p_{\text{SA}}/p_{\text{bulk}}$.

Because the two EFG sapphire samples have different surface-to-volume ratios,
the constraint lines associated with these two measurements have different
slopes. Their intersection marks the only viable $(q_{\text{bulk}}^{-1}$ ,
$q_{\text{SA}}^{-1})$ pair which is consistent with the measured
$q_{\text{sub}}^{-1}$ of both samples. In this way, we distinguish between
bulk loss and surface loss in EFG sapphire and extract
$q_{\text{bulk}}^{-1}=63(8)\times 10^{-9}$, and
$q_{\text{SA}}^{-1}=15(3)\times 10^{-4}$ [19].

Fig. 4a also displays the results of the position sweep on a HEMEX sapphire
sample. The more gradual slope of this data set, compared to the EFG data
sets, indicates that HEMEX sapphire has lower dielectric losses. Having
measured one HEMEX sample, there is only a single contour to constrain the
possible $(q_{\text{bulk}}^{-1}$ , $q_{\text{SA}}^{-1})$ pairs, so the bulk
and surface contributions are not separately resolved. Nonetheless, the
measurement bounds both bulk and surface losses. These bounds are represented
graphically by the $x$ and $y$ intercepts of the HEMEX constraint line in Fig
4b. The inferred bound on bulk loss is $q_{\text{bulk}}^{-1}<19(6)\times
10^{-9}$, consistent with the findings of Ref. [31], and smaller than the bulk
loss tangent of EFG sapphire by a factor of 3 (see Table 2.)

Material | $q^{-1}_{\text{bulk}}$ | $q^{-1}_{\text{SA}}$
---|---|---
EFG sapphire | $63(8)\times 10^{-9}$ | $15(3)\times 10^{-4}$
HEMEX sapphire | $<19(6)\times 10^{-9}$ | $<11(4)\times 10^{-4}$
Table 2: Measurements and bounds of the bulk and surface dielectric loss
tangents for EFG and HEMEX sapphire [19].

##  IV Implications for transmon lifetimes

Having sensitively measured the loss tangents of these forms of sapphire, we
now calculate the current and future coherence limits that they imply for
typical transmon qubits. Assuming a qubit transition frequency of $4$ GHz and
a bulk participation of $p_{\text{bulk}}=80\%$, these measurements imply that
a transmon fabricated on EFG sapphire would be bulk limited to a quality
factor of $Q\leq 20\times 10^{6}$, or $T_{1}\leq 800~{}\upmu$s. Several
comments can be made on this coherence limit, which is illustrated in Fig. 5
alongside the coherence reported in several recent studies of transmons on
sapphire.

Figure 5: Inferred bulk coherence limit and comparison to measured transmon
lifetimes. The coherence limit imposed by EFG sapphire and HEMEX sapphire
assuming a bulk participation of $p_{\text{bulk}}=0.8$, compared to quality
factors of transmons on sapphire from select studies. Comparisons are made in
terms of total loss $Q^{-1}$ and in terms of $T_{1}^{\text{eff}}$; a $T_{1}$
calculated from $Q^{-1}$ assuming a $4$ GHz qubit transition frequency.

First, this limit is consistent with previous studies [12] in which transmon
lifetimes were found to be limited by surface losses, as those studies
featured devices with quality factors up to $4\times 10^{6}$, and therefore
would not have observed a strong effect from a coherence limit at the level of
$20\times 10^{6}$. This is illustrated in Fig. 5 by the EFG sapphire bulk loss
limit being significantly below the total loss rates $Q^{-1}$ observed in that
study.

Second, this limit is consistent with the best-reported lifetimes of transmons
made on EFG sapphire [37, 2, 3], as no such study has yet produced a transmon
with quality factor $Q>20\times 10^{6}$. Interestingly, the quality factors of
the longest lived-transmons are approaching this bound, as seen in Fig. 5.
Supposing that the fabrication process used to make those transmons does
nothing to improve the bulk loss tangent of the substrate, this comparison
suggests those transmons may be significantly bulk limited. For bulk-limited
devices, replacing EFG with HEMEX sapphire will substantially increase
transmon lifetimes. The increased coherence limit is also shown in Fig. 5.
Further, the shared chemistry and lattice constant of these materials implies
that this substitution would not require any modification of the fabrication
process.

A general insight gained from this comparative measurement is that not all
sapphire is equal. As with high-power measurements made at several kelvin [38,
39, 40, 41], growth method has a meaningful impact on the millikelvin-and-low-
power loss tangent of sapphire, and in turn the lifetime of transmons built
upon that sapphire.

Additionally, for completeness we use the measured value of
$q_{\text{SA}}^{-1}$ to calculate the SA coherence limit for a typical
transmon. Assuming $p_{\text{SA}}=1.18\times 10^{-4}$ [12], substrate-air
surface dielectric loss would limit the quality factor of a transmon to $Q\leq
7\times 10^{6}$. In real devices, however, the loss tangent
$q_{\text{SA}}^{-1}$ will likely depend on the details of the substrate
surface preparation. More definite claims can be made about the SA-interface
coherence limit of real devices by leveraging the flexibility of the dipper
technique and measuring dielectric samples that have undergone the same
surface preparation sequence as the substrates used for those devices (for
details on the preparation of samples in this study, see App. D).

Finally, the model used in our analysis has assumed that all sample loss is
dielectric in origin rather than magnetic, but we can extend the model and
allow sample loss to be a combination of dielectric and magnetic
contributions. By considering our results for EFG sapphire in comparison to
performance of striplines from a previous study [42], we can make two bounding
statements about magnetic bulk loss in EFG sapphire. First, the EFG bulk loss
tangent measured in this study must be at least 97% truly dielectric in
origin, and no more than 3% could be from misattributed magnetic loss. This
validates the choice to neglect magnetic sample loss in our analysis. Second,
bulk magnetic loss is not a major source of loss in even the longest-lived
transmons on EFG sapphire, accounting for less than 10% of the total loss.
(for details, see App. H).

##  V Conclusion

Precise knowledge of the decoherence mechanisms in superconducting qubits aids
in all efforts to improve their coherence. Here, we present a method to
precisely measure bulk dielectric loss with $5\times 10^{-9}$ sensitivity, and
apply it in a study of EFG sapphire. At low powers and temperatures, we
measure the bulk loss tangent at microwave frequencies to be $63(8)\times
10^{-9}$. This entails a bulk limit to the quality factor of a typical
transmon on an EFG sapphire substrate of $Q\lesssim 20\times 10^{6}$,
consistent with the longest lifetimes of transmons on sapphire, and suggests
bulk dielectric loss is a major source of loss in those devices. We also
measure HEMEX sapphire, and bound its bulk loss tangent to be less than
$19(6)\times 10^{-9}$. Such a low-loss substrate would reduce one of the
dominant sources of loss by a factor of 3 or more, and could enable longer
transmon lifetimes.

Looking forward, measurements of other materials, such as silicon, quartz, or
lithium niobate, can inform efforts to improve coherence of a variety of
devices, including qubits, on those substrates. We also expect that
correlation of precision dielectric loss measurements with other physical
probes (e.g. x-ray spectroscopy or scattering) will shed light on the
underlying physical mechanisms behind this loss.

We acknowledge N. Ofek, Y. Liu, and P. Reinhold, for their work building the
FPGA firmware and software used in this experiment, C. J. Axline and K. Li for
their contributions in developing earlier versions of the experiment, and N.
P. de Leon, A. Walter, A. Barbour, M. H. Devoret and S. M. Girvin for useful
discussions. Use of facilities was supported by the Yale Institute for
Nanoscience and Quantum Engineering (YINQE), Yong Sun, and the Yale SEAS
cleanroom. We thank MIT Lincoln Laboratory and the Intelligence Advanced
Research Projects Activity (IARPA) for providing a Josephson travelling wave
parametric amplifier. This research, including funding for A.P.R., was
supported by the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, the Co-design Center for Quantum
Advantage (C2QA) under contract number DE-SC0012704. Further support,
including funding for B.J.C., came from the Army Research Office (ARO), under
Grant Number W911NF-18-1-0212. The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing the official parties, either expressed or implied, of the Army
Research Office (ARO) or the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for government purposes
notwithstanding any copyright notation herein.

B.J.C. and R.J.S. conceived the experiment; A.P.R. and B.J.C. designed and
characterized the experimental apparatus, carried out the experiments,
analyzed the data, and wrote the manuscript with input from all authors; L.F.
and R.J.S. supervised the work.

L.F. and R.J.S. are founders and shareholders of Quantum Circuits Inc.

