††thanks: Correspondence and requests for materials should be addressed to NF
(email<EMAIL_ADDRESS>or KDO (email: osborn@lps.umd.edu).††thanks:
Correspondence and requests for materials should be addressed to NF (email:
<EMAIL_ADDRESS>or KDO (email: osborn@lps.umd.edu).

# Dual-Resonator Kinetic-Inductance Detector for Distinction between
Signal and 1/f Frequency Noise

N. Foroozani  Laboratory for Physical Sciences, University of Maryland,
College Park, MD 20740, USA  Department of Physics, University of Maryland,
College Park, MD 20742, USA  B. Sarabi  Laboratory for Physical Sciences,
University of Maryland, College Park, MD 20740, USA  S. H. Moseley  NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  T. Stevenson  NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  E. J. Wollack  NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  O. Noroozian  NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  K. D. Osborn
Laboratory for Physical Sciences, University of Maryland, College Park, MD
20740, USA  Joint Quantum Institute, University of Maryland, College Park, MD
20742, USA

###### Abstract

Astronomical Kinetic Inductance Detectors (KIDs), similar to quantum
information devices, experience performance limiting noise from materials. In
particular, 1/f (frequency) noise can be a dominant noise mechanism, which
arises from Two-Level System defects (TLSs) in the circuit dielectrics and
material interfaces. Here we present a Dual-Resonator KID (DuRKID), which is
designed for improved signal to noise (or noise equivalent power) relative to
1/f-noise limited KIDs. We first show the DuRKID schematic, fabricated
circuit, and we follow with a description of the intended operation, first
measurements, theory, and discussion. The circuit consists of two
superconducting resonators sharing an electrical capacitance bridge of 4
capacitors, each of which hosts TLSs. The device is intended to operate using
hybridization of the modes, which causes TLSs to either couple to one mode or
the other, depending upon which capacitor they reside in. In contrast, the
intended KID signal is directed to an inductor, and due to hybridization this
causes correlated frequency changes in both (hybridized) modes. Therefore, one
can distinguish photon signal from TLS frequency noise. To achieve
hybridization, a TiN inductor is current biased to allow tuning of one bare
resonator mode into degeneracy with the other and measurements show that the
intended resonator modes frequency tune and hybridize as expected. The
interresonator coupling and unintentional coupling of the 2 resonators to
transmission lines are also characterized in measurements. In the theory,
based on a quantum-information-science modes, we calculate the 4-port S
parameters and simulate the 1/f frequency noise of the device. The study
reveals that the DuRKID can exhibit a large and fundamental performance
advantage over 1/f-noise-limited KID detectors.

###### pacs:

##  I I. Introduction

Kinetic Inductance Detectors (KIDs) DayPK ; Mazin ; NoisePropGao ; AIP  for
astronomy contain resonators which experience noise, related to noise in
qubits for quantum information processing (QIP) PRL123 ; Oliver ; NATURE ;
APL2008 ; McRae . For KID-based millimeter and submillimeter-wave astronomical
imaging systems Maloney2010 ; Monfardini2011 ; Galitzki2016  the signal band
of interest has frequency noise which appears as a strong 1/f-like noise
spectrum. This necessitates implementing scan strategies and signal modulation
techniques to mitigate their influence on the final observational data
products in order to achieve the underlying device sensitivity Wright1996 ;
Maino2002 ; Miller2016 . More generally speaking, if unaddressed in such
applications – the presence of residual low frequency variations introduces
correlated noise which can lead to increased effective noise levels and
systematic artifacts that include image striping and amplitude calibration
errors. While significant progress has been made over the past two decades in
reducing low frequency noise in KIDs, developing sensitive detectors with
lower noise remains the primary technical challenge for many current and
future astrophysical observatories omid1 ; omid2 ; omid3 .

A major limitation to the performance in KIDs and qubits arises from the same
defect type, that is a two-level system (TLS) defect residing in a dielectric
or at a material interface within the device. KIDs are perturbed in resonance
frequency by the photon signal, however, TLSs meanwhile induce 1/f frequency
noise which causes difficulties in signal detection. Similarly, in QIP the
qubit transition frequency experiences frequency noise from TLSs PRL123 ;
PRB015 . This noise necessitates recalibration of pulse-driven qubit logic
gates PRA016 ; PRL0052 . Recent studies of TLS noise mechanisms for KIDs and
qubits have shown the importance of a more in-depth understanding of TLSs
PRL123 ; PRB015 ; APL90-2007 ; PRL0052 ; APL92-2008 . Two recent theories
explain that the interaction between TLSs is responsible for 1/f frequency
noise whereby low-frequency thermally fluctuating TLSs influence high-
frequency (near-resonance) TLSs PRB91-2015 ; Alex1 , which in turn interacts
with the high frequency superconducting mode PRL010 ; PRL004 . Traditionally
there is a strategy to minimize the influence of TLSs in KIDs and qubits
through material choices Steffen ; Chu ; Burnett ; Neill ; SRON  and geometry
optimization AIP . While increasing the measurement power lowers the frequency
noise from TLSs, it also generates quasiparticles. This leads to obfuscation
of the KID photon signal, because the signal is also produced by generating
quasiparticles refforQP .

Recognizing that practical limitations to frequency noise in KIDs remain, we
propose to tackle the problem differently. Here we describe a detector circuit
that can allow one to distinguish photon signal from TLS frequency noise. The
circuit uses two resonators coupled with an electrical capacitance bridge. It
is a type of KID as the design allows a shift of an inductor value caused by a
photon signal, similar to a standard KID. Here the inductance shift causes
(correlated) frequency shifts of both resonators. However, in the proposed KID
configuration the TLSs can only cause uncorrelated frequency shifts in the
resonator modes, because it merely assumes that high-frequency TLS noise from
different capacitors will be independent. We use established circuit quantum
electrodynamics (cQED) analysis to model the circuit, which includes TLSs as a
noise source. In the next section we present the circuit schematic for the
device, named a Dual-Resonator KID (DuRKID), which has electrical-bridge
coupling of resonators. Fabrication, details of the DuRKID and the data from
the device, including frequency tuning and hybridization are shown in this
section. In section III, an analysis of the DuRKID is described with
representative TLSs. The analysis is used to extract the dielectric’s loss
from resonator data, and with a new method, extract unintended resonator-to-
transmission line coupling. We conclude in section IV.

##  II II. Device design, method of noise distinction and frequency tuning
measurement

TLSs shift the resonator frequency, as does the photon signal, in standard
KIDs. The former effectively causes a time change in the frequency dependent
dielectric constant, and the latter causes an increase in a superconductor’s
kinetic inductance Mazin2010 . We propose the DuRKID, a device with two
resonator modes in contrast to standard KIDs, to improve the noise sensitivity
that is hampered by TLS-induced frequency shifts. In the DuRKID, we find that
noise from TLSs should be qualitatively different from the signal such that
the two are distinguishable and the effective signal sensitivity is improved.

Figure 1: (a, b) Resonators with two modes (an extra mode relative to a
standard MKID). If the modes are far detuned (in frequency), the modes are
effectively decoupled from each other and also experience different TLS noise
environments. (a) Standard-coupled resonator circuit. When the two modes are
hybridized, TLS noise from both capacitors will equally influence the two
resonator modes: $E_{a}(t)=E_{b}(t)$ and $E_{a}(t)=-E_{b}(t)$. (b) Circuit
schematic of Dual-Resonator KID (DuRKID). An electrical bridge of nominally
equal capacitances ($C_{1}-C_{4}$) creates capacitance for the modes. Spatial
resonator modes A and B, with voltage components $V_{A}$ and $V_{B}$, are
hybridized for equal $L_{A}$ and $L_{B}$ by their stray coupling. For the
hybridized mode where $V_{A}(t)$ and $V_{B}(t)$ are in-phase, the electric
field in capacitors 1 and 3 adds $E_{+}\propto\langle a+b\rangle$, but in 2
and 4 the electric fields are out-of-phase and cancel $E_{-}\propto\langle
a-b\rangle$. For the other hybridized mode, the location of constructive and
destructive interference is swapped. This gives selective sensitivity to the
TLS noise sources according to the capacitor which hosts it. In this schematic
a dc current applied to port 3 may tune inductance $L_{B}$. (c) Optical image
of the fabricated DuRKID with the same (opposite) orientation as Device 1 and
3 (2) in Fig. 3(a). The materials include: Si substrate (black), aluminum
(yellow), TiN (dark red), and silicon nitride dielectric (obscured).

We begin with a discussion of a standard-coupled resonator pair, where LC
resonators are nominally the same frequency (degenerate) prior to coupling.
This reference case is shown in Fig. 1(a), where the resonators are
inductively coupled through M (although they could instead be coupled through
a capacitor for the same effect). In the schematic, each bare (uncoupled)
resonance mode frequency, $f_{r,A}$ and $f_{r,B}$, is created by an individual
inductor, $L_{A}$ and $L_{B}$, respectively. Without resonator coupling, a
given TLS in the capacitor of each resonator will couple solely to the field
of that resonator, and the resonators would act as two separate KIDs. For each
mode, one TLS causes a dispersion (frequency shift): $\Delta
f_{r}=g^{2}/(f_{TLS}-f_{res})$, where $f_{TLS}$ is the TLS frequency. In
contrast, if the resonators are set to the same bare frequency, coupling will
hybridize them and a typical TLS (which is weakly coupled) on either capacitor
will interact with both modes through the same coupling term $g$, and hence
cause correlated noise in the modes. One case that is especially troublesome
is for TLSs between the mode frequencies, since their motion may cause a
correlated but quantitatively different drift in both modes (from the TLS
frequency drift and dispersion). However, a signal fed to one of the inductors
will also cause a correlated change in the frequencies. Thus, in a standard-
coupled resonator pair, hybridized modes will exhibit correlated TLS noise
that provides no obvious method of distinction from signal.

In this work we propose to use an alternative two-resonator device that we
designate a DuRKID. To differentiate the TLS noise from signal, the device
uses an electrical bridge for coupling between the modes (see Fig. 1(b)),
which breaks the degeneracy between the signals of interest and the TLS
induced fluctuation noise. As in the standard-coupled resonator circuit, we
have bare (uncoupled) resonance mode $A$ and $B$ comprised of an individual
inductor, $L_{A}$ and $L_{B}$, respectively, but the bridge provides a shared
set of four capacitors. The capacitors are nominally equal such that both
modes access the same nominal capacitance. The coupling according to the
figure shows that mode $A(B)$ is intentionally coupled to the transmission
line with ports 1 and 2 (3 and 4). However, when hybridized, each mode is
coupled to both transmission lines. To create degeneracy between these bare
modes, $L_{B}$ is designed to be current biased. For example, by using port 3
we can decrease the higher mode frequency of device 2 (see Figs. 1(c) and
2(a)). For the sake of pedagogy, in this work we consider this inductor as
analogous to the KID readout inductor because it can be changed in frequency,
similar to the way a KID has an inductor that is changed in frequency by the
signal.

