# Strong kinetic-inductance Kerr nonlinearity with titanium nitride nanowires

Chaitali Joshi  Moore Laboratories of Electrical Engineering, California
Institute of Technology, Pasadena, CA  Institute of Quantum Information and
Matter, California Institute of Technology, Pasadena, CA  Wenyuan Chen  Moore
Laboratories of Electrical Engineering, California Institute of Technology,
Pasadena, CA  Henry G. LeDuc  Jet Propulsion Laboratory, Pasadena, CA  Peter
K. Day  Jet Propulsion Laboratory, Pasadena, CA  Mohammad Mirhosseini
<EMAIL_ADDRESS>http://qubit.caltech.edu Moore Laboratories of Electrical
Engineering, California Institute of Technology, Pasadena, CA  Institute of
Quantum Information and Matter, California Institute of Technology, Pasadena,
CA

###### Abstract

Thin films of disordered superconductors such as titanium nitride (TiN)
exhibit large kinetic inductance (KI), high critical temperature, and large
quality factors at the single-photon level. KI nonlinearity can be exploited
as an alternative to Josephson junctions for creating novel nonlinear quantum
devices with the potential to operate at higher frequencies and at elevated
temperatures. We study a means of magnifying KI nonlinearity by confining the
current density of resonant electromagnetic modes in nanowires with a small
volume $V\simeq 10^{-4}\text{um}^{3}$. Using this concept, we realize
microwave-frequency Kerr cavities with a maximum Kerr-shift per photon of
$K/2\pi=123.5\pm 3$ kHz and report a nonlinearity-to-linewidth ratio
$K/\gamma=21\%$. With improved design, our devices are expected to approach
the regime of strong quantum nonlinearity in the millimeter-wave spectrum.

##  I Introduction

Circuit quantum electrodynamics (cQED) systems rely on superconducting qubits
based on Josephson junctions. The inductance of a Josephson junction is a
nonlinear function of the current tunneling through it, which formally behaves
as a Kerr nonlinearity and gives rise to discrete energy levels in a qubit.
Alternatively, a current-dependent inductance can be realized via the kinetic
energy of current-carrying cooper pairs in a superconducting wire. This source
of nonlinear electromagnetic response is currently explored for parametric
quantum frequency conversion and amplification in the millimeter-wave band
(100-300 GHz), and at elevated ($>1$ K) temperatures [1, 2], where using (Al-
based) Josephson junctions is not feasible.

While parametric operations are possible with weak nonlinearities, an
interesting possibility is to create a strongly nonlinear response where the
induced self-Kerr shift per photon in a resonator is comparable to its
intrinsic linewidth. Resonators in this regime can be used for synthesizing
quantum states of radiation [3]. Ultimately, when the nonlinearity exceeds the
dissipation rate in a resonator the system effectively behaves as qubit [4,
5]. This concept was recently materialized in transmon qubits based on KI in
oxidized (granular) aluminum films (GrAl) [6, 7]. However, the relatively low
critical temperature of GrAl (2-3 K) makes it unsuitable for application in
the millimeter-wave (mm-wave) band or at elevated temperatures.

Thin films of disordered superconductors such as titanium nitride (TiN),
niobium titanium nitride (NbTiN), and niobium nitride (NbN) provide an
alternative choice of materials. These films exhibit large KI, and high
critical temperature (up to 14.5 K for NbTiN) [8], making them suitable
candidates for developing quantum circuits operating at increased temperatures
and frequencies while circumventing dissipation due to quasiparticle
generation. Additionally, resonators fabricated from these materials can
achieve large intrinsic quality factors factors ($10^{5}-10^{7}$ in the
single-photon regime [9, 10, 11, 12]), and have been widely utilized for high-
impedance quantum circuits, amplifiers, and detectors [13, 14, 15, 16, 17, 18,
19, 20, 21]. Despite these attractive properties, the KI nonlinearity in
nitride films tends to be weaker than GrAl. This is, in part, due to the
distinct underlying sources of KI in homogeneously disordered [22, 23], and
granular materials [24, 25], where in the latter the insulating oxide barriers
in the microstructure of the film act as weak Josephson links.

Figure 1: Nanowire resonator design: a) The resonator consists of a miniature
nanowire with a widths 20 to 40 nm, shunted by coplanar capacitor pads
fabricated from thin film TiN. Inset: Simulated current density in the
nanowire region. The current density is maximum in the nanowire and drops in
the capacitor pad region in the immediate vicinity of the wire. The nanowire
geometry enhances zero-point fluctuations, maximizing the Kerr-shift per
photon. b) Scanning electron microscope (SEM) image of a fabricated nanowire
with width $w$ = 38 nm. c) Circuit model of the nanowire resonator. d) Typical
linear response of a nanowire resonator (see Table 1).

