# Strong backaction on a mechanical resonator by a few photons

Tanmoy Bera  Department of Physics, Indian Institute of Science,
Bangalore-560012 (India)  Vibhor Singh<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science, Bangalore-560012 (India)

###### Abstract

Cavity electromechanical systems, consisting of a mechanical resonator coupled
to an electromagnetic mode, are extensively used for sensing of various forces
and controlling the vibrations of a mechanical mode down to their quantum
limit. In the microwave domain, such devices based on magnetic-flux coupling
have emerged as a promising platform with the potential to reach a single-
photon strong coupling regime. Here, we demonstrate a flux-coupled
electromechanical device using a frequency tunable superconducting transmon
qubit, and a microwave cavity. By tuning the qubit in resonance with the
cavity, the hybridized state (dressed mode) of the qubit and the cavity mode
is used to achieve a magnetic field-dependent electromechanical coupling. It
is established by performing an electromagnetically-induced transparency
(EIT)-like experiment. At the largest applied field, we estimate the single-
photon coupling rate of 60 kHz. Further, in the presence of the pump signal,
we observe backaction, showing both cooling and heating of the mechanical
mode. With a stronger pump, the dressed mode shows the signature of “super-
splitting”, and a strong backaction on the mechanical resonator, reflected in
the broadening of the mechanical linewidth by a factor of 42 while using $<$ 1
photon in the dressed mode.

##  I Introduction

Light carries momentum, and it can be used to control and manipulate the
motion of a mechanical resonator down the quantum regime [1]. Such control
over the motional states is essential for the technological advancement as
well as to probe the fundamental physics [2, 3]. One of the leading platforms
for light-matter interaction is electromechanical devices, where an
electromagnetic(EM) mode is coupled to a mechanical resonator [1, 4]. In the
microwave domain, the vibrations of the mechanical resonators are integrated
into the EM mode using different strategies like the piezoelectric effect, and
the charge modulation. Such devices have been quite successful in
demonstrating the sideband cooling of the mechanical resonator to the ground
state [5], precise measurement beyond standard quantum limit [6], preparation
of non classical states [7, 8, 9, 10], entanglement between two mechanical
oscillators [11, 12], and the parametric ultra-strong coupling [13, 14].

Another fundamentally different way to couple the mechanical resonators to an
EM mode is to use magnetic flux modulation via Josephson inductance [15].
Apart from providing the electromechanical coupling, the nonlinear nature of
the Josephson inductance can be a valuable resource. Indeed, recent
experiments have shown large electromechanical coupling rates [16, 17, 18] and
the near ground state cooling of the mechanical resonator by four-wave-mixing
[19], and by Kerr-enhanced techniques [20]. Further, such flux-coupled
electromechanical devices have been proposed to reach the single-photon strong
coupling regime using its linear scaling with the magnetic field [21, 22, 23].
Thus far, most of these experiments have largely focused on the weak
participation of the Josephson inductance to the total inductance of the EM
mode. In such a limit, the dynamics can largely be understood by treating the
EM mode as a weak-Kerr oscillator, or by considering the Kerr quasi-modes [19,
20].

Here, we demonstrate an electromechanical device using a linear cavity, a
frequency tunable transmon qubit, and a mechanical resonator. The transmon
qubit is realized by dc-SQUID, shunted by a suitable capacitance, and the
mechanical resonator is implemented as a suspending arm of the SQUID loop
[17]. Tuning of the transmon qubit in resonance with the cavity results into
the formation of the new eigenstates due to the designed strong coupling. Both
the new eigenstates show the flux-tuning, and share the nonlinearity of the
Josephson inductance, referred as Jaynes-Cummings nonlinearity [24, 25]. In
the low dissipation limit and at higher powers, the dressed mode can get
further dressed with the drive photons and can exhibit multiple-peaks,
sometimes referred to as “super-splitting”, which at sufficiently large drive
powers converge to the bare linear cavity mode [26, 27]. Thus, at high powers,
the dynamics of the dressed mode is remarkably different than of a weak-Kerr
oscillator, which shows the frequency softening and bifurcation.

The experiment discussed here uses the flux-tunability and the non-trivial
high power response of the dressed mode and investigate its backaction on the
mechanical resonator. We observe interference feature in EIT-like measurement
scheme while having mean occupation of the dressed mode as low as 0.01 photons
and estimate single photon electromechanical rate up to 60 kHz. At high pump
powers, we observe the signature of super-splitting of the dressed mode, and a
strong back-action on the mechanical mode. Further, we conclusively show that
a model based on the Kerr-nonlinearity is not sufficient to explain the
experimental observations.

##  II Device concept

Figure 1: Device concept. (a) A schematic view of a quarter wavelength coaxial
cavity (in red) coupled to a transmon qubit (in yellow). The two capacitance
pads provide the shunting capacitance to the transmon as well as providing
coupling to the fundamental mode of the cavity. The qubit chip is placed such
that its dipole-moment is aligned to the axis of the cavity. The input and
output ports for the microwave measurements are also shown. The axial and
normal components of the magnetic-field are denoted by B∥ and B⟂,
respectively. (b) A false-color scanning electron microscope image of the
SQUID loop with embedded mechanical resonator. The two Josephson junctions and
part of the suspended beam can be seen. Thin Al film and the silicon substrate
are shown in yellow and gray colors, respectively. The selectively etched
region of silicon is shown by the cyan-colored region. The mechanical
resonator has dimensions of 40 $\mu$m$\times$200 nm$\times$28 nm. (c) The
energy-level diagram of the cavity, the transmon qubit, and the vacuum Rabi
split modes. The cavity and transmon frequencies are denoted by $\omega_{c}$
and $\omega_{q}$, respectively. In the resonant limit
$\omega_{q}\simeq\omega_{c}$, the transition frequencies of the vacuum Rabi
split modes are indicated by $\omega_{+}$ and $\omega_{-}$.

Our device consists of a frequency tunable transmon qubit, enabled by a SQUID
loop, and it couples to a coaxial cavity. Fig. 1(a) shows the schematic design
of the device. The coaxial cavity is placed inside a 2-axis vector magnet,
allowing us to control the axial and the normal components of the magnetic
field independently. The cavity is machined from oxygen-free high thermal
conductivity copper, and has a bare resonance frequency of 5.846 GHz. The
qubit is fabricated on an intrinsic silicon chip using electron beam
lithography(EBL), followed by a shadow evaporation of aluminium and in situ
oxidation. To achieve a larger critical in-plane magnetic field, we use 28 nm
thin Al film to fabricate the device [28]. The mechanical resonator is
realized by suspending one of the arms of the SQUID loop by selective
isotropic dry etching of the silicon substrate. The transmon is designed to
have a maximum frequency of 7.4 GHz, which can be tuned by varying the
external magnetic flux through the SQUID loop. All the device parameters are
listed in the Table 1.