As mentioned above, the bridge-coupled resonator pair acts differently than a
standard-coupled resonator pair. A TLS in a bridge of the capacitor is equally
shared by both modes before hybridization. However, once hybridized, one field
mode amplitude is zero in 2 of the 4 capacitors. Likewise, it becomes zero in
the other 2 capacitors for the other hybridized mode. Thus, a given TLS will
only frequency-shift (disperse) one of the hybridized modes in a bridge
coupled resonator. In contrast, a change in the frequency from incident
photons will cause a correlated shift to both modes such that it can be
distinguished from TLS noise.

A fabricated DuRKID is shown in Fig. 1(c). Many DuRKIDs are fabricated
together on a high-resistivity ($>$20 k$\Omega$.cm) silicon wafer. The base
superconducting layer is a TiN-Al bi-layer, with 15 nm thick TiN and a 250 nm
thick Al. This bilayer was patterned followed by the removal of Al to leave
TiN bare in certain locations to provide kinetic inductances $L_{kA},L_{kB}$
and $L_{kC}$ as part of the total inducances $L_{A},L_{B}$ and $L_{C}$,
respectively. The dielectric layer for the capacitor bridge is a high-density
silicon nitride (SiNx) film with a nominal thickness of 275 nm deposited using
PECVD. pike . Then via holes are made using SF6 reactive ion etching. A 250 nm
sputtered Al film forms the counter electrode of the capacitor bridge. After
patterning the counter electrode, excess SiNx is removed from most of the
sample area to allow access to the leads on the base layer. Three devices are
fabricated per chip, as shown in the device layout of Fig. 2(a). Devices 1 and
3 are not rotated relative to Fig. 1(b-c) such that they share galvanic
coupling to port 1. Device 2, the center device, is the only device that is
galvanically coupled to port 3, due to a $\pi$ rotation of the device relative
to Fig. 1(c).

Figure 2: (a) Device layout of three devices on a chip with two feedlines for
each device. One resonator of each device is tunable, using the nonlinear
inductance $L_{B}$. The LC resonances are mainly defined by a 0.6 nH inductor
and trilayer capacitors with C=3.14 pF, containing a 275 nm thick silicon
nitride dielectric film. (b) Measured transmission $|S_{21}|$ and $|S_{43}|$
for devices 1, 2, and 3 at $I_{b}=0$.

The devices are cooled down to 20 mK in a dilution refrigerator. The
transmission data $\left|S_{21}\right|$ and $\left|S_{43}\right|$ for all
three devices are shown in Fig. 2(b) when $I_{b}=0$. The devices were
intentionally designed with differing offsets between the two resonator modes
at zero bias current, $I_{b}$. The offset (detuning) between two modes for
each dual resonator is measured at $\delta_{1}=9$ MHz, $\delta_{2}=97$ MHz and
$\delta_{3}=128$ MHz, respectively. As planned, the tunable resonance at zero
bias was at a higher frequency than the non-tunable resonance in all three
devices. All devices tune with bias current to reach degeneracy. For device 3
the tuning for degeneracy is large $\sim$128 MHz which is enabled by large
kinetic inductance change from the unbiased inductance per square of 56 pH.
The fractional frequency tuning $\delta f_{r}/f_{r}=3.5\%$ is approximately an
order of magnitude higher than achieved with a magnetic-field tuned CPW
resonator Vissers2015 .

Figure 3: Data on device 2. Transmission spectra, $|S_{43}|$, of tunable
resonator with applied bias current up to 210 $\mu$A, measured at 20 mK. (a)
Each color represents one $|S_{43}(f)|$ trace at a separate DC bias. The
frequency of the resonator B, as shown by the $S_{43}$, shifts down from 3.635
GHz to 3.52 GHz, by 120 MHz, when the minimum bias is applied for full
hybridization. (b) $S_{43}(f,I_{b})$ plot that emphasizes resonance frequency
versus bias over the entire applied current range $I_{b}$. (c) Measured
transmission $|S_{43}|$ of device B versus frequency $f_{B}$ and tuning
current $I_{b}$ near degeneracy shows avoided crossing due to hybridization
with the fixed mode at $f_{A}$=3.533 GHz. From the analysis, a coupling of
$\Omega_{AB}/2\pi=5.1$ MHz was obtained.

Data on device 2 transmission $|S_{43}|(f,I_{b})|$ is shown in Fig. 3(a) and
(b), which shows frequency tuning of the resonance versus bias current. This
device has an independent bias line and thus a precisely known bias current.
The applied bias current shifts the higher mode toward the lower resonance and
the tunable resonator disappears at a bias $I_{b,max}=210\mu A$, indicating a
change to the normal state, likely at the wiring vias. Although most of the
geometric inductor length within each resonator is provided by Al, the kinetic
inductance $L_{kA}$ and $L_{kB}$ make for approximately $85\%$ of the total
inductance $L_{A}$ and $L_{B}$. At temperatures much smaller than the
superconducting critical temperature $T_{c}$, the nonlinear response of
$L_{k}$ to $I_{b}$ can be expanded as
$L_{k}(I_{b})=L_{k}(0)(1+(\frac{I_{b}}{I^{\prime}})^{2}+...)$, where
$I^{\prime}$ sets the scale of nonlinearity. By considering up to quadratic
order in $L_{k}(I_{b})$, we extract from a fit to $L_{kB}$ that
$I^{\prime}=789\mu A$ (See Fig. 6 in Appendix A).

$|S_{43}|$ at different $I_{b}$ (Fig. 3(c)) shows an avoided crossing due to
hybridization with the A mode. We also measured the cross coupling from the
input port of one transmission line to the output port of the other, $S_{41}$
and $S_{23}$. The transmission $|S_{41}|$ for hybridization degenerate modes
is on the order of -10dB (not shown), indicating some internal loss in the
resonators but also sufficiently high coupling of the resonators to their
intended transmission lines. The capacitors of the bridge were carefully
designed in layout to be equal. As a result, the mutual inductance is likely
the dominant coupling mechanism. We note that the splitting from mode coupling
must be larger than other line widths for the device to operate as intended.
To further analyze the data quantitatively, we applied the two-resonator
model, to fit the transmission data, $S_{21}$ and $S_{43}$ with the fitting
procedure explained in the Appendix. From Fig. 3(c) we obtained the inter-
resonator coupling of $\Omega_{AB}/2\pi=5.1$ MHz. This coupling agreed with
simulation, within a factor of 2. We then use this as a fixed parameter in the
device model for two separate fitting procedures. In the first fit procedure
we fit both standard transmission data at the same time, according to a
procedure as described in the Appendix B. From this we extract the internal
Q-factor of 2750 and 3100 for resonators A and B respectively. Viewed as a
material loss tangent tan$\delta=1/Q_{i}=(3.4\pm 0.2)\times 10^{-4}$, the
values match our expectations for recipe of the PECVD SiNx used. It has a
lower loss than typical AlOx by approximately an order of magnitude
Martinis2005PRL . In other work, silicon-on-insulator (SOI) is used for higher
Q-factor resonators crystal ; Ed .

##  III III. C-QED model of two resonators coupled to the same TLS

###  III.1 A. Theoretical Model

For a theoretical model of the DURKID, we use a standard quantum information
science method (c.f. Ref. Gardiner ). Fig. 4(a) shows the system diagram for
the model, which has two resonator modes. We describe it with two resonator
modes, each with its own two-port transmission line. Both modes are
simultaneously coupled to a TLS noise source. Only one TLS and coupling is
shown, but the model generally has TLSs which have different effective
couplings to a hybridized resonator mode, due to their symmetry created by
inter-resonator mode coupling $\Omega_{AB}$. $\kappa_{A}$ and $\kappa_{B}$ are
the coupling rates of resonators A and B, respectively, to their intended
2-port transmission line. The coupling rates of the resonators to the other
(unintended) transmission lines are represented by $\kappa_{Ax}$ and
$\kappa_{Bx}$, respectively. In this analysis we focus on TLSs that are far
enough from resonance to not be saturated by the drive field. A coupled TLS
pair could cause a different qualitative noise spectra (c.f. Ref. PRB015 ),
but also the method of protection is qualitative and thus the interaction
within a capacitor should not matter. Generally we consider that the
resonators are coupled to the i-th TLS with a resonance coupling constant of
$g_{i}$. We represent the TLSs by the spin operator $\sigma_{i}^{z}$ due to
the standard analogy between TLS theory and a two state system (with pseudo
spin-$\frac{1}{2}$). The Hamiltonian for the system interacting with two
transmission lines has the form

$\displaystyle H_{sys}$
$\displaystyle=\hbar\omega_{A}a^{\dagger}a+\hbar\omega_{B}b^{\dagger}b+\sum_{i=1}^{N}\varepsilon_{i}\sigma_{i}^{z}+\sum_{j=1}^{M}\varepsilon_{j}\sigma_{j}^{z}$
(1) $\displaystyle+\hbar\varOmega_{AB}(b^{\dagger}a+a^{\dagger}b)$
$\displaystyle-i\hbar\sum_{i=1}^{N}g_{i}^{(n)}(\sigma_{i}^{+}(a+b)+\sigma_{i}^{-}(a^{\dagger}+b^{\dagger}))$
$\displaystyle{-i\hbar\sum_{j=1}^{M}g_{j}^{(m)}(\sigma_{j}^{+}(a-b)+\sigma_{j}^{-}(a^{\dagger}-b^{\dagger})),}$

Figure 4: (a) Theoretical model for the DuRKID, including 2 resonators, 4
ports (fully shown), and TLSs (only 1 TLS shown). In this simulation, the
coupling parameters are $\kappa_{A}/2\pi=\kappa_{B}/2\pi=2$ MHz,
$\gamma_{TLS}/2\pi=2$ MHz, $\varOmega_{AB}/2\pi=5$ MHz and $g/2\pi=1$ MHz.
Resonator A is intentionally coupled to transmission line of port (1) and (2),
and resonator B to port (3) and (4). (b-f) Transmission analysis for
$(1\rightarrow 2$ or $3\rightarrow 4)$ with $Z_{A},Z_{B}\rightarrow 0$ (zero
impedance between transmission line halves). Arrows (red or blue) show
resonator mode frequency (solid curve transmission minima) change due to the
TLS, relative to the mode frequency without the TLS (dashed curve transmission
minima). (b-c) Single-photon transmission spectrum for two different devices
with same parameters, $(\omega_{A}/2\pi=\omega_{B}/2\pi=3.533$ GHz) with a
single degenerate TLS $(\omega_{tls}=\omega_{A}=\omega_{B})$. (b) Standard-
coupled resonators which are hybridized. Note that the frequency shift
(dispersion) occurs equally (correlated) in both modes (blue relative to black
minima, also see blue arrows). (c) DuRKID in intended (hybridized) mode with a
TLS. Here the TLS leaves one of the modes undispersed. (d) DuRKID with a large
drive (classical field) amplitude of $\overline{n}=50$ photons and a TLS
frequency of $\omega_{TLS}/2\pi=3.540$ GHz. Here the dispersion is present,
related to (c), and the TLS is symmetry-protected from saturation. Undispersed
modes occur in DuRKID due to the absence of fields in certain capacitors
according to the symmetry of the hybridized modes (see Figure 1(b)). (e) The
spectra for the bridge-type resonator device in the presence of an added
signal to one inductor (in contrast to other sub-figures which use additional
TLS(s)). Here the hybridization of the modes change as seen in different
resonator transmission amplitudes. The resonator frequency modes shift
downward together (as correlated modes), in contrast to the case of single-TLS
pertubation (c or d). (f) Spectra with two TLSs, where there is one from each
group (where each mainly influences only one of the resonator modes). This
case is used in section III B, for analysis with two TLS noise sources. A TLS
in each TLS group disperses each hybridized mode, but it is distinguishable
from the (correlated-mode) signal. In this special case there is no change of
mode hybridization due to the way that the TLSs oppositely disperse (shift)
the resonator modes.

