# Broadband CPW-based impedance-transformed Josephson parametric amplifier

Bingcheng Qing<EMAIL_ADDRESS>Department of Physics, University of
California, Berkeley, California 94720, USA  Long B. Nguyen
<EMAIL_ADDRESS>Department of Physics, University of California,
Berkeley, California 94720, USA  Computational Research Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Xinyu Liu
Anyon Technologies, Emeryville, California 94662, USA  Hengjiang Ren  Anyon
Technologies, Emeryville, California 94662, USA  William P. Livingston
Current address: Google Quantum AI, Mountain View, CA 94043, USA  Department
of Physics, University of California, Berkeley, California 94720, USA
Computational Research Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Noah Goss  Department of Physics, University
of California, Berkeley, California 94720, USA  Computational Research
Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA
Ahmed Hajr  Department of Physics, University of California, Berkeley,
California 94720, USA  Trevor Chistolini  Department of Physics, University of
California, Berkeley, California 94720, USA  Zahra Pedramrazi  Department of
Physics, University of California, Berkeley, California 94720, USA
Computational Research Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  David I. Santiago  Department of Physics,
University of California, Berkeley, California 94720, USA  Computational
Research Division, Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA  Jie Luo  Anyon Technologies, Emeryville, California 94662, USA
Irfan Siddiqi  Department of Physics, University of California, Berkeley,
California 94720, USA  Computational Research Division, Lawrence Berkeley
National Laboratory, Berkeley, California 94720, USA

###### Abstract

Quantum-limited Josephson parametric amplifiers play a pivotal role in
advancing the field of circuit quantum electrodynamics by enabling the fast
and high-fidelity measurement of weak microwave signals. Therefore, it is
necessary to develop robust parametric amplifiers with low noise, broad
bandwidth, and reduced design complexity for microwave detection. However,
current broadband parametric amplifiers either have degraded noise performance
or rely on complex designs. Here, we present a device based on the broadband
impedance-transformed Josephson parametric amplifier (IMPA) that integrates a
horn-like coplanar waveguide (CPW) transmission line, which significantly
decreases the design and fabrication complexity, while keeping comparable
performance. The device shows an instantaneous bandwidth of 700(200) MHz for
15(20) dB gain with an average saturation power of -110 dBm and near quantum-
limited added noise. The operating frequency can be tuned over 1.4 GHz using
an external flux bias. We further demonstrate the negligible back-action from
our device on a transmon qubit. The amplification performance and simplicity
of our device promise its wide adaptation in quantum metrology, quantum
communication, and quantum information processing.

††preprint: APS/123-QED

Introduction

The parametric process facilitates coherent energy transfer among
electromagnetic waves based on nonlinearities, enabling quantum-limited
parametric amplification Caves (1982). Microwave parametric amplification not
only improves the readout of solid-state Blais  _et al._ (2021); Burkard  _et
al._ (2023); Vijay  _et al._ (2011); Hatridge  _et al._ (2013); Arute  _et
al._ (2019) and nanomechanical Arash  _et al._ (2015) quantum devices, but
also becomes increasingly important in emerging technologies such as hybrid
quantum systems Clerk  _et al._ (2020) and axion dark matter detection
Caldwell  _et al._ (2017); Nitta  _et al._ (2023). It is thus important to
develop amplifiers with outstanding performance: low noise, broad bandwidth,
and high saturation power. Moreover, for easy adaptation of parametric
amplifiers, their design and fabrication should be robust and simple.

In superconducting circuits, parametric amplification is usually achieved by
Josephson parametric amplifiers (JPAs) utilizing multi-wave mixing processes
in nonlinear superconducting resonators, where Josephson junctions provide the
nonlinearity without introducing dissipation Roy and Devoret (2016);
Manucharyan  _et al._ (2007). The nonlinear resonator of the JPA is typically
coupled to a fixed 50-$\Omega$ embedding environment Castellanos-Beltran and
Lehnert (2007); Yamamoto  _et al._ (2008); Bergeal  _et al._ (2010); Hatridge
_et al._ (2011), which prevents tuning its linewidth and saturation power
independently, resulting in a trade-off between saturation power and
bandwidthMutus  _et al._ (2014). Hence, its bandwidth is inherently limited to
10-50 MHz, and its saturation power is typically -120 dBm for a nominal gain
of 20 dB.

There are two strategies to overcome this limitation. In the first approach,
the nonlinear resonator is unpacked into a nonlinear transmission line by
thousands of Josephson junctions, forming a traveling-wave-parametric-
amplifier (TWPA), in which the gain is optimized across a broad range of
frequencies Esposito  _et al._ (2021). Such a device must satisfy two
conditions: (i) the impedance of the amplifier has to match the 50-$\Omega$
environmental impedance, and (ii) the phases of the pump, signal, and idler
tones have to match Esposito  _et al._ (2021). To fulfill these challenging
requirements, 2000 nonlinear unit cells were constructed to form a TWPA with a
20-dB-gain bandwidth over 3 GHz and a saturation input power of -99 dBm
Macklin  _et al._ (2015). However, TWPA noise processes can be complex
Esposito  _et al._ (2021), and the stringent requirements on the phase
matching and the large number of Josephson junctions make the design,
calibration and fabrication difficult for TWPAs Ezenkova  _et al._ (2022).