##  Appendix A Ringdown measurement

To measure the internal energy decay rate $\kappa_{\text{int}}$ of a microwave
cavity with (classical) mode amplitude $a$, the cavity is excited with an
input field $a_{\text{in}}$ (Fig. 6a) and the output field $a_{\text{out}}$ is
monitored to determine the rate at which its energy is lost.

During the pulse, the cavity mode amplitude $a$ rises. After the pulse, $a$
decays exponentially (Fig. 6b). While a portion of the energy from the
excitation pulse makes its way into the cavity mode, the rest is reflected off
the cavity port and propagates to the receiver (for the complete wiring
diagram, see App. J). At the same time, the cavity emits a field proportional
to $a$. The reflected pulse and the cavity emission interfere to create the
output field $a_{\text{out}}$, which is measured at the receiver (Fig. 6c).

Figure 6: Dynamics of a resonant ringdown measurement: (a) the incoming field
$a_{\text{in}}$, (b) cavity mode amplitude $a$, and (c) the outgoing field
$a_{\text{out}}$, each as a function of time. Figure is not to scale.

We determine the total loss of the cavity mode $\kappa_{\text{tot}}$ by
fitting the decay rate of $|a_{\text{\text{out}}}|^{2}$ for times
$t>t_{\text{p}}$, during which $a_{\text{out}}\propto a$. To distinguish
between the effects of dephasing and actual energy decay, the emitted field
must be sampled at a rate faster than the dephasing time of the cavity (see
App. A.2), and data from multiple shots of the experiment must be processed
with a phase-insensitive averaging scheme (see App. A.1).

The external coupling-rate of the cavity mode $\kappa_{\text{ext}}$ is encoded
in the overall amplitude of the emitted field in comparison to the amplitude
and duration of the excitation pulse and can be inferred from the outgoing
field after the beginning ($t=0^{+}$) and after the end ($t=t_{\text{p}}^{+}$)
of the excitation pulse (see App. A.3).

Those two loss rates can then be subtracted to determine the internal energy
decay rate $\kappa_{\text{int}}=\kappa_{\text{tot}}-\kappa_{\text{ext}}$,
which can be divided by the resonant frequency $\omega_{a}$ of the cavity to
yield a dimensionless loss:
$Q^{-1}_{\text{int}}=\kappa_{\text{int}}/\omega_{\text{a}}$.

###  A.1 Decay of a harmonic oscillator with time-varying resonant frequency

In the presence of a fluctuating cavity frequency, the cavity output field
$|\langle a_{\text{out}}\rangle|$ (averaged over many shots) does not decay at
the energy decay rate $\kappa_{\text{tot}}$. Rather, it decays at a rate which
is also affected by cavity dephasing. For this reason, measurements of
$|\langle a_{\text{out}}\rangle|$, such as those from a vector network
analyzer, cannot be used to reliably extract $\kappa_{\text{tot}}$. Instead,
one must measure the ensemble-average of the cavity output power,
$\langle|a_{\text{\text{out}}}|^{2}\rangle$, which is a phase-insensitive
quantity and decays at the same rate, $\kappa_{\text{tot}}$, as the cavity
energy. In many cases, this distinction is irrelevant, but in our experiment,
mechanical vibrations of the high-permittivity dielectric sample cause the
cavity frequency to jitter by as many as 100 line widths (see Fig. 7). When
the dielectric is inserted, accurate measurement of $\kappa_{\text{tot}}$
therefore requires fitting the decay of $\langle|a_{\text{out}}|^{2}\rangle$,
rather than that of $|\langle a_{\text{out}}\rangle|^{2}$. In this section we
describe how the energy decay rate $\kappa_{\text{tot}}$ can be inferred from
ringdown measurements of an oscillator with a fluctuating resonance frequency.

Suppose an oscillator has a time-varying resonant frequency
$\omega_{\text{a}}=\omega_{0}(1+\epsilon(t))$. Assume the fluctuations of the
resonance $\epsilon(t)$ are caused by a stochastic process with zero mean,

$\displaystyle\langle\epsilon(t)\rangle$ $\displaystyle=$ $\displaystyle 0,$
(20)

and characterized by a power spectral density $S(f)$:

$\displaystyle\langle\epsilon(t)^{2}\rangle$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dfS(f)\cos(2\pi ft).$ (21)

where the brackets $\langle\cdot\rangle$ denote ensemble averaging.

The Langevin equation [43] dictates how $a$ evolves in time. In the absence of
an excitation pulse (here, true for $t>t_{\text{p}}$), the measured output
field $a_{\text{out}}$ is proportional to $a$ and governed by an equation of
the same form:

$\displaystyle\dot{a}_{\text{out}}$ $\displaystyle=$
$\displaystyle[i\omega_{0}(1+\epsilon(t))-\kappa_{\text{tot}}/2]a_{\text{out}}.$
(22)

After separation of variables, integration yields

$\displaystyle a_{\text{out}}(t)$ $\displaystyle=$ $\displaystyle
e^{(i[\omega_{0}t+\theta(t)]-\kappa_{\text{tot}}t/2)},$ (23)

with

$\displaystyle\theta(t)$ $\displaystyle\equiv$
$\displaystyle\omega_{0}\int_{0}^{t}\epsilon(t^{\prime})dt^{\prime}.$ (24)

Here we have chosen the constant of integration to reflect an initial
condition of unit amplitude in the oscillator mode.

The average of the output power decays as

$\displaystyle\langle|a_{\text{out}}(t)|^{2}\rangle=e^{-\kappa_{\text{tot}}t}.$
(25)

We identify the decay time of the oscillator as $\tau\equiv
1/\kappa_{\text{tot}}$. $\langle|a_{\text{out}}(t)|^{2}\rangle$ is independent
of $\theta(t)$ and is thus unaffected by the fluctuations in the resonant
frequency.

This is in contrast to the average of the output field,

$\displaystyle|\langle a_{\text{out}}(t)\rangle|$ $\displaystyle=$
$\displaystyle e^{-\kappa_{\text{tot}}t/2}|\langle e^{i\theta(t)}\rangle|,$
$\displaystyle=$ $\displaystyle
e^{-\kappa_{\text{tot}}t/2}\left(1-\frac{1}{2}\langle\theta(t)^{2}\rangle+\mathcal{O}(\theta(t)^{3})\right).$

By our assumption that the fluctuations are zero-mean, the linear term in the
expansion vanishes. (If the noise is Gaussian, the higher moments vanish and
this approximation is exact.) To evaluate the quadratic term, we follow the
treatment given in Ref. [44]:

$\displaystyle\langle\theta(t)^{2}\rangle$ $\displaystyle=$
$\displaystyle\omega_{0}^{2}\int_{0}^{t}\int_{0}^{t}dt^{\prime}dt^{\prime\prime}\langle\epsilon(t^{\prime})\epsilon(t^{\prime\prime})\rangle,$
$\displaystyle=$
$\displaystyle\omega_{0}^{2}\int_{0}^{t}\int_{0}^{t}dt^{\prime}dt^{\prime\prime}\int_{0}^{\infty}dfS(f)\textrm{Re}\\{e^{(2\pi
if(t^{\prime}-t^{\prime\prime}))}\\},$ $\displaystyle=$
$\displaystyle\omega_{0}^{2}\int_{0}^{\infty}dfS(f)W_{0}(f,t),$

where the spectral weight function $W_{0}(f,t)$

$\displaystyle W_{0}(f,t)$ $\displaystyle=$
$\displaystyle\left|\int_{0}^{t}dt^{\prime}e^{2\pi ift^{\prime}}\right|^{2},$
$\displaystyle=$ $\displaystyle\frac{\sin(\pi ft)^{2}}{(\pi f)^{2}},$

low-pass filters the fluctuations.

We therefore have

$\displaystyle|\langle a_{\text{out}}\rangle|$ $\displaystyle=$ $\displaystyle
e^{-\kappa_{\text{tot}}t/2}\times$
$\displaystyle\left(1-\frac{1}{2}\omega_{0}^{2}\int_{0}^{\infty}dfS(f)W_{0}(f,t)+\mathcal{O}(\langle\theta(t)^{3}\rangle)\right).$

The exponential prefactor represents energy decay from the oscillator and the
rest of the expression accounts for the effect of pure dephasing. The form of
this dephasing depends on the noise spectrum $S(f)$.

