# Prospects of cooling a mechanical resonator with a transmon qubit in c-QED
setup

Sourav Majumder  Department of Physics, Indian Institute of Science,
Bangalore-560012 (India)  Tanmoy Bera  Department of Physics, Indian Institute
of Science, Bangalore-560012 (India)  Vibhor Singh  Department of Physics,
Indian Institute of Science, Bangalore-560012 (India)

###### Abstract

Hybrid devices based on the superconducting qubits have emerged as a promising
platform for controlling the quantum states of macroscopic resonators. The
nonlinearity added by a qubit can be a valuable resource for such control.
Here we study a hybrid system consisting of a mechanical resonator
longitudinally coupled to a transmon qubit. The qubit readout can be done by
coupling to a readout mode like in c-QED setup. The coupling between the
mechanical resonator and transmon qubit can be implemented by modulation of
the SQUID inductance. In such a tri-partite system, we analyze the steady-
state occupation of the mechanical mode when all three modes are dispersively
coupled. We use the quantum-noise and the Lindblad formalism to show that the
sideband cooling of the mechanical mode to its ground state is achievable. We
further experimentally demonstrate that measurements of the thermomechanical
motion is possible in the dispersive limit, while maintaining a large coupling
between qubit and mechanical mode. Our theoretical calculations suggest that
single-photon strong coupling is within the experimental reach in such hybrid
devices.

##  I Introduction:

Control over the quantum states of a mechanical resonator by coupling them to
optical modes can have several potential applications in the field of quantum
technologies [1]. The traditional cavity-optomechanics based approach of
coupling a mechanical resonator to an optical mode via the radiation-pressure
interaction has been quite successful [2, 3, 4, 5, 6, 7, 8, 9]. While the
radiation-pressure mediated coupling in such devices is nonlinear, its
magnitude is usually small in most applications. Further, due to the
dispersive interaction, the effects originating from the Kerr-term are
strongly suppressed [10, 11].

To mitigate the limitations of linear cavity optomechanics, hybrid devices
based on the strong nonlinearity of qubits have been proposed and developed
[12, 13, 14]. These proposals explore their performance from the sideband
cooling of the mechanical resonator [15] to the matter-interferometry [16],
while considering a wide range of two-level systems such as superconducting
qubits [15, 17, 18, 19, 20, 21, 22], quantum-dots [23], and nitrogen vacancy
defects in diamond [24]. Particularly, in the microwave domain, experimental
realization of several hybrid devices have been shown using the nonlinearity
of a superconducting qubit [25], Josephson capacitance [26, 27], Josephson
inductance [28, 29, 30, 31], and piezo-electricity [32, 33].

Among these different schemes, the electromechanical coupling stems from
charge or flux modulation, and its tunability is controlled by the external
applied magnetic field. Recently, the magnetic flux-mediated coupling approach
have shown promising experimental results [28]. These systems have
demonstrated large electromechanical coupling [29, 30, 31], four-wave-cooling
of the mechanical resonator to near the quantum ground state [34], and
Lorentz-force induced backaction on the mechanical resonator [35].

Motivated by the progress on flux-mediated approach, here we investigate a
coupled three-mode system consisting of a mechanical mode, transmon qubit, and
a readout cavity. From the practical point of view, the additional readout
cavity is useful ingredient to consider as it allows the quantum non-
demolishing (QND) measurement of qubit mode in circuit-QED setup [36, 37].
While a mechanical mode coupled to a two-level system has been studied
extensively in the past [15, 38, 17, 12, 39], the focus of our investigation
has been on treating the transmon qubit as a weakly anharmonic oscillator. In
addition, we theoretically and experimentally address the readout of the
mechanical mode when transmon is detuned far away from the readout cavity.
This regime is particularly important as large electromechanical coupling with
the qubit mode can be achieved. Using the quantum-Langevin equation of motion
[40], and Lindblad formalism [41], we analyze the possibility of sideband
cooling of the mechanical resonator. Experimentally, we use a two-tone method
to measure the thermo-mechanical motion, and compare it with analytical
results.

This paper is organised as follows: In part II, we discuss the theoretical
model of the three coupled modes. We solve the system’s equations of motion in
part III. The analytical solution of the system is analyzed in part IV, where
we have shown the possibility of cooling the mechanical resonator. In the part
V, we show experimental and analytical results discussing the detection of
mechanical motion in the dispersive regime of the cavity and the qubit mode.
We summarize and conclude our discussion in part VI.

##  II Theoretical model:

Figure 1: (a) A conceptual schematic of the three-mode hybrid device showing a
linear cavity coupled to a qubit which in turn couples to a mechanical
resonator. A direct coupling between the cavity and the mechanical mode is not
considered. (b) A possible implementation using a frequency tunable transmon
qubit, where coupling to mechanical mode is achieved by embedding it the SQUID
loop and by applying a constant magnetic field. A magnetic field perpendicular
to the SQUID loop couples the in-plane mechanical mode to the qubit, while
parallel magnetic field couples the qubit to the out-of-plane mechanical mode.

We consider a coupled system where the mechanical mode modulates the transmon
qubit frequency, therefore resulting in a longitudinal coupling. Such coupling
between transmon qubit and the mechanical resonator can be implemented by
embedding a mechanical resonator into the SQUID loop of the qubit. In
addition, the qubit couples to a linear mode (the readout cavity) transversely
as in the circuit-QED setup. A schematic diagram of the system and a possible
implementation with the equivalent circuit diagram are shown in the Fig. 1 (a)
and (b).

Using the dispersive approximation between the transmon and the readout
cavity, we arrive at the following system Hamiltonian:

$\hat{\mathcal{H}}_{0}=\omega_{c}\hat{a}^{\dagger}\hat{a}+\omega_{q}\hat{c}^{\dagger}\hat{c}-\frac{\alpha_{q}}{2}\hat{c}^{\dagger}\hat{c}^{\dagger}\hat{c}\hat{c}+\omega_{m}\hat{b}^{\dagger}\hat{b}\\\
+\chi~{}\hat{a}^{\dagger}\hat{a}\hat{c}^{\dagger}\hat{c}+g_{0}~{}\hat{c}^{\dagger}\hat{c}(\hat{b}+\hat{b}^{\dagger}),$
(1)

where $\hat{a}$($\hat{a}^{\dagger}$), $\hat{c}$($\hat{c}^{\dagger}$),
$\hat{b}$($\hat{b}^{\dagger}$) are the annihilation(creation) operators for
the cavity, qubit and the mechanical mode of frequency $\omega_{c}$,
$\omega_{q}$, $\omega_{m}$, respectively. The Kerr-nonlinearity of the
transmon is denoted as $\alpha_{q}$. The last two terms are the interaction
terms between the modes, where the dispersive coupling between the qubit and
the cavity is $\chi$. The radiation-pressure type coupling between the
transmon and the mechanical mode is denoted by the single photon coupling rate
$g_{0}$.

