# Level attraction and idler resonance in a strongly driven Josephson cavity

F. Fani Sani<EMAIL_ADDRESS>Kavli institute of Nanoscience,
Delft University of Technology, PO Box 5046, 2600 GA Delft  Institute for
Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
I. C. Rodrigues  Kavli institute of Nanoscience, Delft University of
Technology, PO Box 5046, 2600 GA Delft  D. Bothner  Kavli institute of
Nanoscience, Delft University of Technology, PO Box 5046, 2600 GA Delft
Physikalisches Institut and Center for Quantum Science in LISA+, Universität
Tübingen, 72076 Tübingen, Germany  G. A. Steele  Kavli institute of
Nanoscience, Delft University of Technology, PO Box 5046, 2600 GA Delft

###### Abstract

Nonlinear Josephson circuits play a crucial role in the growing landscape of
quantum information and technologies. The typical circuits studied in this
field consist of qubits, whose anharmonicity is much larger than their
linewidth, and also of parametric amplifiers, which are engineered with
linewidths of tens of MHz or more. The regime of small anharmonicity but also
narrow linewidth, corresponding to the dynamics of a high-$Q$ Duffing
oscillator, has not been extensively explored using Josephson cavities. Here,
we use two-tone spectroscopy to study the susceptibility of a strongly driven
high-$Q$ Josephson microwave cavity. Under blue-detuned driving, we observe a
shift of the cavity susceptibility, analogous to the AC Stark effect in atomic
physics. When applying a strong red-detuned drive, we observe the appearance
of an additional idler mode above the bifurcation threshold with net external
gain. Strong driving of the circuit leads to the appearance of two exceptional
points and a level attraction between the quasi-modes of the driven cavity.
Our results provide insights on the physics of driven nonlinear Josephson
resonators and form a starting point for exploring topological physics in
strongly-driven Kerr oscillators.

## Introduction

Circuit quantum electrodynamics (cQED) Blais et al. (2021), based on nonlinear
superconducting circuits, has laid the foundation for the current tremendous
growth of the superconducting quantum technology field. Microwave circuits
inheriting a large anharmonicity from the nonlinear nature of Josephson
junctions have been used to engineer artificial atoms Krantz et al. (2019),
for Fock and cat state generation Hofheinz et al. (2008); Kirchmair et al.
(2013); Leghtas et al. (2015), for synthesizing arbitrary quantum states of
light Hofheinz et al. (2009), to demonstrate the lasing effect from a
superconducting qubit Astafiev et al. (2007) and for engineering single photon
sources Zhou et al. (2020). Josephson circuits where the nonlinearity does not
exceed the cavity decay rate have been equally successful. These have been
utilized as a platform to investigate the dynamical Casimir effect Wilson et
al. (2011); Lähteenmäki et al. (2013), as systems to perform dispersive
readout of superconducting qubits Lin et al. (2014); Krantz et al. (2016) and
have been recently used for the realization flux-mediated optomechanical
systems Rodrigues et al. (2019); Zoepfl et al. (2020); Schmidt et al. (2020a);
Bothner et al. (2021a) and photon-pressure systems Bothner et al. (2021b);
Eichler and Petta (2018). Also in the context of small nonlinearities, a
common application is the engineering of Josephson Parametric Amplifiers
(JPAs) Yamamoto et al. (2008); Eom et al. (2012); Yaakobi et al. (2013); Zhou
et al. (2014); Castellanos-Beltran and Lehnert (2007); Winkel et al. (2020);
Pogorzalek et al. (2017). Such systems have approached quantum-limited
amplification Bergeal et al. (2010); Macklin et al. (2015) and allowed for
squeezing of vacuum fluctuations Castellanos-Beltran et al. (2008); Zhong et
al. (2013).

Typically the nonlinear circuits utilized for the realization of JPAs are
designed to be highly overcoupled and with a large decay rate in order to
optimize their gain bandwidth ratio. However, there has been a growing
interest lately in strongly driven high-quality-factor (high-$Q$) Josephson
circuits. For example, they have recently been used to cool a nanobeam with a
blue-detuned pump in flux-mediated optomechanical systems Bothner et al.
(2021a), to study the quantum nature of nonlinear damping in microwave
circuits Gely et al. (2021) or for high-sensitivity current detection by
upconversion Schmidt et al. (2020b). Furthermore, circuits with a flux-tunable
nonlinearity have also been used to explore the crossover between a classical
Duffing oscillator and a Kerr parametric oscillator Yamaji et al. (2020);
Andersen et al. (2020).

Bringing a nonlinear oscillator far beyond its bifurcation point is a regime
which has been explored in the context of driven micromechanical resonators
Stambaugh and Chan (2006), optical systems Drummond and Walls (1980);
Khandekar et al. (2015), and has been recently used for the detection of
squeezing of a mechanical mode Huber et al. (2020); Ochs et al. (2021). Past
its bifurcation threshold the system enters a regime of bistability where two
stable solutions exist Dykman and Krivoglaz (2021); Marthaler and Dykman
(2006). In superconducting circuits the switching between these bistable
branches has been explored close to bifurcation for the engineering of the
Josephson bifurcation amplifier Siddiqi et al. (2004); Vijay et al. (2009), a
technology which was later on utilized for fast measurement and single-shot
read-out of superconducting qubits Mallet et al. (2009). Going further beyond
the bifurcation limit, however, is a regime which is typically unexplored
within JPAs as this suppresses the maximum achievable gain of an amplifier
Planat et al. (2019). Nevertheless, superconducting circuits could be a
powerful platform to explore this regime as they grant a large design
flexibility and can be operated in well controlled environments.

Here we explore the physics behind a weakly nonlinear high-$Q$ Josephson
circuit which is driven far beyond its bifurcation point. We study the two-
tone response of the resonator for a pump tone placed above or below the
cavity resonance and investigate the modification to its driven susceptibility
by extracting the pump-modified resonance frequency and decay rates seen by
the probe. While a blue detuned pump gives rise to the well known AC stark
shift Delone and Krainov (1999), a red detuned pump will bring the system
above its bifurcation threshold and place the oscillator in a bistable
operation regime. In addition to a strong modification of the cavity mode
susceptibility, which we later define as signal mode, a second idler mode
exhibiting a net output gain will emerge upon probing the system with a weak
tone. After investigating the characteristics of these two new modes, we focus
on the dependence of the driven cavity state on intracavity photon number and
uncover the existence of level attraction between the signal and idler modes.
Our results provide new insights on the understanding of strongly driven
high-$Q$ Kerr oscillators and reveal the potential for operating these systems
well above their bifurcation threshold.

## Results

The device we study here consists of a microwave Josephson cavity capacitively
coupled to a coplanar waveguide feedline by means of a coupling capacitor
$C_{\textrm{c}}$, and containing a Superconducting QUantum Interference Device
(SQUID). Optical images of the full device and a zoom-in of the
$10~{}\mu\textrm{m}\times 10~{}\mu\textrm{m}$ SQUID including its two
nanobridge Josephson junctions are shown in Figs. 1a and 1b, respectively.

Figure 1: A lumped element Josephson cavity. Panel a shows a micrograph of the
full device and corresponding labeling of the main circuit elements. The
coupling capacitor is $C_{\textrm{c}}=2$ fF, each of the interdigitated
capacitors (IDC) is given by $C=0.5$ pF and the linear inductance contribution
of the circuit by two wires that link the IDCs and the SQUID as $L/2=742$ pH.
The SQUID is visible at the top part of the image, and a zoom-in in which the
two nanobridge Josephson junctions (JJ) are shown, is presented in b. Panel c
shows a schematic representation of the circuit model where
$L_{\textrm{J}}=20$ pH is the SQUID inductance. In panel d the measured
reflection magnitude $|S_{\textrm{11}}|$ and its corresponding fit are shown
as red circles and black line, respectively. From the fit parameters we
extract a resonance frequency of $\omega_{0}=2\pi\cdot 5.8157$ GHz, a total
decay rate $\kappa=2\pi\cdot 157$ kHz and an external decay rate
$\kappa_{\textrm{e}}=2\pi\cdot 12.5$ kHz.

