# A Traveling-Wave Parametric Amplifier and Converter

M. Malnou<EMAIL_ADDRESS>National Institute of Standards and
Technology, 325 Broadway, Boulder, CO 80305, USA  University of Colorado, 2000
Colorado Ave., Boulder, CO 80309, USA  B. T. Miller  National Institute of
Standards and Technology, 325 Broadway, Boulder, CO 80305, USA  University of
Colorado, 2000 Colorado Ave., Boulder, CO 80309, USA  J. A. Estrada  National
Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA
University of Colorado, 2000 Colorado Ave., Boulder, CO 80309, USA  K. Genter
National Institute of Standards and Technology, 325 Broadway, Boulder, CO
80305, USA  University of Colorado, 2000 Colorado Ave., Boulder, CO 80309, USA
K. Cicak  National Institute of Standards and Technology, 325 Broadway,
Boulder, CO 80305, USA  J. D. Teufel  National Institute of Standards and
Technology, 325 Broadway, Boulder, CO 80305, USA  University of Colorado, 2000
Colorado Ave., Boulder, CO 80309, USA  J. Aumentado  National Institute of
Standards and Technology, 325 Broadway, Boulder, CO 80305, USA  F. Lecocq
<EMAIL_ADDRESS>National Institute of Standards and Technology, 325
Broadway, Boulder, CO 80305, USA  University of Colorado, 2000 Colorado Ave.,
Boulder, CO 80309, USA

###### Abstract

High-fidelity qubit measurement is a critical element of all quantum computing
architectures. In superconducting systems, qubits are typically measured by
probing a readout resonator with a weak microwave tone which must be amplified
before reaching the room temperature electronics. Superconducting parametric
amplifiers have been widely adopted as the first amplifier in the chain,
primarily because of their low noise performance, approaching the quantum
limit. However, they require isolators and circulators to route signals up the
measurement chain, as well as to protect qubits from amplified noise. While
these commercial components are wideband and very simple to use, their
intrinsic loss, size, and magnetic shielding requirements impact the overall
measurement efficiency while also limiting prospects for scalable readout in
large-scale superconducting quantum computers. Here we demonstrate a
parametric amplifier that achieves both broadband forward amplification and
backward isolation in a single, compact, non-magnetic circuit that could be
integrated on chip with superconducting qubits. It relies on a nonlinear
transmission line which supports traveling-wave parametric amplification of
forward propagating signals, and isolation via frequency conversion of
backward propagating signals. This kind of traveling-wave parametric amplifier
and converter is poised to reduce the readout hardware overhead when scaling
up the size of superconducting quantum computers.

There are many challenges to achieving a universal fault-tolerant quantum
computer. First, mastering high-fidelity operations over an ever-increasing
number of quantum bits remains one of the main roadblocks. But also, as the
size of these processors grows, scaling the supporting infrastructure becomes
itself a difficult problem. For superconducting qubit systems, these
challenges include scaling the size of the cryogenic system housing the
processor, tackling the delivery of millions of control signals, and, as we
discuss in this work, routing and amplifying the measurement signals back to
room temperature.

A standard method for assessing the state of a superconducting qubit is to
dispersively couple it to a resonator and measure a state-dependent frequency
shift in the resonator’s microwave response [1]. This dispersive interaction
allows for an approximate quantum nondemolition (QND) qubit readout, so long
as the readout power is low enough, on the order of a few photons. This
readout drive must then be amplified over six orders of magnitude to overwhelm
the noise of room-temperature electronics, with minimal degradation of its
signal-to-noise ratio. At the same time, in the dispersive measurement scheme
the qubit frequency becomes dependent on the number of photons in the readout
resonator, leading to a qubit dephasing rate proportional to the cavity
occupancy [2]. This imposes stringent simultaneous requirements on the
amplification chain to have large gain and low added noise in the forward
direction, while maintaining no excess noise in the backward direction.

Typical amplification chains for qubit readout have converged over the last
two decades on the use of parametric amplifiers as the first stage of gain,
thanks to noise performance fundamentally limited by Heisenberg’s uncertainty
principle [3, 4]. However, to enforce directionality and protect the readout
cavity against amplified noise, several microwave circulators or isolators
must be placed between the readout resonator and the amplifier [5]. In
addition to their physical size and high magnetic fields being significant
obstacles for scaling, they also introduce loss which degrades the system
noise performance.

To circumvent these issues, parametric alternatives to the Faraday effect used
in conventional circulators have been investigated [6, 7, 8]. Recent successes
include building parametric circulators and directional amplifiers [9, 10,
11], and applying them to perform efficient qubit measurements [12, 13, 14].
While very promising, these various devices remain relatively narrow-band and
difficult to operate, hindering their large scale deployment in state-of-the-
art quantum computers. Another compelling alternative is to use traveling wave
parametric amplifiers (TWPAs) [15], in which signals are amplified while co-
propagating with a strong pump in a nonlinear transmission line. As such, they
are inherently directional, in addition to having high bandwidth, high
saturation power, and near quantum-limited noise performance [16, 17, 18].
However, in practice, small impedance mismatches are unavoidable, leading to
pump and noise reflections that turn into backward gain, or more generally
excess noise traveling back to the device under test. Consequently, isolators,
along with their drawbacks, are still required between the readout cavities
and traditional TWPAs.

Here we lift this requirement by building a TWPA that incorporates forward
gain with backward isolation (see Fig. 1a). The latter is achieved via
frequency conversion of backward propagating signals [19] instead of the
Faraday effect, making this traveling-wave parametric amplifier and converter
(TWPAC) directly integrable with superconducting qubits. We experimentally
demonstrate the simultaneous forward amplification and backward isolation of
signals over a $500$ MHz usable bandwidth, with a gain of about $7$ dB, and
isolation as low as $-20$ dB. We characterize the noise performance of the
system with a shot-noise tunnel junction (SNTJ). Using the TWPAC as the first
amplifier, we achieve a system-added noise of $5.2$ quanta in the operation
band, on par with the measured performance of commonly used TWPA-based
amplifier chains [16, 20].

Figure 1: The concept of a traveling-wave parametric amplifier and converter
(TWPAC). (a) Signals propagating from left to right (forward direction) are
amplified, while signals propagating from right to left (backward direction)
are converted out of the band. (b) The TWPAC consists of a current-biased
nonlinear lumped-element transmission line. Each unit cell consists of a
series inductor made with a Josephson junction (brown) shorted to ground by a
linear capacitor (green), whose value is periodically modulated (see main
text). Resonators to ground (purple) are added every six cells which, together
with the periodically varying capacitors, enable phase-matching for the two
three-wave mixing parametric processes: (i) from left to right, forward
parametric amplification, whereby a pump (purple arrow) transfers its energy
to a co-propagating signal (blue) and to its corresponding idler (dark blue).
(ii) From right to left, frequency conversion, whereby a pump (green arrow)
mixes a signal (red) up (pink) or down (orange), away from the signal
frequency band. (c) Experimental implementation of a TWPAC in a $2$ cm-long
device, comprising 2640 unit cells. The close-up micrograph shows, in false
color, the junctions (brown), ground capacitors (green) and tank circuits
(purple).

### Concept and design

The TWPAC consists of a nonlinear transmission line that is current-biased to
support two counter-propagating three-wave mixing parametric processes, see
Fig. 1b. In the forward direction, a strong pump at frequency $\omega_{a}$ is
used to amplify a co-propagating signal at frequency $\omega_{s}$ and an idler
at frequency $\omega_{i}=\omega_{a}-\omega_{s}$. In the backward direction,
using a second strong pump at frequency $\omega_{c}$, a signal at frequency
$\omega_{s}$ is either down-converted or up-converted to frequencies
$\omega_{d}=\omega_{s}-\omega_{c}$ or $\omega_{u}=\omega_{s}+\omega_{c}$,
respectively. This frequency conversion effectively swaps any states contained
within the signal band with vacuum states coming from the up- or down-
converted frequency bands, thus providing the isolation property of the TWPAC.
Importantly, both the parametric amplification (PA) and frequency conversion
(FC) processes are directional, because each of these processes must respect a
specific phase-matching condition that is predominantly achieved for signals
co-propagating with the pumps (see supplementary information).

We realize this nonlinear transmission line using Josephson junctions, see
Fig. 1b. Each unit cell within this line is composed of a series inductor from
a single niobium trilayer Josephson junction ($I_{c}\approx 5$ $\upmu$A) and a
capacitor to ground from a parallel-plate using a low-loss amorphous silicon
dielectric ($\tan{\delta}\approx 4\times 10^{-4}$) [10]. The total device
comprises $2650$ unit cells to achieve a $\approx 2$ cm long transmission
line, see Fig. 1c. In the signal band, the line has a $\approx 50$ Ω
characteristic impedance and a phase velocity $\approx 2\%$ of the speed of
light in vacuum. The TWPAC detailed layout is available in the supplementary
information.

To fulfill the PA and FC phase-matching conditions, we engineer the dispersion
relation of the transmission line: (i) every six cells, a tank circuit is
interposed between the capacitor and the ground, and (ii) the capacitors vary
periodically along the line. These features create stopbands in the line’s
transmission where signals cannot propagate [21, 16, 18]. Close to these
stopbands the phase velocity varies strongly with frequency, which we exploit
to obtain phase matching. We calculate the linear response of the line, in the
presence of dielectric loss, to obtain the transmission as a function of
frequency presented in Fig. 2c. The tank circuits create a stopband just below
$14$ GHz, above which we place our PA pump such that signals centered around
the half pump frequency ($\approx 7$ GHz) get amplified (see Fig. 2a). The
periodic variation of the capacitors to ground creates two stopbands, one just
above $5$ GHz, and one around $10$ GHz. The FC pump is placed below the first
stopband. As a result, signals around $7$ GHz can be down-converted to
$\approx 3$ GHz or up-converted to $\approx 12$ GHz (see Fig. 2b). Ideally,
the second stopband suppresses the second-harmonic generation of the
conversion pump at frequency $\omega_{c2}=2\omega_{c}$. A detailed discussion
of the TWPAC design is available in the supplementary information.

Figure 2: The theoretical model supporting the TWPAC’s design. (a) In the
forward (_i.e._ left to right) direction, pump photons at a frequency
$\omega_{a}/2\pi\approx 14$ GHz are used to amplify a signal at frequency
$\omega_{s}/2\pi\approx 7.5$ GHz, while also amplifying an idler at frequency
$\omega_{i}=\omega_{a}-\omega_{s}$. (b) In the backward (_i.e._ right to left)
direction, the signal gets up-converted at $\omega_{u}=\omega_{s}+\omega_{c}$,
or down-converted at $\omega_{d}=\omega_{s}-\omega_{c}$ using pump photons at
$\omega_{c}/2\pi\approx 4.75$ GHz. (c) The TWPAC’s calculated transmission as
a function of frequency is flat and close to $0$ dB, except at distinct
stopbands. The highest-frequency stopband allows us to match the phase of the
PA pump (purple line) with the summed phases of the signal (blue) and idler
(dark blue) that lie in the $6$ to $8$ GHz band. The lowest frequency stopband
allows us to achieve phase matching between the conversion pump (green), the
signal (dashed red) and the down-converted signal (orange). The up-converted
signal (pink) can also be created via frequency conversion, albeit not phase-
matched. Ideally, the second harmonic of the conversion pump (dashed green)
falls into the second stopband. (d) We solve the equations coupling all these
modes (see the supplementary information) to calculate their steady-state
amplitude as a function of input signal frequency. Here the TWPAC is biased
with a dc-current $I_{d}=1.5$ $\upmu$A (the junctions’ critical current is
$I_{c}=5$ $\upmu$A), the amplification pump frequency and power are
$\omega_{a}/2\pi=14.27$ GHz and $P_{a}=-73$ dBm, respectively, and the
conversion pump frequency and power are $\omega_{c}/2\pi=4.7$ GHz and
$P_{c}=-73$ dBm, respectively.