Two additional drive terms of amplitude $\delta$ and $\epsilon$ at frequency
of $\omega_{L}$ (near $\omega_{c}$) and $\omega_{d}$ (near $\omega_{q}$) are
added to the Hamiltonian. We can write the drive Hamiltonian as,

$\hat{\mathcal{H}}_{d}=\delta~{}(\hat{a}~{}e^{+{i}\omega_{L}t}+\hat{a}^{\dagger}~{}e^{-{i}\omega_{L}t})+\epsilon~{}(\hat{c}~{}e^{+{i}\omega_{d}t}+\hat{c}^{\dagger}~{}e^{-{i}\omega_{d}t}).$
(2)

By carrying out rotating frame transformations, given by the unitary operators
$U^{a}=\exp{\left[{i}\omega_{L}\hat{a}^{\dagger}\hat{a}t\right]}$ and
$U^{c}=\exp{\left[{i}\omega_{d}\hat{c}^{\dagger}\hat{c}t\right]}$, the
transformed Hamiltonian can be written as,

$\hat{\mathcal{H}}=-\Delta_{c}\hat{a}^{\dagger}\hat{a}-\Delta_{q}\hat{c}^{\dagger}\hat{c}-\frac{\alpha_{q}}{2}\hat{c}^{\dagger}\hat{c}^{\dagger}\hat{c}\hat{c}+\omega_{m}\hat{b}^{\dagger}\hat{b}+\chi~{}\hat{a}^{\dagger}\hat{a}\hat{c}^{\dagger}\hat{c}\\\
+g_{0}~{}\hat{c}^{\dagger}\hat{c}(\hat{b}+\hat{b}^{\dagger})+\delta~{}(\hat{a}+\hat{a}^{\dagger})+\epsilon~{}(\hat{c}+\hat{c}^{\dagger}),$
(3)

where $\Delta_{c}=\omega_{L}-\omega_{c}$ and
$\Delta_{q}=\omega_{d}-\omega_{q}$. The transformed Hamiltonian is time-
independent in this frame of rotation. For further analysis, we shift the
frame to mean field using the following displacement transformation,

$\mathcal{D}(\alpha,\mu,\beta)=\exp{\left[\alpha(\hat{a}-\hat{a}^{\dagger})+\mu(\hat{c}-\hat{c}^{\dagger})+\beta(\hat{b}-\hat{b}^{\dagger})\right]},$
(4)

where $\alpha$, $\mu$, $\beta$ are real scalar quantities. For a particular
choice of $\alpha=\bar{\alpha}$, $\mu=\bar{\mu}$ and $\beta=\bar{\beta}$, all
the drive terms (terms proportional to $\hat{a}+\hat{a}^{\dagger}$,
$\hat{b}+\hat{b}^{\dagger}$, and $\hat{c}+\hat{c}^{\dagger}$) get cancelled.
After dropping the third and higher order terms, we arrive at the following
effective Hamiltonian,

$\hat{\mathcal{H}^{\prime}}\approx-\tilde{\Delta}_{c}\hat{a}^{\dagger}\hat{a}-\tilde{\Delta}_{q}\hat{c}^{\dagger}\hat{c}-\eta(\hat{c}^{2}+\hat{c}^{\dagger
2})+\omega_{m}\hat{b}^{\dagger}\hat{b}\\\
+J(\hat{a}+\hat{a}^{\dagger})(\hat{c}+\hat{c}^{\dagger})+g(\hat{c}+\hat{c}^{\dagger})(\hat{b}+\hat{b}^{\dagger}),$
(5)

where $\tilde{\Delta}_{c}=\Delta_{c}-\chi\bar{\mu}^{2}$,
$\tilde{\Delta}_{q}=\Delta_{q}+2\alpha_{q}\bar{\mu}^{2}-\chi\bar{\alpha}^{2}-2g_{0}\bar{\beta}$,
$\eta=\frac{\alpha_{q}\bar{\mu}^{2}}{2}$, $J=\chi\bar{\alpha}\bar{\mu}$ and
$g=g_{0}\bar{\mu}$. It might be important to underline here that the coupling
rates $g$ and $J$ as defined above are the scaled coupling rates. They show
the scaling with drive tone amplitude similar to the case in linear
optomechanical device.

##  III Equations of motion:

Dynamics of the system depends on various decay rates associated with
different modes and drive amplitudes. We write the equations of motion for the
field operators while incorporating all the noise operators and decay rates
as,

$\displaystyle\dot{\hat{a}}$
$\displaystyle=-{i}\left[\hat{a},\hat{\mathcal{H}^{\prime}}\right]-\frac{\kappa}{2}\hat{a}+\sqrt{\kappa_{ex}}~{}\hat{a}_{in}+\sqrt{\kappa_{0}}~{}\hat{f}_{in},$
(6a) $\displaystyle\dot{\hat{c}}$
$\displaystyle=-{i}\left[\hat{c},\hat{\mathcal{H}^{\prime}}\right]-\frac{\Gamma}{2}\hat{c}+\sqrt{\Gamma_{ex}}~{}\hat{c}_{in}+\sqrt{\Gamma_{0}}~{}\hat{\xi}_{in},$
(6b) $\displaystyle\dot{\hat{b}}$
$\displaystyle=-{i}\left[\hat{b},\hat{\mathcal{H}^{\prime}}\right]-\frac{\gamma_{m}}{2}\hat{b}+\sqrt{\gamma_{m}}~{}\hat{b}_{in},$
(6c)

where $\hat{a}_{in}$, $\hat{c}_{in}$, $\hat{b}_{in}$, $\hat{f}_{in}$,
$\hat{\xi}_{in}$ are noise operators of cavity, qubit and mechanical mode,
respectively. The mechanical energy dissipation rate is $\gamma_{m}$. The
internal, external and total cavity (qubit) dissipation rates are $\kappa_{0}$
($\Gamma_{0}$), $\kappa_{ex}$ ($\Gamma_{ex}$), and $\kappa$ ($\Gamma$),
respectively.

This set of equations can be easily solved by performing a Fourier
transformation, defined as
$x[\omega]=\mathcal{F}\left[x(t)\right]=\int_{-\infty}^{+\infty}x(t)\mathrm{e}^{{i}\omega
t}\mathrm{d}t$, of the equations. We now define a field vector
$u[\omega]=\begin{bmatrix}\hat{a}[\omega]&(\hat{a}^{\dagger})[\omega]&\hat{c}[\omega]&(\hat{c}^{\dagger})[\omega]&\hat{b}[\omega]&(\hat{b}^{\dagger})[\omega]\end{bmatrix}^{T}$
and evaluate its governing equation of the form,

$u[\omega]=(-{i}\omega\mathbb{1}-A)^{-1}~{}r[\omega]=\mathcal{B}~{}r[\omega],$
(7)

where,

$r[\omega]=\begin{bmatrix}\sqrt{\kappa_{ex}}~{}\hat{a}_{in}[\omega]+\sqrt{\kappa_{0}}~{}\hat{f}_{in}[\omega]\\\
\sqrt{\kappa_{ex}}~{}(\hat{a}_{in}^{\dagger})[\omega]+\sqrt{\kappa_{0}}~{}(\hat{f}_{in}^{\dagger})[\omega]\\\
\sqrt{\Gamma_{ex}}~{}\hat{c}_{in}[\omega]+\sqrt{\Gamma_{0}}~{}\hat{\xi}_{in}[\omega]\\\
\sqrt{\Gamma_{ex}}~{}(\hat{c}_{in}^{\dagger})[\omega]+\sqrt{\Gamma_{0}}~{}(\hat{\xi}_{in}^{\dagger})[\omega]\\\
\sqrt{\gamma_{m}}~{}\hat{b}_{in}[\omega]\\\
\sqrt{\gamma_{m}}~{}(\hat{b}_{in}^{\dagger})[\omega]\end{bmatrix}$ (8)

The matrix $\mathcal{B}$ can be calculated from Eq. (5) and Eq. (6), as

$\mathcal{B}=\begin{bmatrix}1/\chi_{c}&0&{i}J&{i}J&0&0\\\
0&1/\tilde{\chi}_{c}&-{i}J&-{i}J&0&0\\\
{i}J&{i}J&1/\chi_{q}&-2{i}\eta&{i}g&{i}g\\\
-{i}J&-{i}J&2{i}\eta&1/\tilde{\chi}_{q}&-{i}g&-{i}g\\\
0&0&{i}g&{i}g&1/\chi_{m}&0\\\
0&0&-{i}g&-{i}g&0&1/\tilde{\chi}_{m}\end{bmatrix}^{\scalebox{1.0}{-1}}.$ (9)