When an external magnetic field is applied to the device, the Josephson
inductance of the SQUID and therefore the cavity resonance frequency can be
tuned. In the absence of an applied magnetic flux in the loop, i.e. at the
SQUID cavity sweet-spot, the circuit model of the device can be simplified to
a LC resonator, where most of its capacitance $2C$ comes from two symmetric
interdigitated capacitors (IDCs) and its total inductance
$\frac{L}{2}+L_{\textrm{J}}$ arises from a combination of two inductor wires
of inductance $L$ and the sweet-spot Josephson inductance $L_{\textrm{J}}$
representing the SQUID. In the work reported here the cavity will always be
operated at its sweet-spot. A simplified schematic of the circuit is shown in
Fig. 1c, once the device is mounted in a dilution refrigerator with a base
temperature of approximately $T_{\textrm{b}}\approx 15\,\textrm{mK}$.

When probing the system with a weak tone, we observe an absorption dip in the
reflected signal, as shown by the cavity response $|S_{11}|$ in Fig. 1d. From
a fit we extract a resonance frequency $\omega_{0}=2\pi\cdot 5.8157$ GHz, a
total decay rate $\kappa=2\pi\cdot 157$ kHz, an external decay rate
$\kappa_{\textrm{e}}=2\pi\cdot 12.5$ kHz and an internal decay rate
$\kappa_{\textrm{i}}=2\pi\cdot 144.5$ kHz. Furthermore, the ratio between
internal and external decay rates $\kappa_{\textrm{e}}/\kappa_{\textrm{i}}\sim
0.1$ tells us that the power is mostly dissipated inside the cavity, what is
usually considered as an undercoupled detection scheme
($\kappa_{\textrm{e}}/\kappa_{\textrm{i}}<1$).

Figure 2: Observation of the AC stark shift and TLS losses of a Josephson
cavity using two-tone spectroscopy. In panel a the normalized cavity response
$|S_{\textrm{11}}|$ is shown in a colormap as a function of on-chip pump power
$P_{\textrm{p}}$ and probe detuning from the bare cavity resonance
$\Delta_{\textrm{pr}}=\omega_{\textrm{pr}}-\omega_{0}$, for a pump tone
sitting at the point indicated by the dashed black line with detuning
$\Delta_{\textrm{p}}/2\pi=80$ kHz. Here the pump power was increased in steps
of $0.1$ dBm. For each horizontal line of the colormap presented in a we fit
the response spectrum with Eq. (1) and extract parameters such as dressed
resonance frequency $\omega_{\textrm{s}}$, and internal and external
linewidths, $\kappa_{\textrm{i}}$ and $\kappa_{\textrm{e}}$. Additionally, we
estimate the intracavity photon number $n_{\textrm{c}}$ from a second fit of
the cavity response based on the theoretical model of a driven Kerr cavity
(see Supplementary Note 3). Panel b shows the extracted resonance frequency
shift $\delta\omega_{\textrm{0}}=\omega_{\textrm{s}}-\omega_{0}$ versus
$n_{\textrm{c}}$. Together with the data we plot the theoretically predicted
resonance frequency shift (black line) for an anharmonicity
$\mathcal{K}/2\pi=-2.5$ kHz, which arises from the nonlinear inductance of the
circuit. This is consistent with the measured pump line attenuation of $\sim
107$ dB. In panel c we plot the extracted decay rates as function of
$n_{\textrm{c}}$. While the external linewidth $\kappa_{\textrm{e}}$ (blue
circles) stays approximately constant over the presented range
($\kappa_{\textrm{e}}\approx 2\pi\cdot 13$ kHz), the internal loss rate
$\kappa_{\textrm{i}}$ (teal circles) shows a considerable decrease with photon
number, reaching a value of $2\pi\cdot 50$ kHz. Such behavior is well
described by the saturation of dielectric losses from two-level systems
(TLSs), as shown by the fit curve represented by the black line.

In Fig. 2 we present data exploring the cavity response under the excitation
from two microwave tones: a pump tone $\omega_{\textrm{p}}$ which strongly
drives the circuit and a weak probe tone $\omega_{\textrm{pr}}$ that has low
enough power to not drive, but only to measure the response of the device.
Fig. 2a shows a colormap of the reflection magnitude $|S_{\textrm{11}}|$
versus probe detuning $\Delta_{\textrm{pr}}=\omega_{\textrm{pr}}-\omega_{0}$
and on-chip pump power $P_{\textrm{p}}$ for a blue-detuned pump. The pump
frequency $\omega_{\textrm{p}}$ is chosen slightly above the cavity resonance
$\omega_{0}$ with detuning
$\Delta_{\textrm{p}}=\omega_{\textrm{p}}-\omega_{0}<\kappa$, as indicated by
the dashed black line.

The first and most obvious experimental observation is the gradual resonance
frequency shift towards lower values as the pump power is increased. The
origin of this shift emerges from the Josephson nonlinearity of the cavity:
increasing the pump power raises the intracavity photon number
$n_{\textrm{c}}$ and results in a dressed resonance frequency
$\omega_{\textrm{s}}$ that is subsequently detected by the weak probe signal,
analogously to the AC Stark shift in atomic physics Delone and Krainov (1999).
To quantify this, we fit the cavity response $S_{11}$ with the reflection
response function of a linear LC circuit

$S_{11}=1-\kappa_{\textrm{e}}\chi_{\textrm{c}}$ (1)

with the susceptibility
$\chi_{\textrm{c}}^{-1}(\omega)=\frac{\kappa}{2}+i(\omega-\omega_{s})$, and
extract the cavity parameters such as decay rate and resonance frequency. We
note that in this regime our system can still be treated as a linear cavity
with a modified resonance frequency. In Fig. 2b, we plot the extracted
resonance frequency shift between the dressed and the bare resonance frequency
$\delta\omega_{0}=\omega_{\textrm{s}}-\omega_{0}$ as a function of the
intracavity photon number $n_{\textrm{c}}$. The latter was estimated from a
fit using a theoretical model based on the linearized equation of motion of a
Kerr cavity (see Supplementary Note 3). The theoretically predicted frequency
shift for a resonator with a Kerr nonlinearity is plotted simultaneously with
the experimental data as black line and red circles, respectively. For the
theory line we use an anharmonicity value of $\mathcal{K}/2\pi=-2.5$ kHz,
which is consistent with the measured pump line attenuation. Based on this
value, we can model our SQUID cavity as a harmonic oscillator with a weak Kerr
nonlinearity $\mathcal{K}\ll\kappa$. Additionally, we can also analytically
estimate the device anharmonicity based on the circuit parameters
$\mathcal{K}=-\frac{e^{2}}{4C}\left(\frac{2L_{\textrm{J}}}{L+2L_{\textrm{J}}}\right)^{3}=-2\pi\cdot
380~{}$Hz, a considerably smaller value compared to the one extracted from the
measurements. An explanation for this discrepancy could be an asymmetry
between the individual Josephson junctions inductance.

In addition to the frequency shift, a closer look to Fig. 2a shows evidence of
the cavity resonance getting sharper and deeper as the pump power is
increased. This suggests that the internal loss rate of the cavity is
decreasing with increasing pump power. To investigate this, we plot the
extracted external $\kappa_{\textrm{e}}$ and internal $\kappa_{\textrm{i}}$
decay rates versus $n_{\textrm{c}}$ in Fig. 2c.

