††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: Present address: IBM T.J.
Watson Research Center, Yorktown Heights, New York 10598, USA

# Magnetic Field Tolerant Superconducting Spiral Resonators for Circuit QED

Mihirangi Medahinne  University of Rochester, Rochester, NY, 14627 USA  Yadav
P. Kandel  University of Rochester, Rochester, NY, 14627 USA  Suraj Thapa
Magar  University of Rochester, Rochester, NY, 14627 USA  Elizabeth Champion
University of Rochester, Rochester, NY, 14627 USA  John M. Nichol  University
of Rochester, Rochester, NY, 14627 USA  Machiel S. Blok
<EMAIL_ADDRESS>University of Rochester, Rochester, NY, 14627 USA

###### Abstract

We present spiral resonators of thin film niobium (Nb) that exhibit large
geometric inductance, high critical magnetic fields and high single photon
quality factors. These low loss geometric inductors can be a compelling
alternative to kinetic inductors to create high-impedance superconducting
devices for applications that require magnetic fields. By varying the spiral
pitch, we realize resonators with characteristic impedances ranging from
3.25-7.09 k$\Omega$. We measure the temperature and magnetic field dependent
losses and find that the high-impedance resonators maintain an intrinsic
quality factor above $\sim 10^{5}$ for parallel magnetic fields of up to 1 T.
These properties make spiral Nb resonators a promising candidate for quantum
devices that require circuit elements with high impedance and magnetic field
resilience.

##  I Introduction

High-impedance quantum circuit elements consisting of large inductors with
small self-capacitance are an important building block for noise-protected
superconducting qubits, quantum sensors, and hybrid quantum systems. For
example, large-impedance inductors are used to engineer the inductive energy
independent of the charging and Josephson energies leading to delocalized
qubit wavefunctions in phase that are more resilient to flux noise [1, 2].
Furthermore, a resonator with high characteristic impedance has large zero-
point voltage fluctuations, leading to enhanced coupling between microwave
photons and phonons (either using piezoelectric materials [3] or static
electric fields [4]), polar molecules [5] and single electrons [6].

One common approach to making high-impedance inductors is to enhance the
kinetic inductance with disordered superconductors like Nb-N [7], Nb-Ti-N [8,
9] and Ti-N [10], granular aluminum [11], or by forming a large array of
Josephson junctions [12]. However, integration of new materials into devices
can be challenging because they have to be compatible with existing
fabrication processes and experimental operating conditions such as magnetic
fields for coupling to spin qubits. As an alternative, a large geometric
inductance can be achieved through the mutual inductance between the
concentric loops of a planar spiral to realize an impedance that surpasses the
superconducting resistance quantum of $h/(2e)^{2}=6.5\,k\Omega$ [13]. The
impedance of these spiral inductors is determined by their geometry, leading
to a high degree of design control and flexibility. Until now, high-impedance
spiral resonators were made from superconducting aluminum whose critical field
is about 10 mT [14, 15]. This is prohibitively small for most quantum
information protocols that require magnetic fields [8, 16].

Here, we fabricate and characterize high-impedance spiral resonators with Nb
thin films which offer large critical temperature and magnetic field
resilience. We experimentally extract the ratio of kinetic and geometric
inductance to verify that the primary contribution to the resonator inductance
is geometric. We optimize our fabrication recipe to realize a high quality
factor $Q_{\rm int}$ of $\sim 10^{5}$ that is maintained for in-plane magnetic
fields up to $\sim$ 1 T. We analyze the contributions of the noise by
performing measurements at varying photon numbers and temperature. We find
that at low temperatures and low photon numbers, $Q_{\rm int}$ is dominated by
Two-Level System (TLS) losses. With increasing photon number, the TLSs
saturate and the losses are dominated by other, power-independent losses. By
exploring additional cleaning steps and better etch processes in spiral
fabrication we can potentially mitigate these TLS losses [17]. We expect that
$Q_{\rm int}$ can be further improved by performing in-depth analyses of power
independent losses and implementing techniques to reduce them during
measurement and fabrication [17].