In the second approach, the environmental impedance is engineered to enhance
the JPA’s coupling, which leads to enhanced bandwidth and saturation power
Mutus  _et al._ (2014). This has been accomplished by both simple narrowband
impedance transformers Roy  _et al._ (2015); Grebel  _et al._ (2021) and more
complex broadband impedance transformers Mutus  _et al._ (2014); Lu  _et al._
(2022); Ranzani  _et al._ (2022). Narrowband impedance transformers
constructed using $\lambda/4$ and $\lambda/2$ coplanar-waveguides (CPWs) Roy
_et al._ (2015); Grebel  _et al._ (2021) enhances the JPA’s bandwidth to 200
MHz with fixed operating frequency. JPAs with broadband impedance transformers
following Klopfenstein Mutus  _et al._ (2014); Lu  _et al._ (2022) or Ruthroff
Ranzani  _et al._ (2022) taper designs have instantaneous bandwidth in the
range of 600-700 MHz and allow tunable operating frequencies, but their
current implementations either require multi-layer fabrication with lossy
dielectrics Mutus  _et al._ (2014); Ranzani  _et al._ (2022) or careful
simulation and calibration of finely patterned interdigitated capacitors
Ranzani  _et al._ (2022). These requirements increase the complexity of the
amplifiers’ design, calibration, and fabrication.

Fig. 1: Experimental concept. (a) Electrical circuit depiction of the CIMPA.
The nonlinear resonator on the right side is constructed by shunting a SQUID
(yellow) with a capacitor (blue). The structure is galvanically coupled to a
Klopfenstein taper (green) on the left, and inductively coupled to a flux-line
(red) on the right. (b) Parameters of the horn-like CPW impedance transformer,
showing decreasing Zo along the CPW as its center-trace width becomes wider,
varying from 50 $\Omega$ at the left end to 18 $\Omega$ at the right end. (c)
False-color micrograph of the prototype sample with the nonlinear resonator
shown in the inset. From left to right are the horn-like CPW taper (green),
the capacitor (blue) and SQUID (yellow) forming the resonator, and the on-chip
flux line (red). (d) Simplified diagram of the measurement apparatus (details
in Suppl. Mat.). A transmon qubit is utilized to calibrate the input power and
detect possible back action from the CIMPA.

In this Letter, we present a CPW-based broadband impedance-transformed
parametric amplifier (CIMPA) design and showcase its performance. By reducing
the technical requirements of the Klopfenstein taper, we can implement it with
a single horn-like CPW, which keeps the broadband behavior while maintaining
the simplicity of the design and fabrication. The horn-like CPW structure
allows us to leverage streamlined RF design tools Simons (2004) and minimize
the complexity of simulating and fabricating the impedance transformer.
Notably, the CIMPA performs as well as the other IMPAs despite such
simplicity. The amplifier displays an instantaneous bandwidth of 700(200) MHz
for 15(20) dB gain, a 1.4 GHz flux-tunable bandwidth, a saturation input power
of approximately -110 dBm, and no significant back action on the qubit.

Device concept

The CIMPA is constructed from three basic building blocks, as depicted by the
circuit diagram in Fig. 1(a). The core component of the device is a nonlinear
LC resonator that facilitates the necessary parametric process through which
amplification occurs. It consists of a superconducting quantum interference
device (SQUID, yellow) and a capacitor (blue) in parallel. On the right-hand
side, it is inductively coupled to a flux line (red) that allows in-situ
tuning of the resonator frequency and RF flux pumping Grebel  _et al._ (2021);
Yamamoto  _et al._ (2008). On the left-hand side, it is galvanically connected
to an impedance taper (green) that is central to improving its bandwidth and
dynamic range.

The CIMPA operates as follows. When an RF microwave pump is applied to the
nonlinear resonator via the flux line at approximately twice its resonant
frequency $\omega_{p}\approx 2\omega_{0}$, we can apply the rotating wave
approximation (RWA) in the frame rotating at the signal frequency
$\omega_{s}\approx\omega_{0}$ and write the dressed Hamiltonian as Eddins
(2017)

$\hat{\mathcal{H}}/\hbar=\Delta\hat{a}^{\dagger}\hat{a}+\lambda(\hat{a}^{\dagger
2}+\hat{a}^{2})+\hat{O}(\hat{a}^{4}),$ (1)

where $\hat{a}(\hat{a}^{\dagger})$ is the bosonic annihilation (creation)
operator associated with the resonator, $\lambda$ originates from the
modulation of the external flux by the pump, and
$\Delta=\omega_{p}/2-\omega_{0}\sim 0$. If the circuit is coupled to the
environment with strength $\kappa$, its input-output equation of motion is
given by

$\dot{\hat{a}}=\frac{i}{\hbar}[\hat{\mathcal{H}},\hat{a}]-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa}\hat{a}_{\text{in}}=-2i\lambda\hat{a}^{\dagger}-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa}\hat{a}_{\text{in}},$
(2)

where $\hat{a}_{\mathrm{in}}$ and $\hat{a}_{\mathrm{out}}$ are the bosonic
operators associated with incoming and outgoing signals. We then proceed to
define $\bar{a}[\omega]=\int_{-\infty}^{\infty}\text{d}t\ e^{i\omega
t}\hat{a}(t)$ as the frequency-domain bosonic annihilation operator, and use
the boundary condition
$\sqrt{\kappa}\hat{a}=\hat{a}_{\text{in}}+\hat{a}_{\text{out}}$ to obtain

$\begin{split}\bar{a}_{\text{out
}}[\omega]=\left[\kappa\frac{\kappa/2-i\omega}{(\kappa/2-i\omega)^{2}-4|\lambda|^{2}}-1\right]\bar{a}_{\text{in
}}[\omega]\\\
+\left[\frac{-2i\kappa\lambda}{(\kappa/2-i\omega)^{2}-4|\lambda|^{2}}\right]\bar{a}_{\text{in
}}^{\dagger}[-\omega],\end{split}$ (3)

from which the gain is define, $G(\omega)=|\bar{a}_{\text{out
}}/\bar{a}_{\text{in}}|^{2}$.