A false color scanning electron microscope image of the SQUID loop with the
suspended mechanical resonator is shown in Fig. 1(b). The patterned chip is
placed inside the coaxial cavity and cooled down to 20 mK in a dilution
refrigerator. The details of device fabrication, device setup and measurements
are given in Appendix A. Since superconducting films show more resilience to
the in-plane magnetic field [28], we focus on the electromechanical coupling
of the out-of-plane vibrational mode of the mechanical resonator. Further, by
using the control over the normal component of the magnetic field, we can
cancel any out-of-plane component of the magnetic field arising from the
misalignment between the axial field and SQUID plane.

When the qubit is made resonant with the cavity, new eigenstates emerge, which
are the symmetric and anti-symmetric combination of single excitation states
of the qubit and cavity, denoted by $\ket{+}$ and $\ket{-}$ in Fig. 1(c). As
mentioned earlier, these eigenstates retain the flux-tunability of the qubit
and the Josephson nonlinearity is distributed in both the modes. Thus, each
dressed mode can be treated as an effective two-level system and can be used
for the electromechanical coupling. Considering the higher frequency dressed
mode alone, the effective Hamiltonian can be written as,

$H\simeq\omega_{+}\Sigma_{z}+\omega_{m}b^{\dagger}b+g_{+}\Sigma_{z}(b+b^{\dagger}),$
(1)

where $\omega_{+}$, $\omega_{m}$, and $g_{+}$ are dressed mode frequency,
mechanical mode frequency, and single-photon electromechanical coupling rate,
respectively. The ladder operators for the mechanical mode, and the Pauli
operator for the dressed mode are denoted by $b(b^{\dagger})$, and
$\Sigma_{z}$, respectively.

##  III Measurements

Figure 2: Strong-coupling and flux responsivity. (a) Colorplot of the cavity
voltage transmission $|S_{21}|$ when flux through the SQUID loop is varied.
(b) The panel shows two linecuts from (A), corresponding to
$\Phi/\Phi_{0}=0.5$ and $\Phi/\Phi_{0}=0.27$ in yellow and green color,
respectively. The solid-lines are the fitted curves. At $\Phi/\Phi_{0}=0.5$,
the qubit significantly detuned from the cavity, and a nearly “cavity-like”
mode is observed. For $\Phi/\Phi_{0}=0.27$, the qubit and cavity become nearly
resonant, and transmission measurement shows the vacuum Rabi split peaks.

We begin by measuring the voltage transmission $|S_{21}(\omega)|$ through the
cavity while varying the magnetic flux through the SQUID loop. Depending on
the flux passing through the SQUID loop, the qubit frequency can vary from its
maximum value to a minimum value. Fig. 2(a) shows the color plot of the cavity
transmission. As the qubit is brought in resonance with the cavity, the
measured transmission splits into two well-separated dressed modes as shown in
Fig. 2(b), demonstrating the strong coupling. Provided low strength of the
input power, the dressed cavity mode for $\Phi/\Phi_{0}=0.5$, and the vacuum
Rabi-split peaks(VRS) for $\Phi/\Phi_{0}=0.27$ have the characteristic
Lorentzian lineshape. From the separation of VRS, we infer the strength of the
qubit-cavity coupling to be 72 MHz. The reduction in the peak height of VRS in
resonant condition suggests that energy dissipation via the qubit dominates
over the cavity dissipation rate $(\kappa)$, i.e.
$(\Gamma_{1}/2+\Gamma_{\phi})\gg\kappa$, where $\Gamma_{1}$ and
$\Gamma_{\phi}$ are the energy relaxation and the pure dephasing rates of the
qubit, respectively.

For the out-of-plane vibration mode of the mechanical resonator, the single-
photon electromechanical coupling rate $g_{+}$ can be written as
$g_{+}={\alpha}G_{+}B^{\parallel}lx_{zp}$, where $G_{+}=d\omega_{+}/d\Phi$ is
the flux responsivity of the upper dressed mode, $B^{\parallel}$ is the axial
component of the applied magnetic field, $x_{zp}$ is the quantum zero-point
fluctuations of the mechanical resonator and $\alpha$ is a geometrical factor
of order unity and depends on the mechanical modeshape. It is evident that
higher single-photon electromechanical coupling rate can be achieved by
increasing the flux-responsivity and magnetic field $B^{\parallel}$. As
evident from Fig. 2(b), the increased flux-responsivity of the dressed mode
comes at a cost of reduced transmission. Therefore, we choose an optimal
operating point on the upper dressed mode $\omega_{+}/2\pi=5.884$ GHz, which
corresponds to the flux responsivity $G_{+}/2\pi\approx$ 1.85 GHz$/\Phi_{0}$,
and proceed to determine the electromechanical coupling.

##  IV Flux-mediated electromechanical coupling

Figure 3: Electromechanically induced absorption (EIA) and single-photon
coupling rate measurement. (a) Pump and probe scheme for the EIA experiment.
Energy levels of the system and the corresponding drives are shown. The state
$\ket{+,m}$ denotes one excitation is in the dressed mode, and $m$ excitation
in the mechanical resonator. The pump and probe signals are shown by the red
and green arrows, respectively. The pump frequency $\omega_{d}$ is kept fixed
and the probe frequency $\omega_{p}$ is swept to measure $|S_{21}(\omega)|$ of
the dressed mode. (b) $|S_{21}(\omega)|$ measured in the absence of the pump
signal, showing the linear response of the dressed mode. The solid-black line
is a Lorentzian fit to the response. (c) In the presence of a pump signal, an
interference feature appears in the probe transmission. The depth and
linewidth of the interference feature changes with pump strength. The panels
(top to bottom) correspond to mean photon occupation of 3.7$\times$10-2,
1.5$\times$10-2, and 6$\times$10-3, respectively. (d) Variation of
$\Gamma_{m}$ is plotted with mean photon occupation of the dressed mode
$n_{d}$, along with a straight line fit (shown in solid-black line) yielding
the single-photon electromechanical coupling rate $g_{+}\sim 35~{}$kHz. The
experiment is done with a dressed mode frequency $\omega_{+}/2\pi\sim
5.884~{}$GHz and axial magnetic field $B^{\parallel}\sim 27~{}$mT. (e)
Experimentally determined electromechanical coupling rate for three different
values of the applied magnetic field $B^{\parallel}$. While increasing
$B^{\parallel}$, the flux-responsivity is nominally kept constant by adjusting
$B^{\perp}$.