Figure 3: Detection of an idler mode with net output gain in a regime beyond
bifurcation. Panel a shows a colormap of the cavity response
$|S_{\textrm{11}}|$ as a function of on-chip pump power $P_{\textrm{p}}$ and
probe detuning from the bare resonance
$\Delta_{\textrm{pr}}=\omega_{\textrm{pr}}-\omega_{0}$. The pump tone is fixed
at a constant detuning from the bare resonance
$\Delta_{\textrm{p}}/2\pi=-1.16$ MHz as indicated by the black dashed line.
When the pump power is increased above a certain threshold
($P_{\textrm{p}}=-97$ dBm), there is an abrupt jump from a faint dip to a dip-
peak region. A linescan above the switching point ($P_{\textrm{p}}=-96$ dBm)
is shown in b and is indicated by the black arrow. Panel b displays a linescan
showing the dressed cavity mode and a prominent idler-resonance. Red circles
are data and the black line is a theory line used with Eq. (3). The large
amplitude of the pump tone, which is a single frequency point in the
experimental data is ignored in the fit routine. c A schematic diagram is
depicted for a probe tone at $\Delta_{\textrm{pr}}/2\pi\sim-0.5$ MHz,
indicated by the black arrow in b. The cavity density of states is shown as a
blue Lorentzian curve, while the probe and pump tones are represented by the
green solid and black dotted arrows, respectively. The idler
($\textrm{I}_{\textrm{SB}}$) and signal ($\textrm{S}_{\textrm{SB}}$)
sidebands, generated by four-wave mixing, are presented in red and blue
arrows. The cavity nonlinearity is responsible for this mixing mechanism and
when the signal or the idler sideband falls within the cavity linewidth, it
gives rise to parametric amplification.

Interestingly, we observe a strongly power dependent internal loss rate. Such
an observation is common in superconducting resonators Pappas et al. (2011)
and can be attributed to the saturation of dielectric losses due to two-level
systems (TLSs), which are known to be located at material interfaces and film
surfaces Woods et al. (2018). We fit the extracted internal decay rate
$\kappa_{\textrm{i}}$ (teal circles) with a TLS model
$\kappa_{\textrm{i}}=\kappa_{0}+\frac{\kappa_{1}}{\sqrt{1+{n_{\textrm{c}}}/{n_{\textrm{crit}}}}}$,
shown as dark gray line in Fig. 2c. The model describes well the experimental
data for $\kappa_{1}=2\pi\cdot 175$ kHz, $n_{\textrm{crit}}=1$ and the
saturation value $\kappa_{0}=2\pi\cdot 40$ kHz, showing the significant
increase of the cavity internal quality factor when the system is driven by a
strong pump. The external cavity decay rate $\kappa_{\textrm{e}}$ (blue
circles) is found to be approximately constant for all pump powers with an
average value of $2\pi\cdot 13$ kHz (navy blue line).

An intriguing experimental setting is to investigate the cavity response when
a strong pump is placed red-detuned from the bare cavity resonance. The
outcome of this measurement is shown in Fig. 3a, where the reflection
magnitude $|S_{\textrm{11}}|$ is plotted as a function of probe detuning
$\Delta_{\textrm{pr}}$ and on-chip pump power $P_{\textrm{p}}$ for a red-
detuned pump with $\Delta_{\textrm{p}}/2\pi=-1.16$ MHz. Whereas for the case
of a blue-detuned pump the dressed cavity resonance shifts away from the drive
for increasing power, for the red-detuned case the dressed cavity resonance
moves towards the pump, which leads to a further increase in intracavity
photon number. When the driving power reaches a specific threshold, the cavity
hits its bifurcation point and the system subsequently enters a bistable
regime with two possible solutions in response to the pump field: a high-
amplitude and a low-amplitude branch, in which the currents oscillating in the
circuit at the pump frequency have either a low or high amplitude. Crossing
the bifurcation threshold allows the cavity to operate in either of these
regimes and to even punctually switch between them, as visible by the abrupt
jump of the cavity to the other side of the pump around $P_{\textrm{p}}=-97$
dBm (cf. Fig. 3a). This indicates that the cavity is afterwards operated on
the high-amplitude branch and from there onwards, the dressed cavity response
shifts similarly to the case of a blue-detuned pump. Note that although the
cavity is in a nonlinear bifurcated state in response to the strong pump, the
susceptibility seen by the weak probe is still that of a linear response, with
a Lorentzian lineshape that is downshifted in frequency compared to the
unpumped circuit.

The most peculiar outcome of crossing the cavity bifurcation point is the
appearance of an additional mode that has a net output gain at a frequency
symmetric to the dressed cavity resonance with respect to the drive. This is
clearly displayed in Fig. 3b, which shows a linescan of $|S_{\textrm{11}}|$
straight after the switching point, for an on-chip pump power of $-96$ dBm. To
understand the physics behind the appearance of this feature, we additionally
illustrate the intracavity fields in Fig. 3c, for the specific case of a probe
tone frequency indicated by the label (c) in Fig. 3b. Here, the pump and probe
intracavity fields, which are generated by external input signals, are shown
in black dotted and green solid arrows, respectively. In this driving
configuration, the nonlinearity of the cavity generates two additional
intracavity fields (red and blue arrows) via a four-wave mixing process. A
field with the same frequency as the input probe tone, commonly referred as
the signal sideband ($\textrm{S}_{\textrm{SB}}$) and the field arising
symmetrically on the other side of the pump tone, commonly referred to as the
idler sideband ($\textrm{I}_{\textrm{SB}}$) in the nomenclature of parametric
amplifiers. As the amplitude of the internally generated fields depends on the
driven Kerr cavity susceptibility

$\chi_{\mathrm{g}}=\frac{{\chi}_{\mathrm{p}}(0)}{1-\mathcal{K}^{2}n_{\mathrm{c}}^{2}{\chi}_{\mathrm{p}}(0){\chi}_{\mathrm{p}}^{*}(2\Omega_{\textrm{pp}})},$
(2)

with
${\chi}_{\mathrm{p}}^{-1}(\Omega)=\frac{\kappa}{2}+i\left(\Delta_{\textrm{pr}}-2\mathcal{K}n_{\mathrm{c}}+\Omega\right)$
and $\Omega_{\textrm{pp}}=\omega_{\textrm{p}}-\omega_{\textrm{pr}}$, when the
probe scans through the opposite side of the dressed cavity mode, it will also
detect an additional lineshape, which we refer to as idler-resonance. This
happens as the probe field constructively interferes with the signal sideband
generated by four-wave mixing. In fact, the appearance of an idler-resonance
should also happen below the bifurcation threshold. However, the low
intracavity photon number in this regime suppresses the signal sideband field
amplitude below the noise level, and therefore the mode remains experimentally
undetected.

Figure 4: Frequency locking of a Kerr cavity to a strong parametric drive. a
The cavity response $|S_{\textrm{11}}|$ as a function of pump detuning
$\Delta_{\textrm{p}}$ and probe detuning $\Delta_{\textrm{pr}}$ for a on-chip
pump power of $-127$ dBm. The frequency of the pump tone was reduced in steps
of $5$ kHz. Panel b shows a corresponding colormap of the cavity response
computed with Eq. (3). The device parameters were kept constant as extracted
from the analysis of Figs. 1 and 2: Line attenuation of $\sim 107$ dB,
$\kappa_{\textrm{e}}/2\pi\sim 13$ kHz, $\mathcal{K}/2\pi=-2.5$ kHz and the
power-dependent internal loss rate $\kappa_{\textrm{i}}$. In c we plot the
magnitude of the experimentally detected $|S_{11}|$ on resonance with the
signal (blue circles) and idler (red circles) modes, and in d we plot the
signal linewidth versus $\Delta_{\textrm{p}}$. The experimental values were
obtained by fitting the full spectrum with a reflection response function that
contains two individual modes in the region where the idler is detectable (see
Supplementary Note 2). The dotted lines present the analytically estimated
values for the high-amplitude branch (gray for signal and red for idler). The
dashed black line corresponds to the signal mode of the low-amplitude branch
and the full lines are used in the region where there is only a single
solution (gray for signal and red for idler). In panel c, the magnitude of
idler mode for the low-amplitude branch has been omitted for clarity. We point
out that we find a higher level of agreement between the experimental data and
the theory by reducing the parameter $\kappa_{\textrm{1}}/2\pi$ by $15$ kHz.