All $\chi$’s in the matrix represent the susceptibility of the modes, defined
as,

$\chi_{c}[\omega]=\frac{1}{-{i}\omega-{i}\tilde{\Delta}_{c}+\frac{\kappa}{2}};~{}\tilde{\chi}_{c}[\omega]=\frac{1}{-{i}\omega+{i}\tilde{\Delta}_{c}+\frac{\kappa}{2}}$
$\chi_{q}[\omega]=\frac{1}{-{i}\omega-{i}\tilde{\Delta}_{q}+\frac{\Gamma}{2}};~{}\tilde{\chi}_{q}[\omega]=\frac{1}{-{i}\omega+{i}\tilde{\Delta}_{q}+\frac{\Gamma}{2}}$
$\chi_{m}[\omega]=\frac{1}{-{i}\omega+{i}\omega_{m}+\frac{\gamma_{m}}{2}};~{}\tilde{\chi}_{m}[\omega]=\frac{1}{-{i}\omega-{i}\omega_{m}+\frac{\gamma_{m}}{2}}.$

From Eq. 7, we can solve for the field operators. Further, we define the
spectrum of any mode as,

$S_{\mathcal{O}}(\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\langle(\hat{\mathcal{O}}[\omega^{\prime}])^{\dagger}\hat{\mathcal{O}}[\omega]\rangle\mathrm{d}\omega^{\prime}.$
(10)

Eq. 10 and the solution of field operators can be used to get the spectrum of
the modes. The detailed calculations and the correlators of noise operators
are given in Appendix A. The calculated spectrum as follows,

$S_{\text{x}}(\omega)\Big{|}_{\text{x}\epsilon\\{1,3,5\\}}=n_{m}^{i}\gamma_{m}(|\mathcal{B}_{\text{x}5}[\omega]|^{2}+|\mathcal{B}_{\text{x}6}[\omega]|^{2})+\\\
\kappa|\mathcal{B}_{\text{x}2}[\omega]|^{2}+\Gamma|\mathcal{B}_{\text{x}4}[\omega]|^{2}+\gamma_{m}|\mathcal{B}_{\text{x}6}[\omega]|^{2},$
(11)

where $n_{m}^{i}$ is the initial phonon occupation in the mechanical mode. The
indexing $\\{S_{1},S_{3},S_{5}\\}$ maps to the spectrum of cavity, qubit and
mechanics as $\\{S_{a},S_{c},S_{b}\\}$, respectively.

##  IV Spectrum of the qubit and the mechanical mode

Figure 2: Plot of the qubit spectrum for two different values of drive
detunings, $\tilde{\Delta}_{q}=-1.0~{}\omega_{m}$ and
$\tilde{\Delta}_{q}=-1.2~{}\omega_{m}$. The parameters used for the plots:
$\tilde{\Delta}_{c}=0$, $\omega_{m}=2\pi\times 6$ MHz, $J=2\pi\times 0.8$ MHz,
$g=2\pi\times 2$ kHz, $\kappa=2\pi\times 4$ MHz, $\omega_{m}/\Gamma=5$,
$\gamma=2\pi\times 6$ Hz, and $\eta=2\pi\times 2$ MHz.  Figure 3: Cooling of
the mechanical mode:  The spectrum of the mechanical mode is analyzed to
characterize the effect of back action arising from the drive tone near the
qubit frequency $\omega_{q}$. The extracted parameters for effective
mechanical linewidth and shift in the mechanical resonant frequency as the
electromechanical coupling between the qubit and the mechanical mode is
varied, are shown in (a) and (b). Panel (c) shows the final phonon occupancy
of the mechanical mode. It is extracted by calculating the area under the
Lorentzian in mechanical spectrum. For large qubit-mechanics coupling a final
phonon occupation well-below 1 can be achieved for various sideband
parameters. (d) Final phonon occupancy as a function of qubit-mechanics
coupling and scaled detuning between the drive and the qubit frequency for
$\omega_{m}/\Gamma=5$. The parameters used for the plots are:
$\tilde{\Delta}_{c}=0$, $\omega_{m}=2\pi\times 6$ MHz, $J=2\pi\times 0.8$ MHz,
$\eta=2\pi\times 2$ MHz, $\kappa=2\pi\times 4$ MHz, $\gamma=2\pi\times 6$ Hz,
$n_{m}^{i}=300$. For the plot in panel (a), (b), and (c), we use
$\tilde{\Delta}_{q}=-1.2~{}\omega_{m}$ as the detuning.

In this section, we discuss the best cooling scenario of the mechanical
resonator by inspecting the qubit spectrum. Fig. 2 shows the spectrum of the
transmon qubit for two different detuning of the drive
tone,($\tilde{\Delta}_{q}=-1.0~{}\omega_{m}$ and
$\tilde{\Delta}_{q}=-1.2~{}\omega_{m}$). In presence of a the nearly red
detuned drive on qubit mode, its spectrum becomes asymmetric. The cooling rate
is calculated from the asymmetry of the spectrum, which is large for a
specific drive position. In the weak coupling regime ($g<<\Gamma$), the
cooling rate for the mechanical resonator is given by
$\Gamma_{c}=2[g^{2}(S_{c}(\omega_{m})-S_{c}(-\omega_{m}))+\gamma_{m}]$ [17,
12]. The optimum cooling rate, as seen from Fig. 2, is a function of the
position of the drive [17]. Unlike a linear cavity as a bath for cooling, the
cooling rate of a mechanical resonator for an anharmonic oscillator (the
qubit) depends on the position of the cooling tone applied and the
anharmonicity of the resonator mode. This is a direct consequence of the Kerr-
term. In the steady state, the final phonon occupancy can be calculated from
the cooling rate and the qubit spectrum as,

$n_{f}=2\frac{n_{m}^{i}\gamma_{m}}{\Gamma_{c}}+2g^{2}\frac{S_{c}(-\omega_{m})}{\Gamma_{c}}.$
(12)

To further understand the backaction on the mechanical resonator due to a
drive on the qubit mode, we compute the mechanical spectrum $S_{b}(\omega)$.
In the steady state, the mean phonon occupancy of the mechanical mode can be
calculated as $n_{f}=\frac{1}{2\pi}\int S_{b}(\omega)d\omega$, which is the
area under the Lorentzian in the mechanical mode spectrum. While it is
possible to reduce the expression of the mechanical spectrum to a Lorentzian
form, we find it more efficient to compute the spectrum and carry out a
numerical fit to extract the effective linewidth and the effective resonant
frequency. Fig. 3(a) and Fig. 3(b) show the linewidth broadening and resonant
frequency shift of the mechanical mode, for a red detuned
($\tilde{\Delta}_{q}=-1.2~{}\omega_{m}$) qubit drive. The back-action on the
mechanical resonator from the drive on qubit is reflected in the change of
mechanical frequency and an increase in the effective linewidth. The final
phonon occupation is plotted in Fig. 3(c) for different value of sideband
parameter $\omega_{m}/\Gamma$. It is evident from the figure that in the
steady driving of the qubit, the final phonon occupancy strongly depends on
sideband parameter $\omega_{m}/\Gamma$. A larger value of sideband parameter
offers better cooling of the mechanical mode. It is important to underline
here that the cooling to the quantum ground state of the mechanical resonator
is possible well before entering the strong coupling regime,
$g~{}\gtrsim\text{max}~{}(\Gamma,\kappa)$.