In this work, we amplify the KI nonlinearity by engineering the geometry of
planar resonators fabricated from titanium nitride (TiN). Our design maximizes
the nonlinear response by confining the current density of the resonator mode
in nanowires with a small lateral width ($\sim$20-200 nm) patterned on a thin
film (14 nm thickness). Further, we maximize the magnitude of current quantum
zero-point fluctuations by minimizing the resonator impedance, which is
achieved by employing narrow-gap planar capacitors. Using these strategies, we
achieve self-Kerr nonlinearity rates in the range of $2-120$ KHz in nanowire
resonators with intrinsic quality factors reaching $3.5\times{10}^{4}$ in the
best devices. The maximum measured single-photon Kerr-shift to linewidth ratio
in our experiment reaches 0.21, which significantly exceeds the nonlinearities
reported in prior work with homogeneously disordered superconductors [18].
These results motivate future work in optimizing device design, where quantum
nonlinearities are expected to be within reach at higher frequencies.

Figure 2: Film characterization: a) Optical image of two compact, quarter-wave
test resonators. The resonator consists of a meandered wire shunted by a
C-shaped capacitor pad. b) Measured intrinsic quality factor of the compact
resonators as a function of mean cavity photon number $n_{c}$, showing
characteristic behavior of TLS loss. The power dependent quality factor
$Q_{\mathrm{TLS}}$ is $3.2\times 10^{5}$. At powers exceeding $n_{c}=3\times
10^{3}$, the resonator response becomes nonlinear. c) Measurement of critical
temperature $T_{c}$ using temperature-dependent resonant-frequency shift. Fits
to data are from numerical integration of surface-impedance obtained from the
Mattis-Bardeen equations.

##  II Device Design

Our design concept is to realize an electromagnetic resonant mode with the
magnetic energy density confined to a small volume with a large current
density. This can be achieved in a compact microwave resonator, where a
nanowire with a large KI connects the two pads of a planar rectangular
capacitor (see Figure 1a). For weak currents, the KI of the nanowire can be
modeled as a quadratic function $L_{w}/L_{0}=1+\left(I/I_{\ast}\right)^{2}$,
where $I_{\ast}$ is the nonlinear scaling current set by material properties
and wire cross-section [8]. The Hamiltonian of this $LC$-resonator can be
written as

$\displaystyle{H}=\frac{Q^{2}}{2C}+\frac{\phi^{2}}{2L}+\frac{1}{4}\frac{\phi^{4}}{I_{\ast}^{2}}\frac{L_{w}}{L^{4}}.$
(1)

Here, $Q$ and $\phi$ are the charge and flux node variables, and $C$ is the
total capacitance. The parameter $L$ is the total inductance of the resonator,
which includes the KI of the nanowire ($L_{w}$) as well as the contribution
from the non-zero current distribution in the capacitor pads ($L_{p}$). The
first two terms in the Hamiltonian describe a harmonic oscillator with
frequency $\omega=1/\sqrt{LC}$ and the last term models the nonlinear behavior
due to the current-dependent KI in the nanowire. Applying circuit
quantization, [5, 26], Equation 1 can be rewritten in terms of the creation
($\hat{a}^{\dagger}$) and annihilation ($\hat{a}$) operators of the LC
circuit. The nonlinear term in the Hamiltonian can be simplified by expanding
the quadratic current term as
$\hat{I}^{4}=I_{\mathrm{ZPF}}^{4}(\hat{a}+\hat{a}^{\dagger})^{4}$, where
$I_{\mathrm{ZPF}}=\sqrt{\hbar\omega/2L}$ is the current zero-point
fluctuation. Retaining only the photon-number conserving terms, we find

$\displaystyle\hat{H}$
$\displaystyle=\hbar(\omega-K)\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)-\frac{K}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a},$

with the Kerr-shift per photon $K$ defined as

$\displaystyle
K=\frac{3}{2}\omega\frac{L_{w}}{L}\left(\frac{I_{\mathrm{ZPF}}}{I_{\ast}}\right)^{2}.$
(2)

As evident in Equation 2, a unity Kerr-shift to linewidth ratio
($K/\gamma\approx 1$) can be achieved if the zero-point fluctuations of the
current are enhanced sufficiently to reach
$(I_{\mathrm{ZPF}}/I_{\ast})^{2}\approx 1/Q$, where $Q=\omega/\gamma$ is the
quality factor of the resonator. We pursue this goal by minimizing the total
device inductance to achieve a large $I_{\mathrm{ZPF}}$ while maintaining a
sufficiently large wire participation ratio $L_{w}/L$.

The expression for the Kerr coefficient can be further simplified by
substituting the scaling current as a function of material properties [8]

$\displaystyle I_{\ast}=J_{\ast}wt=\sqrt{\frac{\pi
N_{0}\Delta_{0}^{3}}{\hbar\rho_{n}}}wt,$ (3)

where $N_{0}$ is the density of electron states at the Fermi energy and
$\Delta_{0}\approx 1.76k_{B}T_{c}$ is the energy gap at zero temperature. The
parameter $\rho_{n}$ denotes the normal-state resistivity of the film, $w$ is
the wire width, and $t$ is the film thickness. Similarly, the kinetic
inductance of the nanowire can be expressed as a function of geometry and
material properties $L_{w}=\hbar(\rho_{n}/\pi\Delta_{0})(l/wt)$, where $l$ is
the wire length. Finally, defining the participation ratio of KI in the wire
to the total inductance as $\alpha=L_{w}/L$, Equation 2 can be rewritten as

$\displaystyle
K=\frac{3}{4}\frac{\hbar\omega^{2}}{N_{0}\Delta_{0}^{2}}\frac{\alpha^{2}}{V},$
(4)

where $V=w\cdot l\cdot t$ is the volume of the nanowire.