We calculate the theoretical response of the device in the presence of both
pumps by solving the equations of motion for the minimal mode basis
$\\{a,s,i,c,d,u,c_{2}\\}$ with frequencies
$\\{\omega_{a},\omega_{s},\omega_{i},\omega_{c},\omega_{d},\omega_{u},2\omega_{c}\\}$.
We assume here that these modes are only coupled via three-wave mixing
interactions, see the supplementary information. In these coupled-mode
equations (CME), injecting a signal, either in the forward or in the backward
direction allows us to calculate the TWPAC’s forward and backward
transmission, respectively. For a given combination of pump powers, pump
frequencies and dc current bias, we achieve simultaneous forward gain and
backward isolation, within a band centered around $7$ GHz, see Fig. 2d. The
gain profile spans from about $3$ GHz to $12$ GHz. When the signal falls into
one of the stopbands, it cannot propagate and the gain is therefore null at
this frequency. Alternatively, when the idler falls into a stopband, it cannot
propagate either, which results in unity gain for the associated signal. From
$6$ GHz to $8$ GHz, this simple model predicts close to $20$ dB of gain,
concurrently with about $-10$ dB of isolation, dipping as low as $-25$ dB.

### Measurement and analysis

We then test the TWPAC presented in Fig. 1c at cryogenic temperatures, using
the experimental setup presented in Fig. 3a. Notably, this setup is symmetric
in the forward and backward directions, allowing us to measure the TWPAC’s
complete set of scattering parameters. In addition, a pair of switches allows
us to compare in situ the TWPAC’s transmission to that of a through cable.
Remarkably, when current biased, the transmission, shown in black in Fig. 3b,
decreases at most by $\approx 2$ dB when reaching the higher end of the
operation bandwidth ($\approx 8$ GHz), thanks to the low-loss a-Si dielectric.
The forward and backward transmission through the device (referenced to the
cable), with the dc current bias, PA pump, and FC pump turned on, are shown in
blue and red respectively in Fig. 3b. In the forward direction, the TWPAC
generates about $7$ dB of gain, spanning over close to an octave of bandwidth.
Simultaneously, in the backward direction the TWPAC generates at least $5$ dB
of isolation over about $800$ MHz of bandwidth, with sweet-spot frequencies
where the isolation reaches $20$ dB. The TWPAC input and output reflection
coefficients are similar to that of the through cable, and the input $1$ dB
compression point is measured to be $\approx-90$ dBm (see the supplementary
information).

Figure 3: Experimental characterization of the TWPAC’s scattering parameters.
(a) We use two symmetric readout chains, between which the TWPAC is inserted.
Starting from the TWPAC and extending outward in both directions: a pair of
microwave switches allows us to compare the TWPAC’s transmission to that of a
SubMiniature version A (SMA) through connector, a pair of bias tees are used
to deliver the dc current bias $I_{d}$, a pair of attenuators ensure that the
environment is broadly matched to 50 Ω, a pair of $20$ dB directional couplers
deliver the pump and probe tones, a pair of isolators protect from noise and
tones reflected off the HEMTs, and finally a pair of HEMTs at 4 K (not shown)
amplify the transmitted (right) and reflected (left) signals. (b) The TWPAC
transmission as a function of frequency is plotted in the forward (blue) and
backward (red) directions, when $I_{d}=1$ $\upmu$A, $\omega_{a}/2\pi=14.52$
GHz, $P_{a}=-75$ dBm, $\omega_{c}/2\pi=3.15$ GHz and $P_{c}=-80$ dBm (at the
chip inputs), and overlapping the transmission obtained from the time-domain
simulations in the forward (faded blue) and backward (faded red) directions,
for $I_{d}=1$ $\upmu$A, $\omega_{a}/2\pi=14.3$ GHz, $P_{a}=-70$ dBm,
$\omega_{c}/2\pi=3.14$ GHz and $P_{c}=-74$ dBm. We also plot the experimental
(black) and time-domain simulated (faded black) transmission when both pumps
are turned off.

Both the gain and isolation profiles in Fig. 3b qualitatively agree with the
basic model presented in Fig. 2d, but not quantitatively. In fact, we found
the isolation to be optimal when using a FC pump frequency at $3.15$ GHz, in
contradiction with the $4.7$ GHz, predicted by the minimal model described by
the CME. To go beyond the model that guided the TWPAC design, we simulate the
line’s nonlinear response using a time-domain solver (WRspice [22, 23, 24]).
In contrast to the CME, this method grasps real-life non-idealities: it uses
the full Josephson current-phase relation, and accounts for all propagating
modes and mixing processes. The simulated circuit response, shown in faded
colors in Fig. 3b, is in quantitative agreement with the experimental results,
using values for the dc current bias, pump frequencies and pump powers close
to their estimated experimental values. Note however that the transmission
loss, for example coming from the dielectric, cannot be straightforwardly
represented in a time domain simulation[25], and is therefore not included
here, which explains why the simulated unpumped transmission does not fully
agree with the measurement.

In light of the time-domain simulations, we revisit the CME to include all the
four-wave mixing (4WM) terms that couple the modes $a,s,i,c,d,u,c_{2}$ with
each other. These updated CME then capture the observed isolation behavior
when using a FC pump frequency at $3.15$ GHz (see the supplementary
information), which in fact corresponds to the frequency for which the up-
conversion process is phase-matched. Therefore, despite not being phase-
matched, the 4WM terms significantly reduce the efficacy of the down-
conversion process in the TWPAC.

The TWPAC’s maximum achievable gain may be limited by amplification pump
depletion due to these 4WM processes, 3WM amplification of $c_{2}$, and a
spurious interaction producing the high-gain peak (along with its idler)
observed at the right edge of the first stopband. This peak may stem from a
runaway amplification effect due to large impedance mismatch between the TWPAC
and its environment [17, 26]. This is a common issue for TWPAs where the phase
matching is achieved using periodic loading or other resonant techniques [21,
16, 27, 28], but it can be mitigated with more stringent chromatic dispersion,
apodization of the loading profile [29, 23] or non-resonant dispersion-
engineering techniques [30, 31].

Figure 4: Characterization of the TWPAC’s noise performance. (a) The
amplification chain comprises the TWPAC as the first amplifier, and a
calibrated noise source (SNTJ) at the chain’s input. (b) The system-added
noise $N_{\mathrm{sys}}=N_{1}+N_{2}/G$ as a function of frequency diminishes
as the TWPAC’s gain $G$ increases. At very low gain (light blue curve)
$N_{\mathrm{sys}}$ is dominated by $N_{2}$, the effective HEMT-added noise,
which includes the loss of the microwave components placed between the SNTJ
and the HEMT. At the highest operable TWPAC gain (blue curve),
$N_{\mathrm{sys}}$ drops to $5.2$ quanta on average between $5.5$ and $8.5$
GHz. In that band, the contribution of the TWPAC and of its preceding
microwave components is $N_{1}=1.7$ quanta (gray curve). Note that for all
three curves, the noise at the frequency of the second harmonic of the FC pump
($6.3$ GHz) and its corresponding idler ($8.22$ GHz) cannot be properly
measured, and is therefore omitted. (c) We measure $N_{\mathrm{sys}}$ as a
function of the TWPAC gain, plotted here for the frequency bin centered around
$7$ GHz (blue dots). The fitted response (black curve) allows us to obtain
$N_{1}$, the asymptotic system-added noise at high TWPAC gain.

### Noise performance

Irrespective of its isolation properties, a pre-amplifier suitable for qubit
readout must operate close to the quantum limit for added noise. In a separate
cooldown, we measure the noise performance of a chain where the TWPAC is the
first amplifier, using a shot-noise tunnel junction [32, 33, 20] (SNTJ) as a
calibrated noise source at the chain’s input, see Fig. 4a. We operate the
TWPAC with both PA and FC pumps turned on, whose powers and frequencies are
similar to those used to capture its scattering response. Despite modest TWPAC
gain, it is already sufficient to overwhelm most of the noise $N_{2}$
generated by the following elements in the chain, such that the average
system-added noise is $N_{\mathrm{sys}}\approx 5.2$ quanta between 5.5 and 8.5
GHz, see Fig. 4b. Varying the PA pump we measured $N_{\mathrm{sys}}$ at
various TWPAC gains $G$, see Fig. 4c. We then fit the data using
$N_{\mathrm{sys}}=N_{1}+N_{2}/G$ to infer $N_{1}$, the noise added by the
effective first amplifier, comprising the TWPAC, its packaging, and all the
external cabling and components between itself and the SNTJ [20]. On average,
between $5.5$ and $8.5$ GHz, $N_{1}$ approaches $1.7$ quanta extrapolating to
the high gain limit, relatively close to the quantum limit of $0.5$ quanta and
consistent with the presence of loss before the TWPA chip. Note also that the
dependence of $N_{\mathrm{sys}}$ on $G$ does not change when turning off the
FC pump (see the supplementary information), indicating that the FC process
does not alter the TWPAC’s noise performance.

### Conclusion

In this work, we theoretically investigated and experimentally tested a
traveling-wave parametric amplifier that generates backward isolation in
addition to forward amplification. The TWPAC’s bandwidth, noise, and power
handling are on par with traditional TWPAs. In addition, it remains easy to
operate since it only requires two pumps. We achieved quantitative agreement
between the data and time-domain simulation, and qualitative agreement of both
to the refined theory. Future improvements on the dispersion engineering will
leverage the complementary use of coupled mode equations and time-domain
simulations to mitigate spurious pump harmonics, unwanted parametric
processes, and impedance mismatches. With this approach, we expect to achieve
$20$ dB of gain and isolation over more than a gigahertz of bandwidth,
enabling the simultaneous readout of tens of qubits without the use of an
intermediate circulator. Therefore, devices based on this concept are poised
to improve qubit measurement efficiency, and to significantly reduce the
number of conventional microwave circulators and isolators required for the
operation of large-scale quantum computers.

### Acknowledgements

We thank Ryan Kaufman, Connor Denney, Arpit Ranadive, Nicolas Roch, Matthieu
Praquin, and Philippe Campagne-Ibarcq for fruitful discussions. We thank
Manuel Castellanos-Beltran and Cody Scarborough for their feedback on the
manuscript. 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 is
necessarily the best available for the purpose.

