# A three-wave mixing kinetic inductance traveling-wave amplifier with near-
quantum-limited noise performance

M. Malnou<EMAIL_ADDRESS>National Institute of Standards and
Technology, Boulder, Colorado 80305, USA  Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA  M. R. Vissers  National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  J. D. Wheeler
National Institute of Standards and Technology, Boulder, Colorado 80305, USA
J. Aumentado  National Institute of Standards and Technology, Boulder,
Colorado 80305, USA  J. Hubmayr  National Institute of Standards and
Technology, Boulder, Colorado 80305, USA  J. N. Ullom  National Institute of
Standards and Technology, Boulder, Colorado 80305, USA  Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA  J. Gao  National
Institute of Standards and Technology, Boulder, Colorado 80305, USA
Department of Physics, University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

We present a theoretical model and experimental characterization of a
microwave kinetic inductance traveling-wave amplifier (KIT), whose noise
performance, measured by a shot-noise tunnel junction (SNTJ), approaches the
quantum limit. Biased with a dc current, the KIT operates in a three-wave
mixing fashion, thereby reducing by several orders of magnitude the power of
the microwave pump tone and associated parasitic heating compared to
conventional four-wave mixing KIT devices. It consists of a 50 Ω artificial
transmission line whose dispersion allows for a controlled amplification
bandwidth. We measure $16.5^{+1}_{-1.3}$ dB of gain across a 2 GHz bandwidth
with an input 1 dB compression power of -63 dBm, in qualitative agreement with
theory. Using a theoretical framework that accounts for the SNTJ-generated
noise entering both the signal and idler ports of the KIT, we measure the
system-added noise of an amplification chain that integrates the KIT as the
first amplifier. This system-added noise, $3.1\pm 0.6$ quanta (equivalent to
$0.66\pm 0.15$ K) between 3.5 and 5.5 GHz, is the one that a device replacing
the SNTJ in that chain would see. This KIT is therefore suitable to read large
arrays of microwave kinetic inductance detectors and promising for multiplexed
superconducting qubit readout.

##  I INTRODUCTION

Is it possible to build a quantum limited microwave amplifier with enough
gain, bandwidth and power handling to simultaneously read thousands of
frequency-multiplexed superconducting resonators, like those in qubit systems
or microwave kinetic inductance detectors (MKIDs)? When designed with resonant
structures, Josephson junction-based parametric amplifiers have demonstrated
high gain and quantum limited performances Castellanos-Beltran  _et al._
(2008); Bergeal  _et al._ (2010); Roch  _et al._ (2012); Mutus  _et al._
(2013); Zhong  _et al._ (2013); Lecocq  _et al._ (2017); Malnou  _et al._
(2018). However, despite efforts to increase the bandwidth up to a few hundred
megahertz via impedance engineering Mutus  _et al._ (2014); Roy  _et al._
(2015), or to increase the power handling up to a few hundred femtowatts via
Kerr engineering Frattini  _et al._ (2017, 2018); Sivak  _et al._ (2019), they
still cannot read more than a handful of resonators simultaneously. When
designed with nonresonant structures, i.e. transmission lines, Josephson
traveling-wave parametric amplifiers (JTWPAs) have high gain over gigahertz
bandwidth Macklin  _et al._ (2015); White  _et al._ (2015); Planat  _et al._
(2020), but so far still exhibit similar power handling capabilities as their
resonant counterparts. Recent studies Sivak  _et al._ (2020); Zorin (2016,
2019) suggest that a three-wave mixing (3WM) JTWPA with a finely controlled
and canceled Kerr nonlinearity should increase the device’s power handling
tenfold. Compelling experiments have yet to prove the feasibility of this
approach, for which the JTWPA’s design and fabrication increase in complexity.
We propose to tackle this challenge using a different non-linear medium:
starting with the intrinsic broadband and high power handling capabilities of
a kinetic inductance traveling-wave amplifier (KIT) Eom  _et al._ (2012), we
build a near-quantum-limited amplifier.

The current limitations on microwave amplification affect many scientific
endeavors. Although proof-of-principle “quantum supremacy” was demonstrated by
a quantum computer containing a few tens of qubits Arute  _et al._ (2019),
this number has to scale by at least an order of magnitude to run powerful
quantum algorithms Shor (1994); Grover (1997). In the hunt for exoplanets,
cameras with tens of thousands of MKID pixels are being built Szypryt  _et
al._ (2017), and proposals to search for very light warm dark matter also
necessitate the use of a great number of MKID pixels Hochberg  _et al._
(2016a, b). All these applications are either already limited by amplifier
noise, or would greatly benefit from wideband, high gain, high power handling,
quantum limited amplifiers.

The KIT we present in this article is a step toward a practical, quantum-
limited amplifier, whose bandwidth and power handling are compatible with high
channel-count applications. Operating in a 3WM fashion, and fabricated out of
a single layer of NbTiN, it consists of a weakly dispersive artificial
transmission line Chaudhuri  _et al._ (2017); Zobrist  _et al._ (2019), for
which we control the phase matched bandwidth with dispersion engineering. This
limits spurious parametric conversion processes that otherwise degrade the
power handling and noise performance. We measure an average gain of 16.5 dB
over a 2 GHz bandwidth, and a typical 1 dB input compression power of -63 dBm
within that bandwidth. Using a shot-noise tunnel junction (SNTJ) Spietz  _et
al._ (2003, 2006) we measure the added noise of a readout chain with the KIT
as the first amplifier. To quote the true system-added noise of the chain,
i.e. the one that a device replacing the SNTJ in that chain would see, we
develop a novel theoretical framework that accounts for the SNTJ-generated
noise illuminating both the signal and idler ports of the KIT. Failure to
account for the idler port’s noise makes the system-added noise look
significantly better than its true value. We demonstrate a true system-added
noise of $3.1\pm 0.6$ quanta between 3.5 and 5.5 GHz, and estimate that the
KIT alone is near-quantum-limited. It is the first time that the broadband
noise properties of a KIT are fully characterized rigorously.

##  II THEORY AND DESIGN

KITs exploit the nonlinear kinetic inductance of a superconducting line to
generate parametric interaction between pump, signal, and idler photons. In
3WM, a single pump photon converts into signal and idler photons, whereas
four-wave mixing (4WM) converts two pump photons in this fashion. Operating a
KIT with 3WM offers two key advantages over 4WM. First, as the pump frequency
is far detuned from the amplification band, it is easily filtered, which is
often necessary to avoid saturating the following amplifier. Second, it
reduces the rf pump power because energy is extracted from dc power to convert
pump photons, which avoids undesirable heating effects from the strong rf
pump, including those happening in the packaging. More precisely, when biased
with a dc current $I_{d}$, the KIT inductance per unit length L is Vissers
_et al._ (2016):

$L=L_{d}(1+\epsilon I+\xi I^{2}+\mathcal{O}(I^{3})),$ (1)

where $I$ is the rf current, $L_{d}$ is the KIT inductance under dc bias, at
zero rf current, $\epsilon=2I_{d}/(I_{*}^{2}+I_{d}^{2})$ and
$\xi=1/(I_{*}^{2}+I_{d}^{2})$. The current $I_{*}$ sets the scale of the
nonlinearity; it is typically $\sim 10^{3}$ higher than that of Josephson
devices, thereby conferring KITs $\sim 10^{4}-10^{6}$ higher power handling
capabilities than their Josephson equivalents. The term $\epsilon I$ permits
3WM, while $\xi I^{2}$ permits 4WM.

Solving the coupled mode equations (CME) for a pump at frequency $\omega_{p}$,
signal at $\omega_{s}$, and idler at $\omega_{i}$, such that
$\omega_{p}=\omega_{s}+\omega_{i}$ yields the 3WM phase matching condition for
exponential gain:

$\Delta_{k}=-\frac{\xi I_{p0}^{2}}{8}(k_{p}-2k_{s}-2k_{i}),$ (2)

see appendix A.1. Here, $\Delta_{k}=k_{p}-k_{s}-k_{i}$ with $k_{j}$,
$j\in\\{p,s,i\\}$ the pump, signal and idler wavenumbers, and $I_{p0}$ is the
rf pump amplitude at the KIT’s input. In a non-dispersive transmission line
$\Delta_{k}=0$, and thus equation 2 can naturally be fulfilled over a very
wide frequency range in KITs, where $I_{p0}\ll I_{*}$. Although desirable
within the amplification bandwidth, it is undesirable outside of that
bandwidth, where multiple parametric conversion processes take place Vissers
_et al._ (2016); Erickson and Pappas (2017). These processes deplete the pump,
thereby degrading the amplifier’s power handling, and they also induce
multiple conversion mechanisms at each frequency, thereby increasing the
amplifier-added noise.

Figure 1: Schematic of the KIT artificial transmission line. (a) Three cells
in series (not to scale) are arranged in a CPW topology. The highly inductive
central line (red) is flanked by IDC fingers, which match it to 50 Ω. Each
finger constitutes a low-Q, quarter-wave resonator. (b) In the equivalent
electrical circuit, each cell consists of a series inductance $L_{d}$ and two
resonators with inductance $L_{f}$ and capacitance to ground $C/2$.

While in conventional traveling-wave amplifiers dispersion engineering
prevents only pump harmonic generation, we suppress all unwanted parametric
conversion processes by designing our KIT as a weakly dispersive artificial
transmission line. Originally developed to have the KIT matched to 50 Ω
Chaudhuri  _et al._ (2017), this line consists of a series of coplanar
waveguide (CPW) sections, or cells, each with inductance $L_{d}$, flanked by
two interdigitated capacitor (IDC) fingers that form the capacitance to ground
$C$, such that $Z=\sqrt{L_{d}/C}=50$ Ω, see Fig. 1a. Each IDC finger then
constitutes a low-Q quarter-wave resonator, with capacitance $C_{f}=C/2$ and
inductance $L_{f}$ (see Fig. 1b), set by the finger’s length. In practice, we
choose $\omega_{f}=1/\sqrt{L_{f}C_{f}}=2\pi\times 36$ GHz, and
$Q=1/Z\sqrt{L_{f}/C_{f}}=3.3$, to generate a slight dispersion at low
frequencies, where the pump, signal and idler modes lie. The dashed line in
Fig. 2a shows the dispersive part of the wavenumber $k^{*}=k-k_{0}$ with
$k_{0}=\omega\sqrt{L_{d}C}$, as a function of frequency. It is calculated by
cascading the ABCD matrices of the KIT cells, see appendix B. As it slightly
deviates from zero, no triplet $\\{k_{s},k_{i},k_{p}\\}$ can satisfy Eq. 2.