In terms of the material properties, an important observation is that the Kerr
coefficient is independent of the film resistivity (so far the nanowire’s KI
dominates all sources of parasitic inductance). Additionally, the inverse
scaling with the energy gap $\Delta_{0}$ motivates our choice of TiN over
niobium-based nitride films, which in general have higher critical
temperatures. The geometrical scaling is by and large set by the parameter
$V$, which favors nanowires with the minimum possible dimensions. However, in
the presence of a non-zero pad inductance $L_{p}$, reducing the wire length
beyond a certain point reduces the Kerr coefficient via the participation
ratio $\alpha$. Considering this trade-off, choosing optimum dimension
parameters requires the knowledge of film resistivity and the feasible
fabrication geometries for nanowires. In our experiment, we realize $L_{w}$ in
the range of 800 pH - 1.6 nH, requiring a shunt capacitance on the order of
0.3 pF to achieve resonance frequencies in our measurement band (6-8.5 GHz).

Realizing a capacitance of this magnitude with planar geometries is
challenging. A large capacitor footprint translates to increased radiation
loss and the formation of parasitic self-resonances. Alternatively, choosing
narrow gaps and long aspect ratios in interdigitated geometries results in
additional loss from surface two-level defects and a reduced participation
ratio due to increased pad inductance $L_{p}$. Using finite-element method
(FEM) modeling as a guide to bound these effects, we have chosen a symmetric
rectangular capacitor geometry with a total pad area of
$1.14~{}\mathrm{mm}^{2}$ and a narrow gap of $1~{}\mu\mathrm{m}$ (see Figure
1a). As detailed in the following sections, we perform experiments on
fabricated devices to evaluate the range of quality factors within reach for
this extreme geometry.

Figure 3: Nonlinear response of a nanowire resonator: Measured nonlinear
response of a nanowire resonator with resonance frequency $f_{0}$ = 6.3 GHz,
wire width $w$ = 18 $\pm$ 2 nm and wire length $l$ = 460 nm. a) Self-Kerr
response measured at increasing drive powers. The resonator shows
characteristic Duffing-oscillator-type behavior. The intracavity photon number
is swept from 0.25 to 15, with equal steps on a log scale. (b) Two-tone
spectroscopy in the few-photon regime. A radio-frequency drive (blue-detuned
by 5.3 MHz from the resonance) is used to drive the resonator and a weak probe
is used to measure the coherent response. (c) Extracted Kerr-shift per photon
($K/2\pi$ = 123.5 kHz) from the two-tone spectroscopy measurement.

##  III Film characterization

Our goal of maximizing nonlinearity motivates using films with minimum
possible thicknesses. However, reducing the film thickness results in
increasing disorder, which eventually induces a superconducting-to-insulating
transition [9, 27]. As a result, the choice of film thickness requires
balancing the competing requirements for achieving a large kinetic inductance,
a large critical temperature, and a low intrinsic loss tangent. Considering
these factors and previous results in the literature [9] we choose to work
with films in the 10-20 nm thickness range. The TiN samples used in our
experiment were deposited using reactive sputtering on a high-resistivity
($\rho>10~{}\mathrm{k}\Omega$) Si substrate and measured with an ellipsometer
to have a thickness of 14 nm.

To extract the film parameters, we fabricate and measure quarter-wave ‘test’
resonators using the standard single electron-beam lithography followed by
ICP/RIE etching in SF6/Ar (Figure 2a). We cool samples in a dilution
refrigerator with a base temperature of 10 mK and measure coherent reflection
from the devices via a circulator using a vector network analyzer (VNA). We
extract the resonance frequency and the internal quality factors of the
resonators by fitting a Lorentzian lineshape to the measured resonator
response (see Figure 1c). We fit the extracted resonance frequency for
resonators with varying wire lengths to FEM simulation results to calculate
the film sheet KI ($L_{s}=40\pm 1$ pH/$\square$, see Appendix A). Due to the
large wire-cross sections used in these experiments (0.25-2 $\mu$m), the test
resonators exhibit weak nonlinearities ($K/2\pi\approx$ 5-50 Hz), which is
nevertheless observable at high photon numbers. Characterizing the
nonlinearity using using two-tone spectroscopy (detailed in the next section),
we find the nonlinear scaling current density of the film ($J_{\ast}=3.95\pm
0.75\text{ MA}/\text{cm}^{2}$).