As evident from the cavity reflection coefficient greater than unity, the
signature of an idler resonance is accompanied by amplification of the intput
field. This is visible in the colormap shown in Fig. 3a and emphasized in the
linecut presented in Fig. 3b, where the black line shows the magnitude of the
theoretically predicted response

$S_{\textrm{11}}=1-\kappa_{\textrm{e}}\chi_{\mathrm{g}}.$ (3)

This amplification is common in JPAs, where it is typically occurring within
the cavity lineshape. In our case, however, as the cavity is highly
undercoupled, we do not observe output field amplification when the signal is
resonant with the cavity. Instead we see a deeper absorption dip. This occurs
as the intracavity field amplification effectively brings the system closer to
critically coupled, leading to a deeper dip in the response spectrum
$|S_{\textrm{11}}|$. Observing amplification of the output field in the signal
resonance condition would require a sufficiently large intracavity
amplification overcome all internal losses in the device (see Supplementary
Note 6 for an example of such data). This is different, however, for the case
of the idler resonant condition where we detect a net output gain, as show in
Fig. 3b. Here the input signal is not resonant with the cavity and therefore
the interference between the probe and amplified fields is not affected by the
cavity resonance. The frequency matching between the idler sideband and the
cavity resonance, however, provides a high density of states for the idler
photons, a necessary condition for parametric amplification, resulting in the
detection of output field amplification even when the input signal is detuned
by many linewidths outside of the cavity resonance.

Although until now we have only explored the impact of driving power on the
cavity response, we can also investigate the phenomenon of idler-resonance by
playing with another knob, which is sweeping the pump frequency for a fixed
power. In Fig. 4a the cavity response $|S_{\textrm{11}}|$ is shown as a
function of pump and probe detuning from the bare cavity resonance frequency.
The on-chip pump power is chosen to be sufficiently large
($P_{\textrm{p}}=-127$ dBm) to bring the cavity across its bifurcation point.

As we sweep the pump frequency from higher to lower values and approach the
bare cavity resonance, the system experiences a locking mechanism to the
drive. This is visible by the frequency dragging of the signal mode, i.e. the
dressed cavity resonance, with decreasing pump frequency. Moreover, as the
idler mode always emerges mirrored with respect to the drive tone, this one
will also experience a similar dragging as the dressed cavity. Once again,
while the cavity is identified by a dip in the spectrum (blue features in Fig.
4), the idler is visible as a peak above the background level (red features in
Fig. 4). Simultaneously with the locking, also the intracavity pump field
becomes larger as the drive comes nearer the bare cavity lineshape, and
continues to increase as the pump is detuned further down towards negative
values. The increase in drive photons arising from attempting to reduce the
detuning between the pump and the dressed cavity mode will enhance the signal
and idler sidebands, thereby intensifying the signature of both modes upon a
probe reflection measurement and achieving higher intracavity field
amplification.

Similarly to what was observed in Fig. 3, in Fig. 4a we also detect a sharp
modification of the cavity response spectrum for a pump detuning around
$\Delta_{\textrm{p}}/2\pi=-0.38$ MHz, which in this case reveals a transition
from the high-amplitude to the low-amplitude branch solution. Once the cavity
has switched branches and is no longer operated in a metastable regime, the
intracavity field amplification is reduced and the idler mode becomes
experimentally undetectable. We note that in contrast with the data presented
in Fig. 3, here we observe some fast switching between two branches around the
transition point which is most probably triggered by noise.

Fig. 4b shows a colormap of the response based on the analytical model
described in Supplementary Note 3, i.e. on Eq. (3). A clear difference between
the data and the theoretical model is the fact that the switching point is
predicted to occur for a pump detuning below the range of our measurement. In
fact, while in the experiment the cavity switches from a high-amplitude to a
low-amplitude branch, the theoretically predicted transition only happens when
the system reaches the regime of a single-solution. We note though, that the
branch where the system ends up on is highly sensitive to the exact history of
the system and to the noise in the device. Therefore, the experimental
transition point can happen at any frequency when the system is operated above
its bifurcation point. In spite of this somewhat expected mismatch, the
analytical model shows an excellent qualitative agreement with the experiment
as shown in Fig. 4b. Here the horizontal dashed gray line represent the
bifurcation point, which would also be the transition frequency that the
cavity would experience if it had been operated in the low-amplitude branch.

Figure 5: Level attraction between the signal and idler modes. Panel a shows
the response spectrum magnitude $|S_{\textrm{11}}|$ versus probe detuning
$\Delta_{\textrm{pr}}$ and on-chip pump power $P_{\textrm{p}}$. The dashed
horizontal line identifies the frequency of the pump, which is placed slightly
below the cavity bare resonance frequency with $\Delta_{\textrm{p}}/2\pi=-100$
kHz. For every linescan we perform a fit of the full response and extract the
signal and idler resonance frequencies. Simultaneously, for every linecut we
estimate the intracavity photon number $n_{\textrm{c}}$ based on a fit to the
normalized response using the full Kerr model (see Supplementary Note 3). In
panel b we plot the extracted signal ($\delta_{s}$) and idler ($\delta_{i}$)
resonance frequency differences from the pump frequency $\omega_{\textrm{p}}$
versus $n_{\textrm{c}}$ as blue and red circles, respectively. The black line
is a theory curve based on Eq. (4). We observe level attraction between the
two modes and the emergence of two exceptional points (EP). This pinpoints the
appearance of an instability region where the signal and idler overlap in
frequency. A regime which is characterized by the emergence of an additional
imaginary component in the complex resonant solutions
$\tilde{\omega}_{\textrm{i/s}}$, meaning that in this region the modes will
acquire different linewidths. Furthermore, we do not show any experimental
data in the instability region as the presence of the drive and the fast
switching of the cavity between the two sides enclosing the instability region
imposes several challenges in the fitting routine.

At last, we quantitatively investigate the agreement between theory and
experiment by extracting the magnitude of the response spectrum
$|S_{\textrm{11}}|$ on resonance with the signal and idler modes, and the
cavity decay rate. These are compared with the analytically estimated values
in Fig. 4c and d, respectively. While the magnitude of the signal mode
$|S_{\textrm{11}}(\omega_{\textrm{s}})|$ (full black line) is decreased as we
initially reduce the pump detuning, the idler magnitude
$|S_{\textrm{11}}(\omega_{\textrm{i}})|$ (full red line) is increased further
above the background, reaching a maximum output gain of $\sim 2$ dB. We note
that in this region the system still operates in a single solution regime. As
shown in Fig. 2, the total cavity linewidth should be dependent on the
intracavity photon number, thereby decreasing as we reach the bifurcation
point. Past this limit, two new branches emerge. From our theoretical
calculations we see that the cavity remains in the high-amplitude branch for
just a small detuning range after crossing its bifurcation threshold and
shortly after it switches to the low-amplitude branch where the idler mode is
afterwards undetectable. For more details on additional experimental data
where we achieve $\sim 8$ dB gain of output gain at the idler resonance see in
Supplementary Note 6.