In summary, the exponential decay of the phase-insensitive quantity
$\langle|a_{\text{out}}|^{2}\rangle$ yields the time-scale for energy decay
$\tau$, and the phase-sensitive quantity $|\langle a_{\text{out}}\rangle|$ has
a $1/e$ time which is affected by both energy decay and dephasing in the
cavity. This is analogous to the distinction between $T_{1}$ and $T_{2}$ of a
spin ensemble. Fluctuations in the cavity’s resonant frequency can be detected
by comparing $\tau$ to the $1/e$ time of $|\langle
a_{\text{out}}\rangle|^{2}$. By converting these decay rates to quality
factors, they can be checked for consistency with quality factors extracted
from circle fits of spectroscopic measurements [45] (See Fig. 7).

Figure 7: Consistency between ringdown and spectroscopic measurement. When the
sample is withdrawn from the cavity, $Q_{\text{tot}}$ inferred from the decay
rates of $\langle|a(t)|^{2}\rangle$ (blue) and $|\langle a(t)\rangle|^{2}$
(red) are in close agreement with each other, and with $Q_{\text{tot}}$ from
circle fits [45] of measurements taken by a vector network analyzer $\langle
S_{21}(f)\rangle$ (green). As the sample is inserted, energy decay and
frequency jitter tend to reduce $Q_{\text{tot}}$ extracted from $|\langle
a(t)\rangle|^{2}$ or $\langle S_{21}(f)\rangle$, however
$\langle|a(t)|^{2}\rangle$ is not susceptible to frequency jitter (see Eq. 25)
and thus only experiences reduction in $Q_{\text{tot}}$ caused by energy
decay. $Q_{\text{tot}}$ can be, in general, non-monotonic in $p_{\text{bulk}}$
due to frequency-dependent coupling loss, which is measured and subtracted off
before further processing (see App. A.3).

###  A.2 Finite measurement bandwidth

The above arguments can be generalized to account for the effects of finite
measurement bandwidth, such as that imposed by a non-instantaneous
measurement. In short, finite measurement bandwidth causes a reduction in the
contrast of the measurement $\langle|a^{\text{m}}_{\textrm{out}}|^{2}\rangle$,
but does not change the time scale for decay. For completeness, we present
those arguments below, as the suppression of contrast is an effect that must
be controlled in order to accurately measure $Q_{\text{ext}}^{-1}$ (a
measurement discussed in the next subsection, App. A.3).

The result of a finite bandwidth measurement of the output field
$a^{\text{m}}_{\text{out}}(t)$ is the convolution of the signal
$a_{\text{out}}(t)$ with the impulse response $h(t)$ of the detector.

In practice, detector bandwidth is set by the inverse of the period
$t_{\text{m}}$ over which the signal is averaged:

$\displaystyle
a^{\text{m}}_{\text{out}}(t)\approx\frac{1}{t_{\text{m}}}\int_{t-t_{\text{m}}/2}^{t+t_{\text{m}}/2}dt^{\prime}e^{i\theta(t^{\prime})-\kappa_{\text{tot}}t^{\prime}/2}.$
(30)

The resonant frequency $\omega_{0}$ doesn’t appear in this expression because
we assume that detection occurs after demodulation.

To evaluate the effect on the measured coherence time of the cavity, we focus
on the measured average output field

$\displaystyle|\langle a^{\text{m}}_{\text{out}}\rangle|$ $\displaystyle=$
$\displaystyle\left|\langle\frac{1}{t_{\text{m}}}\int_{t-t_{\text{m}}/2}^{t+t_{\text{m}}/2}dt^{\prime}e^{i\theta(t^{\prime})-\kappa_{\text{tot}}/2}\rangle\right|.$

Expanding as in Ref. [44],

$\displaystyle|\langle a^{\text{m}}_{\text{out}}\rangle|$ $\displaystyle=$
$\displaystyle\left|\frac{1}{t_{\text{m}}}\left[\frac{\sinh(\kappa_{\text{tot}}t_{\text{m}}/2)}{(\kappa_{\text{tot}}/2)}e^{-\kappa_{\text{tot}}t/2}-\right.\right.$
$\displaystyle\frac{1}{2}\omega_{0}^{2}\int_{0}^{\infty}dfS(f)\times$
$\displaystyle\
\left.\left.\int_{t-t_{\text{m}}/2}^{t+t_{\text{m}}/2}dt^{\prime}e^{-\kappa_{\text{tot}}t^{\prime}/2}W_{0}(f,t^{\prime})+\mathcal{O}(<\theta(t^{\prime})^{3}>)\right]\right|.$

Comparing this result to eq. (A.1), we can see that the time-averaging has two
effects. The first effect is independent of the frequency fluctuations, as can
be seen by setting $S(f)=0$:

$\displaystyle|\langle
a^{\text{m}}_{\text{out}}\rangle|=\frac{\sinh(\kappa_{\text{tot}}t_{\text{m}}/2)}{(\kappa_{\text{tot}}t_{\text{m}}/2)}e^{-\kappa_{\text{tot}}t/2}.$
(32)

In this limit, the time-dependence is an exponential decay characterized by a
timescale $T_{2}=2/\kappa_{\text{tot}}$. The averaging, however, can reduce
the initial measured amplitude. In the limit of fast measurements
($\kappa_{\text{tot}}t_{\text{m}}\ll 1$), unit amplitude is recovered.

The second effect of time averaging is a redefinition of the spectral weight
function. The new function is a weighted average of $W_{0}$ taken over the
acquisition window.

By a similar argument, we can evaluate the effect of measurement bandwidth on
the measured energy decay rate. The average output power evaluates to

$\displaystyle\langle|a^{\text{m}}_{\textrm{out}}|^{2}\rangle$
$\displaystyle=$ $\displaystyle
e^{-\kappa_{\text{tot}}t}\left(1-\omega_{0}^{2}\int_{0}^{\infty}dfS(f)W_{1}(f)\right)$
$\displaystyle+\mathcal{O}(\langle[\theta(t^{\prime})-\theta(t^{\prime\prime})]^{3}\rangle)+\mathcal{O}(\kappa_{\text{tot}}t_{\text{m}})^{2}.$

with

$\displaystyle W_{1}(\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{1-\textrm{sinc}(\omega
t_{\text{m}}/2)^{2}}{(\kappa_{\text{tot}}/2)^{2}+\omega^{2}}+\mathcal{O}(\kappa_{\text{tot}}t_{\text{m}})^{3}$
(33)

and $\omega=2\pi f$.

From Eq. (A.2), we see that the fluctuations change the contrast, but do not
change the decay rate. The amount by which they change the contrast is
controlled by the spectrum of the fluctuations and the averaging time
$t_{\text{m}}$ because the spectral weight $W_{1}(f)$ acts as a low-pass
filter with corner frequency $1/t_{\text{m}}$.

###  A.3 Measurement of the cavity coupling rate

After measuring the total loss rate $\kappa_{\text{tot}}$ of the system by
fitting $\langle|a_{\text{out}}|^{2}\rangle$ to an exponential decay,
measurement of the coupling loss rate $\kappa_{\text{ext}}$ enables the
isolation of internal loss rate
$\kappa_{\text{int}}=\kappa_{\text{tot}}-\kappa_{\text{ext}}$. We now describe
how the external coupling of the cavity mode can be inferred from a ringdown
experiment. Consider a cavity with classical mode amplitude $a$, frequency
$\omega_{\text{a}}$, and total loss rate $\kappa_{\text{tot}}$ coupled to a
single port with rate $\kappa_{\text{ext}}$. The cavity may be driven through
this port by an incident field $a_{\text{in}}$ with its carrier frequency
$\omega_{\text{d}}$ detuned from the cavity by
$\Delta\equiv\omega_{\text{d}}-\omega_{\text{a}}$. In the rotating frame of
the drive, the Langevin equation [43] is

$\displaystyle\dot{a}=(i\Delta-\kappa_{\text{tot}}/2)a+\sqrt{\kappa_{\text{ext}}}a_{\text{in}}.$
(34)

The general solution for the cavity field is

$\displaystyle
a(t)=-\sqrt{\kappa_{\text{ext}}}\int_{0}^{t}dt^{\prime}a_{\text{in}}(t^{\prime})e^{(i\Delta-\kappa_{\text{tot}}/2)(t-t^{\prime})}.$
(35)

Let the envelope of the incident field be that of a rectangular pulse,

$a_{\text{in}}=\begin{cases}0,&t~{}<0,\\\ a_{0},&0<t<t_{\text{p}},\\\
0,&t_{\text{p}}<t.\end{cases}$

The resulting cavity field is given by

$a(t)=\begin{cases}0,&t~{}<0,\\\ \tilde{a}(t),&0~{}<t<t_{\text{p}},\\\
\tilde{a}(t_{\text{p}})e^{(i\Delta-\kappa_{\text{tot}}/2)(t-t_{\text{p}})},&t_{\text{p}}<t.\end{cases}$

where

$\displaystyle\tilde{a}(t)\equiv\frac{a_{0}\sqrt{\kappa_{\text{ext}}}}{i\Delta-\kappa_{\text{tot}}/2}\left(1-e^{(i\Delta-\kappa_{\text{tot}}/2)t}\right).$
(36)

