# The Superconducting Quasiparticle-Amplifying Transmon: A Qubit-Based Sensor
for meV Scale Phonons and Single THz Photons

C.W. Fink<EMAIL_ADDRESS>Materials Physics and Applications – Quantum, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  C.P. Salemi
<EMAIL_ADDRESS>SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94035, USA  B.A. Young  Santa
Clara University, Santa Clara, CA 95053, USA  D.I. Schuster  Department of
Applied Physics, Stanford University, Stanford, CA 94035, USA  N.A. Kurinsky
<EMAIL_ADDRESS>SLAC National Accelerator Laboratory, Menlo Park,
California 94025, USA  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94035, USA

###### Abstract

With great interest from the quantum computing community, an immense amount of
R&D effort has been invested into improving superconducting qubits. The
technologies developed for the design and fabrication of these qubits can be
directly applied to applications for ultra-low threshold particle detectors,
e.g. low-mass dark matter and far-IR photon sensing. We propose a novel energy
resolving sensor based on the transmon qubit architecture combined with a
signal-enhancing superconducting quasiparticle amplification stage. We refer
to these sensors as SQUATs: Superconducting Quasiparticle-Amplifying
Transmons. We detail the operating principle and design of this new sensor and
predict that with minimal R&D effort, solid-state based detectors patterned
with these sensors can achieve sensitivity to single THz photons, and
sensitivity to $1\,\mathrm{meV}$ phonons in the detector absorber substrate on
the $\mu\mathrm{s}$ timescale.

qubit, dark matter, neutrino, phonon, photon

Introduction—The increasing maturity of superconducting qubits over the past
few decades has allowed the field of superconducting quantum computing to
flourish, producing a massive industry centered on the goal of improving
quantum coherence. Breakthrough studies in the last few years have shown that
environmental radioactivity can induce correlated errors in qubit arrays [1,
2, 3], and subsequent work has demonstrated that charge noise and phonon-
induced drops in coherence time can be mitigated by designing phonon sinks and
reducing photon coupling to films nearest to the qubits [4, 5].

In parallel, advances in detector technology that have built on this wave of
qubit fabrication expertise have shown that this same technology can be
applied to energy sensing at the THz (meV) scale. The first demonstration of
the Quantum Capacitance Detector (QCD) [6, 7] showed that single THz photon
detection can be achieved by utilizing a Cooper pair box coupled to a
resonator – a structure analogous to many early charge-sensitive qubits. This
demonstration and work to understand the radiation sensitivity of qubits have
opened up a new regime of sensing leveraging the single quasiparticle (QP)
sensitivity of qubit-derived structures. The main sensing mechanism comes from
the quantized nature of the qubit and the charge sensitivity of the
transition, such that a single quasiparticle tunneling across the junction is
easily measurable in real time [8, 9, 10, 11]. If energy can be focused into
breaking Cooper pairs near the junction, as is done in Ref [7], then very low
thresholds are achievable.

In this letter we propose a novel sensing technology based on weakly charge-
coupled transmon qubits. The scheme described in this paper differs from the
QCD design [6, 7] and that of conventional superconducting qubits [12, 13, 14,
15, 16, 2] by removing the readout resonator entirely from the architecture
and relying on direct readout of the qubit transition frequency to detect
tunneling events. By removing the readout resonator, we are no longer
sensitive to the quantum capacitance as in the QCD, but rather we are
sensitive to the quantum inductance in the non-linear LC resonator that makes
up the qubit. This architecture change allows for significant reduction in the
overall size of the unit cell, increases in pixel density, reduction of two-
level system noise, and increase in detection efficiency for uniformly
distributed radiation signals. We will show that these devices should
outperform the energy sensitivity of competing technologies, while the RF
based readout scheme allows them to be naturally multiplexed, thus allowing
for highly pixalizable, ultra-low threshold single THz photon and single-
phonon detection.

Operating Principle—Transmon qubits [12] are anharmonic LC oscillators where
the capacitance comes from the proximity of metal islands that are connected
by a Josephson junction (JJ), which provides a nonlinear inductance. The
nonlinearity of the inductance creates unequally spaced energy levels; typical
qubit operation uses only the first two levels, the ground, $\ket{0}$, and
first excited, $\ket{1}$, states.

The energy difference between these states, $E_{01}$, also called the
splitting energy, is dependent on the qubit’s design, parametrized by the
qubit’s charging energy, $E_{C}=e^{2}/2C$, the energy to increment the qubit’s
charge by one electron, and the Josephson energy, $E_{J}=\hbar I_{c}/2e$, the
tunneling energy of a Cooper pair. In the transmon limit, which corresponds to
$\xi=\frac{E_{J}}{E_{C}}\gg 1$, we can model the energy spectrum of the qubit
accurately as [12]

$E_{01}\approx\hbar\omega_{0}+\hbar\chi_{0}\cos(\pi n_{q})\,,$ (1)

where $n_{q}$ is the island charge,

$\hbar\omega_{0}\approx\sqrt{8E_{C}E_{J}}-E_{C}\,,$ (2)

and

$\displaystyle\frac{2\chi_{0}}{\omega_{0}}$ $\displaystyle\approx
e^{-\sqrt{8\xi}}\left[A\xi^{3/4}+B\xi^{1/4}\right],$ (3) $\displaystyle A$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{2^{9}}{8^{1/2}2^{5/4}}\approx 60.7,$
(4) $\displaystyle B$
$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{2^{5}}{8^{1/2}2^{3/4}}\approx 5.37$
(5)

controls the magnitude of the charge dependence of the splitting. We can thus
tune the qubit to a given frequency by careful design of the junction area and
critical current density, as well as the total capacitance within the qubit
geometry. One feature of the nonlinearity of the relationship between $E_{01}$
and $E_{C}$ and $E_{J}$ is that the qubit energy has a periodic dependence on
the number of charges on the island, $n_{q}$. Incrementing this charge by one
electron switches the state parity as the cosine in Eq. 1 changes phase by
$\pi$.

There are two ways to create non-equilibrium quasiparticles in the qubit
islands, as shown in Fig. 1. First, direct energy deposition in the island
produces a hot pair of quasiparticles, which quickly generate a larger
quasiparticle population as they settle to the superconducting energy gap.
This is the case for photon absorption or direct collision of particles with
the qubit island. Alternatively, athermal substrate phonons with energies
large compared to the superconducting gap of the islands will similarly excite
non-equilibrium QPs in the islands. Both processes lead to the same
optimization for collecting the resulting quasiparticles, but the two energy
absorption methods require different considerations for external quantum
efficiency, as described in the following sections.

Figure 1: Schematic of proposed sensor. a) photolithography mask design for
prototype sensor. b) QP energy diagram showing signal measurement process.
From left to right: A Cooper pair is broken by an incident particle with
energy greater than $2\Delta_{\text{island}}$ and creates QPs in an excited
state, these QPs downconvert releasing phonons and lower energy QPs, these QPs
diffuse until becoming trapped in the lower gap trap, the QPs then tunnel
across the junction until they recombine or are trapped by an impurity. c)
Cross-section of the sensor shown on top of a substrate (not to scale). Two
possible signal paths are shown: direct absorption of a photon into the island
(blue) or an athermal phonon from the substrate (black).