The transition between the two amplitude branches of a driven nonlinear system
might appear abrupt and experimentally difficult to predict. Nevertheless,
carefully exploring this transition regime can unravel interesting phenomena.
For this purpose we once again fix the pump on the red side of the cavity, but
this time only slightly detuned from its bare resonance frequency with
$|\Delta_{\textrm{p}}|<\kappa$. After this the pump power is slowly increased
and the probe strength is kept low to avoid triggering the cavity to
prematurely switch. A colormap of the cavity response $|S_{\textrm{11}}|$
versus probe detuning $\Delta_{\textrm{pr}}$ and on-chip pump power
$P_{\textrm{p}}$ is shown in Fig. 5a. Whereas the cavity is initially
attracted to the drive tone, once the power reaches a certain threshold
($P_{\textrm{p}}\sim-129$ dBm) the signal mode jumps onto the other side of
the pump and the idler mode simultaneously emerges. By further increasing the
power, the modes are pushed further away from the pump and continue resonating
symmetrically with respect to the drive tone with complex resonant solutions

$\tilde{\omega}_{\textrm{i/s}}=\omega_{\mathrm{p}}+i\frac{\kappa}{2}\pm\sqrt{\left(\Delta_{\mathrm{p}}-\mathcal{K}n_{\mathrm{c}}\right)\left(\Delta_{\mathrm{p}}-3\mathcal{K}n_{\mathrm{c}}\right)},$
(4)

where the resonance frequency of the idler and signal modes are given by
$\omega_{\textrm{i/s}}=\textrm{Re}~{}(\tilde{\omega}_{\textrm{i/s}})$ and
their linewidths by
$\kappa_{\textrm{i/s}}=2\textrm{Im}~{}(\tilde{\omega}_{\textrm{i/s}})$.

We fit each of the linescans of Fig. 5a with a double-mode reflection response
function (see Supplementary Note 2) and for each of them extract the resonance
frequency of each of the modes. We note that due to the small amplitude of the
idler mode in the linescans below the jump, its experimental resonance
frequency remains unfortunately undetected. However for completeness, in this
region we plot the idler resonances as mirrors of the signal mode as one would
expect from Eq. (4). The frequency difference between the signal/idler modes
and the pump $\delta_{\textrm{s/i}}=\omega_{\textrm{s/i}}-\omega_{\textrm{p}}$
is plotted versus intracavity photon number in Fig. 5b. Besides clearly
depicting the frequency symmetry between the two modes, which had already been
evident in previously presented data, Fig. 5b also unveils a very interesting
regime where both modes overlap in frequency. This has been observed in other
systems Bernier et al. (2014); Eleuch and Rotter (2014); Seyranian et al.
(2005); Bernier et al. (2018); Xu et al. (2016) and it has been refereed to as
level attraction Bernier et al. (2018). The exceptional point (EP) where their
resonance frequencies meet marks the beginning of a instability regime in
which the resonant solutions $\tilde{\omega}_{\textrm{i/s}}$ will develop an
additional imaginary component. This leads to the modes acquiring an inverse
modification to their bare linewidth. In Supplementary Note 4 we show the
theoretical predictions for the change in linewidth of the modes in the level
attraction region. The appearance of such an exceptional point discloses the
high susceptibility of a Kerr cavity to noise inducing fluctuations in its
intracavity photon number, as in this regime the presence of noise can have a
strong impact on its resonance frequency. This is more critical when the
cavity is operated closer to the instability regime, as the bending near to
the exceptional point is more accentuated (see Supplementary Note 5 for more
details).

## Discussion

We have conducted an experimental study of a Josephson cavity operating in the
strong-driving limit and accurately modelled its two-tone response by means of
the linearized equation of motion of a Kerr cavity, including its power
dependent losses. Besides modelling the cavity dispersive shift and its TLS
losses, we have detected the emergence of an idler mode upon approaching and
crossing the bifurcation threshold of the system. We provide an intuitive
physical picture for the origin of this mode and present quantitative
predictions for its amplitude. The appearance of an idler resonance allows us
to amplify signals far outside of the cavity lineshape, with the advantage
that the interference between the probe and the amplified signals is not
affected by the cavity response. While most applications of parametric
amplifiers operate their systems below their bifurcation point, here we show
that for the case of a high-$Q$ resonator, operations above the bifurcation
threshold do not necessarily compromise the maximum achievable output gain.

Furthermore, in this work we have uncovered an interesting phenomenon where
the system hits an exceptional point and there is a level attraction between
the eigenfrequencies of the driven resonator. While this has been studied in
other systems, it has only now been observed between the signal and idler
modes of a weakly damped nonlinear circuit. In the context of Kerr
optomechanics, understanding the appearance of this level attraction could be
extremely valuable for engineering systems that maximize the driving power of
a red-sideband cooling tone. In addition to the expanded possibilities
regarding the manipulation of Josephson circuits beyond their bifurcation
threshold, our results ultimately allow for topological control of strongly
driven Kerr resonators by encircling its exceptional points upon the
implementation of a tunable two-dimensional parameter space Bernier et al.
(2018); Xu et al. (2016).

### Acknowledgements

This research was supported by the Netherlands Organisation for Scientific
Research (NWO) in the Innovational Research Incentives Scheme – VIDI, project
680-47-526. This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 681476 - QOMD) and from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No 732894
- HOT.

## Author contributions

F.F.S. conceived and conducted the experiments. D.B. and I.C.R. designed and
fabricated the device. F.F.S. and I.C.R. performed the data analysis and
prepared the first draft of the manuscript. D.B. formulated the theoretical
model. All authors discussed the data and the manuscript. G.A.S. supervised
the project.

### Competing interest

The authors declare no competing interests.

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## Supplementary Material

### Supplementary Note 1: Device and setup

The device studied here is the SQUID cavity belonging to the photon-pressure
system studied in Ref. Bothner et al. (2021b). The cavity was patterned by
means of Electron Beam Lithography (EBL) and subsequently loaded into a
sputtering machine where a 20 nm layer of Aluminum was deposited. After the
deposition, the chip was placed in the bottom of a beaker containing a small
amount of anisole and inserted in a ultrasonic bath for a few minutes. After
this fabrication process, we proceeded with the patterning of the remaining
structures of the photon-pressure system. A step-by-step description of the
fabrication is given in the Supplementary Material of Ref. Bothner et al.
(2021b).

All the experiments reported here were performed in a dilution refrigerator
operating at a base temperature close to $T_{\textrm{b}}=15$ mK. The printed
circuit board (PCB), onto which the sample was glued and wirebonded, was
placed in a radiation tight copper housing and connected to a coaxial line
used as input/output port. As the device was measured in a reflection
geometry, the input and output signals were split via a directional coupler on
the $15$ mK stage. The output signal subsequently went into a cryogenic HEMT
(High Electron Mobility Transistor) amplifier operating between 4-8 GHz.

Outside of the fridge we used two different setup configurations. For the
single-tone measurements we used a Vector Network Analyzer to characterize the
reflection parameter $S_{11}$. For the two-tone spectroscopy we in addition
utilized a signal generator as pump tone. The tone subsequently went through a
high pass filter and was afterwards combined with the probe signal coming from
the VNA by means of a power combiner. Note that the data presented in
Supplementary Note 6 was taken in an earlier cooldown and there the pump tone
was combined with the probe by means of a room-temperature directional
coupler, where the probe was attached to the coupled port and therefore
attenuated by 10 dB.

### Supplementary Note 2: Fitting routine

### A. Single-mode response spectrum $S_{11}$

The $S_{11}$ response function of a parallel LC circuit capacitively coupled
to a transmission line in a reflection geometry is given by

$S_{11}=1-\frac{2\kappa_{\mathrm{e}}}{\kappa_{\mathrm{i}}+\kappa_{\mathrm{e}}+2i\Delta}$
(S1)

with detuning from the resonance frequency

$\displaystyle\Delta=\omega-\omega_{0}.$ (S2)

We used this expression to model the single-mode linescans of the data
presented in main paper.