We probe the cavity dynamics by measuring the emitted field $a_{\text{out}}$,
which input-output theory relates to the incident field and the cavity mode
[43]

$\displaystyle a_{\text{out}}=\sqrt{\kappa_{\text{ext}}}a+a_{\text{in}},$ (37)

Substituting in the expressions for the input and cavity fields,

$\displaystyle a_{\text{out}}(t)=\begin{cases}0,&t~{}<0,\\\
\sqrt{\kappa_{\text{ext}}}\tilde{a}(t)+a_{0},&0<t<t_{\text{p}},\\\
\sqrt{\kappa_{\text{ext}}}\tilde{a}(t_{\text{p}})e^{(i\Delta-\kappa_{\text{tot}}/2)(t-t_{\text{p}})},&t_{\text{p}}<t.\end{cases}$
(38)

The emitted fields slightly after the start of the pulse ($t=0^{+}$) and
slightly after the end of the pulse ($t=t_{\text{p}}^{+}$) are:

$\displaystyle a_{\text{out}}(0^{+})$ $\displaystyle=$ $\displaystyle a_{0},$
$\displaystyle a_{\text{out}}(t_{\text{p}}^{+})$ $\displaystyle=$
$\displaystyle\sqrt{\kappa_{\text{ext}}}\tilde{a}(t_{\text{p}}).$ (39)

These two points contain all the information required to extract the external
coupling rate $\kappa_{\text{ext}}$. The output field $a_{\text{out}}(t)$ at
these points in time can be converted to phase-insensitive output powers
$\langle|a_{\text{out}}|^{2}\rangle$. Assuming the cavity has been driven on-
resonance ($\Delta=0$),

$\displaystyle\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ $\displaystyle=$
$\displaystyle a^{2}_{0},$ (40)
$\displaystyle\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$
$\displaystyle=$
$\displaystyle\frac{4a^{2}_{0}\kappa_{\text{ext}}^{2}}{\kappa_{\text{tot}}^{2}}\left(1-2e^{-\kappa_{\text{tot}}t_{\text{p}}/2}+e^{-\kappa_{\text{tot}}t_{\text{p}}}\right).$

Solving for the external coupling $\kappa_{\text{ext}}$ in terms of measured
quantities and the experimental parameter pulse length $t_{\text{p}}$,

$\displaystyle\kappa_{\text{ext}}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{\text{tot}}}{2\left(1-e^{-\kappa_{\text{tot}}t_{\text{p}}/2}\right)}\sqrt{\frac{\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle}{\langle|a_{\text{out}}(0^{+})|^{2}\rangle}}.$
(41)

As the pulse is typically very short compared to $1/\kappa_{\text{tot}}$, we
can instead expand Eq. (40) to second order in
$\kappa_{\text{tot}}t_{\text{p}}/2$,

$\displaystyle\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ $\displaystyle=$
$\displaystyle a^{2}_{0},$
$\displaystyle\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$
$\displaystyle\approx$ $\displaystyle
a^{2}_{0}\kappa_{\text{ext}}^{2}t_{\text{p}}^{2},$ (42)

and solve (A.3) for $\kappa_{\text{ext}}$,

$\displaystyle\kappa_{\text{ext}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{t_{\text{p}}}\sqrt{\frac{\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle}{\langle|a_{\text{out}}(0^{+})|^{2}\rangle}}.$
(43)

Although the output field at two precise moments in time is the only
information needed for calculating $\kappa_{\text{ext}}$, a more precise
determination of $\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$ can be
had by fitting the entire ringdown portion of the data ($t<t_{\text{p}}$) to
an exponential decay and extracting the ringdown amplitude.

A similar strategy could be employed to improve the precision of
$\langle|a_{\text{out}}(0^{+})|^{2}\rangle$, but not as lucratively; because
jitter in the system requires the pulse to be short (see App. A.4), there is
not much pulse data ($0<t<t_{\text{p}}$) for the fit. A workaround comes from
creating a scenario where interference from cavity emission is negligible,
thus resulting in $\langle|a_{\text{out}}(t)|^{2}\rangle$ having a constant
amplitude $\langle|a_{\text{out}}(0^{+})|^{2}\rangle$, even for an arbitrarily
long pulse. Assuming the transmission background of the entire measurement
system $|S_{21}|$ is smooth and varies only slowly with frequency,
$\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ can be determined by measuring the
transmitted power of a long pulse far above (perhaps hundreds of linewidths)
and equally far below the cavity frequency $\omega_{\text{a}}$, then averaging
the two for an approximation of $\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ at
$\omega_{\text{a}}$. Pulses played far off-resonant from the cavity will not
excite the cavity, so there will be no emission signal to interfere with the
reflected pulse signal.

If the cavity is frequency-tunable, as it is in the dipper by movement of the
sample, then the off-resonance condition can be met by changing the frequency
of the cavity instead of the frequency of the pulse. If the cavity is moved
away from the frequency of interest, then a single pulse at that frequency can
be used to measure $\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ at that
frequency directly, rather than estimating it by linear interpolation between
two measurements at other frequencies. In practice this is the approach we
take, although both methods give similar results.

Finally, the short pulses ($t_{\text{p}}\approx 5~{}\upmu\text{s}$) and weak
coupling ($Q_{\text{ext}}\approx 2\times 10^{8}$; coupling rate
$\kappa_{\text{ext}}/2\pi\approx 100~{}\text{Hz}$) used in this measurement
mean that, for the same pulse amplitude,
$\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ will be larger than
$\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$ by a few orders of
magnitude. To avoid overdriving the receiver, the transmission measurements of
$\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ can be performed with a pulse
amplitude smaller by some reduction factor to make the
$\langle|a_{\text{out}}(0^{+})|^{2}\rangle$ measurements more comparable in
size to the $\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$
measurement. Before using this reduced-pulse measurement to calculate
$Q_{\text{ext}}$, its result must be re-scaled such as to compensate for not
only that reduction factor, but also for any signal compression on the way to
the cavity. Even if this compression does not affect this smaller pulse for
measuring $\langle|a_{\text{out}}(0^{+})|^{2}\rangle$, it may still affect the
larger pulse used for measuring
$\langle|a_{\text{out}}(t_{\text{p}}^{+})|^{2}\rangle$.

###  A.4 Calculating photon number

A cavity state with mode amplitude $a$ has an average of $n$ photons:

$\displaystyle n(t)=|a(t)|^{2}.$ (44)

To calculate $n$, we can use Eq. (37) which relates the mode amplitude $a$ to
more directly accessible quantities $a_{\text{in}}(t)$ or $a_{\text{out}}(t)$.
To infer cavity photon number from the field emitted by the cavity, we can
start by solving Eq. (37) for $a$ after the pulse has stopped
($a_{\text{in}}(t)=0$):

$\displaystyle a(t)=\frac{a_{\text{out}}(t)}{\sqrt{\kappa_{\text{ext}}}}.$
(45)

Substituting into eq. 44 gives an expression for $n(t)$ in terms of the output
field:

$\displaystyle n(t)=\frac{|a_{\text{out}}(t)|^{2}}{\kappa_{\text{ext}}}.$ (46)

This output-field equation for photon number can alternately be expressed in
terms of an output power
$P_{\text{out}}(t)=\hbar\omega_{\text{a}}|a_{\text{out}}(t)|^{2}$. Solving for
$n(t)$, we have

$\displaystyle
n(t)=\frac{P_{\text{out}}(t)Q_{\text{ext}}}{\hbar\omega_{\text{a}}^{2}}.$ (47)

Using eq. 40, the output-field equation (46) can be converted into an input-
field equation that predicts the number of photons injected by a rectangular
excitation pulse of duration $t_{\text{p}}$ and amplitude $a_{0}$:

$\displaystyle
n(t_{\text{p}})=\frac{4\kappa_{\text{ext}}a^{2}_{0}}{\kappa_{\text{tot}}^{2}}\left(1-2e^{-\kappa_{\text{tot}}t_{\text{p}}/2}+e^{-\kappa_{\text{tot}}t_{\text{p}}}\right).$
(48)

In the limit of a short pulse,

$\displaystyle
n(t_{\text{p}})=\kappa_{\text{ext}}a^{2}_{0}t_{\text{p}}^{2}+\mathcal{O}\left(\kappa_{\text{tot}}t_{p}\right)^{2}.$
(49)

This input-field equation for photon number can alternately be expressed in
terms of an input power
$P_{\text{in}}=\hbar\omega_{\text{d}}|a_{\text{in}}|^{2}=\hbar\omega_{\text{d}}a_{0}^{2}$,
which allows Eq. (49) to be expressed as

$\displaystyle n(t_{\text{p}})=\frac{P_{\text{in}}t_{\text{p}}^{2}}{\hbar
Q_{\text{ext}}}.$ (50)

So far, in this section we have assumed no jitter in the cavity frequency. In
addition to simplifying Eq. (49), a short drive pulse is important for proper
control of photon number in the presence of jitter. In general, the photon
number will be dependent upon the inner product of the cavity spectrum and the
pulse spectrum. If the pulse is short enough that its spectrum is
approximately uniform over the jitter width of the cavity, then that inner
product is approximated well by the above expressions (such considerations are
also essential for accurate measurement of $Q_{\text{ext}}^{-1}$ per App.
A.3).