### Competing interests

M.M., J.D.T., J.A., and F.L. are inventors of a patent related to the subject.

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Supplementary information: A Travelling Wave Parametric Amplifier and
Converter

##  I The TWPAC coupled-mode equations

Here we derive the equations of motion governing the propagation of the few
modes of interest through the TWPAC: amplification pump (frequency
$\omega_{a}$), signal ($\omega_{s}$), idler ($\omega_{i})$, conversion pump
($\omega_{c}$), down-converted signal ($\omega_{d}$), up-converted signal
($\omega_{u}$) and second harmonic of the conversion pump ($\omega_{c_{2}}$).
Our model includes the reflections of these tones at the TWPAC’s port, due to
the imperfect impedance matching between the TWPAC’s (frequency-dependent)
impedance and its environment, closely following the formalism of Kern. et al.
[1].

###  I.1 Equivalent inductance of a dc-biased Josephson junction

We start by deriving the equivalent nonlinear inductance to a dc-biased
Josephson junction (JJ). A JJ’s equivalent inductance is:

$L_{J}=\frac{L_{J0}}{\sqrt{1-\left(\frac{I_{\Sigma}}{I_{c}}\right)^{2}}},$
(S1)

where $L_{J0}=\phi_{0}/I_{c}$ is the JJ’s unbiased inductance, with $\phi_{0}$
the reduced flux quantum, with $I_{c}$ the JJ’s critical current, and where
$I_{\Sigma}$ is the total current traversing the JJ. If we use a fixed dc bias
$I_{d}$ such that $I_{\Sigma}=I_{d}+I$, with $I_{\Sigma}\ll I_{c}$ is a time-
varying oscillating term, we can expand the inductance around the dc working
point:

$L_{J}=L_{d}[1+\epsilon I+\xi I^{2}+\mathcal{O}(I^{3})],$ (S2)

with

$\displaystyle L_{d}$
$\displaystyle=\frac{L_{J0}}{\sqrt{1-\left(\frac{I_{d}}{I_{c}}\right)^{2}}}$
(S3) $\displaystyle\epsilon$ $\displaystyle=\frac{I_{d}}{I_{c}^{2}-I_{d}^{2}}$
(S4) $\displaystyle\xi$
$\displaystyle=\frac{I_{c}^{2}+2I_{d}^{2}}{2(I_{c}^{2}-I_{d}^{2})^{2}}$ (S5)

In practice, the JJ is at the interface between two metallic surfaces, which
therefore add a capacitance $C_{J}$, in parallel to this effective nonlinear
inductance. Below its resonant frequency (called the plasma frequency
$\omega_{p}=1/\sqrt{C_{J}L_{d}}$), the tank circuit is equivalent to an
effective inductance $\tilde{L}_{J}$ such that:

$\tilde{L}_{J}=\frac{L_{J}}{1-C_{J}L_{J}\omega^{2}}=\tilde{L}_{d}[1+\tilde{\epsilon}I+\tilde{\xi}I^{2}+\mathcal{O}(I^{3})],$
(S6)

where

$\displaystyle\tilde{L}_{d}$
$\displaystyle=\frac{L_{d}}{1-\left(\frac{\omega}{\omega_{p}}\right)^{2}}$
(S7) $\displaystyle\tilde{\epsilon}$
$\displaystyle=\frac{\epsilon}{1-\left(\frac{\omega}{\omega_{p}}\right)^{2}}$
(S8) $\displaystyle\tilde{\xi}$
$\displaystyle=\frac{\xi+(\epsilon^{2}-\xi)\left(\frac{\omega}{\omega_{p}}\right)^{2}}{\left[1-\left(\frac{\omega}{\omega_{p}}\right)^{2}\right]^{2}}.$
(S9)

In the following, we drop the $\tilde{}$ notation, but will remember that
$L_{d}$, $\epsilon$ and $\xi$ now all depend on $\omega$, although far from
the plasma frequency, this dependence is negligible. In our case
$\omega_{p}/2\pi\approx 40$ GHz. When ambiguous, we will explicitly write this
dependence.

###  I.2 The coupled-mode equations

Writing the nonlinear inductance up to the second order in $I$ (Eq. S6) allows
us to include the 3WM-type of nonlinearity (which comes from the term
proportional to $\epsilon$), as well as gain compression coming from the 4WM
(_i.e._ Kerr) term proportional to $\xi$. In the first, simple model presented
in Fig. 2, we did not include the effect of the 4WM term (equivalent to taking
$\xi=0$ in the following equations). We then included them when comparing the
measurements to the CME, see Sec. V.

Writing the the telegraphers’ equations in this context, including both 3WM
and 4WM, we obtain the propagation equation [2]

$v_{p}^{2}\frac{\partial^{2}I}{\partial x^{2}}-\frac{\partial^{2}I}{\partial
t^{2}}=\frac{\partial^{2}}{\partial
t^{2}}\left(\frac{\epsilon}{2}I^{2}+\frac{\xi}{3}I^{3}\right),$ (S10)

where $v_{p}=1/\sqrt{L_{d}(\omega)C}$ is the line’s phase velocity, $x$ is the
position along the transmission line (which for us will be in units of cells)
and $t$ is the time.

We now perform a harmonic balance, _i.e._ we assume that the current $I(x,t)$,
solution of the equation of motion Eq. S10, is the sum of the selected
propagating modes of interest, indexed by $n\in\\{a,s,i,c,d,u,c_{2}\\}$. The
modes propagate forward or backward, and we study two situations, depending on
whether we look at the forward or backward response of the device. For the
forward response, we assume that $\\{a,s,i,d,u,c_{2}\\}$ are traveling
forward, whereas $c$ travels backward, and for the backward response, we
assume that $\\{s,i,c,d,u,c_{2}\\}$ are traveling backward, while $a$ travels
forward, which by symmetry is equivalent to considering
$\\{s,i,c,d,u,c_{2}\\}$ as traveling forward, while $a$ travels backward.
Table S1 summarizes the way the modes are propagating. Note that (i) we
account for possible reflections of all these modes at the TWPAC ports
following [1], and (ii) that formally, the only thing that differentiates a
mode traveling forward to a mode traveling backward is that for the backward
traveling mode, we need to do the transformation $k_{n}\leftarrow-k_{n}$,
where $k_{n}$ is the mode wavenumber, in the coupled-mode equations.

| modes traveling forward | modes traveling backward
---|---|---
forward response | $a,s,i,d,u,c_{2}$ | $c$
backward response | $s,i,d,u,c,c_{2}$ | $a$
Table S1: List and propagation orientation of the modes, when looking at the
forward an backward response of the TWPAC.

Thus the current, solution of Eq. S10 is

$\displaystyle I(x,t)$
$\displaystyle=\sum_{n\in\\{a,s,i,c,d,u,c_{2}\\}}\left\\{\frac{1}{2}I_{n}(x)\left[e^{\pm
ik_{n}x}+\Gamma_{n}e^{\pm ik_{n}N}e^{\mp ik_{n}x}+\left(\Gamma_{n}e^{\pm
ik_{n}N}\right)^{2}e^{\pm
ik_{n}x}+...\right]e^{-i\omega_{n}t}+\mathrm{c.c.}\right\\}$ (S11)
$\displaystyle=\sum_{n\in\\{a,s,i,c,d,u,c_{2}\\}}\left[\frac{1}{2}I_{n}(x)t_{n}\left(e^{\pm
ik_{n}x}+\tilde{\Gamma}_{n}e^{\mp
ik_{n}x}\right)e^{-i\omega_{n}t}+\mathrm{c.c.}\right]$
$\displaystyle=\sum_{n\in\\{a,s,i,c,d,u,c_{2}\\}}\frac{1}{2}\alpha_{n}^{\pm}+\frac{1}{2}\alpha_{n}^{*\pm},$

where

$\alpha_{n}^{\pm}=I_{n}(x)t_{n}\left(e^{\pm ik_{n}x}+\tilde{\Gamma}_{n}e^{\mp
ik_{n}x}\right)e^{-i\omega_{n}t},$ (S12)

with the $\pm$ denoting the forward ($+$) or backward ($-$) traveling modes.
For clarity, in what follows we omit the superscript $+$ for the modes
$\\{s,i,d,u,c_{2}\\}$, because they always travel forward (see Tab. S1). Here,
c.c. is the complex conjugate, $I_{n}(x)$ is the complex field amplitude of
mode $n$, and

$\displaystyle t_{n}=\frac{1}{1-\Gamma_{n}e^{i2k_{n}N}}$ (S13)
$\displaystyle\tilde{\Gamma}_{n}=\Gamma_{n}e^{ik_{n}N},$ (S14)

with $N$ the total length of the TWPAC, in units of cell-lengths (so $N$ is
the total number of cells), and $\Gamma_{n}$ is the amplitude reflection
coefficient of mode $n$. In other words, if the TWPAC impedance at
$\omega_{n}$ is $Z_{n}$, and if the TWPAC is embedded in an environment with
impedance $Z_{0}$, we have

$\Gamma_{n}=\frac{\lvert Z_{n}-Z_{0}\rvert}{Z_{n}+Z_{0}}.$ (S15)

We now need to replace $I$ in Eq. S10 with $I(x,t)$ from Eq. S11, and then
collect the terms at each modes of interest. First, we deal with the left-
hand-side (LHS) of Eq. S10:

$\mathrm{LHS}=v_{p}^{2}\sum_{n\in\\{a,s,i,c,d,u,c_{2}\\}}\left[\pm
ik_{n}t_{n}\frac{\partial I_{n}(x)}{\partial x}\left(e^{\pm
ik_{n}x}-\tilde{\Gamma}_{n}e^{\mp
ik_{n}x}\right)e^{-i\omega_{n}t}+\mathrm{c.c.}\right].$ (S16)

Therefore, for any mode $n$ we obtain:

$\mathrm{LHS}\Bigr{|}_{\begin{subarray}{c}\omega=\omega_{n}\end{subarray}}=\pm
i\frac{\omega_{n}^{2}}{k_{n}}t_{n}\frac{\partial I_{n}(x)}{\partial
x}\left(e^{\pm ik_{n}x}-\tilde{\Gamma}_{n}e^{\mp
ik_{n}x}\right)e^{-i\omega_{n}t}+\mathrm{c.c.},$ (S17)

using the fact that $v_{p}=\omega_{n}/k_{n}$.

On the right-hand-side (RHS), we perform a rotating wave approximation: we
only keep the terms whose frequencies are at one of the modes
$n\in\\{a,s,i,c,d,u,c_{2}\\}$. For the three-wave mixing processes, coming
from the term $\partial^{2}I^{2}/\partial t^{2}$ in Eq. S10, we obtain the
coefficients gathered in Tab. S2. For the four-wave mixing processes, coming
from the term $\partial^{2}I^{3}/\partial t^{2}$ in Eq. S10, we obtain the
coefficients gathered in Tab. S3. Note that for both 3WM and 4WM, all of the
permutations of a given process have to be accounted for.