this is similar to the one-cavity Jaynes-Cumming Hamiltonian model, which
unlike the standard Jaynes-Cummings Hamiltonians Walls ; Fink2008 ; Fink2009 ,
includes 2 modes (with operators a and b) with many TLSs (with raising and
lowering operators $\sigma_{i}^{+}$ and $\sigma_{i}^{-}$). The ith TLS has
energy of
$\varepsilon_{i}=\frac{\hbar\omega_{TLS,i}}{2}=\sqrt{\varDelta_{i}^{2}+\Delta_{0,i}^{2}}$,
where $\varDelta_{0}$ and $\varDelta$ are tunneling energy and the asymmetry
energy in the double-well potential model. TLS resonance coupling to the
resonator field is
$g_{i}=\frac{\varDelta_{0,i}}{\varepsilon_{i}}p_{i}cos\theta_{i}\sqrt{\frac{\omega}{2\epsilon_{r}\epsilon_{0}\hbar
V}}$, where $p_{i}$ is the magnitude of the dipole moment, $\theta_{i}$ is the
angle between $p_{i}$ and the applied electric field $\bf E$, V is the
dielectric volume, and $\epsilon_{r}$ is the relative permittivity bahman2 .
Though a given TLS in a capacitor is equally shared by both modes before
hybridization, after hybridization the two mode fields become
$E_{+}{\color[rgb]{0,0,0}\propto}(a+b)$ and
$E_{-}{\color[rgb]{0,0,0}\propto}(a-b)$, which are zero in certain capacitors
as mentioned above. It becomes zero in 2 of the capacitors for one hybridized
mode and similarly zero in the other 2 capacitors of the other hybridized
mode. We find that this approximation is valid even for a TLS throughout the
eigenmode width. For the two modes we thus effectively have only the coupling
to N TLSs (group n) in capacitor pair ${C_{1},C_{3}}$, and M TLSs (group m) in
${C_{2},C_{4}}$. Using a standard theoretical procedure inputout  for the
resonators A and B, interacting with the input and output fields as the heat
bath, the Heisenberg equations of motion can be written as:

$\displaystyle\frac{d}{dt}\left\langle a\right\rangle=$
$\displaystyle-i\omega\left\langle
a\right\rangle=-\frac{i}{\hbar}\left\langle[a,H_{sys}]\right\rangle-(\kappa_{A}+\gamma_{A}+\kappa_{Ax})\left\langle
a\right\rangle$ (2) $\displaystyle+\sqrt{\kappa_{A}}\left\langle
a_{1,in}\right\rangle+\sqrt{\kappa_{A}}\left\langle
a_{2,in}\right\rangle+\sqrt{\kappa_{Ax}}\left\langle b_{3,in}\right\rangle$
$\displaystyle+\sqrt{\kappa_{Ax}}\left\langle
b_{4,in}\right\rangle-\sqrt{\kappa_{A}\kappa_{Bx}}\left\langle
b\right\rangle-\sqrt{\kappa_{Ax}\kappa_{B}}\left\langle b\right\rangle,$

$\displaystyle\frac{d}{dt}\left\langle b\right\rangle=$
$\displaystyle-i\omega\left\langle
b\right\rangle=-\frac{i}{\hbar}\left\langle[b,H_{sys}]\right\rangle-(\kappa_{B}+\gamma_{B}+\kappa_{Bx})\left\langle
b\right\rangle$ (3) $\displaystyle+\sqrt{\kappa_{B}}\left\langle
b_{3,in}\right\rangle+\sqrt{\kappa_{B}}\left\langle
b_{4,in}\right\rangle+\sqrt{\kappa_{Bx}}\left\langle a_{1,in}\right\rangle$
$\displaystyle+\sqrt{\kappa_{Bx}}\left\langle
a_{2,in}\right\rangle-\sqrt{\kappa_{B}\kappa_{Ax}}\left\langle
a\right\rangle-\sqrt{\kappa_{Bx}\kappa_{A}}\left\langle a\right\rangle,$

where $a_{1,(2),in},b_{3,(4),in}$ and $a_{1,(2),out},b_{3,(4),out}$ are the
input and output fields. Assuming a coherent-state approximation for photons
in resonators A and B coupled to input and output fields, the transmission
$S_{21}$ and $S_{43}$ are given by $\frac{\left\langle
a_{2,out}\right\rangle}{\left\langle a_{1,in}\right\rangle}$ and
$\frac{\left\langle b_{4,out}\right\rangle}{\left\langle
b_{3,in}\right\rangle}$, respectively. The boundary conditions which relate
the input and output fields to photon annihilation in each resonator,
$a_{1,(2),out}=\sqrt{\kappa_{A}}a-a_{2,(1),in}+\sqrt{\kappa_{Bx}}b$ and
$b_{3,(4),out}=\sqrt{\kappa_{B}}b-b_{4,(3),in}+\sqrt{\kappa_{Ax}}a$, are used
in the analysis inputout . We assume only one input field $a_{1,in}$ (i.e.
$a_{2,in}=b_{4,in}=b_{3,in}=0)$ is present for $S_{21}$ and $b_{3,in}$ (i.e.
$a_{2,in}=b_{4,in}=a_{1,in}=0)$ for $S_{43}$. The details of theoretical
analysis are described in Appendix B.

We next confine ourselves to degenerate modes
$\omega_{A}=\omega_{B}=\omega_{r}$, and, for simplicity, M-group TLSs (which
belong to the capacitor pair labeled with m). We can see from the Hamiltonian
that for positive $\varOmega_{AB}$ the M TLSs are interacting with the lower
frequency mode. We also reduce the calculation to equal coupling and photon
decay rates in each resonator:
$\kappa_{A}=\kappa_{B}=\kappa_{r},\gamma_{A}=\gamma_{B}=\gamma_{r}$, and
assume each resonator is decoupled from the other transmission line,
$\kappa_{Ax}=\kappa_{Bx}=0$. In the low-temperature and low-drive power limit
$(k_{B}T\ll\hbar\omega$ and $n_{ph}\ll 1)$, the resonator will be
asymptotically close to the ground state and we find the transmission
amplitude of this four-port system as
$S_{43}=S_{21}=1-\kappa_{r}\frac{-i(\omega-\omega^{\prime}_{r})+\kappa_{r}+\gamma_{r}}{-(\omega-\omega^{\prime}_{r})^{2}+(\kappa_{r}+\gamma_{r})^{2}-(i\Omega_{AB}-\sum_{i}^{M}\frac{\tanh(\frac{\hbar\omega}{2k_{B}T})g_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}^{i}}{2}})^{2}-2i(\kappa_{r}+\gamma_{r})(\omega-\omega^{\prime}_{r})},\quad\quad\quad\quad\quad\quad\quad(4)$
where
$\omega^{\prime}_{r}=\omega_{r}-\sum_{i}^{M}\frac{\tanh(\frac{\hbar\omega}{2k_{B}T})g_{i}^{2}}{\omega-\omega_{TLS}+i\frac{\gamma_{TLS}}{2}}$.
In the Appendix, we provide the transmission for TLSs interacting with the
high-frequency mode (Eq. 19) and that of the standard-coupled resonator pair
(Eq. 24), as well as the cross transmission from port 3 to 2 $(S_{23})$ and
from port 1 to 4 $(S_{41})$, with non-zero (finite) $\kappa_{Ax}$ and
$\kappa_{Bx}$ (Eqs. 18, 19).

Figure 5: Noise analysis for $1/f^{\alpha}\sim 1/f$ TLS noise, using time-
domain simulations for two variable-frequency TLSs. (a) Fractional frequency
noise spectral density $S_{\delta\nu/\nu}$ for the two resonator modes. Solid
line is a fit to A/[f /(1Hz)] + B, where here we define A as the 1/f noise
density and B for the white noise density. (b) Complex transmission,
Im[$S_{21}$] versus Re[$S_{21}$], for the DuRKID with two TLSs (black line),
where there is one TLS from the m and n groups, which according to the design
implies that each TLS couples to only one mode. The red dashed line is shown
for the transmission without TLSs (this was numerically obtained with zero TLS
coupling, which is equivalent). The solid and dashed lines are can be compared
to transmission magnitude versus frequency, Fig. 4(f), where there are 4
minima in the case of coupled TLSs. Transmission time-domain simulation at 2
fixed input frequencies (black points in 2 dashed boxes). The 2 sets of
accumulated points are further apart in I than the Q minima (solid line)
because the input frequencies for TLS noise analysis were chosen to be equal
to the case without coupled TLS, and as Fig. 4(f) shows: the resonance minima
for the case with TLS (solid line of Fig. 4(f)) are closer in frequency than
the case without TLS coupling (dashed line Fig. 4(f)). (c) Zoomed view of
transmission points accumulated in time for fixed input frequencies (black
points). The transmission for various frequencies is shown as a grey line.
(upper panel) $I^{\prime}_{A}$ quadrature defined as the tangent of the
transmission versus frequency on the 1st mode. (lower panel) $I^{\prime}_{B}$
quadrature defined as the tangent of the transmission versus frequency on the
2nd mode. Noise in the resonator modes is uncorrelated due to separated noise
sources, unlike the signal in the KID (see Appendix B section 3).

