# Experimental Side-Channel-Free Quantum Key Distribution

Chi Zhang  Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China  Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics, University of
Science and Technology of China, Shanghai 201315, China  Jinan Institute of
Quantum Technology, Jinan, Shandong 250101, China  Xiao-Long Hu  State Key
Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China  Jiu-Peng Chen  Yang Liu  Hefei National
Laboratory for Physical Sciences at Microscale and Department of Modern
Physics, University of Science and Technology of China, Hefei 230026, China
Shanghai Branch, CAS Center for Excellence and Synergetic Innovation Center in
Quantum Information and Quantum Physics, University of Science and Technology
of China, Shanghai 201315, China  Jinan Institute of Quantum Technology,
Jinan, Shandong 250101, China  Weijun Zhang  State Key Laboratory of
Functional Materials for Informatics, Shanghai Institute of Microsystem and
Information Technology, Chinese Academy of Sciences, Shanghai 200050, China
Zong-Wen Yu  State Key Laboratory of Low Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing 100084, China  Data
Communication Science and Technology Research Institute, Beijing 100191, China
Hao Li  Lixing You  Zhen Wang  State Key Laboratory of Functional Materials
for Informatics, Shanghai Institute of Microsystem and Information Technology,
Chinese Academy of Sciences, Shanghai 200050, China  Xiang-Bin Wang  Shanghai
Branch, CAS Center for Excellence and Synergetic Innovation Center in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Shanghai 201315, China  Jinan Institute of Quantum Technology, Jinan,
Shandong 250101, China  State Key Laboratory of Low Dimensional Quantum
Physics, Department of Physics, Tsinghua University, Beijing 100084, China
Qiang Zhang  Hefei National Laboratory for Physical Sciences at Microscale and
Department of Modern Physics, University of Science and Technology of China,
Hefei 230026, China  Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics, University of
Science and Technology of China, Shanghai 201315, China  Jian-Wei Pan  Hefei
National Laboratory for Physical Sciences at Microscale and Department of
Modern Physics, University of Science and Technology of China, Hefei 230026,
China  Shanghai Branch, CAS Center for Excellence and Synergetic Innovation
Center in Quantum Information and Quantum Physics, University of Science and
Technology of China, Shanghai 201315, China

###### Abstract

Quantum key distribution can provide unconditionally secure key exchange for
remote users in theory. In practice, however, in most quantum key distribution
systems, quantum hackers might steal the secure keys by listening to the side
channels in the source, such as the photon frequency spectrum, emission time,
propagation direction, spatial angular momentum, and so on. It is hard to
prevent such kinds of attacks because side channels may exist in any of the
encoding space whether the designers take care of or not. Here we report an
experimental realization of a side-channel-free quantum key distribution
protocol which is not only measurement-device-independent, but also immune to
all side-channel attacks in the source. We achieve a secure key rate of
$4.80\times 10^{-7}$ per pulse through 50 km fiber spools.

## Introduction

The cyber security today is protected by the modern cryptography, which is
based on the computational complexity assumption. This assumption, however,
might be challenged by the progress in algorithm Stevens et al. (2017) or
super computer Shor (1994); Grover (1996). Besides, hackers may steal the
information from side channels instead of decrypting the ciphered message. For
example, one can attack the communication terminals that store the secret bits
by detecting the physical effects like time shift Kocher (1996), power
consumption Kocher et al. (1999), electromagnetic leak van Eck (1985), sound
variation Genkin et al. (2014) and etc.

Guaranteed by basic principles of quantum mechanics Wootters and Zurek (1982),
quantum key distribution (QKD) generates information theoretically secure keys
BENNETT (1984); Gisin et al. (2002); Hwang (2003); Wang (2005); Lo et al.
(2005); Scarani et al. (2009); Xu et al. (2020); Pirandola et al. (2020) even
if hackers have the most powerful attacks that physical laws permit. However,
side channels may appear in practical QKD systems due to device imperfections
Xu et al. (2020), leading to potential security loopholes. Actually, device
imperfections, especially those in the detections, are the most serious threat
to “prepare-and-measure” QKD systems, such as time-shift attack Qi et al.
(2007); Zhao et al. (2008), detector-blinding attack Makarov (2009); Lydersen
et al. (2010), detector-after-gate attack Wiechers et al. (2011) and so on.
Luckily, this can be solved by measurement-device-independent QKD (MDIQKD) Lo
et al. (2012); Braunstein and Pirandola (2012); Tamaki et al. (2012); Wang
(2013); Curty et al. (2014); Xu et al. (2014); Zhou et al. (2016); Yin et al.
(2016), which is immune to all attacks to measurement devices. But the problem
of side channels from the source still exists, leaving potential loopholes.
Though the security is proven with the ideal encoding state, it can still be
undermined when there is difference in the side channels of the emitted
photons. For example, in a protocol using polarization encoding or phase
encoding, there can be imperfections in side-channel space such as the
frequency spectrum, the light emission time, etc. These imperfections are
highly possible, because the encodings in the source inevitably operate on a
larger space. For example, the intensity modulation in the source may also
affect the timing and frequency of the pulse. In such cases, the eavesdropper
may acquire the secure keys by monitoring the side channels only, without
affecting the encoding space. As a simple example, the eavesdropper may
distinguish the intensity by monitoring the wavelength. Thus, the side
channels actually undermine the security of practical QKD systems.

Recently, Wang et al proposed a novel side-channel-free (SCF) protocol Wang et
al. (2019). This protocol is not only immune to all attacks in the side-
channel space of emitted photons, such as the attacks on the imperfections of
the frequency spectrum, emission time, non-ideal propagation direction,
spatial angular momentum, etc, but also closes all potential loopholes in
detection, by adapting the measurement-device-independent architecture, so it
has a higher security level than most QKD protocols before. In the protocol,
coherent state without any phase randomization is used as the source, so the
decoy-state assumption is not required; Alice and Bob only decide on sending
or not-sending the coherent state for encoding, so no more modulations are
needed in the experiment. The theoretical proved side-channel-free
characteristic from the source and the simple operation in encoding are the
essential differences between the SCF-QKDWang et al. (2019) and the twin-field
QKD (TF-QKD) Lucamarini et al. (2018), including the sending-or-not-sending
(SNS) protocol Wang et al. (2018). We note that there are other protocols to
achieve the side-channel-free security Braunstein and Pirandola (2012); Mayers
and Yao (1998); Acín et al. (2007); Scarani and Renner (2008), but this
protocol Wang et al. (2019) is the only one achieved with existing
technologies, which is also possible to achieve a long distribution distance
in practice.

Here, for the first time, we realize the side-channel-free QKD. Secure key
rate of $4.80\times 10^{-7}$ per pulse is achieved over 50 km. Precise
wavelength control and fast phase compensation have been utilized to
accurately control and estimate the phase difference in the single-photon
interference of two independent laser sources. High efficiency detection has
also been applied to meet the demand of estimating phase drift between two
users’ fibers and improve the detection rates of signal pulses simultaneously.

## Protocol

To make the original protocol Wang et al. (2019) more practical, a revised
side-channel-free protocol with phase reference pulses and phase postselection
method is used which includes the following steps:

Step 1. At each time window, Alice (Bob) prepares a nonrandom-phase coherent
state
$|\alpha_{A}\rangle=e^{-\mu/2}\sum_{n=0}^{\infty}\frac{(\sqrt{\mu}e^{i\gamma_{A}})^{n}}{\sqrt{n!}}|n\rangle$
($|\alpha_{B}\rangle=e^{-\mu/2}\sum_{n=0}^{\infty}\frac{(\sqrt{\mu}e^{i\gamma_{B}})^{n}}{\sqrt{n!}}|n\rangle$),
where $\alpha_{A}=\sqrt{\mu}e^{i\gamma_{A}}$
($\alpha_{B}=\sqrt{\mu}e^{i\gamma_{B}}$). With probability $\varepsilon$,
Alice (Bob) decides on sending and she (he) sends out the coherent state
$|\alpha_{A}\rangle$ ($|\alpha_{B}\rangle$) to Charlie and puts down a
classical bit value $1$ ($0$) locally; with probability $1-\varepsilon$, she
(he) decides on not-sending and she (he) does not send out anything and puts
down a classical bit value $0$ ($1$) locally. These pulses (coherent states or
vacuum) are called signal pulses. She (He) also prepares a strong reference
pulse time-multiplexed with the signal pulses. The phases of the reference
pulses are modulated periodically and this will be presented in detail later.
No matter what she (he) decides, she (he) always sends the reference pulse to
Charlie. They define a $\tilde{Z}$ window as a time window when either Alice
or Bob decides on sending and the other decides on not-sending.

Note: Different from the decoy-state method requiring phase-randomized
coherent states, here Alice and Bob are required to use the nonrandom-phase
coherent states. Their initial individual phases ($\gamma_{A}$ and
$\gamma_{B}$) are fixed during the whole experiment. The reference pulse is
introduced only to carry the information about the phase, which is allowed to
be known by Eve according to the protocol. It has nothing to do with either
the bit value or the state of the signal pulse (except the phase). Thus the
introduction of reference pulse doesn’t affect the security.