To gain insight into the spectrum calculation, we consider a simpler case when
qubit anharmonicity is set to zero $\eta=0$, and it is being driven at the
lower mechanical sideband $\tilde{\Delta}_{q}=-\omega_{m}$. With these
parameters and Eq. 10, the mechanical spectrum can be approximately written
as,

$S_{b}(\omega)=\frac{n_{m}^{i}\gamma_{m}\Gamma^{2}/(\Gamma^{2}-8g^{2})}{(\omega-\omega_{m})^{2}+\frac{(4g^{2}+\gamma_{m}\Gamma)^{2}}{4(\Gamma^{2}-8g^{2})}}.$
(13)

From this simplified expression of the mechanical spectrum, we can write the
effective line-width of the mechanical resonator as,
$\gamma_{eff}=\frac{4g^{2}+\gamma_{m}\Gamma}{\sqrt{\Gamma^{2}-8g^{2}}}\simeq\gamma_{m}(C+1)$,
where $C=\frac{4g^{2}}{\gamma_{m}\Gamma}$ is defined as the cooperativity.
Similarly, the final mean phonon occupation can be written as,
$n_{f}=\frac{n_{m}^{i}\gamma_{m}\Gamma^{2}}{4g^{2}+\gamma_{m}\Gamma}\frac{1}{\sqrt{\Gamma^{2}-8g^{2}}}\simeq\frac{n_{m}^{i}}{1+C}$
for $\Gamma\gg g$. We note that in the limit of zero anharmonicity and weak
coupling, the results are consistent with that obtain from linear cavity
optomechanics [2].

For the model Hamiltonian given by Eq. 5, the mean phonon-occupation can also
be obtained by solving Lindblad master equation. Here, we obtain the equations
of motion for the expectation values of mode operators and solve for the
steady-state solutions. From this formalism, we calculate the steady-state
occupancy in the mechanical mode for the various drive detuning
$\tilde{\Delta}_{q}$ and coupling $g$. Fig. 3(d) shows the color plot of the
final phonon occupation for the sideband parameter of $\omega_{m}/\Gamma=5$.
We can see that the optimum cooling can be achieved near the detuning of
$\tilde{\Delta}_{q}\approx-1.2\omega_{m}$. It is important to emphasize here
that the lowest phonon occupation of the mechanical resonator depends on the
device parameters, such as qubit thermal occupation and dissipation rate
$\Gamma$. For the calculations presented in this section, we assumed the
thermal occupation of the qubit and readout cavity to be zero. Another
important parameter that affects the ultimate performance of the sideband
cooling is sideband parameter $\omega_{m}/\Gamma$ [12], and cooling to the
ground state can only be achieved in sideband-resolved limit
$\omega_{m}/\Gamma~{}\gtrsim~{}1$.

##  V Experimental Details

Figure 4:  (a) A SEM-image of the device showing the suspended part of the
SQUID loop and the Josephson junctions. The length and width of the nanowire
is 40 $\mu$m and 200 nm, respectively. The scale bar corresponds to 5 $\mu$m.
(b) Color plot of the cavity transmission $|S_{21}|$ as a function of the
magnetic flux through the SQUID loop. (c) Two-tone measurements spectroscopic
linewidth of the qubit in the dispersive regime.

After discussing the performance of the sideband cooling when the qubit is
dispersively coupled to the readout cavity, we address the next question on
the possibility of the mechanical readout. In the dispersive regime, there is
no direct coupling between the cavity and the mechanical resonator. The
modulation of qubit frequency translates to the cavity mode via dispersive
coupling, and thus creating an effective coupling between the cavity and the
mechanical motion. By tuning the transmon qubit frequency near half flux
quantum, a large electromechanical coupling with the qubit mode can be
obtained. However, when $|\omega_{q}-\omega_{c}|$ is large, the effective
coupling between the cavity and mechanical mode is suppressed. Next, we show
that the addition of cooling tone near the qubit frequency is helpful for the
readout of the mechanical motion.

Figure 5: Experimental Data:  Power spectral density of the cavity mode is
measured while varying the drive detuning from the qubit mode. (a) Schematic
of the measurement process. A drive is present near the qubit mode. The
detuning between the qubit and the drive frequency is being changed in the
measurement. A probe of frequency $\omega_{c}$ is added and its lower and
upper mechanical sidebands are recorded with a spectrum analyzer. (b) The
spectral density is shown for the drive detuning of
$\tilde{\Delta}_{q}=-1.7~{}\omega_{m}$ and
$\tilde{\Delta}_{q}=+1.7~{}\omega_{m}$. We can observe the difference in
spectral height as the detuning change sign. The mechanical resonator has a
frequency of $\omega_{m}/2\pi\approx$ 5.9 MHz and a linewidth
$\gamma_{m}/2\pi=$ 6 Hz. (c) A colorplot of normalized spectral density as a
function of detuning and measurement frequency.  Figure 6:  (a) We have
evaluated the expression for the cavity mode spectrum from the theoretical
model as a function of detuning $\tilde{\Delta}_{q}$ and frequency. Parameters
are taken from the device studied here. (b) Plot of integrated spectrum
$S_{T}=\int S_{a}(\omega)d\omega$ for different detuning is calculated from
the theoretical and experimental results. The square points indicate the
experimental data, plotted as a function of drive detuning
($\tilde{\Delta}_{q}$). The solid curve is plotted for estimated device
parameters from analytical expression. The dashed straight line indicates
noise level of the measurements. The parameters used for the plots:
$\tilde{\Delta}_{c}=0$, $\omega_{m}=2\pi\times 5.9$ MHz, $J=2\pi\times 5.6$
MHz, $g=2\pi\times 3.6$ kHz, $\eta=2\pi\times 2.8$ MHz, $\kappa=2\pi\times 4$
MHz, $\Gamma=2\pi\times 8$ MHz, $\gamma_{m}=2\pi\times 6$ Hz, $n_{m}^{i}$ =
350.

For experimental realization, we use a device consisting of a transmon qubit
with a doubly clamped suspended nanowire embedded in the SQUID loop. For the
qubit readout, we use a 3D copper rectangular waveguide cavity. The scanning
electron microscope (SEM) image of the device is shown in Fig. 4(a). The
transmon, fabricated on a silicon substrate coated with highly stressed SiN,
is designed to have tunable frequency realized via SQUID. One arm of the SQUID
is made suspended to form a nanowire, essentially establishing the mechanical
mode. The silicon substrate is placed inside the readout cavity and cool down
to 20 mK in a dilution refrigerator. A detailed description of the device
fabrication methods and the measurement setup can be found in Ref. [31].

Fig. 4(b) shows the cavity transmission amplitude $|S_{21}|$ as the magnetic
flux through the SQUID loop is varied. When the qubit is brought in resonance
with the cavity mode, the vacuum-Rabi splitting is observed and two hybrid
modes emerge as indicated by the dashed box in Fig. 4(b). From the avoided-
crossing, we determine the qubit-cavity coupling strength to be $75$ MHz. We
measure the dressed cavity frequency to be 6.006 GHz, the maximum qubit
frequency to be 7.8 GHz, and the qubit anharmonicity to be $-130$ MHz. We
apply a magnetic field of $B\approx$ 1.1 mT, perpendicular to the plane of the
SQUID loop. It couples the in-plane motion of the mechanical resonator to the
qubit.

To operate in the dispersive limit, we choose a qubit detuning
$\Delta=\omega_{q}-\omega_{c}$ of $-2\pi\times$900 MHz. A representative two-
tone measurement of the qubit is shown in Fig. 4(c). To record the mechanical
motion at this operating point, we apply two tones to the device, a drive tone
near the qubit frequency and a probe tone near $\omega_{c}$ and record the
mechanical sidebands of the probe tone using a spectrum analyzer. The
positioning of various frequencies and drive tones are shown in Fig. 5(a).

