<EMAIL_ADDRESS>
# Coherent Resonant Coupling between Atoms and a Mechanical Oscillator
Mediated by Cavity-Vacuum Fluctuations

Bo Wang  School of Physics, Sun Yat-sen University, Guangzhou 510275, China
Jia-Ming Hu  School of Physics, Sun Yat-sen University, Guangzhou 510275,
China  Vincenzo Macrì  Theoretical Quantum Physics Laboratory, RIKEN Cluster
for Pioneering Research, Wako-shi, Saitama 351-0198, Japan  Ze-Liang Xiang  [
School of Physics, Sun Yat-sen University, Guangzhou 510275, China  Franco
Nori  Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering
Research, Wako-shi, Saitama 351-0198, Japan  RIKEN Center for Quantum
Computing (RQC), Wako-shi, Saitama 351-0198, Japan  Department of Physics,
University of Michigan, Ann Arbor, Michigan 48109-1040, USA

###### Abstract

We show that an atom can be coupled to a mechanical oscillator via quantum
vacuum fluctuations of a cavity field enabling energy transfer processes
between them. In a hybrid quantum system consisting of a cavity resonator with
a movable mirror and an atom, these processes are dominated by two pair-
creation mechanisms: the counterrotating (atom-cavity system) and dynamical
Casimir interaction terms (optomechanical system). Because of these two pair-
creation mechanisms, the resonant atom-mirror coupling is the result of high-
order virtual processes with different transition paths well described in our
theoretical framework. We perform a unitary transformation to the atom-mirror
system Hamiltonian, exhibiting two kinds of multiple-order transitions of the
pair creation. By tuning the frequency of the atom, we show that photon
frequency conversion can be realized within a cavity of multiple modes.
Furthermore, when involving two atoms coupled to the same mechanical mode, a
single vibrating excitation of the mechanical oscillator can be simultaneously
absorbed by the two atoms. Considering recent advances in strong and
ultrastrong coupling for cavity optomechanics and other systems, we believe
our proposals can be implemented using available technology.

###### pacs:

42.50.Pq, 42.50.Ct

##  I Introduction

Hybrid quantum systems consisting of a mechanical oscillator and two-level
systems [1, 2] are increasingly attracting attention because of their growing
potential for quantum technologies, including the storage and retrieval of
quantum states [3, 4, 5]. One of the key challenges is how to realize and
control the coupling of a mechanical oscillator and an atom at the quantum
level, despite the considerable mass imbalance between them [6]. By reaching
the strong-coupling regime, a quantum interface between the mechanical
oscillator and atoms can be achieved, allowing coherent energy transfers [1,
2]. Moreover, measurement and preparation of micromechanical objects via
state-of-the-art atomic control can be realized [1, 2]. So far, the ongoing
studies of the interaction mechanisms between them mainly include the electric
coupling [7], magnetic coupling [8, 9, 10], dipole-dipole interaction [11],
and the coupling to the potential of optical lattices [12, 13, 14, 15, 16].

Vacuum fluctuations, which is one of the most striking quantum effects
predicted by quantum field theory, have potential value for exploring quantum
technologies [17, 18, 19]. Recently, the heat transfer induced by quantum
fluctuations between two nanomechanical membranes separated by a vacuum gap
has been observed in experiments [20]. It can be attributed to pairs of
virtual photons, acting as messengers to deliver the energy or force between
objects [21, 22, 23, 24, 25, 26, 27, 28, 29]. When electromagnetic quantum
fluctuations interact with a very fast-oscillating boundary condition, e.g.,
the harmonically oscillating mirror, pairwise real excitations can be created
from the vacuum of the electromagnetic field. Such a purely quantum phenomenon
is known as the dynamical Casimir effect (DCE) [30, 31, 32], which has been
experimentally realized in superconducting circuits [19] and Josephson
metamaterials [33].

The so-called counterrotating terms (CRT) in the Rabi Hamiltonian allow the
simultaneous creation or annihilation of an excitation in both atom and cavity
mode. When the system reaches the ultrastrong-coupling regime [34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44], where the coupling rate is comparable to the bare
transition frequencies of the system components, the counterrotating terms can
be responsible for some interesting physical phenomena [45, 46, 47, 48, 49,
50, 51]. Generally, both the DCE and counterrotating terms are responsible for
the amplification mechanisms of vacuum fluctuations, respectively, in cavity-
optomechanics [52] and cavity QED systems [49]. Thus, one may use a cavity
resonator as a bridge to connect these two systems into a hybrid quantum
system, where the interaction between atoms and oscillators through two
fluctuation mechanisms may be investigated.

In this work, we show how a single or multiple cavity-optomechanics modes can
be coupled to an atom by virtual photon pairs, and how mechanical energy can
be coherently converted into a two-level excitation by the amplification
mechanisms of the cavity-vacuum fluctuations (the DCE and the counterrotating
terms). We also develop an analytical method for studying the atom-mirror
coupling, which is in good agreement with the numerical results within a
simple model consisting of a single atom and a single-cavity mode. When the
model involves multiple-cavity modes, it can realize frequency conversion
between different cavity modes. In addition, the single-atom model can be
extended to the two-atom case. By sending a mechanical drive to the movable
mirror, two atoms can be excited even though the mirror and atoms are
spatially separated. These results provide an intriguing platform for
exploring the manipulation of quantum states by quantum fluctuations.

This paper is organized as follows. In Sec. II, we describe a simple model
consisting of a single atom and a single-cavity mode. Then, we derive the
effective system Hamiltonian comparing numerical and analytical results. Last,
we study the system dynamics in this case. In Sec. III, we extend the simple
model in Sec. II from a single-cavity mode to a multiple-cavity mode, to study
photon frequency conversion between the two cavity modes. In Sec. IV, we
extend the single-atom model to a two-atom case. In this case, we show that a
single vibrating excitation of the mechanical oscillator can be absorbed
simultaneously by two atoms. In Sec. V, we provide details on potential
experimental implementations of proposed effects. Finally, a brief discussion
and conclusion are given in Sec. VI.

Figure 1: (a) Schematic of an hybrid quantum system consisting of a single
atom and a cavity resonator with a movable mirror. (b) Relevant energy levels
of the system Hamiltonian versus the ratio between the resonance frequency
$\omega_{c}$ of the cavity mode and the one $\omega_{m}$ of the mirror; the
inset displays the enlarged view of the boxed region showing the avoided-level
crossing between $|g,1,0\rangle$ and $|e,0,1\rangle$, where
$|\psi_{3,4}\rangle\simeq(1/\sqrt{2})(|g,1,0\rangle\mp|e,0,1\rangle)$ are the
symmetric and antisymmetric superposition states at the minimum splitting.

##  II single atom interacting with a cavity-optomechanical mode

In this section we consider a hybrid quantum system consisting of a cavity
resonator with a movable mirror and a single two-level atom [see Fig. 1(a)].
Here, both the cavity field and the displacement of the mirror are treated as
dynamical variables, and a canonical quantization procedure is adopted [52,
53].

###  II.1 Model

We first study the case where only one mechanical mode (cavity mode) [with
resonance frequency $\omega_{m}$ ($\omega_{c}$) and phonon (photon) operators
$\hat{b}$ ($\hat{a}$) and $\hat{b}^{\dagger}$ ($\hat{a}^{\dagger}$)] is
considered. The transition frequency of the atom is $\omega_{a}$, with the
raising and lowering operators $\hat{\sigma}_{+}$ and $\hat{\sigma}_{-}$,
respectively. The system Hamiltonian can be written as
$\hat{H}=\hat{H}_{0}+{\hat{H}}_{\rm I}$, where

$\hat{H}_{0}=\hbar\omega_{c}{\hat{a}}^{\dagger}{\hat{a}}+\hbar\omega_{m}{\hat{b}}^{\dagger}{\hat{b}}+\hbar\omega_{a}{\hat{\sigma}}_{+}{\hat{\sigma}_{-}}$
(1)

is the unperturbed Hamiltonian and ${\hat{H}}_{\rm I}={\hat{H}}_{\rm
I(AF)}+{\hat{H}}_{\rm I(FM)}$ describes the interaction. Here,

${\hat{H}}_{\rm
I(AF)}=\hbar\lambda(\hat{a}+\hat{a}^{\dagger})(\hat{\sigma}_{-}+\hat{\sigma}_{+})$
(2)

is the atom-field interaction, while the field-mirror interaction can be
described as $\hat{H}_{\rm I(FM)}=\hat{V}_{\rm om}+\hat{V}_{\rm DCE}$, where

$\hat{V}_{\rm om}=\hbar
g{\hat{a}}^{\dagger}{\hat{a}}(\hat{b}+{\hat{b}}^{\dagger})$ (3)

is the standard optomechanical interaction and

$\hat{V}_{\rm
DCE}=\frac{\hbar}{2}g({\hat{a}}^{2}+{\hat{a}}^{{\dagger}2})({\hat{b}}+{\hat{b}}^{\dagger})$
(4)

describes the creation and annihilation of photon pairs [53]. The parameter
$\lambda$ is the coupling rate between the two-level atom and the cavity mode,
and $g$ is the optomechanical coupling rate.

In most cavity-optomechanical experiments, the DCE term is always neglected
because the frequency of the oscillating mirror is much smaller than that of
the cavity mode [54, 55, 56, 57, 6], where the standard Hamiltonian of the
field-mirror coupling system [53] is good enough to describe them. However,
when the mechanical frequency is close to the frequency of the cavity mode,
the effect of the DCE terms becomes more important. Such conditions can be
realized in a superconducting circuit consisting of a coplanar transmission
line with a tunable electrical length [19], or using microwave resonators and
ultra-high-frequency mechanical nanoresonators [58, 59].

###  II.2 Analytical Method

From Eq. (3), we know that the transitions between different phonon states
$|k\rangle$ in each n-photon subspace can occur, such as
$|k\rangle\xrightarrow{\hat{V}_{\rm om}}|k+1\rangle\xrightarrow{\hat{V}_{\rm
om}}|k+2\rangle$. Thus, $\hat{V}_{\rm om}$ involves multiple-order
transitions, namely, the transition processes between the different phonon
states in a mechanical oscillator. Such transitions can occur in a high-order
process with a coupling between the cavity and mechanical modes. If another
component, such as an atom, is introduced, then a high-order energy exchange
for multiple mixed modes can occur.

Here we can eliminate the standard optomechanical coupling term $\hat{V}_{\rm
om}$ by performing a unitary transformation with the unitary operator
$\hat{U}=\exp[-\beta\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}-\hat{b})]$,
obtaining the effective system Hamiltonian (details in Appendix A)

$\hat{H}^{U}=\hat{U}^{\dagger}\hat{H}\hat{U}=\hat{H}_{0}^{U}+\hat{H}_{\rm
AFM}^{U}+\hat{H}_{\rm FM}^{U},$ (5)

where

$\begin{split}&\hat{H}_{0}^{U}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{a}\hat{\sigma}_{+}\hat{\sigma}_{-}-\hbar\frac{g^{2}}{\omega_{m}}\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a},\\\
&\hat{H}_{\rm
FM}^{U}\simeq\hbar\frac{g}{2}(\hat{a}^{{\dagger}2}+\hat{a}^{2})(\hat{b}+\hat{b}^{\dagger})\\\
&\qquad+\hbar\frac{g^{2}}{\omega_{m}}\left[(\hat{a}^{{\dagger}2}-\hat{a}^{2})(\hat{b}^{\dagger}-\hat{b})(\hat{b}+\hat{b}^{\dagger})-(\hat{a}^{{\dagger}2}+\hat{a}^{2})\hat{a}^{\dagger}\hat{a}\right],\\\
&\hat{H}_{\rm
AFM}^{U}\simeq\hbar\lambda(\hat{a}^{\dagger}+\hat{a})(\hat{\sigma}_{-}+\hat{\sigma}_{+})\\\
&\qquad\quad+\hbar\left(\frac{g\lambda}{\omega_{m}}\right)(\hat{a}^{\dagger}-\hat{a})(\hat{\sigma}_{-}+\hat{\sigma}_{+})(\hat{b}^{\dagger}-\hat{b}).\end{split}$
(6)

The Hamiltonian $\hat{H}_{0}^{U}$ is now the unperturbed one in the new frame.
The last term in $\hat{H}_{0}^{U}$, originating from the optomechanical
interaction $\hat{V}_{\rm om}$, is related to the energy shift ($\sim\hbar
g^{2}n^{2}/\omega_{m}$) of the cavity mode.

In Eqs. (6), the term $\hat{H}_{\rm FM}^{U}$, which is related to the DCE,
describes the creation and annihilation of photon pairs under the resonance
condition $2{\omega_{c}}=k{\omega_{m}}$, where $k$ is an integer. Moreover,
$\hat{H}_{\rm FM}^{U}$ involves multiple-order transitions, which are related
to the creation and annihilation of multiple vibrating excitations of the
mechanical oscillator. For instance, the term
$\hat{a}^{{\dagger}2}\hat{b}^{2}$ in $\hat{H}_{\rm FM}^{U}$ shows the
production of a photon pair by the annihilation of two excitations of the
mechanical oscillator, indicating a high-order energy exchange between the
cavity mode and the mechanical mode.