Step 2. Charlie measures the signal pulses at the measurement station between
Alice and Bob and announces which detector clicks. If one and only one
detector clicks, this time window is regarded as an effective time window. In
addition, he measures the reference pulses and announces the phase difference
$\delta$ between Alice’s and Bob’s pulses to learn the phase shifts in the
channels. It is shown schematically in Fig. 1.

Figure 1: A schematic illustration of side-channel-free QKD. BS:beam-splitter.

Step 3. According to Charlie’s measurement results, Alice and Bob keep the
bits from all events in which the phase difference $\delta$ satisfies the
condition

$|\delta|<\Delta,$ (1)

where $\Delta$ is the relative phase difference threshold, and they announce
the bit values of other bits, and then discard them. Here $|x|$ means the
degree of the minor angle enclosed by the two rays that enclose the rotational
angle of degree $x$, e.g., $|-15\pi/8|=|15\pi/8|=\pi/8$.

Step 4. Among the preserved bits, Alice and Bob take a random subset, $u$,
through classical communication, to do error test and parameter estimation.
They announce all bit values in set $u$ through classical channel. They
discard the bits from the set $u$ after the error test, and the set of
remaining bits is called the set $v$.

Step 5. Alice and Bob distill (by conducting error correction and privacy
amplification) the effective bits from the set $v$, with the asymptotic key
rate for the number of final bits

$n_{F}=n_{\tilde{Z}}[1-H(\bar{e}^{ph})]-fn_{v}H(E_{v})$ (2)

where $H(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the entropy function;
$n_{\tilde{Z}}$ ($n_{v}$) is the number of remaining effective bits from
$\tilde{Z}$ windows (all time windows) in set $v$; $\bar{e}^{ph}$ is the upper
bound of the phase-flip error rate for bits in effective $\tilde{Z}$ windows
in set $v$; $f$ is the correction efficiency factor which we set $f=1.1$, and
$E_{v}$ is the bit-flip error rate of effective bits in set $v$. The values of
$n_{\tilde{Z}}$ and $\bar{e}^{ph}$ can be obtained by the observed data of the
set $u$ asymptotically.

Equivalently, the key rate (per pulse) can be written as

$R=\frac{1}{N_{t}}\\{n_{\tilde{Z}}[1-H(\bar{e}^{ph})]-fn_{v}H(E_{v})\\},$ (3)

where $N_{t}$ is the total number of signal pulses that Alice (Bob) sends. The
details of the calculation of the key rate are shown in the Supplemental
Material.

## Experiment

To implement the side-channel-free protocol above, the experiment is designed
as in Fig. 2 (a). We use the time-frequency dissemination technology to
accurately control the phase difference in the single-photon interference of
two independent laser sources. Intensity modulators are used to control the
“sending” and “not-sending” encoding. Finally, high performance
superconducting nanowire single-photon detectors are used to meet the
stringent peak counting rate requirement of the reference pulse detection, as
well as high efficiency requirement of the signal pulses. In the following, we
will discuss the experiment in detail.

Figure 2: (a) Experimental setup. Alice and Bob modulate the phase-locked
lasers with a phase modulator (PM) and three intensity modulators (IM1, IM2,
IM3). The PM is used to encode the reference pulses; the intensity modulator
IM1 is used to set “sending” or “not sending” of the signal, while IM2 and IM3
are used to set the intensities between the reference and signal pulses.
Additional attenuators (ATTs) in the secure zones are calibrated to set the
proper output photon intensity. Charlie interferes and measures the light from
Alice and Bob with superconducting nanowire single-photon detectors (SNSPDs).
PC: polarization controller, PBS: polarization beam splitter, CIR: circulator,
BS: beam splitter. (b) Alice and Bob lock the wavelength of their lasers with
a frequency-locking system Liu et al. (2019). The light transmits through an
additional 50 km fiber. AOM: acousto-optic modulator; FM: Faraday mirror; PD:
photodiode; QWP: quarter wave plate; USC: ultra-stable cavity.

First, stable lasers with exact wavelength are required in the remote single-
photon interference in our SCF-QKD experiment. The wavelengths of Alice’s and
Bob’s independent lasers are locked with the time-frequency transmission
technology Liu et al. (2019), through additional 50 km fiber spools shown in
Fig. 2 (b). Alice uses a commercial sub-Hz laser source with a central
wavelength of 1550.1665 nm and it is internally locked into her cavity; Bob
uses a commercial kilo-Hz fiber laser, locked to an ultra-low-expansion(ULE)
glass cavity with Pound-Drever-Hall (PDH) technique Drever et al. (1983);
Pound (1946). The final linewidth is approximately 1 Hz with a central
wavelength of 1550.1674 nm. Obviously the frequency difference of two laser
sources about 112 Megahertz still exists. Therefore, at Bob’s station, we
insert an acoustic-optic modulator (AOM) with a tunable carrier frequency to
compensate the frequency difference in real time. The phase noise via the 50
km fiber spools is cancelled in real time using another AOM with a carrier
frequency of 40 Megahertz by Alice.

With these narrow linewidth coherent light sources prepared, the next step is
to encode. As for signal pulses, only one intensity for “sending” is required.
We use an intensity modulator (IM) to modulate the signal to 1 ns pulse
duration for “sending” and to vacuum for “not sending”. Based on the
experimental conditions, we optimize the coherent state intensity to
$\mu=0.002$, and the probability of “sending” to $\varepsilon=0.021$. The
“sending” and “not sending” are determined by previously prepared quantum
random numbers, with only one intensity modulation required. While the
relative phase between Alice and Bob is not stable: the fluctuation of the
fiber length and refractive index directly affect the relative phase; the
wavelength difference of the sources may also contribute to the phase drift.

In order to correct the relative phase drift, we adopt strong reference pulses
to estimate the relative phase between Alice’s and Bob’s signal pulses Liu et
al. (2019). For every 1 $\mu s$ time interval, 15 signal pulses are encoded in
the first 450 ns as “sending” or “not sending”; then in the following 400 ns,
4 phase encoded reference pulses are sent; the final 150 ns are used as the
recovery time for the superconducting nanowire single-photon detectors
(SNSPDs), with vacuum states sent. Thus the effective signal frequency is 15
MHz considering the reference pulses. The reference intensity is set to
$\mu_{ref}=0.062$ for each 1 ns width.

The relative intensities between signal and reference pulses are modulated
with another two IMs. All three IMs are set to vacuum if the signal state is
“not sending” to increase the extinction ratio. To estimate the relative
phases in the fibers, a phase modulator (PM) in Alice’s station sets the phase
of her reference pulses to {$0$, $\pi$/2, $\pi$, 3$\pi$/2} respectively, while
all of Bob’s phase reference pulses are set to $\pi$. Note that the phase
modulation is only applied to Alice’s and Bob’s reference pulses to estimate
relative phase drift in the fibers; the phases of the signals are always set
to $0$. In another word, the phase modulation works only as an estimation of
the reference frame, thus it would not introduce side channels to the system.

Figure 3: Secure key rates of the SCF-QKD. The dashed blue curve shows the
simulated key rates in standard single mode fiber with the coherent state
intensity $\mu=0.002$, the “sending” probability $\varepsilon=0.021$ and the
phase difference threshold between Alice’s and Bob’s optical pulses
$\Delta=30^{\circ}$. The red cross represents the experimental key rate over
50 km fiber spools. In both simulation and experiment, 10% of the pulses after
phase post-selection are used for error testing and parameter estimating.

The signals from Alice and Bob are transmitted through 25 km fiber spools
respectively to the measurement station, Charlie, with the channel loss of 5.0
dB and 5.2 dB. The light is filtered with circulators to eliminate the SNSPD
back scattered light. Then polarization controllers and polarization beam
splitters are used to correct the polarization before interference at
Charlie’s beam splitter. The additional loss of the optical components are
4.78 dB for Alice and 4.44 dB for Bob. The interference results are measured
with two SNSPDs with detection efficiencies of 60.5% and 62.6%, and recorded
by a time tagger. Here the SNSPDs are designed to achieve high detection
efficiency and high peak counting rates of around 10 MHz simultaneously.

Following the phase estimation procedure Liu et al. (2019) (See Supplemental
Material for the details of phase estimation), Charlie calculates the relative
phase based on the four reference pulses interference results due to the four
corresponding encoded phases. For each effective detection event that one and
only one detector clicks, the interference results of 48 nearest reference
pulses within 12 $\mu$s are collected for estimation. On average, about 45
detections in this time period can be recorded for phase estimation which are
enough for accurate phase estimation.