Figure 2: Influence of the phase matching on the gain profile. (a) The
nonlinear wavenumber $k^{*}$ is calculated as a function of frequency for the
transmission line represented in Fig. 1 (dashed), and for a similar line,
periodically loaded at $80$ Ω (black). The nonlinear part of four triplets
$\\{k_{s},k_{i},k_{p}\\}$ that satisfy the phase matching condition (Eq. 2)
are indicated with colored dots: in purple $\omega_{i}-\omega_{s}=4$ GHz,
green $\omega_{i}-\omega_{s}=3$ GHz, red $\omega_{i}-\omega_{s}=2$ GHz, and
blue $\omega_{i}=\omega_{s}$. Additionally, a gray line indicates a pump
frequency for which phase matching is nowhere fulfilled (b) Solving the CME
(Eqs. 11), the signal power gain profile is calculated (Eq. 15) for the
related pump frequencies (the colors match with panel a). The KIT length is
$N_{c}=6.6\times 10^{4}$ cells.

To retrieve phase-matching over a desired bandwidth, we engineer another
dispersion feature by increasing the line impedance periodically on short
sections. This feature creates a resonance in the line’s phase response (and a
stopband in the line’s amplitude response), at a frequency $\omega_{l}$
controlled by the loading periodicity Chaudhuri  _et al._ (2017); Pozar
(2011). Figure 2a shows $k^{*}$ in a line periodically loaded at $80$ Ω, with
$\omega_{l}=2\pi\times 8.5$ GHz. Because the nonlinear wavenumber close to
resonance varies sharply, there exists values of $k^{*}_{p}$ (above
$\omega_{l}$) for which we can find triplets $\\{k_{s},k_{i},k_{p}\\}$ that
satisfy Eq. 2 (examples of their nonlinear parts are shown in colored dots). A
slight variation of the pump frequency $\omega_{p}$ significantly affects
which pair of signal and idler frequencies is phase matched.

At these matched frequencies, the 3WM gain grows exponentially with $k_{p}$
(in radians per cell), with the KIT length, and with $\delta_{L}$, the
relative inductance modulation amplitude generated by the rf current and
scaling with $I_{d}$. More precisely, the phase matched, small signal power
gain can be written as:

$G_{s}\simeq\cosh^{2}\left(\frac{1}{8}\delta_{L}k_{p}N_{c}\right),$ (3)

where $N_{c}$ is the total number of cells, see appendix A.2. Typically, when
operating our KIT, $\delta_{L}\sim 7\times 10^{-3}$; thus, with $L_{d}\sim 50$
pH/cell (see Sec. III), we need $N_{c}>5\times 10^{4}$ to get $G_{s}>15$ dB at
$\omega_{p}\sim 2\pi\times 9$ GHz.

Since maximum gain is achieved with phase-matching, $\omega_{p}$ influences
the gain profile. To calculate this profile, we insert the dispersion relation
$k(\omega)$ into the CME, and solve them numerically, see appendix B. Figure
2b shows signal power gain profiles, calculated for the pump frequencies
represented in Fig. 2a. As expected, the gain is maximal at the signal and
idler frequencies for which the triplets $\\{k_{s},k_{i},k_{p}\\}$ satisfy Eq.
2. When these frequencies are far apart, there is a region in between where
phase matching is not sufficient and the gain drops. By reducing $\omega_{p}$,
we can lower the distance between phase-matched signal and idler frequencies,
and therefore obtain a wideband, flat gain region. Further reducing
$\omega_{p}$, we reach the value where phase-matched signal and idler
frequencies are equal, beyond what phase matching is nowhere fulfilled
anymore, and the gain drops. Fundamentally, the wideband nature of the gain
depends on the convexity of the dispersion relation, and therefore on the
fingers’ length and capacitance to ground. As $\omega_{f}$ or Q increase,
$k^{*}$ is less convex, and thus closer to a broadband, phase-matched
situation, but at the cost of allowing extra parametric processes to arise.

##  III EXPERIMENTAL REALIZATION

Because the kinetic inductance nonlinearity is weak, in practice a KIT
comprises a transmission line tens of centimeters long. To maximize this
nonlinearity, and to minimize its length, the line as well as IDC fingers are
made 1 $\upmu$m wide, and each unit cell is 5 $\upmu$m long, see Fig. 3a and
b. Fabricated from a 20 nm thick NbTiN layer on 380 $\upmu$m high-resistivity
intrinsic silicon via optical lithography, it yields $I_{*}\sim 7$ mA, and a
sheet inductance $\sim 10$ $\mathrm{pH}/$square. Thus, $L_{d}\sim 50$ pH, and
in order to retrieve $Z=50$ Ω, each finger is made 102 $\upmu$m long. The
loading (Fig.3a) consists of cells with shorter fingers (33.5
$\upmu\mathrm{m}$, $Z=80$ Ω), arranged periodically to generate a resonance at
8.5 GHz, thereby positioning the pump frequency in a way compatible with our
filtering capabilities.

We lay out the line in a spiral topology, on a $2\times 2$ cm chip (Fig. 3c),
which contains $N_{c}=6.6\times 10^{4}$ cells, equivalent to 33 cm. To avoid
spurious chip and microstrip modes, electrical grounding inside the chip is
ensured with spring-loaded pogo pins, descending from the top copper lid of
the packaging, and contacting the chip between the line traces, see appendix
D. The pogo pins also improve the chip-to-packaging thermal link, which
otherwise mostly relies on wire-bonds.

Figure 3: Micrographs of the KIT. (a) The transmission line (false color red)
is periodically loaded with shorter IDC fingers. (b) Line and fingers are 1
$\upmu$m wide, and each cell is 5 $\upmu$m long. (c) The overall KIT is laid
out in a spiral configuration on a $2\times 2$ cm chip, and clamped on a
copper packaging.

In a first experiment, we measure the gain, bandwidth, and power handling of
the KIT when cooled to $\sim 30$ mK. The KIT is mounted as the sole amplifier
in the chain, thereby facilitating comparison of its gain profile to the
theoretical profiles, and revealing the gain ripples of the isolated KIT,
which otherwise also depend on the return loss of external components. Two
mandatory bias tees at the KIT input and output ports combine dc and rf
currents. Figure 4a shows KIT gain profiles, acquired at two different pump
frequencies. The current amplitudes are $I_{d}=1.5$ mA, and $I_{p0}=160$
$\upmu$A ($-29$ dBm in power, calibrated in situ by comparing dc and rf
nonlinear phase shifts, see appendix A.3). For the higher pump frequency, the
gain drops in the middle of the amplification bandwidth. For the lower one,
the gain profile is flatter, with an average value of $16.5^{+1}_{-1.3}$ dB
between 3.5 and 5.5 GHz, where the subscript and superscript denote the
amplitude of the gain ripples. Both profiles agree qualitatively with
behaviors explained in Sec. II. The gain ripples have a $8$ MHz characteristic
frequency (see Fig. 4c), equivalent to $62.5$ cm in wavelength (the phase
velocity being $v_{p}=1/\sqrt{L_{d}C}\sim 1000$ cell per nanosecond), or about
twice the KIT length. We thus attribute their presence to a mismatch in
impedance between KIT and microwave feed structures before and after the KIT.
This mismatch results in a pump standing wave pattern, which influences the
signal amplification, depending on its frequency. Figure 4b shows the gain at
4.5 GHz (obtained at the lower pump power), as a function of $P_{t}$, the
input power of a probe tone. The gain compresses by 1 dB from its small signal
value for $P_{\mathrm{-1dB}}=-63$ dBm, about $7$ dB lower than theoretical
predictions, see appendix C.2. This discrepancy, suggesting substantial room
for improvement, may be due to effects not included in our model, such as
standing wave patterns, or defects in the line, which locally lower $I_{*}$.
Nonetheless, $P_{\mathrm{-1dB}}$ is already about $30$ dB higher than JTWPAs
Macklin  _et al._ (2015); White  _et al._ (2015); Planat  _et al._ (2020), and
about $10$ dB less than 4WM KITs with similar gain (see appendix H). This KIT
is therefore suitable to read thousands of frequency multiplexed MKIDs, that
use drive tone powers typically around $-90$ dBm Zobrist  _et al._ (2019);
Vissers  _et al._ (2016) or even more qubits, whose readout involves powers
substantially less than $-90$ dBm.

Figure 4: Amplification properties of the KIT. (a) The power gain is measured
with a vector network analyzer (VNA), when $\omega_{p}=2\pi\times 8.895$ GHz
(blue), and $\omega_{p}=2\pi\times 8.931$ GHz (red). (b) The gain of a probe
tone at 4.5 GHz compresses as the tone’s power increases. At
$P_{\mathrm{-1dB}}=-63$ dBm, referred to the KIT’s input, the gain has lowered
by 1 dB from its small signal value. It is measured for $\omega_{p}=2\pi\times
8.895$ GHz. (c) At the same pump frequency, a close-up on the small signal
gain around 4.5 GHz shows ripples with 8 MHz characteristic frequency.

As in any phase-insensitive traveling-wave and resonant parametric amplifier,
the practical, usable bandwidth, is half of the presented amplification
bandwidth. It is the bandwidth in which signals coming from microwave devices
can be phase-insensitively amplified. The other half, barring the idler
frequencies, contains a copy of signals in the first half. That is why the
gain in Fig. 4a is nearly symmetric about the half pump frequency ($\sim 4.5$
GHz). The asymmetry - gain and ripples marginally bigger above the half pump
frequency - originates from a frequency dependent saturation power. In fact,
higher frequencies possess a higher saturation power, see appendix C.1. We see
this effect here because we chose a signal power close to $P_{\mathrm{-1dB}}$
in order to maximize the signal-to-noise ratio in this measurement, where the
KIT remains the sole amplifier.

At higher dc current bias (bounded by the dc critical current of the
transmission line, $\sim 2.4$ mA in our device), lower rf pump power can be
used to obtain equivalent small signal gain, at the cost of a reduced 1 dB
compression power. Conversely, reducing $I_{d}$ and increasing $I_{p0}$
improves power handling capabilities, but the gain is then limited by a
superconductivity breaking phenomenon. We suspect the presence of weak links
Bockstiegel  _et al._ (2014), and we are currently investigating the line
breakdown mechanism.

##  IV NOISE PERFORMANCE

The combined gain, bandwidth, and power handling performance are promising,
provided that the KIT also presents competitive noise performance. Measuring
this noise is a topic of current interest Eom  _et al._ (2012); Ranzani  _et
al._ (2018); Zobrist  _et al._ (2019), and we execute the task using a self-
calibrated shot-noise tunnel junction (SNTJ) Spietz  _et al._ (2003, 2006). We
measure the output spectral density, whose power depends on the chain’s gain
and loss. The SNTJ acts as a dynamic variable noise source, allowing for a
continuous sweep in input noise temperature by sweeping its bias voltage, and
our measurement scans the entire KIT bandwidth.