Fig. 5(b) shows the recorded spectrum for two different detunings. The
experimentally measured microwave spectrum $S_{mw}(\omega)$ is normalized and
represented in the units of intra-cavity photons defined as,
$\tilde{S}_{a}=S_{mw}(\omega)/(\hbar\omega_{c}G\kappa_{ex}RBW)$, where $G$ is
the estimated net gain of the output line, $\kappa_{ex}$ is the external
coupling rate of the output port of the cavity, and $RBW$ is the resolution
bandwidth of the spectrum analyzer. Clearly, the spectrum has a larger peak
for negative detuning as compared to the one for the positive detuning. This
asymmetry becomes quite evident as the detuning of qubit drive is varied. Fig.
5(c) shows the colorplot of $\tilde{S}_{a}$ as drive frequency is varied
across the qubit transition.

The mechanical resonator has a frequency of $\omega_{m}/2\pi\approx$ 5.9 MHz
and a linewidth of $\gamma_{m}/2\pi\approx$ 6 Hz. Here, we do not observe any
backaction on the mechanical resonator. Both, the mechanical frequency and
linewidth do not show any measurable change as the detuning
$\tilde{\Delta}_{q}$ is varied across the qubit frequency. This is expected
behavior within the experimental parameters. For these measurements, we
estimated a single-photon coupling rate of $g_{0}/2\pi\approx$ 7.5 kHz, and
measured a qubit linewidth of $\Gamma/2\pi\approx$ 15 MHz. The lower sideband
parameter and single-photon coupling rate reduces the effect of back-action
from the qubit drive.

Another aspect of the measurement is the the enhancement of the transduction
and asymmetry with respect to $\tilde{\Delta}_{q}$. Qualitatively, it can be
understood from the qubit-cavity dispersive coupling and the Kerr-term of the
qubit mode. A drive tone near the qubit frequency acts like a parametric pump
due to the qubit-nonlinearity, resulting in the amplification of the field
fluctuations due to electromechanical coupling. Further, due to the dispersive
interaction between the qubit and the cavity mode, these field fluctuations
result in the modulation of the intracavity probe field, and hence in an
improved transduction. The asymmetry in the response is a direct manifestation
of the weak anharmonicity of the qubit.

To quantitatively understand the enhancement in the transduction and the
asymmetry in spectral density with respect to $\tilde{\Delta}_{q}$, we compute
the cavity spectrum from Eq. 10 as a function of susceptibilities.
Approximately, the cavity spectral density can be written as,

$\displaystyle S_{a}(\omega)$ $\displaystyle\approx
n_{m}^{i}\gamma_{m}(|\chi_{m}|^{2}+|\tilde{\chi}_{m}|^{2})\sigma(\omega),\text{where}$
(14) $\displaystyle\sigma(\omega)$
$\displaystyle=\bigg{|}\frac{gJ\chi_{c}\chi_{q\tilde{q}}(\Delta_{q}-2\eta)}{\Delta_{q}+2{i}\eta^{2}\chi_{q\tilde{q}}+g^{2}\chi_{m\tilde{m}}\chi_{q\tilde{q}}(\Delta_{q}-2\eta)}\bigg{|}^{2}$
(15) $\displaystyle\chi_{q\tilde{q}}$
$\displaystyle=\chi_{q}-\tilde{\chi}_{q}$ (16)
$\displaystyle\chi_{m\tilde{m}}$ $\displaystyle=\chi_{m}-\tilde{\chi}_{m}.$
(17)

Here, we note that the presence of the effective anharmonicity $\eta$ in the
above equation accounts for the asymmetry observed with respect to the
detuning of qubit drive. In the limit $\eta\to 0$, the expression of $\sigma$
becomes symmetric with respect to $\Delta_{q}$ as it enters the expression
through $\chi_{q\tilde{q}}$ only.

Similar to the measurement performed, we analyze the cavity spectral density
as $\tilde{\Delta}_{q}$ is varied. Fig. 6(a) shows theoretically calculated
$\tilde{S}_{a}(\omega)$ using the device parameters. We observe a pattern in
$S_{a}(\omega)$ which is similar to the experimental measurement. For a
quantitative comparison, we define the integrated spectrum as $S_{T}=\int
S_{a}(\omega)d\omega$ and evaluate it for experimental data. Fig. 6(b) shows
the plot of $S_{T}$ from the experimental results shown in Fig. 5(c) and
theoretical calculations. A good match validates the approximation made in
arriving at the effective Hamiltonian in the theoretical calculations.

##  VI Outlook and conclusion

To summarize, this work has investigated a coupled three-mode hybrid system
with a transmon qubit in the presence of external drives. Using the quantum
noise and the Lindblad formalism, we study the possibility of sideband cooling
of the mechanical resonator by the qubit mode. We find that the readout of the
mechanical mode is possible by coupling the transmon qubit to a readout cavity
just like in standard c-QED setup while maintaining a dispersive coupling
between the cavity and the qubit. In addition, we experimentally demonstrate
the applicability of the readout scheme, wherein the experimental results
matches closely to the analytical calculations. In this particular experiment,
we do not observe any cooling of the mechanical resonator due to lower $g_{0}$
and low sideband parameter $(\omega_{m}/\Gamma\approx~{}0.4)$. While the
achieved flux responsivity of the qubit in dispersive limit was high 16
GHz$/\Phi_{0}$, the estimated coupling rate ($g_{0}/2\pi\approx 7.5$ kHz) was
inadequate due to the lower applied magnetic field 1.1 mT.

Looking ahead, the recent experiments have shown promising results for the
transmon linewidth in the parallel magnetic field up to hundreds of mT with no
significant change in the spectroscopic linewidth [42]. In addition, the flux
responsivity of the qubit can be pushed to 40 GHz$/\Phi_{0}$ by increasing the
maximum qubit frequency. With these parameters, the single-photon
electromechanical coupling between qubit and mechanical resonator can be
enhanced up to 10 MHz, bringing the system near to ultra-strong coupling
regime [43]. Such regime opens up the possibilities of observing the photon
blockade effects [10], non-trivial ground state [7] and a path of using low
frequency mechanical resonator in the quantum technologies.

##  VII Acknowledgment

The authors thank G. S. Agarwal and Manas Kulkarni for valuable discussions.
This material is based upon work supported by the Air Force Office of
Scientific Research under award number FA2386-20-1-4003. V.S. acknowledge the
support received under the Core Research Grant by the Department of Science
and Technology (India). The authors acknowledge device fabrication facilities
at CeNSE, IISc Bangalore, and central facilities at the Department of Physics
funded by DST.

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## Appendix A

Spectrum of the cavity mode is calculated from Eq. 10 in the main text. In the
cavity operator, it can be written as,

$S_{a}(\omega)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\langle(\hat{a}[\omega^{\prime}])^{\dagger}\hat{a}[\omega]\rangle\mathrm{d}\omega^{\prime}.$
(18)

Eq. 7 is used to calculate the steady state value of $\hat{a}[\omega]$, which
can be written as

$\hat{a}[\omega]=(\mathcal{B}~{}r[\omega])_{11}=\sum_{j}\mathcal{B}_{1j}(r[\omega])_{j1},$
(19)

where $r[\omega]$ is a column matrix of noise operators of all the modes.