During a 12.0-hour experimental test, $6.04\times 10^{11}$ signal pulses are
sent, in which $7,004,417$ are detected and recorded. For each effective
detection event, the relative phase between Alice’s and Bob’s signal pulses is
calculated with the phase estimation procedure. Next, a threshold of relative
phase difference $\Delta$ is set. Only the data with the relative phase
$|\delta|<\Delta$ are kept as raw keys, as in Eq. (1); for all other
detections, Alice and Bob disclosed the bit values to calculate the state of
the twin-field after phase postselection which is

$\begin{split}\rho^{\prime}=&c_{1}|\alpha,\alpha\rangle\langle\alpha,\alpha|+c_{2}|\alpha,0\rangle\langle\alpha,0|\\\
&+c_{3}|0,\alpha\rangle\langle 0,\alpha|+c_{4}|0,0\rangle\langle
0,0|\end{split}$ (4)

where $c_{1}=0.000442$, $c_{2}=0.020594$, $c_{3}=0.020564$, $c_{4}=0.958400$
according to the estimation (See Supplemental Material for details about the
calculation). After the phase postselection, a portion $p_{t}$ of the bits are
selected as “test bits”. The values of the “test bits” announced by Alice and
Bob, as well as the detections, are then used for calculating the number of
remaining effective bits in $\tilde{Z}$ window $n_{\tilde{Z}}$ and the upper
bound of phase-flip error rate $\bar{e}^{ph}$ (See Supplemental Material for
the theory and the details of parameters). The “test bits” are then discarded.
In our experiment, the threshold of relative phase difference is set to
$\Delta=30^{\circ}$ by optimization and the “test bits” probability is set to
$p_{t}=0.1$. The number of remaining effective bits in $\tilde{Z}$ window and
all time windows after the process of phase post-selection and “test bits”
selection are $n_{\tilde{Z}}=2,207,341$ and $n_{v}=2,248,625$ respectively.
The bit-flip error rate and the upper bound of phase-flip error rate are
calculated to $E_{v}=2.12\%$ and $\bar{e}^{ph}=19.1\%$ respectively. Finally,
we extracted 289,900 bit secure keys which is equivalent to $4.80\times
10^{-7}$ per pulse. The key rate obtained in our experiment and the
theoretical simulation are plotted in Fig. 3. With our experimental
parameters, it is predicted that a more than 80 km distribution distance can
be achieved with our setup.

In conclusion, we have demonstrated the SCF-QKD protocol experimentally and
obtained secure keys over 50 km fiber spools. Our experiment shows that
source-state side-channel-free and measurement-device-independent security can
be simultaneously achieved in the QKD system with matured existing
technologies. The protocol makes no assumptions about the side-channel space
of the quantum state though it assumes a perfect vacuum state in the encoding
space for security proof. This assumption can possibly be loosened in future
work. We also note that instead of post-selecting phase difference between two
fiber channels, active phase feedback can make more use of raw data and
improve key rates of side-channel-free QKD. This shall be studied in the
future.

## Acknowledgments

We would like to thank Feihu Xu for insightful discussions. This work was
supported by the National Key R&D Program of China (Grants No.2017YFA0303900,
2017YFA0304000, 2020YFA0309800), the National Natural Science Foundation of
China, the Chinese Academy of Science (CAS), Shanghai Municipal Science and
Technology Major Project (Grant No.2019SHZDZX01), Key R&D Plan of Shandong
Province (Grant No. 2019JZZY010205, 2020CXGC010105), the Shandong provincial
natural science foundation (Grant No. ZR2020YQ45), the Taishan Scholar Program
of Shandong Province, and Anhui Initiative in Quantum Information
Technologies.

Appendix

##  Appendix A Postselection instead of active phase compensation

In the original protocol in Wang et al. (2018), Charlie is supposed to perform
phase compensation to remove the phases of Alice’s and Bob’s coherent states,
$\gamma_{A}$ and $\gamma_{B}$, and the phase shifts from the channels to
obtain a low phase-flip error rate. But this phase compensation is a bit
difficult to perform in practice. Instead, a revised protocol with phase
reference pulses sent and postselection method can be used. As described in
Step 3 in Sec.II in the main text, Alice and Bob can perform postselection by
only keeping the bits whose phase different $\delta$ satisfies the condition
$|\delta|<\Delta$, using them to distill the final key, and discarding other
bits.

Before postselection, the state of the twin-field sent by Alice and Bob is

$\begin{split}\rho=&\varepsilon^{2}|\alpha,\alpha\rangle\langle\alpha,\alpha|+\varepsilon(1-\varepsilon)|\alpha,0\rangle\langle\alpha,0|\\\
&+(1-\varepsilon)\varepsilon|0,\alpha\rangle\langle
0,\alpha|+(1-\varepsilon)^{2}|0,0\rangle\langle 0,0|\end{split}$ (5)

If the total number of pulse pairs sent is $N$, the numbers of these four
twin-field states are $a_{1}=\varepsilon^{2}N$,
$a_{2}=\varepsilon(1-\varepsilon)N$, $a_{3}=\varepsilon(1-\varepsilon)N$, and
$a_{4}=(1-\varepsilon)^{2}N$, respectively.

According to the measurement results of calibration and $\delta$, Charlie
announces which pulses are available and which are not. Alice and Bob
respectively announce what states they sent in those time windows with
unavailable pulses, according to which they can calculate the new state of the
twin-field after postselection. Specifically, if the numbers of those four
states in unavailable pulses are $b_{1}$, $b_{2}$, $b_{3}$, and $b_{4}$,
respectively, the number of those available pulses is
$N^{\prime}=N-b_{1}-b_{2}-b_{3}-b_{4}$ and the state of them is

$\begin{split}\rho^{\prime}=&c_{1}|\alpha,\alpha\rangle\langle\alpha,\alpha|+c_{2}|\alpha,0\rangle\langle\alpha,0|\\\
&+c_{3}|0,\alpha\rangle\langle 0,\alpha|+c_{4}|0,0\rangle\langle
0,0|\end{split}$ (6)

where $c_{i}=(a_{i}-b_{i})/N^{\prime},i=1,2,3,4$. Then they use effective
events from these $N^{\prime}$ events to do parameter estimation and distill
the final key. Note that Charlie may not be honest about the measurement
results, but it doesn’t affect the security of this revised protocol because
the parameter estimation and key distillation are done by Alice and Bob
themselves.

##  Appendix B Calculation of the key rate

The set of events after postselection is defined as set $V$. Among these
events, Alice and Bob randomly choose $p_{t}=10\%$ from them to obtain the set
$u$, which is used for error test and parameter estimation. The set of
remaining events after postselection is defined as set $v$, i.e. $u\cup v=V$.
They announce all bit values in set $u$ through classical channel, and obtain
the numbers of each twin-field state that was sent, denoted as $N_{ab,u}$, the
numbers of effective events from each state, denoted as $n_{ab,u}$, and the
numbers of effective events causing the detector $i$ clicking, denoted as
$n_{ab,u}^{i}$ with $i=L,R$. In the subscript, $a$ ($b$) can be 0 or $\alpha$,
corresponding to Alice’s (Bob’s) decision of not-sending or sending,
respectively, and “$u$”means the number in set $u$. For example, $n_{\alpha
0,u}^{R}$ denotes the number of effective events in set $u$ when Alice decides
to send and Bob decides not to send, and the right detector clicks. Similarly,
they can define $N_{ab,v}$ as the numbers of each twin-field state that was
sent in set $v$ and calculate these values through Eq. (6) and the value of
$p_{t}$.

With the observed values above, they can calculate the counting rate of each
twin-field state

$S_{ab,u}=n_{ab,u}/N_{ab,u},$ (7)

the counting rate of each twin-field state with detector $i$ clicking

$S_{ab,u}^{i}=n_{ab,u}^{i}/N_{ab,u},$ (8)

the total counting rate of set $u$

$S_{u}=\sum_{a,b}n_{ab,u}/\sum_{a,b}N_{ab,u},$ (9)

and the bit-flip error rate of set $u$

$E_{u}=(n_{00,u}+n_{\alpha\alpha,u})/\sum_{a,b}n_{ab,u},$ (10)

According to the formulas in Ref. Wang et al. (2019), they can calculate the
counting rate and the phase-flip error rate of $\tilde{Z}$ windows through the
data of set $u$ asymptotically. Specifically, in the asymptotic case, the
counting rate of $\tilde{Z}$ windows is

$S_{\tilde{Z}}=\frac{1}{2}(S_{0\alpha,u}+S_{\alpha 0,u}).$ (11)

Thus, the number of $\tilde{Z}$ windows in set $v$ is

$n_{\tilde{Z}}=2\min(N_{0\alpha,v},N_{\alpha 0,v})S_{\tilde{Z}}.$ (12)

Note that as the security proof in Ref. Wang et al. (2019) requires, the state
in $\tilde{Z}$ windows has to be $\rho_{\tilde{Z}}=(|0,\alpha\rangle\langle
0,\alpha|+|\alpha,0\rangle\langle\alpha,0|)/2$, thus we use
$\min(N_{0\alpha,v},N_{\alpha 0,v})$ in Eq. (12). And the upper bound of the
phase-flip error rate of $\tilde{Z}$ windows is Wang et al. (2019)

$e^{ph}\leq\overline{e}^{ph}=\frac{(1+e^{-\mu})\left[\overline{S}_{X_{+}}^{R}-\underline{S}_{X_{+}}^{L}\right]+S_{0\alpha,u}^{L}+S_{\alpha
0,u}^{L}}{2S_{\tilde{Z}}}$ (13)

where

$\begin{split}\overline{S}_{X_{+}}^{R}=\frac{1}{2(1+e^{-\mu})}\\{e^{-\mu}S_{00,u}^{R}+\frac{1}{e^{-\mu}}S_{\alpha\alpha,u}^{R}+\frac{(1-e^{-\mu})^{2}}{e^{-\mu}}\\\
+2\sqrt{S_{00,u}^{R}S_{\alpha\alpha,u}^{R}}+2(1-e^{-\mu})\sqrt{S_{00,u}^{R}}+\frac{2(1-e^{-\mu})}{e^{-\mu}}\sqrt{S_{\alpha\alpha,u}^{R}}\\}\end{split}$
(14)

and

$\begin{split}\underline{S}_{X_{+}}^{L}=\frac{1}{2(1+e^{-\mu})}\bigg{\\{}e^{-\mu}S_{00,u}^{L}+\frac{1}{e^{-\mu}}S_{\alpha\alpha,u}^{L}-\big{[}2\sqrt{S_{00,u}^{L}S_{\alpha\alpha,u}^{L}}\\\
+2(1-e^{-\mu})\sqrt{S_{00,u}^{L}}+\frac{2(1-e^{-\mu})}{e^{-\mu}}\sqrt{S_{\alpha\alpha,u}^{L}}\big{]}\bigg{\\}}.\end{split}$
(15)

Then they can calculate the final key rate with Eq. (3) in the main text.