### B. Double-mode response spectrum $S_{11}$

When the cavity is driven close/beyond its bifurcation point we witness the
appearance of an idler mode. In this regime the modes resonance frequencies
can be extracted by fitting the full response spectrum with a double resonance
response function given by

$S_{11}=1-\frac{2\kappa_{\mathrm{e}}^{\textrm{s}}}{\kappa+2i\Delta_{\textrm{s}}}-\frac{2\kappa_{\mathrm{e}}^{\textrm{i}}}{\kappa+2i\Delta_{\textrm{i}}}$
(S3)

Where $\kappa_{\mathrm{e}}^{s},\kappa_{\mathrm{e}}^{i}$ are the effective
external decay rates of each mode, and where the detuning from the signal and
idler resonance frequencies are given by

$\displaystyle\Delta_{\textrm{s}}=\omega-\omega_{\textrm{s}}$ (S4)
$\displaystyle\Delta_{\textrm{i}}=\omega-\omega_{\textrm{i}}$ (S5)

### C. Real response spectrum $S_{11}$ and fitting routine

When analyzing the measured cavity response, we consider a frequency-dependent
complex-valued reflection background with amplitude and phase modulations
originating from a variety of microwave components in our input and output
lines and possible interfering signal paths

$\displaystyle
S_{11}^{\textrm{back}}=(\alpha_{0}+\alpha_{1}\omega)e^{i(\beta_{1}\omega+\beta_{0})}$
(S6)

Taking into account the impact of this background on the ideal response
functions, the measured response spectrum is given by

$\displaystyle
S_{11}^{\textrm{meas}}=(\alpha_{0}+\alpha_{1}\omega)S_{11}e^{i(\beta_{1}\omega+\beta_{0})}$
(S7)

where we include an additional rotation of the resonance circle with the phase
factor $e^{i\theta}$. The first step in the fitting routine removes the cavity
resonance part from the data curve and fits the remaining background with Eq.
(S6). After removing the background contribution from the full dataset by
complex division, the resonator response is fitted using the ideal response
function Eq. (S1) for the single mode data. In the final step, the full
function is re-fitted to the bare data using as starting parameters the
individually obtained fit numbers from the first two steps. From this final
fit, we extract the final background fit parameters and remove the background
of the full dataset by complex division.

We note that for the data where both signal and idler modes are present, we
calculate the background based on a fit from a single mode linescan and
afterwards divide it off from the data. Posterior to this procedure, the
double-mode normalized data is fitted with Eq. (S3) for the extraction of the
resonance frequencies and with Eq. (3) for the extraction of the intracavity
photon number. For the data presented in Fig. 4, as we do not have access to
the phase information of the signal due to the measurement scheme used for the
acquisition of the data, we exceptionally resorted to a magnitude fitting
routine where the background offset was included in the fit model.

### Supplementary Note 3: Theory of a driven Josephson cavity

###  S0.1 A. Equation of motion

We model the classical intracavity field $\alpha$ of the Josephson cavity
using the equation of motion

$\dot{\alpha}=\left[i(\omega_{0}+\mathcal{K}|\alpha|^{2})-\frac{\kappa}{2}\right]\alpha+i\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{in}}$
(S8)

where $\omega_{0}$ is the bare cavity resonance frequency, $\mathcal{K}$ is
the Kerr nonlinearity, $\kappa$ is the total linewidth, $\kappa_{\mathrm{e}}$
is the external linewidth and $S_{\mathrm{in}}$ is the input field. The
intracavity field is normalized such that $|\alpha|^{2}=n_{\mathrm{c}}$
corresponds to the intracavity photon number $n_{\mathrm{c}}$ and
$|S_{\mathrm{in}}|^{2}$ to the input photon flux (photons per second).

### B. Single-tone response

With a single tone drive field
$S_{\mathrm{in}}=S_{\mathrm{p}}e^{i(\omega_{\mathrm{p}}t+\phi_{\mathrm{p}})}$
and the Ansatz $\alpha_{\mathrm{p}}e^{i\omega_{\mathrm{p}}t}$, where
$S_{\mathrm{p}}$ and $\alpha_{\mathrm{p}}$ are chosen to be real-valued, we
get

$\left[\frac{\kappa}{2}+i\Delta_{\mathrm{d}}\right]\alpha_{\mathrm{p}}-i\mathcal{K}\alpha_{\mathrm{d}}^{3}=i\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{p}}e^{i\phi_{\mathrm{p}}}$
(S9)

with $\Delta_{\mathrm{p}}=\omega_{\mathrm{p}}-\omega_{0}$ the detuning between
the drive and the undriven cavity resonance frequency. From this, by
multiplication with its complex conjugate, we obtain a third order polynomial
for the intracavity photon number $n_{\mathrm{c}}=\alpha_{\mathrm{p}}^{2}$,
which is given by

$\mathcal{K}^{2}n_{\mathrm{c}}^{3}-2\mathcal{K}\Delta_{\mathrm{p}}n_{\mathrm{c}}^{2}+\left(\Delta_{\mathrm{p}}^{2}+\frac{\kappa^{2}}{4}\right)n_{\mathrm{c}}-\kappa_{\mathrm{e}}S_{\mathrm{p}}^{2}=0.$
(S10)

To obtain the full, complex intracavity field with respect to the drive field,
we also need the phase $\phi_{\mathrm{p}}$, which is given by

$\phi_{\mathrm{p}}=\operatorname{atan2}2\left(-\frac{\kappa}{2},\Delta_{\mathrm{p}}-\mathcal{K}n_{\mathrm{c}}\right).$
(S11)

The intracavity field is then given by
$\alpha=\sqrt{n_{\mathrm{c}}}e^{-i\phi_{\mathrm{p}}}$ and the cavity response
spectrum by
$S_{11}=1+i\sqrt{\kappa_{\textrm{e}}}\frac{\alpha}{S_{\textrm{p}}}$.

### C. Data on single-tone power dependence

Figure S1: Observation of the cavity Duffing response through single-tone
spectroscopy. The background corrected cavity response magnitude $|S_{11}|$ is
plotted for different values of on-chip probe power. The power used in the
first curve is $P_{\textrm{pr}}=-141$ dBm and this is increased in steps of
$2$ dB in the subsequent curves. Furthermore, each of the linescans is up-
shifted by $3$ dB for clarity. More details are given in the text.

In Fig. S1 we show the single-tone response of the Josephson cavity for
increased values of VNA probe power. While for low on-chip powers the cavity
exhibits a linear response, i.e. an absorption dip with a Lorentzian
lineshape, for higher powers ($P_{\textrm{pr}}\sim-137$ dBm) the cavity starts
bending towards lowers values and we detect a typical Duffing response where
there is an abrupt jump from the maximum cavity absorption amplitude to the
background level. In addition to the bending of the cavity lineshape, which
arises from the Kerr non-linearity, the response magnitude on resonance also
gets initially deeper. This effect displays the impact of two level system on
the cavity internal quality factor. Once the internal decay rate of the cavity
reaches its saturation value, the cavity anharmonicity is the only mechanism
altering its response and therefore the magnitude on resonance becomes more
shallow with power.

Once the cavity crosses its bifurcation point, Eq. (S10) will have three real
solutions, where two of them are stable (belonging to the high-amplitude and
low-amplitudes branches) and a third unstable solution. Since we scan the
probe from lower to higher frequencies, the cavity is mostly operated in the
low-amplitude branch. Although for the highest powers we curiously do observe
some switching between the branches, which is most probably triggered by noise
in the system.

### C. Two-tone response

If the Josephson cavity is driven by a strong drive field and a weaker second
input field at frequency $\omega_{\mathrm{pr}}$, we write for the total input
field

$S_{\mathrm{in}}=S_{\mathrm{p}}e^{i(\omega_{\mathrm{p}}t+\phi_{\mathrm{p}})}+S_{\mathrm{pr}}e^{i\omega_{\mathrm{pr}}t}$
(S12)

and as Ansatz for the intracavity field we choose

$\alpha=\alpha_{\mathrm{p}}e^{i\omega_{\mathrm{p}}t}+\gamma_{-}e^{i\omega_{\mathrm{pr}}t}+\gamma_{+}e^{i(2\omega_{\mathrm{p}}-\omega_{\mathrm{pr}})t}$
(S13)

with complex-valued amplitudes $\gamma_{-}$ and $\gamma_{+}$.