Figure 5: Schematic of the noise measurement setup. Each component is labeled
with its gain or transmission efficiency, and with its input-referred added
noise. Calibrated noise $N_{\mathrm{in}}^{s}$ ($N_{\mathrm{in}}^{i}$) is
generated by the SNTJ at the signal (idler) frequency. It is routed to the KIT
with transmission efficiency $\eta_{1}^{s}$ ($\eta_{1}^{i}$), i.e. it
undergoes a beamsplitter interaction and part of it is replaced with vacuum
noise whose value $N_{f}=0.5$ quanta. At the KIT’s input, the noise is
$N_{1}^{j}$, $j\in\\{s,i\\}$. On the signal-to-signal path, the KIT then adds
$N_{\mathrm{ex}}^{s}$ quanta of excess noise and has a gain $G$; on the idler-
to-signal path, the KIT adds $N_{\mathrm{ex}}^{i}$ quanta of excess noise and
has a gain $G-1$. Noise $N_{2}^{s}$ at the KIT’s output is then routed with
efficiency $\eta_{2}$ to the HEMT (input noise $N_{3}^{s}$). With gain $G_{H}$
and added noise $N_{H}$, it further directs the noise $N_{4}^{s}$ to room
temperature components. Amplification and loss at room temperature are
excluded from our schematic but not our analysis. The noise reaching the
spectrum analyzer (SA) is $N_{o}^{s}$. The full setup is described in appendix
E. Figure 6: System-added noise measurement of a microwave amplification chain
with a KIT as first amplifier. (a) The gain’s frequency dependence is measured
with a VNA (with a 1 kHz intermediate frequency bandwidth). (b) The output
noise $N_{o}^{s}$ is measured with a SA (5 MHz resolution bandwidth,
comparable to typical resonant amplifier’s bandwidth), while varying the SNTJ
dc voltage bias V. Fitting the whole output noise response, we obtain the
frequency dependent system-added noise $N_{\Sigma}$ and the chain’s signal-to-
signal gain $G_{c}^{ss}$. We divide $N_{o}^{s}$ by $G_{c}^{ss}$ to refer it to
the KIT input, and subtract the zero bias noise value $N_{f}=0.5$, so that
$N_{\Sigma}$ (panel c) visually matches the zero bias value of $N_{o}^{s}$
(see appendix F.3). Three colored curves indicate output noises at 4, 5, and 6
GHz, with fits superimposed in black lines. (c) Data from the output noise
spectra are compiled to form $N_{\Sigma}$. Uncertainties are indicated by the
gray area surrounding the black line. They predominantly come from the fit of
the curves in (b). The quantum limit (QL) on amplifier-added noise is
indicated by the dashed black line.

Because the SNTJ is a wideband noise source, it illuminates the KIT at both
its signal and idler frequency inputs; these input noises are then
parametrically amplified. We thus model the KIT as a three-port network: two
inputs at $\omega_{s}$ and $\omega_{i}$, and one output at $\omega_{s}$.
Figure 5 schematizes the entire amplification chain, where we have labeled the
gain or transmission efficiency, and input-referred added noise of each
component. The KIT power gain on the signal-to-signal path is $G$, while its
power gain on the idler-to-signal path is $G-1$ Caves (1982). The output noise
at the signal frequency (in units of photons) measured on the spectrum
analyzer (SA) can then be written as

$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s})+G_{c}^{si}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i}),$
(4)

where $N_{\mathrm{in}}^{s}$ ($N_{\mathrm{in}}^{i}$) is the SNTJ-generated
noise at the signal (idler) frequency, $G_{c}^{ss}$ ($G_{c}^{si}$) is the
signal-to-signal (idler-to-signal) gain of the entire chain from SNTJ to SA,
and $N_{\mathrm{eff}}^{s}$ ($N_{\mathrm{eff}}^{i}$) is the effective signal-
to-signal (idler-to-signal) path KIT-excess noise, see appendix F.1. When
varying $N_{\mathrm{in}}^{s}$ and $N_{\mathrm{in}}^{i}$ we retrieve
$N_{\mathrm{eff}}^{s}$ and $N_{\mathrm{eff}}^{i}$, equal to zero for a
quantum-limited amplifier.

The system-added noise that a device replacing the SNTJ would see is therefore

$N_{\Sigma}=N_{\mathrm{eff}}^{s}+\frac{G_{c}^{si}}{G_{c}^{ss}}(N_{f}+N_{\mathrm{eff}}^{i}),$
(5)

where $N_{f}=0.5$ quanta is the vacuum noise (provided that the idler port of
this device is cold); for a high gain, phase-insensitive quantum-limited
amplifier $N_{f}$ is the minimum added noise Caves (1982). Failure to account
for the additional change in idler noise at the amplifier’s input leads to an
underestimate of the system-added noise by about a factor two, thereby making
it look significantly better than its true value, given by Eq. 5, see appendix
F.2.

In practice, we measure $N_{o}^{s}$ while varying simultaneously
$N_{\mathrm{in}}^{s}$ and $N_{\mathrm{in}}^{i}$ using the SNTJ voltage bias
(Fig. 6b). We then fit to obtain $G_{c}^{ss}$, $G_{c}^{si}$,
$N_{\mathrm{eff}}^{s}$ and $N_{\mathrm{eff}}^{i}$ (see appendix F.3), and form
$N_{\Sigma}$ with Eq. 5. Figure 6c presents the system-added noise
$N_{\Sigma}$, measured over the KIT’s amplification bandwidth. In this
experimental configuration, the KIT gain is $G=16.6^{+1.8}_{-3.1}$ dB between
3.5 and 5.5 GHz (Fig. 6a), stable over the acquisition time ($\sim 12$ hrs).
In that bandwidth, $N_{\Sigma}=3.1\pm 0.6$ quanta. It is an unprecedented
broadband, true system-added noise performance (see appendix H).

This performance depends on the intrinsic signal (idler) KIT-excess noise
$N_{\mathrm{ex}}^{s}$ ($N_{\mathrm{ex}}^{i}$), but also on the chain’s
transmission efficiencies as well as on the HEMT-added noise $N_{H}$. More
precisely, when $\\{G,N_{H}\\}\gg 1$, Eq. 5 becomes:

$N_{\Sigma}=\frac{N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}}{\eta_{1}^{s}}+\frac{2(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{N_{H}}{\eta_{2}G\eta_{1}^{s}}+N_{f}.$
(6)

From left to right, the first three terms in the right-hand side represent the
contribution to $N_{\Sigma}$ from the KIT alone, from the lossy link between
the SNTJ and the KIT, and from the HEMT-added noise; the last term is the
minimum half quantum of added noise that a quantum-limited amplifier must add
Caves (1982). Measuring the individual loss of the chain’s components, we
estimate the value of the transmission efficiencies to be
$\eta_{1}^{s}=0.57\pm 0.02$ and $\eta_{2}=0.64\pm 0.10$; in addition, by
measuring the system-added noise of the amplification chain with the KIT
turned off, we estimate $N_{H}=8\pm 1$ quanta, see appendix G. With these
additional information, we estimate that the overall KIT-excess noise is
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}=0.77\pm 0.40$ quanta, suggesting that
the KIT alone operates near the quantum limit.

Several strategies can improve the system-added noise. First, increasing the
transmission efficiencies would have a major impact, because all the system-
dependent noise contributions are enlarged by $1/\eta_{1}^{s}$. For example,
$\eta_{1}^{s}=0.8$ would yield $N_{\Sigma}=1.6$ quanta. To that end, we are
currently developing lossless superconducting circuits (bias tees, directional
couplers and low-pass filters) that can be integrated on-chip with the KIT.
Second, increasing the KIT gain $G$ would reduce the HEMT contribution to
$N_{\Sigma}$ (third term in Eq. 6). Here, this contribution is estimated at
$0.5\pm 0.3$ quanta. The solutions to achieve higher gain directly follow from
Eq. 3: higher pump power (i.e. increase $\delta_{L}$), longer line (increase
$N_{c}$), or higher inductance per unit cell (increase $k_{p}$). All represent
non-trivial challenges, starting with better understanding of the line
breakdown mechanism Bockstiegel  _et al._ (2014). If it comes from
imperfections in the line (weak links), a higher resolution fabrication
process, like electron beam lithography, may improve the performance of the
device. Also, running the amplifier at higher gain will require better damping
of the gain ripples, whose amplitude grows with gain. Finally, we are
investigating the origin of the remaining excess noise
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}$. It may be due to parasitic chip
heating or two-level system noise Gao  _et al._ (2008).

##  V CONCLUSION

It is possible to build a microwave amplifier with broadband and near-quantum-
limited performance without sacrificing power handling capability. Engineering
the phase-matched bandwidth is key because it suppresses spurious parametric
conversion processes. We demonstrate this idea on a KIT, whose combined gain,
bandwidth, power handling, and noise performance are fully characterized. In
addition, we develop a theoretical framework adapted to noise measurements
performed with wideband noise sources. Using a SNTJ we evaluate the true
system-added noise of an amplification containing the KIT as the first
amplifier. The KIT itself is estimated to be near-quantum-limited, therefore
it has the potential to initiate a qualitative shift in the way arrays of
superconducting detectors, such as MKIDs, process information, moving them
into a quantum-limited readout regime.

## Acknowledgments

We thank Kim Hagen for his help in the design and fabrication of the KIT
packaging, and Felix Vietmeyer and Terry Brown for their help in the design
and fabrication of room temperature electronics. We acknowledge useful
discussions with John Teufel, Gangqiang Liu and Vidul Joshi. Certain
commercial materials and equipment are identified in this paper to foster
understanding. Such identification does not imply recommendation or
endorsement by the National Institute of Standards and Technology, nor does it
imply that the materials or equipment identified are necessarily the best
available for the purpose. This work was supported by the NIST Innovations in
Measurement Science Program, as well as NASA, under grant NNH18ZDA001N-APRA.

##  Appendix A COUPLED-MODE THEORY OF A DC-BIASED KIT

###  A.1 Coupled-mode equations

The phase matching condition for exponential gain, Eq. 2 is obtained by
solving the CME while pumping in a 3WM fashion, i.e. such that
$\omega_{p}=\omega_{s}+\omega_{i}$, and in the presence of 3WM and 4WM terms
see, Eq. 1. The CME relate the current amplitudes $I_{j}$, $j\in\\{p,s,i\\}$
at the frequencies $\omega_{j}$, $j\in\\{p,s,i\\}$; they are obtained by
injecting equation 1 into the telegrapher’s equations, and by operating the
harmonic balance (HB) with only these three frequencies.