##  Appendix C Signal modulation

Alice and Bob use independent continuous wave (CW) lasers as their light
sources, with their frequencies locked to each other. They use 2 GHz sampling
rate, 14-bit depth AWGs to modulate the CW lasers to generate the signal and
reference pulses. The signal pattern is created with $3.02\times 10^{6}$ pre-
generated quantum random numbers, and loaded to the AWGs prior to the
experiment. Then a start signal from a controlling computer triggers the AWGs
to send signals at the same time. The maximum amplitude generated by the AWGs
is 500 mV, which is then amplified by about 25 dB for the electro-optical
modulators.

In the encoding, the signal and reference pulses are sent in each 1 $\mu$s
time interval. As shown in Fig. 4 and Fig. 5, 15 signal pulses are sent in the
first 450 ns, with 1 ns pulse duration and 30 ns period. Each signal pulse is
either set to $\mu=0.002$ for bit “1”, or completely blocked for bit “0”,
determined by the random signals from the AWGs. In the next 400 ns, 4 phase
reference pulses, with 100 ns pulse width, are sent for estimating the
relative phase between Alice’s and Bob’s fiber channels. Finally, a 150 ns
vacuum state is set, waiting the SNSPDs to recover from high counting region
of reference pulses.

Figure 4: Alice’s encoding pattern.

Figure 5: Bob’s encoding pattern.

For the modulation, a phase modulator (PM) followed by three intensity
modulators (IMs) are used to generate the above mentioned signal pattern. The
phase modulator is set before any intensity modulator so the signal phase will
not affected by the wavelength dependent behavior of the phase modulator. The
phase modulator is used to modulate the phase reference pulses only, setting
Alice’s phase reference to {0, $\pi/2$, $\pi$, $3\pi/2$}, and Bob’s all four
phase references to $\pi$. The phases are set to $0$ for all the signal pulses
and in the 150 ns recovery time.

For the intensity modulation, IM1 is used to modulate the light to 1 ns pulse
width for the “sending” bit, or to vacuum state for “not-sending” bit; IM2 and
IM3 are used to adjust the intensities between the phase reference and the
signal pulses. All the modulators block the light when a vacuum state needs to
be sent. The signals are further attenuated to the single-photon level by an
attenuator.

##  Appendix D Controlling the Relative Phase between Alice and Bob

Due to the single-photon interference requirement, it is essential to
compensate the phase difference between Alice and Bob. The phase difference
originates either from the wavelength difference of the two lasers, or from
the phase drift in the fiber channels. As discussed in the main text and in
Liu et al. (2019), we lock the wavelength of the light sources in real time
with time-frequency-dissemination technology. The phase reference pulses and
their interference results are used to estimate or compensate the relative
phase drift between fibers.

Besides the phase modulation mentioned above, we set the reference pulses
intensity to around 2 MHz counts for each superconducting nanowire single-
photon detector (SNSPD) at Charlie’s station, so that we have enough data for
estimation and it is not too high to affect the SNSPD performance. Charlie
counts the total detections of the two SNSPDs of each phase difference during
this time as $N_{i}$, where $i=\\{1,2,3,4\\}$ represents the modulated phase
difference between Alice and Bob of {$\pi$, $3\pi/2$, 0, $\pi/2$}. Then he
calculates the probability:

$p_{i}=2N_{i}/\Sigma N_{i}$ (16)

and the theoretical probabilities with relative phase induced by the fiber
channel $\Delta\varphi_{T}$:

$p_{Ti}(\Delta\varphi_{T})=\cos^{2}(\frac{\Delta\theta_{i}+\Delta\varphi_{T}}{2})$
(17)

where the phase differences are $\Delta\theta_{i}=${$0$, $\pi/2$, $\pi$,
$3\pi/2$}, for $i=\\{1,2,3,4\\}$ accordingly.

Then he minimizes the error $Err(\Delta\varphi_{T})$ between measured and
theoretical probabilities to estimate $\Delta\varphi_{T}$:

$Err(\Delta\varphi_{T})=\sum_{i}{[p_{i}-p_{Ti}(\Delta\varphi_{T})]^{2}}$ (18)

We set the phase reference statistical time to 12 $\mu$s. On average, 45
counts can be collected for the 4 reference pulses. During this time the
relative phase in fiber may drift about 0.073 rad (or 4.2 degrees) which may
induce less than 3.5% error rate to the system when interfering.

With this phase estimation, we revise the SCF-QKD protocol as mentioned in the
main text, to compensate the phase drifts via post-processing. The phase slice
criterion is used to post-select signals. The phase slice is optimized to
yield a maximum key rate Liu et al. (2019):

$|\delta|<\Delta,$ (19)

where $\delta$ is the estimated values of phase difference for each signal
pulse pair, and $|x|$ means the degree of the minor angle enclosed by the two
rays that enclose the rotational angle of degree $x$, e.g.,
$|-15\pi/8|=|15\pi/8|=\pi/8$. And $\Delta$ is the relative phase difference
threshold which is optimized to $30^{\circ}$ in the experiment. Note that
active phase feedback that compensates the phase difference $\delta$ to around
0 in real time is also possible. This method shall be studied in the future.

##  Appendix E Experimental Parameters

The main parameters used in our experiment are summarized in Tab. 1. The fiber
length between Alice and Charlie and that between Bob and Charlie are set to
the same, the “Fiber Length” in the table is the total fiber length. $\mu$ is
the intensity of the coherent state of signal pulse. The signal pulse width is
1 ns and the phase reference pulse width is 100 ns; as to compare, $\mu_{ref}$
is the reference intensity in 1 ns width. The “sending” probability is denoted
by $p_{S}$. And we calculated the “Output Intensity” at Alice’s (Bob’s)
output. Finally, we listed our QKD system frequency, the equivalent system
frequency (considering phase reference pulses) and the working hours in the
experiment.

Table 1: Experimental parameters. Fiber Length | 50 km
---|---
$\mu$ | 0.002
$\mu_{ref}$ | 0.062
$p_{S}$ | 0.021
Output Intensity (pW) | 3.184
System Frequency (MHz) | 33.3
Equivalent System Frequency (MHz) | 15.0
Working Hours | 12.0

##  Appendix F Experimental Results

We summarized the fiber and the optical elements transmittance and the SNSPD
efficiencies in Tab. 2. The optical elements include the polarization
controllers (PCs), the circulators (CIRs), the polarization beam splitters
(PBSs), and the beam splitter (BS). The results are given for each of the two
inputs (A/B) and outputs (ch0/ch1) as appropriate. The SNSPD efficiency
includes the PC efficiency.

The experimental results are summarized in Tab. 3, including the final key
rate $R$, the total signal pulses sent in the experiment $N_{total}$, the
number of remaining effective bits $n_{\tilde{Z}}$ from $\tilde{Z}$ windows
after the error test, the counting rate $s$, the upper bound of the phase
error rate $\bar{e}^{ph}$ and the bit-flip error rate $E_{v}$ of set $v$, with
the optimized accepted phase difference threshold $\Delta$ (in degrees). For
error testing and parameter estimating, a fraction of signal pulses are
sampled as “test bits” after phase postselection with a portion of $p_{t}$. A
digital gate is applied to select only the central part of the signal pulse,
to avoid imperfect interference due to timing difference. The width of the
gate is labelled as $t_{gate}$ and the fraction of signal in the gate is
denoted as $r_{gate}$.

Table 2: Efficiencies of fibers and measurement station. $\eta_{FiberA}$ | 0.316
---|---
$\eta_{FiberB}$ | 0.310
PC-A | 95.4%
PC-B | 96.7%
CIR-A | 83.4%
CIR-B | 87.0%
PBS-A | 93.4%
PBS-B | 95.6%
BS-A-ch0 | 45.3%
BS-A-ch1 | 43.5%
BS-B-ch0 | 45.5%
BS-B-ch1 | 44.8%
SNSPD-ch0 | 60.52%
SNSPD-ch1 | 62.61%

We summarized the raw data used for the calculations in Tab. 4. The number of
pulses Alice and Bob send is labelled as “Sent-CD”, where “C” (“D”) is “1” or
“0”, indicating the intensity Alice (Bob) for “sending” or “not-sending”. The
number of pulses falling within the accepted phase difference range $\Delta$
is listed as “Sent-CD-$\Delta$”. The number of signal pulses falling within
the accepted range $\Delta$ is listed as “Sent-ABCD-$\Delta$”, where “A” (“B”)
is “S” or “T” indicating signal mode or test mode from Alice (Bob); the number
of detections is listed as “Detected-ABCD-Ch” which means the detections
falling within the accepted difference range $\Delta$, where “Ch” indicates
the detection channel. Note that the correct bits are the effective detections
when Alice decides on “sending” and Bob decides on “not-sending”, or vice
versa.