Phonon coupling—Energetic electrons/holes or optical phonons created from an
inital particle interaction in the substrate will rapidly downconvert to high
energy acoustic athermal phonons. These phonons then undergo anharmonic decay
until their mean free path is on the order of the characteristic size of the
substrate [17]. The phonons will travel ballistically in the substrate until
being absorbed by the active (sensor) and passive (e.g. RF feedline, ground
plane) metal films on the surface of the substrate. From [18], the
characteristic time scale for these phonon event signals can be approximated
by

$\displaystyle\tau_{\text{phonon}}\approx\frac{4V_{\text{det}}}{\sum\left<c_{\text{det}}\right>f_{abs}^{i}A_{i}}$
(6)

where $V_{\text{det}}$ is the volume of the substrate,
$\left<c_{\text{det}}\right>$ is the average sound speed in the substrate,
$A_{i}$ is the area of the $i^{th}$ absorbing material on the detector
surface, and $f_{abs}^{i}$ is the phonon transmission probability between the
substrate and the absorbing material, modeled simply as

$f_{abs}=1-\exp\left[\frac{2t}{c_{s}\tau_{B}}\right],$ (7)

where $t$ is the absorber film thickness, $c_{s}$ is the sound speed in the
film, and $\tau_{B}^{-1}$ is the quasiparticle pair breaking rate, which can
be found in [19] for Al and Nb.

There are two signal efficiency penalties that must be accounted for in this
step. The first is the percentage of ballistic phonons with energy greater
than twice the superconducting bandgap ($\Delta$) of the sensor material,
which is typically $>95\%$ for common materials [20]. Secondly, we must
account for the percentage of phonons that are absorbed by the non-
instrumented areas of the detector surface. This efficiency factor can be
calculated using :

$\varepsilon_{\text{passive}}=\frac{f_{abs}^{\text{active}}A_{\text{active}}}{\sum
f_{abs}^{i}A_{i}}.$ (8)

With our initial design we expect to achieve a fill factor of $\sim 5\%$ for
instrumented sensors. The (passive) coplanar waveguide (CPW) feedlines also
will cover $\sim 5\%$ of the detector surface. Lastly, a (passive) parquet
patterned Nb grid ground plane similar to that used in [21] will cover another
$\sim 5\%$ of the surface. Despite the significantly larger $T_{c}$ of Nb vs.
Al, preliminary studies suggest that a large fraction of the athermal phonon
population in Si will be greater than twice the Nb superconducting gap and
thus the Nb ground plane can become a significant source of phonon loss [22].
Using Eq. 8 we expect a phonon collection efficiency due to passive surfaces
of $\varepsilon_{\text{passive}}\approx 37\%$. From Eq. 6 the phonon
collection time depends linearly on the detector thickness, and for a
$500\,\mu\mathrm{m}$ thick Si wafer substrate, we expect
$\tau_{\text{phonon}}\approx 2\,\mu\mathrm{s}$. If the signal loss from
passive surfaces proves to be a problem, in a future iteration of this design
we propose to use a flip-chip process in which the CPW qubit readout and
control circuitry is on a separate substrate, similar to what is done in [23].
By doing this, nearly 100% of the passive material can be removed from the
detector chip.

Photon coupling— In addition to athermal phonon measurement, these sensors can
be used as single photon sensors via direct absorption in the islands. The
shape of the sensor (see Fig. 1) is designed to maximally collect the induced
quasiparticle signal, as described in the following text; additionally, this
shape naturally lends itself to being engineered into a broadband bow-tie
antenna.

While a detailed simulation of the antenna structure is beyond the scope of
this paper and left for future work, we note that similarly designed antenna
micro-structures have achieved broadband absorption in the 1-10 THz range [24,
25, 26]. For example, [25] has shown with simulations
$10-100\,\mu\mathrm{m}$-size on-chip bowtie antennas with 20% bandwidth around
1 THz with 90% efficiency and [26] has demonstrated a single-structure bowtie
antenna that functions over 1-10 THz. Various studies of THz absorption in
general thin films have also been done, with wide-ranging efficiencies [27,
28, 29, 30]. The QCDs achieved an optical efficiency of 90% in a narrow band
around 1.5 THz using a resonant mesh antenna absorber [7].

The SQUAT photon absorption efficiency can also be improved by using thicker
films, optimizing fabrication methods and material choice, applying surface
treatments, and adding a reflector under the substrate to give photons a
second pass through the device. From the literature, we conservatively
estimate that we can achieve a 50% photon absorption in Al over a 20%
bandwidth at THz frequencies [25, 31, 26].

Quasiparticle Collection Efficiency—The coplanar capacitor islands in our
device are designed to both maximize surface fill factor, and funnel QPs
toward the junction. The funneling effect is achieved in part by the choice of
a geometry that maximizes the QP collection efficiency for a given sensor
area. We can achieve additional collection efficiency through the use of QP
trapping [32, 33, 34], similar to that done with Quasiparticle-trap-assisted
Electrothermal-feedback Transition-edge sensors (QETs) [35], by fabricating
the islands with a material with a larger $T_{c}$ than the junction. For this
proposed device, we choose Al for the athermal phonon collection region. For
the trapping region and junctions, we require a material with a
superconducting gap that is roughly a factor of 10 lower than Al. The simplest
material to use would be AlMn (which we will assume for our calculations
here), since its $T_{c}$ in thin films is easily tunable and it lends itself
well to Josephson junction fabrication, as it naturally creates an aluminum
oxide layer [36, 37, 38, 39].

The full quasiparticle diffusion model is described in detail in Appendix C,
and we highlight the key features here. For a schematic diagram of the
following, see Fig. 1.

  1. 1.

When a phonon or photon of energy greater than $2\Delta_{\text{island}}$ is
absorbed by the island, a Cooper pair will be broken, and the resulting QPs
will be promoted to well above the bandgap energy. These QPs will undergo a
downconversion process to lower energy QPs and phonons, yielding a population
of QPs at the band edge after $\mathcal{O}(1)\,\mathrm{ns}$ [40]. During the
downconversion process, a portion of the initial event energy will be lost as
sub-gap phonons leak into the substrate [19]. For conventional S-wave
superconducting films, incident events with energy $E_{\Omega}\gtrsim
4\,\Delta_{\text{island}}$ will have approximately $60\,\%$ of the event
energy remain in the QP system [41]. We will assume that the events will be in
this energy regime and will adopt a value for the efficiency of the
downconversion process ($\varepsilon_{\text{DC}})$ of $60\,\%$.

  2. 2.

The remaining QPs will settle to a stable population with energy of
$\sim\Delta_{\text{island}}$ and will diffuse until they either recombine into
Cooper pairs over a timescale of $\mathcal{O}(0.1-1)\,\mathrm{ms}$ [42, 43]
(in Al), are trapped by impurities in the film, or ideally get trapped in the
lower-$T_{c}$ superconductor nearby.

  3. 3.

Once the QPs enter the lower-$T_{c}$ material they undergo a second
downconversion process, resulting in an enhancement of QP number that scales
with the ratio of the superconducting gaps of the two materials. These
downconverted QPs have an energy lower than $\Delta_{\text{island}}$ and
become ‘trapped’. This multiplication process results in a signal enhancement,
because the qubit is sensitive to QP _number_ , not energy.

  4. 4.

While the QPs are trapped in the lower $T_{c}$ material, they will tunnel back
and forth across the attached Josephson junction until recombination occurs.
Each measured tunneling event results in a signal.

A plot of the QP collection fraction ($\varepsilon_{\text{QP}}$) as a function
of QP trap and qubit island characteristic lengths, $\ell_{\text{trap}}$ and
$\ell_{\text{island}}$ respectively, for a $500\,\mathrm{nm}$ square JJ can be
seen in Fig. 2. Note that while this device adds an additional tunneling step
beyond that of its QET counterpart [35], this efficiency loss can be made up
by the addition of the QP multiplication and multiple tunnels – allowing for
collection fractions that are greater than unity.

Figure 2: Modeled quasiparticle collection efficiency as a function of
trapping region length and island length for a $500\,\mathrm{nm}$ square
Josephson junction with $100\,\mathrm{nm}$ thick Al and AlMn films. The model
accounts for multiple QP tunneling events across the junction, which in
principle can result in an efficiency greater than unity.