For a suitable value of $\lambda$, the denominator in Eq. 3 approaches zero,
resulting in large signal gain, which remains at finite values in practice
because of higher-order terms that are not included in Eq. 1. Boutin  _et al._
(2017) In addition, large signal power may lead to an ac Stark shift and pump
power depletion, thereby decreasing the effective gain Boutin  _et al._
(2017). The incoming signal power associated with a gain drop of 1 dB is
defined as the saturation input power, or alternatively, the 1-dB compression
point.

Importantly, Eq. 3 shows that the amplifier’s bandwidth is determined by the
coupling strength $\kappa$, and thereby can be improved by enhancing the
coupling between the resonator and the signal. For the circuit shown in Fig.
1(a), this coupling rate is given as $\kappa=(CZ_{0})^{-1}$, while the
saturation power is $P_{\mathrm{sat}}\sim C/Z_{0}$Mutus  _et al._ (2014),
where $Z_{0}$ is the characteristic impedance of the coupled environment.
Thus, by engineering $Z_{0}$, one can increase both the bandwidth and the
dynamic range of the amplifier. Since the signals must come through a nominal
50-$\Omega$ transmission line before and after interacting with the amplifier,
an impedance transformer is necessary for enhanced coupling, shown on the
left-hand side of Fig. 1(a).

Previously, this idea was realized using a Klopfenstein taper with a
characteristic impedance ranging from $50~{}\Omega$ at one end to
$15~{}\Omega$ at the other end, and a maximum return loss of -20 dB above the
cut-off frequency Mutus  _et al._ (2014); Lu  _et al._ (2022). These stringent
requirements may lead to sophisticated designs that are associated with
tedious simulation procedures and complex nanofabrication processes.
Alternatively, a combination of quarter-wavelength and half-wavelength CPW
resonators can be used Roy  _et al._ (2015), with the setback of having a
narrower bandwidth, thereby negating the flux-tunable advantage of the SQUID.

Our design unifies the advantages of these two approaches by relaxing the
condition on maximum return loss to -10 dB. This allows us to engineer a horn-
like CPW transformer with characteristic impedance varying from 50 $\Omega$ to
18 $\Omega$ (Fig. 1(b)). The taper is designed by fixing its gap to 3
$\mathrm{\mu m}$ and varying the center-trace width from 7.8 $\mathrm{\mu m}$
to 218 $\mathrm{\mu m}$. The single CPW structure can be simulated using
simple RF design suites such as TX-Line Simons (2004), and its dimensions are
compatible with single-layer photo-lithography and reactive-ion etching
techniques. As a result, this approach significantly reduces the complexity
overhead in both design and fabrication, which is reflected by the simple
construction of our device shown in Fig. 1(c).

Device characterization

The CIMPA is wire-bonded to a printed circuit board (PCB) and packaged using a
rectangular copper box. The device is then attached to the mixing chamber
stage of a dilution fridge operating at 20 mK. It is connected to room-
temperature microwave apparatus as shown in Fig. 1(d). A transmon qubit is
integrated into the measurement setup to calibrate the signal power and detect
possible back-action from the parametric process. Two QuinStar 4-8 GHz
cryogenic circulators are used to route the signals, and the signals leaving
the transmon are amplified by CIMPA and a high-electron-mobility-transistor
amplifier (HEMT).

Fig. 2: CIMPA’s spectrum. Reflectometry spectroscopy performed using the VNA
shows phase twists across the SQUID’s resonant frequencies. Repeating the
measurement at various flux biases yields the CIMPA’s characteristic frequency
spectrum, which fits the relation given by Eq. 4 with excellent agreement.

In the first step, we extract the parameters of the nonlinear resonator by
measuring the small-signal linear response of the device using a vector
network analyzer (VNA). As the probe signal goes through $\omega_{0}/(2\pi)$,
the reflected signal phase is twisted, indicating a resonant condition. When
we adjust the external flux bias $\Phi_{e}$ of the SQUID, the resonant
frequency changes according to

Fig. 3: CIMPA’s performance. (a) Instantaneous signal gain, (b) noise
temperature, and (c) saturation input power of the CIMPA operated at
$\Phi_{e}/\Phi_{o}=0.3$ and $f_{p}=13.41~{}\mathrm{GHz}$. (d) Signal gain at
different CIMPA frequencies tuned by varying $\Phi_{e}$.