To probe the electromechanical coupling between the dressed mode and the
mechanical resonator, we use the technique utilized for electromagnetic
induced transparency (EIT) experiment in the atomic physics [29]. Its
mechanical analog in a cavity optomechanical devices, known as
optomechanically induced transparency (OMIT) [30]. Fig. 3(a) shows the
schematic of the different transitions involved in the measurement. In this
technique, two coherent signals are sent to the device, i.e. a pump signal at
$(\omega_{+}-\omega_{m})$ which drives the
$\ket{0,m+1}\leftrightarrow\ket{+,m}$ transition, and a weak probe signal at
$\omega_{p}$ (near $\omega_{+}$) to measure the transmission.

Without the pump, the voltage transmission $|S_{21}(\omega_{p})|$ has a linear
response with characteristic Lorentzian lineshape as shown in Fig. 3(b). In
the presence of the pump, an absorption feature appear in the transmission
spectrum, as shown in Fig. 3(c). It arises from the destructive interference
of the probe field from two different pathways [31, 30]. We emphasize that due
to a high anharmonicity of the dressed mode and low pump strengths used in the
experiment, the interference feature effectively arises dominantly from the
participation of ground and first excited states of the upper dressed mode,
denoted by $\ket{0}$ and $\ket{+}$, respectively. The shape of the absorption
feature is determined by the response of the mechanical resonator and can be
used to extract its resonant frequency and effective dissipation rate. We
estimate the mechanical resonant frequency to be $\omega_{m}/2\pi\approx 3.97$
MHz.

The mechanical resonator further experiences back-action in the presence of
pump signal, and it manifests in the enhanced linewidth of the interference
feature. Fig. 3(d) shows the change in the linewidth of interference feature
for different strengths of the pump signal. The effective linewidth of the
absorption signal is given by $\Gamma_{m}=\gamma_{m}(1+C)$ [32], where
$\gamma_{m}$ is intrinsic mechanical linewidth,
$C=4g_{+}^{2}n_{d}/(\kappa\gamma_{m})$ is the optomechanical cooperativity and
$\kappa$ is the energy decay rate of the dressed mode. The quantity $n_{d}$
represents the mean circulating power in the dressed mode in the units of
$\hbar\omega_{d}$ equivalently, the mean photon occupation of the dressed
mode. It can be calibrated to a good accuracy by tuning the qubit in the
dispersive regime, and performing ac-Stark shift measurements(see Appendix
B.1).

As shown in Fig. 3(d), the plot of the effective mechanical linewidth
$\Gamma_{m}$ with $n_{d}$ yields the slope $4g_{+}^{2}/\kappa$, and thus
$g_{+}$ (see Appendix C for additional details). From the slop, we estimated
the single-photon electromechanical coupling rate $g_{+}/2\pi\sim~{}35~{}$kHz,
obtained at the axial field of $B^{\parallel}\sim 27~{}$mT. The intercept of
the straight line fit in Fig. 3(d) gives the intrinsic mechanical linewidth
$\gamma_{m}/2\pi\sim 13$ Hz. Further, at the lower magnetic fields and lower
flux responsive operating points, we measure an intrinsic mechanical linewidth
as low as $\gamma_{m}/2\pi\sim 6$ Hz (see Appendix B.4 for additional
details).

Since the single-photon electromechanical coupling rate is linearly
proportional to the axial magnetic field $B^{\parallel}$, we carry out similar
interference experiments at different magnetic fields, and estimate the single
photon electromechanical coupling rates. Fig. 3(e) shows the plot of
experimentally determined $g_{+}$ for different values of the magnetic field.
A straight-line extrapolation of the data suggest our system achieves a very
high coupling rate $g_{+}/2\pi\sim 60$ kHz for a magnetic field
$B^{\parallel}\sim 45~{}$mT, which was the maximum applied magnetic field in
our experiment. As discussed in the next section, such a high coupling can be
a valuable resource in the active sideband cooling of the mechanical resonator
using just a few photons in the dressed mode.

##  V Strong backaction and sideband cooling: Beyond the linear response

Figure 4: Dynamical backaction on the mechanical resonator. Optomechanical
damping and spring-shift is plotted with pump detuning. From left to right
panel the pump power is increased, which results into stronger backaction. For
these measurements, the dressed mode is tuned to $\omega_{+}/2\pi\sim 5.8734$
GHz and the axial magnetic field is set to $B^{\parallel}\sim 18~{}$mT. These
parameters correspond to a coupling strength of $g_{+}/2\pi\sim 17$ kHz. The
solid circles are the experimental data points. The dashed lines are the
results from the theoretical calculations based on a Kerr-like oscillator. The
solid-black lines are the results from similar calculations except the
circulating power in the dressed mode is obtained from the experiment. The
details of both the calculations are given in Appendix D. The blue-color
gradient and gray-shaded region denote the cooling and unstable response of
the mechanical resonator.

To further investigate the effect of dynamical backaction on the mechanical
resonator, we send a pump signal, and measure the power spectral density(PSD)
of the output microwave signal while varying the pump detuning for different
pump powers. Depending on the pump detuning, an imbalance between the up- and
down-scattering rates of microwave photons by the mechanical resonator is
achieved, which leads to its cooling or heating. Initially, we operate at a
lower magnetic field of $B^{\parallel}\sim 18~{}$mT, and at a reduced flux
responsivity of $G_{+}/2\pi\sim$ 1.1 GHz$/\Phi_{0}$, which corresponds to
$g_{+}/2\pi\sim 15~{}$kHz. Fig. 4 shows the measurements of the effective
mechanical linewidth and the shift in the mechanical frequency as the pump
detuning and amplitude are varied. As expected, we observe a broadening of the
mechanical linewidth for negative detunings and an unstable response for the
positive detunings. Further, as the strength of the pump signal is increased,
the backaction effects becomes enhanced, and the onset of unstable response
shifts towards negative detuning due to the Josephson nonlinearity.