Inserting these into the equation of motion, going to the frame rotating with
the signal $\omega_{\mathrm{pr}}$, linearizing the solution by dropping all
terms not linear in $\gamma_{-},\gamma_{+}$ and sorting the equation by
frequency components yields three individual equations

$\displaystyle\left[\frac{\kappa}{2}+i(\Delta_{\mathrm{p}}-\mathcal{K}n_{\mathrm{c}})\right]\alpha_{\mathrm{p}}=i\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{p}}e^{i\phi_{\mathrm{p}}}$
(S14)
$\displaystyle\left[\frac{\kappa}{2}+i(\Delta_{\mathrm{pr}}-2\mathcal{K}n_{\mathrm{c}})\right]\gamma_{-}-i\mathcal{K}n_{\mathrm{c}}\gamma_{+}^{*}=i\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{pr}}$
(S15)
$\displaystyle\left[\frac{\kappa}{2}+i(\Delta_{\mathrm{pr}}-2\mathcal{K}n_{\mathrm{c}}+2\Omega_{\mathrm{pp}})\right]\gamma_{+}-i\mathcal{K}n_{\mathrm{c}}\gamma_{-}^{*}=0.$
(S16)

where $\Omega_{\mathrm{pp}}=\omega_{\mathrm{p}}-\omega_{\mathrm{pr}}$ and
$\Delta_{\mathrm{pr}}=\omega_{\mathrm{pr}}-\omega_{0}$. The first of these
equations is exactly the same as the one we obtained for single-tone driving.
With the procedure described in the previous section, the intracavity field
$\alpha_{\mathrm{p}}$, the intracavity photon number $n_{\mathrm{c}}$ and the
phase $\phi_{\mathrm{p}}$ can be determined. Having solved for
$n_{\mathrm{c}}$ allows then to solve also for $\gamma_{-}$ and $\gamma_{+}$.

We write the second and third equations as

$\displaystyle\frac{\gamma_{-}}{\chi_{\mathrm{p}}(0)}-i\mathcal{K}n_{\mathrm{c}}\gamma_{+}^{*}$
$\displaystyle=$ $\displaystyle i\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{pr}}$
(S17)
$\displaystyle\frac{\gamma_{+}}{\chi_{\mathrm{p}}(2\Omega_{\mathrm{pp}})}-i\mathcal{K}n_{\mathrm{c}}\gamma_{-}^{*}$
$\displaystyle=$ $\displaystyle 0$ (S18)

where we defined

$\chi_{\mathrm{p}}(\Omega)=\frac{1}{\frac{\kappa}{2}+i\left(\Delta_{\mathrm{pr}}-2\mathcal{K}n_{\mathrm{c}}+\Omega\right)}.$
(S19)

We solve for $\gamma_{+}$ and get by complex conjugation

$\gamma_{+}^{*}=-i\mathcal{K}n_{\mathrm{c}}\chi_{\mathrm{p}}^{*}(2\Omega_{\mathrm{pp}})\gamma_{-}$
(S20)

Inserting this into the equation for $\gamma_{-}$ gives

$\displaystyle\gamma_{-}$ $\displaystyle=$ $\displaystyle
i\frac{\chi_{\mathrm{p}}(0)}{1-\mathcal{K}^{2}n_{\mathrm{c}}^{2}\chi_{\mathrm{p}}(0)\chi_{\mathrm{p}}^{*}(2\Omega_{\mathrm{dp}})}\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{pr}}$
(S21) $\displaystyle=$ $\displaystyle
i\chi_{\mathrm{g}}(0)\sqrt{\kappa_{\mathrm{e}}}S_{\mathrm{pr}}$ (S22)

where in the last step we defined

$\chi_{\mathrm{g}}(\Omega)=\frac{\chi_{\mathrm{p}}(\Omega)}{1-\mathcal{K}^{2}n_{\mathrm{c}}^{2}\chi_{\mathrm{p}}(\Omega)\chi_{\mathrm{p}}^{*}(\Omega+2\Omega_{\mathrm{pp}})}.$
(S23)

#### Cavity response function

The reflection of the driven cavity for a single input probe tone is given by

$\displaystyle S_{11}$ $\displaystyle=$ $\displaystyle
1+i\sqrt{\kappa_{\mathrm{e}}}\frac{\gamma_{-}}{S_{\mathrm{pr}}}$ (S24)
$\displaystyle=$ $\displaystyle 1-\kappa_{\mathrm{e}}\chi_{\mathrm{g}}$ (S25)

#### Signal and idler modes

To find the probe tone resonances of the driven Kerr oscillator, we solve for
the complex frequencies $\tilde{\omega}_{\mathrm{p}}$, for which
$\chi_{\mathrm{g}}^{-1}=0$. This is equivalent to

$1-\mathcal{K}^{2}n_{\mathrm{c}}^{2}\chi_{\mathrm{p}}(0)\chi_{\mathrm{p}}^{*}(2\tilde{\Omega}_{\mathrm{pp}})=0$
(S26)

After multiplying out and sorting for terms with
$\tilde{\omega}_{\mathrm{p}}$, we can write down the two complex solutions as

$\displaystyle\tilde{\omega}_{\textrm{i/s}}=\omega_{\mathrm{p}}+i\frac{\kappa}{2}\pm\sqrt{\left(\Delta_{\mathrm{p}}-\mathcal{K}n_{\mathrm{c}}\right)\left(\Delta_{\mathrm{p}}-3\mathcal{K}n_{\mathrm{c}}\right)},$
(S27)

where the resonance frequency of the idler and signal modes are given by
$\omega_{\textrm{i/s}}=\textrm{Re}~{}(\tilde{\omega}_{\textrm{i/s}})$ and
their linewidths by
$\kappa_{\textrm{i/s}}=2\textrm{Im}~{}(\tilde{\omega}_{\textrm{i/s}})$.

### Supplementary Note 4: Level attraction

### A. Modification to the bare decay rates

As described in the main paper, upon entering the instability regime
characteristic of level attraction, i.e. the regime where the signal and idler
mode overlap in frequency, the modes also acquire different decay rates. This
is captured by the emergence of an additional imaginary component in the
complex resonant solutions given by Eq. (4).

Our major experimental resource to study this regime is the cavity response
$S_{11}$ and within the range enclosed by the two exceptional points we
observe that the system displays a single resonance dip in the spectrum. Even
though this resonance lineshape is modified due to the frequency overlap of
the two modes, as the total response of the system still remains a single
resonance dip, for a small difference between their decay rates the result of
a standard $S_{11}$ fit becomes inaccurate. Furthermore, as we the cavity and
idler modes simultaneously overlap with the pump tone, fitting the cavity
response becomes challenging. Nevertheless, by using the parameters resulting
from the fit of the data presented in Fig. 5 we can plot the theoretical
estimated values for the change in linewidth of the signal and idler modes.
Note that here we did not take into account the power dependent internal
losses but only the effects arising from the level attraction.

As shown in Fig. S2, outside of the instability regime, the signal and idler
modes should have identical decay rates. However, as the systems enters this
range enclosed by the two exceptional points, the idler (red line) should have
a reduction in total decay rate while the signal (blue line) should acquire a
larger linewidth. For the cavity parameters used in Fig. 5 we estimate that
the signal and idler modes should have a maximum modification to their bare
decay rates of $\sim 2\pi\cdot 120$ kHz.

Figure S2: Modification of the signal and idler decay rates within the regime
of level attraction. The red/blue lines describe the modification to the
idler/signal decay rates versus intracavity photon number. The black line
describes the region where their linewidths overlap. The lines are the result
of analytical calculations for the parameter regime of Fig. 5. Outside of the
region enclosed by the two exceptional points (EP) the signal and idler modes
display identical linewidths and distinct resonance frequencies. For the
instability region where the modes overlap in frequency, they also acquire
different linewidths. While the idler decay rate decreases, the signal decay
rate is enhanced. In the operation regime of Fig. 5 of the main paper, the
maximum modification to their bare linewidths is predicted to reach $\sim
2\pi\cdot 120$ kHz.