The last term $\hat{H}_{\rm AFM}^{U}$ in Eqs. (6) describes multiple couplings
involving the atom, the cavity mode, and the mechanical mode. Importantly,
$\hat{H}_{\rm AFM}^{U}$ can be used to investigate the counterrotating terms,
which takes into account the coupling between the atom and the mechanical
oscillator for the resonance condition

$q\omega_{m}\simeq\omega_{c}+\omega_{a},$ (7)

where $q$ is an integer. Indeed, the term
$\hat{a}^{\dagger}\hat{\sigma}_{+}\hat{b}$ in $\hat{H}_{\rm AFM}^{U}$ is
responsible for the production of an excitation pair by the annihilation of an
excitation of the mechanical oscillator. It is analogous to the DCE, where a
photon pair can be created by annihilating an excitation of the mechanical
oscillator. Moreover, $\hat{H}_{\rm AFM}^{U}$ involves multiple-order
transitions, indicating the high-order energy exchange, such as the term
$\hat{a}^{\dagger}\hat{\sigma}_{+}\hat{b}^{2}$ (see Appendix A).

The Hamiltonians of the DCE and the counterrotating terms describe the pair-
creation phenomena, involving the annihilation of vibrating excitations of the
mechanical oscillator. However, the intrinsic pair-creation mechanisms of
these two Hamiltonians are different and involve different transition paths
for a coupling between the atom and the mirror.

For the unperturbed Hamiltonian $\hat{H}_{0}^{U}$, the eigenstates are
described by $|j,k,n\rangle=|j\rangle\otimes|k\rangle\otimes|n\rangle$, where
$|k\rangle$ ($|n\rangle$) denotes the Fock state of the mechanical mode
(cavity mode) and $|j\rangle$ ($j=g,e$) denotes the atom state, with
eigenvalues

$E_{j,k,n}=\hbar\omega_{c}n+\hbar\omega_{m}k+\hbar\omega_{a}\langle
j|e\rangle-\hbar g^{2}n^{2}/\omega_{m}.$ (8)

Figure 2: (a) Sketch of the processes giving the main contribution to the
effective coupling between the bare states $|g,1,0\rangle$ and
$|e,0,1\rangle$, via the DCE (red arrows) and the counterrotating terms (blue
arrows). (b) Comparison between the numerical calculated normalized level
splitting (red dots) and the corresponding analytical calculations (blue solid
curve), obtained using perturbation theory.

###  II.3 Numerics and analysis

We begin our study by numerically diagonalizing the Hamiltonian $\hat{H}$ in a
truncated finite-dimensional Hilbert space. In Fig. 1(b), the blue curves
display the energy differences $E_{i}-E_{0}$ ($E_{0}$ is the ground-state
energy) of the total system Hamiltonian versus $\omega_{c}/\omega_{m}$, with
the atom resonance frequency $\omega_{a}=0.4\omega_{m}$, optomechanical
coupling rate $g=0.03\omega_{m}$, and atom-cavity coupling rate
$\lambda=0.005\omega_{m}$. Compared to the dashed gray curves, which denote
the lowest energy-level $E_{j,k,n}$ of the unperturbed $\hat{H}_{0}^{U}$,
small energy shifts and an avoided-level crossing can be observed around
$\omega_{c}/\omega_{m}\sim 0.6$ (blue curve).

This avoided-level crossing, induced by the coupling between states
$|g,1,0\rangle$ and $|e,0,1\rangle$, shows that the vibrating mirror with one
phonon can directly excite one atom and radiate one photon, simultaneously, as
shown by the inset of Fig. 1(b). At the minimum energy-level splitting
$2\Omega_{e01}^{g10}=1.83\times 10^{-3}\omega_{m}$, the resulting states are
well approximated by

$|\psi_{3,4}\rangle\simeq(1/\sqrt{2})(|g,1,0\rangle\mp|e,0,1\rangle).$ (9)

In this coupling region, the resonance condition for the DCE is
$2\omega_{c}=\omega_{m}$, which means the creation of photon pairs. Since one
of these photon pairs is converted into one atomic excitation, the resonance
condition of the whole system can be written as

$\omega_{c}+\omega_{a}=\omega_{m}.$ (10)

Note that the result of this process is the same as the resonance condition of
the multiple-couplings, $q\omega_{m}\simeq\omega_{c}+\omega_{a}$ with $q=1$.

Using perturbation theory, the coupling between states $|g,1,0\rangle$ and
$|e,0,1\rangle$ can be described by the DCE and counterrotating terms, as
shown in Fig. 2(a). This indicates that the coupling between the atom and the
mirror is induced by quantum fluctuations of a cavity field. In particular,
one of the processes is dominated by the DCE, as shown in the red inclined
double-arrow. If $|g,1,0\rangle$ is the initial state, then it can reach the
final state $|e,0,1\rangle$ through the virtual transition involving the out-
of-resonance intermediate state $|g,0,2\rangle$. Such transition processes are
dominated by the first terms of both $\hat{H}_{\rm AFM}^{U}$ and $\hat{H}_{\rm
FM}^{U}$ in Eqs. (6). The other path originates from the counterrotating
terms, as shown by the blue horizontal double-arrow. In the interaction
picture, when the counterrotating terms (e.g.,
$\hat{a}^{\dagger}\hat{\sigma}_{+}\exp[i(\omega_{c}+\omega_{a})t]$) are on
resonance with the vibrating mirror (e.g., $\hat{b}\exp[-i\omega_{m}t]$), the
coupling between states $|g,1,0\rangle$ and $|e,0,1\rangle$ can be effectively
triggered. Such a phenomenon of multiple-couplings can be justified by the
term
$(g\lambda/\omega_{m})(\hat{a}^{\dagger}\hat{\sigma}_{+}\hat{b}+\hat{a}\hat{\sigma}_{-}\hat{b}^{\dagger})$
in $\hat{H}_{\rm AFM}^{U}$. Note that the influence from the counterrotating
terms grows while the detuning between the frequencies of the atom and the
cavity mode increases. This can be indicated by the ratio

$\frac{\omega_{a}-\omega_{c}}{\omega_{a}+\omega_{c}}=\frac{1}{1+(2\omega_{c}/\bigtriangleup)}$
(11)

with the detuning $\bigtriangleup=\omega_{a}-\omega_{c}$. When the ratio
increases, the influence from the counterrotating terms grows.

Therefore, the effective coupling rate between the states $|g,1,0\rangle$ and
$|e,0,1\rangle$ becomes,

$\begin{split}\Omega_{e01}^{g10}=\,&\,\frac{\langle e,0,1|\hat{H}_{\rm
I}^{U}|g,0,2\rangle\langle g,0,2|\hat{H}_{\rm
I}^{U}|g,1,0\rangle}{E_{g,1,0}-E_{g,0,2}}\\\ \,&\,+\langle e,0,1|\hat{H}_{\rm
AFM}^{U}|g,1,0\rangle\\\ =\,&\,\frac{\lambda
g\left(\sqrt{2}-\frac{\sqrt{2}g^{2}}{2\omega_{m}^{2}}\right)\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}g^{2}}{\omega_{m}^{2}}\right)}{\omega_{m}-2\omega_{c}+4\frac{g^{2}}{\omega_{m}}}-\frac{\lambda
g}{\omega_{m}}+\frac{\lambda g^{3}}{2\omega_{m}^{3}},\end{split}$ (12)

where $\hat{H}_{\rm I}^{U}=\hat{H}_{\rm AFM}^{U}+\hat{H}_{\rm FM}^{U}$ is the
effective perturbative Hamiltonian. In calculating the effective coupling
rate, we keep the terms of the expansion of the effective Hamiltonian ($H_{\rm
FM}^{U}$ and $H_{\rm AFM}^{U}$ in Appendix A) up to the third order. The
effective coupling rate in Eq. 12 versus $g/\omega_{m}$ is in good agreement
with the numerical results when the normalized optomechanical coupling is not
too strong, as shown in Fig. 2(b).

Note that, while the optomechanical coupling is increasing to the ultra-
strongly coupling regime, the term $\hat{V}_{\rm DCE}$ can no longer be
regarded as a perturbation because this term will cause a nonnegligible
energy-shift. For avoided-level crossings at higher energy with the resonance
condition $\omega_{a}+\omega_{c}=2\omega_{m}$, a ladder of increasing level
splittings and its corresponding analytical solution are presented in Appendix
A. When this condition is met, the higher-order DCE and counterrotating terms
in the Hamiltonians $\hat{H}_{\rm FM}^{U}$ and $\hat{H}_{\rm AFM}^{U}$
respectively, are necessary for studying energy splittings.

###  II.4 System dynamics

To investigate the system dynamics, a master equation in the Lindblad form is
used (details in Appendix A). For temperature $T=0$, by numerically solving
the master equation, we can obtain the time evolution of the mean photon
number $\langle{\hat{A}^{\dagger}}\hat{A}\rangle$, the mean phonon number
$\langle{\hat{B}^{\dagger}}\hat{B}\rangle$, and the mean atom-excitation
number $\langle{\hat{X}^{\dagger}}\hat{X}\rangle$. The dressed photon, phonon
and atom lowering operators $\hat{O}=\hat{A},\hat{B},\hat{X}$ are defined in
terms of their bare counterparts $\hat{o}=\hat{a},\hat{b},\hat{\sigma}$
(details in Appendix A).

####  II.4.1 Weak-coupling regime

Here, we first study the dynamical evolution when the effective coupling rate
(between states $|g,1,0\rangle$ and $|e,0,1\rangle$) is smaller than the loss
rates of the system, defined as the weak-coupling regime following the
terminology of cavity QED. Here we assume the transition frequency of the atom
$\omega_{a}=0.5\omega_{m}$ and set the atom, photonic, and mechanical loss
rate as $\eta=\kappa=\gamma=C$, where $C$ is a constant. To properly describe
the system dynamics, a single-tone mechanical drive $F(t)=A\cos(\omega_{d}t)$,
with $\omega_{d}=\omega_{m}$, and a weak excitation $A=0.8\gamma$ is used.
Initially, we consider the system in its ground state.

Figure 3(a) is obtained using $g{\rm{=}}0.01{\omega_{m}}$,
$\lambda{\rm{=}}0.01{\omega_{m}}$, and $C={\omega_{m}}/200$ (with a minimum
energy splitting $2\Omega_{e01}^{g10}=8.37\times{10^{-3}}\omega_{m}$). Figure
3(b) using $g{\rm{=}}0.03{\omega_{m}}$, $\lambda{\rm{=}}0.01{\omega_{m}}$, and
$C=3{\omega_{m}}/200$ (with
$2\Omega_{e01}^{g10}=2.46\times{10^{-2}}{\omega_{m}}$). Figure 3(c) using
$g{\rm{=}}0.03{\omega_{m}}$, $\lambda{\rm{=}}0.005{\omega_{m}}$, and
$C=3{\omega_{m}}/200$ (with
$2\Omega_{e01}^{g10}=2.34\times{10^{-2}}{\omega_{m}}$). The three plots show
that the mechanical and atom excitations almost reach a same intensity, which
implies that the excitations transfer between them. In Fig. 3, all the results
show that the atom and the photon mean excitation number are different. This
is because if the state $|g,0,2\rangle$, which is the out-of-resonance
intermediate state between states $|g,1,0\rangle$ and $|e,0,1\rangle$ [see
Fig. 2(a)], decays to the state $|g,0,1\rangle$, the remaining photon will not
reach the state $|e,0,1\rangle$ and come back to the state $|g,1,0\rangle$.
Thus, the excitation number of photon is larger than that of the atom, as
shown in Fig. 3(a).

Furthermore, the greater the coupling rate $g$, the more important is the
proportion of the transition path induced by the DCE [Fig. 2(a)], resulting in
a greater difference of excitation number between the atom and the photon, as
shown in Figs. 3(a) and (b).

Moreover, the greater the coupling rate $\lambda$, the less important is the
proportion of the transition path induced by the DCE, resulting in a smaller
difference of the excitation number between the atom and the photon, as shown
in Figs. 3(b) and (c).

Figure 3: System dynamics for $\omega_{c}\backsimeq 0.5\omega_{m}$ with
different normalized optomechanical coupling rates and cavity-atom coupling
rates, under a sinusoidal drive of the vibrating mirror. The blue and red
solid curves denote the mean phonon numbers
$\langle\hat{B}^{\dagger}\hat{B}\rangle$ and mean atom-excitation numbers
$\langle\hat{X}^{\dagger}\hat{X}\rangle$, respectively, while the black dashed
curve describes the mean photon numbers
$\langle\hat{A}^{\dagger}\hat{A}\rangle$ Figure 4: System dynamics for
$\omega_{c}\backsimeq 0.6\omega_{m}$ after the pulse arrival. The blue and red
solid curves denote the mean phonon number
$\langle\hat{B}^{\dagger}\hat{B}\rangle$ and the mean atom-excitation number
$\langle\hat{X}^{\dagger}\hat{X}\rangle$, respectively, while the black dashed
curve describes the mean photon number
$\langle\hat{A}^{\dagger}\hat{A}\rangle$. (a) The pulse amplitude is
$A=0.25\pi$. (b) The Fourier transform of the mean atom-excitation number of
panel (a). (c) The pulse amplitude is $A=0.40\pi$. (d) The Fourier transform
of the mean atom-excitation number of panel (c).