$r[\omega]=\begin{bmatrix}\sqrt{\kappa_{ex}}~{}\hat{a}_{in}[\omega]+\sqrt{\kappa_{0}}~{}\hat{f}_{in}[\omega]\\\
\sqrt{\kappa_{ex}}~{}(\hat{a}_{in}^{\dagger})[\omega]+\sqrt{\kappa_{0}}~{}(\hat{f}_{in}^{\dagger})[\omega]\\\
\sqrt{\Gamma_{ex}}~{}\hat{c}_{in}[\omega]+\sqrt{\Gamma_{0}}~{}\hat{\xi}_{in}[\omega]\\\
\sqrt{\Gamma_{ex}}~{}(\hat{c}_{in}^{\dagger})[\omega]+\sqrt{\Gamma_{0}}~{}(\hat{\xi}_{in}^{\dagger})[\omega]\\\
\sqrt{\gamma_{m}}~{}\hat{b}_{in}[\omega]\\\
\sqrt{\gamma_{m}}~{}(\hat{b}_{in}^{\dagger})[\omega]\end{bmatrix}$ (20)

The noise operators in the frequency domain satisfy the following relations,

$\langle\hat{a}_{in}[\omega](\hat{a}_{in}[\omega^{\prime}])^{\dagger}\rangle=2\pi\delta(\omega-\omega^{\prime});~{}\langle(\hat{a}_{in}[\omega])^{\dagger}\hat{a}_{in}[\omega^{\prime}]\rangle=0$
(21a)
$\langle\hat{f}_{in}[\omega](\hat{f}_{in}[\omega^{\prime}])^{\dagger}\rangle=2\pi\delta(\omega-\omega^{\prime});~{}\langle(\hat{f}_{in}[\omega])^{\dagger}\hat{f}_{in}[\omega^{\prime}]\rangle=0$
(21b)

$\langle\hat{c}_{in}[\omega](\hat{c}_{in}[\omega^{\prime}])^{\dagger}\rangle=2\pi\delta(\omega-\omega^{\prime});~{}\langle(\hat{c}_{in}[\omega])^{\dagger}\hat{c}_{in}[\omega^{\prime}]\rangle=0$
(21c)
$\langle\hat{\xi}_{in}[\omega](\hat{\xi}_{in}[\omega^{\prime}])^{\dagger}\rangle=2\pi\delta(\omega-\omega^{\prime});~{}\langle(\hat{\xi}_{in}[\omega])^{\dagger}\hat{\xi}_{in}[\omega^{\prime}]\rangle=0$
(21d)
$\langle\hat{b}_{in}[\omega](\hat{b}_{in}[\omega^{\prime}])^{\dagger}\rangle=2\pi(n_{m}^{i}+1)\delta(\omega-\omega^{\prime})$
(21e)
$\langle(\hat{b}_{in}[\omega])^{\dagger}\hat{b}_{in}[\omega^{\prime}]\rangle=2\pi
n_{m}^{i}\delta(\omega-\omega^{\prime}),$ (21f)

where $n_{m}^{i}$ is the thermal phonon occupancy of the mechanical mode. We
can expand the Eq. 19 and write the solution of $\hat{a}[\omega]$ as,

$\hat{a}[\omega]=\sqrt{\kappa_{ex}}\mathcal{B}_{11}[\omega]~{}\hat{a}_{in}[\omega]+\sqrt{\kappa_{0}}\mathcal{B}_{11}[\omega]~{}\hat{f}_{in}[\omega]+\sqrt{\kappa_{ex}}\mathcal{B}_{12}[\omega]~{}(\hat{a}_{in}^{\dagger})[\omega]\\\
+\sqrt{\kappa_{0}}\mathcal{B}_{12}[\omega]~{}(\hat{f}_{in}^{\dagger})[\omega]+\sqrt{\Gamma_{ex}}\mathcal{B}_{13}[\omega]~{}\hat{c}_{in}[\omega]+\sqrt{\Gamma_{0}}\mathcal{B}_{13}[\omega]~{}\hat{\xi}_{in}[\omega]\\\
+\sqrt{\Gamma_{ex}}\mathcal{B}_{14}[\omega]~{}(\hat{c}_{in}^{\dagger})[\omega]+\sqrt{\Gamma_{0}}\mathcal{B}_{14}[\omega]~{}(\hat{\xi}_{in}^{\dagger})[\omega]+\sqrt{\gamma_{m}}\mathcal{B}_{15}[\omega]~{}\hat{b}_{in}[\omega]\\\
+\sqrt{\gamma_{m}}\mathcal{B}_{16}[\omega]~{}(\hat{b}_{in}^{\dagger})[\omega].$
(22)

By using the identity $(x^{\dagger})[\omega]=(x[-\omega])^{\dagger}$, we can
re-write the solution of $\hat{a}[\omega]$.

$\hat{a}[\omega]=\sqrt{\kappa_{ex}}\mathcal{B}_{11}[\omega]~{}\hat{a}_{in}[\omega]+\sqrt{\kappa_{0}}\mathcal{B}_{11}[\omega]~{}\hat{f}_{in}[\omega]+\sqrt{\kappa_{ex}}\mathcal{B}_{12}[\omega]~{}(\hat{a}_{in}[-\omega])^{\dagger}\\\
+\sqrt{\kappa_{0}}\mathcal{B}_{12}[\omega]~{}(\hat{f}_{in}[-\omega])^{\dagger}+\sqrt{\Gamma_{ex}}\mathcal{B}_{13}[\omega]~{}\hat{c}_{in}[\omega]+\sqrt{\Gamma_{0}}\mathcal{B}_{13}[\omega]~{}\hat{\xi}_{in}[\omega]\\\
+\sqrt{\Gamma_{ex}}\mathcal{B}_{14}[\omega]~{}(\hat{c}_{in}[-\omega])^{\dagger}+\sqrt{\Gamma_{0}}\mathcal{B}_{14}[\omega]~{}(\hat{\xi}_{in}[-\omega])^{\dagger}+\sqrt{\gamma_{m}}\mathcal{B}_{15}[\omega]~{}\hat{b}_{in}[\omega]\\\
+\sqrt{\gamma_{m}}\mathcal{B}_{16}[\omega]~{}(\hat{b}_{in}[-\omega])^{\dagger}.$
(23)

From the above equation and Eq. 21f, we can calculate
$\langle(\hat{a}[\omega^{\prime}])^{\dagger}\hat{a}[\omega]\rangle$,

$\langle(\hat{a}[\omega^{\prime}])^{\dagger}\hat{a}[\omega]\rangle=2\pi\kappa_{ex}\mathcal{B}_{12}^{\ast}[\omega^{\prime}]\mathcal{B}_{12}[\omega]~{}\delta(\omega-\omega^{\prime})\\\
+2\pi\kappa_{0}\mathcal{B}_{12}^{\ast}[\omega^{\prime}]\mathcal{B}_{12}[\omega]~{}\delta(\omega-\omega^{\prime})\\\
+2\pi\Gamma_{ex}\mathcal{B}_{14}^{\ast}[\omega^{\prime}]\mathcal{B}_{14}[\omega]~{}\delta(\omega-\omega^{\prime})\\\
+2\pi\Gamma_{0}\mathcal{B}_{14}^{\ast}[\omega^{\prime}]\mathcal{B}_{14}[\omega]~{}\delta(\omega-\omega^{\prime})\\\
+2\pi
n_{m}^{i}\gamma_{m}\mathcal{B}_{15}^{\ast}[\omega^{\prime}]\mathcal{B}_{15}[\omega]~{}\delta(\omega^{\prime}-\omega)\\\
+2\pi(n_{m}^{i}+1)\gamma_{m}\mathcal{B}_{16}^{\ast}[\omega^{\prime}]\mathcal{B}_{16}[\omega]~{}\delta(\omega-\omega^{\prime}).$
(24)

Substituting this to Eq. 10, the spectrum of the cavity mode can be written
as,

$S_{a}(\omega)=n_{m}^{i}\gamma_{m}(|\mathcal{B}_{15}[\omega]|^{2}+|\mathcal{B}_{16}[\omega]|^{2})+\kappa|\mathcal{B}_{12}[\omega]|^{2}+\Gamma|\mathcal{B}_{14}[\omega]|^{2}+\gamma_{m}|\mathcal{B}_{16}[\omega]|^{2},$
(25)

where $\kappa$, $\Gamma$ and $\gamma_{m}$ are total dissipation rates of the
cavity, qubit and mechanical mode respectively. $n_{m}^{i}$ is the initial
mechanical mode occupancy. The terms $\mathcal{B}_{12}[\omega]$,
$\mathcal{B}_{14}[\omega]$, $\mathcal{B}_{15}[\omega]$,
$\mathcal{B}_{16}[\omega]$ are calculated using Wolfram Mathematica.