###  III.2 B. Theoretical Results

In order to investigate the effect of TLS noise on the two resonator modes, we
use the analytical results derived from Eq. (4) for both transmission lines in
the limit of a weakly coupled TLS where $g\ll\varOmega_{AB}$. The system we
consider is shown in Fig. 4(a). Figs. 4(b) and (c) show the spectra of
transmission $|S_{21}|$ for a TLS with the uncoupled modes in the standard-
coupled design and the DuRKID, respectively. As shown, a given TLS in the
DuRKID generally only shifts one of the two hybridized modes (Fig. 4(c)), as
expected. This is true even for a TLS that is detuned from the coupled modes
(related to hybridized mode symmetries). In contrast, a TLS in either
capacitor in the standard degenerate design will couple with equal strength to
the two modes and cause a frequency shift to both (see Fig. 4(b)). The
difference is evident from the TLS-resonator interaction term in Eq. (1) which
differs from the standard-coupled resonators (see Eq. 24 in the Appendix).
Thus the DuRKID has single-mode noise for a single TLS, whereas a standard-
coupled TLS has correlated-mode noise (see Appendix B for more data on the
off-resonance TLS case). We also consider the case of strong readout tone from
port 1 or 3 (see also Appendix A). In Fig. 4(d) the transmission spectra is
shown in the presence of strong pump drive with
$H_{d}=i\hbar\sqrt{\kappa_{A}}A(t)a^{\dagger}$, where $\left\langle
a_{1,in}(t)\right\rangle=A(t)$ and $\left\langle
a_{2,in}(t)\right\rangle=\left\langle b_{3,in}(t)\right\rangle=\left\langle
b_{4,in}(t)\right\rangle=0$. As shown, a given TLS in the bridge capacitor
design, even in the presence of the strong pump drive, only shifts one of the
two hybridized modes. Figure 4(e) shows the simulated signal on both resonance
modes in the bridge resonator design viewed from transmission spectra
$1\rightarrow 2$ (with $Z_{A},Z_{B}\rightarrow 0$ ). Specifically, the solid
line shows the transmission with a signal that increases $L_{B}$, as expected
for a KID. We see relative to the unperturbed transmission (the dashed line)
that the transmission spectra has a shift in both resonance modes to lower
frequency, as expected from earlier arguments. In addition, the relative size
of the observed transmission notches are changed due to the way the modes are
changed qualitatively – one mode is more A-like, just as the other mode is
more B-like. In summary, one DuRKID mode will experience frequency shifts from
one set of capacitors (m-group TLSs in equation 4) that will shift a single
mode; this is distinguishable from two correlated modes caused by the signal.

Figure 4(f) shows the transmission spectra of the DuRKID in the presence of
two TLSs, one from each group (capacitor pair). Next this arrangement with two
TLSs is used to calculate the cross-spectral density with TLSs and the two
noise sources. For this we simulate the TLS-induced frequency noise of two
resonator modes split by $\Omega_{AB}$, with bare resonance frequencies of
$\omega_{0}/2\pi=3.533$ GHz. The frequency noise spectral density
$S_{\delta\nu/\nu}(f)$ is simulated with one TLS from each TLS group (m or n),
by setting a 1/f noise spectrum on each. In Fig. 5(a), we show the simulated
frequency-noise spectral density for resonators A and B. TLS frequency
fluctuations in turn create fluctuations on the resonator modes, according to
their effective couplings. The TLS with higher energy is coupled to the higher
resonator mode while the lower energy TLS is coupled to the lower mode. These
conditions create uncorrelated 1/f frequency noise in the resonator modes (see
Appendix B). Im[S${}_{21}]$ versus Re[S${}_{21}]$ is shown in Fig. 5(b) and
(c) without any TLSs interaction, $g=0$, (red line) and the TLSs interaction
(grey line). Even though we have not analyzed more TLSs, from the principle of
superposition we can expect the uncorrelated noise feature will continue for a
typical TLS distribution – TLSs noise should not be correlated in separate
capacitors. This is distinct from correlated noise in the modes, which exists
only where expected (See Appendix).

###  III.3 C. Discussion

The most straightforward analysis of the system uses one feedline, e.g., ports
1 and 2. This corresponds to simulated data of $S_{21}$ shown in Fig. 5(c).
The advantage of the method is sufficient for TLS noise distinction and in a
future version of the DuRKID one may choose to eliminate the second feedline.
However, one could instead measure with 2 feedlines, where each feedline
measures one resonator. In this case, when a signal perturbs the frequencies,
each resonator signal could be equal in magnitude as each mode takes on an
emphasized character of one of the physical resonator modes. We also measured
unintended leakage (crosstalk) between the two different feedlines at
$|S_{14}|\sim 10^{-4}$ (this was measured with resonators detuned to see the
stray coupling near either the A or B resonator mode. From this we extracted
the couplings of the resonators to the unintended feed lines as
$\kappa_{ax}=\kappa_{bx}\sim 20$ kHz (see Appendix).

We specified a TLS noise mitigation technique for ideal fabrication, where the
hybridized mode noise is perfectly uncorrelated. However, noise sources can
become correlated if the capacitor bridge is asymmetric, or if the inter-
resonator coupling becomes too small to create distinction of the modes. In
practice, there will be some correlation between the components even in an
optimal experimental circuit due to measurement noise and factors that were
not addressed above. Thus in the future, one may want to construct an optimal
linear filter based on the correlations and other noise mechanisms (white
noise from amplifiers included, for example). In one approach for this, the
optimal linear filter can be computed by measuring the 4-component noise and
then computing the cross-spectral density matrix (see Eqs. 13 and 16 in Ref.
Stevenson ).

##  IV IV. CONCLUSION

In summary, we have proposed a new KID detector prototype consisting of two
superconducting circuit resonators. We first described the fabricated device
and schematic consisting of two resonators that both share an electrical
capacitance bridge. We then showed the basic tranmission and fittings of the
device data along with frequency tuning of one mode which is necessary for
device operation. 1/f noise will naturally arise from TLSs in the dielectric
of the capacitors, as found in previous studies. In our study the capacitor
configuration in the device allows the two resonators to share common TLSs
generally, but this changes once the resonators are tuned to degeneracy, where
a given TLS only perturbs one of the two hybridized resonator modes due to its
residency in one of two TLS groups (n or m). This leads to uncorrelated noise
in the two resonator modes since they experience separate noise sources.

We performed transmission measurements on the DuRKID and found experimentally
that the coupling between the modes is larger than the coupling of a TLS to a
resonator mode. Furthermore, to achieve degeneracy, we tuned a resonators by
up to 120 MHz with an applied current. To show that any given TLS will be
coupled to only one of the two fully hybridized modes, we then analyzed the
device schematic theoretically using a model with 4 ports coupled to 2
resonators, where a representative TLS of each TLS group resides in the
bridge. The simulation shows that TLS noise is different than the intended KID
signal. From the transmission data we obtain the TLS loss tangents, inter-
resonator coupling, and the intended and unintended couplings to transmission
lines.

In the future, we plan to demonstrate the uncorrelated noise in the device
from TLSs. In principle the measurement can be made by monitoring the lack of
correlation over time while the resonator pair is near the fully hybridized
state. We specify the TLS noise mitigation technique in terms of 2 quadratures
of resonator transmission measurements of amplitude and phase at 2 frequencies
(4 components) of the DuRKID. The noise from the hybridized state can be
compared when the resonators are non-hybridized, where the noise from TLSs
should appear as correlated noise on the resonators.

##  V acknowledgments

The authors thank F. C. Wellstood, C. Richardson, R. Ruskov and W. Wustmann
for scientific discussions. This work was funded through a NASA Science
Innovation Fund (SIF) award and the Intelligence Community Postdoctoral
Fellowship Award.

## Appendix A: Nonlinearity of the kinetic inductance and the fitting
procedure

##  Appendix A 1\. Nonlinearity of the kinetic inductance

As described in the main text, the inductance of the LC resonators is made
tunable with DC bias. We measured the nonlinearity explicitly in device 2 at
milliKelvin temperature with a direct applied bias current (note that device 1
and 3 share a single dc-input current in parallel). The relationship was given
previously for inductance as a quadratic function of bias current, but here we
describe the nonlinear inductance. The TiN inductor width is 5 mm wide. Fig.
6(a) shows frequency of the resonator as a function of bias current squared.
The fractional resonance frequency shift can be written in terms of current or
inductance as:

$\frac{\delta f_{r}}{f_{r}}=-\frac{I_{b}^{2}}{2I^{\prime 2}}=-\frac{\delta
L_{r}}{2L_{r}}.$ (4)

Figure 6: a) Frequency and b) kinetic inductance dependence on $I_{b}$.

We fit to a quadratic model as a function of bias current $I_{b}$ as shown in
Fig. 6(a) and from this extracted $I^{\prime}$= 789 $\mu$A. In Fig. 6(b) we
show the same type of fit, except using inductance as the independent variable
rather than the frequency.

##  Appendix B 2\. The fitting procedure

In order to fit the measured complex transmission data, $S_{21}$, $S_{43}$ and
$S_{31}$ we use Eqs. (18), (19) and (20). Although general formulas can be
made to directly fit for the many parameters simultaneously, such multi-
parameter fitting problems can be extremely sensitive to the initial values
due to the local optimization of the parameters, and different amounts of
noise in different data sets. Fitting a small subset of the data leads to
under determined constraints and inaccuracies due to covariances. To resolve
this issue, we utilize an iterative procedure through which our fitting
problem, with 10-parameters, is broken down into several independent fitting
problems with carefully chosen fixed parameters. Each fit contains only a few
(two or three) parameters at a time. First we extract the cross coupling
$\varOmega_{AB}$ from the data of Fig. 3(c). We have previously performed
resonator fittings from transmission data (e.g., $S_{21}$ ) using the diameter
correction method (DCM) (for a single resonator) Khalil2012 . This technique
uses a Least-Squares Monte Carlo (LSM) method to find the minimum error,
$\chi^{2}$. To fit both $S_{21}$ and $S_{43}$ we developed a fitting sequence
using a sequence of different LSM fittings in multi-step procedure. We start
by using LSM fitting primarily with $S_{21}$ data, using device model
(equation (18)), except that an extra multiplicative factor $e^{i\theta}$ is
applied to the right hand side of Eq. (18) to account for the transmission
line length. Initial guesses must be placed for all variables, including
$\omega_{A}$, $\omega_{B},$ $\kappa_{A}$, $\kappa_{B}$, $\gamma_{A}$,
$\gamma_{B}$, $\kappa_{Ax}$, $\kappa_{Bx}$ . Within the LSM fitting a Monte
Carlo guess is made in the form $x=x_{0}e^{\zeta\xi}$ for each fit parameter,
where $x_{0}$ is the previous (initial) guess for the fit parameter $x$,
$\zeta$ is a randomly generated number between -1 and 1, and $\xi$ is a
parameter smaller than one, that determines the MC guess domain.