##  II Design and Fabrication

Figure 1:  Experimental setup of the device. (a) Schematic diagram of the
measurement device with a spiral inductively coupled to the CPW feed line.
Around its fundamental resonance frequency, a spiral resonator can be modeled
as an LC circuit where the total inductance L has contributions from the
geometric inductance $L_{g}$ and the kinetic inductance $L_{k}$. (b) An
optical microscope image of the measurement device (in grayscale). (c-d)
Scanning electron microscope (SEM) images of one of the spiral resonators on
the measurement device. (e) Magnitude (top) and phase (bottom) of a measured
spiral resonator (R0 in the text) at 60 mK. Measured data are shown in green.
The magnitude of the measurement has a Lorentzian-shaped dip, and phase has a
2$\pi$ shift. We extract the resonator parameters by fitting a theoretical
model for $S_{11}$, shown in black.

In this work, we design Nb spiral resonators where characteristic impedance
$Z_{C}$ is varied by adjusting the mutual inductance and capacitance between
the concentric loops. At the fundamental resonance frequency $f_{g}$, a planar
spiral can be modeled as an LC oscillator, $Z_{C}=\sqrt{L/C}=2\pi f_{0}L$,
where the total inductance $L$ has both kinetic and geometric contributions
and $C$ is the capacitance of the spiral [Figure 1(a)]. Therefore, by
increasing the mutual inductance between the concentric loops of the spiral,
we can enhance $Z_{C}$.

An analytic expression for the geometric contribution to the spiral inductance
$L_{g}$, derived with the current sheet method is [18, 13]

$L_{g}=\tfrac{\mu_{0}n^{2}d_{\rm
av}}{2}\left(ln\left(2.5/\rho\right)+0.2\rho^{2}\right),$ (1)

where $\mu_{0}$ is the vacuum permeability, $n$ is the number of turns,
$d_{\rm av}$ is the average of the inner diameter $d_{\rm in}$ and the outer
diameter $d_{\rm out}$, and $\rho$ is the fill ratio of the spiral and it is
given as $\rho=\left(d_{\rm out}-d_{\rm in}\right)/\left(d_{\rm out}+d_{\rm
in}\right)$ [Figure 1(a)]. As $d_{\rm out}$ scales linearly with the number of
turns $n$, it can be seen from Equation 1 that $L_{g}$ scales with the number
of turns n as $L_{g}\sim n^{3}$. For planar spirals, the analytical formula
for the fundamental resonance frequency $f_{g}$ is [19, 13]

$f_{g}=\xi\tfrac{c_{0}}{\sqrt{\epsilon_{eff}}}\tfrac{2p}{\pi\left(d_{\rm
in}+2np\right)^{2}},$ (2)

where $\xi$ is a shape-dependent constant with $\xi=0.81$ for planar spirals,
$c_{0}$ is the speed of light in vacuum, ${\epsilon_{\rm eff}}$ is the
effective dielectric constant, and the pitch $p$ is the wire width plus the
gap between adjacent turns of the spiral. Here we can see that $f_{g}$ scales
with the pitch p and number of turns $n$ as $f_{0}\sim n^{-2}p^{-1}$ and the
impedance $Z_{C}=2\pi f_{0}L$ scales as $Z_{C}\sim p^{-1}n$ [13]. Because
there is no simple geometric parameter that can tune the impedance
independently of frequency, we chose to design high-impedance resonators by
decreasing the spiral pitch, while adjusting the number of turns to maintain
resonance frequencies $f_{g}$ in the 4-8 GHz range. Smallest pitch used in
this work is 300 nm and is limited by our nano-fabrication tools.

We design the spirals with a pitch ranging from 300 nm to 1000 nm with a
characteristic impedance between 3-7 k$\Omega$ (Table 1). Four spiral
resonators at different frequencies are measured in reflection by inductively
coupling them to a shared shorted transmission line as shown in Figure 1(b).
In addition to the four spiral resonators, our design features a co-planar
waveguide (CPW) resonator with width = 20 $\mu$m and gap = 10 $\mu$m that we
use to benchmark our design and the fabrication process.

The devices are fabricated on a 50 nm Nb film deposited on an intrinsic
silicon substrate (Star Cryoelectronics). Figures 1(c-d) show SEM images of a
spiral resonator with a 500 nm pitch (See supplementary materials for more
details).