##  Appendix B Participation simulations

To calculate the participation ratios $p_{j}$, we use finite-element analysis
software (Ansys HFSS) to solve for the eigenmodes of Maxwell’s equations in
our cavity. The participation ratios are defined in terms of material
parameters and the field solutions $\mathbf{E}$ and $\mathbf{H}$ for the
fundamental cavity mode. We consider four participations in our model: the
bulk dielectric participation $p_{\text{bulk}}$, the substrate-air surface
dielectric participation $p_{\text{SA}}$, the metal-air surface dielectric
participation $p_{\text{MA}}$, and the conductor participation
$p_{\text{cond}}$ (equal to the kinetic inductance ratio in bulk
superconductors [46]). We define these participations as

$\displaystyle p_{\text{bulk}}$ $\displaystyle\equiv$
$\displaystyle\frac{U_{\text{bulk}}}{U_{\text{tot}}},$ $\displaystyle
p_{\text{SA}}$ $\displaystyle\equiv$
$\displaystyle\frac{U_{\text{SA}}}{U_{\text{tot}}},$ $\displaystyle
p_{\text{MA}}$ $\displaystyle\equiv$
$\displaystyle\frac{U_{\text{MA}}}{U_{\text{tot}}},$ $\displaystyle
p_{\text{cond}}$ $\displaystyle\equiv$
$\displaystyle\frac{U_{\text{cond}}}{U_{\text{tot}}},$ (51)

where the energies $U_{j}$ in Eq. (51) are defined as [24, 12]

$\displaystyle U_{\text{tot}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\text{Re}\left\\{\int_{V_{\text{tot}}}\mathbf{E}\cdot\mathbf{D}^{\ast}\
dV\right\\},$ $\displaystyle=$
$\displaystyle\frac{1}{2}\text{Re}\left\\{\int_{V_{\text{tot}}}\mathbf{H}\cdot\mathbf{B}^{\ast}\
dV\right\\},$ $\displaystyle U_{\text{bulk}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\text{Re}\left\\{\int_{V_{\text{sub}}}\mathbf{E}\cdot\mathbf{D}^{\ast}\
dV\right\\},$ $\displaystyle U_{\text{SA}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}t_{\text{SA}}\text{Re}\left\\{\int_{S_{\text{sub}}}\mathbf{E}\cdot\mathbf{D}^{\ast}\
dA\right\\},$ $\displaystyle U_{\text{MA}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}t_{\text{MA}}\text{Re}\left\\{\frac{\epsilon_{\text{0}}}{\epsilon_{\text{MA}}}\int_{S_{\text{cav}}}\mathbf{E}\cdot\mathbf{D}^{\ast}\
dA\right\\},$ $\displaystyle U_{\text{cond}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\lambda_{\text{L}}\text{Re}\left\\{\int_{S_{\text{cav}}}\mathbf{H}\cdot\mathbf{B}^{\ast}\
dA\right\\}.$ (52)

Here, $V_{\text{sub}}$ and $S_{\text{sub}}$ refer to the volume and surface of
the dielectric substrate under study, $S_{\text{cav}}$ is the surface of the
aluminum cavity, and $V_{\text{tot}}$ is the volume of the entire system.

When solving for the eigenmodes of our system, we account for the
birefringence of sapphire by assigning a tensorial relative permittivity
$\overleftrightarrow{\epsilon_{\text{bulk}}}$ to the sample region. We assume
a literature value for relative permittivity of 11.35 parrallel to the
$c$-axis, and 9.27 perpendicular to the $c$-axis [47].

For energies in material interface regions; $U_{\text{SA}}$, $U_{\text{MA}}$
and $U_{\text{cond}}$, volume integrals have been reduced to surface integrals
by assuming field uniformity over the depth of the interface and integrating
over the depth to yield a corresponding length prefactor. This simplification
allows these interface regions with distinct material properties to be
represented by interfaces between regions defined in software, rather than by
regions of their own. This amounts to a significant saving in processing time,
as accurate field solutions in high aspect ratio volumes requires fine
meshing, which can be computationally intensive. By this simplification,
interface regions are treated as perturbations on the system, as the assumed
material properties and thickness of these interfaces do not feed back and
affect the field solutions acquired by the simulation.

The participations quoted in this work assume the surface dielectric thickness
parameters $t_{\text{SA}}$ and $t_{\text{MA}}$ to be $3$ nm. We emphasize that
this is a choice made for ease of comparison with surface loss tangents on
silicon [26], and is not based on any direct knowledge of the surface
thicknesses in our system. The London penetration depth $\lambda_{\text{L}}$
is set to $50$ nm, which we confirm to be approximately accurate by fitting
temperature sweeps of the cavity resonance frequency and quality factor to
Mattis-Bardeen theory (see Table 8 in App. F).

For calculation of $U_{\text{SA}}$, it is crucial to perform the integral to
one side of the substrate-air (SA) interface (rather than directly over it) to
avoid ambiguity about the permittivity assigned by the software. The
expression for $U_{\text{SA}}$ found in eq. (LABEL:eqn:p_energies) is able to
take on its compact form because of two choices made in our analysis. For one,
we chose to perform the integral over the substrate-side of the SA-interface.
Additionally, we assume the permittivity of the sample surface
$\epsilon_{\text{SA}}$ to be the same as that of the sample bulk
$\epsilon_{bulk}$, in which the integration was performed. The expression for
$U_{\text{SA}}$ can be generalized to relax this equal permittivity
assumption. This is done by decomposing the integrand into its components
normal and tangent to the SA interface and invoking continuity of $\mathbf{E}$
and $\mathbf{D}$ where appropriate:

$\displaystyle\mathbf{E}\cdot\mathbf{D}^{\ast}$ $\displaystyle\rightarrow$
$\displaystyle{E}_{\parallel}\left(\frac{\epsilon_{\parallel}^{\text{SA}}}{\epsilon_{\parallel}^{\text{bulk}}}{D}_{\parallel}\right)^{\ast}+\left(\frac{\epsilon_{\perp}^{\text{bulk}}}{\epsilon_{\perp}^{\text{SA}}}{E}_{\perp}\right){D}_{\perp}^{\ast}.$
(53)

After this generalizing substitution, $U_{\text{SA}}$ is still expressed in
terms of the fields on the substrate-side of the SA interface. However,
similar attention to boundary conditions permits $U_{\text{SA}}$ to be
alternately expressed in terms of the fields in the vacuum immediately outside
the sample, in case it is preferable to evaluate the surface integral over
that side of the SA interface instead.

Unlike the SA interface, the MA interface exists at the outer boundary of the
simulation. This means when calculating $U_{\text{MA}}$, there is no ambiguity
in the value of permittivity assigned by the software; nor is there any choice
about the side of the interface over which to perform the integration—the
integral must be evaluated over the vacuum side. Again invoking continuity of
$\mathbf{E}$ and $\mathbf{D}$, fields in the MA interface region, along with
the resulting $U_{\text{MA}}$, can be expressed in terms of the fields in the
vacuum, where the field solutions are available. Unlike for the SA interface,
however, this transformation is simplified by the conductor at the MA
interface boundary, which requires all electric fields to be normal to the
surface. With no tangential term in the integrand, the permittivity factor
multiplying the normal term can be factored out of the integral. The result is
that $U_{\text{MA}}$ uniformly scales inversely with $\epsilon_{\text{MA}}$,
which we assume to be equal to $10$.