We characterize the film loss tangent by fitting the change in the internal
quality factor of the resonators as a function of the number of photons in the
cavity. As evident in Figure 2b, the internal $Q$-factors shows a saturation
behavior that is characteristic of TLS loss [10, 12]. We perform $Q$-factors
measurements over a sample size of seven test resonators and find an average
(standard deviation) of $2.1\times{10}^{5}$ ($9\times{10}^{4}$) at sub-single-
photon powers. The $Q$-factors at high photon numbers do not show excessive
quasi-particle loss despite the thin film thickness, and are comparable to
reported values in previous work with films deposited via atomic layer
deposition (ALD) [27]. Importantly, we did not observe a statistically
significant trend with respect to the wire width of the test resonators.

Finally, we measure the critical temperature of the TiN film by measuring the
temperature dependence of the compact resonator frequency (Figure 2c). The
temperature-dependence of the surface impedance $X_{s}$ results in a change in
the resonator frequency, which is given by [28, 29],

$\displaystyle\frac{\delta
f_{0}}{f_{0}}=-\frac{\alpha}{2}\frac{X_{s}(T)-X_{s}(0)}{X_{s}(0)}$ (5)

where $\alpha$ is the KI inductance fraction of the resonator (extracted from
FEM simulation of the device) and $\delta f_{0}$ is the change in the cavity
resonance frequency. We use surface impedance calculated from numerical
integration of the Mattis-Bardeen equations ([28, 30]) to fit the fractional
change in resonance frequency $\delta f_{0}/f_{0}$ and extract $T_{c}=2.9$ K.

##  IV Nonlinear response of nanowire resonators

Having established our film parameters, we aim to realize nanowire resonators
with the smallest feasible dimensions. To achieve this, we have optimized our
fabrication procedure and achieved wire widths as small as 20 nm with run-to-
run repeatability of about 5 nm. We have measured the nonlinear response of
fabricated resonators with varying wire dimensions (Table 1). The dimensions
of the capacitor pads were chosen to be identical for all resonators, and the
wire length was adjusted, for each width, to achieve resonance frequencies in
the range of 6-8.5 GHz.

We perform linear spectroscopy and find that the nanowire resonators exhibit
the characteristic shape of Duffing oscillators (see Figure 3a), consistent
with a Kerr nonlinear response. As expected, beyond a critical circulating
power in the resonator, the system undergoes bifurcation and exhibits
bistability [31]. In addition to the power-dependent frequency shift, we also
observe an increase in the internal linewidth of the resonator with the
increasing number of photons in the cavity. Given the very small photon
numbers used in these measurements ($n_{c}\sim 1-10$), the broadening is
unlikely to be caused by classical nonlinear loss processes. Instead, we
attribute this effect to dephasing caused by quantum fluctuations in the
number of interactivity photons (i.e., the photon shot noise). This
explanation is further substantiated by observing a linear dependence of the
linewidth from the fit to the square root of the intracavity photon number
(see Appendix C). Despite the rich underlying physics, which verifies the
strong nonlinear response of our system, this photon-number dependent cavity
linewidth makes it challenging to extract the self-Kerr coefficient ($K$) from
a direct fit to the Duffing oscillator model.

As an alternative way of finding $K$, we perform two-tone spectroscopy. In
these measurements, a detuned coherent pump tone is applied to the cavity
while measuring the spectral response to a weak resonant probe (see Figure
3b). The power of the probe tone is set such that the cavity population from
the resonant drive remains two orders of magnitude lower than that of the pump
tone. Crucially, the large detuning of the pump tone from the cavity in this
measurement scheme leads to a much faster fluctuation time-scale in the number
of intracavity photons, effectively eliminating broadening from photon shot
noise [32]. As evident in Figure 3c, varying the number of photons from the
pump does not lead to any significant change in the resonator linewidth,
further verifying the role of photon-shot noise.

To fit the data, we calibrate the power delivered to the chip precisely by
measuring the frequency-dependent input line attenuation using thermometry of
a cryogenic 50 $\Omega$ termination (see Appendix B). From linear fits to the
cavity-frequency shift, we extract the self-Kerr coefficient of each resonator
(see Figure 3c). The measurement results are summarized in Table 1. We find
Kerr-shift to linewidth ($K/\gamma$) ratios in a range of 0.7% to 21%, and
internal $Q$-factors in the range of $0.2-3.5\times 10^{4}$ (Table 1). The
measured self-Kerr shift is largest for the wire with smallest mode volume
$V=1.2\times 10^{-4}\mu\mathrm{m}^{3}$. After that, we observe a monotonic
decrease in Kerr-shift for increasing wire volume, in agreement with the
expected $1/V$ scaling behavior (see Figure 4). Using a fit to the model
(Equation 4), we extract participation ratios $\alpha$ in the range of 0.7 -
0.97. These participation ratios are larger than those expected using our
measured value of sheet inductance ($L_{s}$) and the wire dimensions obtained
from imaging the devices. We attribute this discrepancy to contributions to
the nonlinear response from the current distribution in the capacitor pads,
which are not included in our lumped-element model.