Table 1: Designed and experimentally extracted parameters of the spiral resonators. We calculate $f_{g}$ and $L_{g}$ from Equations 1 and 2. The $Z_{c}(=2\pi f_{g}L_{g})$ calculation is described in the main text. $f_{0}$ and $\alpha$ are the fit results from the QP analysis with Equation 3. $Q_{\rm{TLS,0}}$ is the fit result from the TLS analysis with Equations 5 and 6. For $f_{0}$, $\alpha$, and $Q_{\rm{TLS,0}}$, the first value is from the fit to the internal quality factor $Q_{\rm{int}}$ , and the second value is from the fit to the resonance frequency $f_{0}$.  label | p($\mu$m) | $Z_{c}(k\Omega)$ |  $L_{g}$(nH) |  $f_{g}$(GHz) |  $f_{0}$(GHz) | $\alpha(\%)$ |  $L_{k}$(nH) | $Q_{\rm{TLS,0}}(\times 10^{5})$
---|---|---|---|---|---|---|---|---
R0 | 0.3 | 7.09 | 244 | 4.61 | 4.04 | 3.2-0.5 | 8.0-1.2 | 0.36 - 3.2
R1 | 0.3 | 6.67 | 205 | 5.19 | 4.53 | 1.5-0.1 | 3.2-0.3 | 1.2 - 7.4
R2 | 0.5 | 5.00 | 142 | 5.61 | 5.00 | none-8.0 | none-12 | none-2.7
R3 | 1.0 | 3.25 | 78 | 6.67 | 5.95 | 6.0-5.5 | 5.0-4.5 | 2.3 - 7.2

##  III Measurements and Characterization

The fabricated chips are mounted on a sample board and cooled down in a
dilution refrigerator at milli-Kelvin temperatures. Figure 1(e) shows a
typical reflection measurement of a spiral resonator with a 2$\pi$ phase shift
indicative of the overcoupled regime ($Q_{\rm{int}}>Q_{\rm{e}}$). From a fit
to the complex $S_{11}$ signal we extract an internal quality factor
$Q_{\rm{int}}$ and the fundamental resonance frequency. Next, we will
characterize the spirals’ geometric inductance and main loss mechanisms from
reflection measurements at different powers and temperatures. We will then
test their magnetic field resilience in a parallel magnetic field.

###  III.1 Quantifying the Total Inductance

The geometric inductance of the spiral resonators can be extracted by
measuring how the fundamental resonance frequency
$f_{0}=\frac{1}{2\pi\sqrt{LC}}$ shifts with temperature due to changes in the
inductance. The total inductance is $L=L_{g}+L_{k}$ where $L_{g}$ and $L_{k}$
are the geometric inductance and the kinetic inductance of the superconducting
spiral wires respectively. At higher temperatures, the environment has more
energy to break up Cooper pairs, leading to a reduced Cooper pair density. As
a result, the kinetic inductance increases with temperature while the geometry
inductance remains constant. The resulting temperature-dependent frequency
shift can be modeled by Bardeen–Cooper–Schrieffer theory [20, 13]:

Figure 2: Shifts in the spiral frequency and quality factor characteristic of
QPs. (Top) Frequency shift $\delta f$, and (bottom) internal qualify factor
$Q_{\rm int}$ of the resonators as temperature is varied. Solid lines are the
fits to Equation 3. Error bars of $\delta f$ and $Q_{\rm int}$ at each data
point denote the statistical error, which is estimated by fitting a
theoretical model for $S_{11}$ as shown in Figure 1(e).

$\displaystyle\frac{\delta f(T)}{f_{0}(0)}$
$\displaystyle=-\frac{\alpha\gamma}{2}\frac{\delta\sigma_{2}(T,\Delta)}{\sigma_{2}(T,\Delta)}$
(3a) and $\displaystyle\delta Q^{-1}_{\text{int}}(T)$
$\displaystyle=\alpha\gamma\frac{\delta\sigma_{1}(T,\Delta)}{\sigma_{2}(T,\Delta)},$
(3b)

where $\delta f(T)=f_{0}(T)-f_{0}(0)$ is the relative fundamental resonance
frequency shift; $\delta Q^{-1}_{\rm int}=Q^{-1}_{\rm int}(T)-Q^{-1}_{\rm
int}(0)$ is the relative internal quality factor shift;
$\alpha=L_{k}/\left(L_{k}+L_{g}\right)$ is the fraction of the kinetic
inductance $L_{k}$ to total inductance $L_{k}+L_{g}$; $\gamma$ is a material
and film dependent parameter (for thin film Nb, $\gamma$ = -1) [21];
$\sigma_{1}$ and $\sigma_{2}$ are the real and imaginary parts of the total
conductivity $\sigma$ of the superconducting film respectively; and
$\Delta=1.76k_{B}T_{c}$ [21] is the superconducting gap of Nb where $k_{B}$
and $T_{c}$ are the Boltzmann constant and the critical temperature of the Nb
film respectively.