##  Appendix C Measurement sensitivity in a polynomial basis

To gain intuition about what sets the sensitivities plotted in Fig. 2, we can
use the mapping between $z$ and $p_{\text{bulk}}(z)$ (Fig. 1b) to express
$p_{\text{cond}}(z)$, $p_{\text{MA}}(z)$, and $Q^{-1}(z)$ as polynomials in
$p_{\text{bulk}}$. In practice, 2nd-order polynomials closely approximate the
system, as shown in Fig. 8:

$\displaystyle Q^{-1}$ $\displaystyle=$ $\displaystyle
y_{0}+y_{1}p_{\text{bulk}}+y_{2}p_{\text{bulk}}^{2},$ $\displaystyle
p_{\text{cond}}$ $\displaystyle=$ $\displaystyle x_{0}^{\text{cond}},$
$\displaystyle p_{\text{bulk}}$ $\displaystyle=$ $\displaystyle
p_{\text{bulk}},$ $\displaystyle p_{\text{MA}}$ $\displaystyle=$
$\displaystyle
x_{0}^{\text{MA}}+x_{1}^{\text{MA}}p_{\text{bulk}}+x_{2}^{\text{MA}}p_{\text{bulk}}^{2}.$
(54)

Recasting the problem in this fashion is convenient because conductor and
substrate losses are orthogonal in the $p_{\text{bulk}}$-polynomial basis; the
resulting matrix equation,

$\displaystyle\left(\begin{array}[]{c}y_{0}\\\ y_{1}\\\ y_{2}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}x^{\text{cond}}_{0}&0&x^{\text{MA}}_{0}\\\
0&1&x^{\text{MA}}_{1}\\\ 0&0&x^{\text{MA}}_{2}\\\
\end{array}\right)\left(\begin{array}[]{c}q_{\text{cond}}^{-1}\\\
q_{\text{sub}}^{-1}\\\ q_{\text{MA}}^{-1}\\\ \end{array}\right),$ (64)

is nearly diagonal.

Figure 8: Polynomial representation: (a) the measured inverse internal quality
factors, (b) the simulated MA participation, and (c) the simulated conductor
participation of the HEMEX sample, each as a function of bulk participation.
All lines are 2nd-order polynomial fits per Eq. (54).

Conveniently, it yields a single equation for $q_{\text{sub}}^{-1}$:

$\displaystyle q_{\text{sub}}^{-1}=y_{1}-x_{1}^{\text{MA}}q_{\text{MA}}^{-1}$
(65)

Error propagation on Eq. 65 gives the desired expression for the sensitivity:

$\displaystyle\sigma_{q_{\text{sub}}^{-1}}$ $\displaystyle=$
$\displaystyle\sigma_{y_{1}}+|x_{1}^{\text{MA}}|\sigma_{q_{\text{MA}}^{-1}}+\sigma_{x_{1}^{\text{MA}}}q_{\text{MA}}^{-1}.$
(66)

The terms in Eq. 66 reveal what sets the measurement sensitivity, and how it
can be improved. To estimate the relative importance of the terms in Eq. 66,
we calculate their magnitudes assuming a hypothetical lossless substrate with
the same permittivity as sapphire [47]. Neglecting errors in alignment and
simulation, the first term
$\sigma_{y_{1}}\approx\sqrt{\frac{1}{N}}\frac{\sigma_{Q^{-1}}}{\Delta
p_{\text{bulk}}}\approx 1$ ppb. The next two terms can be approximated using
the coefficient $x_{1}^{\text{MA}}$ listed in Table 3.

$k$ | $10^{9}~{}y_{k}$ | $10^{6}~{}x_{k}^{\text{cond}}$ | $x_{k}^{\text{bulk}}$ | $10^{9}~{}x_{k}^{\text{MA}}$
---|---|---|---|---
0 | 9.18(0.05) | 43.92(0.01) | 0 | 249(1)
1 | 26(3) | 0 | 1 | -300(60)
2 | 240(40) | 0 | 0 | 15800(900)
Table 3: Polynomial coefficients $\mathbf{y}$ and $\mathbf{x}^{j}$ extracted
from the fits to $Q^{-1}$ and $p_{\text{j}}$ in Fig. 8.

The two summands are then each approximately $2\times 10^{-9}$. If this
analysis is repeated with polynomial order of 3 or 4, the $x_{1}^{\text{MA}}$
term grows to approximately 5 ppb. Taken together, these considerations
suggest that the sensitivity floor estimated from this polynomial analysis is
several parts per billion, and is limited by our ability to separate MA losses
from those caused by the substrate. This baseline is consistent with the
sensitivity shown in Fig. 2b.

This sensitivity of several parts per billion, when compared with the assumed
measurement error of approximately 0.1 ppb indicates that input errors are
amplified by a factor of roughly 100 during the regression. This factor is
much less than the condition number of the participation matrix, which is
roughly $10^{7}$ for the samples considered in this work. Such a discrepancy
indicates the value of using the full covariance matrix.

Note that the above argument neglects the 1% fractional change in
$p_{\text{cond}}$ and the 3% fractional change in the cavity frequency that
occur when the sample is inserted, which make conductive losses slightly
dependent on insertion. Accounting for these effects, such as by replacing
$p_{\text{cond}}$ with $\tilde{p}_{\text{cond}}$, results in a small but non-
vanishing $x_{1}^{\text{cond}}$. Our analysis omits these effects as they
change the cavity loss $Q^{-1}$ by less than $10^{-10}$.

##  Appendix D Sample preparation

All samples are sourced from double-side-polished $c$-plane wafers (i.e., the
extraordinary axis normal to the plane of the wafer). The EFG wafers are
sourced from Kyocera, and the HEMEX wafers are sourced from GT Advanced
Technologies (now called Crystal Systems). Samples are cut from the wafers,
using a LatticeGear FlexScribe, to have dimensions of 6 mm x 45 mm. The
samples are then cleaned with isopropanol and blown dry with nitrogen gas.
Sample dimensions are verified by a wafer micrometer and Zeta-20 optical
profilometer.

##  Appendix E Power sweeps

###  E.1 Defining the low-power regime

We define the low-power regime as the range of cavity photon number $n$ which
is sufficiently small such that total loss in the system $Q^{-1}$ is nearly
desaturated and approximately the same as it would be at single-photon powers
$Q^{-1}_{\text{lp}}$. This deviation of $Q^{-1}$ from single-photon behavior
can be compared to the total saturable loss in the system to determine the
saturation fraction $F$ of loss in the system:

$\displaystyle F\equiv\frac{Q^{-1}_{\text{lp}}-Q^{-1}}{Q^{-1}_{\text{sat}}}$
(67)

This value is identically zero when system behavior matches
$Q^{-1}_{\text{lp}}$. Using Eq. (18), the saturation fraction can be expressed
in terms of $n$, $n_{\text{c}}$ and $\alpha$.

$\displaystyle
F=1-\frac{1}{\sqrt{1+\left(\frac{n}{n_{\text{c}}}\right)^{\alpha}}}$ (68)

Solving for $n$ indicates a maximum allowable photon number, given a maximum
allowable saturation fraction and the parameters $n_{\text{c}}$ and $\alpha$
from measurement of the system.

$\displaystyle\frac{n}{n_{\text{c}}}<\left(\frac{1}{(1-F)^{2}}-1\right)^{1/\alpha}$
(69)

We define the boundary of the low power regime as the $n$ for which $F<0.1$.
With $\alpha\approx 1/2$, this requires $n/n_{\text{c}}<5\times 10^{-2}$.
(Smaller values of $\alpha$ put less stringent bounds on $n$). To be
conservative, we choose $n=10^{5}$ such that $n/n_{\text{c}}<10^{-3}$. Low-
power measurements are made with a Josephson travelling wave parametric
amplifier [48].

###  E.2 Power sweep bulk participations

Table 4 shows the bulk participations for the inserted and withdrawn power
sweeps on all samples.

Material | withdrawn $p_{\text{bulk}}$ | inserted $p_{\text{bulk}}$
---|---|---
100 $\upmu$m EFG | $3.6\times 10^{-5}$ | $1.7\times 10^{-2}$
460 $\upmu$m EFG | $1.2\times 10^{-4}$ | $5.66\times 10^{-2}$
440 $\upmu$m HEMEX | $4.9\times 10^{-4}$ | $7.1\times 10^{-2}$
Table 4: Bulk participations for inserted and withdrawn power sweeps

For power sweeps of the EFG samples, these $p_{\text{bulk}}$ are chosen to
match that of the endpoints of the position sweeps in Fig. 4. This is done for
ease of cross-comparison of power sweeps and position sweeps (as both have
measurements with equal $p_{\text{bulk}}$ and $n=10^{5}$).