####  II.4.2 Strong-coupling regime

Now, we study the system dynamics in the strong-coupling regime, where the
effective coupling rate $2\Omega_{e01}^{g10}$ is larger than the loss rate of
the system. We consider an ultrafast resonant pulse [60]
$F(t)=AG(t-t_{0})\cos(\omega_{d}t)$, where $G(t)$ is the Gaussian function and
$\omega_{d}=\omega_{m}$ is the central driving frequency, directly impinging
on the mirror. Here, we set the transition frequency of the atom
$\omega_{a}=0.4\omega_{m}$, while the resonance frequency of the cavity mode
is fixed at the value providing the minimum level splitting. With the coupling
rates $g=0.03\omega_{m}$ and $\lambda=0.01\omega_{m}$, we obtain a minimum
energy splitting $2\Omega_{e01}^{g10}=3.64\times 10^{-3}\omega_{m}$. In this
case, we assume the loss rate $\eta=\kappa=\gamma=\omega_{m}/1000$ and a
standard deviation $\sigma=(15\Omega_{e01}^{g10})^{-1}$.

After the pulse arrival, Fig. 4 shows the system dynamics in the strong-
coupling regime and the Fourier transform of the mean atom-excitation number,
obtained with pulse amplitudes, $A=0.25\pi$ for Fig. 4(a) and $A=0.40\pi$ for
Fig. 4(c). In particular, Fig. 4(a) shows sinusoidal-like oscillations,
exhibiting that the excitations can be reversibly transferred between the
mirror and the atom (or photon), at a rate
$\omega_{3,4}=E_{4}-E_{3}=2\Omega_{e01}^{g10}$.

Note that, at a weak excitation level, the only desired states
$|\psi_{3,4}\rangle=(1/\sqrt{2})(|g,1,0\rangle\mp|e,0,1\rangle)$ are excited,
as shown in Fig. 4(b). When the amplitude of the pulse increases to
$A=0.40\pi$, the system dynamics is no longer a sinusoidal-like oscillation
[Fig. 4(c)]. This is because the higher-energy state (originating from the
coupling between states $|g,2,1\rangle$ and $|e,1,2\rangle$) is excited,
indicated by a peak at ${\omega_{16,17}}={E_{17}}-{E_{16}}$ in Fig. 4(d). This
process contributes to the system dynamics. The other states indicated by
${\omega_{7,8}}$, ${\omega_{10,11}}$, and ${\omega_{12,13}}$ in Fig. 4(d), do
not significantly contribute to the dynamics, since the minimum splitting of
these states is not provided at the resonance frequency of the cavity mode,
such that they have not been effectively excited.

##  III Single atom interacting with multiple cavity-optomechanical modes

In this section, we extend the previous model from a single cavity-
optomechanical mode to multiple cavity-optomechanical modes. In this case, we
will show how to realize frequency conversion of cavity modes.

###  III.1 Model

In the single-atom model with a single cavity-optomechanical mode, the
resonance condition ${\omega_{c}}+{\omega_{a}}={\omega_{m}}$ implies that the
photons in the cavity resonator can be affected and even mediated by an atom
with adjustable resonance frequency. By introducing more cavity modes into the
system, one can transfer the excitations between two or more of them. Here, we
study such a case where two cavity modes ($\omega_{c1}$ and $\omega_{c2}$)
interact with the atom and mechanical mode, fulfilling the condition of
frequency conversion: $2\omega_{c2}=\omega_{a}+\omega_{c1}$. The acquisition
of this condition will be discussed in the next subsection. The atom is
assumed to mainly interact with the mode $\omega_{c1}$. Considering the DCE
and counterrotating terms, the total Hamiltonian can be described as [53]
$\hat{H}=\hat{H}_{0}+\hat{H}_{\rm I}$, where

$\begin{split}\hat{H}_{0}&=\sum_{i=1}^{2}\hbar\omega_{ci}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{a}\hat{\sigma}_{+}\hat{\sigma}_{-},\\\
\hat{H}_{\rm I}&=\frac{\hbar
g}{2}\left[(\hat{a}_{1}^{2}+\hat{a}_{1}^{{\dagger}2}+2\hat{a}_{1}^{\dagger}\hat{a}_{1})+\frac{\omega_{c2}}{\omega_{c1}}(\hat{a}_{2}^{2}+\hat{a}_{2}^{{\dagger}2}+2\hat{a}_{2}^{\dagger}\hat{a}_{2})\right.\\\
&\left.-2\sqrt{\frac{\omega_{c2}}{\omega_{c1}}}(\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})\right](\hat{b}+\hat{b}^{\dagger})\\\
&+\hbar\lambda(\hat{a}_{1}+\hat{a}_{1}^{\dagger})(\hat{\sigma}_{-}+\hat{\sigma}_{+}).\end{split}$
(13)

The second and third lines of Eqs. (13) originate from the DCE and imply that
the mechanical oscillator can create photon pairs in different cavity modes.
By defining the quantized boson operators for hybridized cavity modes,
$\hat{H}$ can be transformed into a similar form as that in the scenario of
the single-atom interacting with a single cavity-optomechanical mode (details
in Appendix B). Consequently, one can analytically study the coupling
mechanism of this system (details in Appendix B).

###  III.2 Frequency conversion

The mechanism of the frequency conversion in the model of the multiple cavity-
optomechanical modes can be introduced by following two processes. In the
first process, the energy transfer between the cavity mode $\omega_{c2}$ and
the mechanical mode $\omega_{m}$ can occur with resonance condition
$2\omega_{c2}=k\omega_{m}$. The second process is based on the atom-mirror
coupling system. In this process, the energy transfer between the cavity mode
$\omega_{c1}$ and the mechanical mode $\omega_{m}$ can be mediated by an atom
at the resonance condition $\omega_{a}+\omega_{c1}=q\omega_{m}$. Since these
two processes utilize the same mechanical mode, we can obtain the relation
$2\omega_{c2}=\omega_{a}+\omega_{c1}$.

Figure 5: Frequency conversion. (a) Lowest-energy levels of the system
Hamiltonian $\hat{H}$ versus the ratio between the transition frequency
$\omega_{a}$ of the atom and the mechanical frequency $\omega_{m}$ of the
mirror. (b) Time evolution of the mean photon numbers
$\langle{\hat{A}_{1}^{\dagger}}\hat{A}_{1}\rangle$ and
$\langle{\hat{A}_{2}^{\dagger}}\hat{A}_{2}\rangle$. Here, the dressed photon
lowering operators $\hat{O}=\hat{A}_{1},\hat{A}_{2}$ are defined in terms of
their bare counterparts $\hat{o}=\hat{a}_{1},\hat{a}_{2}$ (details in Appendix
A) . The detuning
$\delta\omega_{a}=\omega_{a}-\omega_{a}^{0}=0.18\omega_{c}^{0}$, where the
$\omega_{a}^{0}$ is the value at the minimum splitting, is displayed by a gray
dashed curve. Such detuning can be realized by using a superconducting quantum
interference device if the two-level system is an $LC$ superconducting qubit
[61].

For instance, we consider the case where two cavity modes are
$\omega_{c1}=0.65\omega_{m}$ and $\omega_{c2}=0.5\omega_{m}$, with coupling
rates $g=0.02{\omega_{m}}$ and $\lambda=0.01{\omega_{m}}$. The lowest-energy
levels of the system Hamiltonian versus ${\omega_{a}/\omega_{m}}$ are plotted
in Fig. 5(a). The eigenstates are $|j,k,n_{1},n_{2}\rangle$, where the
$|n_{i}\rangle$ is the Fock state of the cavity mode $\omega_{ci}$
($i=1,2$)(details in Appendix B). Note that there are two states
$(1/\sqrt{2})(|g,0,0,2\rangle\pm|g,1,0,0\rangle)$, which originate from the
minimum splitting between $|g,1,0,0\rangle$ and $|g,0,0,2\rangle$. These
superposition states indicate the energy exchange between the cavity mode
$\omega_{c2}$ and mechanical mode $\omega_{m}$. If the atom is present and
satisfies the relation $2\omega_{c2}=\omega_{a}+\omega_{c1}$, then the energy
of the cavity mode $\omega_{c2}$ will be first transferred to the mechanical
oscillator and then to the cavity mode $\omega_{c1}$ and the atom.
Specifically, when the avoided-level crossing between states $|e,0,1,0\rangle$
and $(1/\sqrt{2})(|g,1,0,0\rangle\pm|g,0,0,2\rangle)$ appears, the frequency
conversion of two cavity modes can occur. The corresponding effective coupling
rate $\Omega$ can be calculated by using perturbation theory (details in
Appendix B).

Now, we use the master equation in the Lindblad form to study the dynamical
evolution of this hybrid system in the presence of dissipation (details in
Appendix A). The initial state is
$(1/\sqrt{2})(|g,1,0,0\rangle-|g,0,0,2\rangle)$, which can be prepared by
utilizing the DCE [52], with $\omega_{a}$ sufficiently detuned from the
frequency-conversion resonance $2\omega_{c2}=\omega_{a}+\omega_{c1}$. After
preparation, the atom is quickly tuned into resonance for half a Rabi
oscillation period and then tuned off immediately. This scheme is commonly
used for state transfer between resonators (or qubits) in circuit QED [61, 62,
63, 64, 65]. With $\kappa_{1}=\kappa_{2}=\gamma=\eta=\Omega/60$ for photonic
(cavity modes $\omega_{c1}$ and $\omega_{c2}$), mechanical, and atomic loss
rates, Fig. 5(b) displays the energy exchange between cavity modes
$\omega_{c1}$ and $\omega_{c2}$.

##  IV Two-atom interacting with single cavity-optomechanical mode

In this section, we extend the single-atom model to the two-atom case. We will
show that a single vibrating excitation of the mechanical oscillator can be
absorbed simultaneously by two atoms.

###  IV.1 Model

A hybrid quantum system consisting of a cavity resonator with a movable mirror
and the two atoms is considered. The system Hamiltonian can be written as

$\begin{split}\hat{H}=\,&\,\hbar{\omega_{c}}{\hat{a}^{\dagger}}\hat{a}+\hbar{\omega_{m}}{\hat{b}^{\dagger}}\hat{b}+\hbar{\omega_{a1}}\hat{\sigma}_{+}^{(1)}{\hat{\sigma}_{-}^{(1)}}+\hbar{\hat{\omega}_{a2}}\hat{\sigma}_{+}^{(2)}{\hat{\sigma}_{-}^{(2)}}\\\
\,&\,+\hbar
g{\hat{a}^{\dagger}}\hat{a}(\hat{b}+{\hat{b}^{\dagger}})+\frac{\hbar}{2}g({\hat{a}^{2}}+{\hat{a}^{{\dagger}2}})(\hat{b}+{\hat{b}^{\dagger}})\\\
\,&\,+\hbar{\lambda_{1}}(\hat{a}+{\hat{a}^{\dagger}})({\hat{\sigma}_{+}^{(1)}}+\hat{\sigma}_{-}^{(1)})\\\
\,&\,+\hbar{\lambda_{2}}(\hat{a}+{\hat{a}^{\dagger}})({\hat{\sigma}_{+}^{(2)}}+\hat{\sigma}_{-}^{(2)}).\end{split}$
(14)

The two-atom model can be analytically solved by performing a unitary
transformation (details in Appendix C), which is similar to the case of
single-atom with single cavity-optomechanical mode. For the unperturbed
Hamiltonian $\hat{H}_{0}^{U}$ shown in Appendix C, the eigenstate is described
as $|j_{1},j_{2},k,n\rangle$, where $|j_{1}\rangle$ and $|j_{2}\rangle$
($j=g,e$) denote the states of the atoms $\omega_{a1}$ and $\omega_{a2}$,
respectively. For the coupling between the ground state $|g,g,1,0\rangle$ and
the excited state $|e,e,0,0\rangle$, the effective coupling rate
$\Omega_{ee00}^{gg10}$ can be calculated by using perturbation theory (details
in Appendix C). By study this coupling, we show that the cavity-vacuum field
couples the vibrating mirror via the complete exchange of virtual photons to
the two atoms.

Figure 6: The dynamical evolution of the hybrid quantum system, (a) in the
weak-coupling regime with a sinusoidal drive of the vibrating mirror, and (b)
in the strong-coupling regime with an ultrafast resonant pulse for the mirror,
where the inset displays the Fourier transform (FT) of the mean excitation
number of atom 1. The black and blue solid curves describe the mean
intracavity photon number $\langle\hat{A}^{\dagger}\hat{A}\rangle$ and mean
phonon number $\langle\hat{B}^{\dagger}\hat{B}\rangle$, respectively, while
the red and green dotted curves describe the mean atom-excitation number
$\langle\hat{X}_{1}^{\dagger}\hat{X}_{1}\rangle$ and
$\langle\hat{X}_{2}^{\dagger}\hat{X}_{2}\rangle$, respectively. Here, the
dressed photon, phonon and atom lowering operators
$\hat{O}=\hat{A},\hat{B},\hat{X}_{1},\hat{X}_{2}$ are defined in terms of
their bare counterparts
$\hat{o}=\hat{a},\hat{b},\hat{\sigma}_{-}^{(1)},\hat{\sigma}_{-}^{(2)}$
(details in Appendix A)

###  IV.2 System dynamics: One phonon being absorbed simultaneously by two
atoms

By using a master equation approach, we study the excitation transfer between
the mirror and the two atoms (details in Appendix A). For this purpose, a
coherent external drive is applied to the movable mirror. The driving
Hamiltonian is written as $\hat{H}_{d}(t)=F(t)(\hat{b}+\hat{b}^{\dagger})$,
where $F(t)$ is proportional to the external force on the mirror. Moreover,
$\eta_{1}$, $\eta_{2}$, $\kappa$, and $\gamma$ are the loss rates of the atom
$\omega_{a1}$, atom $\omega_{a2}$, cavity mode, and mechanical mode
respectively.