### Supplementary Note 5: Limitations of red-sideband driving in Kerr
optomechanics

Figure S3: Level attraction dependence on $\Delta_{\textrm{p}}$. In panel a we
plot the frequency difference between the cavity/idler mode and the pump, for
different values of bare pump detuning
$\Delta_{\textrm{p}}=\omega_{\textrm{p}}-\omega_{0}$. Here
$\Delta_{\textrm{p1}}=-2\pi\cdot 10$ MHz, $\Delta_{\textrm{p2}}=-2\pi\cdot 20$
MHz, $\Delta_{\textrm{p3}}=-2\pi\cdot 40$ MHz and
$\Delta_{\textrm{p4}}=-2\pi\cdot 60$ MHz. The horizontal red dashed line
corresponds to the point where the signal mode and the pump are detuned by one
mechanical frequency $\Omega_{\textrm{m}}=2\pi\cdot 2$ MHz. The black dots
represent the exceptional point for each pump detuning. The intracavity photon
number necessary to hit the red-sideband driving condition is
$n_{\textrm{c}}=1255$, $n_{\textrm{c}}=2626$, $n_{\textrm{c}}=5313$ and
$n_{\textrm{c}}=7985$ for $\Delta_{\textrm{p1}}$, $\Delta_{\textrm{p2}}$,
$\Delta_{\textrm{p3}}$ and $\Delta_{\textrm{p4}}$, respectively. In panel b we
show the pump photon responsivity of the signal mode versus normalized photon
number for each $\Delta_{\textrm{p}}$, where the normalization factor is the
intracavity photon number of their respective exceptional point
$n_{\textrm{EP}}$. As expected, the responsivity versus normalized photon
number is independent of $\Delta_{\textrm{p}}$. Details are given in text.

A question that arises upon being aware of the instability region
characteristic of the level attraction between the signal and the idler modes,
is how does this impacts a system which relies on a strong drive field with a
fixed frequency distance to the dressed mode. For example, in the field of
optomechanics a strong drive tone is crucial to cool mechanical degrees of
freedom far below their thermal bath temperatures. As the cooling rate
increases with intracavity photon number, the community strives for large pump
powers to enhance the multi-photon coupling rates. Furthermore, in cooling
protocols the detuning between the cavity and pump is not arbitrary and has to
accurately meet the mechanical resonance frequency $\Omega_{\textrm{m}}$, i.e.
$\Delta_{\textrm{p}}=\omega_{\textrm{p}}-\omega_{0}=-\Omega_{\textrm{m}}$. The
further away the pump is from the red-sideband frequency, the larger is the
reduction in the up-scattering rate of pump photons, thereby also reducing the
removal of thermal excitations from the mechanical mode.

In Kerr optomechanics this protocol becomes more challenging as the cavity
frequency will shift with power, thereby generating an undesired frequency
offset from the cavity red-sideband. An idea to counter this problem would be
to initiate the protocol with a pump tone further detuned from the cavity red-
sideband and afterwards increase its power, thereby maximizing the intracavity
photon number for the right detuning between the signal mode and the drive.

We explore this possibility in Fig. S3. Here the frequency difference between
the signal/idler mode and the pump is plotted for a variety of pump detunings.
Note that in the case of a Kerr cavity the pump detuning $\Delta_{\textrm{p}}$
refers to the frequency difference between the bare cavity resonance and the
pump tone. From a look to Fig. S3a it might appear that the right detuning
between the signal mode and the pump can be achieved for arbitrary values of
photon numbers, just by altering the bare detuning $\Delta_{\textrm{p}}$.
However this ideal scenario is disturbed by the appearance of the exceptional
point (EP) already described in Fig. 5. In fact, as we increase the detuning
$\Delta_{\textrm{p}}$, we need to operate the system closer and closer to this
exceptional point in order to preserve the red-sideband driving condition,
meaning that the signal mode responsivity to changes in the intracavity photon
number will be enhanced.

In Fig. S3b we plot the pump photon responsivity of the signal mode versus
normalized photon number. The normalization factor of each of the curve is
given by the intracavity photon number of their respective exceptional point,
as this one also depends on $\Delta_{\textrm{p}}$. Fig. S3b confirms that the
signal mode responsivity increases as we approach the exceptional point. Here,
the plotted stars correspond to the pump photon responsivity of the signal
mode for the four points described in Fig. S3a, i.e. where
$\omega_{\textrm{p}}-\omega_{\textrm{s}}=-\Omega_{\textrm{m}}$.

From this figure it becomes clear how the signal mode becomes more susceptible
to changes in the intracavity photon number in a cooling protocol using larger
values of $\Delta_{\textrm{p}}$. While for an initial detuning between the
bare cavity and the pump of $-10$ MHz one photon changes the signal frequency
by $10$ kHz, when $\Delta_{\textrm{p}}/2\pi=-60$ MHz one photon shifts the
signal frequency by $75$ kHz, a value in the order of magnitude as typical
decay rates of microwave cavities. Experimentally this can impose several
challenges on the cooling of mechanical modes, as the system becomes highly
susceptible to fluctuations of the cavity bare resonance frequency, which
subsequently leads to a modification of $\Delta_{\textrm{p}}$ and changes in
the intracavity photon number. For certain detunings, these fluctuations can
lead to frequency shifts on the order of a linewidth or even trigger the
system to switch between the two sides of the instability regime confined by
the two exceptional points (cf. Fig. 5).

Figure S4: Observation of net output gain in the signal and idler modes. Panel
a shows a colormap of the normalized cavity response spectrum $|S_{11}|$ (dB)
versus probe detuning $\Delta_{\textrm{pr}}$ and pump detuning
$\Delta_{\textrm{p}}$. The on-chip pump power used here was
$P_{\textrm{p}}=-147$ dBm. As the pump is swept from higher to lower values
the intracavity photon number is enhanced and also the amount of parametric
amplification in the device. Note that, contrarily to the data presented in
Fig. 4, here the cavity remains in the high-amplitude branch for the whole
range of detunings. In panel b we show a linescan corresponding to the lowest
detuning of panel a. Red points are data, the black line is a fit using
Eq.(3). The dashed vertial line represents the pump. Here both the signal and
idler modes appear as peaks that go above the background level, reveling a
regime where the effective internal decay rate of the system is been reduced
below zero.

### Supplementary Note 6: Maximizing the cavity output gain

Fig. S4 shows a similar dataset to the one of Fig. 4. Here the pump frequency
is once again swept from higher to lower values, in such way that it crosses
the bare cavity resonance frequency. Fig. S4a shows the normalized cavity
response spectrum $|S_{11}|$ versus probe detuning $\Delta_{\textrm{pr}}$ and
pump detuning $\Delta_{\textrm{p}}$. In contrast to the data presented in the
main paper, in this dataset the cavity remained operating in the high-
amplitude branch for the whole range of the measurement. The fact that the
cavity did not jump sooner to the low-amplitude branch comes from the fact
that we used a smaller probe power compared to Fig. 4. Furthermore, this data
was taken during an earlier cooldown where the device was mounted inside a
magnetic shield and possibly there was less noise in measurement setup that
could trigger the cavity to switching branches. As the system lingered in this
operation regime for a wide range of pump detunings, the intracavity photon
number strongly exceeded the maximum amount of drive photons of Fig. 4, which
in that case was achieved immediately prior to the cavity switching point. As
shown in Fig. S4, here not only the idler emerges as a peak above the
background, but also the signal mode has turned into a peak. This happens as
we reach enough parametric amplification to overcome the internal losses of
the cavity.

Fig. S4b shows a linescan corresponding to the lowest pump detuning
$\Delta_{\textrm{p}}$ in Fig. S4a. Here the signal maximum output gain is
approximately $3$ dB and the idler reaches a value as high as $8$ dB. This is
the largest output gain obtained with this device. This limit could in
principle be surpassed in future devices by engineering a higher bare external
decay rate.