However, to optimize $S_{21}$ in the first fit we use two different $\xi$ for
the different parameters. For parameters associated with resonator B, we use
fast iterations e.g. using $\xi=0.1$ for $\omega_{A}$, $\kappa_{A}$,
$\gamma_{A}$ , $\kappa_{Ax}$, $\theta_{21}$ to optimize it quickly. Meanwhile,
for parameters associated with resonator A, we use slow iterations, e.g.,
using $\xi=10^{-5}$ for $\omega_{B},$ $\kappa_{B}$, $\gamma_{B}$,
$\kappa_{Bx}$, $\theta_{43}$ , to leave the parameters with only slight
changes. After this step, the updated values are used to fit $S_{43}$
following the same procedure, with the difference that this time the variation
of resonator A parameters are quickly optimized and resonator B parameters are
are mainly left unchanged. Next the $S_{43}$ fit parameters are determined,
such that $S_{21}$ could be fit, and this completes one cycle of our
procedure. The previous 2 steps are repeated in sequence: for example by using
the fit results from $S_{21}$ ($S_{43}$) we then fit $S_{43}$ ($S_{21}$). The
cycles of fitting are then repeated iteratively for many times until
convergence is obtained. The standard resonator coupling to the intended
transmission line is $Q_{c}^{A,B}=\frac{\omega_{A,B}}{2\kappa_{A,B}}$. The
internal loss from one resonator is often desired, which is
$Q_{i}=\frac{\omega}{\gamma}$ in our devices, where the resonator loss
$1/Q_{i}$ is due to the sole dielectric in each resonator. Since we have dual
transmission line setup, it is useful to define the apparent internal quality
factor for each resonator
$Q_{i,app}^{A,B}=\frac{\omega_{A,B}}{\gamma_{A,B}+2\kappa_{Ax,Bx}}$, which not
only depends on the internal loss rate of the resonator to its internal loss
$\gamma_{a,b}$ , but also the coupling loss to the unintended transmission
line $2\kappa_{Ax,Bx}$, respectively. Figure 7 (a) and (b) show an example of
such a fit to two resonance lineshapes for reasonators A and B which are
coupled to transmission lines 1-2 and 3-4 respectively for -140 dB power at
device. The extracted fitting parameters from $S_{21}$ and $S_{43}$ are used
as an initial guess to fit the cross transmission from port 1 to 4, $S_{41}$.

Figure 7: Transmission data in the absence of bias for resonator A (a) and
resonator B (b). The red curves show the fit of data to the model. The fit
yielded $\omega_{A}=3.5357$ GHz, $\gamma_{A}=0.88$ MHz, $\kappa_{A}=0.9$ MHz,
$\kappa_{Ax}=20$ kHz, and $\omega_{B}=3.6335$ GHz, $\gamma_{B}=0.77$ MHz,
$\kappa_{B}=1.01$ MHz, $\kappa_{Bx}=19.7$ kHz, $Q_{i}^{A}=2741$,
$Q_{i}^{B}=3026$ and $Q_{c}^{A}=1930$, $Q_{c}^{B}=1800$. (c) Crosstalk
coupling (unintended coupling) between port 1 and 4 of device two in the
absence of bias. Fitting data gives 20.3 kHz of cross coupling between port 1
and 4 with $\omega_{A}=3.534$ GHz, $\gamma_{A}=0.91$ MHz, $\kappa_{A}=0.97$
MHz, $\kappa_{Ax}=20.3$ kHz, and $\omega_{B}=3.6335$ GHz, $\gamma_{B}=0.77$
MHz, $\kappa_{B}=1.01$ MHz, $\kappa_{Bx}=19.5$ kHz. Agreement between data and
4-port model confirms appropriateness of the fit method and the many parameter
model.

## Appendix B: Derivation of the transmission for two resonances interacting
with TLS

1\. LC resonator pair with an internal electrical bridge of capacitors

The system we consider in Fig. 1(a) consists of two resonators and TLSs,
described by the Hamiltonian $H_{sys}$ (Eq. 2). It is coupled to 2
transmission lines that are accessible at their ends via 4 ports. The system
has resonator modes a and b. According to the standard theory of input and
output for quantum dissipative systems Gardiner , the transmission lines can
be modeled as a heat bath. Additionally, the couplings can be specified
between transmission lines and the intended couplings, $\kappa_{A}$ and
$\kappa_{B}$, as well as the unintended couplings, $\kappa_{Ax}$ and
$\kappa_{Bx}$. The Heisenberg equation of motion for the bare resonator modes
in terms of the system and input-output fields can be written as Eq. (2), (3),

$\displaystyle\frac{d}{dt}\left\langle a\right\rangle=$
$\displaystyle-i\omega\left\langle
a\right\rangle=-\frac{i}{\hbar}\left\langle[a,H_{sys}]\right\rangle+\kappa_{A}\left\langle
a\right\rangle-\sqrt{\kappa_{A}}\left\langle
a_{1,out}\right\rangle-\sqrt{\kappa_{A}}\left\langle a_{2,out}\right\rangle$
$\displaystyle-\sqrt{\kappa_{Ax}}\left\langle
b_{3,out}\right\rangle-\sqrt{\kappa_{Ax}}\left\langle
b_{4,out}\right\rangle{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}-\gamma_{A}\left\langle
a\right\rangle+\kappa_{Ax}\left\langle
a\right\rangle+\sqrt{\kappa_{A}\kappa_{Bx}}\left\langle
b\right\rangle+\sqrt{\kappa_{Ax}\kappa_{B}}\left\langle b\right\rangle,}}$

and

$\displaystyle\frac{d}{dt}\left\langle b\right\rangle=$
$\displaystyle-i\omega\left\langle
b\right\rangle=-\frac{i}{\hbar}\left\langle[b,H_{sys}]\right\rangle+\kappa_{B}\left\langle
b\right\rangle-\sqrt{\kappa_{B}}\left\langle
b_{3,out}\right\rangle-\sqrt{\kappa_{B}}\left\langle b_{4,out}\right\rangle$
$\displaystyle-\sqrt{\kappa_{Bx}}\left\langle
a_{1,out}\right\rangle-\sqrt{\kappa_{Bx}}\left\langle
a_{2,out}\right\rangle{\color[rgb]{0,0,0}{\color[rgb]{0,0,0}-\gamma_{B}\left\langle
b\right\rangle+\kappa_{Bx}\left\langle
b\right\rangle+\sqrt{\kappa_{B}\kappa_{Ax}}\left\langle
a\right\rangle+\sqrt{\kappa_{Bx}\kappa_{A}}\left\langle a\right\rangle}}.$

We write the equation of motion for the TLS operator $\sigma_{i}^{-}$ for TLS
‘i’ using the Bloch equations approximation for relaxation and decoherence

$\displaystyle\frac{d}{dt}\left\langle\sigma_{i}^{-}\right\rangle=$
$\displaystyle-i\omega\left\langle\sigma_{i}^{-}\right\rangle=-i\omega_{TLS,i}\left\langle\sigma_{i}^{-}\right\rangle{\color[rgb]{0,0,0}-\frac{\gamma_{TLS,i}}{2}\left\langle\sigma_{i}^{-}\right\rangle+2g\left\langle\sigma_{i}^{z}a\right\rangle+2g\left\langle\sigma_{i}^{z}b\right\rangle},$
(7)

where the TLS decoherence rate is $\gamma_{TLS,i}=k_{1,i}+2k_{2,i}$, where
typically $k_{1,i}=A_{1}$$\varDelta_{0,i}^{2}$coth
($\frac{\hbar\omega}{2k_{B}T}$) and $k_{2,i}\approx A_{2}$T2. The rates
$k_{1,i}$ and $k_{2,i}$ describe TLS $i$ relaxation and phase decoherence
rates associated with TLS-photon and TLS-TLS interactions, respectively and
$A_{1}$ and $A_{2}$ are material related constants Bahmanthesis . In the low-
temperature and low-drive power limit ($k_{B}T\ll\hbar\omega$ and $n_{ph}\ll
1$), the resonator will be asymptotically close to the ground state. This
suggests that we can replace the spin operator
$\left\langle\sigma_{i}^{z}\right\rangle$ in Eq. (6) with the ground state
value $-\frac{1}{2}$. With these assumptions we obtain a closed system of
linear equations which can be solved for the TLS operators:

$\left\langle\sigma_{i}^{-}\right\rangle=\frac{2g(\left\langle
a\right\rangle+\left\langle
b\right\rangle)\left\langle\sigma_{i}^{z}\right\rangle}{i(\omega_{TLS,i}-\omega)+\frac{\gamma_{TLS,i}}{2}}.$
(8)

For thermally excited TLS, one can use a mean field approach replacing the
operator $\left\langle\sigma^{z}_{i}\right\rangle$ in Eq. (6) with its
thermodynamic average value, i. e.
$\left\langle\sigma_{i}^{z}{}_{th}\right\rangle=-\frac{1}{2}\tanh(\frac{\hbar\omega}{2k_{B}T}).$
This approach is consistent with previous analysis of sound and microwave
absorption by TLSs Hunklinger1986 ; Classen1994 . The average value of
$\left\langle\sigma_{i}^{z}\right\rangle$ at higher photon regime is
calculated in the next section.

Figure 8: a) Simulated transmission magnitude $|S_{21}|$ (with
$Z_{B}\rightarrow 0$ ) for a single TLS off resonace coupled to a+b field in
the bridge design. The TLS frequency of $\omega_{tls}=3.52$ GHz is used here
and the rest of parameters are the same as before. Note that dispersion is not
caused at the neighboring frequency mode, but at the mode that has the proper
coupling for the TLSs capacitor. b) The impact of a single TLS on resonace
coupled to a+b field on both hybridized modes. c) The impact of non zero
unintended coupling of each resonator to the other transmission line,
$\kappa_{Ax}=\kappa_{Bx}=20$kHz for data presented in (a).

To proceed, we now substitute the solution for
$\left\langle\sigma_{i}^{-}\right\rangle$ (Eq. 8) into Eqs. (2-4) and obtain
the solution for resonator mode $a$ and $b$:

$L_{A,+}\left\langle a\right\rangle+L_{AB,+}\left\langle
b\right\rangle=\sqrt{\kappa_{A}}\left\langle
a_{1,in}\right\rangle+\sqrt{\kappa_{A}}\left\langle
a_{2,in}\right\rangle+\sqrt{\kappa_{Ax}}\left\langle
b_{3,in}\right\rangle+\sqrt{\kappa_{Ax}}\left\langle b_{4,in}\right\rangle,$
(9) $L_{A,-}\left\langle a\right\rangle+L_{AB,-}\left\langle
b\right\rangle=-\sqrt{\kappa_{A}}\left\langle
a_{1,out}\right\rangle-\sqrt{\kappa_{A}}\left\langle
a_{2,out}\right\rangle-\sqrt{\kappa_{Ax}}\left\langle
b_{3,out}\right\rangle-\sqrt{\kappa_{Ax}}\left\langle b_{4,out}\right\rangle,$
(10) $L_{B,+}\left\langle b\right\rangle+L_{AB,+}\left\langle
a\right\rangle=\sqrt{\kappa_{B}}\left\langle
b_{3,in}\right\rangle+\sqrt{\kappa_{B}}\left\langle
b_{4,in}\right\rangle+\sqrt{\kappa_{Bx}}\left\langle
a_{1,in}\right\rangle+\sqrt{\kappa_{Bx}}\left\langle a_{2,in}\right\rangle,$
(11) $L_{B,-}\left\langle b\right\rangle+L_{AB,-}\left\langle
a\right\rangle=-\sqrt{\kappa_{B}}\left\langle
b_{3,out}\right\rangle-\sqrt{\kappa_{B}}\left\langle
b_{4,out}\right\rangle-\sqrt{\kappa_{Bx}}\left\langle
a_{1,out}\right\rangle-\sqrt{\kappa_{Bx}}\left\langle a_{2,out}\right\rangle,$
(12)

where,

$L_{A,\pm}=-i\omega+i\omega_{A}+\gamma_{A}\pm\kappa_{A}\pm\kappa_{Ax}-\sum_{i=1}^{N}\frac{2sg_{i}^{2}}{i(\omega_{TLS,i}-\omega)+\frac{\gamma_{TLS,i}}{2}},$
(13)
$L_{AB,\pm}=i\Omega_{AB}\pm\sqrt{\kappa_{A}\kappa_{Bx}}\pm\sqrt{\kappa_{Ax}\kappa_{B}}-\sum_{i=1}^{N}\frac{2sg_{i}^{2}}{i(\omega_{TLS,i}-\omega)+\frac{\gamma_{TLS,i}}{2}},$
(14)
$L_{B,\pm}=-i\omega+i\omega_{B}+\gamma_{B}\pm\kappa_{B}\pm\kappa_{Bx}-\sum_{i=1}^{N}\frac{2sg_{i}^{2}}{i(\omega_{TLS,i}-\omega)+\frac{\gamma_{TLS,i}}{2}}.$
(15)