Qubit Tuning — In order to optimize the readout of these sensors, we must
first tune the SQUAT design. Here, we focus on three main design parameters:

  1. 1.

The undressed resonance frequency, $f_{0}$;

  2. 2.

The maximum frequency separation of the even and odd parity states,
$2\chi_{0}$;

  3. 3.

The total quality factor of the qubit coupled to the feed line, $Q$;

We model the qubits as notch filters, assuming an ideal transmision line, as
[44],

$S_{21}(f,Q,f_{q})\approx
1-\left[1+2iQ\left(\frac{f-f_{q}}{f_{q}}\right)\right]^{-1}$ (9)

where the qubit resonance frequency $f_{q}$ is a function of qubit charge
$n_{q}$ and state spacing $2\chi_{0}$. Converting $E_{01}$ from Eq. 1 into
frequency, we see that (in the transmon limit), $f_{q}$ is well approximated
by the sinusoidal function

$f_{q}(\chi_{0},n_{q})\approx\chi_{0}\cos(\pi n_{q})+f_{0}\,.$ (10)

We can choose a charge operating point $n_{q}$ by applying a charge bias along
the readout line or using a dedicated charge gate. Effectively this charge
bias tunes the state frequency spacing, represented in Eq. 1 as
$\chi=\chi_{0}\cos(\pi n_{q})$. When a quasiparticle tunneling event imparts a
fixed charge shift of 1 electron (shifting the overall phase by $\pi$) the
magnitude of the shift of $f_{q}$ is a fraction of the maximum shift set by
the operating point (see Fig. 3, right).

From Eqs. 2 and 3, we can see that $f_{0}$ and $\chi_{0}$ are both determined
by $E_{J}$ and $E_{C}$, and thus these parameters must be co-optimized. We
first choose $f_{0}$ to be in the operating range of standard RF electronics,
e.g. $\sim 4-8$ GHz (this is also informed by available quantum-limited
readout). We must also assure that the state separation (determined solely by
$E_{J}/E_{C}$) is in the weakly-charge sensitive limit,
$\chi_{0}\sim\mathcal{O}(1-10)$. Note that we will return to the precise
optimization of $\chi_{0}$ in the next section.

With $f_{0}$ and $\chi_{0}$ chosen, we turn to the question of resonance
bandwidth, which is given by $\mathrm{BW}\approx f_{0}/Q$. The sensors can be
readily fabricated to have high-enough fidelity that the limiting quality
factor is determined by that of the qubit-to-feedline coupling, $Q\approx
Q_{c}$. The optimal bandwidth depends on the readout method, as discussed
later in this section and in Appendix A. In order to effectively resolve
phonon events at the timescale of $1\,\mu\mathrm{s}$ we must achieve a
bandwidth of $\mathrm{BW}\approx 1\,\mathrm{MHz}$, implying $Q_{c}\approx
1000$.

Using simulations in HFSS we have demonstrated that generating designs within
the optimal parameter space is easily achievable; an example of a SQUAT design
tuned for phase readout is shown in Fig. 1a. The fact that an optimization is
possible can be made intuitive by thinking about how each parameter depends on
the geometry:

  * •

$Q\approx Q_{c}$ depends almost entirely on the capacitive coupling between
the qubit and the feedline. So, tuning $Q$ is achieved by changing the island-
feedline separation.

  * •

$E_{C}$ is inversely dependent on the effective capacitance between the
islands. Although the capacitance between the islands and the feedline
contributes to this total, it is subdominant to the direct island-to-island
capacitance.

  * •

$E_{J}$ depends on junction parameters including the junction area and
thickness. Although the junction effectively acts as a parallel plate
capacitor, this capacitance is also less than the direct island-to-island
capacitance that dominates $E_{C}$.

  * •

To choose $\chi_{0}$, one can fix the $E_{J}/E_{C}$ ratio.

  * •

Then, with fixed ratio, $E_{J}$ and $E_{C}$ can be varied together to get an
appropriate $f_{0}$.

The optimal $\chi_{0}$ and $Q$ are determined by the need to optimize readout
signal-to-noise ratio (SNR).

In order to read out the SQUAT, a signal tone of fixed frequency is pumped
through the feedline; we will discuss which frequency is optimal later. We
will focus on a transmission measurement, although a reflection measurement is
also possible in principle. Given an input readout tone $V_{r}$, the signal is
described by a complex voltage,

$V_{s}=S_{21}V_{r}\,.$ (11)

While we measure voltage directly, we convert the voltage signal to power and
phase, which allows us to factor out impedance sources in our sensitivity
calculations. The measured power transmitted is thus

$P_{s}=|S_{21}|^{2}P_{r}$ (12)

with a phase shift of

$\tan(\theta)=\frac{\mathrm{Im}(S_{21})}{\mathrm{Re}(S_{21})},$ (13)

where $P_{r}$ is the input tone power. This also allows us to determine the
power dissipated in the device, with a precision determined by the degree to
which attenuation and gain in our readout system are known.

There are two fundamental modes in which we can read out the SQUATs in this
two-dimensional basis. The first mode is to observe a shift in the amplitude
of the transmitted signal, with phase shift fixed at zero; the second is to
observe a shift in the signal phase with amplitude unchanged. Both correspond
to states separated by the diameter of the resonator circle in the IQ space,
but amplitude readout is less sensitive to the precise frequency dispersion of
the qubit, while properly-optimized phase readout is less sensitive to readout
frequency. These cases are discussed in more detail in Appendix A, and
summarized qualitatively here to focus on readout optimization results for the
SQUAT.

In the amplitude readout mode, the optimal signal would be one in which
$S_{21}$ equals 1 or 0 for the different parity states, so that the signal
tone completely appears ($V_{s}=V_{r}$) or disappears ($V_{s}=0$) during a
tunneling event. This situation can be achieved if the readout tone is within
the bandwidth of one of the parity states and there is effectively no overlap
of the two states, e.g. the state separation is much larger than the resonator
width:

$2\chi\gg f_{0}/Q\,.$ (14)

In the second readout mode, in which we wish to measure distance between
states in the complex plane, a different optimization of $Q$ and $\chi$ is
necessary. Consider placing the readout tone at a frequency between the parity
state frequencies, such that the transmitted magnitude is constant, but a
relative phase shift is observed. As we derive in detail in Appendix A, the
measured phase shift is maximized when

$2\chi\sim f_{0}/Q\,.$ (15)

The main advantage of this second readout scheme is, as shown in the appendix,
that a sensor optimized in this way will produce a fixed shift in the IQ plane
for any readout frequency between the two parity states. This means we don’t
need to worry about precise readout frequency optimization. In this case, in
which noise is uncorrelated in phase, we will not see a large difference in
readout fidelity as a function of readout frequency. For noise primarily in
the phase or amplitude direction, we can adjust readout frequency to minimize
the amount of noise projected onto our readout vector.

For both of these cases, we show in Appendix B that the qubit fidelity is just
defined by the noise temperature and readout power according to the equation

$SNR^{-1}=\sigma^{2}=\frac{2P_{n}}{\epsilon_{r}P_{r}}=\frac{2k_{B}T_{n}f_{bw}\eta\left(\frac{hf_{0}}{k_{B}T}\right)}{P_{r}\left[1-\exp\left(-\frac{\hbar
f_{0}^{2}}{Q_{c}P_{r}}\right)\right]}\,.$ (16)

where $f_{bw}$ is the readout bandwidth, $T$ is the system noise temperature,
and $f_{0}$ is the readout frequency of the SQUAT. For $T_{n}\sim$1.8 K and
$f_{0}\sim$6 GHz, Eq. 16 dictates that we can achieve high readout fidelity
above -135 dBm readout power for an integration time corresponding to
$f_{bw}=10^{4}\,\mathrm{Hz}$. With a quantum-limited readout, we find the
reduced equation

$SNR=\frac{P_{r}\epsilon_{r}}{2hf_{0}f_{bw}}\,.$ (17)

where high fidelity is achieved above $P_{r}\approx-140\,\textrm{dBm}$ at
$f_{bw}\sim 10^{5}$ Hz. These readout powers have already been used to
demonstrate high readout fidelity for state-of-the-art qubits. Real devices
will need to map out the tradeoffs between parasitic pair-breaking of the
readout tone and noise-limited readout, and may benefit from stimulated
emission from the SQUAT at higher readout powers.