More precisely, the telegrapher’s equations in a lossless transmission line
relate current I and voltage V as:

$\displaystyle-\frac{\partial I}{\partial x}$ $\displaystyle=C\frac{\partial
V}{\partial t}$ (7) $\displaystyle-\frac{\partial V}{\partial x}$
$\displaystyle=L\frac{\partial I}{\partial t},$

with x a length per unit cell. Injecting Eq. 1 into Eqs. 7, we obtain:

$v_{p}^{2}\frac{\partial^{2}I}{\partial x^{2}}-\frac{\partial^{2}I}{\partial
t^{2}}=\frac{\partial}{\partial t^{2}}\left(\frac{1}{2}\epsilon
I^{2}+\frac{1}{3}\xi I^{3}\right),$ (8)

with $v_{p}=1/\sqrt{CL_{d}}$ the phase velocity. To solve Eq. 8 we perform the
HB: we assume that the current in the transmission line is a sum of three
terms at three different frequencies:

$\displaystyle I=\frac{1}{2}\big{(}$ $\displaystyle
I_{p}(x)e^{i(k_{p}x-\omega_{p}t)}$ (9) $\displaystyle+$ $\displaystyle
I_{s}(x)e^{i(k_{s}x-\omega_{s}t)}$ $\displaystyle+$ $\displaystyle
I_{i}(x)e^{i(k_{i}x-\omega_{i}t)}+c.c\big{)},$

and we then keep only the mixing terms from Eq. 8 that emerge at these
frequencies. This approach is valid in our case, because the phase matching
bandwidth is limited by dispersion engineering (see appendix B), and thus
mostly these three frequencies are able to mix together. Under the slow-
varying envelope approximation, $\lvert d^{2}I_{j}/dx^{2}\rvert\ll\lvert
k_{j}dI_{j}/dx\rvert$ for $j\in\\{p,s,i\\}$, the left hand side of Eq. 8
yields:

$v_{p}^{2}\frac{\partial^{2}I}{\partial x^{2}}-\frac{\partial^{2}I}{\partial
t^{2}}=v_{p}^{2}\sum_{j=p,s,i}ik_{j}\frac{dI_{j}}{dx}e^{i(k_{j}x-\omega_{j}t)}+c.c.$
(10)

Using $\omega_{p}=\omega_{s}+\omega_{i}$, we collect terms at $\omega_{j}$,
$j\in\\{p,s,i\\}$ in the right hand side (rhs) and form the CME:

$\displaystyle\frac{dI_{p}}{dx}$
$\displaystyle=\frac{ik_{p}\epsilon}{4}I_{s}I_{i}e^{-i\Delta_{k}x}+\frac{ik_{p}\xi}{8}I_{p}(\lvert
I_{p}\rvert^{2}+2\lvert I_{s}\rvert^{2}+2\lvert I_{i}\rvert^{2})$ (11)
$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}e^{i\Delta_{k}x}+\frac{ik_{s}\xi}{8}I_{s}(2\lvert
I_{p}\rvert^{2}+\lvert I_{s}\rvert^{2}+2\lvert I_{i}\rvert^{2})$
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}e^{i\Delta_{k}x}+\frac{ik_{i}\xi}{8}I_{i}(2\lvert
I_{p}\rvert^{2}+2\lvert I_{s}\rvert^{2}+\lvert I_{i}\rvert^{2}),$

with $\Delta_{k}=k_{p}-k_{s}-k_{i}$. The phase matching condition, Eq. 2, is
found for a strong pump, where $\\{\lvert I_{s}\rvert,\lvert
I_{i}\rvert\\}\ll\lvert I_{p}\rvert$. Assuming the pump undepleted, $\lvert
I_{p}(x)\rvert=I_{p0}$, Eqs. 11 rewrite:

$\displaystyle\frac{dI_{p}}{dx}$
$\displaystyle=\frac{ik_{p}\xi}{8}I_{p}I_{p0}^{2}$ (12)
$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}e^{i\Delta_{k}x}+\frac{ik_{s}\xi}{4}I_{s}I_{p0}^{2}$
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}e^{i\Delta_{k}x}+\frac{ik_{i}\xi}{4}I_{i}I_{p0}^{2},$

which results in $I_{p}(x)=I_{p0}\exp{(i\xi k_{p}I_{p0}^{2}x/8)}$. Signal and
idler amplitudes are then searched with the form:
$I_{j}(x)=\tilde{I}_{j}(x)\exp{(i\xi I_{p0}^{2}k_{j}x/4)}$, $j\in\\{s,i\\}$.
Equations 12 then yield:

$\displaystyle\frac{d\tilde{I}_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p0}\tilde{I}_{i}^{*}e^{i\Delta_{\beta}x}$
(13) $\displaystyle\frac{d\tilde{I}_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p0}\tilde{I}_{s}^{*}e^{i\Delta_{\beta}x},$

with $\Delta_{\beta}=\Delta_{k}+\frac{\xi I_{p0}^{2}}{8}(k_{p}-2k_{s}-2k_{i})$
and $\Delta_{k}=k_{p}-k_{s}-k_{i}$. The system of equations 13 has known
solutions Boyd (2019). In particular, when phase matching is achieved, i.e.
$\Delta_{\beta}=0$, we obtain:

$\displaystyle\tilde{I}_{s}$ $\displaystyle=\cosh{(g_{3}x)\tilde{I}_{s0}}$
(14) $\displaystyle\tilde{I}_{i}$
$\displaystyle=i\sqrt{\frac{k_{i}}{k_{s}}}\sinh{(g_{3}x)\tilde{I}_{s0}},$

with $g_{3}=\frac{\epsilon I_{p0}}{4}\sqrt{k_{i}k_{s}}$, and with initial
conditions $I_{s}(0)=I_{s0}$ and $I_{i}(0)=0$. The signal power gain

$G_{s}(x)=\left\lvert\frac{I_{s}(x)}{I_{s0}}\right\rvert^{2}$ (15)

is then exponential with $x$: $G_{s}=\cosh^{2}(g_{3}x)$.

###  A.2 3WM gain

We can re-write $g_{3}$ as a function of more meaningful quantities. In fact,
the linear inductance $L$ exposed in Eq. 1 also writes:

$\displaystyle L$
$\displaystyle=L_{0}\left(1+\frac{I_{d}^{2}}{I_{*}^{2}}\right)\left(1+\frac{2I_{d}I}{I_{*}^{2}+I_{d}^{2}}+\frac{I^{2}}{I_{*}^{2}+I_{d}^{2}}\right)$
(16) $\displaystyle\simeq
L_{d}\left(1+\frac{2I_{d}I}{I_{*}^{2}+I_{d}^{2}}\right),$

when $I\ll I_{d}$, and with $L_{d}=L_{0}(1+I_{d}^{2}/I_{*}^{2})$. Here,
$L_{0}$ is the bare linear inductance. It is the one directly derived from the
sheet kinetic inductance and the geometry of the line, while $L_{d}$ is the
inductance per unit length under dc bias. Because $I_{d}^{2}\ll I_{*}^{2}$,
for design purposes $L_{d}\sim L_{0}$. In the strong pump regime $I=I_{p0}$,
therefore, $2I_{d}I/(I_{*}^{2}+I_{d}^{2})=\epsilon I_{p0}$; assuming
$k_{s}=k_{i}\simeq k_{p}/2$, we therefore get:

$G_{s}(x)\simeq\cosh^{2}\left(\frac{1}{8}\delta_{L}k_{p}x\right),$ (17)

where

$\delta_{L}=\frac{L-L_{d}}{L_{d}}=\frac{2I_{d}I_{p0}}{I_{*}^{2}+I_{d}^{2}}$
(18)

is the relative inductance variation due to $I_{p0}$. With $I_{*}=7$ mA, and
typical values: $I_{d}=1.5$ mA (limited by the dc critical current, $\sim 2.4$
mA, value specific to our NbTiN film and to the line’s width) and
$I_{p0}=I_{*}/60$, we get $\delta_{L}=6.8\times 10^{-3}$.

###  A.3 Pump phase shift

From the phase matching condition, Eq. 2, it is clear that only the 4WM term
$\xi$ creates a dispersive phase shift of pump, signal and idler. In other
words, in a pure 3WM case, $\xi=0$ and the phase matching condition becomes
$\Delta_{k}=0$, naturally fulfilled in a dispersion-less transmission line.
While detrimental for noise properties (see Sec. IV), we can use the continued
presence of 4WM to our advantage, because it allows us to calibrate the pump
power, down to the KIT input.

In fact, in such a situation $I_{d}$ and $I_{p0}$ influence the pump tone
phase shift, which we can measure unambiguously (i.e. not $\mod 2\pi$) with a
VNA. More precisely, although the phase $\phi=\arg(S_{21})$ read by a VNA is
$2\pi$-wrapped, its shift $\delta_{\phi}=\phi-\phi_{0}$ from an initial value
$\phi_{0}$ can be continuously monitored when $I_{d}$ and $I_{p0}$ vary, and
thus unambiguously determined. This phase shift in turn translates into a
wavenumber variation $\delta_{p}=-\delta_{\phi}/N_{c}$. If initially at zero
dc bias and small rf pump amplitude, then $\delta_{p}=\beta_{p}-k_{p}$, with
$\beta_{p}$ the pump wavenumber, dependent on $I_{d}$ and $I_{p0}$, and
$k_{p}$ the linear wavenumber. When a single pump tone travels along the line
(no input signal), we are by default under the strong pump approximation, and
the first equation of Eqs. 12 gives $I_{p}(x)=I_{p0}\exp{(i\xi
k_{p}I_{p0}^{2}x/8)}$. In addition, the current I in the line then writes as
$I=1/2\\{I_{p}(x)\exp[i(k_{p}x-\omega_{p}t)]+c.c\\}$, and thus the pump
wavenumber is $\beta_{p}=\xi k_{p}I_{p0}^{2}/8+k_{p}$, which leads to
$\delta_{p}=\xi k_{p}I_{p0}^{2}/8$. Because $k_{p}=\omega_{p}\sqrt{L_{d}C}$,
we can rewrite:

$\delta_{p}=\frac{1}{8}\frac{I_{p0}^{2}}{I_{*}^{2}}\omega_{p}\sqrt{L_{0}C}\sqrt{1+\frac{I_{d}^{2}}{I_{*}^{2}}},$
(19)

therefore $I_{p0}$ and $I_{d}$ induce similar phase shift in the line. Knowing
$I_{d}$ (room temperature parameter), we thus get $I_{p0}$ at the KIT input.