Table 3: Experimental results Fiber Length | 50 km
---|---
$R$ | $4.799\times 10^{-7}$
$N_{total}$ | $6.0396\times 10^{11}$
$n_{\tilde{Z}}$ | $2.21\times 10^{6}$
$s$ | $2.77\times 10^{-4}$
$\bar{e}^{ph}$ | 19.1%
$E_{v}$ | 2.12%
$p_{t}$ | 0.1
$\Delta$ | $30^{\circ}$
Signal Pulse width (ns) | 1
$t_{gate}$ (ns) | 0.85
$r_{gate}$ | 0.67
Table 4: Raw data Sent-00 | 578835000000 | Detected-SS00-ch0 | 1285
---|---|---|---
Sent-01 | 12438400000 | Detected-SS00-ch1 | 1220
Sent-10 | 12420000000 | Detected-SS01-ch0 | 546658
Sent-11 | 266800000 | Detected-SS01-ch1 | 545018
Sent-00-$\Delta$ | 206492000000 | Detected-SS10-ch0 | 554790
Sent-01-$\Delta$ | 4436985713 | Detected-SS10-ch1 | 554207
Sent-10-$\Delta$ | 4430718614 | Detected-SS11-ch0 | 42800
Sent-11-$\Delta$ | 95191234 | Detected-SS11-ch1 | 2647
Sent-SS00-$\Delta$ | 185843000000 | Detected-TT00-ch0 | 149
Sent-SS01-$\Delta$ | 3993295035 | Detected-TT00-ch1 | 126
Sent-SS10-$\Delta$ | 3987675420 | Detected-TT01-ch0 | 60690
Sent-SS11-$\Delta$ | 85674748 | Detected-TT01-ch1 | 60926
Sent-TT00-$\Delta$ | 20649175977 | Detected-TT10-ch0 | 62121
Sent-TT01-$\Delta$ | 443690678 | Detected-TT10-ch1 | 61683
Sent-TT10-$\Delta$ | 443043194 | Detected-TT11-ch0 | 4728
Sent-TT11-$\Delta$ | 9516486 | Detected-TT11-ch1 | 315

Additionally, we also calculated the $QBERs$ when Alice and Bob both sent
coherent states with different detection counts according to different phase
difference thresholds $\Delta$. Finally we extracted the optimized secure key
rates with the parameter values. These are all listed in Tables. 5.

Table 5: QBERs and Detections for both Alice and Bob sent coherent states and Key Rate for different $\Delta$. Results$\mid$$\Delta$ | $2^{\circ}$ | $5^{\circ}$ | $8^{\circ}$ | $10^{\circ}$ | $12^{\circ}$ | $15^{\circ}$ | $30^{\circ}$ | $45^{\circ}$
---|---|---|---|---|---|---|---|---
QBERs | 3.4% | 3.4% | 3.6% | 3.8% | 4.0% | 4.1% | 5.9% | 8.5%
Detections | 4401 | 9169 | 14223 | 17522 | 20937 | 25981 | 50490 | 74728
Key rates | $7.19\times 10^{-8}$ | $1.26\times 10^{-7}$ | $2.29\times 10^{-7}$ | $2.26\times 10^{-7}$ | $3.29\times 10^{-7}$ | $2.95\times 10^{-7}$ | $4.80\times 10^{-7}$ | $4.73\times 10^{-7}$

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# Dielectric Loss due to Charged-Defect Acoustic Phonon Emission

Mark E. Turiansky<EMAIL_ADDRESS>Materials Department,
University of California, Santa Barbara, CA 93106-5050, U.S.A.  Chris G. Van
de Walle<EMAIL_ADDRESS>Materials Department, University of
California, Santa Barbara, CA 93106-5050, U.S.A.

###### Abstract

The coherence times of state-of-the-art superconducting qubits are limited by
bulk dielectric loss, yet the microscopic mechanism leading to this loss is
unclear. Here we propose that the experimentally observed loss can be
attributed to the presence of charged defects that enable the absorption of
electromagnetic radiation by the emission of acoustic phonons. Our explicit
derivation of the absorption coefficient for this mechanism allows us to
derive a loss tangent of $7.2\times 10^{-9}$ for Al2O3, in good agreement with
recent high-precision measurements [A. P. Read et al., Phys. Rev. Appl. 19,
034064 (2023)]. We also find that for temperatures well below $\sim$0.2 K, the
loss should be independent of temperature, also in agreement with
observations. Our investigations show that the loss per defect depends mainly
on properties of the host material, and a high-throughput search suggests that
diamond, cubic BN, AlN, and SiC are optimal in this respect.

##  I Introduction

Superconducting qubits are currently the main hardware for realizing quantum
computing [1]. A key figure of merit is the coherence time, which describes
how long the qubit can retain quantum information. Extending coherence times
is imperative to enable scaling up beyond hundreds of qubits and realizing
universal quantum computing. Progress has been made by feats of engineering,
such as exploring different device architectures, implementing control
protocols, and learning to work with noise [2, 3]. The coherence times are now
reaching the fundamental limits set by the materials used to construct the
devices [4, 3]. Improvements in materials are thus essential to extend
coherence times.

Identifying the microscopic loss mechanisms that limit coherence times is
challenging [5, 6]. Surfaces and interfaces have been implicated as a source
of loss [7, 8] Judicious use of cleaning techniques [9] and device
architectures that minimize surface participation [10] have led to marked
improvements, to the point where bulk dielectric loss is now limiting state-
of-the-art devices based on Al2O3 [11], which is the most commonly used
substrate for current qubits [5].

The microscopic mechanism that leads to this bulk dielectric loss is, however,
still unclear. Two-level systems (TLSs) have received the most attention, both
experimentally and theoretically [12]. However, TLSs give rise to resonances,
while the spectrum of the dominant loss is broad and featureless. Intrinsic
bulk dielectric loss is expected to be exceedingly low in a perfect crystal
[13]. However, even the highest-quality crystals contain some concentration of
defects (we use the term “defect” to refer to both intrinsic point defects and
extrinsic impurities). Point defects are of course known to be able to absorb
light or infrared radiation [14], but the ability for defects to absorb
microwave radiation has been overlooked.

Here we explore a mechanism that can explain absorption of microwave radiation
by charged defects via the emission of acoustic phonons [15, 13]. We develop a
formalism that allows for quantitative evaluation of absorption coefficients
and loss tangents, and show that this mechanism can explain the observed
dielectric loss in high-quality Al2O3 crystals. Furthermore, the temperature
dependence (or lack thereof) agrees with experimental observations.

Given the ability of our formalism to describe experimentally observed loss,
we implement a high-throughput search for alternative materials with low loss.
The results show that diamond, cubic BN, AlN, and SiC are promising host for
limiting these dielectric losses and thus further improving the coherence
times of superconducting qubits.

##  II Theory

Dielectric loss must be due to an absorption mechanism in which energy is
dissipated, e.g., in the form of phonons. Superconducting qubits are typically
operated between 1-10 GHz [2]. Optical phonon modes, with typical frequencies
on the order of THz, are too far off resonance to contribute to absorption at
these frequencies. Acoustic phonon modes, on the other hand, form a continuum
with frequencies ranging from zero to THz. They are therefore prime candidates
to lead to absorption in the microwave regime, provided there is a mechanism
to couple to radiation.

Consider first a perfect crystal in the absence of defects. Photons carry
(approximately) no momentum, meaning that only zone-center phonon modes can be
excited. Therefore, in a perfect crystal with translational symmetry, acoustic
phonon modes cannot lead to absorption to first order [16].

The introduction of defects into the crystal has two important consequences.
First, assuming the defects are randomly distributed, the translational
symmetry of the crystal will be broken. This relaxes the zone-center
requirement, as all phonon modes can be thought of as being folded back to the
zone center in the imperfect crystal; a phonon mode from anywhere in the
Brillouin zone can then in principle contribute to absorption if it is on
resonance. Second, if the defect is charged, the displacement of charge
produces a dipole that can couple to electromagnetic radiation.

We want to calculate the absorption coefficient $a(\omega)$ at photon
frequency $\omega$ in the presence of a charged defect. The absorption
coefficient is related to the attenuation of the electromagnetic radiation in
the material by [14]

$I(z)=I(0)\,e^{-a(\omega)z}\;,$ (1)

where $I$ is the intensity at coordinate $z$ along the direction of
propagation. $a(\omega)$ can be related to the absorption cross section per
defect $\sigma(\omega)$ by

$a(\omega)=N_{\rm def}\,\sigma(\omega)\;.$ (2)

where $N_{\rm def}$ is the density of defects assumed to be distributed
homogeneously throughout the material.