To theoretically understand the experimental observations, we model the
dressed mode as a weakly nonlinear oscillator by including a Kerr-term.
Effectively, the total Hamiltonian of the system can be written as
$H=-\Delta\hat{a}^{\dagger}\hat{a}-\alpha_{c}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}/2+\omega_{m}\hat{b}^{\dagger}\hat{b}+g_{+}~{}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})+\epsilon~{}(\hat{a}+\hat{a}^{\dagger})$,
where $\alpha_{c}$ is the Kerr-nonlinearity, $\epsilon$ is the pump amplitude,
$\Delta$ is the pump detuning from $\omega_{+}$, and
$\hat{a}(\hat{a}^{\dagger})$ is the lowering (raising) operator of the upper
dressed mode. For simplicity, we neglect the lower dressed mode due to its
large detuning. To compute the backaction effects on the mechanical resonator,
we solve the equations of motion for the coupled modes and obtain the
expressions of the effective linewidth and the frequency shift of the
mechanical resonator. Details of these calculations are included in Appendix
D. The dashed lines in Fig. 4 show the theoretically calculated results using
a Kerr nonlinearity of $\alpha_{c}/2\pi\sim~{}5.1$ MHz/photon (see Appendix E
for details). In addition, for the calculations of effective linewidth and
frequency shift, we can also use the experimentally determined value of the
circulating power, estimated from mode transmission $|S_{21}(\omega)|$. Since
the input line attenuation, the input/output port couplings, and net output
gain of the cavity( see section Appendix B.2) can be calibrated, it becomes
straightforward to compute the circulating power $n_{d}\hbar\omega_{d}$ in the
dressed mode. Solid black lines in Fig. 4 are theoretical results based on
experimentally determined $n_{d}\hbar\omega_{d}$. It is evident from Fig. 4
that a weak Kerr oscillator model does not predict the backaction effects
accurately at higher pump powers while the calculation taking into account the
actual mode transmission matches better with the experimental results.

Figure 5: Strong Backaction. (a) The response of the dressed mode at large
axial magnetic field $B^{\parallel}\sim 45~{}$mT. The top panel shows the
linear response at low input power of -24 dBm which is a Lorentzian at the
resonant frequency of $\omega_{+}/2\pi\sim 5.884~{}$GHz. The middle panel
shows the dressed mode response at relatively higher power of -4 dBm where
multiple overlapping peaks arising from the super-splitting of dressed mode
can be seen. With further increase in the power to +10 dBm, the bare cavity
response with Lorentzian lineshape is recovered as shown in the bottom panel.
(b) Shows the backaction results on the mechanical resonator for the pump
power corresponding to the middle panel of (a). The solid black curves are the
results from theoretical calculations based on the experimentally determined
circulating power in the dressed mode. The blue color gradient and gray shadow
area denotes the cooling and the unstable response of the mechanical
resonator. The inset shows the thermal spectrum of the mechanical resonator
corresponding to the maximum cooling at a pump detuning of
$\Delta\approx-3.5\omega_{m}$.

As mentioned earlier, a Kerr-nonlinear oscillator shows a Duffing response
before showing bifurcations, however, the power response of the vacuum dressed
split modes is very different. In the dispersive limit
$|\omega_{q}-\omega_{c}|\gg g$, the high power response has been explained by
considering the unconfined states of the transmon qubit [33, 34]. In the
weakly dissipative resonant limit, it is explained by the “dressing” of the
dressed mode with the drive photons, usually referred to as “super-splitting”.
It results in a series of additional peaks arising from the multi-photon
transitions [24]. At sufficiently high powers, the multi-photon transitions
are spaced closely to the bare cavity transition due to enhanced loss or
temperature, and thus results in a classical bare cavity response [27].

We observe a non-trivial response of the dressed mode, when we tune the
dressed mode to 5.884 GHz, and $B^{\parallel}$ to $45$ mT to explore strong
back action on the mechanical resonator. This is due to the fact that the
dressed mode has a higher Kerr-nonlinearity $\alpha_{c}/\pi\sim~{}13.2$
MHz/photon at this detuning. As shown in Fig. 5(a), the linear response (top
panel) of the dressed mode becomes quite complex and shows multiple peaks
(middle panel) when the pump is increased to -4 dBm. These peaks arise from
the super-splitting of the dressed mode , and are not well-resolved due to
larger qubit dissipation and dephasing rates. At higher power, the response
again becomes Lorentzian showing the bare-cavity mode (lowest panel)

At this operating point, the estimated single-photon coupling rate
$g_{+}/2\pi$ becomes 60 kHz. Consequently, it results in enhanced backaction
on the mechanical resonator as shown in Fig. 5(b). In addition, the dressed
mode’s high nonlinearity causes the boundary of the unstable response to shift
further towards the negative pump detunings. To understand the experimental
results, we again use the transmission measurements $|S_{21}(\omega)|$
performed at a fixed input power, and compute the circulating power in the
microwave mode. The solid black curves in Fig. 5(b) show the calculated
results. We see that the theoretical calculations show minor deviation from
the experimental results. To fully understand this aspect, a treatment of the
backaction considering the Jaynes-Cummings nonlinearity might be required.

The inset of Fig. 5(b), we show the mechanical resonator’s spectrum when the
backaction is strongest. We observe that the mechanical linewidth broadens to
550 Hz at $\Delta\approx-3.5\omega_{m}$, which is an enhancement of 42 times
over the intrinsic mechanical linewidth. The circulating power in the
microwave mode is remarkably low, and it is equivalent to the mean occupation
of 0.7 photons. We emphasize here that due to large electromechanical
coupling, we observe such a strong backaction with less than one photon in the
dressed mode.

We next try to estimate the effective mechanical phonon occupation
corresponding to the effective linewidth of 550 Hz. We first establish the
effective mechanical temperature by measuring the thermal power spectral
density using a weak pump signal at zero detuning such that backaction effects
can be neglected. We estimate that the mechanical mode is thermalized to 70
mK, corresponding to an average phonon occupation of approx 365 (details in
Appendix B.3). We then extend our analysis based on the estimation of the
circulating power from the transmission measurement. Assuming no multi-photon
loss or microwave gain, we convert the thermo-mechanical power spectral
density to the mean phonon occupation. We estimated the mean phonon occupancy
of 8.6 for the maximum cooling of the mechanical resonator.

##  VI Conclusion and outlook

To summarize, the performance of the flux-coupled electromechanical device has
been enhanced using the strong nonlinearity of the superconducting transmon
qubit. This has been largely made possible by fabricating the device with thin
film of aluminium ($28~{}$nm), which provides strong resilience against the
magnetic field and also results in lower mass of the mechanical resonator
($\sim 1~{}$pg). The combined effects result in large single-photon coupling
rate $g_{+}/2\pi\sim 60~{}$kHz, which is an order of magnitude improvement
over the previously demonstrated flux coupled devices[16, 17, 18, 20]. Due to
such a large single-photon electromechanical coupling rate, we could observe
the electromechanically-induced interference feature at extremely low
($<10^{-2}$) photon occupations, which is lower by several orders of magnitude
as compared to all the previous implementations of electromechanical devices.
We further observe the signature of the super-splitting of the dressed modes,
and utilize their nonlinearity to observe a strong backaction on the
mechanical oscillator.