3WM process | coefficient
---|---
$\omega_{s}+\omega_{i}=\omega_{a}$ | $-\omega_{a}^{2}\frac{1}{2}\alpha_{s}\alpha_{i}$
$\omega_{a}-\omega_{i}=\omega_{s}$ | $-\omega_{s}^{2}\frac{1}{2}\alpha_{a}^{\pm}\alpha_{i}^{*}$
$\omega_{a}-\omega_{s}=\omega_{i}$ | $-\omega_{i}^{2}\frac{1}{2}\alpha_{a}^{\pm}\alpha_{s}^{*}$
$\omega_{c}+\omega_{d}=\omega_{s}$ | $-\omega_{s}^{2}\frac{1}{2}\alpha_{c}^{\mp}\alpha_{d}$
$\omega_{s}-\omega_{d}=\omega_{c}$ | $-\omega_{c}^{2}\frac{1}{2}\alpha_{s}\alpha_{d}^{*}$
$\omega_{s}-\omega_{c}=\omega_{d}$ | $-\omega_{d}^{2}\frac{1}{2}\alpha_{s}\alpha_{c}^{*\mp}$
$\omega_{u}-\omega_{c}=\omega_{s}$ | $-\omega_{s}^{2}\frac{1}{2}\alpha_{u}\alpha_{c}^{*\mp}$
$\omega_{u}-\omega_{s}=\omega_{c}$ | $-\omega_{c}^{2}\frac{1}{2}\alpha_{u}\alpha_{s}^{*}$
$\omega_{s}+\omega_{c}=\omega_{u}$ | $-\omega_{u}^{2}\frac{1}{2}\alpha_{s}\alpha_{c}^{\mp}$
$\omega_{c}+\omega_{c}=\omega_{c_{2}}$ | $-\omega_{c_{2}}^{2}\frac{1}{4}\alpha_{c}^{\mp}\alpha_{c}^{\mp}$
$\omega_{c_{2}}-\omega_{c}=\omega_{c}$ | $-\omega_{c}^{2}\frac{1}{2}\alpha_{c_{2}}\alpha_{c}^{*\mp}$
$\omega_{u}-\omega_{d}=\omega_{c_{2}}$ | $-\omega_{c_{2}}^{2}\frac{1}{2}\alpha_{u}\alpha_{d}^{*}$
$\omega_{u}-\omega_{c_{2}}=\omega_{d}$ | $-\omega_{d}^{2}\frac{1}{2}\alpha_{u}\alpha_{c_{2}}^{*}$
$\omega_{c_{2}}+\omega_{d}=\omega_{u}$ | $-\omega_{u}^{2}\frac{1}{2}\alpha_{c_{2}}\alpha_{d}$
Table S2: List of all the 3WM processes and their coefficients when restricting to the mode basis $\\{a,s,i,c,d,u,c_{2}\\}$. For clarity, each permutations of a given process are grouped together. For the PA pump (mode $a$) and FC pump (mode $c$), we distinguish the forward ($+$) and backward ($-$) traveling modes. 4WM process | coefficient
---|---
$\omega_{a}+\omega_{a}-\omega_{a}=\omega_{a}$ | $-\omega_{a}^{2}\frac{3}{8}\alpha_{a}^{\pm}\lvert\alpha_{a}^{\pm}\rvert^{2}$
$\omega_{a}+\omega_{c}-\omega_{c}=\omega_{a}$ | $-\omega_{a}^{2}\frac{6}{8}\alpha_{a}^{\pm}\lvert\alpha_{c}^{\mp}\rvert^{2}$
$\forall n\in\\{s,i,d,u,c_{2}\\},\hskip 2.84544pt\omega_{a}+\omega_{n}-\omega_{n}=\omega_{a}$ | $-\omega_{a}^{2}\frac{6}{8}\alpha_{a}^{\pm}\lvert\alpha_{n}\rvert^{2}$
$\omega_{c}+\omega_{c}-\omega_{c}=\omega_{c}$ | $-\omega_{c}^{2}\frac{3}{8}\alpha_{c}^{\mp}\lvert\alpha_{c}^{\mp}\rvert^{2}$
$\omega_{c}+\omega_{a}-\omega_{a}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{c}^{\mp}\lvert\alpha_{a}^{\pm}\rvert^{2}$
$\forall n\in\\{s,i,d,u,c_{2}\\},\hskip 2.84544pt\omega_{c}+\omega_{n}-\omega_{n}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{c}^{\mp}\lvert\alpha_{n}\rvert^{2}$
$\forall m\in\\{s,i,d,u,c_{2}\\}$
$\omega_{m}+\omega_{m}-\omega_{m}=\omega_{m}$ | $-\omega_{m}^{2}\frac{3}{8}\alpha_{m}\lvert\alpha_{m}\rvert^{2}$
$\omega_{m}+\omega_{a}-\omega_{a}=\omega_{m}$ | $-\omega_{m}^{2}\frac{6}{8}\alpha_{m}\lvert\alpha_{a}^{\pm}\rvert^{2}$
$\omega_{m}+\omega_{c}-\omega_{c}=\omega_{m}$ | $-\omega_{m}^{2}\frac{6}{8}\alpha_{m}\lvert\alpha_{c}^{\mp}\rvert^{2}$
$\forall n\in\\{s,i,d,u,c_{2}\\}\hskip 2.84544pt\&\hskip 2.84544ptn\neq m,\hskip 2.84544pt\omega_{m}+\omega_{n}-\omega_{n}=\omega_{m}$ | $-\omega_{m}^{2}\frac{6}{8}\alpha_{m}\lvert\alpha_{n}\rvert^{2}$
$\omega_{u}-\omega_{d}-\omega_{c}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{u}\alpha_{d}^{*}\alpha_{c}^{*\mp}$
$\omega_{u}-\omega_{c}-\omega_{c}=\omega_{d}$ | $-\omega_{d}^{2}\frac{3}{8}\alpha_{u}(\alpha_{c}^{*\mp})^{2}$
$\omega_{d}+\omega_{c}+\omega_{c}=\omega_{u}$ | $-\omega_{u}^{2}\frac{3}{8}\alpha_{d}(\alpha_{c}^{\mp})^{2}$
$\omega_{c}+\omega_{s}-\omega_{d}=\omega_{c_{2}}$ | $-\omega_{c_{2}}^{2}\frac{6}{8}\alpha_{c}^{\mp}\alpha_{s}\alpha_{d}^{*}$
$\omega_{c}+\omega_{s}-\omega_{c_{2}}=\omega_{d}$ | $-\omega_{d}^{2}\frac{6}{8}\alpha_{c}^{\mp}\alpha_{s}\alpha_{c_{2}}^{*}$
$\omega_{c_{2}}-\omega_{c}+\omega_{d}=\omega_{s}$ | $-\omega_{s}^{2}\frac{6}{8}\alpha_{c_{2}}\alpha_{c}^{*\mp}\alpha_{d}$
$\omega_{c_{2}}-\omega_{s}+\omega_{d}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{c_{2}}\alpha_{s}^{*}\alpha_{d}$
$\omega_{u}-\omega_{s}+\omega_{c}=\omega_{c_{2}}$ | $-\omega_{c_{2}}^{2}\frac{6}{8}\alpha_{u}\alpha_{s}^{*}\alpha_{c}^{\mp}$
$\omega_{u}-\omega_{c_{2}}+\omega_{c}=\omega_{s}$ | $-\omega_{s}^{2}\frac{6}{8}\alpha_{u}\alpha_{c_{2}}^{*}\alpha_{c}^{\mp}$
$\omega_{c_{2}}-\omega_{c}+\omega_{s}=\omega_{u}$ | $-\omega_{u}^{2}\frac{6}{8}\alpha_{c_{2}}\alpha_{c}^{*\mp}\alpha_{s}$
$\omega_{c_{2}}-\omega_{u}+\omega_{s}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{c_{2}}\alpha_{u}^{*}\alpha_{s}$
$\omega_{a}-\omega_{i}+\omega_{c}=\omega_{u}$ | $-\omega_{u}^{2}\frac{6}{8}\alpha_{a}^{\pm}\alpha_{i}^{*}\alpha_{c}^{\mp}$
$\omega_{a}-\omega_{u}+\omega_{c}=\omega_{i}$ | $-\omega_{i}^{2}\frac{6}{8}\alpha_{a}^{\pm}\alpha_{u}^{*}\alpha_{c}^{\mp}$
$\omega_{u}-\omega_{a}+\omega_{i}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{u}\alpha_{a}^{*\pm}\alpha_{i}$
$\omega_{u}-\omega_{c}+\omega_{i}=\omega_{a}$ | $-\omega_{a}^{2}\frac{6}{8}\alpha_{u}\alpha_{c}^{*\mp}\alpha_{i}$
$\omega_{a}-\omega_{i}-\omega_{c}=\omega_{d}$ | $-\omega_{d}^{2}\frac{6}{8}\alpha_{a}^{\pm}\alpha_{i}^{*}\alpha_{c}^{*\mp}$
$\omega_{a}-\omega_{d}-\omega_{c}=\omega_{i}$ | $-\omega_{i}^{2}\frac{6}{8}\alpha_{a}^{\pm}\alpha_{d}^{*}\alpha_{c}^{*\mp}$
$\omega_{a}-\omega_{d}-\omega_{i}=\omega_{c}$ | $-\omega_{c}^{2}\frac{6}{8}\alpha_{a}^{\pm}\alpha_{d}^{*}\alpha_{i}^{*}$
$\omega_{d}+\omega_{i}+\omega_{c}=\omega_{a}$ | $-\omega_{a}^{2}\frac{6}{8}\alpha_{d}\alpha_{i}\alpha_{c}^{\mp}$
Table S3: List of all the 4WM processes and their coefficients when
restricting to the mode basis $\\{a,s,i,c,d,u,c_{2}\\}$. For clarity, family
of processes are grouped together.