The absorption cross section $\sigma(\omega)$ can be obtained by application
of Fermi’s golden rule,

$\sigma(\omega)=Z_{\rm eff}^{2}{\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{4\pi^{2}\alpha}{n_{r}}\times\\\
\sum_{I,F}p_{I}{\lvert\braket{\Phi_{F}\lvert{\bm{\xi}}\cdot\hat{\bm{u}}_{d}\rvert\Phi_{I}}\rvert}^{2}\Omega_{IF}\delta(\omega-\Omega_{IF})\;,$
(3)

where $\alpha$ is the fine-structure constant, $n_{r}$ is the index of
refraction, $p_{I}$ is the thermal occupation factor, and $\hbar\Omega_{IF}$
is the energy difference between the initial and final states. The prefactor
$(\mathcal{E}_{\rm eff}/\mathcal{E}_{0})^{2}$ represents local-field effects,
where $\mathcal{E}_{\rm eff}$ is the effective electric field at the defect
and $\mathcal{E}_{0}$ the bulk value. This factor captures the fact that the
electric field at the defect is different from that in bulk due to scattering
of light at the defect [14]. Various models [17, 14] have been proposed to
model the effect. In the Lorentz-Lorenz model [18], $\mathcal{E}_{\rm
eff}/\mathcal{E}_{0}=(\varepsilon+2)/3$ where $\varepsilon$ is the dielectric
permittivity. In the Onsager model [19], $\mathcal{E}_{\rm
eff}/\mathcal{E}_{0}=3\varepsilon/(2\varepsilon+1)$. The Lorentz-Lorenz model
is generally considered to be an overestimate of local-field effects; in the
present work we therefore use the Onsager model, recognizing that it probably
represents an underestimate of the local-field enhancement.

${\bm{\xi}}$ is the polarization vector of the incoming radiation. $Z_{\rm
eff}$ is the Born effective charge; for coupling to an acoustic phonon mode
this corresponds to the charge on the defect [20] in the usual treatment of
charge states of defects [21]. $\hat{\bm{u}}_{d}$ is the displacement operator
for the defect site, whose evaluation will be discussed shortly.
$\ket{\Phi_{J}}$ are the many-body phonon wavefunctions, which are given by
direct products of the usual harmonic oscillator wavefunctions
$\ket{\phi_{k,n_{k}}}$:

$\ket{\Phi_{J}}=\prod_{k=1}^{3N_{a}}\ket{\phi_{k,n_{k}}}\;,$ (4)

where $N_{a}$ is the number of atoms in the crystal and $n_{k}$ is the
occupation number of the $k$th phonon mode. Superconducting qubits typically
operate at mK temperatures; we will assume that all phonon modes are in the
$n_{k}=0$ state in the initial state. We can then remove the summation over
$I$ in Eq. (3) and set $p_{0}=1$.

We are concerned with a purely vibronic transition (schematically depicted in
Fig. 1), but it can be useful to draw an analogy with the case of an
electronic transition. For an electronic transition, the vibronic
wavefunctions $\Phi_{J}$ is replaced with the electronic wavefunctions
$\Psi_{J}$, and the defect displacement operator $\hat{\bm{u}}_{d}$ is
replaced with the position operator $\hat{\bm{r}}$. The dipole for an
electronic transition would then be
$e\lvert\braket{\Psi_{F}\lvert{\bm{\xi}}\cdot\hat{\bm{r}}\rvert\Psi_{I}}\rvert$,
compared to $eZ_{\rm
eff}\lvert\braket{\Phi_{F}\lvert{\bm{\xi}}\cdot\hat{\bm{u}}_{d}\rvert\Phi_{I}}\rvert$
for a vibronic transition.

Figure 1:  The potential energy surface of a phonon mode in the harmonic
approximation. The probability densities of the vibronic wavefunctions are
shaded in color. At low $T$ only the $n=0$ harmonic oscillator level is
occupied and microwave radiation (wavy red arrow) drives a transition to the
$n=1$ level.

The displacement $\hat{\bm{u}}_{d}$ is not a normal mode of the system, but
can be expanded in the basis of normal modes,

${\bm{\xi}}\cdot\hat{\bm{u}}_{d}=\sum_{j=1}^{3N_{a}}\chi_{j}\,\sqrt{\frac{\hbar}{2M_{j}\omega_{j}}}\left(\hat{a}_{j}^{\dagger}+\hat{a}_{j}\right)\;,$
(5)

where $a_{j}$ ($a_{j}^{\dagger}$) is the annihilation (creation) operator for
the phonon mode with frequency $\omega_{j}$ and modal mass $M_{j}$, and
$\chi_{j}$ are the expansion coefficients. Since the initial state has
$n_{k}=0$, the annihilation operator in Eq. (5) does not contribute, and only
final states that differ from the initial state by the occupation of a single
phonon mode can contribute. Equation (3) then reduces to

$\sigma(\omega)=Z_{\rm eff}^{2}{\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{4\pi^{2}\alpha}{n_{r}}\sum_{j=1}^{3N_{a}}\frac{\hbar}{2M_{j}}{\lvert\chi_{j}\rvert}^{2}\delta(\omega-\omega_{j})\;.$
(6)

We are left with the task of determining the expansion coefficients
$\chi_{j}$.

In principle, the expansion coefficients could be obtained directly from a
supercell calculation using, e.g., density functional perturbation theory
[22]. While this is practical for optical phonons, this approach is infeasible
for acoustic phonons due to their extended nature. For example, since the
speed of sound in silicon is $\sim$8400 m/s, an acoustic phonon with a 4 GHz
frequency has a wavelength of 2.1 $\mu$m. It is impossible to perform a first-
principles calculation for a supercell of this size. A model calculation is
called for, and we follow the approach of Dawber and Elliott [23, 24] who
studied a “mass defect”, where only the mass of the defect atom is changed and
force constants retain their bulk values. For the case of a monatomic cubic
crystal they found

${\lvert\chi_{j}\rvert}^{2}=\frac{1}{N_{a}}{\left\\{\pi^{2}\epsilon^{2}z^{2}\nu^{2}(z)+{\left[1+\epsilon
z\;\mathcal{P}\\!\\!\int\frac{d\mu\,\nu(\mu)}{\mu-z}\right]}^{2}\right\\}}^{-1}\;,$
(7)

where $\mu=\omega^{2}$, $z=\omega_{j}^{2}$, and $\mathcal{P}$ indicates that
we take the Cauchy principal value of the integral.
$\epsilon=(M-M^{\prime})/M$ is a measure of the defect mass $M^{\prime}$ with
respect to the atomic mass $M$ of the lattice species. $\nu(\mu)$ is the
density of phonon states per squared unit frequency with normalization

$\int d\mu\,\nu(\mu)=1\;.$ (8)

[I.e., $\nu(\omega^{2})=\rho(\omega)/2\omega$, where $\rho(\omega)$ is the
usual density of phonon states per unit frequency.]

In the low frequency limit ($z=\omega_{j}^{2}\rightarrow 0$) relevant for
acoustic phonons, Eq. (7) reduces to

${\lvert\chi_{j}\rvert}^{2}=\frac{1}{3N_{a}}\;.$ (9)

One could have obtained this result by noting that the displacement of the
defect occurs on a single site, while the acoustic phonons are delocalized
over all $N_{a}$ atoms of the crystal. Indeed acoustic phonons are insensitive
to the microscopic details and are effectively a bulk material property. We
thus expect this to be a robust result, independent of the model or
calculation method. In particular, the following results are not restricted to
the case of a monatomic cubic crystal.

We can now evaluate the absorption cross section, Eq. (6). Since the phonons
make up a continuum we convert the summation over final states to an integral,

$\sum_{j=1}^{3N_{a}}\rightarrow 3N_{a}\int d\omega_{j}\,\rho(\omega_{j})\;,$
(10)

where $\rho(\omega_{j})$ is the phonon density of states normalized to unity.
With Eq. (9) we can then write the cross section as

$\sigma(\omega)=Z_{\rm eff}^{2}{\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{4\pi^{2}\alpha}{n_{r}}\frac{\hbar}{2M}\rho(\omega)\;,$
(11)

where we have replaced the modal mass $M_{j}$ by the average atomic mass of
the host lattice $M$, which is appropriate for acoustic phonons.

Finally, we use the fact that the phonon density of states for acoustic
phonons is well described by the Debye model:

$\rho(\omega)=\frac{3\omega^{2}}{\omega_{m}^{3}}\;.$ (12)

$\omega_{m}$ is the Debye frequency, which is the maximum possible phonon
frequency in the model [i.e., $\rho(\omega>\omega_{m})=0$], and is given by

$\omega_{m}=2\pi{\left(\frac{3N_{s}}{4\pi}\right)}^{1/3}v_{s}\;,$ (13)

where $N_{s}$ is the site density and $v_{s}$ is the sound velocity. The sound
velocity can be obtained from the elastic constants of the material,

$\displaystyle
v_{s}^{(t)}=\sqrt{\frac{G}{\varrho}}\;,v_{s}^{(l)}=\sqrt{\frac{K+4G/3}{\varrho}}\;,$
(14)

where $K$ is the bulk modulus, $G$ is the shear modulus, and $\varrho$ is the
mass density. The superscript indicates whether the speed corresponds to the
transverse $(t)$ or longitudinal $(l)$ mode. Our goal is to reproduce the low-
frequency phonon density of states, and we find from empirically fitting the
full phonon density of states that the transverse component is close to the
fitted value. Thus we will use the transverse speed of sound in subsequent
calculations.