$\omega_{0}=\left[C(L_{\text{J,eff}}\left|\cos(\pi\Phi_{e}/\Phi_{0})\right|^{-1}+L_{\text{geo}})\right]^{-\nicefrac{{1}}{{2}}},$
(4)

where $L_{\text{geo}}$ is the parasitic geometric inductance originating from
the leads, $L_{\text{eff}}$ is the effective inductance of the SQUID,
$\Phi_{0}$ is the magnetic flux quantum. Measuring the phase response at each
flux point gives us the spectrum shown in Fig. 2. By fitting the measured
spectrum, we extract the following parameters:
$L_{\text{J,eff}}=69~{}\text{pH},~{}C=4~{}\text{pF}$, and
$L_{\text{geo}}<1~{}\text{fH}$. The spectrum also allows us to estimate the
asymmetry of the SQUID to be less than 1%.

We proceed to characterize the CIMPA’s gain profile, noise performance, and
dynamic range. We set the external flux bias to $\Phi_{e}=0.3\Phi_{0}$, and
apply a microwave pump tone at frequency
$\omega_{p}/(2\pi)=13.41~{}\text{GHz}$ with 3.41 dBm power outside the fridge.
we measure the microwave response using the VNA, observing over 20 (15) dB
gain over a 200 (700)-MHz bandwidth (Fig. 3(a)). Such a high increase in the
bandwidth cannot be explained solely by increasing the resonator linewidth
Mutus  _et al._ (2014). As discussed in Ref. Mutus  _et al._ (2014), the weak
frequency dependence of the environment’s characteristic impedance strongly
assists the enhancement of the bandwidth, but hinders the comparison between
the theory and experiments without the careful calibration of the
environment’s characteristic impedance.

The noise performance of the device can be characterized via the signal-to-
noise ratio (SNR) improvement $\text{SNR}_{\text{imprv}}$ measured by a
spectrum analyzer (SA) Mutus  _et al._ (2014); Macklin  _et al._ (2015); Roy
_et al._ (2015); Aumentado (2020). In an amplification chain consisting of a
CIMPA and a HEMT, the SNR improvement due to the CIMPA is given as

$\text{SNR}_{\text{imprv}}=\left(\frac{T_{\text{CIMPA}}}{T_{\text{HEMT}}}+\frac{1}{G}\right)^{-1},$
(5)

where $T_{\text{CIMPA}}$ and $T_{\text{HEMT}}$ indicate the noise temperatures
of the CIMPA and the HEMT, respectively. The extracted noise temperature with
$T_{\text{HEMT}}=2.3$ K - $2.9$ K is shown in Fig. 3(b), which approaches the
quantum noise limit. The uncertainty here originates from the estimated
$T_{\text{HEMT}}$ range. According to the convention used in Ref. Mutus  _et
al._ (2014), we define the bandwidth of our device to be 700 MHz as the
frequency range over which the device approaches the quantum noise limit. The
noise increase beyond this range is mainly because the CIMPA’s gain is not
large enough to compensate for the noise at the HEMT.

Presently, we characterize the saturation power of the device as follows.
First, the relation between the actual signal power going into the CIMPA and
the power set on the VNA must be established. By measuring the ac Stark shift
and measurement-induced dephasing of the transmon qubit at various VNA powers,
we can calibrate the cable attenuation Macklin  _et al._ (2015). Then, we
gradually increase the power until the gain falls off by 1 dB, and convert
this using the established attenuation. The result in Fig. 3(c) shows an
average 1-dB compression point of approximately -110 dBm. Compared with the
saturation power of simple JPAs coupled to a $50~{}\Omega$ environment (with
$Q\approx 15$), which is less than -120 dBm Yamamoto  _et al._ (2008); Mutus
_et al._ (2014); Ranzani  _et al._ (2022), our saturation power is increased
by around $10$ dB. This can be explained by the reduction in the environment’s
characteristic impedance in our design (with $Q\approx 4$), as
$P_{\mathrm{sat}}\sim C/Z_{0}$ Mutus  _et al._ (2014).

In addition to the instantaneous bandwidth, we can flux-tune the device and
operate it at other frequencies. To showcase this capability, we calibrate the
CIMPA’s parameters at five different external flux points corresponding to a
frequency range of 1.4 GHz, then characterize the gain performance of the
device across this broad range. Figure 3 (d) shows that the combination of the
engineered broadband impedance transformer and the inherent flux-tunability
allows the device to amplify signals with frequency ranging from 6 GHz to 7.4
GHz and at least 15 dB of maximum gain.

Importantly, possible back-action from the parametric process can undermine
the integration of an amplifier into a qubit measurement setup. For example,
spurious microwave radiation from the CIMPA can populate the readout resonator
connected to the qubit, leading to photon-noise dephasing Gambetta  _et al._
(2006); Kindel  _et al._ (2016). To investigate the possible back-action of
the CIMPA prototype, we measure the echo time $T_{2,\text{echo}}$ of the qubit
with and without amplification. To turn off the CIMPA, we remove the pump tone
and flux-tune its frequency away from the qubit’s readout resonator, such that
the amplifier acts as a passive mirror. When the CIMPA is turned on, it
provides a gain of 21 dB for the readout signal frequency at 6.633 GHz. The
$T_{2,\text{echo}}$ measurement is repeated continuously over 201 iterations
in each setting to provide reliable statistics despite temporal fluctuations.
We attribute the wider fluctuation range associated with the measurement when
the CIMPA is off to the approximately ten times longer data acquisition time.
As shown in Fig. 4, there is no significant degradation of the qubit’s
coherence time beyond random fluctuation when the CIMPA is operated.