For HEMEX (measured by an earlier version of our protocol), we measured the
inserted power sweeps with the sample in contact with the post, rather than
some small distance ($z\approx 30$ $\upmu$m) away from the post. This was done
with the intent of maximizing the sensitivity to the sample loss, but had the
drawback of removing direct cross-comparability between the power sweep and
position sweep of HEMEX. Based on the trend in $Q^{-1}$ measured as a function
of $p_{\text{bulk}}$ (Fig. 4), the large participation of the inserted power
sweep is insufficient to explain the large $Q^{-1}$ measured at the lowest
excitation powers (Fig. 3). We posit that this discrepancy was caused by
measuring the inserted HEMEX power sweep with the sample in contact with the
top of the post, which could make the cavity mode more sensitive to surface
losses associated with surface roughness or material non-homogeneity, neither
of which are captured by our simulations.

The different condition under which HEMEX was measured does not invalidate the
low-power verification done by the inserted power sweep, as further insertion
of the sample tends to reduce the critical photon number $n_{\text{c}}$. This
can be seen by comparing Table 1 to Table 5. The result of this effect is to
give a stricter criterion for the boundary of the low-power regime, making the
verification process conservative.

Sample |  $n_{\text{c}}$ (billions) | $\alpha$
---|---|---
$100$ $\upmu$m EFG | $7(3)$ | $0.5(0.1)$
$460$ $\upmu$m EFG | $8(1)$ | $0.4(0.16)$
$440$ $\upmu$m HEMEX | $14(4)$ | $0.32(0.03)$
Table 5: Fit results for the withdrawn (orange) power sweeps in Fig. (3) using
the TLS model in Eq. (18).

###  E.3 Constraining the position sweep fit

When the sample is withdrawn, the total internal loss of the system $Q^{-1}$
consists predominantly of two loss channels: conductivity loss
$Q^{-1}_{\text{cond}}$ and metal-air interface dielectric loss
$Q^{-1}_{\text{MA}}$. Power sweeps of the cavity provide some information
about these individual underlying loss mechanisms. This information can be
translated into bounds on the low-power material loss factors $q^{-1}_{j}$
associated with the cavity, and those bounds can be applied to the analysis of
the position sweep described in Sec. II.

As with any resonator measurement, $Q^{-1}$ can act as an upper bound on the
loss of each underlying loss channel. For instance, the low-power value of
$Q^{-1}_{\text{MA}}$ can be bounded from above by the low-power total
$Q^{-1}$, and that upper-bound can be converted into a bound on the low-power
value of $q^{-1}_{\text{MA}}$. Additional information about
$Q^{-1}_{\text{MA}}$ comes from assuming that conductivity loss
$Q^{-1}_{\text{cond}}$ is power-independent up to $n=10^{14}$ (an assumption
consistent with observations up to $n=10^{10}$ in prior studies of aluminum
cylindrical cavities [20]). If $Q^{-1}_{\text{cond}}$ is power-independent,
then any power-dependence observed in $Q^{-1}$ must be attributed to
$Q^{-1}_{\text{MA}}$, and this attribution can be converted into a lower-bound
on $q^{-1}_{\text{MA}}$ (see eq. 19).

An upper-bound on $Q^{-1}_{\text{cond}}$ can be inferred in the same way, but
assuming that $Q^{-1}_{\text{cond}}$ is power-independent implies a slightly
stricter bound. If the low-power and high-power values of
$Q^{-1}_{\text{cond}}$ are assumed to be the same, then the high-power
$Q^{-1}$ (or any power $Q^{-1}$) can serve as an upper-bound on the low-power
$Q^{-1}_{\text{cond}}$, and converted into an upper-bound on
$q^{-1}_{\text{cond}}$ (see eq. 19).

The precise bounds obtained for each sample are listed in Table 6. These
bounds are obtained separately for each cooldown, to control for potential
temporal variations in the bare quality factor of the cavity. Over the course
of the measurements presented in this study, however, no such variations were
observed. Consequently, the bounds in Table 6 are relatively similar for all
three measurements.

Sample material | Cavity loss factor bounds
---|---
100 $\upmu$m EFG | $8.62\times 10^{-3}\leq q_{\text{MA}}^{-1}\leq 38.2\times 10^{-3}$
| $\hskip 41.25641pt0\leq q_{\text{cond}}^{-1}\leq 1.66\times 10^{-4}$
460 $\upmu$m EFG | $9.18\times 10^{-3}\leq q_{\text{MA}}^{-1}\leq 37.6\times 10^{-3}$
| $\hskip 41.25641pt0\leq q_{\text{cond}}^{-1}\leq 1.60\times 10^{-4}$
440 $\upmu$m HEMEX | $9.33\times 10^{-3}\leq q_{\text{MA}}^{-1}\leq 38.3\times 10^{-3}$
| $\hskip 41.25641pt0\leq q_{\text{cond}}^{-1}\leq 1.62\times 10^{-4}$
Table 6: Constraints on cavity loss factors

When performing the insertion analysis in Fig. 4, the three measurements are
fit jointly, which has the advantage that it removes four free parameters from
the fit, reducing the risk of over-fitting. The “background” loss rates
extracted in this way are shown in Table 7, and are consistent with other
measurements of etched 4N aluminum cavities [20, 49]. (The quoted value for
conductor loss is given at the withdrawn frequency of 4.55 GHz.) We could
repeat the insertion analysis with the cavity loss factors constrained using
the bounds from Table 6, but this has no affect on the fit, as the
unconstrained fit already yields loss rates that fall within the bounds in
Table 6. This demonstrates consistency between the power analysis from Sec.
IIIA and the insertion analysis from Sec. IIIB.

Loss factor | Extracted value
---|---
$q^{-1}_{\text{MA}}$ | $33(9)\times 10^{-3}$
$q^{-1}_{\text{cond}}$ | $2(5)\times 10^{-5}$
$R_{\text{s}}$ | $4(9)\times 10^{-8}~{}\Omega$
$q^{-1}_{\text{sub}}$ (thin EFG) | $170(13)\times 10^{-9}$
$q^{-1}_{\text{sub}}$ (thick EFG) | $88(6)\times 10^{-9}$
$q^{-1}_{\text{sub}}$ (HEMEX) | $19(6)\times 10^{-9}$
$q^{-1}_{\text{bulk}}$ (EFG) | $63(8)\times 10^{-9}$
$q^{-1}_{\text{SA}}$ (EFG) | $15(3)\times 10^{-4}$
Table 7: Summary of fit results.

##  Appendix F Temperature dependence

Conductor loss from the aluminum walls of the cavity is temperature dependent.
Near the vicinity of the superconducting transition temperature
$T_{\text{c}}$, it is much greater than the metal-air (MA) surface dielectric
loss from the cavity oxide layer. We utilize this fact to experimentally
confirm that insertion of the dielectric does not change the conductor
participation of the cavity mode.

Fig. 9 shows measurements of cavity loss rate $Q^{-1}$ as a function of the
temperature $T$ measured at the mixing chamber plate of the dilution
refrigerator. Measurements of $Q^{-1}$ are performed with the sample fully
withdrawn (orange traces) and fully inserted (green traces).

Figure 9: Temperature sweeps. Measured internal loss $Q^{-1}$ with the sample
withdrawn (orange) and inserted (green) as a function of the mixing-chamber
plate temperature, for (a) $100$ $\upmu$m thick EFG sapphire, (b) $460$
$\upmu$m thick EFG sapphire, and (c) $440$ $\upmu$m thick HEMEX sapphire.
Orange lines are fits to Mattis-Bardeen theory [50], as described in the
supplement of Ref. [36].

Two distinct regimes are visible in the measurements. For temperatures at or
near $T_{\text{c}}$, conductor losses dominate, and system behavior can be
described by Mattis-Bardeen theory [50], which we fit to the temperature
dependence of the withdrawn temperature sweeps (orange traces). The results
are shown in Table 8.

Material | Thickness |  $T_{\text{c}}$ (K) |  $\lambda_{\text{L}}$ (nm)
---|---|---|---
EFG sapphire |  $100$ $\upmu$m | $1.13$ | $135$
EFG sapphire |  $460$ $\upmu$m | $1.19$ | $90$
HEMEX sapphire |  $440$ $\upmu$m | $1.13$ | $90$
Table 8: Cavity material properties according to fit results for withdrawn
(orange) temperature sweeps in Fig. (9) using Mattis-Bardeen theory [50].

Near $T_{\text{c}}$, conductivity loss grows to be orders of magnitude larger
than the other loss channels, making total loss approximately proportional to
participation in the conductor: $Q^{-1}\approx
p_{\text{cond}}q_{\text{cond}}^{-1}$. A pair of high-temperature measurements
with the sample withdrawn and inserted may therefore serve as a probe of the
fractional change in $p_{\text{cond}}$ at the two measured positions.