Plugging Eq. (11) into Eq. (2) with the Debye model for the density of states
[Eq. (12)] leads to an absorption coefficient

$a(\omega)=\underbrace{N_{\rm def}Z_{\rm
eff}^{2}\vphantom{\frac{\omega^{2}}{\omega_{m}^{3}}}}_{\text{defect}}\underbrace{{\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{6\pi^{2}\alpha}{n_{r}}\frac{\hbar}{M}\frac{\omega^{2}}{\omega_{m}^{3}}}_{\text{host}}\;,$
(15)

where the brackets highlight the fact that there are two separate
contributions: (i) a defect-specific contribution described by the density of
defects $N_{\rm def}$ and effective charge $Z_{\rm eff}$, and (ii) a host-
specific contribution described by the index of refraction $n_{r}$, average
atomic mass $M$, and Debye frequency $\omega_{m}$. In this light,
$\sigma(\omega)/Z_{\rm eff}^{2}$ can be viewed as the “intrinsic” cross
section that only depends on the host material parameters. We can define a
characteristic parameter $\mathcal{A}_{\rm c}$, given by

$\mathcal{A}_{c}=\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\sqrt{\frac{6\pi^{2}\alpha}{n_{r}}\frac{\hbar}{M}\frac{1}{\omega_{m}^{3}}}\;,$
(16)

which depends only on the host material and has units of an absement
(m$\cdot$s). The final absorption coefficient then takes the convenient form

$a(\omega)=N_{\rm def}Z_{\rm eff}^{2}\mathcal{A}_{c}^{2}\omega^{2}\;.$ (17)

Experimental measurements typically report a loss tangent $\tan(\delta)$,
which is the ratio of the imaginary part of the dielectric permittivity
$\epsilon(\omega)$ (which is related to the absorption coefficient) to the
real part:

$\tan(\delta)=\frac{{\rm Im}\,\epsilon(\omega)}{{\rm
Re}\,\epsilon(\omega)}=\frac{c}{n_{r}\omega}a(\omega)\;,$ (18)

where we have used ${\rm Re}\,\epsilon(\omega)\approx n_{r}^{2}$. When
$\delta$ is small, $\tan(\delta)\approx\delta$. Utilizing Eqs. (13) and (15),
we obtain

$\tan(\delta)=\frac{1}{4\pi\epsilon_{0}n_{r}^{2}}{\left(\frac{\mathcal{E}_{\rm
eff}}{\mathcal{E}_{0}}\right)}^{2}\frac{N_{\rm def}Z_{\rm
eff}^{2}e^{2}}{\varrho v_{s}^{3}}\omega\;.$ (19)

This equation is very similar to the result from Garin [15], despite starting
from different equations. Garin started from an elastic continuum model,
focusing only on the acoustic phonon modes. Garin included the effect of
spatial correlations between the defects, obtaining a dependence that can be
viewed as a correction factor to our result. Here we assume that defect
positions are random and uncorrelated, which is consistent with dilute defect
concentrations and means no correction is necessary. Furthermore, Garin
neglected local-field effects.

For purposes of distinguishing this loss mechanism from others, it is useful
to look at the scaling as a function of various parameters. Eq. (19) clearly
shows that the loss is proportional to the frequency $\omega$. While we did
not explicitly include the temperature dependence in the formulas, we can make
some comments. When the temperature is low relative to the phonon frequency
($kT\ll\hbar\omega$), we can assume that the phonons are in their ground state
and our derivation holds. Therefore, as long as $kT$ stays well below
$\hbar\omega$, the absorption will be independent of temperature. For
$\omega/(2\pi)=4$ GHz, this corresponds to a temperature of 192 mK.

##  III Results

Now that we have explicit equations we can turn to quantitatively evaluating
the loss. In principle, the values that are necessary to evaluate Eq. (16)
could be obtained from experiments. However, we would like to assess a wide
range of materials, and a curated experimental database that contains all the
necessary parameters does not exist to our knowledge. Towards this end, we
will utilize high-throughput first-principles calculations based on density
functional theory. We will utilize the database from the Materials Project
[25], which at the writing of this manuscript contains 154,718 materials.

The characteristic parameter $\mathcal{A}_{c}$ depends on the average mass
$M$, which is trivially obtained from the host crystal structure, and the
Debye frequency $\omega_{m}$, which is obtained from the elastic constants of
the material [Eqs. (13) and (14)]. The index of refraction can be
approximately obtained from the dielectric response of the material:

$n_{r}=\sqrt{{\rm Tr}[\overleftrightarrow{\varepsilon}]/3}\;,$ (20)

where $\overleftrightarrow{\varepsilon}$ is the dielectric tensor of the
material. In principle, the index of refraction should be evaluated at the
frequency of interest, but the frequency-dependent value is not easily
accessible. Using the low-frequency value of the dielectric tensor is expected
to be a good approximation; indeed, experimental values for dielectric
permittivities of Al2O3 at 8 GHz (Ref. 26) are in very good agreement with
values obtained at audio frequencies (Ref. 27). All these parameters are
amenable to high-throughput simulation, since they can be obtained with
reasonable accuracy using a semi-local functional. Elastic properties are
known to be well described by semi-local functionals. While the band gap is
underestimated by semi-local functionals, the deviations for dielectric
properties from experiment are within a reasonable uncertainty.

We first examine the case of Al2O3, for which experimental results are
available [11]. Read et al. [11] performed highly sensitive measurements of
the loss tangent in high-grade Al2O3 grown with the heat-exchanger method
(HEMEX), measuring a loss tangent of $19\times 10^{-9}$ at 4.5 GHz. The
dominant impurity in HEMEX Al2O3 has been identified to be Si, which is
present in concentrations on the order of $10^{18}$ cm-3 [28, 29, 30]. Silicon
prefers to incorporate on the Al site in Al2O3 and acts as a donor in a +1
charge state [31]; charge neutrality then requires acceptors to be present in
similar concentrations. Alumininum vacancies are the most likely compensating
species, acting as deep acceptors predominantly in the $-3$ charge state [32].
Thus we will use $Z_{\rm eff}=+1$ and $N_{\rm def}=10^{18}$ cm-3 for Si,
$Z_{\rm eff}=-3$ and $N_{\rm def}=1/3\times 10^{18}$ cm-3 for $V_{\rm Al}$,
and add their contributions. With the values for the Debye frequency and index
of refraction calculated from first principles for Al2O3, we find a loss
tangent of $\tan(\delta)=7.2\times 10^{-9}$. Experimentally a value of
$19\times 10^{-9}$ (Ref. 11) was found.

There are several reasons why our calculated value is expected to be an
underestimate of the experimental value. First, contributions from other
defects are likely to be present. Indeed, Refs. 28 and 29 reported a number of
other impurities to be present in concentrations on the order of $10^{17}$
cm-3, and each charged impurity will be accompanied by compensating point
defects such as Al vacancies. Second, as discussed in Sec. II, our treatment
of local-field effects in the Onsager model likely represents an
underestimate, possibly by as much as an order of magnitude. Within the
Onsager model, the enhancement factor $(\mathcal{E}_{\rm
eff}/\mathcal{E}_{0})^{2}$ for Al2O3 is equal to 2.04, while in the Lorentz-
Lorenz model the factor would be 15.3.

We also note that Read et al. found the loss tangent to be independent of
temperature below 200 mK, as predicted by our model. Overall, the agreement
found here validates our hypothesis that absorption by charged defects is the
limiting factor in bulk loss in these samples.

The concentration of defects will depend on the growth method used. The HEMEX
sample examined in Ref. 11 was actually the highest grade Al2O3 available.
Read et al. also measured Al2O3 produced by edge-defined film-fed growth (EFG)
and found the loss tangent to be a factor of three higher, $63\times 10^{-9}$
[11]. While we were not able to find measurements of impurity concentrations
in EFG samples, it is plausible that defect concentrations are a factor of 3
higher in EFG Al2O3 compared to HEMEX. Later studies found that annealing EFG
Al2O3 at 1473 K resulted in loss tangents similar to that of HEMEX material
[33]. Annealing at such temperatures may allow defects to migrate and
recombine or form complexes, thus lowering the loss.

If absorption by charged defects is the limiting factor in bulk loss, then
controlling this loss is of paramount importance for improving coherence times
of superconducting qubits. Since the loss is proportional to the defect
concentration, controlling the incorporation of defects using high-quality and
high-purity crystal growth techniques is essential. However, our derivation of
the defect-induced absorption [Eq. (17)] also makes clear that the loss
depends on properties of the host material. It is therefore fruitful to
identify materials that may have lower loss than Al2O3, based on a high-
throughput search. Of the materials in the Materials Project database, we
focus on those that have a finite band gap when using the semi-local
functional of Perdew, Burke, and Ernzerhof (PBE) [34], as well as having their
elastic [35] and dielectric properties [36] calculated. Furthermore, we
exclude magnetic materials from the search to avoid the presence of spins that
may contribute magnetic noise.