From $L_{g}$ we calculate the characteristic impedance $Z_{c}=2\pi f_{g}L_{g}$
and find $Z_{c}=3.25$ k$\Omega$ for the spiral resonator with the largest
pitch and $Z_{c}=7.09$ k$\Omega$ for the spiral resonator with the smallest
pitch (Table 1). We extract the ratio of kinetic to total inductance,
$\alpha$, by fitting the $f_{0}$ data to Equation 3a with $f_{0}(0),\alpha$
and $T_{c}$ as the fitting parameters and by fitting the $Q_{\rm int}$ data to
Equation 3b with $Q_{\rm int}(0),\alpha$ and $T_{c}$ as the fitting parameters
(Table 1). We find $\alpha$ values ranging from 0.1-8$\%$ as expected for
resonators with inductances primarily due to geometric inductance. R2 shows
reflection measurements that could not be modeled with a $S_{11}$ Lorentzian
dip (Supplementary Figure 1). We hypothesize that this is due to impedance
mismatches that cannot be corrected in reflection measurements [17].
Therefore, the R2 $Q_{\rm int}$ values are not included in Figure 2.

Now we compare the extracted $\alpha$ and $T_{c}$ values obtained with
Equation 3a to those obtained with Equation 3b [Supplementary Figure 3(a)]. We
find that the $f_{0}$ fit results show $\alpha$ values of 0.5$\%$, 0.1$\%$,
8.0$\%$, and 5.5$\%$ for R0, R1, R2 and R3 respectively. We independently
extract $\alpha$ from $Q_{\rm int}$ fits and find $\alpha$’s between 1.5-6$\%$
(Table 1). We attribute the discrepancy between the two methods to the
sensitivity of $Q_{\rm int}$ to two-level systems. At lower temperatures, the
changes in $f_{0}$ and $Q_{\rm int}$ with temperature are due to the effects
of TLSs, which are not accounted for in Equation 3. If this low-temperature
change is small compared with the overall change in $f_{0}$ and $Q_{\rm int}$
caused by the high-temperature quasiparticles (QPs), the TLS effect on $f_{0}$
and $Q_{\rm int}$ is masked by the QP losses, and so the fits are robust to
the low-temperature changes. We hypothesize that for R0 and R1, the fits are
not as robust to TLS effects as they are for R3, thus giving different results
for $\alpha$ and $T_{c}$ for the $f_{0}$ and $Q_{\rm int}$ fits. Despite these
uncertainties, we can conclude that for our resonators the majority of the
inductance is due to the geometric contribution.

Next we compare the extracted $T_{c}$ to four-point measurements
[Supplementary Figure 3(a)]. We find that the $f_{0}$ fit results show $T_{c}$
values of 3.5 K, 2.1 K, 7.9 K, and 7.9 K for R0, R1, R2 and R3 respectively.
For $Q_{\rm int}$ fits, these values are 5.4 K, 3.9 K, and 8.3 K (for R0, R1
and R3 respectively). We attribute the deviations in $T_{c}$ between the two
fits to the above mentioned effects of low-temperature TLSs. Four-point
measurement $T_{c}$ values are 7.5 K, 7.9 K, and 8.1 K for 50 $\mu$m long wire
with wire width of 150 nm (designed wire width for R0 and R1), 500 nm
(designed wire width for R2), and 1 $\mu$m (designed wire width for R3)
respectively. We attribute the differences in $T_{c}$ to imperfections in
spiral fabrication, making the spiral wire dimensions different from the
designed wire widths. Additionally, for small features such as our spirals,
the value of $T_{c}$ is sensitive to geometry, making the $T_{c}$ values of
four-point measurements different from the spiral $T_{c}$ values. To extract
the characteristic impedance from the measurements, we estimate the kinetic
inductance $L_{k}$ for each resonator by combining the fitted $\alpha$ with
our analytical estimate for the geometric inductance based on the designed
dimensions (Equation 1). We find kinetic inductances on the order of a few nH
and calculate geometric inductances between 78 and 244 nH (Table 1).