To study one coherent input, we then set $\left\langle
a_{2,in}\right\rangle=\left\langle b_{4,in}\right\rangle=\left\langle
b_{3,in}\right\rangle=0$ to calculate $S_{21}$ and $\left\langle
a_{2,in}\right\rangle=\left\langle b_{4,in}\right\rangle=\left\langle
a_{1,in}\right\rangle=0$ to calculate $S_{43}$. The solution for the resonator
modes $a$ and $b$ takes the form

$\left\langle
a\right\rangle=\frac{\sqrt{\kappa_{A}}-\frac{L_{AB,+}\sqrt{\kappa_{Bx}}}{L_{B,+}}}{L_{A,+}-\frac{L_{AB,+}^{2}}{L_{B,+}}}\left\langle
a_{1,in}\right\rangle,$ (16)

and

$\left\langle b\right\rangle=\frac{\sqrt{\kappa_{Bx}}}{L_{B,+}}\left\langle
a_{1,in}\right\rangle-\frac{L_{AB,+}}{L_{B,+}}\frac{\sqrt{\kappa_{A}}-\frac{L_{AB,+}\sqrt{\kappa_{Bx}}}{L_{B,+}}}{L_{A,+}-\frac{L_{AB,+}^{2}}{L_{B,+}}}\left\langle
a_{1,in}\right\rangle.$ (17)

Using the boundary conditions,
$a_{1,(2),out}=\sqrt{\kappa_{A}}a-a_{2,(1),in}+\sqrt{\kappa_{Bx}}b$ and
$b_{3,(4),out}=\sqrt{\kappa_{B}}b-b_{4,(3),in}+\sqrt{\kappa_{Ax}}a$, and Eq.
(14) and (15), we obtain $S_{21}=\frac{\left\langle
a_{2,out}\right\rangle}{\left\langle a_{1,in}\right\rangle}$,
$S_{43}=\frac{\left\langle b_{4,out}\right\rangle}{\left\langle
b_{3,in}\right\rangle}$ and $S_{41}=\frac{\left\langle
b_{4,out}\right\rangle}{\left\langle a_{1,in}\right\rangle}$ as

$\displaystyle S_{21}$
$\displaystyle=1-\frac{\kappa_{A}}{-i(\omega-\omega_{A})+\kappa_{A}+\gamma_{A}-\sum_{i=1}^{N}\frac{2Sg_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}^{i}}{2}}-\frac{(i\Omega_{AB}-\sum_{i=1}^{N}(\frac{2g_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}^{i}}{2}}))^{2}}{-i(\omega-\omega_{B})+\kappa_{B}+\gamma_{B}-\sum_{i=1}^{N}\frac{2Sg_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}^{i}}{2}}}},$
(18)
$S_{43}=1-\frac{\kappa_{B}}{-i(\omega-\omega_{B})+\kappa_{B}+\gamma_{B}-\sum_{i=1}^{N}\frac{2Sg_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}^{i}}{2}}-\frac{(i\Omega_{AB}-\sum_{i=1}^{N}(\frac{2g_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}}{2}}))^{2}}{-i(\omega-\omega_{A})+\kappa_{A}+\gamma_{A}-\sum_{i=1}^{N}\frac{2Sg_{i}^{2}}{-i(\omega-\omega_{TLS}^{i})+\frac{\gamma_{TLS}}{2}}}},$
(19)

and

$S_{41}=\frac{\sqrt{\kappa_{Bx}\kappa_{B}}}{L_{B,+}}-\frac{L_{AB,+}(L_{B,+}\sqrt{\kappa_{B}\kappa_{A}}-L_{AB,+}\sqrt{\kappa_{B}\kappa_{Bx})}}{L_{B,+}(L_{B,+}L_{A,+}-L_{AB,+}^{2})}+\frac{\sqrt{\kappa_{Ax}\kappa_{A}}-\frac{\sqrt{\kappa_{Ax}}L_{AB,+}\sqrt{\kappa_{Ax}\kappa_{Bx}}}{L_{B,+}}}{L_{A,+}-\frac{L_{AB,+}^{2}}{L_{B,+}}}.$
(20)

Reducing the calculation to equal coupling and photon decay rates for each
resonator: $\kappa_{A}=\kappa_{B}=\kappa_{r}$,
$\gamma_{A}=\gamma_{B}=\gamma_{r}$, and assuming no (cross) coupling to the
unintended transmission line, $\kappa_{Ax}=\kappa_{Bx}=0$, the transmission
properties of this four-port system coupling to a single TLS through the $a+b$
field is given by

$S_{43}=S_{21}=1-\kappa_{A}\frac{-i(\omega-\omega^{\prime}_{r})+\kappa_{r}+\gamma_{r}}{-(\omega-\omega^{\prime}_{r})^{2}+(\kappa_{r}+\gamma_{r})^{2}-(i\Omega_{AB}-\frac{\tanh(\frac{\hbar\omega}{2k_{B}T})g^{2}}{-i(\omega-\omega_{TLS})+\frac{\gamma_{TLS}}{2}})^{2}-2i(\kappa_{r}+\gamma_{r})(\omega-\omega^{\prime}_{r})},$
(21)

where
$\omega^{\prime}_{r}=\omega_{r}-\frac{g^{2}}{\omega-\omega_{TLS}+i\frac{\gamma_{TLS}}{2}}.$
Similarly, coupling to a single TLS through the $a-b$ field can be obtained.

2\. Standard-coupled two-resonator

Considering the standard-coupled two-resonator with Hamiltonian of

$\displaystyle H_{sys}$
$\displaystyle=\hbar\omega_{A}a^{\dagger}a+\hbar\omega_{B}b^{\dagger}b+\hbar\varOmega_{AB}(b^{\dagger}a+a^{\dagger}b)+\varepsilon_{TLS}\sigma^{z}$
(22) $\displaystyle-i\hbar
g_{A}(\sigma_{A}^{+}a+\sigma_{A}^{-}a^{\dagger})-i\hbar
g_{B}(\sigma_{B}^{+}b+\sigma_{B}^{-}b^{\dagger}),$ (23)

and assuming $g_{A}=g_{B}=g,$ we obtain

$S_{43}=S_{21}=1-\kappa_{A}\frac{-i(\omega-\omega^{\prime}_{r})+\kappa_{r}+\gamma_{r}}{-(\omega-\omega^{\prime}_{r})^{2}+(\kappa_{r}+\gamma_{r})^{2}+\Omega_{AB}^{2}-2i(\kappa_{r}+\gamma_{r})(\omega-\omega^{\prime}_{r})}.$
(24)

3\. Coherent drive in the Hamiltonian (Treatment of the many-photon case)

We consider the case of an incoming coherent radiation from the left-hand side
of transmission line, port 1 or 3. The presence of such coherent drive is
accounted for by an effective Hamiltonian

$H=H_{sys}+H_{d},$ (25)

where the drive appears as a term $H_{d}=i\hbar Ja^{\dagger}+H.c.$, and
$J=\sqrt{\Gamma_{ext}^{a}}A(t),$ and $\Gamma_{ext}^{a}=\kappa_{A}.$ This form
is derived by assuming $\left\langle a_{1,in}(t)\right\rangle=A(t)$ and
$\left\langle a_{2,in}(t)\right\rangle=\left\langle
b_{3,in}(t)\right\rangle=\left\langle b_{4,in}(t)\right\rangle=0$. The
interaction of a single TLS with the resonators can be described by the
Jaynes-Cummings Hamiltonian in a frame rotating at the pump frequency
$\omega_{p}$ :

$\displaystyle H$
$\displaystyle=\hbar(\omega_{p}-\omega_{A})a^{\dagger}a+\hbar(\omega_{p}-\omega_{B})b^{\dagger}b+\frac{\hbar(\omega_{p}-\omega_{TLS})}{2}\sigma_{z}$
$\displaystyle+\hbar\varOmega_{AB}(b^{\dagger}a+a^{\dagger}b)-i\hbar
g(\sigma^{-}(a^{\dagger}+b^{\dagger})-\sigma^{+}(a+b))+i\hbar
J(a^{\dagger}-a)$

The corresponding dissipation can be described by the Lindblad master equation
Capelle2020 :

$\displaystyle\frac{d\rho}{dt}$
$\displaystyle=-i\hbar[H_{sys},\rho]+\Gamma_{\downarrow\uparrow}(n_{th}+1)\mathcal{D}_{\sigma}(\rho)+\frac{\Gamma_{\phi}}{2}\mathcal{D}_{\sigma
z}(\rho)$
$\displaystyle+\Gamma_{\downarrow\uparrow}n_{th}\mathcal{D}_{\sigma^{\dagger}}(\rho)+\Gamma_{ext}^{a}\mathcal{D}_{a}(\rho)+\Gamma_{ext}^{b}\mathcal{D}_{b}(\rho),$
(27)

where the occupation number of TLS is $n_{th}=1/(e^{\hbar\omega/kT}-1)$, the
damping of resonator A and B in the absence of TLS is
$\Gamma_{ext}^{a}$=$\kappa_{A}$ and $\Gamma_{ext}^{b}=\kappa_{B}$, the TLS
dephasing rate is $\Gamma_{\phi}$, its rate at zero temperature
$\Gamma_{\downarrow\uparrow}$ , and $\mathcal{D}_{A}(\rho)=A\rho
A^{\dagger}-\frac{1}{2}(A^{\dagger}A\rho+\rho A^{\dagger}A).$ Using
$\left\langle A\right\rangle=Tr\left\langle A\rho\right\rangle$ and
$(d/dt)\left\langle A\right\rangle=Tr\left\langle A(d/dt)\rho\right\rangle$,
we can compute the Maxwell-Bloch equations