##  Appendix B ABCD TRANSFER MATRICES

The dispersion relations, Fig. 2a are calculated by cascading the ABCD
matrices of the KIT cells, a method suitable for any periodic loading pattern.
We then compute the KIT $S_{21}$ scattering parameter as
$S_{21}=2/(A+B/Z_{0}+CZ_{0}+D)$ Pozar (2011), where $Z_{0}=50$ Ω is the input
and output ports impedance, and finally get
$k=-\mathrm{unwrap}[\arg(S_{21})]/N_{c}$.

In the unloaded case, represented in Fig. 1, the ABCD matrix cell is:

$\bm{T_{c}}=\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}.$
(20)

All the cells being identical, the KIT’s ABCD matrix is simply
$\bm{T_{K}}=\bm{T_{c}}^{N_{c}}$. In Fig. 2a we used $N_{c}=6.6\times 10^{4}$
to match the length of our fabricated KIT, and $L_{d}=45.2$ pH, $C=18.8$ fF,
and $L_{f}=1.02$ nH, values that match our design (fingers are $102$ $\upmu$m
long and $1$ $\upmu$m wide).

In the loaded case, some cells have shorter fingers, see Fig. 3a. In these
cells, the capacitance to ground is $C_{l}=L_{d}/Z_{l}^{2}$, where $Z_{l}$ is
the load impedance, and a finger’s inductance is $L_{l}$. To compute the KIT
scattering parameter, we form the ABCD matrix of the repetition pattern
comprised with unloaded and loaded cells:

$\displaystyle\bm{T_{\mathrm{sc}}}=$
$\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}^{N_{u}/2}$
(21) $\displaystyle\times$ $\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C_{l}\omega}{2-L_{l}C_{l}\omega^{2}}&1-\frac{2L_{d}C_{l}\omega^{2}}{2-L_{l}C_{l}\omega^{2}}\end{bmatrix}^{N_{l}}$
$\displaystyle\times$ $\displaystyle\begin{bmatrix}1&iL_{d}\omega\\\
\frac{i2C\omega}{2-L_{f}C\omega^{2}}&1-\frac{2L_{d}C\omega^{2}}{2-L_{f}C\omega^{2}}\end{bmatrix}^{N_{u}/2},$

where $N_{u}$ is the number of unloaded cells and $N_{l}$ is the number of
loaded cells in the pattern, which we call a supercell. As before, to get the
KIT’s ABCD matrix, we simply form
$\bm{T_{K}}=\bm{T_{\mathrm{sc}}}^{N_{\mathrm{sc}}}$, where
$N_{\mathrm{sc}}=N_{c}/(N_{u}+N_{l})$ is the number of supercells in the KIT.
Here, $N_{u}=60$ (equivalent to $300\upmu$m), $N_{l}=6$ (equivalent to
$30\upmu$m), $N_{c}=66000$ (equivalent to $33$ cm), therefore
$N_{\mathrm{sc}}=1000$. The plain line in Fig. 2a shows the wavenumber $k^{*}$
thus found, with $Z_{l}=80$ Ω, and $L_{l}=335$ pH, as the finger length in a
loaded cell is $33.5$ $\upmu$m .

To compute the signal power gain, Fig. 2b, we inject the expression of
$k(\omega)$ for a periodically loaded KIT (from $\bm{T_{K}}$) in the CME, Eqs.
11. We solve them numerically for different pump frequencies. For $8.8812$ GHz
(blue curve), $8.8992$ GHz (red), $8.9256$ GHz (green) and $8.9736$ GHz
(purple), phase matched signal and idler are detuned from the half pump
frequency by $0$, $1$, $1.5$ and $2$ GHz respectively. For $8.855$ GHz (gray
curve), phase matching is nowhere achieved. We used $N_{c}=6.6\times 10^{4}$,
$I_{*}=7$ mA, $I_{d}=1.5$ mA, and the initial conditions $I_{p0}=I_{*}/60$,
$I_{s0}=I_{p0}/100$, and $I_{i0}=0$, close to experimental values.

##  Appendix C KIT SATURATION

###  C.1 Strong signal gain profile asymmetry

When the input signal power amounts to a significant fraction of the pump
power, parametric amplification depletes the pump. It surprisingly generates
asymmetry in the signal gain profile, with respect to the half pump frequency.
Figure 7a shows signal gain profiles, calculated when phase matching is
achieved at exactly half the pump frequency, i.e. for $\omega_{s}=\omega_{i}$
(corresponding to $\omega_{p}=8.8812$ GHz), at various initial signal powers
$P_{s0}$. They are obtained by solving the CME 11, which incorporate pump
depletion effects. As $P_{s0}$ increases, the gain diminishes, and the
originally flat profile tilts, with higher frequencies presenting higher gain.

Figure 7: Theory of KIT saturation, calculated by solving the full CME 11. (a)
Distortion of the gain profile, at four different input signals:
$I_{s0}=I_{p0}/100$ (blue curve), corresponding to a small signal regime,
$I_{s0}=I_{p0}/12$ (red), $I_{s0}=I_{p0}/8$ (green), and $I_{s0}=I_{p0}/6$
(purple). The KIT length, is $N_{c}=6.6\times 10^{4}$ cells. Vertical lines
indicate three frequencies: 3.4406 GHz (long dashed), 4.4406 GHz (plain),
equal to the half pump frequency, and 5.4406 GHz (short dashed). (b) The gain
of a probe tone is calculated at these three frequencies, as a function of the
probe tone power. (c) The 1 dB compression power is shown as a function of
frequency.

Fundamentally, this asymmetry stems from the fact that when solving the CME in
the case where $I_{p}$ varies along the KIT transmission line, the initial
conditions (ICs) vary as a function of frequency. More precisely, the second
and third equations in Eqs. 11, govern the evolution of $I_{s}$ and $I_{i}$
respectively; in a simplified version, they write as

$\displaystyle\frac{dI_{s}}{dx}$
$\displaystyle=\frac{ik_{s}\epsilon}{4}I_{p}I_{i}^{*}$ (22)
$\displaystyle\frac{dI_{i}}{dx}$
$\displaystyle=\frac{ik_{i}\epsilon}{4}I_{p}I_{s}^{*}.$

We dropped the second terms in the rhs of Eqs. 11, representing the 4WM
conversion processes, as the asymmetry still holds when $\xi=0$, and we
assumed perfect phase matching in 3WM, $\Delta_{k}=0$, i.e. a dispersionless
line.

In the undepleted pump regime $I_{p}(x)=I_{p0}$, and we can decouple the
equations on $I_{s}$ and $I_{i}$. Deriving with respect to $x$ Eqs. 22, we get

$\frac{d^{2}I_{j}}{dx^{2}}=g_{3}^{2}I_{j},$ (23)

with $j\in\\{s,i\\}$, and $g_{3}=\frac{\epsilon I_{p0}}{4}\sqrt{k_{i}k_{s}}$,
as defined in appendix A.1. Using the ICs $I_{s}(0)=I_{s0}$ and
$dI_{s}/dx(0)=0$ (because $I_{i}^{*}(0)=0$), $I_{s}=\cosh(g_{3}x)I_{s0}$, as
seen in Eqs. 14. Signal and idler wavenumbers appear as a product in this
solution, hence for any $x$ the signal amplitude $I_{s}$ is symmetric with
respect to the half pump frequency.

In the soft pump regime, where $I_{p}(x)$ is not constant, Eqs. 22 cannot be
decoupled. We can however write them in a canonical form, deriving with
respect to $x$:

$\frac{d^{2}I_{j}}{dx^{2}}-\frac{1}{I_{p}}\frac{dI_{p}}{dx}\frac{dI_{j}}{dx}-\frac{k_{s}k_{i}\epsilon^{2}}{16}\lvert
I_{p}\rvert^{2}I_{j}=0,$ (24)

with $j\in\\{s,i\\}$. Here, the interplay between $I_{s}$ and $I_{i}$ comes
from $I_{p}$, which contains the product $I_{s}I_{i}$ (see Eqs. 11). Although
signal and idler wavenumbers also appear only as a product in Eqs. 24, these
CME lead to an asymmetric signal amplitude profile, because out of the five
ICs required to solve them, one changes with frequency: $I_{p}(0)=I_{p0}$,
$I_{s}(0)=I_{s0}$, $I_{i}(0)=0$, $dI_{s}/dx(0)=0$, and $dI_{i}/dx(0)=i\epsilon
I_{p0}I_{s0}k_{i}/4$. This last IC depends on $k_{i}$, which depends on the
signal frequency. In the small signal limit, $dI_{i}/dx(0)\xrightarrow{}0$,
and we recover a symmetric gain profile with respect to the half pump
frequency.

###  C.2 Compression power calculation

This asymmetry produces higher power handling capabilities at frequencies
above $\omega_{p}/2$, compared to below $\omega_{p}/2$. Thus, considering that
only half the bandwidth is usable for resonator readout, it is more
advantageous to have these lie above $\omega_{p}/2$. Figure 7b shows the gain
as a function of a probe tone power $P_{t}$, calculated from the CME 11 at
three frequencies: one at $\omega_{p}/2$, and two at $\omega_{p}/2\pm 1$ GHz.
Because phase matching is set to be optimal at
$\omega_{s}=\omega_{i}=\omega_{p}/2$, gain is maximal at this frequency. The
small signal gain is identical for $\omega_{p}/2\pm 1$ GHz, however it visibly
compresses at higher tone power for $\omega_{p}/2+1$ GHz. Figure 7c presents
the 1 dB compression power $P_{\mathrm{-1dB}}$, calculated in the interval
$[\omega_{p}/2-1,\omega_{p}/2+1]$ gigahertz. As expected, $P_{\mathrm{-1dB}}$
is a few dB higher when $\omega>\omega_{p}/2$. This phenomenon is reminiscent
of gain distortion, seen in JPAs Malnou  _et al._ (2018). Effects not included
in the CME 11, such as standing wave patterns, or defects in the line, which
locally lower $I_{*}$, may cause the discrepancy between these theoretical
calculations and the measurements (see Sec. III)

##  Appendix D KIT PACKAGING

There are three main concerns when packaging a KIT: first, the package should
be matched to 50 Ω. Any mismatch will result in reflections, creating gain
ripples (see Sec. III). Second, the package should ensure good electrical
grounding of the transmission line. Otherwise, given the fairly large chip
size, spurious chip modes can appear within the frequency range of interest.
Third, the package should ensure good thermalization inside the chip. Because
the pump power remains high for millikelvin operations, any inhomogeneous rise
in temperature may trigger a hot spot near a weak-link, and possibly break
superconductivity. We implemented a series of technologies to address these
concerns.