$\leavevmode\resizebox{460.35788pt}{}{$\mathcal{B}_{12}[\omega]=-\frac{J^{2}\chi_{c}\tilde{\chi}_{c}(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))}{-4{i}\eta^{2}\chi_{q}\tilde{\chi}_{q}+J^{2}(\chi_{c}-\tilde{\chi}_{c})(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))+{i}(1+g^{2}(\chi_{q}-\tilde{\chi}_{q})(\chi_{m}-\tilde{\chi}_{m}))+4g^{2}\eta\chi_{q}\tilde{\chi}_{q}(\chi_{m}-\tilde{\chi}_{m})}$}$
(26a)
$\leavevmode\resizebox{460.35788pt}{}{$\mathcal{B}_{14}[\omega]=\frac{J\chi_{c}(1+2{i}\eta\chi_{q})\tilde{\chi}_{q}}{-4{i}\eta^{2}\chi_{q}\tilde{\chi}_{q}+J^{2}(\chi_{c}-\tilde{\chi}_{c})(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))+{i}(1+g^{2}(\chi_{q}-\tilde{\chi}_{q})(\chi_{m}-\tilde{\chi}_{m}))+4g^{2}\eta\chi_{q}\tilde{\chi}_{q}(\chi_{m}-\tilde{\chi}_{m})}$}$
(26b)

$\leavevmode\resizebox{460.35788pt}{}{$\mathcal{B}_{15}[\omega]=-\frac{gJ\chi_{c}(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))\chi_{m}}{-4{i}\eta^{2}\chi_{q}\tilde{\chi}_{q}+J^{2}(\chi_{c}-\tilde{\chi}_{c})(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))+{i}(1+g^{2}(\chi_{q}-\tilde{\chi}_{q})(\chi_{m}-\tilde{\chi}_{m}))+4g^{2}\eta\chi_{q}\tilde{\chi}_{q}(\chi_{m}-\tilde{\chi}_{m})}$}$
(26c)
$\leavevmode\resizebox{460.35788pt}{}{$\mathcal{B}_{16}[\omega]=-\frac{gJ\chi_{c}(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))\tilde{\chi}_{m}}{-4{i}\eta^{2}\chi_{q}\tilde{\chi}_{q}+J^{2}(\chi_{c}-\tilde{\chi}_{c})(-{i}\tilde{\chi}_{q}+\chi_{q}({i}+4\eta\tilde{\chi}_{q}))+{i}(1+g^{2}(\chi_{q}-\tilde{\chi}_{q})(\chi_{m}-\tilde{\chi}_{m}))+4g^{2}\eta\chi_{q}\tilde{\chi}_{q}(\chi_{m}-\tilde{\chi}_{m})}$}.$
(26d)

## Appendix B

From the Lindblad formalism the time-domain master equation of the density
operator $\dot{\hat{\rho}}(t)$ is written as,

$\dot{\hat{\rho}}={i}[\hat{\rho},\tilde{\mathcal{H}}]+\kappa(n_{c}^{i}+1)\mathcal{D}[\hat{a}]\hat{\rho}+\kappa
n_{c}^{i}\mathcal{D}[\hat{a}^{\dagger}]\hat{\rho}+\Gamma(n_{q}^{i}+1)\mathcal{D}[\hat{c}]\hat{\rho}+\Gamma
n_{q}^{i}\mathcal{D}[\hat{c}^{\dagger}]\hat{\rho}+\frac{\Gamma_{\phi}}{2}\mathcal{D}[\hat{c}^{\dagger}\hat{c}]\hat{\rho}\\\
+\gamma_{m}(n_{m}^{i}+1)\mathcal{D}[\hat{b}]\hat{\rho}+\gamma_{m}n_{m}^{i}\mathcal{D}[\hat{b}^{\dagger}]\hat{\rho}.$
(27)

Here $\kappa$ and $\gamma_{m}$ are energy relaxation rates of cavity and
mechanical mode. Qubit relaxation and pure dephasing are represented as
$\Gamma$ and $\Gamma_{\phi}$. The initial thermal occupancy of the cavity,
qubit and the mechanical modes are $n_{c}^{i}$, $n_{q}^{i}$, and $n_{m}^{i}$
respectively. For our calculation we have considered $\Gamma_{\phi}=0$.
$\mathcal{D}[\hat{\mathcal{O}}]$ is the Lindblad super-operator written as,

$\mathcal{D}[\hat{\mathcal{O}}]\hat{\rho}:=\hat{\mathcal{O}}\hat{\rho}\hat{\mathcal{O}}^{\dagger}-\frac{1}{2}\hat{\mathcal{O}}^{\dagger}\hat{\mathcal{O}}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{\mathcal{O}}^{\dagger}\hat{\mathcal{O}}.$
(28)

We write down the equation of motion from the Hamiltonian in Eq. (5). This is
to calculate expectation values of different operators. The coupled linear
equations are written in the matrix form,

$\dot{\boldsymbol{d}}=\mathcal{M}\boldsymbol{d}+\mathcal{N},$ (29)

where $\boldsymbol{d}$ is the column matrix consisting of the expectations
values.

$\boldsymbol{d}=\begin{bmatrix}\langle\hat{a}^{\dagger}\hat{a}\rangle\\\
\langle\hat{b}^{\dagger}\hat{b}\rangle\\\
\langle\hat{c}^{\dagger}\hat{c}\rangle\\\ \langle\hat{a}^{2}\rangle\\\
\langle\hat{a}^{\dagger 2}\rangle\\\ \langle\hat{b}^{2}\rangle\\\
\langle\hat{b}^{\dagger 2}\rangle\\\ \langle\hat{c}\rangle\\\
\langle\hat{c}^{\dagger 2}\rangle\\\ \langle\hat{a}\hat{b}\rangle\\\
\langle\hat{a}^{\dagger}\hat{b}^{\dagger}\rangle\\\
\langle\hat{a}^{\dagger}\hat{b}\rangle\\\
\langle\hat{a}\hat{b}^{\dagger}\rangle\\\ \langle\hat{c}\hat{b}\rangle\\\
\langle\hat{c}^{\dagger}\hat{b}^{\dagger}\rangle\\\
\langle\hat{c}^{\dagger}\hat{b}\rangle\\\
\langle\hat{c}\hat{b}^{\dagger}\rangle\\\ \langle\hat{a}\hat{c}\rangle\\\
\langle\hat{a}^{\dagger}\hat{c}^{\dagger}\rangle\\\
\langle\hat{a}^{\dagger}\hat{c}\rangle\\\
\langle\hat{a}\hat{c}^{\dagger}\rangle\\\ \end{bmatrix}$ (30)