Looking ahead, given that transmon qubits have been shown to operate in the
magnetic fields as high as 1 T [28], one can envision devices that can operate
in a regime where the single-photon coupling rate becomes comparable to the
mechanical resonant frequency. While the sideband cooling performance in the
current experiment is limited due to the lower sideband-resolution parameter
$\omega_{m}/\kappa\approx 0.25$. It can easily be improved by making smaller
length mechanical resonator, resulting in higher mechanical resonant
frequency. With such promising futuristic parameters, it is feasible to
achieve ground state cooling via sideband driving below the level of a single
photon, and it further suggests ways to prepare non-classical mechanical
states including Schrodinger-cat [22, 23]. Even with the existing parameters,
solely by improving qubit and cavity coherence, it is possible to prepare
mechanical Fock and cat states [35]. Finally, such techniques can be extended
to the low frequency flux-family superconducting qubits to realize transverse
electromechanical couplings, and thus extending the toolbox available with the
flux-coupled devices [36].

###### Acknowledgements.

T.B. thanks S. Majumder for the valuable discussion in developing the
theoretical model. T.B. also thanks Harsh Vardhan Upadhyay for the insights in
QuTip calculation. This work is supported by the Air Force Office of
Scientific Research under Award No. FA2386-20-1-4003. The authors acknowledge
the support under the CoE-QT by MEITY and QuEST program by DST, Govt. of
India. The authors acknowledge device fabrication facilities at CeNSE, IISc
Bangalore.

---
Device parameters | Symbol | Value
Bare cavity frequency | $\omega_{c}/2\pi$ | 5.846 GHz
Maximum qubit frequency | $\omega_{q}^{0}/2\pi$ | 7.38 GHz
Qubit-cavity coupling rate | $J/2\pi$ |  $72~{}$MHz
Measured transmon anharmonicity | $\alpha/2\pi$ |  $-284~{}$MHz
SQUID tunnel resistance after deposition | $R_{n}$ | 8.9 k$\Omega$
SQUID tunnel resistance after annealing | $R_{n}$ | 5 k$\Omega$
SQUID tunnel resistance after etching | $R_{n}$ | 5.9 k$\Omega$
Josephson inductance of SQUID | $L_{J}$ | 7 nH
Mechanical resonator length | $l$ |  $\sim 40~{}\mu$m
Mechanical resonator width | $b$ |  $\sim$250 nm
Mechanical resonator thickness | $d$ |  $\sim$28 nm
Total mass of the mechanical resonator | $m$ |  $\sim$0.75 pg
Mechanical resonator frequency | $\omega_{m}/2\pi$ |  $\sim$3.97 MHz
Maximum applied axial magnetic field | $B^{\parallel}_{max}$ |  $\sim 45~{}$mT
Table 1: Summary of the key parameters of the sample studied.

##  Appendix A Device fabrication and experimental setup

The device is fabricated on a cleaned 2.5 mm$\times$7 mm silicon-100
substrate. A single-step electron beam lithography(EBL) is used to pattern the
substrate with a bi-layer resist stack of MMA-EL-11 and PMMA-950-A4.
Subsequently, aluminum(Al) films are deposited using shadow evaporation
technique with an intermediate in-situ oxidation step to form the tunnel
junction. We found the evaporated films to be under compressive strain after
the deposition. To convert it to the tensile strain, the chip is annealed at
180$\degree$C for 15 min in the ambient environment. It leads to a change in
the tunnel resistance of the SQUID at room temperature, as mentioned in Table
1.

Next, we carry out electron beam lithography using a single layer of PMMA
resist and pattern a rectangular window surrounding the nanowire. It is
followed by a reactive ion etching (RIE) process, where the silicon underneath
the wire is etched out. The etching process is done in two steps, using SF6
gas only. In the first step, silicon is vertically etched where the RIE
chamber pressure is kept low($\approx 9$ mT). It is followed by isotropic
silicon etch at higher chamber pressure($\approx 95$ mT). The isotropic etch
step removes silicon underneath the nanobeam and makes it suspended.
Subsequently, PMMA ashing is done in to remove any residual resist on the
substrate. Fig. 6(a) shows the optical image of the qubit, fabricated on the
silicon chip. The etching process further affects the tunnel resistance of the
SQUID. We consistently observe a reduction in the room temperature resistance
by $40-45$ percent while annealing the substrate and an increase in the
resistance by $15-20$ percent in the etching process. To adjust for these
changes, the oxidation parameters during the junction fabrication are tuned to
get the target junction resistance after the final step.

Finally, the chip is placed inside a coaxial cavity, and it is placed inside a
home-built vector magnet setup. We use two layers of concentric shielding cans
to protect from infrared radiation and stray magnetic fields. The inner can is
coated with IR absorbing layer is made radiation tight, and the outer can is
made of cryo-perm, which helps in reducing the magnetic field fluctuations at
the sample. It can be seen in Fig. 6(b). All the assembly is mounted to the
mixing chamber plate of dilution refrigerator and cooled down to $20~{}$mK for
the measurement. Fig. 6(c) shows the schematic of the complete measurement
setup used in the experiment.

Figure 6: Device details and the measurement setup. (a) An optical microscope
image of the qubit device patterned on a silicon substrate. The white
rectangles in the white are the qubit antenna pads. In between the pads, there
is SQUID loop with suspended nanowire. (b) Device assembly, which is put
inside a two-layer shield, is attached to the mixing chamber plate of the
dilution refrigerator. (c) Shows the schematic of the measurement setup,
showing input and output lines with the attenuation, amplifier, and filters.