Firstly, we study the coherent energy transfer between the two atoms and the
mirror in the weak-coupling regime, showing how the excitations of the mirror
can be absorbed by the two atoms. A single-tone mechanical drive
$F(t)=A\cos(\omega_{d}t)$ with a weak ($A=2\gamma$) resonance excitation of
the vibrating mirror ($\omega_{d}=\omega_{m}$) is adopted. The system is
initially in its ground state and the evolution can be obtained, as shown in
Fig. 6(a), with the transition frequency of two atoms
$\omega_{a1}=\omega_{a2}=0.5\omega_{m}$, the normalized optomechanical
coupling rate $g=0.007\omega_{m}$, the normalized atom-cavity coupling rate
$\lambda_{1}=\lambda_{2}=0.007\omega_{m}$, loss rates
$\eta_{1}=\eta_{2}=\kappa=\gamma=1\times 10^{-4}\omega_{m}$, and
$\omega_{c}=0.55\omega_{m}$.

Figure 6(a) shows a remarkable energy transfer from the vibrating mirror to
the two atoms via quantum vacuum fluctuations. Note that, at the beginning of
the energy transfer, the energy of the two atoms rises simultaneously. At the
steady state, the mean atom-excitation number are detectable, even though the
optomechanical coupling is not very strong. Furthermore, the photon population
[black curve in Fig. 6(a)], is negligible throughout time, implying that the
energy transfer is mediated by virtual photons pairs.

Then, we study the strong-coupling regime. An ultrafast resonant pulse
$F(t)=AG(t-t_{0})\cos(\omega_{d}t)$ is applied on the movable mirror to
produce a coherent mechanical state. We consider a pulse with standard
deviation $\sigma=(30\Omega_{ee00}^{gg10})^{-1}$, amplitude $A=0.25\pi$, and
central driving frequency $\omega_{d}=\omega_{m}$. The system is initially
prepared in its ground state with parameters
$\kappa=\gamma=\eta_{1}=\eta_{2}=1\times 10^{-5}\omega_{m}$,
$g/\omega_{m}=0.03$, $\lambda_{1}=\lambda_{2}=0.005\omega_{m}$, and
$\omega_{c}\simeq 0.55\omega_{m}$. Figure 6(b) displays a coherent energy
exchange between the mechanical oscillator and the two atoms after the pulse
arrival. The result shows that the energy can be reversibly transferred via
quantum fluctuations at a rate
$\omega_{5,6}=E_{6}-E_{5}=2\Omega_{ee00}^{gg10}$, as indicated by the peak in
the inset of Fig. 6(b).

##  V Experimental platform for the observation of proposed effects

To demonstrate these results, circuit optomechanics using ultra-high-frequency
mechanical resonators [58, 59] provide a promising platform [52, 21]. An
optomechanical system with a radiation-pressure interaction Hamiltonian and a
cavity-atom system with Rabi Hamiltonian must be introduced into such a
platform.

The radiation-pressure interaction with a reasonable coupling strength can be
obtained by considering a tripartite system consisting of an electromagnetic
resonator, an ultra-high-frequency mechanical resonator, and a superconducting
charge qubit. In this system, the interaction between the former two parts is
mediated by the qubit, which can strongly enhance the optomechanical coupling
[66, 67]. Specifically, considering one coupling between a generic mechanical
oscillator and a charge qubit through a capacitor, the interaction Hamiltonian
can be written as [21, 67]

$\hat{H}_{\text{qm}}=g_{m}(\hat{b}+\hat{b^{{\dagger}}})\hat{\sigma}^{(0)}_{z},$
(15)

where $g_{m}$ is the coupling strength. Note that this coupling induces a
qubit energy shift depending on the mechanical displacement so that
$\omega_{q}\longrightarrow\omega_{q}+2g_{m}(\hat{b}+\hat{b}^{{\dagger}})$,
where $\omega_{m}$ is the mechanical frequency and $\omega_{q}$ is the
transition frequency of the qubit. On the other hand, following Refs. [66] and
[67], we consider the additional interaction of the qubit with an
electromagnetic resonator, described by
$H_{\text{qc}}=g_{c}(\hat{a}+\hat{a}^{{\dagger}})\hat{\sigma}^{(0)}_{x}$. In
the dispersive regime, the qubit-cavity interaction can be approximately
described by [68]

$\hat{H}_{\text{qm}}=\frac{g^{2}_{c}}{2\varDelta}(\hat{a}+\hat{a}^{{\dagger}})^{2}\hat{\sigma}^{(0)}_{z},$
(16)

where $\varDelta=\omega_{q}-\omega_{c}$ is the detuning frequency. By
considering the energy shift of the qubit, the detuning can be rewritten as
$\varDelta=\omega_{q}+2g_{m}(\hat{b}+\hat{b}^{{\dagger}})-\omega_{c}$,
depending on the mechanical displacement. If we assume a small displacement
and consider the qubit in its ground state, then we can obtain the following
optomechanical interaction from Eq. (16),

$\hat{H}_{\text{qm}}=\frac{g}{2}(\hat{a}+\hat{a}^{{\dagger}})^{2}(\hat{b}+\hat{b}^{{\dagger}}),$
(17)

with

$g=\frac{2g_{m}g^{2}_{c}}{(\omega_{q}-\omega_{c})^{2}}.$ (18)

Assuming that the optomechanical system depicted above can be coupled through
a capacitor to a superconducting qubit, the effects discussed in this work
could be effectively observed. The interaction Hamiltonian can be described by

$\hat{H}_{\text{I}}=\frac{g}{2}(\hat{a}+\hat{a}^{{\dagger}})^{2}(\hat{b}+\hat{b}^{{\dagger}})+\lambda(\hat{a}+\hat{a}^{{\dagger}})\hat{\sigma}_{x}.$
(19)

The optomechanical coupling rate $g$ and the cavity-qubit coupling rate
$\lambda$ can reach the ultra-strong coupling limit [35, 36], which should
allow us to observe the phenomena predicted in this work.

If the above electromagnetic resonator consists of a coplanar waveguide
resonator, which can be designed to trap atomic ensembles [69, 70], then it is
possible to realize the integration of ultracold atoms (or molecules) with
circuit optomechanics. This integration could achieve the atom-mirror coupling
via quantum fluctuations. To obtain a moderate scale of the coupling rate
$\lambda$, a rather effective way is to employ an ensemble of $N$ atoms
instead of a single atom [71, 72, 73, 74, 69, 75] to enhance the atom-cavity
coupling by a factor of $\sqrt{N}$.

In addition, to observe the dynamical evolution of the system, we can apply a
coherent driving field to the mechanical resonator through the piezoelectric
material, which converts the electrical signal into mechanical vibrations [58,
76]. In the weak-coupling regime, as in the previous discussion, one can
employ the continuous-wave electric field as the coherent drive [77, 78, 79].
For the system dynamics in the strong-coupling regime, the required ultrafast
resonant pulse, like the Gaussian pulse, can be achieved by mixing the fields
from the arbitrary waveform generator and a continuous-wave coherent drive
[77].

Table 1: The coupling between the atoms and the mirror via cavity-vacuum
fluctuations with three models described in this work.

Model${}^{\text{a}}$ |  $N_{a}=1$, $N_{c}=1$, $N_{m}=1$ |  $N_{a}=1$, $N_{c}=2$, $N_{m}=1$ |  $N_{a}=2$, $N_{c}=1$, $N_{m}=1$
---|---|---|---
Main feature | The creation of an excitation | Frequency conversion | The creation of an excitation
| in both 1 atom and 1 photon | between two cavity modes | in both atoms
Resonant condition | $N\omega_{m}=\omega_{c}+\omega_{a}$ | $2\omega_{c2}=\omega_{a}+\omega_{c1}$ | $N\omega_{m}=\omega_{a1}+\omega_{a2}$
Dominant effect${}^{\text{b}}$ | DCE and CRT | DCE, CRT, and scattering effect | DCE and CRT

${}^{\text{a}}$ $N_{a}$, $N_{c}$, and $N_{m}$ refer to the number of atoms,
optomechanical-cavity modes, and mechanical modes, respectively.

${}^{\text{b}}$ DCE and CRT refer to the dynamical Casimir effect and the
counterrotating terms, respectively.

##  VI Discussion and Conclusion

We have demonstrated that the coupling between the atoms and a mechanical
oscillator can be induced by vacuum fluctuations of a cavity field, which are
amplified by the DCE and counterrotating terms. The novel coupling mechanism
has the potential to couple the atom and the mechanical oscillator with a
considerable mass imbalance.

Table 1 lists the three models described in this work. By comparing the zero-
atom case [52] with the single-atom and the two-atoms ones, we demonstrate
that the DCE photons can be transformed to the atomic excitations from one
atom to two atoms. It is possible to detect the DCE effect by collecting the
readout signal of the quantum state of the atomic system. This approach can be
immune to some spurious ambient emission processes for the measured photons
and, therefore, can be used as an auxiliary means to estimate the DCE effect.
Compared to the two-atom case, the single-atom one may be more suitable for
this task because the former (single-atom) requires more high-order processes
than the latter.

For the two-atom case, the conditional quantum-state transfer is a possible
application. One atom could be encoded with information as a superposition of
the ground and excited states. The second atom can be used as a switch to
control whether the information is transmitted to the membrane. If the second
atom’s transition frequency satisfies the resonant condition of the system
(e.g., $N\omega_{m}=\omega_{a1}+\omega_{a2}$), then a single phonon will be
created in a membrane that contains the encoded information.

For the two cavity-optomechanical modes model, from the resonant condition of
the photon frequency conversion $2\omega_{c2}=\omega_{a}+\omega_{c1}$, this
model can realize frequency up-conversion and down-conversion of optical
signals by mediating the atom’s transition frequency. In addition, it is also
possible to realize wideband wavelength conversion at the expense of requiring
a high optomechanical coupling rate and a cavity-qubit coupling rate.

One model with two mechanical modes can also be explored. In this case, we can
consider a hybrid quantum system consisting of a cavity resonator with two
movable mirrors and two atoms. When one of the two mirrors is driven and
encoded with quantum information, the atoms can absorb the excitations. Before
reaching the other mirror, we can store these excitations by controlling the
atom’s transition frequency to get away from the resonant condition of the
system. Thus, controlling the arrival time of the excitations between two
mirrors is possible. Such a model can also be analytically studied by using
the method of Sec. II.

###### Acknowledgements.

We thank Mauro Cirio for a critical reading of the manuscript. This work was
supported by the National Natural Science Foundation of China (Grant No.
11874432), the National Key R&D Program of China (Grant No. 2019YFA0308200),
and the China Postdoctoral Science Foundation (Grant No. 2021M693682). F.N. is
supported in part by: Nippon Telegraph and Telephone Corporation (NTT)
Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap
Flagship Program (Q-LEAP) program, and the Moonshot R&D Grant No. JPMJMS2061],
the Japan Society for the Promotion of Science (JSPS) [via the Grants-in-Aid
for Scientific Research (KAKENHI) Grant No. JP20H00134], the Asian Office of
Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069),
and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-
IAF19-06.

##  Appendix A Single atom interacting with single cavity-optomechanical mode

###  A.1 Effective system Hamiltonian

By performing a unitary transformation with the unitary operator
$\hat{U}=\exp[-\beta{\hat{a}^{\dagger}}\hat{a}({\hat{b}^{\dagger}}-\hat{b})]$
for $\hat{H}$ in Sec. II of the main text, we can obtain an effective system
Hamiltonian

$\hat{H}^{U}=\hat{U}^{\dagger}\hat{H}\hat{U}=\hat{H}_{0}^{U}+\hat{H}_{\rm
FM}^{U}+\hat{H}_{\rm AFM}^{U},$ (20)

where

$\hat{H}_{0}^{U}=\hat{U}^{\dagger}(\hat{H}_{0}+\hat{V}_{\rm
om})\hat{U}=\hat{H}_{0}-\hbar\frac{g^{2}}{\omega_{m}}\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a},$
(21) $\begin{split}\hat{H}_{\rm FM}^{U}=\,&\,\hat{U}^{\dagger}\hat{V}_{\rm
DCE}\hat{U}\\\ =\,&\,\sum_{n=0}^{\infty}\frac{1}{n!}\hbar
g\left(\frac{g}{\omega_{m}}\right)^{n}2^{n-1}\left\\{\left[\hat{a}^{{\dagger}2}+(-1)^{n}\hat{a}^{2}\right]\right.\\\
\,&\,\left.\times(\hat{b}^{\dagger}-\hat{b})^{n}(\hat{b}^{\dagger}+\hat{b})-n\left[\hat{a}^{{\dagger}2}+(-1)^{n-1}\hat{a}^{2}\right]\right.\\\
\,&\,\left.\times\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}-\hat{b})^{n-1}\right\\},\end{split}$
(22)

and

$\begin{split}\hat{H}_{\rm AFM}^{U}=\,&\,\hat{U}^{\dagger}\hat{H}_{\rm
I(AF)}\hat{U}\\\
=\,&\,\hbar\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{g}{\omega_{m}}\right)^{n}\lambda\left[\hat{a}^{\dagger}+(-1)^{n}\hat{a}\right](\hat{\sigma}_{-}+\hat{\sigma}_{+})\\\
\,&\,\times(\hat{b}^{\dagger}-\hat{b})^{n}.\end{split}$ (23)

###  A.2 Analytical approach to the high-order resonance condition
$\omega_{a}+\omega_{c}=2\omega_{m}$

Figure 7: (a) Relevant energy levels of the system Hamiltonian $\hat{H}$
versus the ratio between the resonance frequencies of the cavity mode
$\omega_{c}$ and the vibrating mirror $\omega_{m}$. (b) The enlarged view of
the boxed region in panel (a) shows the avoided-level crossing between
$|g,2,0\rangle$ and $|e,0,1\rangle$, which is induced by the DCE and the
counterrotating terms. The resulting coupling involves the high-order
resonance condition ${\omega_{a}}+{\omega_{c}}=2{\omega_{m}}$.