$\displaystyle\frac{1/Q^{\text{I}}-1/Q^{\text{W}}}{1/Q^{\text{W}}}\approx\frac{p_{\text{cond}}^{\text{I}}-p_{\text{cond}}^{\text{W}}}{p_{\text{cond}}^{\text{W}}}.$
(70)

(Here the superscripts indicate whether a measurement is made with the sample
withdrawn or inserted). With this argument, high-temperature measurements with
the sample withdrawn and inserted bound the fractional change in conductor
participation to less than 1%, in agreement with the simulations shown in App.
C.

For temperatures $T\ll T_{\text{c}}$, conductor losses are only weakly
dependent upon temperature, so we attribute the observed temperature
dependence to the dielectrics in the system. Fig. 9d-f shows this regime in
greater detail. Weak temperature dependence is visible in both the inserted
and withdrawn sweeps, indicating some saturability of the dielectric losses.
However, subtracting the withdrawn and inserted measurements, as shown in Fig.
9g-i, reveals that inserting the sample does little to change this temperature
dependence, indicating that the temperature dependence arises primarily from
the metal-air interface of the cavity, rather than from the sample under
study. In the future, modifications to the experimental design could improve
the contrast of such temperature-dependence comparisons by better thermalizing
the dielectric substrate to the mixing chamber, or by constructing the cavity
from a superconductor with higher transition temperatures, such as niobium.

##  Appendix G Repeatability and cross-wafer variation

As a control test to check for measurement reproducibility and wafer-to-wafer
variation in nominally identical samples, we repeated the position sweep of
$460$ $\upmu$m thick EFG sapphire on a second control sample cut from a
different wafer. The results are shown in Fig. 10.

Figure 10: Position sweep on nominally identical samples. The position sweep
in Fig. 4 was repeated with a second piece of $460$ $\upmu$m thick EFG
sapphire denoted as a control. The control was prepared in nominally the same
way, but cut from a different wafer of the same material.

The two measurements are quite similar, suggesting that the dipper is a
measurement technique with high reproducibility and that the dielectric
properties of these two samples are the same, up to the resolution of the
measurement.

##  Appendix H Bounding magnetic bulk loss

Implicit in our analysis is the assumption that the origin of loss from the
sample is purely dielectric and none of the loss is magnetic. In this appendix
we relax that assumption, and modify our model to consider the possibility
that some of the loss from the substrate could be magnetic in origin. We then
compare our measurement results of EFG sapphire with quality factors of
stripline resonators fabricated on the same material. From this comparison, we
can make bounding statements about the composition of bulk loss in our sample,
and about the prevalence of bulk magnetic loss in superconducting qubits on
EFG sapphire.

Just as substrate loss can be decomposed into bulk and surface contributions
(See Eq. 4 and Fig. 4b), we can further decompose loss from the bulk of our
sample $Q_{\text{bulk}}^{-1}$ into contributions from bulk dielectric loss
$Q_{\text{bulk(E)}}^{-1}$ and bulk magnetic loss $Q_{\text{bulk(H)}}^{-1}$.
These losses can be treated with the same participation formalism and
attributed to an associated dielectric loss tangent $q_{\text{bulk(E)}}^{-1}$
and a magnetic loss tangent $q_{\text{bulk(H)}}^{-1}$, respectively:

$\displaystyle
Q_{\text{bulk}}^{-1}=p_{\text{bulk(E)}}q_{\text{bulk(E)}}^{-1}+p_{\text{bulk(H)}}q_{\text{bulk(H)}}^{-1}.$
(71)

Because of the quasi-linear relationship between bulk dielectric participation
$p_{\text{bulk(E)}}$ and bulk magnetic participation $p_{\text{bulk(H)}}$ in
our geometry (see Fig. 11a), our measurement has limited ability to
distinguish between these two loss channels but it does constrain them to a
contour in the space of all ordered pairs
$(q_{\text{bulk(E)}}^{-1},q_{\text{bulk(H)}}^{-1})$. In close analogy to the
method of bulk-surface loss separation performed in Fig. 4b, an additional
measurement with a different ratio of participations
$p_{\text{bulk(E)}}/p_{\text{bulk(H)}}$ would suggest another constraint line
with a different slope, and further constrain the accessible
$(q_{\text{bulk(E)}}^{-1},q_{\text{bulk(H)}}^{-1})$ pairs. Because our
measurements probe loss in our sample with the fringe field of a charge
antinode, our measurement is preferentially sensitive to dielectric loss, and
achieves $p_{\text{bulk(E)}}/p_{\text{bulk(H)}}\approx 200$. To acquire
another constraint line, we can consider the performance of a stripline
fabricated on EFG sapphire from another study [42]. A stripline of this design
can achieve $Q\approx 8\times 10^{6}$ and has
$p_{\text{bulk(E)}}/p_{\text{bulk(H)}}=0.40/0.31\approx 1.3$; producing a
contour with a much more gradual slope which we pair with our steeper
measurement contour (See Fig. 11b). Because the $Q$ of the stripline is the
result of a multitude of possible loss channels, the associated contour
represents an inequality:

$\displaystyle Q^{-1}\geq
p_{\text{bulk(E)}}q_{\text{bulk(E)}}^{-1}+p_{\text{bulk(H)}}q_{\text{bulk(H)}}^{-1}.$
(72)

This means that the intersection of the two contours cannot be used to extract
the true value of $(q_{\text{bulk(E)}}^{-1},q^{-1}_{\text{bulk(H)}})$
associated with our sample, however upper and lower bounds can be inferred on
both these loss tangents using the region where our measurement contour
overlaps the region bounded from above by the stripline contour.

Figure 11: Bounding magnetic bulk loss. (a) Bulk magnetic participation
$p_{\text{bulk(H)}}$ as a function of bulk dielectric participation
$p_{\text{bulk(E)}}$. (b) Contours representing viable decompositions of bulk
substrate loss into dielectric and magnetic components in our experiment (Eq.
71) and in a stripline resonator [42] (Eq. 72). After our model is generalized
to allow losses from the substrate bulk to be a combination of dielectric and
magnetic loss (purple contour), the relative contributions of these losses can
be constrained by the performance of a stripline fabricated on the same
substrate (orange bounding contour). To be consistent with stripline $Q$, it
must be that at least $97\%$ of the sample loss observed in our measurement
must be dielectric in origin. These constraints also suggest that magnetic
loss is at most a $10\%$ contributor to the decoherence of transmons on EFG
sapphire.

The contours bound the bulk dielectric loss tangent to $6.1\times 10^{-8}\leq
q_{\text{bulk(E)}}^{-1}\leq 6.3\times 10^{-8}$, meaning that even considering
the possibility that there may be magnetic losses associated with the sample
bulk, this can account for no more than $3\%$ of the sample loss detected in
our measurement, and at least $97\%$ of the bulk loss we detect must be
dielectric in origin. This validates the analysis in the main text, which
neglects magnetic losses. Although prior studies of HEMEX sapphire dielectric
resonators have observed tunable loss due to coupling with electron spins of
magnetic impurities brought into resonance by tesla-scale magnetic fields [51,
52, 53], our result shows that dielectric loss is the dominant source of
dissipation in EFG sapphire at low magnetic fields.

The contours also bound the bulk magnetic loss tangent to
$q_{\text{bulk(H)}}^{-1}\leq 3.3\times 10^{-7}$. This bound is much looser
than the bounds for $q_{\text{bulk(E)}}^{-1}$, largely due to the relative
insensitivity of our measurement to magnetic losses. Fortunately, a typical
transmon qubit, because of its high kinetic inductance, has
$p_{\text{bulk(H)}}\approx 2.5\%$ (relatively small compared to the 31% of a
typical stripline), so this bound on $q_{\text{bulk(H)}}^{-1}$ can still be
used to infer a useful transmon coherence limit of
$Q_{\text{bulk(H)}}>1.2\times 10^{8}$. This high coherence limit suggests that
even in today’s longest-lived transmons on sapphire, bulk magnetic loss
accounts for less than $10\%$ of total loss, making bulk magnetic loss at most
a minor contributor to their decoherence.

##  Appendix I Comparison to other studies

Fig. 12 compares the loss tangents measured in this work to those reported in
other studies on the microwave loss of sapphire at low temperatures and low
powers.

Figure 12: Comparison of low-power and low-temperature loss-tangent
measurements of sapphire.

##  Appendix J Microwave measurement circuit

Fig. 13 shows the microwave wiring diagram of the measurement system.

Figure 13: Microwave measurement circuit. Microwave cavity (DUT) is cooled to
$20$ mK and measured with heterodyne detection, employing switched amplifier
and attenuator banks for improved dynamic range.

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