##  Appendix B Calibration of the input attenuation, net-output gain, and
mechanical mode temperature

###  B.1 Calibration of the input line attenuation

Figure 7: Calibration. (a) Plot of the experimentally determined mean photon
occupation with the input probe power. The data points show the normalized
shift in the qubit frequency while performing ac-Stark shift measurement. The
solid black line is the fit that relates the injected microwave power to the
mean photon occupation of the mode. (b) It shows the plot of the transmitted
power $P_{d}$ at $\omega_{+}$ with mean dressed mode occupation $n_{d}$. The
dashed line is the linear fit, and its slope is used to estimate the net
output gain. The inset shows the output signal, recorded using a spectrum
analyzer, when a pump signal at $\omega_{+}$ is sent. (c) The output microwave
power at the sideband peak $S_{VV}$ is plotted with mean dressed mode
occupation. The dashed line is a linear fit, and its slope is used to estimate
mean thermal phono occupation. Inset of (c) shows the mechanical resonator’s
spectrum. It is the PSD of the lower mechanical sideband for a pump at
$\omega_{+}$. (d) PSD of the mechanical resonator when axial magnetic field is
18 mT, and dressed mode frequency is 5.864 GHz. From the Lorentzian fit (black
line) to the data, we estimate the mechanical resonator frequency
$\omega_{m}/2\pi\sim 3.97~{}$MHz and intrinsic linewidth $\gamma_{m}/2\pi\sim
6~{}$Hz.

To estimate the total attenuation in the input line, we use the ac-stark shift
measurement. We tune the qubit to $\omega_{q}/2\pi\sim 5.325~{}$GHz where it
couples dispersively to the cavity. Using the two-tone spectroscopy technique,
we measure the qubit spectrum while probe power is varied. With the increase
in probe power, the qubit frequency $\omega_{q}$ shifts from its maximum
frequency $\omega_{q}^{0}$. The shift in the qubit transition frequency is
given by $\omega_{q}=\omega_{q}^{0}-2n_{d}\chi$, where $n_{d}$ and $\chi$ are
mean intracavity probe photon occupation and the dispersive shift of the
qubit, respectively. In a separate measurement, we estimate the dispersive
shift $\chi$ of the qubit. The experimentally computed intracavity photon
$(\omega_{q}^{0}-\omega_{q})/2\chi$ is plotted with the input probe power in
Fig. 7(a). Thus, it allows us to estimate the product of the total input line
attenuation and the coupling rate of the input port.

###  B.2 Calibration of the net output gain

To calibrate the net output gain, we send a pump signal at frequency
$\omega_{+}$ and record the transmitted power $P_{d}$ at the same frequency.
The Inset of Fig. 7(b) shows the measured power spectral density recorded
using a spectrum analyzer. The transmitted power at pump frequency is given by
$P_{d}/\hbar\omega_{+}=A_{P}(1/2+n_{add}+n_{d}\kappa_{e})$, where $A_{P}$ is
the net output power gain, $n_{add}$ is the noise added by the amplifier
chain, $n_{d}$ is the mean occupation of the dressed mode by the coherent pump
photons, and $\kappa_{e}$ is the coupling rate of the output port. We change
the pump signal strength and plot measured $P_{d}$ with mean dressed mode
occupation $n_{d}$, as shown in Fig. 7(b). From the slope of the linear fit,
we estimate the net gain. For this measurement, the dressed mode frequency is
tuned to 5.864 GHz and $B^{\parallel}\sim 18$ mT is applied. The quantity
$n_{d}$ is estimated using known pump power, the input line attenuation, and
the dressed mode decay rate $\kappa/2\pi\sim 9.7~{}$MHz which can be measured
separately in a transmission measurement. By performing reflection
measurements at room temperature, we estimate the output coupling rate to be
$\kappa_{e}/2\pi\sim 6.25~{}$MHz. Using all these parameters, we estimate the
net gain of the output line to be $A_{P}=58.52$ dB.

###  B.3 Estimation of the effective mechanical mode temperature

A pump signal, tuned to the dressed mode frequency $\omega_{+}$ produces two
sidebands due to the thermo-mechanical motion of the mechanical resonator at
$\omega=\omega_{+}\pm\omega_{m}$. The output microwave power spectral density
(PSD) of the sideband peak is given by,

$S_{VV}/\hbar\omega=A_{P}\left(\frac{1}{2}+n_{add}+\frac{k_{e}}{\gamma_{m}}\frac{16g_{+}^{2}n_{d}n_{m}}{(k^{2}+4\omega_{m}^{2})}\right).$

We pump the dressed mode at zero detuning and record the PSD of the lower
mechanical sideband using a spectrum analyzer. In the inset of Fig. 7(c), we
show a representative measurement of the microwave PSD showing the mechanical
mode. Fig. 7(c) shows the plot the microwave output power at the sideband peak
$S_{VV}(\omega_{+}-\omega_{m})$ with the mean pump photon occupation $n_{d}$
in the dressed mode. From the slope of a linear line fit, we estimate the mean
thermal occupation $n_{m}$ of the mechanical resonator. For this experiment,
the dressed mode frequency is tuned to $w_{+}/2\pi=5.884$ GHz and axial
magnetic field $B^{\parallel}\sim 18~{}$mT is applied. These parameter
corresponds to an electromechanical coupling of $g_{+}/2\pi\sim 22$ kHz, which
was measured separately in OMIA experiment as described in the main text. The
cavity decay rate $\kappa/2\pi=11.5$ MHz is extracted from dressed mode
transmission $|S_{21}(\omega)|$. With all these parameters, we estimate the
thermal phonon occupation of the mechanical resonator to be $n_{m}\sim 365$,
which corresponds to a mode temperature of $70$ mK.

###  B.4 Mechanical resonator’s linewidth in lower magnetic fields

As mentioned in the main text, we have observed narrower mechanical linewidths
at lower values of the axial magnetic fields. Fig. 7(d) shows the PSD of the
lower mechanical sideband for a pump signal sent at $\omega_{+}/2\pi\sim
5.864~{}$GHz. By doing a Lorentzian fit on the spectrum, we get the intrinsic
linewidth of $\gamma_{m}/2\pi\sim 6$ Hz. For these measurements, we use a
smaller magnetic field ($B^{\parallel}\sim 18~{}$mT), an operating point with
the lower flux responsivity of the dressed mode ($G_{+}/2\pi=0.55$
GHz$/\Phi_{0}$), and a low pump strength to avoid any backaction. We suspect
that the broadening to mechanical linewidth could be due to the increase in
the mode temperature at higher magnetic fields. It is important to remark here
that to generate 18 mT of axial magnetic field, the electromagnet requires
nearly 400 mA of current.

##  Appendix C Analysis of the electromechanically induced absorption(EIA)
measurements

Figure 8: Fitting of EIA. Plot of the real (orange), imaginary (green) and the
absolute (blue) value of the transmission $S_{21}$ signal in an EIA
measurement. The solid lines are the fits.