Fig. 4: Transmon’s coherence time statistics for CIMPA on (pumped) and off
(unpumped), with the mean values equal to 14.78 $\mathrm{\mu s}$ and 16
$\mathrm{\mu s}$, and the standard deviations equal to 1.41 $\mathrm{\mu s}$
and 5.45 $\mathrm{\mu s}$, respectively. The statistics consist of 201
individual measurements in each setting.

Discussion and outlook

In conclusion, we show that the CIMPA’s design is as simple as the JPA’s, yet
its performance is comparable to earlier IMPA implementations. Therefore, it
fills the technological gap between JPAs and TWPAs, with potential
applications ranging from qubit readout to axion dark matter detection.
Compared to typical TWPAs, the prototype device reported here has a narrower
bandwidth and lower dynamic range. However, we note that, on one hand, a
majority of quantum experiments do not require a 3-GHz-bandwidth, and on the
other hand, we can integrate SQUID array White  _et al._ (2023) or
superconducting nonlinear asymmetric inductive elements (SNAIL) Ezenkova  _et
al._ (2022) into the CIMPA design to increase its power handling capability.

Acknowledgments

We thank Larry Chen and Kan-Heng Lee for providing the transmon sample. This
work was supported by the Quantum Testbed Program of the Advanced Scientific
Computing Research Division, Office of Science of the U.S. Department of
Energy under Contract No. DE-AC02-05CH11231.

Competing interest

X.L., H.R., and J.L. are Anyon Technologies’ employees.

Author contributions

J.L. and B.Q. conceived and organized the project. X.L., B.Q., and H.R.
fabricated the device. B.Q., L.B.N., and X.L. performed the measurement and
acquired the data. B.Q. analyzed the results. N.G. assisted with the
measurement apparatus. A. H. and Z.P. assisted with the cryogenic setup. W.P.L
and T.C. assisted with initiating the project. H.R., J.L., D.I.S., and I.F.
supervised the effort. B.Q. and L.B.N. wrote the manuscript with input from
all authors.