The characteristic parameters $\mathcal{A}_{c}$ of the remaining 1,821
materials are shown in Fig. 2. Values for selected materials are shown in
Table 1, along with the loss tangent $\tan(\delta)$ calculated assuming
$Z_{\rm eff}=1$ and $N_{\rm def}=10^{18}$ cm-3. As expected from Eq. (16),
there is a strong dependence on the Debye frequency $\omega_{m}$ [Fig. 2(a)].
Indeed this explains why diamond has the lowest $\mathcal{A}_{c}$ of any
material in the database. We also show the dependence of $\mathcal{A}_{c}$ on
the band gap of the material [Fig. 2(b)]. Since it is known that semi-local
functionals underestimate the band gap [37], we utilize the empirical
correction of Morales-García et al. [38],

$E_{g}=1.355\times E_{g}^{\rm PBE}+0.916\;,$ (21)

where $E_{g}^{\rm PBE}$ is the band gap computed at the PBE level. There is no
explicit dependence of $\mathcal{A}_{c}$ on the band gap in our equations;
however, the band gap can be an important selection criterion for evaluating
material choices. For example, silicon was chosen as a barrier material for
the recently proposed FinMET design [39] because the smaller band gap would
enable thicker barriers to be used. We will further discuss the importance of
the band gap in the following section.

Figure 2:  High-throughput calculations of the characteristic parameter $\mathcal{A}_{c}$ [Eq. (16)] as a function of (a) the Debye frequency $\omega_{m}$ [Eq. (13)] and (b) the band gap $E_{g}$ [Eq. (21)].  Table 1:  Debye frequency $\omega_{m}$, index of refraction $n_{r}$, characteristic parameter $\mathcal{A}_{c}$, and loss tangent $\tan(\delta)$ for selected materials from the high-throughput search. $\tan(\delta)$ is calculated assuming $\omega/(2\pi)=4.5$ GHz, $N_{\rm def}=10^{18}$ cm-3, and $Z_{\rm eff}=1$.  Material | Space Group | Centrosymmetric? |  $\omega_{m}/(2\pi)$ [THz] | $n_{r}$ |  $\mathcal{A}_{c}$ [m$\cdot$s] | $\tan(\delta)$
---|---|---|---|---|---|---
C | $Fd\bar{3}m$ | Y | 42.3 | 2.41 | $9.8\times 10^{-27}$ | $3.4\times 10^{-10}$
BN | $F\bar{4}3m$ | N | 35.8 | 2.64 | $1.2\times 10^{-26}$ | $4.6\times 10^{-10}$
SiC | $R3m$ | N | 21.8 | 3.29 | $1.8\times 10^{-26}$ | $8.6\times 10^{-10}$
BP | $F\bar{4}3m$ | N | 20.2 | 3.05 | $2.1\times 10^{-26}$ | $1.2\times 10^{-9}$
BeO | $P6_{3}mc$ | N | 23.5 | 2.71 | $2.2\times 10^{-26}$ | $1.6\times 10^{-9}$
MgO | $Fm\bar{3}m$ | Y | 16.9 | 3.28 | $2.7\times 10^{-26}$ | $1.8\times 10^{-9}$
Al2O3 | $R\bar{3}c$ | Y | 17.3 | 3.12 | $2.6\times 10^{-26}$ | $1.8\times 10^{-9}$
AlN | $P6_{3}mc$ | N | 17.4 | 2.95 | $2.6\times 10^{-26}$ | $2.0\times 10^{-9}$
Si3N4 | $P6_{3}/m$ | Y | 17.6 | 2.86 | $2.7\times 10^{-26}$ | $2.1\times 10^{-9}$
BAs | $F\bar{4}3m$ | N | 12.7 | 3.15 | $2.8\times 10^{-26}$ | $2.2\times 10^{-9}$
GaN | $P6_{3}mc$ | N | 11.4 | 3.30 | $3.3\times 10^{-26}$ | $2.8\times 10^{-9}$
Si | $Fd\bar{3}m$ | Y | 11.8 | 3.61 | $3.7\times 10^{-26}$ | $3.2\times 10^{-9}$
Ga2O3 | $C2/m$ | Y | 9.8 | 3.34 | $4.4\times 10^{-26}$ | $4.9\times 10^{-9}$
LiF | $Fm\bar{3}m$ | Y | 13.5 | 2.95 | $4.9\times 10^{-26}$ | $6.8\times 10^{-9}$
ZnO | $P6_{3}mc$ | N | 7.5 | 3.29 | $6.4\times 10^{-26}$ | $1.1\times 10^{-8}$
SiO2 | $P3_{2}21$ | N | 11.2 | 2.18 | $5.8\times 10^{-26}$ | $1.3\times 10^{-8}$
NaCl | $Fm\bar{3}m$ | Y | 5.6 | 2.56 | $1.3\times 10^{-25}$ | $5.3\times 10^{-8}$
KBr | $Fm\bar{3}m$ | Y | 3.1 | 2.20 | $2.3\times 10^{-25}$ | $2.0\times 10^{-7}$

##  IV Discussion

Silicon has also been used in constructing superconducting qubits [4, 40, 39].
Our predicted loss tangent for Si is slightly higher than for Al2O3, assuming
equivalent defect concentrations. However, given the importance of Si for
semiconductor technology, defect control is very advanced and defect
concentrations far lower than the $10^{18}$ cm-3 value assumed in Table 1 are
readily achievable. Thus one would expect the dielectric loss from charged
defects in Si to be far less than that in Al2O3, and it could potentially be
reduced further in specially prepared samples.

Surprisingly, measurements on devices based on Si measure loss tangents an
order of magnitude greater than those for Al2O3 [8, 41]. Recent measurements
of loss tangents in high-resistivity Si using high-Q superconducting
resonators found a value two orders of magnitude greater than that of Al2O3
[42]. Checchin et al. [42] attribute the dominant loss to residual
conductivity arising from variable-range hopping between defects. This
highlights the importance of the band gap as a design criterion [Fig. 2(b)].
Narrow band-gap materials tend to host shallow defects, which have loosely
bound hydrogenic wavefunctions that are easy to ionize. Ultra-wide band-gap
materials, on the other hand, tend to not have shallow defects and instead are
dominated by deep defects, which have tightly bound wavefunctions and are
harder to ionize. (In other words, this is the difference between a
semiconductor and an insulator.) We can hypothesize that in the absence of
this conduction mechanism, Si would have had lower loss due to better control
over deep defects.

Piezoelectricity has also been invoked as a potential mechanism that may lead
to loss [43]. In a similar way to the presence of charged defects,
piezoelectricity enables the conversion of electromagnetic radiation into
acoustic phonons. To our knowledge, there is no rigorous confirmation of this
proposal, except empirical evidence that switching from wurtzite to zincblende
AlN improved coherence times in some devices [44]. Zincblende AlN is still
piezoelectric but has zero coupling for certain orientations of the electric
field. Thus one an engineer around piezoelectricity in some crystal phases.
For this reason, and because of the lack of evidence that piezoelectricity is
an important loss mechanism, we do not exclude any materials based on their
piezoelectricity from our search. However, we do provide the space group and
whether or not it is centrosymmetric in Table 1, as it may be a consideration
when designing a device.

From our survey, we can suggest materials that may be fruitful to explore.
Materials with a large Debye frequency will be best at suppressing the loss
tangent. We would also recommend choosing materials with a large band gap to
potentially avoid the type of loss mechanisms that were observed in Si.
Diamond is by far the best choice: it has the largest Debye frequency and
lowest loss tangent as a result. Furthermore, it has a large band gap,
avoiding the type of losses observed in Si, and is centrosymmetric, avoiding
piezoelectric contributions. Cubic BN is second best, its main drawback being
piezoelectricity, which as we noted can be engineered around. However, these
materials may be difficult to work with and incorporate in devices. An
alternative to cubic BN may be found by considering the group-III nitrides
(including InN, GaN, and AlN), a mature materials platform due to their use in
solid-state lighting [45]. AlN has a band gap comparable to cubic BN and lower
loss than Al2O3 (assuming identical defect chemistry). There have been initial
efforts to produce all-nitride qubits [44, 40]. Similarly, SiC has one of the
lowest loss values within our model, yet is easier to work with for devices.

##  V Conclusions

In conclusion, we have studied the impact of charged defects on dielectric
loss in a variety of materials. Charged defects break translational symmetry
and enable the absorption of electromagnetic radiation by acoustic phonons,
and we derived the absorption coefficient and loss tangent for this process.
Our calculated loss tangent of $7.2\times 10^{-9}$ is in good agreement with
measurements on high-grade Al2O3, which found a value of $19\times 10^{-9}$
[11]. Furthermore, we expect the loss tangent to be independent of temperature
for this process, in agreement with the observations. This demonstrates that
charged defects are likely limiting the bulk loss in such samples and
controlling their incorporation is necessary to improve coherence times. In
addition, the loss is highly dependent on properties of the host material, in
particular the Debye frequency. We implemented a high-throughput search on a
database of first-principles calculations with the goal of identifying
alternative low-loss materials. Based on our results, we suggest that diamond,
cubic BN, AlN, and SiC are promising materials to explore.

###### Acknowledgements.

We gratefully acknowledge fruitful discussions with C. E. Dreyer, N. P. de
Leon, and R. J. Schoelkopf. This work was supported by the U.S. Department of
Energy, Office of Science, National Quantum Information Science Research
Centers, Co-design Center for Quantum Advantage (C2QA) under contract number
DE-SC0012704. The research used resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231 using NERSC award BES-ERCAP0021021.

## Data Availability

The data that support the findings of this study are openly available in
Zenodo at https://doi.org/[DOI-TO-BE-INSERTED].

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