In an EIA measurement, the presence of a pump signal at $\omega_{d}$ leads to
an interference feature near $(\omega_{d}+\omega_{m})$ in dressed mode’s
transmission. Depending on pump frequency, the position of the interference
feature could be anywhere within the bandwidth of the dressed mode. Therefore,
in general, the interference signal could be asymmetric with respect to the
probe frequency. Here, we use a generic complex formula to fit the
experimental data given by,

$S_{21}(\omega)=x_{0}+\mathbf{i}y_{0}-\alpha\frac{e^{\mathbf{i}\phi}}{1-\mathbf{i}\frac{(\omega-\omega_{0})}{\Gamma_{m}/2}},$

where $x_{0},~{}y_{0},~{}\alpha,~{}\phi,~{}\omega_{0}$ and $\Gamma_{m}$ are
the fit parameter. Fig. 8 shows the real, imaginary, and absolute components
of the transmission measurement related to the response of the mechanical
resonator. To reduce the noise in the signal, we record the data three times
and do post-averaging. It is particularly helpful at low pump power, where the
signal to noise ratio(SNR) is low. The extracted $\Gamma_{m}$ is used to
calculate the electromechanical coupling $g_{+}$, as described in the main
text.

##  Appendix D Calculation of the backaction using a weakly nonlinear Kerr
mode

In a frame rotating at the pump frequency, the Hamiltonian of the system can
be written as,

$\mathcal{H}=-\Delta\hat{a}^{\dagger}\hat{a}-\frac{\alpha_{c}}{2}\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}+\omega_{m}\hat{b}^{\dagger}\hat{b}\\\
+g_{+}~{}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})+\epsilon~{}(\hat{a}+\hat{a}^{\dagger}),$
(2)

where $\alpha_{c}$ is the Kerr-nonlinearity of the dressed mode, $\epsilon$ is
the drive strength of the pump, $\Delta$ is the drive detuning, and the rest
of the symbols have their usual meaning.

We write the quantum Langevin equations of the system as

$\dot{\hat{a}}=-{i}\left[\hat{a},\mathcal{H}\right]-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa_{ex}}~{}\hat{a}_{in}+\sqrt{\kappa_{0}}~{}\hat{f}_{in},$
(3a)
$\dot{\hat{b}}=-{i}\left[\hat{b},\mathcal{H}\right]-\frac{\gamma}{2}\hat{b}+\sqrt{\gamma_{m}}~{}\hat{b}_{in},$
(3b)

where $\hat{a}_{in}$, $\hat{f}_{in}$ are noise operators of the dressed mode
and $\hat{b}_{in}$ is noise operator of the mechanical mode. $\kappa$,
$\gamma_{m}$ are the decay rate of the dressed mode and the mechanical mode,
and $\kappa_{ex}$, $\kappa_{0}$ are the external and internal decay rates of
the dressed mode, respectively. Using the Hamiltonian in Eq. 2 and Eq. 3b, we
can obtain the equations of motion (EOM) of both the modes as,

$\dot{\hat{a}}=({i}\Delta-\frac{\kappa}{2})\hat{a}+{i}\alpha_{c}\hat{a}^{\dagger}\hat{a}\hat{a}-{i}g_{0}\hat{a}(\hat{b}+\hat{b}^{\dagger})\\\
+\epsilon+\sqrt{\kappa_{ex}}~{}\hat{a}_{in}+\sqrt{\kappa_{0}}~{}\hat{f}_{in}$
(4a)
$\dot{\hat{b}}=(-{i}\omega_{m}-\frac{\gamma_{m}}{2})\hat{b}-{i}g_{0}\hat{a}^{\dagger}\hat{a}+\sqrt{\gamma_{m}}~{}\hat{b}_{in}$
(4b)

By defining the mean field occupation of the dressed mode and the mechanical
mode as $\alpha$ and $\beta$, we arrive at the following semi-classical
equations of motion,

$\dot{\alpha}=({i}\Delta-\frac{\kappa}{2})\alpha+{i}\alpha_{c}\left|\alpha\right|^{2}\alpha-{i}g_{0}\alpha(\beta+\beta^{\ast})+\epsilon$
(5a)
$\dot{\beta}=(-{i}\omega_{m}-\frac{\gamma_{m}}{2})\beta-{i}g_{0}\left|\alpha\right|^{2}.$
(5b)

In the steady state of the system, these equations become $\dot{\alpha}=0$,
$\dot{\beta}=0$. Solving these, we find the steady state field amplitude
$\alpha$ and $\beta$.

Now, we assume an ansatz solution of Eq. 4b, where we write down the fields as
a combination of steady state amplitude and a fluctuation term, i.e.
$\hat{a}=\alpha+\delta\hat{a}$, and $\hat{b}=\beta+\delta\hat{b}$.
Substituting these ansatz to Eq. 4b, and ignoring the higher order fluctuation
terms, we get the EOMs as,

$\dot{\delta\hat{a}}=\left[{i}\tilde{\Delta}-\frac{\kappa}{2}\right]\delta\hat{a}+{i}\eta\delta\hat{a}^{\dagger}-{i}G(\delta\hat{b}+\delta\hat{b}^{\dagger})+\sqrt{\kappa_{ex}}~{}\hat{a}_{in}\\\
+\sqrt{\kappa_{0}}~{}\hat{f}_{in},$ (6a)
$\dot{\delta\hat{a}}^{\dagger}=\left[-{i}\tilde{\Delta}-\frac{\kappa}{2}\right]\delta\hat{a}^{\dagger}-{i}\eta^{\ast}\delta\hat{a}+{i}G^{\ast}(\delta\hat{b}+\delta\hat{b}^{\dagger})+\sqrt{\kappa_{ex}}~{}\hat{a}_{in}^{\dagger}\\\
+\sqrt{\kappa_{0}}~{}\hat{f}_{in}^{\dagger},$ (6b)
$\dot{\delta\hat{b}}=\left[-{i}\omega_{m}-\frac{\gamma_{m}}{2}\right]\delta\hat{b}-{i}G^{\ast}\delta\hat{a}-{i}G\delta\hat{a}^{\dagger}+\sqrt{\gamma_{m}}~{}\hat{b}_{in}.$
(6c)
$\dot{\delta\hat{b}}^{\dagger}=\left[{i}\omega_{m}-\frac{\gamma_{m}}{2}\right]\delta\hat{b}^{\dagger}+{i}G\delta\hat{a}^{\dagger}+{i}G^{\ast}\delta\hat{a}+\sqrt{\gamma_{m}}~{}\hat{b}_{in}^{\dagger},$
(6d)

where
$\tilde{\Delta}=\Delta+2\alpha_{c}\left|\alpha\right|^{2}-g_{0}(\beta+\beta^{\ast})$,
$G=g_{0}\alpha$ and $\eta=\alpha_{c}\alpha^{2}$.