We do a self-consistency check for the extracted spiral parameters by
comparing the resonance frequency $f_{0}$ to the designed analytical frequency
$f_{g}$. The analytical model, which only considers the geometric inductance,
can be adjusted to include the kinetic inductance:
$f_{0}=f_{g}\sqrt{L_{g}/(L_{k}+L_{g})}$ [13]. For each spiral, we calculate
$f_{g}$ by substituting $L_{k}$, $L_{g}$, and the fitted $f_{0}$ in this
equation. We find these estimates of $f_{g}$ to be 7 - 13$\%$ lower than the
$f_{g}$ calculated from the designed dimensions (Equation 2). One possible
explanation for the discrepancy is variations in spiral dimensions due to an
imperfect fabrication processes. When the fabricated spirals do not have the
designed gap ($s$) to wire ($w$) ratio of 1:1, the current-sheet method
(Equation 1) gives a maximum of 8$\%$ error in $L_{g}$ for $s\leq 3w$ [18]. We
confirm that the fabricated spirals satisfy this geometric condition with SEM
images. Other possible sources for the discrepancy might be the change in
self-capacitance due to deviations in the $s:w$ ratio of the fabricated
spirals or parasitic capacitance and inductance to the transmission line and
ground plane.

###  III.2 Characterization of Loss Mechanisms

Having determined the impedance of the fabricated spiral resonators, next we
characterize their dissipation mechanisms. We distinguish three common sources
of loss for superconducting resonators arising from TLSs, QPs, and other
temperature and power-independent losses for example radiative decay or
packaging. First, we measure the power-dependent quality factor which shows
behavior characteristic of TLS loss: the quality factor saturates to a single
photon level at low powers and the TLS bath becomes saturated by the photons
in the resonator as evidenced by a rise in $Q_{\text{int}}$ in the high power
regime [Figure 3(b)]. We fit the data into a simplified model for TLS-induced
dielectric loss [21, 22]:

$Q_{\text{int}}^{-1}=Q_{\text{TLS,0}}^{-1}\frac{\tanh{\left(\frac{hf_{0}}{2k_{B}T}\right)}}{(1+\frac{\langle
n\rangle}{n_{c}})^{\beta}}+Q_{\text{other}}^{-1},$ (4)

where $Q_{\rm{TLS,0}}$ is a temperature-independent TLS quality factor,
$\langle n\rangle$ is the average number of photons in the resonator, $n_{c}$
is the critical number of photons in the resonator, the exponent $\beta$ is a
parameter to account for TLS interactions and $Q_{\text{other}}$ takes into
account all non-TLS related loss. A typical power dependence curve is shown in
Figure 3(b) and we find single photon quality factors $Q_{\text{int}}$ between
$0.9\text{x}10^{5}$ and $2.1\text{x}10^{5}$ for our spiral devices
(Supplementary Figure ).

Figure 3: Shifts in the spiral frequency and quality factor characteristic of
TLS. (a) (Top) Frequency shift $\delta f$, (bottom) internal qualify factor
$Q_{\rm int}$ of the resonator (R3 in the text) as temperature is varied.
Solid lines represent fits to Equation 5. (b) $Q_{\rm int}$ as resonator
average photon number is varied. The solid line represents the fit to Equation
4.

Next, we study the temperature- and power-dependent noise, which requires a
more comprehensive noise model to account for the temperature dependence of
the TLS loss tangent and QPs. Following [23], we will model the resonance
frequency shift and the internal quality factor shift:

$\displaystyle\frac{\delta f(T)}{f_{0}(0)}$
$\displaystyle={\left({\frac{\delta
f(T)}{f_{0}(0)}}\right)}_{\text{TLS}}{+}{\left({\frac{\delta
f(T)}{f_{0}(0)}}\right)}_{\text{QP}}$ (5a) and $\displaystyle
Q_{\text{int}}^{-1}$ $\displaystyle=Q_{\text{TLS}}(\langle
n\rangle,T)^{-1}+Q_{\text{QP}}(T)^{-1}+Q_{\text{other}}^{-1}.$ (5b)

Here the TLS resonance frequency shift and the quality factor are [24, 23]