In a ladder of increasing level splitting, the corresponding effective
coupling rate can also be calculated by using perturbation theory. For the
coupling between states $|g,2,0\rangle$ and $|e,0,1\rangle$ in Fig. 7, with
parameters $g{\rm{=}}0.06{\omega_{m}}$, $\lambda{\rm{=}}0.01{\omega_{m}}$, and
${\omega_{a}}{\rm{=}}0.8{\omega_{m}}$, the transition paths are shown in Fig.
8(a). The top two transition paths involve the creation (or annihilation) of
excitation pairs with the high-order processes of the DCE and the
counterrotating terms, while the last path shows the standard resonant
condition of the DCE. With these transition paths, the effective coupling rate
can be obtained

$\begin{split}\Omega_{e01}^{g20}=\,&\,\frac{\langle e,0,1|\hat{H}_{\rm
I}^{U}|g,0,2\rangle\langle g,0,2|\hat{H}_{\rm
I}^{U}|g,2,0\rangle}{E_{g,2,0}-E_{g,0,2}}\\\ \,&\,+\frac{\langle
e,0,1|\hat{H}_{\rm I}^{U}|g,1,2\rangle\langle g,1,2|\hat{H}_{\rm
I}^{U}|g,2,0\rangle}{E_{g,2,0}-E_{g,1,2}}\\\ \,&\,+\langle
e,0,1|\hat{H}_{AFM}^{U}|g,2,0\rangle\\\
=\,&\,\frac{-\frac{g^{2}\lambda}{\omega_{m}}\left(\sqrt{2}-\frac{\sqrt{2}g^{2}}{2\omega_{m}^{2}}\right)}{\omega_{m}-\omega_{c}+\frac{2g^{2}}{\omega_{m}}}+\frac{\frac{g^{2}\lambda}{\omega_{m}}\left(\sqrt{2}-\frac{\sqrt{2}g^{2}}{2\omega_{m}^{2}}\right)}{\omega_{m}-2\omega_{c}+\frac{4g^{2}}{\omega_{m}}}\\\
\,&\,+\frac{\sqrt{2}\lambda g^{2}}{2\omega_{m}^{2}}.\end{split}$ (24)

Figure 8: (a) Sketch of the processes giving the main contribution to the
effective coupling between the states $|g,2,0\rangle$ and $|e,0,1\rangle$, via
the DCE (red arrows) and the counterrotating terms (blue arrows). (b)
Comparison between the numerical (red dots) and analytical (blue solid curve)
results of the normalized level splitting.

where $\hat{H}_{\rm I}^{U}=\hat{H}_{\rm AFM}^{U}+\hat{H}_{\rm FM}^{U}$ is the
effective perturbative Hamiltonian after the unitary transformation. For
calculating the effective coupling rate, we keep the terms of the expansion of
the effective Hamiltonian ($\hat{H}_{\rm FM}^{U}$ and $\hat{H}_{\rm AFM}^{U}$)
up to the third order. In Fig. 8(b), the numerical and the analytical results
of the normalized splitting ($2\Omega_{e01}^{g20}/{\omega_{m}}$) are in good
agreement. This shows that a perturbative approach is still applicable if the
normalized optomechanical coupling is not too strong.

###  A.3 Master equation approach

To study the system dynamics, a master equation in the Lindblad form is used.
Here, the Born-Markov approximation without the post-trace rotating-wave
approximation [52, 80] is considered and the resulting master equation for the
density-matrix operator $\hat{\rho}(t)$ at temperature $T=0$ [42] can be
described by

$\begin{split}\dot{\hat{\rho}}(t)=\,&\,\frac{i}{\hbar}[\hat{\rho}(t),\hat{H}+\hat{H}_{d}(t)]\\\
\,&\,+\kappa D[\hat{A}]\hat{\rho}(t)+\gamma
D[\hat{B}]\hat{\rho}(t)+{\eta}D[{\hat{X}}]\hat{\rho}(t),\end{split}$ (25)

where the $\kappa$, $\gamma$, and ${\eta}$ are photonic, mechanical, and atom
loss rates, respectively. The driving Hamiltonian is written as $\hat{H}_{\rm
d}(t)=F(t)(\hat{b}+\hat{b}^{\dagger})$, where $F(t)$ is proportional to the
external force on the mirror. The dressed photon, phonon and atom lowering
operators $\hat{O}=\hat{A},\hat{B},\hat{X}$ are defined in terms of their bare
counterparts $\hat{o}=\hat{a},\hat{b},\hat{\sigma}_{-}$ as

$\hat{O}=\sum\limits_{E_{n}>E_{m}}{\langle\psi_{m}|\hat{o}+\hat{o}^{\dagger}|\psi_{n}\rangle}|\psi_{m}\rangle\langle\psi_{n}|,$
(26)

and the superoperator D is defined as

$D[\hat{O}]\hat{\rho}=\frac{1}{2}(2\hat{O}\hat{\rho}{\hat{O}^{\dagger}}-{\hat{O}^{\dagger}}\hat{O}\hat{\rho}-\hat{\rho}{\hat{O}^{\dagger}}\hat{O}).$
(27)

Moreover, we label the $|\psi_{n}\rangle$ eigenvectors of $\hat{H}$ and the
$E_{n}$ corresponding eigenvalues, such that $E_{n}>E_{m}$ for $n>m$.

##  Appendix B Single atom interacting with multiple cavity-optomechanical
modes

###  B.1 Effective system Hamiltonian

In this case, the system Hamiltonian is described as [53]
$\hat{H}=\hat{H}_{0}+\hat{H}_{\rm I}$, where

$\hat{H}_{0}=\sum_{i=1}^{2}\hbar\omega_{ci}\hat{a}_{i}^{\dagger}\hat{a}_{i}+\hbar\omega_{m}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{a}\hat{\sigma}_{+}\hat{\sigma}_{-}$
(28)

is the unperturbed Hamiltonian and

$\begin{split}\hat{H}_{\rm I}&=\frac{\hbar
g}{2}\left[(\hat{a}_{1}^{2}+\hat{a}_{1}^{{\dagger}2}+2\hat{a}_{1}^{\dagger}\hat{a}_{1})+\frac{\omega_{c2}}{\omega_{c1}}(\hat{a}_{2}^{2}+\hat{a}_{2}^{{\dagger}2}+2\hat{a}_{2}^{\dagger}\hat{a}_{2})\right.\\\
&\left.-2\sqrt{\frac{\omega_{c2}}{\omega_{c1}}}(\hat{a}_{1}\hat{a}_{2}+\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}+\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})\right](\hat{b}+\hat{b}^{\dagger})\\\
&+\hbar\lambda(\hat{a}_{1}+\hat{a}_{1}^{\dagger})(\hat{\sigma}_{-}+\hat{\sigma}_{+})\end{split}$
(29)

describes the interaction.

To conveniently analyze this quantum system, we need to perform a
transformation for the system Hamiltonian. Here, we define the quantized boson
operators for the hybridized cavity modes

$\begin{split}\hat{p}=\frac{c\hat{a}_{1}+\hat{a}_{2}}{\sqrt{1+c^{2}}}\ \
\text{and}\ \
\hat{q}=\frac{\hat{a}_{1}-c\hat{a}_{2}}{\sqrt{1+c^{2}}},\end{split}$ (30)

where $c=\sqrt{\omega_{c2}/\omega_{c1}}$. They satisfy these relations
$[\hat{p},\hat{q}]=0$, $[\hat{p},\hat{q}^{\dagger}]=0$,
$[\hat{p},\hat{p}^{\dagger}]=1$, and $[\hat{q},\hat{q}^{\dagger}]=1$. The
system Hamiltonian can be rewritten as
$\hat{\tilde{H}}=\hat{\tilde{H}}_{0}+\hat{\tilde{H}}_{\rm I}$, where

$\begin{split}\hat{\tilde{H}}_{0}\,&\,=\hbar\tilde{\omega}_{c1}{\hat{p}^{\dagger}}\hat{p}+\hbar\tilde{\omega}_{c2}{\hat{q}^{\dagger}}\hat{q}+\hbar{\omega_{m}}{\hat{b}^{\dagger}}\hat{b}+\hbar{\omega_{a}}{\hat{\sigma}_{+}}\hat{\sigma}_{-}\end{split}$
(31)

is the unperturbed Hamiltonian and $\hat{\tilde{H}}_{\rm
I}=\hat{\tilde{V}}_{\rm DCE}+\hat{\tilde{V}}_{\rm om}+\hat{\tilde{H}}_{\rm
I(AF)}+\hat{\tilde{H}}_{\rm I(FF)}$ describes the interaction. Here,

$\begin{split}\hat{\tilde{V}}_{\rm DCE}=\frac{{\hbar
G}}{2}(\hat{q}^{2}+\hat{q}^{{\dagger}2})(\hat{b}^{\dagger}+\hat{b}),\end{split}$
(32) $\begin{split}\hat{\tilde{V}}_{\rm om}\,&\,=\hbar
G\hat{q}^{\dagger}\hat{q}(\hat{b}^{\dagger}+\hat{b}),\end{split}$ (33)
$\begin{split}\hat{\tilde{H}}_{\rm
I(AF)}=\frac{\hbar\lambda}{\sqrt{1+c^{2}}}(c\hat{p}^{\dagger}+c\hat{p}+\hat{q}^{\dagger}+\hat{q})(\hat{\sigma}_{-}+{\hat{\sigma}_{+}}),\end{split}$
(34)

and

$\begin{split}\hat{\tilde{H}}_{\rm I(FF)}=\frac{\Delta
c\hbar}{1+c^{2}}(\hat{p}^{\dagger}\hat{q}+\hat{q}^{\dagger}\hat{p}),\end{split}$
(35)

where $G=(1+c^{2})g$ is the effective optomechanical coupling rate for the
hybridized cavity modes and $\Delta=\omega_{c1}-\omega_{c2}$ is the detuning
between the cavity mode 1 and mode 2.

$\displaystyle\tilde{\omega}_{c1}=\omega_{c2}+\frac{\Delta c^{2}}{1+c^{2}}\ \
\text{and}\ \ \tilde{\omega}_{c2}=\omega_{c2}+\frac{\Delta}{1+c^{2}}$

are the effective frequency of the hybridized cavity mode 1 and mode 2,
respectively. $\hat{\tilde{H}}_{\rm I(FF)}$ describes the scattering
interaction between the hybridized cavity modes. Note that the system
Hamiltonian $\hat{\tilde{H}}$ is analogous to the scenario of a single-atom
with a single cavity-optomechanical mode, so we can utilize the analytical
method of Sec. II to this case.

Applying a unitary transformation with the unitary operator
$\hat{U}=\exp[-(G/\omega_{m}){\hat{q}^{\dagger}}\hat{q}({\hat{b}^{\dagger}}-\hat{b})]$
for $\hat{\tilde{H}}$, we obtain

$\hat{\tilde{H}}_{\rm 0}^{U}={\hat{U}^{\dagger}}(\hat{\tilde{H_{\rm
0}}}+\hat{\tilde{V}}_{\rm om})\hat{U}=\hat{\tilde{H_{\rm
0}}}-\hbar\frac{{{G^{2}}}}{{{\omega_{m}}}}{\hat{q}^{\dagger}}\hat{q}{\hat{q}^{\dagger}}\hat{q},$
(36) $\begin{split}\hat{\tilde{H}}_{\rm
FM}^{U}=\,&\,{\hat{U}^{\dagger}}\hat{\tilde{V}}_{\rm DCE}\hat{U}\\\
=\,&\,\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}\hbar
G{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}{2^{n-1}}\left\\{\left[{\hat{q}^{{\dagger}2}}+{(-1)^{n}}{\hat{q}^{2}}\right]\right.\\\
\,&\,\left.\times{({\hat{b}^{\dagger}}-\hat{b})^{n}}({\hat{b}^{\dagger}}+\hat{b})-n\left[{\hat{q}^{{\dagger}2}}+{(-1)^{n-1}}{\hat{q}^{2}}\right]\right.\\\
\,&\,\left.\times{\hat{q}^{\dagger}}\hat{q}{({\hat{b}^{\dagger}}-\hat{b})^{n-1}}\right\\},\end{split}$
(37) $\begin{split}\hat{\tilde{H}}_{\rm
AFM}^{U}=\,&\,{\hat{U}^{\dagger}}\hat{\tilde{H}}_{\rm AF}\hat{U}\\\
=\,&\,\hat{\tilde{H}}_{\rm
AF}+\frac{\hbar\lambda}{\sqrt{1+c^{2}}}\sum\limits_{n=1}^{\infty}{\frac{1}{{n!}}}{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}\\\
\,&\,\times\left[{\hat{q}^{\dagger}}+{(-1)^{n}}\hat{q}\right](\hat{\sigma}_{-}+{\hat{\sigma}_{+}}){({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(38)

and

$\begin{split}\hat{\tilde{H}}_{\rm
FFM}^{U}=\,&\,{\hat{U}^{\dagger}}\hat{\tilde{H}}_{\rm FF}\hat{U}\\\
=\,&\,\frac{\Delta
c\hbar}{1+c^{2}}\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}\left[{\hat{q}^{\dagger}\hat{p}}+{(-1)^{n}}\hat{q}\hat{p}^{\dagger}\right]\\\
\,&\,\times{({\hat{b}^{\dagger}}-\hat{b})^{n}}.\end{split}$ (39)

Here, $\hat{\tilde{H}}_{\rm FFM}^{U}$ describes the scattering interaction
between the hybridized cavity modes $\tilde{\omega}_{c1}$ and
$\tilde{\omega}_{c2}$, involving the multiple-order transition processes
induced by $\hat{\tilde{V}}_{\rm om}$.