Finally, we join the LHS and RHS together, to get the coupled-mode equations
(and their complex conjugate, not shown):

$\displaystyle\begin{aligned} \frac{\partial I_{a}}{\partial
x}=\pm&i\frac{\epsilon(\omega_{a})}{4}\frac{k_{a}}{t_{a}}\mathcal{F}^{si}_{\pm
a}t_{s}t_{i}I_{s}I_{i}\\\
\pm&i\frac{\xi(\omega_{a})}{8}k_{a}I_{a}\left(\mathcal{F}^{\pm a\pm a\pm
a^{*}}_{\pm a}\lvert t_{a}\rvert^{2}\lvert I_{a}\rvert^{2}+2\mathcal{F}^{\pm
a\mp c\mp c^{*}}_{\pm a}\lvert t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{s,i,d,u,c_{2}\\}}2\mathcal{F}^{\pm amm^{*}}_{\pm
a}\lvert t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
\pm&i\frac{\xi(\omega_{a})}{8}\frac{k_{a}}{t_{a}}\left(2\mathcal{F}^{u\mp
c^{*}i}_{a}t_{u}t_{c}^{*}t_{i}I_{u}I_{c}^{*}I_{i}+2\mathcal{F}^{di\mp
c}_{a}t_{d}t_{i}t_{c}I_{d}I_{i}I_{c}\right)\end{aligned}$ (S18)
$\displaystyle\begin{aligned} \frac{\partial I_{s}}{\partial
x}=&i\frac{\epsilon(\omega_{s})}{4}\frac{k_{s}}{t_{s}}\left(\mathcal{F}^{\pm
ai^{*}}_{s}t_{a}t_{i}^{*}I_{a}I_{i}^{*}+\mathcal{F}^{\mp
cd}_{s}t_{c}t_{d}I_{c}I_{d}+\mathcal{F}^{u\mp
c^{*}}_{s}t_{u}t_{c}^{*}I_{u}I_{c}^{*}\right)\\\
+&i\frac{\xi(\omega_{s})}{8}k_{s}I_{s}\left(\mathcal{F}^{sss^{*}}_{s}\lvert
t_{s}\rvert^{2}\lvert I_{s}\rvert^{2}+2\mathcal{F}^{s\pm a\pm a^{*}}_{s}\lvert
t_{a}\rvert^{2}\lvert I_{a}\rvert^{2}+2\mathcal{F}^{s\mp c\mp c^{*}}_{s}\lvert
t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{i,d,u,c_{2}\\}}2\mathcal{F}^{smm^{*}}_{s}\lvert
t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
+&i\frac{\xi(\omega_{s})}{8}\frac{k_{s}}{t_{s}}\left(2\mathcal{F}^{c_{2}\mp
c^{*}d}_{s}t_{c_{2}}t_{c}^{*}t_{d}I_{c_{2}}I_{c}^{*}I_{d}+2\mathcal{F}^{uc_{2}^{*}\mp
c}_{s}t_{u}t_{c_{2}}^{*}t_{c}I_{u}I_{c_{2}}^{*}I_{c}\right)\end{aligned}$
(S19) $\displaystyle\begin{aligned} \frac{\partial I_{i}}{\partial
x}=&i\frac{\epsilon(\omega_{i})}{4}\frac{k_{i}}{t_{i}}\mathcal{F}^{\pm
as^{*}}_{i}t_{a}t_{s}^{*}I_{a}I_{s}^{*}\\\
+&i\frac{\xi(\omega_{i})}{8}k_{i}I_{i}\left(\mathcal{F}^{iii^{*}}_{i}\lvert
t_{i}\rvert^{2}\lvert I_{i}\rvert^{2}+2\mathcal{F}^{i\pm a\pm a^{*}}_{i}\lvert
t_{a}\rvert^{2}\lvert I_{a}\rvert^{2}+2\mathcal{F}^{i\mp c\mp c^{*}}_{i}\lvert
t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{s,d,u,c_{2}\\}}2\mathcal{F}^{imm^{*}}_{i}\lvert
t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
+&i\frac{\xi(\omega_{i})}{8}\frac{k_{i}}{t_{i}}\left(2\mathcal{F}^{\pm
au^{*}\mp c}_{i}t_{a}t_{u}^{*}t_{c}I_{a}I_{u}^{*}I_{c}+2\mathcal{F}^{\pm
ad^{*}\mp
c^{*}}_{i}t_{a}t_{d}^{*}t_{c}^{*}I_{a}I_{d}^{*}I_{c}^{*}\right)\end{aligned}$
(S20) $\displaystyle\begin{aligned} \frac{\partial I_{c}}{\partial
x}=\mp&i\frac{\epsilon(\omega_{c})}{4}\frac{k_{c}}{t_{c}}\left(\mathcal{F}^{sd^{*}}_{\mp
c}t_{s}t_{d}^{*}I_{s}I_{d}^{*}+\mathcal{F}^{us^{*}}_{\mp
c}t_{u}t_{s}^{*}I_{u}I_{s}^{*}+\mathcal{F}^{c_{2}\mp c^{*}}_{\mp
c}t_{c_{2}}t_{c}^{*}I_{c_{2}}I_{c}^{*}\right)\\\
\mp&i\frac{\xi(\omega_{c})}{8}k_{c}I_{c}\left(\mathcal{F}^{\mp c\mp c\mp
c^{*}}_{\mp c}\lvert t_{c}\rvert^{2}\lvert I_{c}\rvert^{2}+2\mathcal{F}^{\mp
c\pm a\pm a^{*}}_{\mp c}\lvert t_{a}\rvert^{2}\lvert
I_{a}\rvert^{2}+\sum_{m\in\\{s,i,d,u,c_{2}\\}}2\mathcal{F}^{\mp cmm^{*}}_{\mp
c}\lvert t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
\mp&i\frac{\xi(\omega_{c})}{8}\frac{k_{c}}{t_{c}}\Big{(}2\mathcal{F}^{ud^{*}\mp
c^{*}}_{\mp
c}t_{u}t_{d}^{*}t_{c}^{*}I_{u}I_{d}^{*}I_{c}^{*}+2\mathcal{F}^{c_{2}s^{*}d}_{\mp
c}t_{c_{2}}t_{s}^{*}t_{d}I_{c_{2}}I_{s}^{*}I_{d}+2\mathcal{F}^{c_{2}u^{*}s}_{\mp
c}t_{c_{2}}t_{u}^{*}t_{s}I_{c_{2}}I_{u}^{*}I_{s}\\\ +&2\mathcal{F}^{u\pm
a^{*}i}_{\mp c}t_{u}t_{a}^{*}t_{i}I_{u}I_{a}^{*}I_{i}+2\mathcal{F}^{\pm
ad^{*}i^{*}}_{\mp
c}t_{a}t_{d}^{*}t_{i}^{*}I_{a}I_{d}^{*}I_{i}^{*}\Big{)}\end{aligned}$ (S21)
$\displaystyle\begin{aligned} \frac{\partial I_{d}}{\partial
x}=&i\frac{\epsilon(\omega_{d})}{4}\frac{k_{d}}{t_{d}}\left(\mathcal{F}^{s\mp
c^{*}}_{d}t_{s}t_{c}^{*}I_{s}I_{c}^{*}+\mathcal{F}^{uc_{2}^{*}}_{d}t_{u}t_{c_{2}}^{*}I_{u}I_{c_{2}}^{*}\right)\\\
+&i\frac{\xi(\omega_{d})}{8}k_{d}I_{d}\left(\mathcal{F}^{ddd^{*}}_{d}\lvert
t_{d}\rvert^{2}\lvert I_{d}\rvert^{2}+2\mathcal{F}^{d\pm a\pm a^{*}}_{d}\lvert
t_{a}\rvert^{2}\lvert I_{a}\rvert^{2}+2\mathcal{F}^{d\mp c\mp c^{*}}_{d}\lvert
t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{s,i,u,c_{2}\\}}2\mathcal{F}^{dmm^{*}}_{d}\lvert
t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
+&i\frac{\xi(\omega_{d})}{8}\frac{k_{d}}{t_{d}}\left(\mathcal{F}^{u\mp
c^{*}\mp c^{*}}_{d}t_{u}t_{c}^{*2}I_{u}I_{c}^{*2}+2\mathcal{F}^{\mp
csc_{2}^{*}}_{d}t_{c}t_{s}t_{c_{2}}^{*}I_{c}I_{s}I_{c_{2}}^{*}+2\mathcal{F}^{\pm
ai^{*}\mp
c^{*}}_{d}t_{a}t_{i}^{*}t_{c}^{*}I_{a}I_{i}^{*}I_{c}^{*}\right)\end{aligned}$
(S22) $\displaystyle\begin{aligned} \frac{\partial I_{u}}{\partial
x}=&i\frac{\epsilon(\omega_{u})}{4}\frac{k_{u}}{t_{u}}\left(\mathcal{F}^{s\mp
c}_{u}t_{s}t_{c}I_{s}I_{c}+\mathcal{F}^{c_{2}d}_{u}t_{c_{2}}t_{d}I_{c_{2}}I_{d}\right)\\\
+&i\frac{\xi(\omega_{u})}{8}k_{u}I_{u}\left(\mathcal{F}^{uuu^{*}}_{u}\lvert
t_{u}\rvert^{2}\lvert I_{u}\rvert^{2}+2\mathcal{F}^{u\pm a\pm a^{*}}_{u}\lvert
t_{a}\rvert^{2}\lvert I_{a}\rvert^{2}+2\mathcal{F}^{u\mp c\mp c^{*}}_{u}\lvert
t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{s,i,d,c_{2}\\}}2\mathcal{F}^{umm^{*}}_{u}\lvert
t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
+&i\frac{\xi(\omega_{u})}{8}\frac{k_{u}}{t_{u}}\left(\mathcal{F}^{d\mp c\mp
c}_{u}t_{d}t_{c}^{2}I_{d}I_{c}^{2}+2\mathcal{F}^{c_{2}\mp
c^{*}s}_{u}t_{c_{2}}t_{c}^{*}t_{s}I_{c_{2}}I_{c}^{*}I_{s}+2\mathcal{F}^{\pm
ai^{*}\mp c}_{u}t_{a}t_{i}^{*}t_{c}I_{a}I_{i}^{*}I_{c}\right)\end{aligned}$
(S23) $\displaystyle\begin{aligned} \frac{\partial I_{c_{2}}}{\partial
x}=&i\frac{\epsilon(\omega_{c_{2}})}{4}\frac{k_{c_{2}}}{t_{c_{2}}}\left(\frac{1}{2}\mathcal{F}^{\mp
c\mp
c}_{c_{2}}t_{c}^{2}I_{c}^{2}+\mathcal{F}^{ud^{*}}_{c_{2}}t_{u}t_{d}^{*}I_{u}I_{d}^{*}\right)\\\
+&i\frac{\xi(\omega_{c_{2}})}{8}k_{c_{2}}I_{c_{2}}\left(\mathcal{F}^{{c_{2}}{c_{2}}{c_{2}}^{*}}_{c_{2}}\lvert
t_{c_{2}}\rvert^{2}\lvert I_{c_{2}}\rvert^{2}+2\mathcal{F}^{{c_{2}}\pm a\pm
a^{*}}_{c_{2}}\lvert t_{a}\rvert^{2}\lvert
I_{a}\rvert^{2}+2\mathcal{F}^{{c_{2}}\mp c\mp c^{*}}_{c_{2}}\lvert
t_{c}\rvert^{2}\lvert
I_{c}\rvert^{2}+\sum_{m\in\\{s,i,d,u\\}}2\mathcal{F}^{{c_{2}}mm^{*}}_{c_{2}}\lvert
t_{m}\rvert^{2}\lvert I_{m}\rvert^{2}\right)\\\
+&i\frac{\xi(\omega_{c_{2}})}{8}\frac{k_{c_{2}}}{t_{c_{2}}}\left(2\mathcal{F}^{\mp
csd^{*}}_{c_{2}}t_{c}t_{s}t_{d}^{*}I_{c}I_{s}I_{d}^{*}+2\mathcal{F}^{us^{*}\mp
c}_{c_{2}}t_{u}t_{s}^{*}t_{c}I_{u}I_{s}^{*}I_{c}\right)\end{aligned}$ (S24)

where we explicitly added $\pm$ or $\mp$ in front of the terms whose sign
changes between the forward ($+$ or $-$) and backward ($-$ or $+$) response.
Here,