$\displaystyle\begin{aligned} &{\left({\frac{\delta
f(T)}{f_{0}(0)}}\right)}_{\text{TLS}}=\frac{1}{\pi
Q_{\text{TLS,0}}}{\text{Re}}\left[\Psi\left(\frac{1}{2}{+}i\frac{hf_{0}}{2\pi
k_{B}T}\right)\right.\\\ &\qquad\qquad\qquad-{\text{ln
}\left(\frac{hf_{0}}{2\pi
k_{B}T}\right)}\left.\vphantom{\frac{1}{2}}\right]\end{aligned}$ (6a) and
$\displaystyle Q_{\text{TLS}}(\langle
n\rangle,T)=Q_{\text{TLS,0}}\frac{\sqrt{1+(\frac{\langle
n\rangle^{\beta_{2}}}{DT^{\beta_{1}}})\tanh{\left(\frac{hf_{0}}{2k_{B}T}\right)}}}{\tanh{\left(\frac{hf_{0}}{2k_{B}T}\right)}}.$
(6b)

Now the TLS quality factor has temperature and photon number dependence. Here
$\Psi$ is the complex digamma function; and $D$, $\beta_{1}$ and $\beta_{2}$
are parameters that characterize the TLS saturation. ${\left({\frac{\delta
f(T)}{f_{0}(0)}}\right)_{\rm QP}}$ and $Q_{\rm QP}$ are the resonance
frequency shift and the quality factor associated with QP losses given in
Equation 3.

We extract $Q_{\rm{TLS,0}}$ by fitting the $Q_{\rm int}$ data to Equation 5b
with $Q_{\rm{TLS,0}}$, $\alpha$, $T_{c}$, $D$, $\beta_{1}$, $\beta_{2}$ and
$Q_{\rm other}$ as fitting parameters; and by fitting the $f_{0}$ data with
Equation 5a with $Q_{\rm{TLS,0}}$, $\alpha$, $T_{c}$, and $f_{0}(0)$ as
fitting parameters. At low temperatures, the internal quality factor is
limited by the TLSs for low photon number, while at high enough excitation
power, the TLS bath is saturated and the quality factor is dominated by
residual losses $(Q_{\rm other}$). At higher temperatures, the losses are
dominated by QPs independent of power. Therefore we hypothesize that at low
temperatures the temperature and power sweeps, modeled with Equation 5, do not
provide accurate $\alpha$ and $T_{c}$.

We compare $Q_{\rm{TLS,0}}$ of the $f_{0}$ and $Q_{\rm int}$ fits
[Supplementary Figure 3(b)]. We find that $Q_{\rm{int}}$ fit results are
significantly lower than $f_{0}$ fits for R0, R1, and R3 respectively. We
hypothesize that masking of the TLS-induced $f_{0}$ shift by the QP-induced
$f_{0}$ shift, along with having fewer data points than $Q_{\rm int}$ fits due
to the power independence of the $f_{0}$ shift, cause these deviations between
the two fits.

###  III.3 Magnetic Field Compatibility

Next, we measure the spiral resonators’ characteristics in the presence of an
applied magnetic field.

Figure 4: Performance of the spiral resonators in the external magnetic field
$B_{\|}$. For comparison CPW resonator performances are included. (Top) The
internal quality factor $Q_{\rm int}$ plotted against $B_{\|}$. The solid
lines are guides to the eye. (Bottom) The resonance frequency $f_{0}$ plotted
against $B_{\|}$. Quadratic fits for each spiral resonator are shown with
solid lines. Vertical dashed lines indicate where $Q_{\rm int}$ starts to
degrade drastically due to vortex effects. Figure 5: Performance of spiral
resonators in external magnetic fields $B_{\|}$ near the ESR dip. (a) The
internal quality factor $Q_{\rm int}$ of the resonators near the electron spin
resonance condition. (b) $B_{\|}$ of each resonator where the $Q_{\rm int}$
dip occurs, plotted against its respective resonance frequency $f_{0}$. The
solid blue line is the fit to $hf_{0}=g\mu_{B}B_{\|}$ as described in the
text.