$\frac{d\left\langle
a\right\rangle}{dt}=(-i(\omega_{p}-\omega_{a})-\frac{\Gamma_{ext}^{a}}{2})\left\langle
a\right\rangle+g_{a}\left\langle\sigma\right\rangle+J$ (28)
$\frac{d\left\langle
b\right\rangle}{dt}=(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2})\left\langle
b\right\rangle+g_{b}\left\langle\sigma\right\rangle$ (29)
$\frac{d\left\langle\sigma\right\rangle}{dt}=(-i(\omega_{p}-\omega_{TLS})-\Gamma_{2})\left\langle\sigma\right\rangle+g_{a}\left\langle
a\sigma_{z}\right\rangle+g_{b}\left\langle b\sigma_{z}\right\rangle$ (30)
$\frac{d\left\langle\sigma_{z}\right\rangle}{dt}=-2g_{a}(\left\langle
a^{\dagger}\sigma\right\rangle+\left\langle
a\sigma^{\dagger}\right\rangle)-2g_{b}(\left\langle
b^{\dagger}\sigma^{-}\right\rangle+\left\langle
b\sigma^{+}\right\rangle)-\Gamma_{1}(\left\langle\sigma_{z}\right\rangle-\left\langle\sigma_{z}\right\rangle_{th}),$
(31)

where we define
$\Gamma_{2}=(\Gamma_{\downarrow\uparrow}/2)(1+2n_{th})$+$\Gamma_{\phi},$
$\Gamma_{1}=\Gamma_{\downarrow\uparrow}(1+2n_{th})$,
$\left\langle\sigma_{z}\right\rangle_{th}=-1/(1+2n_{th})=-$tanh
($\hbar\omega/2k_{B}T$). To transform this system Eqs. (28)-(31) into a closed
set of equations, we neglect the correlations and factorize the products
$\left\langle a\sigma_{z}\right\rangle=$$\left\langle
a\right\rangle\left\langle\sigma_{z}\right\rangle$, $\left\langle
a^{\dagger}\sigma_{z}\right\rangle=$$\left\langle
a^{\dagger}\right\rangle\left\langle\sigma_{z}\right\rangle$, $\left\langle
b\sigma_{z}\right\rangle=$$\left\langle
b\right\rangle\left\langle\sigma_{z}\right\rangle$, $\left\langle
b^{\dagger}\sigma_{z}\right\rangle=$$\left\langle
b^{\dagger}\right\rangle\left\langle\sigma_{z}\right\rangle$. We first
determine the solution for the cavity field, $\left\langle a\right\rangle$ and
$\left\langle b\right\rangle$, using the approximation $\left\langle
a\right\rangle$=$\alpha+\delta\alpha(t)e^{-i(\omega_{p}-\omega_{a})t}$,
$\left\langle
b\right\rangle$=$\beta+\delta\beta(t)e^{-i(\omega_{p}-\omega_{b})t},$$\left\langle\sigma\right\rangle$=$\sigma_{0}+\delta\sigma(t)e^{-i(\omega_{p}-\omega_{TLS})t}$
and $\left\langle\sigma_{z}\right\rangle=\sigma_{z0},$ where
$\delta\alpha(t)$, $\delta\beta(t)$ and $\delta\sigma(t)$ are slowly varying
complex functions Capelle2020 . The equation for the stationary components
are:

$0=(-i(\omega_{p}-\omega_{a})-\frac{\Gamma_{ext}^{a}}{2})\alpha+g_{a}\sigma_{0}-i\Omega_{AB}\beta+J$
(32)
$0=(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2})\beta+g_{b}\sigma_{0}-i\Omega_{AB}\alpha$
(33)
$0=(-i(\omega_{p}-\omega_{TLS})-\Gamma_{2})\sigma_{0}+g_{a}\alpha\sigma_{z0}+g_{b}\beta\sigma_{z0}$
(34)
$0=-2g_{a}(\alpha^{*}\sigma_{0}+\alpha\sigma_{0}^{*})-2g_{b}(\beta^{*}\sigma_{0}+\beta\sigma_{0}^{*})-\Gamma_{1}(\sigma_{z0}-\left\langle\sigma_{z}\right\rangle_{th}).$
(35)

Grouping terms by terms related to $\alpha$ and $\beta$, one finds

$\sigma_{0}=\frac{g_{a}\alpha\sigma_{z0}+g_{b}\beta\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}.$
(36)

The solution for the outfield is obtained from the boundary conditions
$a_{1,(2),out}=\sqrt{\Gamma_{ext}^{a}}a-a_{2,(1),in}$ with inserting the
assumed form of the input field $\left\langle a_{1,in}(t)\right\rangle=A(t)$.
From Eq. (33) and replacing $\sigma_{0}$ from (36) we have

$0=(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2})\beta+\frac{g_{b}g_{a}\alpha\sigma_{z0}+g_{b}g_{b}\beta\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}-i\Omega_{AB}\alpha$
(37)
$0=(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\beta+\frac{g_{b}g_{a}\alpha\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}-i\Omega_{AB}\alpha$
(38)
$-(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\beta=\frac{g_{b}g_{a}\alpha\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}-i\Omega_{AB}\alpha$
(39)
$-(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\beta=(\frac{g_{b}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}-i\Omega_{AB})\alpha.$
(40)

Finally we obtain

$\beta=\frac{-\frac{g_{b}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}+i\Omega_{AB}}{(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}\alpha$
(41)

To proceed to the output fields, we next substitute the solution for $\beta$
from Eq. (41) into Eq. (32)

$0=(-i(\omega_{p}-\omega_{a})-\frac{\Gamma_{ext}^{a}}{2}+\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\alpha+(-i\Omega_{AB}+\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\beta+J$
(42)
$(i(\omega_{p}-\omega_{a})+\frac{\Gamma_{ext}^{a}}{2}-\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\alpha+(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\beta=J$
(43)
$(i(\omega_{p}-\omega_{a})+\frac{\Gamma_{ext}^{a}}{2}-\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\alpha+(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})\frac{-\frac{g_{b}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}}+i\Omega_{AB}}{(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}\alpha+J$
(44)
$[(i(\omega_{p}-\omega_{a})+\frac{\Gamma_{ext}^{a}}{2}-\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})+\frac{(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}{(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}]\alpha=\sqrt{\Gamma_{ext}^{a}}a_{1,in}.$
(45)

The solution for the resonator $\alpha$ field is

$\alpha=\frac{\sqrt{\Gamma_{ext}^{a}}}{[(i(\omega_{p}-\omega_{a})+\frac{\Gamma_{ext}^{a}}{2}-\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})+\frac{(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})^{2}}{(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}]}a_{1,in}.$
(46)

Using this result one can find transmission and reflection coefficient using
the boundary condition $a_{2,out}=\sqrt{\Gamma_{ext}^{a}}a-a_{1,in}$. The
transmission, $S_{21}=\frac{a_{2,out}}{a_{1,in}}$ can be written as

$S_{21}=1-\frac{\Gamma_{ext}^{a}}{[(i(\omega_{p}-\omega_{a})+\frac{\Gamma_{ext}^{a}}{2}-\frac{g_{a}g_{a}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})+\frac{(i\Omega_{AB}-\frac{g_{a}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})^{2}}{(-i(\omega_{p}-\omega_{b})-\frac{\Gamma_{ext}^{b}}{2}+\frac{g_{b}g_{b}\sigma_{z0}}{i(\omega_{p}-\omega_{TLS})+\Gamma_{2}})}]},$
(47)

where

$\sigma_{z0}=\left\langle\sigma_{z}\right\rangle_{th}[1-\frac{\Gamma_{2}^{2}\overline{n}/n_{s}}{(\omega_{TLS}-\omega_{p})+\Gamma_{2}^{2}(1+\overline{n}/n_{s})}],$
(48)

and $\overline{n}=$$|A(t)|^{2}$ is the mean photon number in the cavity and
$n_{s}^{-1}=4g^{2}/\Gamma_{1}\Gamma_{2}$ the number of photons required to
saturate the TLS transition.

3\. Correlation amplitude:

Figure 9: Correlation amplitude $C_{xy}$ between noise quantities: 1/f-noise
TLS frequencies and resonator transmission quadratures ($I_{A}$, $Q_{A}$,
$I_{B}$ and $Q_{B}$). 1/f noise sources from m-group TLS and n group TLS are
not correlated as there is no significant TLS-TLS coupling mechanism. There is
near perfect noise correlation between $I_{A}$ and n-group TLS, and $I_{B}$
and m-group TLS. There is no significant normalized correlation between
resonator quadratures $I_{A}$ and $I_{B}$ because the noise fluctuations occur
as a m- or n- group TLS source on a corresponding resonator quadrature $I_{A}$
or $I_{B}$. This implies that noise is not significantly correlated through
the resonator-resonator coupling. $Q_{A}$ and $Q_{B}$ have slightly larger
normalized correlation amplitude, but also this is of no consequence because
these quadratures have small noise powers.

TLS frequency (1/f) noise gives noise in resonator transmission (e.g.
$S_{21})$, where the transmission changes in phase and amplitude generally for
one input frequency. The noise is mainly seen in a particular direction in IQ
space as fluctuations in [Re $(S_{21}(f_{0A}))$, Im $(S_{21}(f_{0A}))]$ and
[Re $(S_{21}(f_{0B}))$, Im $(S_{21}(f_{0B}))]$ which are tangent to the
complex transmission plot, which we use (Eq. (18)) to define the quadratures
$[Q^{\prime}_{A},I^{\prime}_{A}]$ and $[Q^{\prime}_{B},I^{\prime}_{B}]$ shown
in Fig. 5(c). As in a standard KID, the fluctuation quadrature is thus
frequency noise. However, now we have two input frequencies and two TLS noise
sources. The TLSs induce noise in the resonators, where $I^{\prime}_{A}$ and
$I^{\prime}_{B}$ in Fig. 5(c) are defined to be tangent to the transmission
versus frequency plot near the probe frequencies [$f_{A}$, $f_{B}$]. In Fig. 9
we show the correlation amplitude (the normalized cross spectral density)

$C_{xy}=\frac{S_{\nu}^{xy}}{(S_{\nu}^{xx}S_{\nu}^{yy})^{1/2}},$ (49)

between different quadratures and TLS groups. Here, x and y are the different
possible aforementioned TLS and quadrature noise types. $S_{\nu}^{xy}$ is the
cross-spectral density between x and y, and $S_{\nu}^{xx}$ and $S_{\nu}^{yy}$
are the autospectral density of x and y respectively. The result plotted in
Fig. 9 shows that the 1/f noise on each mode and its own TLS fluctuations are
highly correlated, but the noise induced on the measurement quadratures of the
two resonator modes is uncorrelated.