## References

  * Caves (1982) C. M. Caves, Physical Review D 26, 1817 (1982).
  * Blais  _et al._ (2021) A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Rev. Mod. Phys. 93, 025005 (2021).
  * Burkard  _et al._ (2023) G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Rev. Mod. Phys. 95, 025003 (2023).
  * Vijay  _et al._ (2011) R. Vijay, D. Slichter, and I. Siddiqi, Physical review letters 106, 110502 (2011).
  * Hatridge  _et al._ (2013) M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, _et al._ , Science 339, 178 (2013).
  * Arute  _et al._ (2019) F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, _et al._ , Nature 574, 505 (2019).
  * Arash  _et al._ (2015) B. Arash, J.-W. Jiang, and T. Rabczuk, Applied Physics Reviews 2, 021301 (2015), https://pubs.aip.org/aip/apr/article-pdf/doi/10.1063/1.4916728/14572831/021301_1_online.pdf .
  * Clerk  _et al._ (2020) A. Clerk, K. Lehnert, P. Bertet, J. Petta, and Y. Nakamura, Nature Physics 16, 257 (2020).
  * Caldwell  _et al._ (2017) A. Caldwell, G. Dvali, B. Majorovits, A. Millar, G. Raffelt, J. Redondo, O. Reimann, F. Simon, and F. Steffen (MADMAX Working Group), Phys. Rev. Lett. 118, 091801 (2017).
  * Nitta  _et al._ (2023) T. Nitta, T. Braine, N. Du, M. Guzzetti, C. Hanretty, G. Leum, L. J. Rosenberg, G. Rybka, J. Sinnis, J. Clarke, I. Siddiqi, M. H. Awida, A. S. Chou, M. Hollister, S. Knirck, A. Sonnenschein, W. Wester, J. R. Gleason, A. T. Hipp, P. Sikivie, N. S. Sullivan, D. B. Tanner, R. Khatiwada, G. Carosi, N. Robertson, L. D. Duffy, C. Boutan, E. Lentz, N. S. Oblath, M. S. Taubman, J. Yang, E. J. Daw, M. G. Perry, C. Bartram, J. H. Buckley, C. Gaikwad, J. Hoffman, K. W. Murch, M. Goryachev, E. Hartman, B. T. McAllister, A. Quiskamp, C. Thomson, M. E. Tobar, J. A. Dror, H. Murayama, and N. L. Rodd (ADMX Collaboration), Phys. Rev. Lett. 131, 101002 (2023).
  * Roy and Devoret (2016) A. Roy and M. Devoret, Comptes Rendus Physique 17, 740 (2016).
  * Manucharyan  _et al._ (2007) V. Manucharyan, E. Boaknin, M. Metcalfe, R. Vijay, I. Siddiqi, and M. Devoret, Physical Review B 76, 014524 (2007).
  * Castellanos-Beltran and Lehnert (2007) M. Castellanos-Beltran and K. Lehnert, Applied Physics Letters 91 (2007).
  * Yamamoto  _et al._ (2008) T. Yamamoto, K. Inomata, M. Watanabe, K. Matsuba, T. Miyazaki, W. D. Oliver, Y. Nakamura, and J. Tsai, Applied Physics Letters 93 (2008).
  * Bergeal  _et al._ (2010) N. Bergeal, F. Schackert, M. Metcalfe, R. Vijay, V. Manucharyan, L. Frunzio, D. Prober, R. Schoelkopf, S. Girvin, and M. Devoret, Nature 465, 64 (2010).
  * Hatridge  _et al._ (2011) M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke, and I. Siddiqi, Phys. Rev. B 83, 134501 (2011).
  * Mutus  _et al._ (2014) J. Y. Mutus, T. C. White, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, _et al._ , Applied Physics Letters 104 (2014).
  * Esposito  _et al._ (2021) M. Esposito, A. Ranadive, L. Planat, and N. Roch, Applied Physics Letters 119 (2021).
  * Macklin  _et al._ (2015) C. Macklin, K. O’brien, D. Hover, M. Schwartz, V. Bolkhovsky, X. Zhang, W. Oliver, and I. Siddiqi, Science 350, 307 (2015).
  * Ezenkova  _et al._ (2022) D. Ezenkova, D. Moskalev, N. Smirnov, A. Ivanov, A. Matanin, V. Polozov, V. Echeistov, E. Malevannaya, A. Samoylov, E. Zikiy, _et al._ , Applied Physics Letters 121 (2022).
  * Roy  _et al._ (2015) T. Roy, S. Kundu, M. Chand, A. Vadiraj, A. Ranadive, N. Nehra, M. P. Patankar, J. Aumentado, A. Clerk, and R. Vijay, Applied Physics Letters 107 (2015).
  * Grebel  _et al._ (2021) J. Grebel, A. Bienfait, É. Dumur, H.-S. Chang, M.-H. Chou, C. Conner, G. Peairs, R. Povey, Y. Zhong, and A. Cleland, Applied Physics Letters 118 (2021).
  * Lu  _et al._ (2022) Y. Lu, W. Xu, Q. Zuo, J. Pan, X. Wei, J. Jiang, Z. Li, K. Zhang, T. Guo, S. Wang, _et al._ , Applied Physics Letters 120 (2022).
  * Ranzani  _et al._ (2022) L. Ranzani, G. Ribeill, B. Hassick, and K. C. Fong, in  _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (IEEE, 2022) pp. 314–319.
  * Simons (2004) R. N. Simons,  _Coplanar waveguide circuits, components, and systems_ (John Wiley & Sons, 2004).
  * Eddins (2017) A. W. Eddins,  _Superconducting Circuits for Quantum Metrology with Nonclassical Light_ (University of California, Berkeley, 2017).
  * Boutin  _et al._ (2017) S. Boutin, D. M. Toyli, A. V. Venkatramani, A. W. Eddins, I. Siddiqi, and A. Blais, Physical Review Applied 8, 054030 (2017).
  * Aumentado (2020) J. Aumentado, IEEE Microwave magazine 21, 45 (2020).
  * Gambetta  _et al._ (2006) J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Physical Review A 74, 042318 (2006).
  * Kindel  _et al._ (2016) W. F. Kindel, M. Schroer, and K. Lehnert, Physical Review A 93, 033817 (2016).
  * White  _et al._ (2023) T. White, A. Opremcak, G. Sterling, A. Korotkov, D. Sank, R. Acharya, M. Ansmann, F. Arute, K. Arya, J. C. Bardin, _et al._ , Applied Physics Letters 122 (2023).
  * Pozar (2011) D. M. Pozar,  _Microwave engineering_ (John wiley & sons, 2011).
  * Dolan (1977) G. Dolan, Applied Physics Letters 31, 337 (1977).
  * Potts  _et al._ (2001) A. Potts, G. Parker, J. Baumberg, and P. De Groot, IEE Proceedings-Science, Measurement and Technology 148, 225 (2001).

Supplementary Materials

##  I Impedance Transformer Parameters

To construct the impedance transformer, we tailor a CPW with its gap fixed at
3 $\mathrm{\mu m}$ and its center trace width tapered from 7.8 $\mathrm{\mu
m}$ to 218 $\mathrm{\mu m}$, forming a horn-like shape. The characteristic
impedance along this horn-like CPW line follows the Klopfenstein taper profile
Pozar (2011), which has a high-pass feature that suppresses reflection above a
cutoff frequency. The impedance transformer here is designed to have a maximum
reflection of -10 dB above 2 GHz. We note that it can be shortened by half if
one increases the cutoff frequency to 4 GHz, which further reduces the
complexity and footprint size of the CIMPA. The Klopfenstein profile is
calculated using MATLAB, and the CPW characteristic impedance is calculated
using TX-Line, a convenient industrial-level transmission line calculator
Simons (2004). To verify the property of the impedance transformer, we
simulate the transmission and reflection coefficients (S21 and S11,
respectively) of the horn-like CPW line with ANSYS HFSS. The result in Fig. S1
shows low reflection (below -10 dB) and high transmission (insertion loss
below 0.3 dB) above the intended 2-GHz cutoff frequency.

Fig. S1: S-matrix simulation result of the horn-like CPW line.