Figure 8: KIT packaging. (a) The KIT is clamped on a gold plated copper box
and wire-bonded to PCBs and to the box itself. Although not very sensitive to
magnetic field, the KIT is shielded with aluminum and A4K cryogenic shielding.
Two SMA connectors protrude on both side of the packaging. (b) A close-up on
the central region of the KIT shows the periodically loaded transmission line,
with gold strips deposited between its spiral arms. (c) The top copper lid
contains spring-loaded pogo pins inserted half-way into the box. They are
arranged to contact the chip between the KIT trace. (d) A close-up on the pins
shows their top thinner part, which can retract inside the body of the pin.
They are fixed on the copper lid with dried silver paint.

Figure 8a presents the chip, clamped onto the bottom part of the copper
packaging. The chip is wire-bonded on both sides to printed circuit boards
(PCBs). They convert the on-chip CPW layout to microstrip, and then the
central pin of sub-miniature version A (SMA) connectors are soldered onto the
microstrip. We suspect imperfect PCBs, with measured impedance close to 52 Ω,
play a role in creating gain ripples. When designing the spiral, we carefully
adjusted the radius of the turns, in conjunction with the unit cell length, to
have these turns match the straight sections’ inductance and capacitance per
length.

Electrical grounding inside the chip is ensured with pogo pins inserted in the
top lid of the packaging, see Fig. 8c and d. When closing the box, these pins
contact the chip between the line traces. If absent, we have measured spurious
resonant modes with harmonics at gigahertz frequencies. Pins are $140$
$\upmu$m in diameter, and each applies a 20 g force to the chip.

Figure 9: Full schematic of the noise measurement experimental setup. The KIT
is represented by a spiral, enclosed in a square. A permanent magnet
suppresses the Josephson effect in the SNTJ. A magnetic shield protects other
microwave components from the effect of this magnet. The 4-12 GHz isolator is
from Quinstar technology, model CWJ1015-K13B. The dashed purple line
represents the bypass.

These pins also act as thermal links to the packaging. In addition, we
deposited gold strips onto the NbTiN layer, inside the spiral, between the KIT
line traces, and near the chip edges, see Fig. 8b. These strips contact the
pins. Absent the pogo pins, superconductivity breaks down before high gain can
be reached. Finally, we gold-bond the chip ground plane (from the deposited
gold) to the copper box (instead of using standard aluminum bonding): gold
remains a normal metal at millikelvin temperatures, thereby better
thermalizing the chip.

##  Appendix E NOISE MEASUREMENT EXPERIMENTAL SETUP

Figure 9 presents a schematic of the full experimental setup used to measure
the system-added noise of a readout chain using a KIT as a pre-amplifier. In
total, noise generated by the SNTJ travels through three amplifiers: the KIT,
a HEMT at 4K, and a room temperature amplifier, before finally being recorded
with a SA.

The SNTJ is packaged with a permanent magnet suppressing the Josephson effect,
and a magnetic shield protects other elements from this magnetic field. In
addition, a bias tee routes SNTJ-generated rf noise to the microwave readout
chain, while at the same time allowing for dc bias. In fact, the SNTJ is
biased with an arbitrary waveform generator (AWG). It outputs a low frequency
(50 Hz) triangular voltage wave on a $10$ kΩ current limiting resistor,
thereby creating a current $I_{\mathrm{AWG}}$, varying between $\pm
12$$\upmu$A, which sweeps the SNTJ-generated noise value.

An oscilloscope reads the SNTJ voltage in situ while the AWG outputs a known
current, allowing the computation of the SNTJ impedance
$Z_{\mathrm{SNTJ}}=54\pm 4$ Ω, and with it, the SNTJ voltage bias
$V=Z_{\mathrm{SNTJ}}I_{\mathrm{AWG}}$.

Noise from the SNTJ is combined with rf tones (pump, and probe from a VNA to
measure the gain profile) via a 20 dB directional coupler (DC) connected to
the KIT input. Additionally, a 7 GHz low-pass filter placed between the SNTJ
and the DC prevents the rf pump from leaking back to the SNTJ. Because the KIT
requires a fairly high pump power ($-29$ dBm), we only attenuate the pump line
by 10 dB at 4K. Then, an 8 GHz high-pass filter at 30 mK rejects noise at
frequencies within the KIT amplification band, while allowing the pump to
pass. A bias tee at the KIT input port combines rf signals (including noise
from the SNTJ) with the KIT dc current bias, and a second bias tee at the KIT
output separates dc from rf. The rf signal then passes through a $4-12$ GHz
isolator, and a 7 GHz low-pass filter (Pasternack PE87FL1015, $\sim 45$ dB
rejection at 9 GHz), preventing the pump tone from saturating the HEMT.

The SA is operated in a zero-span mode, its acquisition triggered by the AWG.
This measures the output noise at a single frequency, over a 5 MHz resolution
bandwidth (RBW), and directly traces out the curves of Fig. 6b. Varying the SA
center frequency, we obtain the system-added noise over the full 3-6.5 GHz
bandwidth, Fig. 6c.

##  Appendix F NOISE THEORY

###  F.1 System-added noise

When propagating through the experimental setup presented in Fig. 9, noise
generated by the SNTJ undergoes loss and amplification. Both processes affect
the effective noise at each amplifier input, and therefore also, the noise
reaching the SA. While the overall system-added noise $N_{\Sigma}$ encompasses
microwave loss, we derive its complete expression as a function of signal
($N_{\mathrm{ex}}^{s}$) and idler ($N_{\mathrm{ex}}^{i}$) amplifier-excess
noise, gain, and transmission efficiencies to estimate
$N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}$.

Figure 5 represents the lossy amplification chain, with the transmission
efficiencies, gain and added noise associated with each interconnection and
amplification stages. We have:

$\displaystyle N_{1}^{j}$
$\displaystyle=\eta_{1}^{j}\left[N_{\mathrm{in}}^{j}+\frac{N_{f}(1-\eta_{1}^{j})}{\eta_{1}^{j}}\right]$
(25) $\displaystyle N_{2}^{s}$
$\displaystyle=G(N_{1}^{s}+N_{\mathrm{ex}}^{s})+(G-1)(N_{1}^{i}+N_{\mathrm{ex}}^{i})$
(26) $\displaystyle N_{3}^{s}$
$\displaystyle=\eta_{2}\left[N_{2}^{s}+\frac{N_{f}(1-\eta_{2})}{\eta_{2}}\right]$
(27) $\displaystyle N_{4}^{s}$ $\displaystyle=G_{H}(N_{3}^{s}+N_{H})$ (28)
$\displaystyle N_{o}^{s}$ $\displaystyle=G_{r}N_{4}^{s}.$ (29)

Here, $N_{1}^{j}$ with $j\in\\{s,i\\}$ is the KIT-input noise at the signal
and idler frequency respectively; then, at the signal frequency, $N_{2}^{s}$
is the KIT-output noise, $N_{3}^{s}$ is the HEMT-input noise, $N_{4}^{s}$ is
the HEMT-output noise, and $N_{o}^{s}$ is the noise measured by the SA. Note
that the beamsplitter interaction between the KIT and the HEMT assumes the
loss to be mostly cold, which is the case in our setup where the lossy
components before the HEMT are at the 30 mK stage. Refer to table 1 for the
other variable definitions. From Eqs. 25-29 we can derive:

$\displaystyle N_{o}^{s}$
$\displaystyle=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s})+G_{c}^{si}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i})$
(30)
$\displaystyle=G_{c}^{ss}\left[N_{\mathrm{in}}^{s}+N_{\mathrm{eff}}^{s}+\frac{G-1}{G}\frac{\eta_{1}^{i}}{\eta_{1}^{s}}(N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{i})\right],$
(31)

where

$\displaystyle G_{c}^{ss}$ $\displaystyle=G_{r}G_{H}\eta_{2}G\eta_{1}^{s}$
(32) $\displaystyle G_{c}^{si}$
$\displaystyle=G_{r}G_{H}\eta_{2}(G-1)\eta_{1}^{i},$ (33)

and where

$\displaystyle N_{\mathrm{eff}}^{s}$
$\displaystyle=\frac{N_{\mathrm{ex}}^{s}+(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{(1-\eta_{2})N_{f}+N_{H}}{\eta_{2}G\eta_{1}^{s}}$
(34) $\displaystyle N_{\mathrm{eff}}^{i}$
$\displaystyle=\frac{N_{\mathrm{ex}}^{i}+(1-\eta_{1}^{i})N_{f}}{\eta_{1}^{i}}.$
(35)

The system-added noise is defined as the part of the output noise in Eq. 31
that is not due to the input $N_{\mathrm{in}}^{s}$, and by assuming a cold
input at the idler port, i.e. assuming $N_{f}=N_{\mathrm{in}}^{i}$. We thus
find

$N_{\Sigma}=N_{\mathrm{eff}}^{s}+\frac{G-1}{G}\frac{\eta_{1}^{i}}{\eta_{1}^{s}}(N_{f}+N_{\mathrm{eff}}^{i}).$
(36)

This equation is equivalent to Eq. 5, where we did not simplify the ratio
$G_{c}^{si}/G_{c}^{ss}$.

Figure 10: System-added noise measurement of a microwave amplification chain
with the HEMT as first amplifier. (a) The transmission through the whole noise
setup (KIT un-pumped), when the SNTJ has been replaced by a transmission line
capacitively coupled to an array of resonators (not used in the present
experiment). It is performed in two situations: without (black curve) and with
(purple curve) the bypass. (b) Examples of
$T_{o}^{s^{\prime}}=N_{o}^{s^{\prime}}\hbar\omega/k_{B}$ are shown for the SA
centered at 4, 5 and 6 GHz, with fits superimposed in black lines. The SA RBW
is 5 MHz. (c) From the fits, we extract the system-added noise temperature
$T_{\Sigma}^{\prime}=N_{\Sigma}^{\prime}\hbar\omega/k_{B}$ as a function of
frequency without (black curve) and with (purple curve) the bypass.
Uncertainties are indicated by the gray area surrounding the lines.

Assuming $\\{G,N_{H}\\}\gg 1$ and inserting Eqs. 34 and 35 into Eq. 36 we get

$N_{\Sigma}=\frac{N_{\mathrm{ex}}^{s}+N_{\mathrm{ex}}^{i}}{\eta_{1}^{s}}+\frac{2(1-\eta_{1}^{s})N_{f}}{\eta_{1}^{s}}+\frac{N_{H}}{\eta_{2}G\eta_{1}^{s}}+N_{f}.$
(37)

which is Eq. 6 presented in the main text.