In practice, the difference in SNR between phase and amplitude readout will be
determined by the level of phase noise in the system (which we elaborate on in
Appendix B.1), and will be assessed during first device tests. We have the
freedom to tune state separation to optimize for either readout mode.

Figure 3: Simulation of the signal readout with $\chi=10\,\mathrm{MHz}$ and an
effective quality factor of $Q=1\,\mathrm{k}$. Left) Magnitude of $S_{21}$
showing the frequency separation between the odd and even parity states.
Right) Charge dispersion of the sensor normalized to the readout frequency,
showing the even and odd states. Shown in grey is the target dispersion
($\chi_{0}$). Bottom) Noisy simulation of single QP tunneling background
events with an example dark count rate of 1 Hz.

Estimated Sensor Performance—We can combine all of our known efficiency
factors to predict the final energy sensitivity of the device to both direct
absorption of photons and measurement of athermal phonons in the substrate.
Recall that we are measuring total QP number
$n_{\text{qp}}\approx\eta_{\text{qp}}/\Delta_{\text{island}}$, where
$\eta_{\text{qp}}$ is the total energy in the QP system. The energy in the QP
system can be written as the product of the impinging particle energy
($E_{\text{particle}}$), the absorption efficiency
($\varepsilon_{\text{abs}}$) and the efficiency from the downconversion
process ($\varepsilon_{\text{DC}}$). Thus, our measured number of QP tunneling
events ($N_{\text{QP}}$) will be

$N_{\text{QP}}=E_{\text{particle}}\times\varepsilon_{\text{abs}}\times\varepsilon_{\text{DC}}\times\frac{1}{\Delta_{\text{Al}}}\times
F_{\text{collect}},$ (18)

where $F_{\text{collect}}$ is the QP collection fraction given by Eq. 62, and
$\varepsilon_{\text{abs}}$ is the efficiency of the particle absorption in the
sensor, i.e. the antenna coupling for photons and the signal loss due to
passive surfaces for phonons.

Considering first the athermal phonon detection, taking the worst case
scenario that all the phonons are above the Nb gap, our phonon collection
efficiency due to passive surface losses is roughly
$\varepsilon_{\text{passive}}\approx 37\,\%$. Note that this value is
relatively independent of individual sensor size, and the total sensor number
can be changed to keep the surface coverage the same. Reading out the detector
such that a single QPs can be counted, we calculate what the expected athermal
phonon energy sensitivity threshold is of the entire detector, shown in Fig.
4. From the figure, we can see that for various geometries the SQUAT can
easily achieve single meV phonon thresholds. They dynamic range of the
measurement can span multiple order of magnitude in energy with minimal signal
loss and is a function of of the measurement readout bandwidth, discussed in
Appendix D.

Figure 4: Left Simulated phonon energy resolution for single QP sensitivity as
a function of island and trap length assuming negligible change in fill factor
as a function of island length. Shown in the hatched region is twice the
superconducting gap of Al, and the blacked out regions represent non-physical
design parameters. Middle Simulated QP tunneling number $N_{\mathrm{QP}}$
across the junction as a function of incident photon frequency and island
length for a $4\,\mu\mathrm{m}$ trap. Note, we are not factoring in the
antenna effect when changing the Island length at this point – simply assuming
that a $\sim\,50\%$ photon absorption efficiency can be measured. Right
Comparison of the SQUAT with a $4\,\mu\mathrm{m}$ trap vs a device of the same
geometry, but with no trapping region. Shown in purple on the right axis is
the sensitivity advantage of including QP trapping.

The photon energy sensitivity is expected to be similar to the phonon scenario
since $\varepsilon_{\text{abs}}$ is very close for both cases. A full
simulation of the antenna structure is left for future work, but from initial
conservative estimates we expect that sensitivity to single sub-THz photons is
reasonably achievable. A simulation of the expected sensor response as a
function of photon energy and island length is shown in Fig. 4.

In order to illustrate the incredible gain in sensitivity that the QP trapping
stage adds, we have plotted the simulated QP collection efficiency of the
SQUAT sensor with a $4\,\mu\mathrm{m}$ trap compared to an equivalent geometry
sensor but without the trap in Fig. 4 right. As can be seen, the trapping
region adds multiple orders of magnitude of phonon or photon sensitivity to
the detectors.

Discussion—In this letter we have presented the design sketch and modeling of
a novel detector concept based on superconducting qubit technology, which if
properly optimized, would allow for sensitivities to particle interactions at
the meV scale. While the majority of our quasiparticle model we have developed
in Appendix C is based on previously measured data, we have made a few
(conservative) assumptions, which we plan to measure in future work. Over the
next few years, our R&D path towards realizing these devices is to:

  * •

Fabricate devices of the geometry presented in this paper using only Al for
the islands and junctions (no QP trapping) to understand the optimization of
the readout and qubit parameters.

  * •

Optimize the fabrication of separate AlMn (and similar materials) junctions to
understand the QP tunneling probability of the junctions.

  * •

Fabricate the complete SQUAT sensor as described in this paper to measure the
QP trapping probability at the Al/AlMn overlap.

With the AlMn trapping and tunneling parameters measured, a further
optimization of the sensors will be possible. Once the sensitivity is
maximized, the SQUATs have a large number of use cases, primarily in the low-
energy, rare-event regime. The sensor offers two different pathways for event
detection:

  1. 1.

The antenna structure of the sensor itself allows this device to be sensitive
to single THz photons, allowing the devices to used in searches for QCD axion
DM such as BREAD [45]. Additionally, this device is sensitive to
electromagnetically-interacting fermionic light dark matter (LDM) via direct
absorption or scattering in the island itself as proposed in [46].

  2. 2.

The incredible sensitivity to athermal phonons in the device makes it well-
suited to detect LDM at the $\mathcal{O}(\mathrm{MeV})$ mass scale and sub-eV
bosonic DM such as dark photons. Absorption of LDM and far-IR photons is
possible in Si via a multiphonon excitation process [47, 48]. The LDM or
photon coupling strength can be increased by using a polar crystal substrate
[49, 50] or a low-bandgap semiconductor [51, 52]. In all of these cases,
sensitivity to acoustic phonons at the $\mathcal{O}(1-10\,\mathrm{meV})$ scale
is required.

While there already exists a large amount of competition in the space of
superconducting sensors for low-energy rare-event measurement [53, 54, 55, 56,
57, 58, 59], the device we have proposed potentially offers multiple orders of
magnitude of improvement in terms of absolute energy sensitivity beyond these
currently existing detectors. Furthermore, because each qubit sensor is
naturally read out individually, they inherently have a powerful background
discrimination tool by determining the location in the detector in which the
event occurred.

Acknowledgements—Thanks to Sasha Anferov for several useful conversations
about qubit modeling. This work was supported in part by the US Department of
Energy Early Career Research Program (ECRP) under FWP 100872. Caleb Fink is
supported by the Laboratory Directed Research and Development program of Los
Alamos National Laboratory. C. P. Salemi is supported by the Kavli Institute
for Particle Astrophysics and Cosmology Porat Fellowship. This work was
supported in part by the DOE Office of Science High Energy Physics QuantISED
program.