## References

  * (1) P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, A broadband superconducting detector suitable for use in large arrays, Nature 425, no. 6960, 817–821, (2003).
  * (2) B. A. Mazin, Ph.D. thesis, Microwave kinetic inductance detectors, Caltech, (2004).
  * (3) J. Gao, J. Zmuidzinas, B. A. Mazin, H. G. LeDuc, and P. K. Day, Noise properties of superconducting coplanar waveguide microwave resonators, Appl. Phys. Lett. 90, 102507 (2007).
  * (4) Omid Norooziana, Jiansong Gaob, Jonas Zmuidzinasa, Henry G. LeDucc, and Benjamin A. Mazind, Two‐level system noise reduction for Microwave Kinetic Inductance Detectors, Conference Proceedings 1185, 148 (2009).
  * (5) Steffen Schlör, Jürgen Lisenfeld, Clemens Müller, Alexander Bilmes, Andre Schneider, David P. Pappas, Alexey V. Ustinov, and Martin Weides, Correlating decoherence in transmon qubits: low frequency noise by single fluctuators, Phys. Rev. Lett. 123, 190502 (2019).
  * (6) Oliver Dial et al., Bulk and surface loss in superconducting transmon qubits, Supercond. Sci. Technol. 29 044001 (2016).
  * (7) J. Burnett et al., Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator. Nat Commun 5, 4119 (2014). https://doi.org/10.1038/ncomms5119
  * (8) J. Gao, M. Daal, A. Vayonakis, S. Kumar, J. Zmuidzinas, B. Sadoulet, B. A. Mazin, P. K. Day, and H. G. Leduc, Experimental evidence for a surface distribution of two-level systems in superconducting lithographed microwave resonators, Appl. Phys. Lett. 92, 152505 (2008), https://doi.org/10.1063/1.2906373
  * (9) C.R.H. McRae, H. Wang, J. Gao, M. R. Vissers, T. Brecht, A. Dunsworth, D. P. Pappas, and J. Mutus, Materials loss measurements using superconducting microwave resonators, Review of Scientific Instruments 91, 091101 (2020) https://doi.org/10.1063/5.0017378
  * (10) P. R. Maloney, N. G. Czakon, P. K. Day, T. P. Downes, R. Duan and J. Gao, J. Glenn, S. R. Golwala, M. I. Hollister, H. G. LeDuc, B. A. Mazin, S. G. McHugh, O. Noroozian. H. T. Nguyen, J. Sayers, J. A. Schlaerth, S. Siegel, J. E. Vaillancourt, A. Vayonakis, P. Wilson and J. Zmuidzinas, Millimeter, submillimeter, and far-infrared detectors and instrumentation for astronomy V, MUSIC for sub/millimeter astrophysics, SPIE publisher, 124-134, (2010).
  * (11) A. Monfardini et al., A dual-band millimeter-wave kinetic inductance camera for the IRAM 30 m telescope, 2011 ApJS 194 24 (2011).
  * (12) N. Galitzki et al., Millimeter, submillimeter, and far-infrared detectors and instrumentation for astronomy VII, International Society for Optics and Photonics, 108-118, (2016).
  * (13) E. L. Wright, G. Hinshaw, C. L. Bennett, Producing Megapixel Cosmic Microwave Background from Differential Radiometer Data, Astrophysical Journal Letters v.458, p.L53 (1996).
  * (14) D. Maino, C. Burigana, K. M. Górski, N. Mandolesi, M. Bersanelli, Removing 1/f noise stripes in cosmic microwave background anisotropy observations, Astronomy and Astrophysics, v.387, p.356-365 (2002).
  * (15) N. J. Miller et al., Recovery of large angular scale CMB polarization for instruments employing variable-delay polarization modulators, ApJ 818 151 (2016).
  * (16) Steven Hailey-Dunsheath, Reinier M. J. Janssen, Jason Glenn, Charles M. Bradford, Joanna Perido, Joseph Redford, Jonas Zmuidzinas, Kinetic inductance detectors for the Origins Space Telescope, J. of Astronomical Telescopes, Instruments, and Systems, 7(1), 011015 (2021).
  * (17) Johnson, B.R., Flanigan, D., Abitbol, M.H. et al. Development of multi-chroic MKIDs for next-generation CMB polarization studies. J Low Temp Phys 193, 103–112 (2018).
  * (18) J. van Rantwijk, M. Grim, D. van Loon, S. Yates, A. Baryshev and J. Baselmans, ”Multiplexed readout for 1000-pixel arrays of microwave kinetic inductance detectors,” in IEEE Transactions on Microwave Theory and Techniques, vol. 64, no. 6, pp. 1876-1883, June 2016, doi: 10.1109/TMTT.2016.2544303.
  * (19) Clemens Müller, Jürgen Lisenfeld, Alexander Shnirman, and Stefano Poletto, Interacting two-level defects as sources of fluctuating high-frequency noise in superconducting circuits, Phys Rev B 92, 035442 (2015).
  * (20) J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Lucero, M. Neeley, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, and John M. Martinis, Scalable in situ qubit calibration during repetitive error detection, Phys Rev A 94, 032321 (2016)
  * (21) John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann, K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C. Yu, Decoherence in Josephson Qubits from Dielectric Loss, PRL 95, 210503 (2005)
  * (22) Jiansong Gao and Jonas Zmuidzinas, Noise properties of superconducting coplanar waveguide microwave resonators, App Phys Lett 90, 102507 (2007)
  * (23) Jiansong Gao, Miguel Daal, John M. Martinis, Anastasios Vayonakis, Jonas Zmuidzinas, Bernard Sadoulet, Benjamin A. Mazin, Peter K. Day, and Henry G. Leduc, A semiempirical model for two-level system noise in superconducting microresonators, Appl Phys Lett 92 212504 (2008)
  * (24) L. Faoro and L. B. Ioffe, Interacting tunneling model for two-level systems in amorphous materials and its predictions for their dephasing and noise in superconducting microresonators, Phys Rev B 91, 014201 (2015).
  * (25) M Schechter et al., Nonuniversality and strongly interacting two-level systems in glasses at low temperatures, New J. Phys. 20 063048 (2018).
  * (26) Yoni Shalibo, Ya’ara Rofe, David Shwa, Felix Zeides, Matthew Neeley, John M. Martinis, and Nadav Katz, Lifetime and Coherence of Two-Level Defects in a Josephson Junction, Phys. Rev. Lett. 105, 177001(2010).
  * (27) R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and John M. Martinis, Decoherence in Josephson Phase Qubits from Junction Resonators, Phys. Rev. Lett. 93, 077003 (2004)
  * (28) M. Steffen, M. Sandberg, S. Srinivasan, Recent research trends for high coherence quantum circuits, Supercond. Sci. Technol. 30, 030301 (2017).
  * (29) Chu, Y. et al. Suspending superconducting qubits by silicon micromachining. Phys. Lett. 109, 112601 (2016).
  * (30) Burnett, J. et al. Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator. Nat. Commun. 5, 4119 (2014).
  * (31) Neill, C. et al. Fluctuations from edge defects in superconducting resonators. Appl. Phys. Lett. 103, 072601 (2013).
  * (32) Stephen J. C. Yates, Juan Bueno, Akira Endo, Andrey M. Baryshev, Lorenza Ferrari, Vignesh Murugesan, David J. Thoen, Jochem J. A. Baselmans, On the design and performance of very large MKID arrays, Proceedings Volume 11453, Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy X; 114530C (2020).
  * (33) De Visser, P.J., Quasiparticle dynamics in aluminium superconducting microwave resonators, Delft University of Technology (2014).
  * (34) Benjamin A. Mazin, Kieran O’Brien, Sean McHugh, Bruce Bumble, David Moore, Sunil Golwala, Jonas Zmuidzinas, ARCHONS: a highly multiplexed superconducting optical to near-IR camera, Proceedings Volume 7735, Ground-based and Airborne Instrumentation for Astronomy III; 773518 (2010) https://doi.org/10.1117/12.856440
  * (35) Hanhee Paika) and Kevin D. Osborn, Reducing quantum-regime dielectric loss of silicon nitride for superconducting quantum circuits, Appl. Phys. Lett. 96, 072505 (2010).
  * (36) M. R. Vissers, J. Hubmayr, M. Sandberg, S. Chaudhuri, C. Bockstiegel, and J. Gao, Frequency-tunable superconducting resonators via nonlinear kinetic inductance, Appl. Phys. Lett. 107, 062601 (2015).
  * (37) J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Decoherence in josephson qubits from dielectric loss, Phys. Rev. Lett. 95, 210503 (2005).
  * (38) S. J. Weber, K. W. Murch, D. H. Slichter, R. Vijay, and I. Siddiqia, Single crystal silicon capacitors with low microwave loss in the single photon regime, Appl. Phys. Lett. 98, 172510 (2011).
  * (39) Cataldo, G., Wollack, E.J., et al., Analysis and calibration techniques for superconducting resonators, Review of Scientific Instruments, Vol. 86:1 ; Iss. 1, 013103 (2015).
  * (40) C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation, Phys. Rev. A 31, 3761 (1985).
  * (41) D. F. Walls and G. Milburn, Quantum Optics (Springer, Berlin, 2008).
  * (42) J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system, Nature 454, 315-318 (2008).
  * (43) J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, Dressed collective qubit states and the Tavis-cummings model in circuit QED, Phys. Rev. Lett. 103, 083601 (2009).
  * (44) B. Sarabi, A. N. Ramanayaka, A. L. Burin, F. C. Wellstood, and K. D. Osborn, Cavity quantum electrodynamics using a near-resonance two-level system: Emergence of the Glauber state, Appl. Phys. Lett. 106, 172601 (2015).
  * (45) Juha Leppäkangas, Jan David Brehm, Ping Yang, Lingzhen Guo, Michael Marthaler, Alexey V. Ustinov, and Martin Weides, Resonance inversion in a superconducting cavity coupled to artificial atoms and a microwave background, Phys. Rev. A 99, 063804 (2019).
  * (46) T. R. Stevenson, Limits on the sensitivity of spherical gravitational wave detectors and on the accuracy of reconstructed signals, Phys. Rev. D 56, 564 (1997).
  * (47) M. S. Khalil, M. J. A. Stoutimore, F. C. Wellstood, and K. D. Osborn, An analysis method for asymmetric resonator transmission applied to superconducting devices, Journal of Applied Physics 111, 054510 (2012).
  * (48) B. Sarabi, Cavity quantum electrodynamics of nanoscale two-level systems, University of Maryland, (2014).
  * (49) S. Hunklinger and A. K. Raychaudhuri, Thermal and elastic anomalies in glasses at low temperatures. Prog. Low Temp. Phys., 9:267 344, (1986).
  * (50) J. Classen, C. Enss, C. Bechinger, G. Weiss, and S. Hunklinger, Low frequency acoustic and dielectric measurements on glasses. Annalen der Physik, 506(5):315335, (1994).
  * (51) T. Capelle et al., Probing a two-level system bath via the frequency shift of an off-resonantly driven cavity, Phys. Rev. Applied 13, 034022 (2020).