Now, by substituting the Eqs. (30) to the Eqs. (36)-(39), we can obtain the
effective system Hamiltonian $\hat{H}^{U}=\hat{H}_{0}^{U}+\hat{H}_{\rm
FM}^{U}+\hat{H}_{\rm AFM}^{U}+\hat{H}_{\rm FFM}^{U}$. Here,

$\begin{split}\hat{H}_{0}^{U}=\,&\,\hat{H}_{0}-\frac{\hbar
g^{2}}{\omega_{m}}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}+2c^{2}a_{1}^{\dagger}\hat{a}_{1}\hat{a}_{2}^{\dagger}\hat{a}_{2}\right.\\\
\,&\,\left.+c^{4}a_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2}+c^{2}a_{1}^{\dagger}\hat{a}_{1}\hat{a}_{2}\hat{a}_{2}^{\dagger}+c^{2}a_{2}^{\dagger}\hat{a}_{2}\hat{a}_{1}\hat{a}_{1}^{\dagger}\right)\end{split}$
(40)

is the unperturbed Hamiltonian.

$\begin{split}\hat{H}_{\rm
FM}^{U}=\,&\,\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}\hbar
G{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}{2^{n-1}}\left[\frac{1}{1+c^{2}}L_{1}{({\hat{b}^{\dagger}}-\hat{b})^{n}}\right.\\\
\,&\,\left.\times({\hat{b}^{\dagger}}+\hat{b})-\frac{n}{(1+c^{2})^{2}}L_{2}L_{3}{({\hat{b}^{\dagger}}-\hat{b})^{n-1}}\right],\end{split}$
(41)

where

$\begin{split}L_{1}=\,&\,\left[\hat{a}_{1}^{{\dagger}2}+(-1)^{n}\hat{a}_{1}^{2}\right]+c^{2}\left[\hat{a}_{2}^{{\dagger}2}+(-1)^{n}\hat{a}_{2}^{2}\right]\\\
\,&\,-2c\left[\hat{a}_{1}^{{\dagger}}\hat{a}_{2}^{{\dagger}}+(-1)^{n}\hat{a}_{1}\hat{a}_{2}\right],\end{split}$
(42)
$\begin{split}L_{2}=\,&\,\left[\hat{a}_{1}^{{\dagger}2}+(-1)^{n-1}\hat{a}_{1}^{2}\right]+c^{2}\left[\hat{a}_{2}^{{\dagger}2}+(-1)^{n-1}\hat{a}_{2}^{2}\right]\\\
\,&\,-2c\left[\hat{a}_{1}^{{\dagger}}\hat{a}_{2}^{{\dagger}}+(-1)^{n-1}\hat{a}_{1}\hat{a}_{2}\right],\end{split}$
(43)

and

$\begin{split}L_{3}={\hat{a}_{1}^{\dagger}}\hat{a}_{1}-c{\hat{a}_{1}^{\dagger}}\hat{a}_{2}-c{\hat{a}_{2}^{\dagger}}\hat{a}_{1}+c^{2}{\hat{a}_{2}^{\dagger}}\hat{a}_{2},\end{split}$
(44)

describes the creation (or annihilation) of photon pairs with different cavity
modes, involving the multiple-order transitions. This

$\begin{split}\hat{H}_{\rm
AFM}^{U}=\,&\,\hbar\lambda({\hat{a}_{1}}+\hat{a}_{1}^{\dagger})(\hat{\sigma}_{-}+{\hat{\sigma}_{+}})\\\
\,&\,+\frac{\hbar\lambda}{\sqrt{1+c^{2}}}\sum\limits_{n=1}^{\infty}{\frac{1}{{n!}}}{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}\left\\{\left[{\hat{a}_{1}^{\dagger}}+{(-1)^{n}}\hat{a}_{1}\right]\right.\\\
\,&\,\left.-c\left[{\hat{a}_{2}^{\dagger}}+{(-1)^{n}}\hat{a}_{2}\right]\right\\}(\hat{\sigma}_{-}+{\hat{\sigma}_{+}}){({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(45)

describes the photon-atom-phonon coupling with different cavity modes,
involving multiple-order transitions.

$\begin{split}\hat{H}_{\rm FFM}^{U}\,&\,=\hat{H}_{\rm FF}+\frac{\Delta
c\hbar}{(1+c^{2})^{2}}\sum\limits_{n=1}^{\infty}{\frac{1}{{n!}}}{\left(\frac{G}{{{\omega_{m}}}}\right)^{n}}\\\
\,&\,\times\left\\{\left[1+(-1)^{n}\right]c({\hat{a}_{1}^{\dagger}}\hat{a}_{1}-{\hat{a}_{2}^{\dagger}}\hat{a}_{2})+\left[1+(-1)^{n-1}c^{2}\right]\right.\\\
\,&\,\left.\times\left[{\hat{a}_{1}^{\dagger}}\hat{a}_{2}+(-1)^{n}{\hat{a}_{1}}\hat{a}_{2}^{\dagger}\right]\right\\}{({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(46)

where

$\begin{split}\hat{H}_{\rm
FF}=\,&\,-\frac{g^{2}\hbar}{\omega_{m}}\left[-(c{\hat{a}_{1}^{\dagger}}\hat{a}_{1}+c^{3}{\hat{a}_{2}^{\dagger}}\hat{a}_{2})({\hat{a}_{1}^{\dagger}}\hat{a}_{2}+{\hat{a}_{2}^{\dagger}}\hat{a}_{1})\right.\\\
\,&\,\left.-({\hat{a}_{1}^{\dagger}}\hat{a}_{2}+{\hat{a}_{2}^{\dagger}}\hat{a}_{1})(c{\hat{a}_{1}^{\dagger}}\hat{a}_{1}+c^{3}{\hat{a}_{2}^{\dagger}}\hat{a}_{2})\right.\\\
\,&\,\left.+c^{2}({\hat{a}_{1}^{{\dagger}2}}\hat{a}_{2}^{2}+{\hat{a}_{2}^{{\dagger}2}}\hat{a}_{1}^{2})\right],\end{split}$
(47)

describes the scattering interaction between the cavity modes. The scattering
effect involves the multiple-order transitions between different phonon states
in a mechanical oscillator.

For the unperturbed Hamiltonian $\hat{H}_{0}^{U}$, the eigenstates can be
written as
$|j,k,n_{1},n_{2}\rangle=|j\rangle\otimes|k\rangle\otimes|n_{1}\rangle\otimes|n_{2}\rangle$
with the eigenvalue

$\begin{split}{E_{j,k,n}}=\,&\,\hbar{\omega_{c1}}n+\hbar{\omega_{c2}}n+\hbar{\omega_{m}}k+\hbar{\omega_{a}}\langle
j|e\rangle\\\ \,&\,-\frac{\hbar
g^{2}}{\omega_{m}}\left[n_{1}^{2}+2c^{2}{n_{1}n_{2}}+c^{4}{n_{2}^{2}}\right.\\\
\,&\,\left.+c^{2}{n_{1}(n_{2}+1)}+c^{2}{n_{2}(n_{1}+1)}\right].\end{split}$
(48)

Figure 9: (a) Sketch of the processes giving the main contribution to the
effective coupling between the states
$(1/\sqrt{2})(|g,1,0,0\rangle-|g,0,0,2\rangle)$ and $|e,0,1,0\rangle$, via DCE
(red arrows), counterrotating terms (blue arrows), and scattering effect
(green arrows). (b) Comparison between the numerical (red dots) and analytical
(blue solid curve) results of the normalized level splitting.

###  B.2 Analytical approach to the resonance condition
$2\omega_{c2}=\omega_{a}+\omega_{c1}$

For the coupling between states $|e,0,1,0\rangle$ and
$(1/\sqrt{2})(|g,1,0,0\rangle-|g,0,0,2\rangle)$ shown in Fig. 5(a) in the main
text, the transition paths are shown in Fig. 9(a). If the state
$(1/\sqrt{2})(|g,1,0,0\rangle-|g,0,0,2\rangle)$ is the initial state and
prepared, then it reaches the final state $|e,0,1,0\rangle$ through virtual
transition involving the intermediate states $|g,0,2,0\rangle$ by the DCE, or
it directly reaches the final state by the counterrotating terms. Besides the
two effects, the scattering effect can also contribute to the coupling, as
shown in Fig. 9(a). Thus, the effective coupling rate can be obtained by using
perturbation theory.

$\begin{split}\Omega=\,&\,\frac{\langle e,0,1,0|\hat{H}_{\rm
I}^{U}|g,0,2,0\rangle\langle g,0,2,0|\hat{H}_{\rm
I}^{U}|\psi_{i}\rangle}{{E_{\rm i}}-{E_{g,0,2,0}}}\\\ \,&\,+\langle
e,0,1,0|\hat{H}_{\rm AFM}^{U}|\psi_{i}\rangle\\\
=\,&\,\frac{-\lambda\left(1-\frac{(1+c^{2})g^{2}}{2\omega_{m}^{2}}\right)\left(\frac{2g^{2}c^{2}}{\omega_{m}}+\frac{\sqrt{2}g}{2}+\frac{(1+c^{2})^{2}g^{3}\sqrt{2}}{\omega_{m}^{2}}\right)}{2(\omega_{c2}-\omega_{c1})-\frac{4g^{2}(c^{4}-1)}{\omega_{m}}}\\\
\,&\,-\frac{1}{\sqrt{2}}\left(-\frac{g\lambda}{\omega_{m}}+\frac{(1+c^{2})^{2}\lambda
g^{3}}{2\omega_{m}^{3}}\right),\end{split}$ (49)

where $\hat{H}_{\rm I}^{U}={\hat{H}_{\rm AFM}^{U}+\hat{H}_{\rm
FM}^{U}+\hat{H}_{\rm FFM}^{U}}$ is the effective perturbative Hamiltonian
after the unitary transformation and
$|\psi_{i}\rangle=(1/\sqrt{2})(|g,1,0,0\rangle-|g,0,0,2\rangle)$ is the
initial state. In this case, the transition path of the scattering effect is
dominated by the last line of Eq. (47). Specifically,

$\displaystyle\begin{split}\langle g,0,2,0|\hat{H}_{\rm
FFM}^{U}|g,0,0,2\rangle=\,&\,\langle g,0,2,0|\hat{H}_{\rm
FF}|g,0,0,2\rangle\\\ =\,&\,\frac{-2g^{2}c^{2}}{\omega_{m}}.\end{split}$

This means that the scattering effect is a second-order process in the
coupling. Compared to the DCE and counterrotating terms, the contribution of
the scattering effect in the effective coupling rate is relatively small.
Figure 9(b) shows a comparison between the numerical and the analytical
results of the normalized splitting ($2\Omega/{\omega_{m}}$). The agreement is
good if the normalized optomechanical coupling is not too strong.

Figure 10: Sketch of the processes of the photon frequency conversion. (a) The
path of the energy flow. (b) The paths with the interaction (indicated as the
green arrow) between the cavity mode $\omega_{c2}$ and the atom.

Note that, in the above case, the interaction between the cavity mode
$\omega_{c2}$ and the atom has not been considered, as shown by Fig. 10(a). If
this interaction is included [indicated by green arrow in Fig. 10(b)], then
the unfavorable energy exchange will occur in the photon frequency conversion
and then the conversion efficiency will be decreased. To avoid this condition,
one can make the dipole moment vector of the atom perpendicular to the field
vector of cavity mode $\omega_{c2}$ or keep the atom away from the field
distribution of the cavity mode $\omega_{c2}$.

##  Appendix C Two-atom interacting with single cavity-optomechanical mode:

###  C.1 Analytical approach to the resonance condition
$\omega_{a1}+\omega_{a2}=\omega_{m}$

As the description of the two-atom model in Sec. IV of the main text, the
system Hamiltonian can be written as

$\begin{split}\hat{H}=\,&\,\hbar{\omega_{c}}{\hat{a}^{\dagger}}\hat{a}+\hbar{\omega_{m}}{\hat{b}^{\dagger}}\hat{b}+\hbar{\omega_{a1}}\hat{\sigma}_{+}^{(1)}{\hat{\sigma}_{-}^{(1)}}+\hbar{\hat{\omega}_{a2}}\hat{\sigma}_{+}^{(2)}{\hat{\sigma}_{-}^{(2)}}\\\
\,&\,+\hbar
g{\hat{a}^{\dagger}}\hat{a}(\hat{b}+{\hat{b}^{\dagger}})+\frac{\hbar}{2}g({\hat{a}^{2}}+{\hat{a}^{{\dagger}2}})(\hat{b}+{\hat{b}^{\dagger}})\\\
\,&\,+\hbar{\lambda_{1}}(\hat{a}+{\hat{a}^{\dagger}})({\hat{\sigma}_{+}^{(1)}}+\hat{\sigma}_{-}^{(1)})\\\
\,&\,+\hbar{\lambda_{2}}(\hat{a}+{\hat{a}^{\dagger}})({\hat{\sigma}_{+}^{(2)}}+\hat{\sigma}_{-}^{(2)}).\end{split}$
(50)

Figure 11: (a) Relevant energy levels of the system Hamiltonian versus ratio
between the transition frequency of the atom $\omega_{a2}$ and the frequency
of the vibrating mirror $\omega_{m}$. (b) The enlarged view of the boxed
region in panel (a) showing the avoided-level crossing between
$|g,g,1,0\rangle$ and $|e,e,0,0\rangle$, induced by the DCE and the CRE. The
resonance condition in this case is $\omega_{a1}+\omega_{a2}=\omega_{m}$.
Here, the transition frequency of the first atom is
$\omega_{a1}=0.45\omega_{m}$, and the normalized optomechanical coupling rate
is $g=0.01\omega_{m}$

.