Figure 4: Geometric dependence of nanowire nonlinearity: Measured Kerr-shift
for range of wire volumes varying by two orders of magnitude. The largest
Kerr-shift is obtained for the wire with smallest mode volume $V$ = 1.2
$\times 10^{-4}$ $\mu\text{m}^{3}$ and decreases to 2.2 kHz for a volume of
$V=2.3\times 10^{-2}\mu\text{m}^{3}$. The red dotted curve is a fit to the
model $K=a/V$, with $a=2\times 10^{-2}$ kHz-$\mu\mathrm{m}^{3}$.

We find the internal $Q$-factors of the nanowires resonators to be smaller (by
about an order of magnitude) than those found in lumped-element test
resonators used for film characterization. Importantly, we do not observe
significant wire-width dependence for the internal $Q$-factor; the device with
the largest nonlinearity and narrowest wire width has an internal $Q$-factors
of 1.08 $\times 10^{4}$. We have verified that the measured $Q$-factors are
dominated by energy loss, despite the presence of a slow dephasing mechanism
(see Appendix D). Below, we discuss the dominant loss mechanisms in our
devices and potential avenues for improving the $K/\gamma$ ratio.

| $f_{0}$ | Width | Length | Linewidth | Intrinsic | Kerr shift | $K/\gamma$
---|---|---|---|---|---|---|---
| (GHz) | (nm) | (nm) | ($\gamma$, kHz) |  $Q$-factors | ($K/2\pi$, kHz) | ($\%$)
1 | 6.30 | 18 $\pm$ 2 | 460 | 580 | $1.08\times 10^{4}$ | 123.5 $\pm$ 3 | 21
2 | 8.48∗ | 38 $\pm$ 2 | 600 | 1600 | $0.53\times 10^{4}$ | 88 $\pm$ 5.3 | 5.5
3 | 8.07 | 38 $\pm$ 2 | 800 | 800 | $1\times 10^{4}$ | 72.5 $\pm$ 4.5 | 9
4 | 7.03 | 36 $\pm$2 | 1450 | 310 | $2.3\times 10^{4}$ | 46 $\pm$ 6 | 15
5 | 7.12∗ | 38 $\pm$ 2 | 1200 | 1000 | 0.71 $\times 10^{4}$ | 42 $\pm$ 5.4 | 4.2
6 | 6.50 | 44 $\pm$ 2 | 1800 | 2300 | $0.2\times 10^{4}$ | 15.5 $\pm$ 1.5 | 0.7
7 | 7.70 | 225 $\pm$ 2 | 7250 | 220 | $3.5\times 10^{4}$ | 2.2 $\pm$ 0.15 | 1

Table 1: Summary of results. The uncertainty in calculating the Kerr
coefficient is dominated by the uncertainties in the measurement of the
frequency-dependent input line attenuation. The internal linewidth $\gamma$ of
all resonators was extracted from the complex amplitude response at single-
photon ($n_{c}\approx 0.3$) drive levels. ∗ These devices were on a separate
chip (see text).

##  V Dissipation mechanisms

The narrow-gap capacitor geometry and the resulting large electric fields in
our designs make our devices sensitive to TLS loss at interfaces [33, 34].
Common techniques for extracting the TLS-induced loss involves saturating the
TLS bath via by a coherent external drive or by elevating the measurement
temperature. However, in our current experiment the small bistability
threshold set by the strongly nonlinear response of the cavities does not
allow for directly applying these techniques. Additionally, while these
measurements are, in principle, possible in test structures with identical
electric field distributions but with a diluted nonlinearity (e.g. via wider
wires), the inevitable reduction in kinetic inductance leads to much higher
resonance frequencies beyond our setup’s measurement band. Beyond TLS loss, we
expect two primary sources of dissipation in our devices. The large KI
fraction of the nanowire resonators makes them susceptible to dissipation from
excess quasiparticles generated by stray radiation [14, 35, 36, 37]. In
addition, the large size of the capacitor pads together with a small gap
results in a large effective dipole, making the resonators susceptible to
radiative loss. Here, we provide order-of-magnitude estimates for loss
contributions from these sources.

The rate of dissipation due to a nonequilibium quasiparticle density
$n_{\text{qp}}$ can be predicted from the Mattis-Bardeen equations for the
complex conductivity, and is given by [38],

$\displaystyle\Gamma_{\text{qp}}=\frac{\alpha\omega_{0}}{\pi}\sqrt{\frac{2\Delta_{0}}{\hbar\omega_{0}}}\frac{n_{\text{qp}}}{D(E_{F})\Delta_{0}}$
(6)

where $\alpha$ is the kinetic inductance fraction, $\omega_{0}$ is the
resonance frequency, $n_{\text{qp}}$ is the excess quasiparticle density, and
$D(E_{F})$ is the density of Cooper pairs at Fermi energy. We evaluate
Equation 6 using parameters reported in literature for TiN thin films
($D(E_{F})=2\times 10^{10}/\mu\text{m}^{3}-eV$ [39],
$n_{\text{qp}}=10/\mu\text{m}^{3}$ [38, 37]), our measured critical
temperature, and the simulated KI fraction $\alpha$. As a verification of the
calculation, we find the QP-limited $Q$-factors of the test resonators used
for film characterization (in Section III) ($4.8\times 10^{5}$), which is
found to be in close agreement with measured $Q$-factors at high powers
($2.1\times 10^{5}$). For the nanowire resonators, the calculation gives a
similar decay rate in the range of 12-14 kHz, which is much smaller than the
observed values in the experiment. Interestingly, the calculated quasi-
particle limited decay rates are about an order of magnitude smaller than the
largest measured single-photon Kerr shifts, pointing to the intrinsically low-
loss nature of KI as a source of nonlinearity.