Next, we perform Fourier transform of Eq. 6d by defining the transformation as
$x[\omega]=\mathcal{F}\left[x(t)\right]=\int_{-\infty}^{+\infty}x(t)\mathrm{e}^{{i}\omega
t}\mathrm{d}t$. Using the identities
$(x^{\dagger})[\omega]=(x[-\omega])^{\dagger}$, and
$\mathcal{F}[\dot{x}(t)]=-{i}\omega\mathcal{F}[x(t)]$, the new set of
equations of motion in frequency domain become,

$\begin{bmatrix}1/\chi_{c}&-{i}\eta\\\
{i}\eta^{\ast}&1/\tilde{\chi}_{c}\end{bmatrix}\begin{bmatrix}\delta\hat{a}[\omega]\\\
(\delta\hat{a}^{\dagger})[\omega]\end{bmatrix}=-{i}\begin{bmatrix}G&G\\\
-G^{\ast}&-G^{\ast}\end{bmatrix}\begin{bmatrix}\delta\hat{b}[\omega]\\\
(\delta\hat{b}^{\dagger})[\omega]\end{bmatrix}+\\\
\begin{bmatrix}\sqrt{\kappa_{ex}}~{}\hat{a}_{in}[\omega]+\sqrt{\kappa_{0}}~{}\hat{f}_{in}[\omega]\\\
\sqrt{\kappa_{ex}}~{}(\hat{a}_{in}^{\dagger})[\omega]+\sqrt{\kappa_{0}}~{}(\hat{f}_{in}^{\dagger})[\omega]\end{bmatrix}$
(7)

$\begin{bmatrix}1/\chi_{m}&0\\\
0&1/\tilde{\chi}_{m}\end{bmatrix}\begin{bmatrix}\delta\hat{b}[\omega]\\\
(\delta\hat{b}^{\dagger})[\omega]\end{bmatrix}=-{i}\begin{bmatrix}G^{\ast}&G\\\
-G^{\ast}&-G\end{bmatrix}\begin{bmatrix}\delta\hat{a}[\omega]\\\
(\delta\hat{a}^{\dagger})[\omega]\end{bmatrix}\\\
+\begin{bmatrix}\sqrt{\gamma_{m}}~{}\hat{b}_{in}[\omega]\\\
\sqrt{\gamma_{m}}~{}(\hat{b}_{in}^{\dagger})[\omega]\end{bmatrix},$ (8)

where
$\chi_{c}[\omega]=1/(-{i}(\omega+\tilde{\Delta})+\kappa/2),\tilde{\chi}_{c}[\omega]=1/(-{i}(\omega-\tilde{\Delta})+\kappa/2)$
are dressed mode’s susceptibilities and
$\chi_{m}[\omega]=1/(-{i}(\omega-\omega_{m})+\gamma_{m}/2),~{}\tilde{\chi}_{m}[\omega]=1/(-{i}(\omega+\omega_{m})+\gamma_{m}/2)$
are the mechanical susceptibilities.

To find the effective dynamics of the mechanical resonator, we first solve Eq.
7, and substitute the solution of $(\hat{a}[\omega]$ and
$(\hat{a}^{\dagger})[\omega]$ in Eq. 8. This leads to the simplified equations
of the mechanical mode as,

$\begin{bmatrix}-{i}(\omega-\omega_{m})+\frac{\gamma_{m}}{2}+\Sigma_{c}[\omega]&\Sigma_{c}[\omega]\\\
-\Sigma_{c}[\omega]&-{i}(\omega+\omega_{m})+\frac{\gamma_{m}}{2}-\Sigma_{c}[\omega]\end{bmatrix}\\\
\times\begin{bmatrix}\delta\hat{b}[\omega]\\\
(\delta\hat{b}^{\dagger})[\omega]\end{bmatrix}=\sqrt{\gamma_{m}}\begin{bmatrix}\hat{B}_{in}[\omega]\\\
(\hat{B}_{in}^{\dagger})[\omega]\end{bmatrix}.$ (9)

The quantity defined as
$\Sigma_{c}[\omega]=2{i}|G|^{2}[\tilde{\Delta}-|\eta|][1/\chi_{c}\tilde{\chi}_{c}-|\eta|^{2}]^{\scalebox{1.0}{-1}}$
is the modification in mechanical resonator’s dynamics due to nonlinear Kerr
mode. The frequency shift and the optomechanical damping of the mechanical
resonator is given by, $\delta\omega_{m}[\omega]=$ Im($\Sigma_{c}[\omega]$)
and $\delta\gamma_{m}[\omega]=2\times$Re($\Sigma_{c}[\omega]$), respectively.

##  Appendix E Flux responsivity and Kerr nonlinearity of the dressed mode

Figure 9: Flux responsivity and Kerr nonlinearity. Plot of the flux
responsivity ($G_{+}$) and Kerr nonlinearity ($\alpha_{c}$) of the upper
dressed mode. While the $G_{+}$ is estimated from the direct cavity
measurements, the anharmonicity of the dressed mode $|\alpha_{c}|$ is computed
using QuTip and using the experimentally determined device parameter i.e. bare
cavity frequency $\omega_{c}/2\pi\sim 5.846~{}$GHz, transmon anharmonicity
$\alpha/2\pi\sim 284~{}$MHz, and a qubit-cavity coupling rate $J/2\pi\sim
72~{}$MHz.

From the cavity transmission ($|S_{21}(\omega)|$) near the vacuum Rabi
splitting, as shown in Fi. 2(a) of main text, we can calculate the flux
responsivity of the dressed modes. First, we estimate the mode frequency
$\omega_{+}$ at each flux bias point by doing a Lorentzian fit on
$|S_{21}(\omega)|$. Subsequently, we do a polynomial fit of $\omega_{+}$ with
respect to flux bias ($\Phi$) and calculate its first derivative, i.e., the
flux responsivity. In Fig. 9, we plot the flux responsivity $G_{+}$ of the
upper dressed mode with the mode frequency.

Next, we try to estimate the Kerr nonlinearity of the dressed modes
theoretically. Using the QuTip package, we compute the eigenenergies of the
system while varying the qubit’s frequency. While defining the system
Hamiltonian, we used realistic device parameters given in Table. 1.
Subsequently, we estimate the Kerr nonlinearity of the dressed mode by
calculating the difference between different energy levels, and the result is
plotted in red curve of Fig. 9.

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