$\mathcal{F}^{\pm m\pm n}_{\pm p}=\frac{(e^{\pm
ik_{m}x}+\tilde{\Gamma}_{m}e^{\mp ik_{m}x})(e^{\pm
ik_{n}x}+\tilde{\Gamma}_{n}e^{\mp ik_{n}x})}{e^{\pm
ik_{p}x}-\tilde{\Gamma}_{p}e^{\mp ik_{p}x}}$ (S25)

and

$\mathcal{F}^{\pm m\pm n\pm p}_{\pm q}=\frac{\left(e^{\pm
ik_{m}x}+\tilde{\Gamma}_{m}e^{\mp ik_{m}x}\right)\left(e^{\pm
ik_{n}x}+\tilde{\Gamma}_{n}e^{\mp ik_{n}x}\right)\left(e^{\pm
ik_{p}x}+\tilde{\Gamma}_{p}e^{\mp ik_{p}x}\right)}{e^{\pm
ik_{q}x}-\tilde{\Gamma}_{q}e^{\mp ik_{q}x}}$ (S26)

are the terms that account for the linear phase mismatch and for reflections
for the 3WM and 4WM processes, respectively. In other words, when there is no
reflection and when the linear process is phase matched, the corresponding
term is equal to unity. For example, if $k_{a}=k_{s}+k_{i}$, and
$\tilde{\Gamma}_{a}=\tilde{\Gamma}_{s}=\tilde{\Gamma}_{i}=0$, we have
$\mathcal{F}^{si}_{\pm a}=1$. Note that these CME include soft pump effects.

###  I.3 The phase-matching condition

The 4WM mixing terms (generated by $\xi\neq 0$), affect the ideal, trivial 3WM
phase-matching (PM) conditions we would otherwise have for the parametric
amplification (PA) and frequency conversion (FC) processes,
$k_{a}=k_{s}+k_{i}$ and $k_{d}=k_{s}-k_{c}$ (or $k_{u}=k_{s}+k_{c}$ for the up
conversion), respectively. To find the modified PM conditions, one has to look
for solutions of the CME in the strong pump case, where $\lvert
I_{a}\rvert\gg\\{\lvert I_{s}\rvert,\lvert I_{i}\rvert\\}$ on the one hand,
and $\lvert I_{c}\rvert\gg\\{\lvert I_{s}\rvert,\lvert I_{d}\rvert\\}$ (or
$\lvert I_{c}\rvert\gg\\{\lvert I_{s}\rvert,\lvert I_{u}\rvert\\}$ for the up
conversion) on the other hand. The two PM conditions can be solved for
independently. Including the effect of reflections, one can show that [1]:

$\Delta\beta_{PA}=\Delta
k_{PA}+\chi_{PA}(1+\Gamma_{a}^{2})(k_{a}-2k_{s}-2k_{i})=k_{a}-k_{s}-k_{i}+\chi_{a}(1+\Gamma_{a}^{2})(k_{a}-2k_{s}-2k_{i})=0,$
(S27)

is the 3WM PM condition for the PA process, where

$\chi_{a}=\frac{\xi\lvert I_{a}(x=0)\rvert^{2}}{8}=\frac{\xi\lvert
I_{a0}\rvert^{2}}{8}$ (S28)

is the PA pump self-phase modulation. Similarly,

$\Delta\beta_{FC}=\Delta
k_{FC}+\chi_{c}(1+\Gamma_{c}^{2})(k_{c}+2k_{d}-2k_{s})=k_{c}+k_{d}-k_{s}+\chi_{FC}(1+\Gamma_{c}^{2})(k_{c}+2k_{d}-2k_{s})=0,$
(S29)

is the 3WM PM condition for the FC process, where

$\chi_{c}=\frac{\xi\lvert I_{c}(x=0)\rvert^{2}}{8}=\frac{\xi\lvert
I_{c0}\rvert^{2}}{8}$ (S30)

is the FC pump self-phase modulation. In practice, the two PM conditions, Eqs.
S27 and S29, are only satisfied for two selected triplets of frequencies
$\\{\omega_{a},\omega_{s},\omega_{i}\\}$ and
$\\{\omega_{c},\omega_{s},\omega_{d}\\}$, respectively, because the line is
designed to be dispersive, see Sec. II. Varying the frequencies of the pumps
allows us to move the signal, idler and down-converted tone frequencies at
which PM takes place, but we optimize the TWPAC design to obtain the PM
conditions with signal and idler tones detuned by
$\Delta\omega_{\mathrm{PM}}=1$ GHz. It is a somewhat arbitrary choice (and the
design does not sensitively depend on it), which ensures us to have a wideband
amplifier response.

###  I.4 The amplification and conversion gain

We numerically solve the CME (Eqs. S18-S24) in the forward and backward
directions, using a set of initial conditions, where we specify an input PA
pump amplitude $I_{a0}$, an input FC pump amplitude $I_{c0}$, and a small
signal $I_{s0}\ll\\{I_{a0},I_{c0}\\}$. Any other mode amplitude is initially
set to zero. Once the response has been solved, we obtain the current
amplitude for each mode in the forward ($+$) and backward ($-$) directions, as
a function of the position within the line. We then compute the gain response
for all the modes, and in particular for the signal we have:

$G_{s}^{\pm}(x)=\left(\frac{I_{s}^{\pm}(N)}{I_{s0}}\right)^{2}T_{s},$ (S31)

where $N$ is the total number of cells within the TWPAC and where

$T_{s}=(1-\Gamma_{s}^{2})^{2}\lvert t_{s}\rvert^{2}$ (S32)

is the (power) transmission coefficient, that accounts for the possible
impedance mismatch between the TWPAC and its environment [1].

##  II The TWPAC linear response: dispersion engineering

Dispersion engineering (DE) is the technique by which we accomplish phase-
matching (PM) for the PA and FC processes (Eqs. S27 and S29, respectively). In
a linear transmission line, a pure 3WM process is naturally phase-matched,
however, as shown in Eqs. S27 and S29, the presence of the higher order
nonlinearity (the Kerr) affects the PM. Also, in our case the line is
naturally weakly dispersive, due to the capacitances in parallel with each JJ.
It creates the so-called the chromatic dispersion, a low-Q resonance at the
plasma frequency $\omega_{p}/2\pi\approx 40$ GHz, which affects the dispersion
relation at lower frequencies. That is why we need to engineer additional
resonances: by placing the pump frequencies close to these resonances, we
retrieve the PM conditions.

We aim to have $f_{a}=14$ GHz (the 3WM PA pump frequency), so that the
amplification band is centered around $f_{a}/2=7$ GHz. We therefore engineer a
resonant phase-matching (rpm) circuit: a resonator to ground (see Fig. 1b),
composed of an inductance $L_{\mathrm{rpm}}$ in parallel with a capacitor
$C_{\mathrm{rpm}}$, with resonant frequency $f_{\mathrm{rpm}}=13.51$ GHz,
slightly lower than the desired $f_{a}$. It is inserted every 6 cells, and its
mode impedance is $20$ Ω, both of which are a compromise between the resonance
quality factor (dictating how sensitive the pump placement will be) and the
resonator footprint on chip, which depends on the amorphous silicon thickness
($t=300$ nm) and dielectric constant ($\epsilon_{r}=9.2$).

Figure S1: The TWPAC unbiased linear response, using the design parameters
presented in Tab. S4. (a) The transmission as a function of frequency (black),
calculated from the ABCD matrices of the TWPAC’s components, quantitatively
agrees to the measured transmission (gray). (b) Similarly, the calculated
nonlinear part of the wavenumber $k^{*}$ (black) quantitatively agrees with
the measured one (gray). (c) The calculated TWPAC’s impedance as a function of
frequency has been optimized to be close to $50$ Ωbetween $6$ and $8$ GHz.

We also aim to have $f_{c}=4.7$ GHz (the FC pump frequency), so that the up-
and down- converted tones are far-enough detuned from the bandwidth of
interest. Concurrently, we aim to suppress the second harmonic at $2f_{c}$ by
placing it into a stopband. Using resonators to create resonances at such low
frequencies is inconvenient, because it would lead to having very big
capacitor pads, which would complicate the TWPAC layout. That is why we opted
for periodically loading the line: we sinusoidally modulate the value of the
ground coupling capacitor $C_{c}(x)$ along the line, _i.e._ we modulate the
line’s impedance. We overlap two modulations, one that creates the stopband
and $f_{c}$, and one creates the stopband around $2f_{c}$:

$Z_{\mathrm{pl}}=Z_{m}\left(1+\delta_{c}\cos{\frac{2\pi}{N_{0}}x}+\delta_{c2}\cos{\frac{4\pi}{N_{0}}x}\right).$
(S33)

Here, $Z_{m}$ is a mean impedance, which adjusted to $47$ Ω in order to have
the TWPAC impedance be as close to $50$ Ω as possible in the bandwidth of
interest, around $7$ GHz. It is an effect of the first periodic loading at
$4.7$ GHz, which slightly increases the nearby impedances above resonance.
$N_{0}$ is the number of cells within a the smallest repetition pattern within
the line, composed of both modulations and of the rpm circuits, which we call
a supercell. In other words, we chose $N_{0}$ so that the periodicities of all
the repetition patterns are commensurate within one supercell. We adjusted
both modulation depths, $\delta_{c}$ and $\delta_{c2}$, to achieve PM for the
FC process, and at the same time have $2f_{c}$ fall into the second stopband.

Figure S2: Design of the ground coupling capacitor $C_{c}$. (a) the rpm
circuit (purple) as well as the Josephson capacitor $C_{J}$ (red) slightly
affect the design of $C_{c}$ (black). Both the rpm circuit and the static part
of the capacitively shunted Josephson junction can be modeled (below their
resonance) as effective inductors, $\tilde{L}_{\mathrm{rpm}}$ and
$\tilde{L}_{d}$ respectively. (b) The value of $C_{c}$ within a supercell
containing $66$ cells is modulated according to Eq. S36. The black dots
indicate the presence of an rpm circuit.