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##  Appendix A Resonator Readout Modes

When measuring the qubits, we can operate in two distinct modes: Amplitude and
phase readout. Recall that our voltage signal is the transmission signal of a
complex pulsed readout tone $V_{r}$

$V_{s}=S_{21}V_{r}.$ (19)

Amplitude Readout—Since $S_{21}$ can only take on values between zero and one,
it is clear to see that the maximum and minimum amplitudes that the signal
voltage could ever take on would be $|V_{s}|=|V_{r}|$ and $|V_{s}|=0$
respectively. This is accomplished in practice by designing a state separation
$2\chi$ that is much larger than the bandwidth of the readout ($2\chi\gg
f_{0}/Q$) such that one parity state is on resonance and the other state is
outside the BW of the readout. This can be seen visually on the IQ circle in
Fig. 5.

Phase Readout—The opposite readout technique would be to consider only the
phase of the signal tone. The maximum phase separation between the two states
occurs where the amplitude of the readout signal is the same, or rather

$|S_{21,e}|=|S_{21,o}|.$ (20)

The optimization problem becomes finding the $\chi$ to maximize the phase
separation between $S_{21,e}$ and $S_{21,o}$. Let us first recall the
definition of $S_{21}$ when limited by the quality factor of the coupling

$S_{21}(f,Q,f_{q})\approx
1-\left[1+2iQ_{c}\left(\frac{f-f_{q}}{f_{q}}\right)\right]^{-1}$ (21)

where the qubit resonance frequency is given by

$f_{q}(\chi_{0},n_{q})\approx\chi+f_{0}$ (22)

recalling that the effective dispersion $\chi=\chi_{0}\cos(\pi n_{q})$.

For simplicity, let us write $S_{21}$ as

$\displaystyle S_{21}$ $\displaystyle=\frac{-i\alpha}{1-i\alpha}$ (23)
$\displaystyle=\frac{\alpha^{2}-i\alpha}{1+\alpha^{2}}$ (24)

where $\alpha$ can be written for the even and odd states as

$\alpha_{e/o}=2Q\left(\frac{\pm\chi}{f_{0}\pm\chi}\right).$ (25)

Separating $S_{21}$ into IQ space we get that

$\displaystyle\text{I}\equiv\operatorname{Re}(S_{21})$
$\displaystyle=\frac{\alpha^{2}}{1+\alpha^{2}}$ (26)
$\displaystyle\text{Q}\equiv\operatorname{Im}(S_{21})$
$\displaystyle=\frac{-\alpha}{1+\alpha^{2}}$ (27)

and the phase angle is

$\tan(\phi)={\frac{\text{Q}}{\text{I}}}$ (28)

Figure 5: Left: Schematic of the two qubit readout scenarios shown on the IQ
circle. The Green dots represent the case of amplitude readout, in which the
state separation is much larger than the bandwidth of the readout such that
$S_{21}$ can take on the values of 1 or 0. The purple dots represent the case
of phase readout, in which the amplitude of $S_{21}$ is equal for the odd and
even parity states but the phase difference between the states is maximized.
Right: The corresponding points shown in frequency space, keeping the readout
frequency constant.

Considering two points on the IQ unit circle (see Fig. 5), the maximum
separation (maximum signal) of phase between the points occurs when they are
separated by a phase shift of $\pi/2$. Since the signal is symmetric about the
I axis, the maximum corresponds to $\tan{\pm\pi/4}=\pm 1$.

$\displaystyle\pm 1$ $\displaystyle=\frac{\text{Q}}{\text{I}}$ (29)
$\displaystyle\pm 1$ $\displaystyle=-\frac{1}{\alpha}$ (30) $\displaystyle\pm
1$ $\displaystyle=\frac{f_{0}\pm\chi}{2Q_{c}(\pm\chi)}$ (31) $\displaystyle 1$
$\displaystyle\approx\frac{f_{0}}{2Q_{c}\chi}$ (32)

since $f_{0}\gg\chi$. Finally, this gives us the optimum dispersion for a
given bandwidth,

$2\chi\approx\frac{f_{0}}{Q_{c}}.$ (33)

##  Appendix B Readout Fidelity

Given the readout modes described in Appendix A, we are prepared now to talk
about readout fidelity for this type of sensor design. The readout in either
case is a measurement of transmitted power across the resonator given a known
input tone power and phase. Alternatively, we can frame the signal amplitude
as the distance between even and odd parity states in IQ space. In the general
case, the distance between two points in IQ space is

$|\Delta S_{21}|^{2}=(I_{1}-I_{0})^{2}+(Q_{1}-Q_{0})^{2}$ (34)

For the amplitude case, measured power transmitted through the qubit $A$ for
an input power $P_{r}$ is

$A=|\Delta S_{21}|^{2}P_{r}=\left[(1-0)^{2}+(0-0)^{2}\right]P_{r}=P_{r}$ (35)

and for phase readout, where the signal is constant in I and only the sign of
Q changes, we find

$A=|\Delta
S_{21}|^{2}P_{r}=\left[\left(\frac{1}{2}-\frac{1}{2}\right)^{2}+\left(\frac{1}{2}+\frac{1}{2}\right)^{2}\right]P_{r}=P_{r}.$
(36)

Thus in terms of signal amplitude, there’s no difference between these readout
modalities.

Let’s now consider the general case for the optimization of the phase readout
(in which we assume that we have access to both the phase and amplitude of
$S_{21}$). Our optimization ensures that the even and odd states lie opposite
each other on the resonance circle. For a rotation angle of $\theta$ about the
point (I,Q) = (1/2,0), we find that

$\displaystyle I$ $\displaystyle=\frac{1}{2}\cos(\theta+\Omega)+\frac{1}{2}$
(37) $\displaystyle Q$ $\displaystyle=\frac{1}{2}\sin(\theta+\Omega)$ (38)

where $\Omega=\pm\pi/2$ for the even/odd state. We thus find that

$\displaystyle\Delta I$
$\displaystyle=\frac{1}{2}\left[\left(\cos\left(\theta+\frac{\pi}{2}\right)+1\right)-\left(\cos\left(\theta-\frac{\pi}{2}\right)+1\right)\right]$
(39)
$\displaystyle=\frac{1}{2}\left[-\sin(\theta)-\sin(\theta)\right]=-\sin(\theta)$
(40) $\displaystyle\Delta Q$
$\displaystyle=\frac{1}{2}\left[\sin\left(\theta+\frac{\pi}{2}\right)-\sin\left(\theta-\frac{\pi}{2}\right)\right]$
(41)
$\displaystyle=\frac{1}{2}\left[\cos(\theta)-(-\cos(\theta))\right]=\cos(\theta)$
(42)

which tells us that for the general case of a frequency shift from the optimal
readout frequency, for an optimized design,

$\displaystyle|\Delta S_{21}|^{2}$ $\displaystyle=(\Delta I)^{2}+(\Delta
Q)^{2}$ (43) $\displaystyle=\sin(\theta)^{2}+\cos(\theta)^{2}=1$ (44)

Thus the design optimized for phase readout will have the same signal
amplitude regardless of the readout frequency, as the two points will be
shifted an equal amount around the resonator circle but maintain unit distance
between them. Contrast this with amplitude readout in the non-optimized case,
where maximum signal is only achieved when the readout frequency is exactly
the frequency of one of the parity states.

One important caveat to this argument is that, unlike a lumped-element
resonator, the SQUAT cannot interact with more than one photon at a time. We
can account for this by adding a readout efficiency term to the signal part of
our sensitivity calculation, which accounts for the photons that are unable to
interact with the SQUAT after the initial photon has been absorbed. This
efficiency is given by the equation

$\epsilon_{r}=1-\exp\left(-\frac{\hbar f_{0}^{2}}{Q_{c}P_{r}}\right)$ (45)

The frequency dependence comes both from calculating the decay rate for a
given coupling factor $Q_{c}$ and from converting readout power into photon
number, leading to the conclusion that SQUAT efficiency improves for lower Q
or higher readout frequency. This also shows that for high readout power we
suffer from the limited number of photons interacting with the SQUAT.