##  II Device fabrication

The device is fabricated on an intrinsic high-resistivity Si wafer with
resistivity $\rho>$ 10 $\mathrm{k\Omega}\text{-}\mathrm{cm}$. After sputtering
a 100-nm-Al film, the horn-like CPW’s features are defined using photo-
lithography and reactive-ion etching. Notably, the fabrication process is
compatible with the photo-lithography technique thanks to the 3-$\mathrm{\mu
m}$ gap. Then, the capacitor is fabricated using the lift-off e-beam
lithography technique followed by e-beam evaporation of aluminum oxide and
aluminum electrodes. We note that the size of the capacitor is approximately
39$\times$39 $\mathrm{\mu m}^{2}$, which can also be made using photo-
lithography. The SQUID structure is defined using e-beam lithography and
double-angle e-beam evaporation of aluminum films following the standard Dolan
bridge technique Dolan (1977). The galvanic contact between the SQUID and the
capacitor electrodes is formed using the Argon ion-milling band-aid process
Potts  _et al._ (2001). Finally, a layer of methyl methacrylate (MMA) resist
is used to protect the wafer before dicing. The chips are diced and then
cleaned using N-methylpyrrolidone (NMP) before being packaged into a copper
box.

##  III Device measurement setup and calibration

The CIMPA is integrated into a transmon qubit measurement setup and
characterized inside a dilution fridge. The detailed fridge wiring diagram is
shown in Fig. S2.

Fig. S2: The wiring diagram of the fridge

A Rohde & Schwarz SGT100A vector RF source (Control) is used to generate qubit
control pulses. An Agilent Technologies E8257D signal generator (RO) generates
the readout signal at the readout resonator frequency. By mixing the readout
signal with a Hewlett Packard 8671B local oscillator (LO), the readout signal
information is converted to an intermediate frequency of 50 MHz for the
AlazarTech digitizer (Alazar) to record the signal. The readout signal and
control signal are pulse-controlled using HMC-C058 SPDT switches. A Tektronix
5014C arbitrary waveform generator generates the pulse sequences to control
the qubit, readout the resonator signals, and trigger the Alazar. The DC bias
current to the CIMPA is provided by a Yokogawa GS200 DC voltage/current
source, and the AC pump is supplied by a Hewlett Packard 83732B signal
generator. The CIMPA gain and noise performance are measured using an Agilent
Technologies N9010A signal analyzer (SA) and an Agilent Technologies N5230C
network analyzer (VNA).

Fig. S3: The calibration of the measurement setup. (a) Fitting of the readout
resonator. (b) Fitting of the ac Stark shift. (c) Fitting of the measurement-
induced dephasing. (d) Calibration between the power going into CIMPA and the
power at the top of the fridge

To precisely measure the input power of the amplifier, we need to calibrate
the attenuation along the input line and convert the power sent out of the VNA
to the power fed into CIMPA. According to the method mentioned in Ref. Macklin
_et al._ (2015), we can calibrate the input signal power by extracting the
intra-cavity photon number occupation $\bar{n}$ inside the readout resonator.
The power of the signal leaving the qubit and feeding into CIMPA is given by

$P=\hbar\omega_{r}\kappa_{r}\bar{n},$ (S1)

where $\omega_{r}$ and $\kappa_{r}$ are the angular frequency and linewidth of
the readout resonator, respectively. The resonator linewidth and frequencies
are extracted by directly fitting the resonator reflection spectrum, as shown
in Fig. S3(a).

The intra-cavity photon number is measured using the ac Stark shift and
measurement-induced dephasing of the qubit, which can be extracted by fitting
the Ramsey measurement. Considering a qubit dispersively coupled to a readout
resonator with a dispersive shift $\chi$, the ac Stark shift caused by the
photon population in the readout resonator is $\Delta_{ac}=2\chi\bar{n}$, and
the measurement-induced dephasing is given by
$\Gamma_{\phi}=8\chi^{2}\bar{n}/\kappa_{r}$Gambetta  _et al._ (2006).
Utilizing the quadratic dependence of the dephasing rate and the linear
dependence of the frequency shift on $\chi$, we can extract $\chi$ and
$\bar{n}$. As we vary the VNA power outside the fridge, we can extract the
intra-cavity photon number of the readout resonator by the Ramsey spectrum of
the qubit, and then calibrate the corresponding signal power at the CIMPA. The
ac Stark shift and measurement-induced dephasing at various VNA powers are
shown in Fig. S3(b, c). The extracted correspondence between the signal sent
into CIMPA and the VNA power outside the fridge is shown in Fig. S3 (d). All
extracted parameters are listed in Table 1.

Parameter |  $\omega_{q}/(2\pi)$ (GHz) |  $\omega_{r}/(2\pi)$ (GHz) |  $\kappa_{r}/(2\pi)$ (kHz) |  $T_{1}$ ($\mathrm{\mu s}$) |  $T_{2,\text{echo}}$ ($\mathrm{\mu s}$)
---|---|---|---|---|---
Value |  5.350 |  6.633 |  309 |  45 |  16
Table 1: Qubit and resonator parameters

##  IV Derivation of CIMPA Hamiltonian and gain

The theoretical derivation in this section is based on Ref. Eddins (2017) The
gain of the amplifier originates from the 3-wave mixing process enabled by the
SQUID in the nonlinear LC resonator. If we DC bias the SQUID at a specific
flux point and pump it using a small AC flux tone
$\Phi_{e}=\frac{\Phi_{0}}{\pi}(F+\delta f\cos(\omega_{p}t))$, we can write
down its Hamiltonian as