We estimate radiation loss contributions by modeling the near-field coupling
of the nanowire resonators to the packaging box modes. Our device is packaged
within a copper box with air-gaps below and above the chip, with the lowest
order resonant mode located near $f_{b}$ = 17 GHz. We use a conservative
$Q$-factors for the microwave package mode ($\kappa/2\pi=35$ MHz) [40] and use
analytical expressions for the electric field of coplanar capacitor plates to
estimate an upper bound for radiative loss rate of $\Gamma/2\pi\approx 60$ kHz
(see Appendix E). While significant, this source of loss does not adequately
account for the observed linewidths. Ruling out QP-dissipation and radiative
loss, we suspect the device performance is currently limited by TLS loss.

##  VI Conclusion

In conclusion, we have investigated the limits of KI Kerr nonlinearity by
optimizing the geometry of resonators fabricated from thin-film TiN. Confining
current density in miniaturized nanowires, we have observed a nonlinearity-to-
loss ratio of $K/\gamma=0.21$. While the achieved ratio is not sufficient to
make a qubit based on this concept in the microwave frequency band, the
quadratic scaling of nonlinearity with frequency (Equation 2) promises future
opportunities: At a frequency of 100 GHz, this scaling translates to Kerr-
shift coefficients in the range of 20-30 MHz per photon. Intrinsic quality
factors as large as $3\times 10^{4}$ ($\gamma/2\pi=3$ MHz) have been
previously demonstrated using NbN films at this frequency [1]. Achieving our
scaled Kerr-shift coefficients in devices with comparable quality factors
would translate to the observation of a strong quantum nonlinear response.
This motivates future studies of loss mechanisms in TiN, and a broader set of
disordered superconductors, in the mm-wave band. Additionally, the improved
Kerr coefficients in our devices may be exploited for frequency conversion
from microwave to mm-wave frequencies, where operation with lower pump powers
and stronger nonlinearities can reduce absorption heating. Using the measured
and scaled Kerr-shifts at microwave and millimeter-wave frequencies we
estimate a cross Kerr shift of
$K_{\mathrm{uw,mm}}=2\left({K_{\mathrm{uw}}K_{\mathrm{mm}}}\right)^{1/2}\approx
2\pi\cdot 3$ MHz [41]. Finally, we note the possibility of further
improvements by going beyond planar geometries by using parallel-plate
capacitors, where large quality factors ($10^{5}$) have been previously
demonstrated [42].

## acknowledgments

This work was supported by startup funds from the Caltech EAS division, a
Braun trust grant, and the National Science Foundation (grant No. 1733907).
C.J. gratefully acknowledges support from the IQIM/AWS Postdoctoral
Fellowship. We acknowledge Niv Drucker from Quantum Machines for software
support while performing the cavity ringdown measurements.

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##  Appendix A Film characterization: sheet inductance

We estimate the sheet inductance of the TiN thin films by fitting the
extracted resonance frequency of the quarter-wave resonators with wire width
$w=2~{}\mu$m and varying wire lengths (Figure 5). A capacitance of $C=60$ fF
was extracted for the quarter-wave resonator geometry (Figure 2a) using finite
element method simulations. From the slope $m=4\pi^{2}L_{s}C/w$ of the fit
shown in Figure 5, we obtain a sheet inductance of $L_{s}=40\pm 1$
pH/$\square$.

Figure 5: Calibration of sheet inductance: Measured $1/f_{0}^{2}$ vs. wire
length for quarter-wave resonators with wire width $w=2\mu$m. From the slope
of the fit $m=4\pi^{2}L_{s}C/w$ and capacitance extracted from simulations (C
= 60 fF), we extract a sheet kinetic inductance of
$L_{s}=40\text{pH}/\square$.

$f$ | Net Transmission | Gain | Attenuation
---|---|---|---
(GHz) | ($S_{\text{{IO}}}$, dB) | ($G_{C}$, dB) | ($A_{IN}$, dB)
4.4 | -5.5 | 68.3 $\pm$ 0.4 | 73.8 $\pm$ 0.4
6.30 | -11.50 | 64.6 $\pm$ 0.1 | 76.1 $\pm$ 0.1
6.56 | -11.95 | 64.5 $\pm$ 0.4 | 76.5 $\pm$ 0.4
7.03 | -13.95 | 63.6 $\pm$ 0.6 | 77.6 $\pm$ 0.6
7.70 | -16.90 | 62.9 $\pm$ 0.3 | 79.8 $\pm$ 0.3
8.07 | -18.30 | 61.4 $\pm$ 0.3 | 79.7 $\pm$ 0.3

Table 2: Line attenuation calibration Figure 6: Shot-noise induced dephasing
in self-Kerr response: Measured cavity linewdith vs photon number for the
self-Kerr data shown in Figure 3a. The increase in linewidth can be attributed
to dephasing due to shot noise in the cavity photon number when the cavity is
resonantly driven.