$\mathcal{M}=\leavevmode\resizebox{346.89731pt}{}{{\hbox{\setcounter{MaxMatrixCols}{21}\begin{bmatrix}-\kappa&0
&0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &{i}J &-{i}J &-{i}J &{i}J \\\ 0
&-\gamma_{m} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &{i}g &-{i}g &{i}g &-{i}g &0 &0
&0 &0 \\\ 0 &0 &-\Gamma&0 &0 &0 &0 &-2{i}\eta&2{i}\eta&0 &0 &0 &0 &{i}g &-{i}g
&-{i}g &{i}g &{i}J &-{i}J &{i}J &-{i}J \\\ 0 &0 &0 &1/\chi_{\hat{a}\hat{a}} &0
&0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &-2{i}J &0 &0 &-2{i}J \\\ 0 &0 &0 &0
&1/\chi_{\hat{a}\hat{a}}^{\ast} &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &2{i}J
&2{i}J &0 \\\ 0 &0 &0 &0 &0 &1/\chi_{\hat{b}\hat{b}} &0 &0 &0 &0 &0 &0 &0
&-2{i}g &0 &-2{i}g &0 &0 &0 &0 &0 \\\ 0 &0 &0 &0 &0 &0
&1/\chi_{\hat{b}\hat{b}}^{\ast} &0 &0 &0 &0 &0 &0 &0 &2{i}g &0 &2{i}g &0 &0 &0
&0 \\\ 0 &0 &4{i}\eta&0 &0 &0 &0 &1/\chi_{\hat{c}\hat{c}} &0 &0 &0 &0 &0
&-2{i}g &0 &0 &-2{i}g &-2{i}J &0 &-2{i}J &0 \\\ 0 &0 &-4{i}\eta&0 &0 &0 &0 &0
&1/\chi_{\hat{c}\hat{c}}^{\ast} &0 &0 &0 &0 &0 &2{i}g &2{i}g &0 &0 &2{i}J &0
&2{i}J \\\ 0 &0 &0 &0 &0 &0 &0 &0 &0 &1/\chi_{\hat{a}\hat{b}} &0 &0 &0 &-{i}J
&0 &-{i}J &0 &-{i}g &0 &0 &-{i}g \\\ 0 &0 &0 &0 &0 &0 &0 &0 &0 &0
&1/\chi_{\hat{a}\hat{b}}^{\ast} &0 &0 &0 &{i}J &0 &{i}J &0 &{i}g &{i}g &0 \\\
0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &1/\chi_{\hat{a}^{\dagger}\hat{b}} &0 &{i}J &0
&{i}J &0 &0 &-{i}g &-{i}g &0 \\\ 0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0
&1/\chi_{\hat{a}^{\dagger}\hat{b}}^{\ast} &0 &-{i}J &0 &-{i}J &{i}g &0 &0
&{i}g \\\ 0 &-{i}g &-{i}g &0 &0 &-{i}g &0 &-{i}g &0 &-{i}J &0 &-{i}J &0
&1/\chi_{\hat{c}\hat{b}} &0 &2{i}\eta&0 &0 &0 &0 &0 \\\ 0 &{i}g &{i}g &0 &0 &0
&{i}g &0 &{i}g &0 &{i}J &0 &{i}J &0 &1/\chi_{\hat{c}\hat{b}}^{\ast} &0
&-2{i}\eta&0 &0 &0 &0 \\\ 0 &{i}g &-{i}g &0 &0 &{i}g &0 &0 &-{i}g &{i}J &0
&{i}J &0 &-2{i}\eta&0 &1/\chi_{\hat{c}^{\dagger}\hat{b}} &0 &0 &0 &0 &0 \\\ 0
&-{i}g &{i}g &0 &0 &0 &-{i}g &{i}g &0 &0 &-{i}J &0 &-{i}J &0 &2{i}\eta&0
&1/\chi_{\hat{c}^{\dagger}\hat{b}}^{\ast} &0 &0 &0 &0 \\\ -{i}J &0 &-{i}J
&-{i}J &0 &0 &0 &-{i}J &0 &-{i}g &0 &0 &-{i}g &0 &0 &0 &0
&1/\chi_{\hat{a}\hat{c}} &0 &0 &2{i}\eta\\\ {i}J &0 &{i}J &0 &{i}J &0 &0 &0
&{i}J &0 &{i}g &{i}g &0 &0 &0 &0 &0 &0 &1/\chi_{\hat{a}\hat{c}}^{\ast}
&-2{i}\eta&0 \\\ -{i}J &0 &{i}J &0 &-{i}J &0 &0 &{i}J &0 &0 &-{i}g &-{i}g &0
&0 &0 &0 &0 &0 &2{i}\eta&1/\chi_{\hat{a}^{\dagger}\hat{c}} &0 \\\ {i}J &0
&-{i}J &{i}J &0 &0 &0 &0 &-{i}J &{i}g &0 &0 &{i}g &0 &0 &0 &0 &-2{i}\eta&0 &0
&1/\chi_{\hat{a}^{\dagger}\hat{c}}^{\ast} \\\ \end{bmatrix}}}}$ (31)

Various susceptibilities are defined below.

$\chi_{\hat{a}\hat{a}}=\frac{1}{2{i}\Delta_{c}-\kappa}$ (32a)

$\chi_{\hat{b}\hat{b}}=\frac{1}{-2{i}\omega_{m}-\gamma_{m}}$ (32b)

$\chi_{\hat{c}\hat{c}}=\frac{1}{2{i}\Delta_{q}-\Gamma-\Gamma_{\phi}}$ (32c)

$\chi_{\hat{a}\hat{b}}=\frac{1}{{i}(\Delta_{c}-\omega_{m})-\frac{1}{2}(\kappa+\gamma_{m})}$
(32d)

$\chi_{\hat{a}^{\dagger}\hat{b}}=\frac{1}{-{i}(\Delta_{c}+\omega_{m})-\frac{1}{2}(\kappa+\gamma_{m})}$
(32e)

$\chi_{\hat{c}\hat{b}}=\frac{1}{{i}(\Delta_{q}-\omega_{m})-\frac{1}{2}(\Gamma+\frac{\Gamma_{\phi}}{2}+\gamma_{m})}$
(32f)

$\chi_{\hat{c}^{\dagger}\hat{b}}=\frac{1}{-{i}(\Delta_{q}+\omega_{m})-\frac{1}{2}(\Gamma+\frac{\Gamma_{\phi}}{2}+\gamma_{m})}$
(32g)

$\chi_{\hat{a}\hat{c}}=\frac{1}{{i}(\Delta_{c}+\Delta_{q})-\frac{1}{2}(\Gamma+\frac{\Gamma_{\phi}}{2}+\kappa)}$
(32h)

$\chi_{\hat{a}^{\dagger}\hat{c}}=\frac{1}{-{i}(\Delta_{c}-\Delta_{q})-\frac{1}{2}(\Gamma+\frac{\Gamma_{\phi}}{2}+\kappa)}$
(32i)

$\mathcal{N}=\begin{bmatrix}n_{c}^{i}\kappa\\\ n_{m}^{i}\gamma_{m}\\\
n_{q}^{i}\Gamma\\\ 0\\\ 0\\\ 0\\\ 0\\\ 2{i}\eta\\\ -2{i}\eta\\\ 0\\\ 0\\\ 0\\\
0\\\ -{i}g\\\ {i}g\\\ 0\\\ 0\\\ -{i}J\\\ {i}J\\\ 0\\\ 0\end{bmatrix}$ (33)

The steady-state solution of $\boldsymbol{d}$ matrix can be written as,

$\boldsymbol{d}=-\mathcal{M}^{-1}\mathcal{N}.$ (34)

From Eq. 34 we have calculated the final mechanical occupation $n_{f}$ as a
function of the device parameters. The plot of $n_{f}$ as a function of
coupling $g$ and detuning ($\tilde{\Delta}_{q}$) is shown in Fig. 4d.