For all the spirals, the extracted internal quality factor $Q_{\rm int}$ is
$\sim 10^{5}$ at the zero magnetic field. The spiral with larger pitch (R3)
retains this higher $Q_{\rm int}$ up to higher in-plane field than those with
smaller pitch (R0 and R1). We benchmark the performance of the spirals in the
external magnetic field by comparing them to simultaneous measurements of the
CPW resonator. We see that the spirals have better magnetic field performance
than the CPW resonator. We attribute this to the suppression of vortices with
decreasing device width. We note, however, that despite having a larger wire
width than R0 and R1, R3 has better magnetic field resilience. We attribute
this to the fact that smaller wire widths are more sensitive to impurities,
leading to a decrease in critical magnetic field strength independent of
vortex suppression [25]. Finally, as discussed above, the low-magnetic field
reflection measurements for R2 do not show a Lorentzian dip, which we
attribute to impedance mismatches. We do, however, observe a Lorentzian dip
for magnetic fields larger than $\sim$1 T, permitting the extraction of
$Q_{\rm{int}}$ [Figure 4 (top)]. We interpret this as the impedance mismatch
being reduced in the presence of a strong magnetic field.

One feature apparent in Figure 4 is a sharp dip in $Q_{\rm int}$ around 100 -
200 mT. We investigated this feature by first warming the sample up above Nb
transition temperature and cooling it back down, thus eliminating any magnetic
field hysteresis effects caused by the previous magnetic field sweep. We then
measured a higher-resolution magnetic field sweep in that region and show the
results in Figure 5. We attribute this dip to the electron spin resonance
(ESR) caused by the resonator coupling to the magnetic impurities in the
substrate and the other interfaces [9]. In the presence of a static magnetic
field, these impurities will have a Zeeman splitting with an energy difference
of $g\mu_{B}B_{\|}$, where $g$ and $\mu_{B}$ are the Land$\rm\acute{e}$
g-factor and the Bohr magneton respectively. As shown in Figure 5(a) , at the
ESR condition of $hf_{0}=g\mu_{B}B_{\|}$ where $h$ is the Planck constant,
there is a sharp increase in the resonator losses indicated by the $Q_{\rm
int}$ minima. In Figure 5(b) we plot the relationship between the magnetic
field strengths $B_{\|}$ at which these minima occur and the corresponding
resonance frequency $f_{0}$ of each resonator.

We fit this data to the Zeeman resonance condition $hf_{0}=g\mu_{B}B_{\|}$ and
obtain a Land$\rm\acute{e}$ g-factor of 1.97$\pm$0.01, consistent with the
hypothesis that the dips are due to the resonators coupling to free electron
spins. The spirals show an enhancement in $Q_{\rm int}$ with an increasing
magnetic field following the ESR dip. There are two hypotheses regarding this
phenomenon: the Abrikosov vortices trap the lossy QPs or there is a
contribution from magnetic impurities to the resonator losses before the ESR
dip [26].

The bottom graph of Figure 4 shows the shift in $f_{0}$ against $B_{\|}$. At
lower magnetic fields, the decrease in $f_{0}$ with increasing $B_{\|}$ is due
to the increase in $L_{k}$ as a consequence of a decreasing Cooper pair
density. This can be fitted to a quadratic function as shown by the solid
lines in Figure 4 [9, 8]. At higher magnetic fields, the frequency shift
deviates from the quadratic trend when the resonator’s dominant loss mechanism
shifts from QPs to vortex losses [26, 25, 27]. The magnetic field response of
the vortices is characterized by a complex resistivity, with the real part
contributing to vortex-related losses and the imaginary part corresponding to
their reactive response [27]. When the frequency shift is dominated by
vortices but not by the QPs generated by the external magnetic field, the
frequency shift deviates from the quadratic form. Here, we have fitted the
data to the quadratic model until each resonator’s quality factor starts to
decrease drastically, as indicated by vertical grey lines in Figure 4. We see
that the frequency shift deviates from the quadratic function once the spirals
start to decay, confirming that the losses arise from the vortices. In
contrast to the spirals, we see that $Q_{\rm int}$ of the CPW resonator
decreases drastically with increasing magnetic field from the beginning. This
indicates that the dominant losses arise from the vortices from the start. We
confirm this further by observing that the decrease in $f_{0}$ cannot be
fitted to a quadratic function in a similar manner to spirals [Figure
4(bottom)].