The junction’s parallel capacitance $C_{J}$ as well as the rpm circuit
slightly affect the value of $C_{c}$, see Fig. S2a. In fact, the impedance of
a symmetric unit cell can be calculated from its ABCD matrix coefficients:
$Z_{\mathrm{uc}}=\sqrt{B_{\mathrm{uc}}/C_{\mathrm{uc}}}$, which in the case of
the TWPAC gives:

$Z_{\mathrm{uc}}=\sqrt{\frac{\tilde{L}_{d}}{C_{c}}\left[2-C_{c}\omega^{2}(2\tilde{L}_{\mathrm{rpm}}-\tilde{L}_{d})\right]},$
(S34)

where $\tilde{L}_{d}$ is given by Eq. S7, and where

$\tilde{L}_{\mathrm{rpm}}=\frac{L_{\mathrm{rpm}}}{1-L_{\mathrm{rpm}}C_{\mathrm{rpm}}\omega^{2}}$
(S35)

if the cell contains a resonator, or $\tilde{L}_{\mathrm{rpm}}=0$ if it does
not. Therefore, inverting Eq. S34 and taking
$Z_{\mathrm{uc}}=Z_{\mathrm{pl}}(x)$ we obtain:

$C_{c}(x)=\frac{2\tilde{L}_{d}}{Z_{\mathrm{pl}}(x)^{2}+\tilde{L}_{d}\omega^{2}(2\tilde{L}_{\mathrm{rpm}}+\tilde{L}_{d})}.$
(S36)

Figure S2b shows $C_{c}(x)$, Tab. S4 summarizes all the TWPAC design
parameters and Fig. shows the TWPAC linear response, calculated by cascading
the ABCD matrices of its various components. The overall TWPAC transmission is
calculated from the overall ABCD matrix coefficients:

$S_{\mathrm{21}}=\frac{2}{A+B/Z_{0}+CZ_{0}+D},$ (S37)

where $Z_{0}=50$ Ω is the TWPAC’s environment impedance. The overall TWPAC
nonlinear wave number is $k^{*}=k-\omega/v_{p}$, where
$v_{p}=1/\sqrt{L_{d}\braket{C_{c}(x)}_{x}}$ is the TWPAC phase velocity (at
$\omega=0$) with $\braket{}_{x}$ the space averaging, and where

$k=\frac{1}{N}\mathrm{unwrap}\left[\Im\left(\operatorname{arccosh}{\frac{A+D}{2}}\right)\right].$
(S38)

The overall TWPAC impedance is $Z=\sqrt{B/C}$.

parameter | value
---|---
Junctions critical current |  $I_{c}\approx 5$ $\upmu$A
Junctions plasma frequency |  $\omega_{p}/2\pi\approx 40$ GHz
Josephson capacitor |  $C_{J}=240.5$ fF
mean periodic loading impedance |  $Z_{m}=47$ Ω
fundamental impedance modulation depth | $\delta_{c}=0.1$
second harmonic impedance modulation depth | $\delta_{c2}=0.12$
rpm spacing |  $\Delta_{\mathrm{rpm}}=6$ cells
rpm circuit frequency |  $f_{\mathrm{rpm}}=13.51$ GHz
rpm circuit mode impedance |  $Z_{\mathrm{rpm}}=20$ Ω
rpm circuit coupled quality factor | $Q_{\mathrm{crpm}}=578$
rpm circuit inductance |  $L_{\mathrm{rpm}}=230$ pH
rpm circuit capacitance |  $C_{\mathrm{rpm}}=557$ fF
dc biasing current |  $I_{d}=1.5$ $\upmu$A
static inductance under dc bias (at $\omega_{a}/2$) |  $\tilde{L}_{d}(\omega_{a}/2)=71.4$ pH
first order nonlinearity (at $\omega_{a}$) |  $\epsilon(\omega_{a})=0.075$ $\upmu$A-1
second order nonlinearity (at $\omega_{a}$) |  $\xi(\omega_{a})=0.033$ $\upmu$A-2
total number of cells | $N=2640$
number or supercells | $N_{\mathrm{sc}}=40$
number of cells per supercell | $N_{0}=66$
mean coupling capacitance to ground |  $\braket{C_{c}(x)}_{x}=32.2$ fF
phase velocity (at $\omega=0$) |  $v_{p}=670$ cell.ns-1
cell length |  $l_{\mathrm{uc}}=7$ $\upmu$m
design signal-to-idler detuning at PM |  $\Delta\omega_{\mathrm{PM}}=1$ GHz
PA pump frequency |  $\omega_{a}/2\pi=14.5$ GHz
PA pump chip-input power |  $P_{a}=-73.4$ dBm
FC pump frequency |  $\omega_{c}/2\pi=4.7$ GHz
FC pump chip-input power |  $P_{a}=-72.2$ dBm
phase-matched signal frequency |  $\omega_{s}/2\pi=7.65$ GHz
phase-matched idler frequency |  $\omega_{i}/2\pi=6.65$ GHz
phase-matched down-converted frequency |  $\omega_{d}/2\pi=2.95$ GHz
signal chip-input power |  $P_{a}=-133$ dBm
a-Si thickness |  $t=300$ nm
a-Si dielectric constant | $\epsilon_{r}=9.2$
loss tangent | $\tan\delta=4\times 10^{-4}$
Table S4: List of the TWPAC design parameters.

##  III The TWPAC layout

Figure S3 shows one TWPAC supercell. It is composed of $66$ cells, each
containing one Josephson junction and one capacitor to ground. The resonators
to ground are placed every six cells.

Figure S3: False color scanning-electron microscope picture of one TWPAC
supercell. The color coding is identical to that of Fig. 1: the Josephson
junctions are colored in brown, the capacitors to ground are colored in green,
and the resonators to ground are colored in purple.

##  IV Losses in the TWPAC

Our original TWPAC design did not account for any microwave loss. However,
upon comparing the TWPAC experimental linear response (_i.e._ unpumped,
unbiased) to the theoretical one, see Fig. S1, a quantitative agreement was
found when assuming a loss tangent $\tan\delta=4\times 10^{-4}$, a value that
is in agreement with previous results of devices fabricated via the same
process [5]. Thus, the harmonic balance simulations presented in the main text
(Fig. 2) as well as those presented throughout the supplementary information
all assume $\tan\delta=4\times 10^{-4}$. Modeling this loss channel as an
admittance in parallel with the coupling capacitance $C_{c}$ and with the rpm
capacitance $C_{\mathrm{rpm}}$, we account for this loss by performing the
transformation:

$C_{x}\rightarrow C_{x}(1-i\tan\delta),$ (S39)

where $C_{x}\in\\{C_{c},C_{\mathrm{rpm}}\\}$.

It generates a microwave loss that is current bias dependent. In fact, in
general the propagation constant $\gamma$ for a lossy transmission line, whose
loss is modeled by a resistance $R$ in series with the inductor $L$ and an
admittance $G$ in parallel with the capacitor $C$ is [8]

$\gamma=\alpha+i\beta=\sqrt{(R+i\omega L)(G+i\omega C)},$ (S40)

where $\\{\alpha,\beta\\}\in\mathcal{R}$. When $\alpha\neq 0$, signals
attenuate along the line. Assuming that $R=0$, we can eliminate $\beta$ and
solve for $\alpha$ to find:

$\alpha=\frac{1}{2}G\sqrt{\frac{L}{C}}=\frac{GZ}{2}.$ (S41)

In our situation with the TWPAC, when increasing the current bias we
effectively increase $L$, which increases the attenuation constant $\alpha$
(note that $S_{21}=e^{-2\alpha N}$ and $\arg{S_{21}}=-\beta N$). In other
words, when increasing the current bias we increase the TWPAC effective
length, and therefore we increase its attenuation accordingly.

However, it does not account for all the loss observed when the TWPAC is
current biased, see Fig. S5. This additional loss channel could be due to a
bias dependent quasi-particle current, that would be modeled as a current-
dependent resistor $R_{\mathrm{qp}}$ in parallel with each JJ, with
$R_{\mathrm{qp}}(I_{d}=0)=0$. The origin of this additional loss channel
remains unclear, so we chose to not account for it in the TWPAC modelling.

The time-domain simulations do not account for any loss, neither dielectric
loss nor this additional loss channel, because WRspice does not
straightforwardly include the possibility of defining complex capacitors, nor
does it allow to define a current-dependent resistor.

Figure S4: The TWPAC transmission as a function of frequency, when biased with
a $1$ $\upmu$A dc current, calculated from the ABCD matrices (black) and
measured (gray).

##  V Simulation when matching to the up conversion process

When designing the TWPAC, we chose a FC pump frequency $\omega_{c}$ that
phase-matches the down-conversion process: $\omega_{d}=\omega_{s}-\omega_{c}$,
and we did not include the effect of the 4WM processes. For a signal frequency
$\omega_{s}/2\pi=7.65$ GHz, it yields $\omega_{c}/2\pi=4.7$ GHz, which
dictates the position of the second stopband, placed to prevent the
propagation of the second harmonic of this pump tone (at $2\omega_{c}$).
However, in practice the TWPAC only demonstrates backward isolation when
pumped at a lower FC pump frequency, around $\omega_{c}/2\pi=3.15$ GHz, which
corresponds to matching the up-conversion process,
$\omega_{u}=\omega_{s}+\omega_{c}$. In fact, if we solve the coupled-mode
equations with $\omega_{c}/2\pi=3.15$ GHz, and if we include the 4WM processes
(see Eqs. S18 to S24), we find a backward transmission profile that is
qualitatively similar to the one obtained experimentally, as well as the one
obtained from the time-domain simulations, see Fig. S5.

Figure S5: The TWPAC transmission as a function of frequency, when biased with
a $1$ $\upmu$A dc current, and pumped with a forward PA pump at frequency
$f_{a}=14.34$ GHz and power $P_{a}=-71$ dBm, and backward FC pump at frequency
$f_{c}=3.15$ GHz and power $P_{c}=-76$ dBm. Here, the CME include all the 3WM
and 4WM terms, presented in Eqs. S18 to S24.

The inability to pump the down-conversion process may be due to the fact that
the FC pump frequency that matches this process is too close to the first
stopband, and therefore experiences too much of an impedance mismatch when
entering the TWPAC (see Fig. S1c). This mismatch would (i) prevent the FC pump
from efficiently entering the TWPAC and (ii) make the TWPAC resonant at this
particular frequency, preventing the directional FC process. Also, it is
possible that the FC pump frequency that matches the down-conversion process
generates too much of 4WM amplification in the $6$ to $8$ GHz band of
interest.

##  VI The time-domain simulations

We closely follow the work of Gaydamachenko et al. [3] to perform the time-
domain simulations. We use WRspice [4], and the circuit netlist is defined
from the circuit parameters presented in Tab. S4, and using the periodically
varying coupling capacitance to ground $C_{c}$ presented in Fig. S2. The dc
current bias is modeled with a dc current source placed between the first and
last nodes of the TWPAC and the rf pumps are modeled with sinusoidal sources.
The TWPAC environment is set using two $Z_{0}=50$ Ω shunt resistors, at its
input and output. We perform two sets of simulations: when simulating the
forward response, the PA pump is placed at the TWPAC input, together with a
small input signal (sinusoidal source with amplitude $0.05$ $\upmu$A), while
the FC pump is placed at the TWPAC output. When simulating the backward
response, the PA and FC pumps are swapped. We sweep the signal frequency
between $20$ MHz and $12$ GHz, stepping it every $20$ MHz. For each signal
frequency we perform a forward and a backward time-domain simulation.