##  Appendix B Line attenuation calibration

Figure 7: Telegraph noise: a) Single-shot measurements of the frequency
response of the resonator with the narrowest linewidth ($f_{0}$ = 7.7 GHz).
The resonator exhibits switching between two states. b) Measured time-
dependent phase response, showing characteristics of a telegraph process. c)
Auto-correlation of the time-dependent phase of the resonator. We extract a
switching time of $\tau$ = 1.93 s.

To calculate accurately the power delivered to the chip, we calibrate the
frequency-dependent attenuation of our input line using thermometry. The input
line contains cryogenic attenuation at each stage of the cryostat to minimize
thermal noise conducted from room temperature electronics to the cold stages
via the input lines[43]. In addition, the input line consists of a crogenic
circulator (Low noise factory, LNF-CIC4$\\_$12A) and a fan-out switch. The
output line consists of a HEMT amplifier (LNF-LNC4$\\_$8C) thermalized to the
4 K stage and a room temperature amplifier. We first measure the overall
frequency-dependent transmission of the input and output line using an RF tone
sent from a signal generator. We then calibrate the total gain of the output
line using thermometry. A 50 $\Omega$ cryogenic terminator is thermalized to
the mixing (MXC) stage of the cryostat. The MXC stage temperature is raised by
reducing cooling power by turning off the turbo to reduce 3He/4He mixture
flow, and applying heat using the MXC stage heater. With no external input
power, we measure the output power from the amplifier chain with a spectrum
analyzer at different mixing stage temperatures $T_{\text{MXC}}$. The measured
output has contributions from the thermal noise of the resistor at the MXC
stage, and the HEMT noise characterized by a fixed noise temperature
$T_{\text{HEMT}}$. The output power measured in an IF bandwidth
$\Delta\nu_{IF}$ on the spectrum analyzer is equal to the sum of the Johnson-
Nyquist noise from the two sources and is given by,

$\displaystyle
P_{\text{OUT}}=\Delta\nu_{\text{IF}}k_{\text{B}}G_{C}\left(T_{\text{MXC}}+T_{\text{HEMT}}\right)$
(7)

Here, $G_{C}$ is the net absolute gain factor of the output line from the
chip. At $T_{\text{MXC}}=10$ mK, the measured output power is dominated by
HEMT noise. The output gain $G_{C}$ can then be calculated by subtracting the
contribution of the HEMT noise from the total output power measured at various
$T_{\text{MXC}}$. We perform this measurement at 4 different MXC temperatures
between 730 mK and 1.05 K. The mean and standard deviation of the four
measurements is used to obtain $G_{C}$. We use a factor
$\eta=(h\nu/kT)/\left(\exp{\left(h\nu/kT\right)}-1\right)$ to account for
corrections to Equation 7 due to the Bose-Einstein distribution in the regime
$h\nu\sim k_{B}T$ [44]. The input line attenuation can be calculated using,

$\displaystyle A_{IN}[\text{dB}]+G_{C}[\text{dB}]=S_{IO}[\text{dB}].$ (8)

Here, $S_{IO}$ is the net transmission of the input-output lines measured at
different frequencies and $G_{C}[\text{dB}]$ is the calibrated output gain in
dB. The results are summarized in Table 2.

##  Appendix C Shot-noise induced dephasing

The measured self-Kerr response of the nanowire resonators shows an increased
linewidth when the cavity is populated with few photons (Figure 3a). The
relatively large ratios of Kerr shift to linewidth create the possibility of
dephasing due to shot noise when the cavity is populated via a resonant drive.
As shown in Figure 6, the measured linewidth increases proportional to
$\sqrt{n_{c}}K$. Additionally , we extract a slope of 80 kHz for this linear
dependence. which is close to the measured self-Kerr coefficient for this
resonator (123.5 kHz). In contrast, we do not observe a significant linewidth
broadening in the two-tone spectroscopy measurement, where the cavity is
driven off-resonantly (Figure 3b). In this case, the cavity is only virtually
populated, and we estimate the increase in linewidth to be proportional to
$nK^{2}/\Delta$, which is of the order of 10 kHz [32].

##  Appendix D Telegraph noise and $T_{1}$ lifetime measurement

Figure 8: Cavity-ring down spectroscopy: Time-dependent energy decay of the
resonator after turning-off of the drive pulse. We measure a $T_{1}$ lifetime
of 1.06 $\mu$s, corresponding to an instantaneous linewidth of 150 kHz for the
7.7 GHz resonator.