$\mathcal{H}=4E_{c}n^{2}-E_{\text{J,eff}}\left|\cos(F+\delta
f\cos(\omega_{p}t))\right|\cos(\phi),$ (S2)

where $E_{c}=e^{2}/(2C)$ is the single electron capacitive energy,
$E_{\text{J,eff}}=\Phi_{0}^{2}/(4\pi^{2}L_{\text{J,eff}})$ is the effective
Josephson energy of the SQUID, $\Phi_{0}$ is the magnetic flux quantum, $n$
and $\phi$ are two conjugate variables of the circuit: Cooper-pair number on
the capacitor and gauge-invariant phase across the SQUID, respectively. After
the quantization of the Hamiltonian, one can write down the Hamiltonian in the
rotating frame with the bosonic modes $\hat{a}$ of the CIMPA and apply
rotating wave approximation,

$\hat{\mathcal{H}}/\hbar=\Delta\hat{a}^{\dagger}\hat{a}+\lambda(\hat{a}^{\dagger
2}+\hat{a}^{2})+\hat{O}(\hat{a}^{4}),$ (S3)

where $\lambda=\delta
fE_{\text{J,eff}}\sqrt{\sin(F)\tan(F)E_{c}/(8E_{\text{J,eff}})}$ originates
from flux modulation, $\Delta=\omega_{p}/2-\omega_{0}$ is the detuning between
the half pump frequency and the CIMPA resonance frequency, which will be
assumed to be zero in the following discussion. Now we can use the input-
output theory to write down the equation of motion for the CIMPA coupled to a
transmission line with decay rate $\kappa$,

$\dot{\hat{a}}=\frac{i}{\hbar}[\hat{\mathcal{H}},\hat{a}]-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa}\hat{a}_{\text{in}}=-2i\lambda\hat{a}^{\dagger}-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa}\hat{a}_{\text{in}}.$
(S4)

We can solve this equation of motion in the frequency domain by defining
$\bar{a}[\omega]=\int_{-\infty}^{\infty}\text{d}t\ e^{i\omega t}\hat{a}(t)$ as
the frequency-domain bosonic annihilation operator, as shown in Eq. S5. One
should notice here that $\omega$ is the detuning from the CIMPA resonance
frequency because we work in the rotating frame,

$\bar{a}[\omega]=\sqrt{\kappa}\left(\frac{(\kappa/2-i\omega)\bar{a}_{\text{in
}}[\omega]-2i\lambda\bar{a}_{\text{in
}}^{\dagger}[-\omega]}{|\kappa/2-i\omega|^{2}-4|\lambda|^{2}}\right).$ (S5)

Finally, with the input-output boundary condition
$\sqrt{\kappa}\hat{a}=\hat{a}_{\text{in}}+\hat{a}_{\text{out}}$, we can obtain
the relation between the input signal tone $\hat{a}_{\text{in}}$ and output
signal tone $\hat{a}_{\text{out}}$,

$\begin{split}\bar{a}_{\text{out
}}[\omega]=\left(\kappa\frac{\kappa/2-i\omega}{|\kappa/2-i\omega|^{2}-4|\lambda|^{2}}-1\right)\bar{a}_{\text{in
}}[\omega]\\\
+\left(\frac{-2i\kappa\lambda}{|\kappa/2-i\omega|^{2}-4|\lambda|^{2}}\right)\bar{a}_{\text{in
}}^{\dagger}[-\omega],\end{split}$ (S6)

from which we can identify the gain $G(\omega)$ as the ratio between the power
of output and input signals with the same frequency,

$G(\omega)=\left|\kappa\frac{\kappa/2-i\omega}{|\kappa/2-i\omega|^{2}-4|\lambda|^{2}}-1\right|^{2}.$
(S7)

##  V Dependence on pump power and signal power

CIMPA offers gain near the instability point where the denominator of the gain
formula shown in Eq. S7 approaches zero. Therefore, a small change in the
denominator will greatly affect the gain. There are usually three factors
influencing the denominator of the gain profile: pump frequency, pump power,
and signal power. The pump frequency and power appear in the denominator
formula, and the signal power introduces an ac Stark shift which effectively
detunes the signal and decreases the gain. With a normal industrial-level
signal generator, the precision of pump frequency can be as high as 1 Hz, but
the pump power and signal power can fluctuate more. Therefore, it is worth
examining the sensitivity of CIMPA gain performance with respect to the pump
power and signal power. Figure S4(a) shows the gain profile at different
signal frequencies while sweeping the power of the pump signal. The pump
frequency is fixed at 13.41 GHz and the device is DC biased at
$\Phi_{e}=0.3\Phi_{0}$. We observe reasonable gain over around 1 dBm range of
the pump power.

The sensitivity of the gain profile with respect to the signal power indicates
the saturation effect. The gain profile is expected to be almost unchanged at
small signal power to offer a linear gain and decrease at large signal power
due to saturation. Figure S4(b) shows the gain of a signal at 6.665 GHz as we
increase the signal power. The device is operated with 13.41-GHz pump
frequency and $\Phi_{e}=0.3\Phi_{0}$ DC flux bias point. We observe an almost
constant signal gain below the 1-dB compression point of -111.5 dBm, and a
fast drop in gain after this value. The saturation input power is similarly
extracted for different operation parameters.

Fig. S4: Gain performance with different (a) pump power and (b) signal power.