Having established that our signal amplitude is always equal to our readout
power, we now consider the noise contribution to our signal in order to
determine readout fidelity. Assuming we are limited by Johnson-Nyquist noise
(which is uncorrelated to the signal), we find a noise power in an impedance
matched network of

$P_{n}=k_{B}Tf_{bw}\eta\left(\frac{hf}{k_{B}T}\right)$ (46)

where

$\eta(x)=\frac{x}{\exp(x)-1}+\frac{x}{2},$ (47)

$f_{bw}$ is the readout bandwidth of the measurement, and $T$ is the physical
temperature of the system. The role of $\eta$ is to introduce quantum
corrections from vacuum fluctuations that break the strict temperature
dependence of the noise; for low-enough physical system temperature, this is
reduced to the simpler, quantum-limited expression

$P_{n}\approx f_{bw}\frac{hf}{2}$ (48)

If we assume that the noise and signal are uncorrelated, we can continue to
use readout power as our signal to write down the signal-to-noise ratio, or
equivalently the inverse normalized signal variance, as

$SNR^{-1}=\sigma^{2}=\frac{2P_{n}}{\epsilon_{r}P_{r}}=\frac{2k_{B}Tf_{bw}\eta\left(\frac{hf}{k_{B}T}\right)}{P_{r}\left[1-\exp\left(-\frac{\hbar
f_{0}^{2}}{Q_{c}P_{r}}\right)\right]}.$ (49)

Here the factor of 2 comes from the fact that we have defined our signal as a
difference; we thus need two measurements to establish that difference, and we
pick up two noise contributions as a result. Note that given our signal in
this case is a difference, the power noise can thus can be treated as a
variance; this is different from an absolute measurement where noise power is
a strictly positive number. As discussed in the main body of the paper,
correlated noise such as TLS noise will modify the SNR, effectively by making
the noise level dependent on the operating point on the IQ circle.

Figure 6: Schematic of readout fidelity calculation.

We can now consider the fidelity $\mathcal{F}$ for our readout. Assuming
random non-stationary noise, we can calculate the fidelity utilizing the
square root of the overlap of two normal distributions (see Fig. 6) with
$\mu=0,1$ and variance defined in Eq. 49,

$\mathcal{F}=1-\exp\left(-\frac{SNR}{4}\right).$ (50)

This fidelity measure is the joint probability that the two states overlap,
and goes to zero for low SNR, 1 for high SNR. Repeated readouts increase SNR,
and thus in general, reduced readout bandwidth can be used to increase
fidelity. A plot of the readout fidelity as a function of readout power and
qubit resonance frequency for both a quantum limited and HEMT limited readout
can be seen in Fig. 7.

Figure 7: Expected readout fidelity ($\mathcal{F}_{\text{readout}}$) of sensor
as a function of readout power and resonance frequency for a quantum limited
readout (Top) and an HEMT limited readout, assuming a noise temperature of
$T_{n}=2\,\mathrm{K}$ (Bottom).

###  B.1 Sources of Phase Noise

In the main body of the text and in this appendix (Appendix B), we have
assumed the ideal case of perfect qubit stability and stationary, uncorrelated
noise additive with I and Q. In reality, there are two major noise sources we
have to worry about: charge noise and two-level system (TLS) noise. Charge
noise affects the point on the dispersion curve and thus the state separation
(we can think of this as a differential-mode phase noise). TLS noise affects
the mean frequency of the device; it can be thought of as a common-mode phase
noise.

In the case of charge noise, we can use the gate capacitance to estimate the
effect of a known gate voltage noise on the system; TLS noise will be
determined by the occupation and coupling of oxide states near the qubit to
the qubit islands.

##  Appendix C Quasiparticle Diffusion Model

This diffusion model describes the dynamics of a non-equilibrium quasiparticle
population in the qubit islands and how they are measured. As described in the
text above, this quasiparticle population arises from an impinging particle,
e.g. a phonon or photon, with energy greater than twice the superconducting
bandgap of the island. The incident particle breaks a Cooper-pair into highly
energetic quasiparticles, which then undergo a downconversion process into
phonons and lower energy quasiparticles until a quasi-stable population of
quasiparticles exists at the superconducting band edge [19]. A schematic
diagram of the full process can be seen in Fig. 1.

###  C.1 QP Diffusion in Island

Similar to what is done in [18, 60], we model the QP transport in the island
using a 2D diffusion model. We start with the diffusion equation

$\displaystyle\frac{\partial}{\partial
t}n(\mathbf{x},t)=D_{\text{island}}\nabla^{2}n(\mathbf{x},t)-\frac{n(\mathbf{x},t)}{\tau_{\text{island}}}+s\,,$
(51)

where $n(\mathbf{x},t)$ is the QP number density, $D_{\text{island}}$ and
$\tau_{\text{island}}$ are the diffusion coefficient and finite QP lifetime in
the island (Al in this case), and $s$ is a source of particles. This
differential equation can be solved in two dimensions; from [61], the fraction
of QPs that get trapped in the overlapping region is given by

$F_{c}^{2D}=\frac{2\rho_{1}}{\rho_{2}^{2}-\rho_{1}^{2}}\frac{I_{1}(\rho_{2})K_{1}(\rho_{1})-I_{1}(\rho_{1})K_{1}(\rho_{2})}{I_{1}(\rho_{2})\left[K_{0}(\rho_{1})+\lambda_{a}K_{1}(\rho_{1})\right]+K_{1}(\rho_{2})\left[I_{0}(\rho_{1})-\lambda_{a}I_{1}(\rho_{1})\right]},$
(52)

where $I_{i}$ and $K_{i}$ are modified Bessel functions of the first and
second kind and

$\displaystyle\rho_{1}\equiv\frac{R_{1}}{L_{d}},\quad\rho_{2}\equiv\frac{R_{2}}{L_{d}},\quad\lambda_{a}\equiv\frac{L_{a}}{L_{d}}.$
(53)

$R_{1}$ and $R_{2}$ are the effective radii of the island/trapping regions,
and $L_{d}$ and $L_{a}$ are two competing length scales: the diffusion length
and absorption length, given by

$\displaystyle
L_{d}\equiv\sqrt{D_{\text{island}}\tau_{\text{island}}}\approx\alpha_{\text{Al}}h_{\text{island}}$
(54) $\displaystyle
L_{a}\equiv\frac{D_{\text{island}}}{\nu_{abs}}\approx\frac{1}{p_{abs}}\frac{h_{\text{island}}^{2}}{\ell_{\text{trap}}},$
(55)

where $\alpha_{\text{island}}$ is a film specific scalar for the case that the
diffusion length is limited by the film thickness, $h_{\text{island}}$ is the
island thickness, $\ell_{\text{trap}}$ is the length of the overlapping region
between the two materials, we refer to $\nu_{abs}$ as the effective QP
absorption velocity, and $p_{abs}$ is the per QP transmission probability at
the interface.

We must estimate $p_{abs}$, which is a function of the island material, the
trapping material, and the thickness of the trap. For this model, we propose
to use AlMn for the trap and junction - since its $T_{c}$ can be tuned with
the amount of Mn added and AlOx layers can easily be formed on it to create
junctions [38]. For the simplest model of QP trapping, we model the change in
energy between the Al and AlMn trap as a step function between
$\Delta_{\text{Al}}$ and $\Delta_{\text{AlMn}}$. In reality, this step
function will be slightly smeared by the proximity effect. We assume that the
QPs that enter the trap are all at energy $\Delta_{\text{Al}}$ (since the QP
downconversion timescale is orders of magnitude faster than the average time
for a QP to reach the trap). The question then becomes: what is the
probability that the QP undergoes a scattering process and loses enough energy
such that it cannot travel back into the Al island? We are targeting a $T_{c}$
for the AlMn of roughly 10 times less than that of the Al islands. The QP
scattering lifetime was measured in AlMn by [37] to be
$\sim\,65\,\mathrm{ns}$. Additionally, in [62] it was shown that increasing
the disorder of Al films by adding Mn decreased the Al QP lifetime. Thus
depending on the percent of Mn added, this scattering time could be even
smaller.