When the KIT is not pumped, we consider it as a lossless, noiseless, passive
element. We therefore have $G=1$, $N_{\mathrm{ex}}^{s}=0$ and
$N_{\mathrm{ex}}^{i}=0$. In that situation

$N_{o}^{s^{\prime}}=G_{c}^{ss^{\prime}}(N_{\mathrm{in}}^{s}+N_{\Sigma}^{\prime}),$
(38)

where $G_{c}^{ss^{\prime}}=G_{r}G_{H}\eta_{2}\eta_{1}^{s}$. From Eq. 36 and 34
we thus get

$N_{\Sigma}^{\prime}=\frac{(1-\eta_{2}\eta_{1}^{s})N_{f}+N_{H}}{\eta_{2}\eta_{1}^{s}}$
(39)

as the chain’s system-added noise with the HEMT as first amplifier.

###  F.2 Discarding the idler port input noise

In a simple case where $G-1\simeq G$, and $\eta_{1}^{s}\simeq\eta_{1}^{i}$,
Eq. 30 becomes

$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{\mathrm{in}}^{i}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}).$
(40)

Thus, varying the SNTJ bias i.e. $N_{\mathrm{in}}^{s}$ and
$N_{\mathrm{in}}^{i}$ simultaneously, the y-intercept gives
$N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$, equal to zero for a quantum-
limited amplifier. Also, in that simple case, the system-added noise is
$N_{\Sigma}=N_{f}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$, as shown by Eq.
36.

On the other hand, if the calibrated noise (coming from the SNTJ or any other
wideband noise source, such as a hot/cold load, or a variable temperature
stage) only illuminates the signal port of the amplifier, or if the noise at
the idler port is wrongly discarded from the analysis, we get
$N_{\mathrm{in}}^{i}=N_{f}$ and Eq. 30 becomes

$N_{o}^{s}=G_{c}^{ss}(N_{\mathrm{in}}^{s}+N_{f}+N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}).$
(41)

Usually, no distinction is made between $N_{\mathrm{eff}}^{s}$ and
$N_{\mathrm{eff}}^{i}$, and the effective excess noise is simply
$N_{\mathrm{eff}}=N_{\mathrm{eff}}^{s}+N_{\mathrm{eff}}^{i}$. The sum
$N_{f}+N_{\mathrm{eff}}$ is then what is commonly defined as the system-added
noise. Varying the SNTJ bias, i.e. $N_{\mathrm{in}}^{s}$, the y-intercept
gives $N_{f}+N_{\mathrm{eff}}$, equal to half a quantum for a quantum-limited
amplifier.

In practice, we fit Eqs. 40 or 41 to find the chain’s gain and the
y-intercept. Fitting a situation correctly described by Eq. 40 with Eq. 41
leads to an underestimate of the system-added noise. In fact, assuming
$N_{\mathrm{in}}^{s}\simeq N_{\mathrm{in}}^{i}$, Eq. 40 can be rewritten into
a form comparable to Eq. 41:

$N_{o}^{s}\simeq
2G_{c}^{ss}\left(N_{\mathrm{in}}^{s}+\frac{N_{\mathrm{eff}}}{2}\right).$ (42)

The interpretation of the y-intercept is crucial. Here, the fit yields
$N_{\mathrm{eff}}/2$ as the y-intercept, and $2G_{c}^{ss}$ as the chain’s
gain. The true system-added noise is then twice the y-intercept value plus a
half quantum, $N_{\Sigma}=N_{\mathrm{eff}}+N_{f}$, and the true chain’s gain
is half of the fitted slope. Conversely, assuming Eq. 41 leads to the
conclusion that the y-intercept value is already the system-added noise, and
that the slope is the chain’s gain. The true system-added noise is thus more
than twice as high, and the chain’s true gain is half as much (i.e. 3 dB
lower, a mistake usually hard to detect).

###  F.3 Shot-noise fit

The SNTJ generates a known noise power Spietz  _et al._ (2006). The amount of
noise $N_{\mathrm{in}}^{j}$, with $j\in\\{s,i\\}$ delivered to the 50 Ω
transmission line can be written as Lecocq  _et al._ (2017)

$\displaystyle N_{\mathrm{in}}^{j}=\frac{k_{B}T}{2\hbar\omega_{j}}\bigg{[}$
$\displaystyle\frac{eV+\hbar\omega_{j}}{2k_{B}T}\coth\left(\frac{eV+\hbar\omega_{j}}{2k_{B}T}\right)$
(43) $\displaystyle+$
$\displaystyle\frac{eV-\hbar\omega_{j}}{2k_{B}T}\coth\left(\frac{eV-\hbar\omega_{j}}{2k_{B}T}\right)\bigg{]},$

where T is the physical temperature of the SNTJ, and V the SNTJ voltage bias.
In practice, the AWG has a slight voltage offset, which we include as a fit
parameter: we write $V-V_{\mathrm{off}}$ instead of V in Eq. 43. In a first
step we fit the asymptotes of the output noise response, for which $\lvert
eV/(2\hbar\omega)\rvert>3$ quanta. In that case, Eq. 43 reduces to
$N_{\mathrm{in}}^{j}=eV/(2\hbar\omega_{j})$, and thus Eq. 30 reduces to

$\displaystyle N_{o}^{s}=$ $\displaystyle G_{c}^{ss}\left(\frac{eV-
V_{\mathrm{off}}}{2\hbar\omega_{s}}+N_{\mathrm{eff}}^{s}\right)$ (44)
$\displaystyle+$ $\displaystyle G_{c}^{si}\left(\frac{eV-
V_{\mathrm{off}}}{2\hbar\omega_{i}}+N_{\mathrm{eff}}^{i}\right).$

We thus get $V_{\mathrm{off}}$, $G_{c}^{ss}$ and $G_{c}^{si}$. In a second
step, we fix $V_{\mathrm{off}}$, $G_{c}^{ss}$ and $G_{c}^{si}$ to their values
derived in the first step and fit the central region (where $\lvert
eV/(2\hbar\omega)\rvert\leq 3$ quanta) to Eq. 30 to get
$N_{\mathrm{eff}}^{s}$, $N_{\mathrm{eff}}^{i}$ and T.

The spectrum analyzer (SA) records a power spectrum $P_{o}^{s}$ (in Watts),
which we convert into a number of photons:
$N_{o}^{s}=P_{o}^{s}/(B\hbar\omega)$, where B is the SA resolution bandwidth.
Dividing by $G_{c}^{ss}$, we then refer $N_{o}^{s}$ to the chain’s input.
Finally, we can subtract $N_{f}$ (in quanta), to visually read $N_{\Sigma}$ at
$V=0$ on the SNTJ curves of Fig. 6b. In fact,

$\frac{N_{o}^{s}(0)}{G_{c}^{ss}}-N_{\mathrm{in}}^{s}(0)-\frac{G_{c}^{si}}{G_{c}^{ss}}N_{\mathrm{in}}^{i}(0)+N_{f}\frac{G_{c}^{si}}{G_{c}^{ss}}=N_{\Sigma}.$
(45)

At $V=0$,

$N_{\mathrm{in}}^{j}(0)=\frac{1}{2}\coth{\left(\frac{\hbar\omega_{j}}{k_{B}T}\right)},$
(46)

where $j\in\\{s,i\\}$, therefore if $\hbar\omega_{j}\gg k_{B}T$,
$N_{\mathrm{in}}^{j}(0)\simeq N_{f}=0.5$. Considering that $G_{c}^{ss}\simeq
G_{c}^{si}$, Eq. 45 yields:

$\frac{N_{o}^{s}(0)}{G_{c}^{ss}}-N_{f}\simeq N_{\Sigma}.$ (47)

###  F.4 List of variables

Table 1 lists the variables used throughout Sec. IV and appendix F.1.

variable name | definition
---|---
$N_{\mathrm{in}}^{s}$ | SNTJ-generated noise at the signal frequency
$N_{\mathrm{in}}^{i}$ | SNTJ-generated noise at the idler frequency
$N_{f}$ | vacuum (or thermal) noise, set by the refrigerator temperature
$N_{\mathrm{ex}}^{s}$ | signal-to-signal path KIT-excess noise
$N_{\mathrm{ex}}^{i}$ | idler-to-signal path KIT-excess noise
$N_{\mathrm{eff}}^{s}$ | signal-to-signal path effective KIT-excess noise, Eq. 34
$N_{\mathrm{eff}}^{i}$ | idler-to-signal path effective KIT-excess noise, Eq. 35
$N_{H}$ | HEMT-added noise
$N_{o}^{s}$ | noise measured by the SA at the signal frequency, Eq. 30
$N_{o}^{s^{\prime}}$ | noise measured by the SA when the KIT is off, Eq. 38
$N_{\Sigma}$ | system-added noise, Eq. 36
$N_{\Sigma}^{\prime}$ | system-added noise when the KIT is off, Eq. 39
$\eta_{1}^{s}$ | transmission efficiency between SNTJ and KIT at the signal frequency
$\eta_{1}^{i}$ | transmission efficiency between SNTJ and KIT at the idler frequency
$\eta_{2}$ | transmission efficiency between KIT and HEMT at the signal frequency
$G$ | KIT signal power gain
$G_{H}$ | HEMT signal power gain
$G_{r}$ | room-temperature signal power gain
$G_{c}^{ss}$ | signal-to-signal chain’s gain, Eq. 32
$G_{c}^{si}$ | idler-to-signal chain’s gain, Eq. 33
Table 1: List of the variables used in the noise theory. All the variables
designating a noise quantity are in units of quanta. All the transmission
efficiencies are dimensionless. All the gains are in linear units.

##  Appendix G LOSS BUDGET IN THE NOISE MEASUREMENT SETUP

To quote the KIT-excess noise terms $N_{\mathrm{ex}}^{s}$ and
$N_{\mathrm{ex}}^{i}$, it is necessary to account for the transmission
efficiencies (and therefore the loss) in the measurement setup,
$\eta_{1}^{s}$, $\eta_{1}^{i}$ and $\eta_{2}$. Note that $\eta_{1}^{i}$ is
simply the transmission efficiency at the idler frequency
$\omega_{i}=\omega_{p}-\omega_{s}$, symmetric of the signal frequency
$\omega_{s}$ with respect to the half-pump frequency $\omega_{p}/2$. Thus, a
broadband measurement of $\eta_{1}^{s}$ on both sides of $\omega_{p}/2$
provides both $\eta_{1}^{s}$ and $\eta_{1}^{i}$. We estimate these
efficiencies in this appendix and give an overall loss budget per component
between the SNTJ and the HEMT. Knowing where the loss comes from also provides
guidance on how to improve the amplifier because many lossy components could
be optimized, particularly by integrating them on-chip with the KIT.