To derive the effective system Hamiltonian to analyze this model, we can use
the method, which is presented in the model of single-atom with single cavity-
optomechanical mode. Then, we have

$\begin{split}\hat{H}_{\rm
0}^{U}=\,&\,\hbar{\omega_{c}}{\hat{a}^{\dagger}}\hat{a}+\hbar{\omega_{m}}{\hat{b}^{\dagger}}\hat{b}+\hbar{\omega_{a1}}\hat{\sigma}_{+}^{(1)}{\hat{\sigma}_{-}^{(1)}}+\hbar{\omega_{a2}}\hat{\sigma}_{+}^{(2)}{\hat{\sigma}_{-}^{(2)}}\\\
\,&\,-\hbar\frac{{{g^{2}}}}{{{\omega_{m}}}}{\hat{a}^{\dagger}}\hat{a}{\hat{a}^{\dagger}}\hat{a}{\rm{}},\end{split}$
(51)

$\begin{split}\hat{H}_{\rm{A^{(1)}}FM}^{U}=\,&\,\hbar\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}{\left(\frac{g}{{{\omega_{m}}}}\right)^{n}}{\lambda_{1}}\left[{\hat{a}^{\dagger}}+{(-1)^{n}}\hat{a}\right]\\\
\,&\,\times({\hat{\sigma}_{-}^{(1)}}+\hat{\sigma}_{+}^{(1)}){({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(52)

$\begin{split}\hat{H}_{\rm{A^{(2)}}FM}^{U}=\,&\,\hbar\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}{\left(\frac{g}{{{\omega_{m}}}}\right)^{n}}{\lambda_{2}}\left[{\hat{a}^{\dagger}}+{(-1)^{n}}\hat{a}\right]\\\
\,&\,\times({\hat{\sigma}_{-}^{(2)}}+\hat{\sigma}_{+}^{(2)}){({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(53)

and

$\begin{split}\hat{H}_{\rm
FM}^{U}=\,&\,\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}\hbar
g{\left(\frac{g}{{{\omega_{m}}}}\right)^{n}}{2^{n-1}}\left\\{\left[{\hat{a}^{{\dagger}2}}+{(-1)^{n}}{\hat{a}^{2}}\right]\right.\\\
\,&\,\left.\times{({\hat{b}^{\dagger}}-\hat{b})^{n}}({\hat{b}^{\dagger}}+\hat{b})-n\left[{\hat{a}^{{\dagger}2}}+{(-1)^{n-1}}{\hat{a}^{2}}\right]\right.\\\
\,&\,\left.\times{\hat{a}^{\dagger}}\hat{a}{({\hat{b}^{\dagger}}-\hat{b})^{n-1}}\right\\}.\end{split}$
(54)

In this model, the resonant condition is
$\omega_{a1}+\omega_{a2}=k\omega_{m}$, where $k$ is an integer. This formula
implies the exchange of virtual excitations between the atoms and a mirror.
For the unperturbed Hamiltonian $\hat{H}_{0}^{U}$, the eigenstate is described
as $|j_{1},j_{2},k,n\rangle$, where $|j_{1}\rangle$ and $|j_{2}\rangle$
($j=g,e$) denote the states of the atom $\omega_{a1}$ and $\omega_{a2}$,
respectively, with eigenvalue

$\begin{split}E_{j_{1}.j_{2},k,n}=\,&\,\hbar\omega_{c}n+\hbar\omega_{m}k+\hbar\omega_{a}\langle
j_{1}|e\rangle+\hbar\omega_{a}\langle j_{2}|e\rangle\\\ \,&\,-\hbar
g^{2}n^{2}/\omega_{m}.\end{split}$ (55)

Here, we assume that the two atoms are nonidentical, the mode frequency of the
cavity mode is ${\omega_{c}}=0.7{\omega_{m}}$, and that the normalized atom-
cavity coupling rates are $\lambda_{1}=0.014\omega_{m}$ and
$\lambda_{2}=0.022\omega_{m}$.

Figure 12: (a) Sketch of the processes giving the main contribution to the
effective coupling between the states $|g,g,1,0\rangle$ and $|e,e,0,0\rangle$,
via the DCE (red arrows) and the counterrotating terms (blue arrows). (b)
Comparison between the numerical (red dots) and the analytical (blue solid
curve) results of the normalized level splitting.

Figure 11 displays the lowest energy levels (${E_{j}}-{E_{0}}$) of the system
Hamiltonian versus the ratio between the transition frequency of the atom
$\omega_{a2}$ and the mechanical frequency of the mirror $\omega_{m}$. The
avoided-level crossing in Fig. 11(b) originates from the coherent coupling
between the states $|g,g,1,0\rangle$ and $|e,e,0,0\rangle$. At the minimum
energy splitting, the resulting states are well approximated by
$|{\varphi_{4,5}}\rangle\simeq\frac{1}{{\sqrt{2}}}(|g,g,1,0\rangle\mp|e,e,0,0\rangle)$.
Obviously, this splitting occurs in a resonant condition
$\omega_{a1}+\omega_{a2}=\omega_{m}$.

In Fig. 12(a), the transition paths for a coupling between states
$|g,g,1,0\rangle$ and $|e,e,0,0\rangle$ are dominated by the DCE and the
counterrotating terms. If $|g,g,1,0\rangle$ is the initial state of the
system, then it reaches the final state $|e,e,0,0\rangle$ through the virtual
transition involving the out-of-resonance intermediate states
$|g,g,0,2\rangle$ or $|e,g,0,1\rangle$ ($|g,e,0,1\rangle$). Such a transition
process is a detuned resonance transition. This means that when one system
satisfies the resonance condition $\omega_{a1}+\omega_{a2}=k\omega_{m}$ even
if $\omega_{a1}\neq\omega_{a2}$, a single vibrating excitation of the
mechanical oscillator can be absorbed by the two atoms. Based on the above
effective system Hamiltonian, the effective coupling rate can be obtained by
using perturbation theory,

$\begin{split}\Omega_{ee00}^{gg10}=\,&\,\frac{{\frac{g\lambda_{1}\lambda_{2}}{\omega_{m}}\left({1-\frac{g^{2}}{2\omega_{m}^{2}}}\right)\left(-1+\frac{{{g^{2}}}}{{2\omega_{m}^{2}}}\right)}}{{\left({\omega_{m}}-{\omega_{c}}-{\omega_{a1}}+\frac{{{g^{2}}}}{{{\omega_{m}}}}\right)}}\\\
\,&\,+\frac{{\frac{g\lambda_{1}\lambda_{2}}{\omega_{m}}\left({1-\frac{g^{2}}{2\omega_{m}^{2}}}\right)\left(-1+\frac{{{g^{2}}}}{{2\omega_{m}^{2}}}\right)}}{{\left({\omega_{m}}-{\omega_{c}}-{\omega_{a2}}+\frac{{{g^{2}}}}{{{\omega_{m}}}}\right)}}\\\
\,&\,+\frac{{2g\lambda_{1}\lambda_{2}\left({1-\frac{g^{2}}{2\omega_{m}^{2}}}\right)^{2}\left(\frac{1}{2}+\frac{g^{2}}{\omega_{m}^{2}}\right)}}{{\left({\omega_{m}}-{\omega_{c}}-{\omega_{a1}}+\frac{{{g^{2}}}}{{{\omega_{m}}}}\right)\left({\omega_{m}}-{2\omega_{c}}+\frac{{{4g^{2}}}}{{{\omega_{m}}}}\right)}}\\\
\,&\,+\frac{{2g\lambda_{1}\lambda_{2}\left({1-\frac{g^{2}}{2\omega_{m}^{2}}}\right)^{2}\left(\frac{1}{2}+\frac{g^{2}}{\omega_{m}^{2}}\right)}}{{\left({\omega_{m}}-{\omega_{c}}-{\omega_{a2}}+\frac{{{g^{2}}}}{{{\omega_{m}}}}\right)\left({\omega_{m}}-{2\omega_{c}}+\frac{{{4g^{2}}}}{{{\omega_{m}}}}\right)}}.\\\
\end{split}$ (56)

In Fig. 12(b), it is in good agreement between the numerical and the
analytical results of the normalized splitting
($2\Omega_{ee01}^{gg10}\omega_{m}$) versus the normalized optomechanical
coupling ($g/{\omega_{m}}$). This demonstrates that two atoms can couple to
the mirror via a complete exchange of virtual excitations.

###  C.2 System dynamics for two nonidentical atoms with different coupling
rates

Figure 13: The dynamical evolution of the hybrid quantum system, (a) in the
weak-coupling regime (loss rate $\eta_{1}=\eta_{2}=\kappa=\gamma=1.3\times
10^{-4}\omega_{m}$) with a single-tone mechanical drive
$F(t)=A\cos(\omega_{d}t)$ (a weak resonance excitation $A=2\gamma$ and
$\omega_{d}=\omega_{m}$), and (b) in the strong-coupling regime (
$\kappa=\gamma=\eta_{1}=\eta_{2}=1\times 10^{-5}\omega_{m}$) with an ultrafast
resonant pulse $F(t)=AG(t-t_{0})\cos(\omega_{d}t)$ applied on the movable
mirror (standard deviation $\sigma=(30\Omega_{ee00}^{gg10})^{-1}$, amplitude
$A=0.25\pi$, and central driving frequency $\omega_{d}=\omega_{m}$). The
system is initially prepared at the ground state.

As in the previous discussion, from the resonance condition
$\omega_{a1}+\omega_{a2}=k\omega_{m}$, we know that the two atoms do not need
to be identical. In this subsection, we study the system dynamics in a more
general case, where the two atoms are nonidentical with different coupling
rates $\lambda_{i}$.

Figure 13 displays the system dynamics [corresponding to the minimum level
splitting shown in Fig. 11(b)] of the mean photon number
$\langle\hat{A}^{\dagger}\hat{A}\rangle$, the mean phonon number
$\langle\hat{B}^{\dagger}\hat{B}\rangle$, the mean atom-excitation number
$\langle\hat{X}_{1}^{\dagger}\hat{X}_{1}\rangle$, and
$\langle\hat{X}_{2}^{\dagger}\hat{X}_{2}\rangle$. These are done in the weak-
coupling regime with the continuous-wave mechanical drive and the strong-
coupling regime with the ultrafast resonant mechanical pulse. Figure 13 shows
that, at the beginning of the dynamics, the energy of two atoms rises
simultaneously after the mechanical oscillator is driven. This phenomenon is
similar to the case shown in Fig. 6, where the two atoms and corresponding
coupling rates $\lambda$ are all identical. The difference is that, after a
short time, the excitations of the two atoms have not coincided with each
other in Fig. 13. This can be attributed to the differences of proportion of
the transition paths [Fig. 12(a)], which are induced by the differences
between the coupling rates of the cavity-atom.

In addition, we observe that the single-atom excitation
$\langle\hat{X}_{i}^{\dagger}\hat{X}_{i}\rangle$ is close to the two-atom
correlation $G_{a}^{(2)}=\langle
X_{1}^{{\dagger}}X_{2}^{{\dagger}}X_{2}X_{1}\rangle$. By excluding the effect
of system decay, we have also calculated $G_{a}^{(2)}$ and found that
$\langle\hat{X}_{i}^{\dagger}\hat{X}_{i}\rangle$ and $G_{a}^{(2)}=\langle
X_{1}^{{\dagger}}X_{2}^{{\dagger}}X_{2}X_{1}\rangle$ almost coincide over
time. This result is a clear signature of joint excitation: if one atom
becomes excited, the probability of the other atom being excited is very close
to one.

##  Appendix D Additional discussions

Very recently, we noticed a work considering the case when the mechanical
oscillator can excite an atom through a cavity-vacuum field [81]. The cavity-
atom part of this work is based on the single-photon Rabi model and the two-
photon Rabi model. Reference [81] gives the analytical solution of the two-
photon Rabi model by using the method of Ref [52]. However it neglects the
transition processes between the different phonon states of one mechanical
oscillator, which is essential in this method. In the single-photon Rabi
model, Ref. [81] only presents the numerical results. Actually, the method of
Ref. [52] is not suitable for the single-photon Rabi model because the energy
transfer between an atom and a mechanical oscillator is at the single-photon
level. In this section, we will compare the analytical methods used in our
works and in Ref. [52] for studying the coupling between an atom and a
mechanical oscillator, with the single-photon Rabi model or two-photon Rabi
model.