Regarding the validity of the step-function model of the Al/AlMn interface,
there are two factors that we must consider. The overlapping regions between
the Island and the trap will have a global $T_{c}$ shift from the proximity
effect in the Z direction, and the boundary of the overlapping region in the
XY plane will have a local $T_{c}$ gradient from the longitudinal proximity
effect [63]. While the proximity effect in AlMn has not been well studied, at
least for Al in the dirty limit the coherence length has been measured [64] to
be on the order of $\xi_{\text{dirty}}\sim 100\,\mathrm{nm}$, which is a
reasonable order of magnitude estimate for AlMn. Since our films for both the
Island and trap will be $\mathcal{O}(100\,\mathrm{nm})$ thick, we expect this
region to fully proximitized and we do not need to consider any spacial
variation in the Z direction. Further, given the QP scattering time for AlMn,
this is equivalent to
$\lambda_{\text{qp}}\sim\mathcal{O}(100\,\mu\mathrm{m})$. Thus for our initial
designs with $\ell_{\text{trap}}\sim\mathcal{O}(1\,\mu\mathrm{m})$, we have
the situation where

$\xi_{\text{dirty}}\ll\ell_{\text{trap}}\ll\lambda_{\text{qp}}.$ (56)

For both of these reasons, the trapping interface can be well approximated as
a step function.

The scattering rate can be used to estimate the trapping probability for a
given trap thickness from Eq. 7, repeated below for completeness,

$p_{abs}=1-\exp\left[\frac{2t}{c_{s}\tau_{s}}\right],$ (57)

where $c_{s}$ is the sound speed in the trap, $t$ is the thickness and
$\tau_{s}$ is the QP-phonon scattering rate. Using a trap thickness of
$t=100\,\mathrm{nm}$ and scattering rate of $\tau_{s}=65\,\mathrm{ns}$ from
[37], the trapping probability for our Al/AlMn devices should be
$p_{abs}\approx 6\times 10^{-4}$. It is interesting to note that this value is
very close to that measured by [65, 66], who found that the trapping
probability for Al/W traps is $p_{abs}\approx 10^{-4}$. For this model we will
thus take the conservative approach and use $p_{abs}\approx 10^{-4}$ in our
calculations.

###  C.2 Quasiparticle Multiplication

When the quasiparticles enter the trap they will have energy of roughly
$\Delta_{\text{island}}$ (see Fig. 4 b). Once in the lower $T_{c}$ trap
however, these QPs will undergo a downconversion step into a lower energy QP
population. While some portion of the energy is lost to sub-gap phonons, there
is ultimately a QP number enhancement given roughly by [32]

$\displaystyle
G_{QP}\approx\frac{\Delta_{\text{island}}}{1.7\Delta_{\text{trap}}}\approx
7\,.$ (58)

###  C.3 Quasiparticle Diffusion in Trap

At this point, a stable population of QPs has developed in the trap. We then
model the trap QP transport using a 1D diffusion model, which is justified in
this case since the length of this region is much larger than the width or
film thickness. Solving Eq. 51 in one dimension, the fraction of QPs tunneling
across the junction is given by

$F_{c}^{1D}\approx\left[1+\frac{\ell_{\text{trap}}}{\ell_{JJ}}\frac{1}{\alpha_{\text{trap}}^{2}p_{\text{tunnel}}}\right]^{-1}\,,$
(59)

where $\alpha_{\text{trap}}$ is a scalar quantifying the diffusion in the thin
film thickness limited case, and $p_{\text{tunnel}}$ is the per-QP tunneling
probability at the junction. Since the relative amount of Mn in the AlMn
should be very low, the QP scattering diffusion length should be close to that
of a typical Al thin film. As such, we use a value of
$\alpha_{\text{AlMn}}\approx 570$ as measured in [66]. From [67] we estimate
the QP tunneling probability to be $p_{tunnel}\approx 1-10\,(\times 10^{-6})$
for pure Al. For the AlMn in this model, we adopt the conservative lower bound
of $1\times 10^{-6}$.

This step is where the true utility of this design comes into fruition. While
the QP tunneling probability across the Josephson junction is very small -
requiring a QP on average to impinge upon the junction $10^{6}$ times on
average before it tunnels, this setback is counteracted by confining the QP’s
to a volume that is at least 5 orders of magnitude smaller than the islands.

###  C.4 Signal Gain from Multiple Tunneling Events

Since there is no potential difference across the junction, QPs near the
junction will have multiple chances to tunnel before recombining. Using simple
geometric arguments, we estimate the characteristic time it takes a QP in the
trap to reach the junction as

$\displaystyle\tau_{qp\to
JJ}\approx\frac{4\text{V}_{\text{trap}}}{\ell_{\text{JJ}}v_{QP}}$ (60)

where $\text{V}_{\text{trap}}$ is the volume of the trap and
$\ell_{\text{JJ}}$ is the length of a single side of the square Josephson
junction, and $v_{QP}$ is the QP velocity at the operating temperature. We can
then define the average number of tunnels per-QP as

$\left<n_{\text{tunnel}}\right>\approx
p_{\text{tunnel}}\frac{\tau_{\text{life}}}{\tau_{qp\to JJ}}$ (61)

where $\tau_{\text{life}}$ is the QP lifetime in the junction material. While
this term depends on many material parameters, it is reasonable to expect each
QP will be measured $\mathcal{O}(10-100)$ times for an
$\mathcal{O}(1-10\,\mu\mathrm{m})$ sized trapping region. However, since this
effect needs to be studied, for this current model we make the conservative
assumption that $\left<n_{\text{tunnel}}\right>=1$.

###  C.5 Total Quasiparticle Collection Efficiency

The final fraction of collected QPs starting with a stable QP population in
the island is thus given by the product of Eqs. [52] [58] [59] [61]

$F_{\text{collect}}=F^{2D}_{\text{island}}\times G_{QP}\times
F^{1D}_{\text{trap}}\times\left<n_{\text{tunnel}}\right>.$ (62)

And example plot of how this collection fraction depends on the model
parameters is shown in Fig. 2 for the parameters given in the main text. Fig.
8 shows how the final energy resolution depends on the trapping and tunneling
probabilities.

Figure 8: Simulated phonon energy resolution for single QP sensitivity as a
function of island and trap length assuming negligible change in fill factor
as a function of island length. Shown in the hatched region is twice the
superconducting gap of Al, and the blacked out regions represent non-physical
design parameters. Top, Resolution as a function of $p_{\text{trap}}$ with
fixed $p_{\text{tunnel}}=1\times 10^{-6}$ for $p_{\text{trap}}=[1\times
10^{-3},1\times 10^{-4},1\times 10^{-5}]$ from left to right. Bottom,
Resolution as a function of $p_{\text{tunnel}}$ with fixed
$p_{\text{trap}}=1\times 10^{-4}$ for $p_{\text{tunnel}}=[5\times
10^{-7},1\times 10^{-6},5\times 10^{-6}]$ from left to right.

##  Appendix D Sensor Bandwidth

The energy measurement efficiency of the SQUAT will be ultimately limited by
the total sensor bandwidth. For the case of direct photon absorption, there
are two relevant timescales that need to be considered: the inverse readout
bandwidth $\tau_{\text{BW}}$, and the average time between quasiparticle
tunneling events $\tau_{\text{tunnel}}$. While the bandwidth is fixed, we note
that the average tunneling time is energy dependent. For the case of phonon
absorption there is an additional time constant of the average phonon
collection time $\tau_{\text{phonon}}$ as given by Eq. 6.