###  G.1 System-added noise temperature with un-pumped KIT

First, we measured the system-added noise temperature of the amplification
chain $T_{\Sigma}^{\prime}=N_{\Sigma}^{\prime}\hbar\omega/k_{B}$ (with $\hbar$
the reduced Planck’s constant and $k_{B}$ the Boltzmann’s constant) when the
KIT is off, i.e. with the HEMT as the first amplifier. In that case,
$N_{\Sigma}^{\prime}$ is given by Eq. 39. We measured $N_{\Sigma}^{\prime}$ in
two situations: one for the measurement setup presented in Fig. 9, and one
where the KIT surrounded by its two bias tees, the directional coupler and the
low pass filter next to it have been bypassed, i.e. replaced by a microwave
cable. In the limit where $N_{f}\ll N_{\Sigma}^{\prime}$, the ratio in
$N_{\Sigma}^{\prime}$ between these two situations gives direct access to the
bypassed components’ insertion loss (IL) $\mathcal{I}_{\mathrm{BP}}$ (the
exact transmission efficiency ratio from which we calculate
$\mathcal{I}_{\mathrm{BP}}$ is equivalent to finding $\eta_{2}\eta_{1}^{s}$ in
Eq. 39). Figure 10c shows $T_{\Sigma}^{\prime}$ as a function of frequency,
obtained from fitting curves like those presented in Fig. 10b, obtained with
and without the bypass. There are ripples with 130 MHz characteristic
frequency, likely due to reflections in coaxial cables between the SNTJ and
the HEMT. Without the bypass $T_{\Sigma}^{\prime}=5.1\pm 1.4$ K between 3.5
and 5.5 GHz, while with the bypass $T_{\Sigma}^{\prime}=2.9\pm 1$ K. Thus, the
ratio gives $\mathcal{I}_{\mathrm{BP}}=2.5$ dB.

###  G.2 Component loss

Second, we replaced the SNTJ (including the bias tee) with a transmission
line, to measure the transmission of a probe tone through the setup (with a
VNA). This transmission line is capacitively coupled to an array of
resonators, whose resonant frequencies span between $4$ and $5$ GHz, and is
part of a future experiment. These resonances are not relevant for the current
characterization. We measured the transmission with and without the bypass,
shown in Fig. 10a. Once again, the transmission ratio between these two
situations gives direct access to the bypassed components’ IL. We find
$\mathcal{I}_{BP}=2.4\pm 0.6$ dB, in agreement with the system-added noise
measurements of Fig. 10c.

authors | mixing |  | pump
---
power [dBm]
| gain
---
[dBm]
| bandwidth
---
[GHz]
| saturation
---
[dBm]
| noise
---
bandwidth
| system
---
noise [K]
this work | 3WM | $-29$ | $16.5^{+1}_{-1.3}$ | 2 | $-63$ |  $3$-$6.5$ GHz | $0.66\pm 0.15$
Eom et.al. Eom  _et al._ (2012) | 4WM | $-8$ | $10^{+3}_{-3}$ | $4$ | $-52$ |  $1$ Hz |  $1.5$111idler noise input not accounted for
Bockstiegel et.al. Bockstiegel  _et al._ (2014) | 4WM | $-10$ | $20^{+3}_{-3}$ | $8$ | - | - | -
Vissers et.al. Vissers  _et al._ (2016) | 3WM | $-10$ | $20^{+5}_{-5}$ | $4$ |  $-45$222at $10$ dB gain | - | -
Chaudhuri et.al. Chaudhuri  _et al._ (2017) | 4WM | $-10$ | $15^{+3}_{-3}$ | $3$ | - | - | -
Ranzani et.al. Ranzani  _et al._ (2018) | 3WM | $-30$ | $10^{+2}_{-2}$ | $4$ | - |  $6$-$10$ GHz |  $1.5$-$5^{\mathrm{\ref{ftn:ftn2}}}$
Zobrist et.al. Zobrist  _et al._ (2019) | 3WM | $-23$ |  $15$333data quoted but not shown in the paper | $5^{\mathrm{\ref{ftn:ftn1}}}$ | $-53^{\mathrm{\ref{ftn:ftn1}}}$ | 1 MHz | ${0.58^{+0.2}_{-0.03}}^{\mathrm{\ref{ftn:ftn2}}}$
Table 2: Comparison of our KIT to other published KIT results. As discussed in
appendix F.2, failure to account for the noise at the idler input results in
an underestimate of the system-added noise temperature by about a factor of
two.

We also measured the individual transmissions of the chain’s components at 4K:
bias tee (BT), filter (LPF), directional coupler (DC) and isolator (ISO).
Figure 11a shows the IL of these four components. Between $3.5$ and $5.5$ GHz,
$\mathcal{I}_{\mathrm{BT}}=0.3\pm 0.04$ dB, $\mathcal{I}_{\mathrm{LPF}}=0.2\pm
0.1$ dB, $\mathcal{I}_{\mathrm{DC}}=0.2\pm 0.04$ dB, and
$\mathcal{I}_{\mathrm{ISO}}=0.7\pm 0.6$ dB. Its IL increases below $4$ GHz as
we leave its operating band, which degrades the system-added noise performance
in the same manner as reducing the KIT gain would, see Eq. 34. The SNTJ
packaging loss, including the bias tee, has been previously reported to be
$\mathcal{I}_{\mathrm{SNTJ}}=1$ dB (transmission efficiency of $0.8$) Chang
_et al._ (2016).

Figure 11: Insertion loss (IL) from the microwave components used in the noise
measurement setup (see Fig. 9). The ILs (a) are measured at 4K: the bias tee,
Anritsu K250 (red curve), the low pass filter, Pasternack PE87FL1015 (green
curve), the directional coupler, Pasternack PE2204-20 (blue curve), and the
isolator, Quinstar CWJ1015-K13B (purple curve). (b) Combining the IL from the
components before and after the KIT, we estimate the transmission efficiencies
$\eta_{1}^{s}$ (black curve) and $\eta_{2}$ (purple curve) respectively.

###  G.3 HEMT-added noise, transmission efficiencies, KIT-excess noise

We can estimate the HEMT-added noise temperature
$T_{H}=N_{H}\hbar\omega/k_{B}$ from eq. 39, because we measured
$T_{\Sigma}^{\prime}$ (see Fig. 10c), and because we can have an estimation of
the total IL $\mathcal{I}_{T}$ from the SNTJ to the HEMT. In fact, without the
bypass

$\mathcal{I}_{T}=\mathcal{I}_{\mathrm{SNTJ}}+\mathcal{I}_{\mathrm{BP}}+\mathcal{I}_{\mathrm{ISO}}+\mathcal{I}_{\mathrm{LPF}}$
(48)

(in dB), which gives $\mathcal{I}_{T}=4.3\pm 0.6$ dB. Then,
$\eta_{2}\eta_{1}^{s}=10^{-\mathcal{I}_{T}/10}$, and we get $T_{H}=1.8\pm 0.2$
K (i.e. $N_{H}=8\pm 1$ quanta) between 3.5 and 5.5 GHz, in agreement with the
HEMT data sheet, which gives $T_{H}=1.6\pm 0.3$ K in that frequency range.

Subtracting $\mathcal{I}_{\mathrm{LPF}}$, $\mathcal{I}_{\mathrm{DC}}$ and
$2\times\mathcal{I}_{\mathrm{BT}}$ from $\mathcal{I}_{\mathrm{BP}}$, the KIT’s
packaging (see Fig. 8a) is responsible for about
$\mathcal{I}_{\mathrm{KIT}}=1.4\pm 0.6$ dB of IL. This loss may be decreased
in future optimization, for example by coating the PCBs and the SMA pins with
superconducting material.

Finally, we can separately estimate $\eta_{1}^{s}$ and $\eta_{2}$ by adding
(in dB) the IL of the chain’s components, and then use Eqs. 34 and 35 to
estimate the KIT-excess noise $N_{\mathrm{ex}}^{s}$ and $N_{\mathrm{ex}}^{i}$.
Between the SNTJ and the KIT,

$\mathcal{I}_{\eta_{1}^{s}}=\mathcal{I}_{\mathrm{SNTJ}}+\mathcal{I}_{\mathrm{LPF}}+\mathcal{I}_{\mathrm{DC}}+\mathcal{I}_{\mathrm{BT}}+\frac{\mathcal{I}_{\mathrm{KIT}}}{2},$
(49)

which gives $\mathcal{I}_{\eta_{1}}=2.4\pm 0.1$ dB, and between the KIT and
the HEMT

$\mathcal{I}_{\eta_{2}}=\frac{\mathcal{I}_{\mathrm{KIT}}}{2}+\mathcal{I}_{\mathrm{BT}}+\mathcal{I}_{\mathrm{ISO}}+\mathcal{I}_{\mathrm{LPF}},$
(50)

which gives $\mathcal{I}_{\eta_{2}}=1.9\pm 0.6$ dB. Figure 11b shows
$\eta_{1}^{s}=10^{-\mathcal{I}_{\eta_{1}}/10}$ and
$\eta_{2}=10^{-\mathcal{I}_{\eta_{2}}/10}$ thus obtained. Therefore, between
3.5 and 5.5 GHz, $\eta_{1}^{s}=\eta_{1}^{i}=0.57\pm 0.02$ (both equal since
the frequency band is nearly symmetric with respect to $\omega_{p}/2=4.4$
GHz), and $\eta_{2}=0.64\pm 0.10$.

##  Appendix H KIT PERFORMANCE COMPARISON

Table 2 compares the performance of our KIT with previous results on KITs. The
pump power (third column) is the power at the KIT’s input. The gain (fourth
column) is the average gain observed over the KIT amplification bandwidth
(fifth column). The amplitude of gain ripples over that bandwidth is indicated
next to the gain value. The saturation power (sixth column) corresponds to the
input $1$ dB compression point. The noise bandwidth (seventh column) is the
bandwidth over which a noise measurement was performed. Finally, column eight
reports the measured system noise temperature
$T_{\Sigma}=N_{\Sigma}\hbar\omega/k_{B}$ in that bandwidth. In our case,
$N_{\Sigma}=3.1\pm 0.6$ quanta, which translates into $T_{\Sigma}=0.66\pm
0.15$ K. Note that Eom et.al Eom  _et al._ (2012), Ranzani et.al. Ranzani  _et
al._ (2018) and Zobrist et.al. Zobrist  _et al._ (2019) used a wideband noise
source (a hot/cold load) but did not account for the effect of the idler
port’s input noise. Therefore, their true system-added noise is about twice of
what they reported (see appendix F.2). We do not compare the amplifier-added
noise or amplifier-excess noise, because it is subject to approximations in
the amount of loss present in the noise measurement setup. In addition, the
inferred added noise usually does not include the loss of mandatory microwave
components used when operating the amplifier, and as such it is an under-
estimation of the true, useful amplifier-added noise.

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