##  IV Conclusions

In this work we have fabricated high-impedance Nb geometric resonators with
impedances of 3-7 k$\Omega$. In the single photon limit these resonators show
a quality factor of $\sim 10^{5}$, which they retain in external in-plane
magnetic fields up to $\sim$ 1 T. Further in the study, we probe resonator
loss mechanisms by varying the temperature and photon number and measuring the
resonator response. We confirm that at low temperatures, the losses are
dominated by TLS losses. At high temperatures and powers the TLSs saturate,
and therefore no longer contribute to the resonator losses. In this regime we
identify QPs as the dominant loss mechanism.

Our results demonstrate the feasibility of operating high-impedance
superconducting resonators in strong magnetic fields. Because our fabrication
methods are compatible with existing material platforms and architectures, we
expect that these resonators could enable hybrid quantum devices. In
particular we propose leveraging the high impedance and high magnetic field
resilience realized in our spiral resonators to coherently couple microwave
photons to electron spin qubits. Furthermore, high-impedance Nb resonators can
be easily integrated into more conventional circuit QED systems to explore
novel qubit architectures intrinsic noise protection and high coherence.

##  V Acknowledgment

This work is supported by the National Science Foundation under Grant No.
2210309.

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## Supplementary Information

##  I Chip Fabrication

A 50 nm Nb film was deposited on a 4” double-sided polished intrinsic silicon
wafer by Star Cryogenics. Then, the wafer was diced into 10 mm$\times$ 10 mm
square chips. Then a diced chip was covered with $\sim$ 450 nm thick MMA EL 11
spun at 4200 rpm. After spinning the resist, the chip was baked at
$140-150^{\circ}$ C for 10 minutes. The spirals along with the CPW resonator
and the on-chip 2D transmission lines were patterned with Elionix ELS-S50EX.
After the exposure, the chip was developed in MIBK/IPA 1:3 developer at
$2^{\circ}$ C for 1 minute and then rinsed with IPA to stop the development.
Next, the reactive iron etch was done with South Bay Technology Reactive Ion
Etcher RIE-2000. The etching was done with $\rm{Ar/SF_{6}}$ chemistry for 1-2
minutes at 80 W forward power. The resist was removed with hot Remover PG,
followed by a sonication and an IPA wash. Then, the chip was mounted to a
copper box with GE Varnish, wire bonded to a printed circuit board and cooled
down.

##  II Reflection measurements of R2

The supplementary Figure 1 shows the reflection measurement of R2 at 60 mK and
at 2.5 K at zero external magnetic field. We see that at 60 mK the shape of
the reflection measurement is not a dip but with increasing temperature, the
shape transforms into a Lorentzian dip which we can successfully fit to get
$Q_{\rm int}$. Similar behavior is observed with increasing magnetic field at
60 mK.

##  III spiral characteristic shift due to the effect of TLS

The main text shows R3 resonator characteristic shift with temperature and
power due to TLS. The supplementary Figure 2 shows R0, R1 and R2
characteristic shifts due to TLS which are not shown in the main text.

##  IV Comparison of $Q_{\rm int}$ and $f_{0}$ fit results

The supplementary Figure 3(a) shows the extracted $\alpha$ and $T_{c}$ results
of the fits shown in the main text Figure 2. Four-point measurement results of
Nb wires with wire widths as same as the spiral wire widths are also included
in Figure 3(a). The R2 results are excluded as it is giving a non-positive
number for $Q_{\rm int}$. The supplementary Figure 3(b) shows the extracted
$Q_{\rm TLS,0}$ results of the fits shown in the main text Figure 3 and the
supplementary Figure 2.

Figure 6: Magnitude (top) and phase (bottom) of R2 at (a) 60 mK and (b) 2.5 K.
Measured data are shown in pastel blue. The magnitude of the measurement has a
Lorentzian-shaped dip at 2.5 K. A theoretical model for $S_{11}$ is shown in
black.

Figure 7: Frequency shift $\delta f$ and internal qualify factor $Q_{\rm int}$
(top) as temperature is varied and (bottom) average number of photons in the
resonator is varied of (a) R0, (b) R1, and (c) R2. Solid lines represent fits
to (top) Equation 5 and (bottom) Equation 4 in the main text. Figure 8: The
extracted parameters from the spiral resonator fits. (a) $\alpha$ and $T_{c}$
are obtained with Equation 3 in the main text. The $T_{c}$ values obtained
with four-point measurements are also included in the graph (pentagon-shaped
markers). (b) $Q_{\rm TLS,0}$ obtained with Equations 5 and 6 in the main
text.