For each simulation, we measure the voltage to ground as a function of time,
$V_{\mathrm{in}}(t)$ and $V_{\mathrm{out}}(t)$, at the first and last node
respectively, as well as the time currents entering ($I_{\mathrm{in}}(t)$) and
exiting ($I_{\mathrm{out}}(t)$) the TWPAC. After Fourier transforming these
four time vectors, we form the transmission scattering parameter:

$S_{\mathrm{21}}=\frac{V_{\mathrm{out}}(\omega_{s})+Z_{0}I_{\mathrm{out}}(\omega_{s})}{V_{\mathrm{in}}(\omega_{s})+Z_{0}I_{\mathrm{in}}(\omega_{s})},$
(S42)

where $\omega_{s}$ is the signal frequency. We then calculate the gain in dB
as $G_{\mathrm{dB}}=10\log\lvert S_{\mathrm{21}}\rvert^{2}$.

Note that we chose the RSJ junction model of WRspice (the so-called level=1),
and set the shunt resistance to zero (using the option rtype=0). Otherwise,
the WRspice junction model assumes a default shunt resistance [4].

##  VII The TWPAC response to a single pump tone

Figure S6 shows the forward and backward transmission through the TWPAC, when
only the PA pump is turned on (Fig. S6a) or when only the FC pump is turned on
(Fig. S6b). It demonstrates that both parametric processes can be activated
independently of each other. The pumps’ powers and frequencies are similar to
that used in Fig. 3b. When only the PA pump is turned on, the backward
transmission is slightly higher than when the PA pump is off, in particular in
the $6$ to $8$ GHz band. It shows that in this conventional operating
situation where the TWPAC functions as a conventional TWPA, there is some non-
negligible reverse gain.

Figure S6: The gain and isolation of the TWPAC, when either the PA pump (a) or
the FC pump (b) is turned on. The TWPAC biasing parameters are similar to that
of Fig. 3b: $I_{d}=1$ $\upmu$A, $\omega_{a}/2\pi=14.52$ GHz, $P_{a}=-75$ dBm,
$\omega_{c}/2\pi=3.15$ GHz and $P_{c}=-80$ dBm (at the chip inputs).

##  VIII Signals reflected off the TWPAC

Figure S7: The full two-port scattering parameters of the TWPAC, in the
frequency window where we observe both PA and FC. The TWPAC biasing parameters
are similar to that of Fig. 3b. The input (a) and output (b) reflections of
the through cable (black) and of the TWPAC (green) with the FC conversion pump
turned on are plotted as a function of frequency. (c) The transmission as a
function of frequency is the same as presented in Fig. 3b. (d) The input
reflection without (dashed line) and with (plain line) the FC pump turned on.

In the main text we presented the forward and backward transmission of the
TWPAC. More precisely, we presented $10\log\lvert S_{\mathrm{21}}|^{2}$ and
$10\log\lvert S_{\mathrm{12}}|^{2}$, respectively the forward and backward
power transmission in decibels. Figures S7a and b show the raw input
reflections (i.e. measured by the VNA, uncorrected from the chain’s gain and
attenuation) $10\log\lvert S_{\mathrm{11}}|^{2}$ and output reflections
$10\log\lvert S_{\mathrm{22}}|^{2}$, around the frequency window where the
TWPAC presents forward amplification and backward isolation (Fig. S7c), when
both the PA and FC pump are turned on. They are as good as that of the through
cable. Also, in that same band the input reflection with the FC pump turned
off is degraded, compared to when it is on (Fig. S7d).

##  IX The TWPAC saturation

###  IX.1 The $1$ dB compression point

Figure S8 shows the input $1$ dB compression point, $P_{1\mathrm{dB}}$, and
the corresponding small signal TWPAC gain, here defined as the ratio of the
VNA transmission when the dc bias and the pumps are turned on, to the
transmission when the dc bias and the pumps are turned off. The
$P_{1\mathrm{dB}}$ is the point at which the gain drops by $1$ dB when
increasing the VNA probe power. Between $5.5$ and $9$ GHz,
$P_{1\mathrm{dB}}\approx-90$ dBm on average, with a corresponding average
TWPAC gain $G\approx 7.3$ dB. Assuming that $P_{1\mathrm{dB}}$ will decrease
with increasing gain, it is on par with conventional Josephson-based TWPAs [6,
7], for which $P_{1\mathrm{dB}}$ is $\approx-100$ dBm at $\approx 20$ dB gain.
Note also that when the FC pump is turned off, $P_{1\mathrm{dB}}$ stays at the
same level, whereas the gain increases to $\approx 7.9$ dB on average between
$5.5$ and $9$ GHz, indicating that for the same gain, turning on the FC
marginally diminishes $P_{1\mathrm{dB}}$, by $\approx 0.6$ dB.

Figure S8: The TWPAC saturation power, focusing on the frequency window where
we observed both PA and FC. (a) The TWPAC gain as small VNA probe tone power
is shown as a function of frequency with (blue) and without (yellow) the FC
pump turned on. Here, it is calculated as the ratio of the transmission when
the dc bias and the pumps are turned on, to the transmission when the dc bias
and the pumps are turned off. (b) The TWPAC-input $1$ dB compression point is
shown with (blue) and without (yellow) the FC pump turned on. The cut portion
around $8.8$ GHz corresponds to a situation where $P_{1\mathrm{dB}}$ was not
yet reached at the highest VNA probe tone power.

The input attenuation of the line, from the VNA (emitting the probe tone) to
the TWPAC input is calibrated as follows: we first perform a noise measurement
using the SNTJ, with the TWPAC off (pumps off, no dc bias), from which we
obtain the system-added noise and the gain of the chain
$G_{c}^{\mathrm{off}}$, from the SNTJ’s reference plane to the spectrum
analyzer (SA). We then measure the output power
$P_{\mathrm{VNA}}^{\mathrm{out}}$ on the SA when sending a VNA probe tone,
varying its frequency (TWPAC still off). Finally, we divide the probe tone
power emitted by the VNA $P_{\mathrm{VNA}}^{\mathrm{in}}$ by the tone power,
referred to the SNTJ’s reference plane
$P_{\mathrm{VNA}}^{\mathrm{out}}/G_{c}^{\mathrm{off}}$, which gives us a good
estimate of the line’s input attenuation $A$, from the VNA emitting port to
the TWPAC input:

$A=\frac{P_{\mathrm{VNA}}^{\mathrm{in}}}{P_{\mathrm{VNA}}^{\mathrm{out}}/G_{c}^{\mathrm{off}}}.$
(S43)

Note that this attenuation includes the loss between the SNTJ and the TWPAC,
as such it is an over-estimate of the attenuation between the VNA emitting
port and the TWPAC chip-input. As a consequence, the input $1$ dB compression
point is slightly under-estimated.

###  IX.2 The TWPAC output spectrum

Experimentally, the FC pump frequency $\omega_{c}$ being lower than what was
expected from the original design, its second harmonic
$\omega_{c_{2}}=2\omega_{c}$ lies within the amplification bandwidth, together
with its corresponding idler $\omega_{a}-2\omega_{c}$, where $\omega_{a}$ is
the PA pump frequency. One can wonder whether these modes limit the TWPAC
power handling. In Fig. S9a and S9b we show the TWPAC output spectrum measured
on the SA, when sending a probe tone at $\omega_{s}/2\pi=7$ GHz, at low power
(a) and at the power $P_{1\mathrm{dB}}$ for that frequency (b). We identify
the modes of interest at $\omega_{c}$, $\omega_{s}$, $2\omega_{c}$, and
$\omega_{a}-2\omega_{c}$. Evidently, there are some other peaks, in particular
we notice side-bands on each of these main modes, detuned by $\pm 87$ MHz,
which might come from noise in the dc biasing line of the TWPAC. At high tone
power (Fig. S9b), more peaks arise from the spectrum noise floor compared to
the low power case, because more parametric processes become non-negligible,
and the input probe tone power get scattered onto various modes. This is what
saturates the amplifier.

In Fig. S9c we plot the output power of the few selected modes of interest,
referred to the TWPAC output, as a function of the $7$ GHz input probe tone
power (referred to the TWPAC input). Taken individually, none of these tones
seem to be predominantly driving the TWPAC saturation, because their power
remain much below the TWPAC saturation power $P_{1\mathrm{dB}}$ at that
particular frequency.

Figure S9: The evolution of the TWPAC output spectrum, measured on the SA,
when increasing a CW probe tone at $7$ GHz. (a) The output spectrum at low
probe tone power presents $5$ major peaks at $\omega_{c}$,
$\omega_{c_{2}}=2\omega_{c}$, $\omega_{s}$,
$\omega_{i}=\omega_{a}-\omega_{s}$, and $\omega_{a}-2\omega_{c}$. We cut the
spectrum at the first TWPAC stopband, because within the stopband the gain of
the chain cannot be accurately measured, therefore the power of any bin within
the stopband cannot be accurately compared to the other bins. (b) The output
spectrum when sending a probe tone power $P_{1\mathrm{dB}}=-86$ dBm contains
more peaks, that correspond to activating more parametric processes. (c) The
output power of the few selected modes, referred to the TWPAC output is
plotted as a function of the input probe tone power, referred to the TWPAC
input. The vertical line indicates $P_{1\mathrm{dB}}$ for the CW tone at
$\omega_{s}=7$ GHz. The input probe tone power calibration was the same as in
Sec. IX.1, and we refer the output power to the TWPAC output by dividing the
power measured on the SA by the chain’s gain, measured with the SNTJ, with the
TWPAC off.

##  X Effect of the frequency conversion on noise measurement

In Fig. 4 we presented the noise measurement when the TWPAC is driven with
both the PA and FC pumps. Here we show that the FC pump does not degrade the
TWPAC noise performance. In fact, Fig. S10a shows the measured TWPAC noise as
a function of frequency, when the FC is on and off. On average, when the FC is
off the noise between $5.5$ and $8.5$ GHz is $4.8$ quanta, slightly lower than
when the FC is on ($5.2$ quanta). However, this discrepancy is perfectly
explained by the slight gain variation existing between the two situations FC
on and FC off (in both situations the PA pump is kept at the same power and
frequency). In fact, when plotting the added noise $N_{\mathrm{add}}$ as a
function of the TWPAC gain for the frequency bin centered at $7$ GHz (Fig.
S10b), both situations (FC on and FC off) overlap on the same curve describing
the model $N_{\mathrm{add}}=N_{1}+N_{2}/G$ (where $N_{1}$ is the noise added
by the TWPAC and by all the lossy elements preceding it, up to the SNTJ, and
where $N_{2}$ is the noise added by the elements in the chain after the
TWPAC). Therefore, the intrinsic added noise of the TWPAC stays the same,
regardless of having the FC pump turned on or off.

Figure S10: Effect of the FC pump on the added noise. (a) The added noise is
plotted as a function of frequency, when the FC pump is off (yellow) and when
it is on (blue). (b) The added noise of the bin centered at $7$ GHz is plotted
as a function of the TWPAC gain, when the FC pump is off (yellow) and when it
is on (blue). The black line represents the model
$N_{\mathrm{add}}=N_{1}+N_{2}/G$.

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