We consider first the average time per quasiparticle tunnel. While this is
technically a two stage process, the QPs must travel through the island before
getting trapped and then must tunnel across the junction, the trapping time is
at least four orders of magnitude faster than the tunneling time. As such,
since the process is Poissonian, we can model the sensor response to an
impulse of energy with a single pole exponential with a time constant set by
$\tau_{\text{tunnel}}$. For a single quasiparticle in the trap, the
characteristic time that it will take to diffuse to the trap is given by Eq.
60. The average time between tunneling events ($\tau_{\text{tunnel}}$) will
scale with both the total number of QPs $N_{\text{QP}}$ and the average number
of tunnels per QP $\left<n_{\text{tunnel}}\right>$, as

$\displaystyle\tau_{\text{tunnel}}=$ $\displaystyle\frac{\tau_{qp\to
JJ}}{N_{\text{QP}}(E)\left<n_{\text{tunnel}}\right>}$ (63) $\displaystyle=$
$\displaystyle\frac{\tau_{qp\to
JJ}^{2}}{p_{\text{tunnel}}N_{\text{QP}}(E)\tau_{\text{life}}}.$ (64)

Since the total number of quasiparticles is a function of the deposited
energy, as given by Eq. 18, this means that the average time between tunneling
events must also be energy dependent. We thus get that the fraction of
quasiparticles in the trap that tunnel across the junction per unit time is
described by

$N_{\text{tunnel}}=\frac{1}{\tau_{\text{tunnel}}}e^{\left(-\frac{t}{\tau_{\text{tunnel}}}\right)}.$
(65)

In the limit of infinite readout bandwidth, the fraction of measured tunneling
events would be equal to the fraction of quasiparticles created. However, the
SQUAT has a finite readout bandwidth; in our current design
$\tau_{\text{BW}}=1\,\mu\mathrm{s}$. In the worst case scenario, the
consequence of this is that any consecutive tunneling events occurring in a
shorter time interval than $2\tau_{\text{BW}}$ will not be accurately measured
and we will get on average a measurement rate of $2\tau_{\text{BW}}$. In this
case we get that the fraction of measured QPs is

$N(E)_{\text{measured}}=\int_{2\tau_{BW}}^{\infty}N(t,E)_{\text{tunnel}}\,dt,$
(66)
$N(E)_{\text{measured}}=\begin{cases}\exp{\left(-\frac{2\tau_{\text{BW}}}{\tau(E)_{\text{tunnel}}}\right)},\quad&\tau_{\text{tunnel}}>2\tau_{\text{BW}}\\\
\exp{\left(-1\right)},\quad&\tau_{\text{tunnel}}\leq
2\tau_{\text{BW}}\end{cases}$ (67)

The above expression describes the response of the sensor to a direct energy
absorption into the island, e.g. from a photon, and a plot of the measurement
efficiency (taking into account the total quasiparticle efficiency model
described in App. C) versus particle energy can be seen in Fig. 9. For the
case of phonon absorption, we must take also into account the pulse shape of
the phonon energy. The fraction of QP’s tunneling across the junction is now
given by the convolution of Eq. 65 and Eq. 6. In this case we get that the
total fraction of measured quasiparticles is given by

$N(E)^{\text{phonon}}_{\text{measured}}=A\left(\frac{e^{-2\lambda_{2}\tau_{\text{BW}}}}{\lambda_{2}}-\frac{e^{-2\lambda_{1}\tau_{\text{BW}}}}{\lambda_{1}}\right),$
(68)

where

$\displaystyle A=$
$\displaystyle\frac{\lambda_{1}\lambda_{2}}{\lambda_{1}-\lambda_{2}},$ (69)
$\displaystyle\lambda_{1}=$ $\displaystyle\frac{1}{\tau_{\text{phonon}}},$
(70) $\displaystyle\lambda_{2}=$
$\displaystyle\frac{1}{\tau_{\text{tunnel}}},\quad\tau_{\text{tunnel}}>2\tau_{\text{BW}}$
(71) $\displaystyle\lambda_{2}=$
$\displaystyle\frac{1}{\tau_{\text{BW}}},\quad\tau_{\text{tunnel}}\leq
2\tau_{\text{BW}}$ (72) . (73)

From Fig. 9 we can see that in the case that the phonon pulse is slower than
the tunneling time and inverse readout bandwidth, the SQUAT readout efficiency
will saturate. A simulated typical phonon pulse is shown in Fig. 10 as well as
a depiction of what the measurement will physically look like in terms of the
tunneling rate observed by the sensor.

Figure 9: Estimated measurement efficiency due to the finite readout bandwidth
for a geometry of $\ell_{\text{island}}=100\,\mu\mathrm{m}$ and
$\ell_{\text{trap}}=4\,\mu\mathrm{m}$, assuming
$\tau_{\text{phonon}}=2\,\mu\mathrm{s}$. Figure 10: Example of characteristic
event signal. a) Analytic components of a typical phonon event. Note the
component pulses are arbitrarily scaled. b) Total collected quasiparticles as
a function of integration time. c) Analytic log scaled pulse shape for a
typical event. d) Simulated parity switching for the same pulse shown above.
e) Zoomed in plot of simulated parity switching.

##  Appendix E SQUAT geometric tuning and simulation

To achieve the optimal device parameters for a given readout method (as
detailed in Appendix B), we must tune the geometry of the qubit and its
coupling to the feedline. In this appendix we show how tuning various
geometric parameters affects the readout parameters as simulated with ANSYS
HFSS, a finite element RF modeling software [68].

Each qubit is measured with a modal network simulation type. Just as the
SQUATs will be in actual measurements, they are stimulated with a transmission
measurement along the feedline. This is achieved in simulation with ports that
inject modes at swept frequencies across the qubit resonance. An example
electric field plot for a qubit driven on resonance is shown in Fig. 11.

The trends in frequency, quality factor, and charge dispersion resulting from
the HFSS simulations are shown in Fig. 12. The uncertainties shown there
reflect how much variation in those parameters we expect based on a single
simulation method, and do not include systematic uncertainties associated with
differences between simulation and the real device. We expect the frequencies
and quality factors to be fairly accurate, but the dispersion’s exponential
dependence on $E_{J}/E_{C}$ (see Eq. 3) means that small discrepancies in the
simulated versus actual $E_{J}$ and $E_{C}$ can lead to large differences in
$2\chi$.

To see the magnitude of this effect, we ran an ANSYS Maxwell simulation to get
another estimate for $E_{C}$ for some of the data points shown in the plots.
The difference in $E_{C}$ was up to a factor of $\sim 2$, leading to shifts of
$2\chi$ by up to two orders of magnitude. However, although the two simulation
methods give different results, the overall dispersion trends with geometry
should be consistent between the methods. Thus the $2\chi$ curves in Fig. 12
should be interpreted for the geometric trends rather than for the values
themselves. Once devices have been fabricated and tested, we will benchmark
our simulations to the measured values.

Figure 11: Electric fields of a SQUAT that is driven at the qubit resonance as
simulated in ANSYS HFSS.

Figure 12: SQUAT parameters vary with the device geometry. In each plot, only
one geometric parameter was varied. The uncertainty bands reflect the level of
variation within one simulation method. See the text for further discussion of
systematic uncertainty. Top left, variation of the coupling distance between
the qubit and the feedline and ground plane. Top right, variation of the gap
between the qubit islands. Bottom left, variation of the size of the qubit
islands. Bottom right, variation of the Josephson inductance, which is a
function of the junction size and thickness.