We observe telegraph noise in the frequency response of the resonator with the
narrowest linewidth (Device #7, see Table 1). We perform single-shot VNA
measurements to capture the fast dephasing of this resonator. The resonator
exhibits switching between two states, as shown in Figure 7a. To measure the
characteristic transition time $\tau$ between the two states, we measure the
phase response of the resonator continuously using the zero-span mode of the
VNA. As shown in Figure 7b, the phase response is reminiscent of the telegraph
process. The switching time can be extracted from the autocorrelation function
of the telegraph process, given by
$\langle\phi(t)\phi(t^{\prime})\rangle\propto\exp{{-2|t-t^{\prime}|}/{2\tau}}$,
where $\phi(t)$ is the time-dependent phase shown in Figure 7b. We extract a
switching time of $\tau=1.93$s.

Figure 9: E-field distribution of coplanar capacitor pads: (a) Cross-section
of the coplanar capacitor pads. (b) E-field magnitude of coplanar capacitor
pads in the vicinity of the capacitor gap, calculated from analytical
expressions using conformal mapping. The field distribution resembles that of
a dipole. (c) In-plane ($E_{x}$) field magnitude, which results in the
dominant coupling to the TE011 box mode.

We perform cavity ring-down spectroscopy on this resonator to extract it’s
$T_{1}$ lifetime. The resonator was driven using a modulated signal generated
from an FPGA, upconverted by frequency mixing with an RF signal generator
acting as the local oscillator (LO). The resonator is driven using a $5\mu
s$-long pulse, sufficiently long to ensure that the cavity population is
saturated. The drive is then turned off abruptly. The drive power was set to
be sufficiently low such that the resonator response is in the linear regime.
The time-dependent decay of the cavity population is shown in Figure 8. The
measured $T_{1}$ lifetime is 1.06 $\mu$s, corresponding to a decay rate of 150
kHz. Comparing this value with the measured resonator linewidth with long
averages (220 kHz), bounds the contributions from dephasing to approximately
70 kHz.

Telegraph noise has been previously observed in kinetic inductance resonators
[45, 46, 47, 48]. Potential mechanisms for the origin of this noise are strong
coupling to TLS [45, 47, 48] or quasiparticle burst noise from high-energy
particles [46], and requires further investigation which is beyond the scope
of this study.

##  Appendix E Radiation loss calculation

We estimate the radiation loss of the resonator to the box using using the
Purcell decay expression in the far-detuned regime, where the resonator decay
rate can be estimated as,

$\displaystyle\Gamma=\frac{\kappa g^{2}}{\Delta^{2}}.$ (9)

Here, we assume $\Delta\gg\kappa,g$. The parameter $\Delta/2\pi=f_{b}-f_{0}$
is the detuning between the resonator and the box mode $f_{b}$ and
$\kappa/2\pi=f_{b}/Q$ is the decay rate of the box mode. The coupling rate $g$
between the resonator and the box mode is related to the electric field
overlap between the cavity and the box modes, and is given by [49, 50],

$\displaystyle\hbar
g=\sqrt{\dfrac{\hbar\omega_{r}}{2\epsilon_{0}V_{r}}}\sqrt{\dfrac{\hbar\omega_{b}}{2\epsilon_{0}V_{b}}}\int
dV\epsilon\epsilon_{0}|\mathbf{a_{r}^{\ast}}\cdot\mathbf{a_{b}}|$ (10)

where $V_{r}$ and $V_{b}$ are the effective mode volumes of the resonator and
box mode respectively. The coefficients $\mathbf{a_{r}}$ and $\mathbf{a_{b}}$
are normalized, dimensionless amplitudes such that the electric-field of
$\mathbf{E_{i}}$ is given by
$\mathbf{E_{i}}=\sqrt{{\hbar\omega_{i}}/{2\epsilon_{0}V_{i}}}\mathbf{a_{i}},i\in[r,b]$.
The integral is carried over the volume of the box. The field distribution
$\mathbf{E_{r}}$ of the resonator mode can be approximated from analytical
expressions for coplanar capacitor pads obtained using conformal mapping [51,
52]. The electric field distribution for the resonator mode is shown in Figure
9. As expected, the field distribution resembles that of a dipole due to the
large electric field in the small capacitor gap. Using FEM simulations, the
lowest order box mode is found to be a TE011 mode at $f_{b}$ = 17 GHz.
Assuming the resonator is placed in the center of the box where the E-field of
the box mode peaks, we evaluate Equation 10 numerically and determine a
coupling rate $g/2\pi=410$ MHz. For a typical resonator frequency $f=7$ GHz
and assuming a typical package mode Q $\approx$ 500, we obtain a radiation
loss rate of $\Gamma/2\pi=57$ kHz using Equation 9. The degenerate TE101 mode
does not couple to the resonator mode due to it’s orthogonal polarization. The
immediate higher order modes at higher frequencies also do not contribute
significantly as they have E-field nodes at the center of the box, and the
decay rate is further suppressed due to their large detuning $\Delta$ relative
to the resonator mode.