###  D.1 Single-photon Rabi model

Figure 14: Sketch of the processes giving the main contribution to the
effective coupling between the states $|g,1_{0},0\rangle$ and
$|e,0_{1},1\rangle$, via the DCE (red arrows), using the method of Ref. [52].

Using the method presented in Ref. [52] for the case of single atom
interacting with a single cavity-optomechanical mode,
$\hat{H}_{0}+\hat{V}_{\rm om}$ (details in Sec. II) is considered as an
unperturbed Hamiltonian and can be analytically diagonalized by defining the
displacement operator $D(n\beta)=\exp[n\beta(b-{b^{\dagger}})]$ with
$\beta=g/{\omega_{m}}$. Therefore, for the unperturbed Hamiltonian
$\hat{H}_{0}+\hat{V}_{\rm om}$, the system eigenstates can be written as

$\begin{split}|j,k_{n},n\rangle=|j\rangle\otimes
D(n\beta)|k\rangle\otimes|n\rangle,\end{split}$ (57)

where $|k\rangle$ ($|n\rangle$) denotes the Fock state of the mechanical
(cavity) mode and $|j\rangle$ ($j=g,e$) denotes the atom states, with the
eigenvalue

${E_{j,k,n}}=\hbar{\omega_{c}}n+\hbar{\omega_{m}}k+\hbar{\omega_{a}}\langle
j|e\rangle-\hbar{{\rm{g}}^{2}}{n^{2}}/{\omega_{m}}.$ (58)

The perturbed Hamiltonian is described by

$\begin{split}\hat{H}_{\rm p}=\dfrac{\hbar
g}{2}(\hat{a}^{2}+\hat{a}^{{\dagger}2})(\hat{b}+\hat{b}^{{\dagger}})+\hbar\lambda(\hat{a}+\hat{a}^{{\dagger}})(\hat{\sigma}_{-}+\hat{\sigma}_{+}).\end{split}$
(59)

According to this analytical model, the coupling between states
$|g,1_{0},0\rangle$ and $|e,0_{1},1\rangle$ can only be analyzed by one
transition path, as shown in Fig. 14, which is dominated by the DCE. The
effective coupling rate can be written as,

$\begin{split}\Omega_{p}=\,&\,\frac{\langle e,0_{1},1|\hat{H}_{\rm
p}|g,0_{2},2\rangle\langle g,0_{2},2|\hat{H}_{\rm
p}|g,1_{0},0\rangle}{E_{g,1,0}-E_{g,0,2}}\\\ =\,&\,\frac{\sqrt{2}\lambda
D_{0,0}(\beta)[\frac{\sqrt{2}}{2}gD_{0,0}(-2\beta)+gD_{0,2}(-2\beta)]}{\omega_{m}-2\omega_{c}+\frac{4g^{2}}{\omega_{m}}},\end{split}$
(60)

where the matrix elements of the displacement operators can be expressed in
terms of associated Laguerre polynomials:

$D_{k^{\prime},k}(\alpha)=\sqrt{k!/k^{\prime}!}\alpha^{k^{\prime}-k}e^{-|\alpha|^{2}}L^{k^{\prime}-k}_{k}(|\alpha|^{2}).$
(61)

Actually, the counterrotating terms in the cavity-atom system Hamiltonian can
also contribute to the coupling if it can combine with the annihilation of one
vibrating excitation of the mechanical oscillator, which is involved in
$V_{\rm om}$. However, $V_{\rm om}$ is not in the perturbed Hamiltonian.

Performing a unitary transformation to eliminate $V_{\rm om}$ can better deal
with this problem, because the influence of this term $V_{\rm om}$ on both
$H_{0}$ and $H_{\rm p}$ can then be revealed (details in Appendix A). By
defining a unitary operator $U=\exp[-\beta{a^{\dagger}}a({b^{\dagger}}-b)]$,
the eigenvalue equation can be described as

$\begin{split}E_{j,k,n}\,&\,=\langle j,k_{n},n|UU^{\dagger}(H_{0}+V_{\rm
om})UU^{\dagger}|j,k_{n},n\rangle\\\ \,&\,=\langle
j,k,n|U^{\dagger}(H_{0}+V_{\rm om})U|j,k,n\rangle,\end{split}$ (62)

Then, for the unperturbed Hamiltonian $U^{\dagger}(H_{0}+V_{\rm om})U$, the
eigenstate is described by $|j,k,n\rangle$. In this method, the coupling
between states $|g,1,0\rangle$ and $|e,0,1\rangle$ can be analyzed by two
transition paths, which is dominated by the DCE and the counterrotating terms
independently, as shown in Fig. 2(a).

Figure 15: Comparison between the numerical calculated normalized level
splitting (red dots) and the corresponding analytical calculations
[2$\Omega_{e01}^{g10}/\omega_{m}$ (blue curve), 2$\Omega_{p}/\omega_{m}$
(black dotted curve), 2$\tilde{\Omega}_{e01}^{g10}/\omega_{m}$ (green dotted
curve)], obtained using perturbation theory. These results are obtained by
using the parameters of Fig. 2(b)

Figure 15 shows the comparison between the numerical calculated normalized
level splitting and the three kinds of analytical calculations, where
$\Omega_{e01}^{g10}$ and $\Omega_{p}$ are respectively described by Eq. (12)
and Eq. (60). We have also presented the effective coupling rate,

$\begin{split}\tilde{\Omega}_{e01}^{g10}=\,&\,\frac{\langle e,0,1|\hat{H}_{\rm
I}^{U}|g,0,2\rangle\langle g,0,2|\hat{H}_{\rm
I}^{U}|g,1,0\rangle}{E_{g,1,0}-E_{g,0,2}}\\\ =\,&\,\frac{\lambda
g\left(\sqrt{2}-\frac{\sqrt{2}g^{2}}{2\omega_{m}^{2}}\right)\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}g^{2}}{\omega_{m}^{2}}\right)}{\omega_{m}-2\omega_{c}+4\frac{g^{2}}{\omega_{m}}},\end{split}$
(63)

using the method of Sec. II, without calculating the transition path of the
counterrotating term [Fig. 2(a)]. The results show that the effect of the
counterrotating term has an important contribution to the coupling between the
atom and the vibrating mirror by comparing $\Omega_{e01}^{g10}$ with
$\Omega_{p}$ ($\tilde{\Omega}_{e01}^{g10}$). In addition, the two curves
$\Omega_{p}$ and $\tilde{\Omega}_{e01}^{g10}$ almost coincide at the range
$g/\omega_{m}\leq 0.1$, illustrating that the method used in Sec. II can also
exactly describe the DCE effect.

###  D.2 Two-photon Rabi model

Reference [81] utilizes the method of Ref. [52] to analyze its model.
Specifically, the unperturbed Hamiltonian is $H_{0}+V_{\rm om}$, while the
perturbed one is written as

$\begin{split}\hat{H}_{\rm p}=\dfrac{\hbar
g}{2}(\hat{a}^{2}+\hat{a}^{{\dagger}2})(\hat{b}+\hat{b}^{{\dagger}})+\hbar\lambda(\hat{a}^{2}+\hat{a}^{{\dagger}2})(\hat{\sigma}_{-}+\hat{\sigma}_{+}).\end{split}$
(64)

The system eigenstates are described by $|j,k_{n}n\rangle=|j\rangle\otimes
D(n\beta)|k\rangle\otimes|n\rangle$, where $|k\rangle$ ($|n\rangle$) denotes
the Fock state of the mechanical (cavity) mode and $|j\rangle$ ($j=g,e$)
denotes the atom states. With the two-photon Rabi model, the energy transfer
between an atom and a mechanical oscillator occurs in the form of a two-photon
process, as shown in Fig. 16. This is a special case where the form of the
energy transfer matches the creation of photon pairs induced by the DCE [or
the effect of counterrotating terms of cavity-atom system Hamiltonian in Eq.
(64)]. Thus, the method of Ref. [52] can analyze this case. However, in Ref.
[81], all these states shown in Fig. 16 are treated as the bare states
$|j,k,n\rangle$. It means that Ref. [81] neglects the transition processes
between the phonon states within a $n$-photon subspace, which are indicated by
the matrix elements of the displacement operator, $\langle
k_{0}^{{}^{\prime}}|D(n\beta)|k_{0}\rangle=D_{k^{{}^{\prime}},k}(n\beta)$.
Thus, the analytical solution for the effective coupling rate in Ref. [81] is
obtained by incorrectly using the method in Ref. [52]. Actually, the effective
coupling rate should be written as

$\begin{split}{\Omega_{\rm eff}}=\,&\,\dfrac{\biggl{[}g\lambda
D_{0,0}(2\beta)-2g\lambda\beta
D_{0,1}(2\beta)\biggr{]}D_{1,1}(-2\beta)}{-\omega_{a}-2\omega_{c}+4g^{2}/\omega_{m}}\\\
\,&\,+\dfrac{\lambda
gD_{0,0}(2\beta)D_{0,0}(-2\beta)}{\omega_{m}-2\omega_{c}+4g^{2}/\omega_{m}}.\end{split}$
(65)

Figure 16: Sketch of the processes giving the main contribution to the
effective coupling between the states $|g,1_{0},0\rangle$ and
$|e,0_{0},0\rangle$ [81], using the method of Ref. [52].

Note that Ref. [81] utilizes the two photons produced by the counterrotating
term of the two-photon Rabi Hamiltonian in Eq. (64) to transfer energy between
the mirror and the atom, as indicated by the transition path (blue arrows) in
Fig. 16. In our works, the atom and photon are treated as a whole, which
interacts with the mirror. This is a significant difference between this work
and Ref. [81]. Moreover, because the transition path (blue arrows) in Fig. 16
also utilizes such a counterrotating term $a^{2}b$ in Eq. (64), the
contribution of this path to the effective coupling rate $\Omega_{\rm eff}$ is
too small.

Importantly, the two-photon Rabi model can also be analyzed in our method. We
can eliminate the term $\hat{V}_{\rm om}$ by performing a unitary
transformation with the unitary operator
$\hat{U}=\exp[-\beta\hat{a}^{\dagger}\hat{a}(\hat{b}^{\dagger}-\hat{b})]$.
Then, we can obtain the effective system Hamiltonian:

$\hat{H}_{0}^{U}={\hat{H}_{0}}-\hbar\frac{{{g^{2}}}}{{{\omega_{m}}}}{\hat{a}^{\dagger}}\hat{a}{\hat{a}^{\dagger}}\hat{a}{\rm{}},$
(66) $\begin{split}\hat{H}_{\rm
AF_{(2)}M}^{U}=\,&\,{\hat{U}^{\dagger}}{\left[\hbar\lambda(\hat{a}^{2}+\hat{a}^{{\dagger}2})(\hat{\sigma}_{-}+\hat{\sigma}_{+})\right]}\hat{U}\\\
=\,&\,\hbar\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}\lambda{\left(\frac{g}{{\omega_{m}}}\right)^{n}}2^{n}\left[{\hat{a}^{{\dagger}2}}+{(-1)^{n}}\hat{a}^{2}\right]\\\
\,&\,\times(\hat{\sigma}_{-}+{\hat{\sigma}_{+}}){({\hat{b}^{\dagger}}-\hat{b})^{n}},\end{split}$
(67)

and

$\begin{split}\hat{H}_{F_{(2)}M}^{U}=\,&\,\sum\limits_{n=0}^{\infty}{\frac{1}{{n!}}}\hbar
g{\left(\frac{g}{{{\omega_{m}}}}\right)^{n}}{2^{n-1}}\left\\{\left[{\hat{a}^{{\dagger}2}}+{(-1)^{n}}{\hat{a}^{2}}\right]\right.\\\
\,&\,\left.\times{({\hat{b}^{\dagger}}-\hat{b})^{n}}({\hat{b}^{\dagger}}+\hat{b})-n\left[{\hat{a}^{{\dagger}2}}+{(-1)^{n-1}}{\hat{a}^{2}}\right]\right.\\\
\,&\,\left.\times{\hat{a}^{\dagger}}\hat{a}{({\hat{b}^{\dagger}}-\hat{b})^{n-1}}\right\\}.\end{split}$
(68)

For the unperturbed Hamiltonian $\hat{H}_{0}^{U}$, the eigenstates are
described by $|j,k,n\rangle=|j\rangle\otimes|k\rangle\otimes|n\rangle$, with
the eigenvalue

${E_{j,k,n}}=\hbar{\omega_{c}}n+\hbar{\omega_{m}}k+\hbar{\omega_{a}}\langle
j|e\rangle-\hbar{{\rm{g}}^{2}}{n^{2}}/{\omega_{m}}.$ (69)

In this analytical method, the transition processes indicated by
$D_{k^{{}^{\prime}},k}(n\beta)$ can be directly exhibited in $\hat{H}_{\rm
AF_{(2)}M}^{U}$ and $\hat{H}_{F_{(2)}M}^{U}$. Then, the effective coupling
rate in our method can be written as

$\Omega_{\rm eff}=\dfrac{\lambda g-6\frac{\lambda
g^{3}}{\omega_{m}^{2}}}{-\omega_{a}-2\omega_{c}+\frac{4g^{2}}{\omega_{m}}}+\frac{\lambda
g+\frac{2\lambda
g^{3}}{\omega_{m}^{2}}}{\omega_{m}-2\omega_{c}+\frac{4g^{2}}{\omega_{m}}}.$
(70)

In summary, the analytical method used in this work is more suitable for
studying the coupling between the atoms and a mechanical oscillator induced by
quantum vacuum fluctuations.

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