# Nonreciprocal Bundle Emissions of Quantum Entangled Pairs

Qian Bin  School of Physics and Institute for Quantum Science and Engineering,
Huazhong University of Science and Technology, and Wuhan Institute of Quantum
Technology, Wuhan 430074, China  Hui Jing  Key Laboratory of Low-Dimensional
Quantum Structures and Quantum Control of Ministry of Education, Department of
Physics and Synergetic Innovation Center for Quantum Effects and Applications,
Hunan Normal University, Changsha 410081, China  Ying Wu  School of Physics
and Institute for Quantum Science and Engineering, Huazhong University of
Science and Technology, and Wuhan Institute of Quantum Technology, Wuhan
430074, China  Franco Nori  Theoretical Quantum Physics Laboratory, Cluster
for Pioneering Research, RIKEN, Wakoshi, Saitama 351-0198, Japan  Center for
Quantum Computing, RIKEN, Wakoshi, Saitama 351-0198, Japan  Physics
Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
Xin-You Lü<EMAIL_ADDRESS>School of Physics and Institute for Quantum
Science and Engineering, Huazhong University of Science and Technology, and
Wuhan Institute of Quantum Technology, Wuhan 430074, China  School of Physics
and Institute for Quantum Science and Engineering, Huazhong University of
Science and Technology, and Wuhan Institute of Quantum Technology, Wuhan
430074, China  Key Laboratory of Low-Dimensional Quantum Structures and
Quantum Control of Ministry of Education, Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha 410081, China  School of Physics and Institute for
Quantum Science and Engineering, Huazhong University of Science and
Technology, and Wuhan Institute of Quantum Technology, Wuhan 430074, China
Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research,
RIKEN, Wakoshi, Saitama 351-0198, Japan  Center for Quantum Computing, RIKEN,
Wakoshi, Saitama 351-0198, Japan  Physics Department, The University of
Michigan, Ann Arbor, Michigan 48109-1040, USA<EMAIL_ADDRESS>School of
Physics and Institute for Quantum Science and Engineering, Huazhong University
of Science and Technology, and Wuhan Institute of Quantum Technology, Wuhan
430074, China

###### Abstract

Realizing precise control over multiquanta emission is crucial for quantum
information processing, especially when integrated with advanced techniques of
manipulating quantum states. Here, by spinning the resonator to induce the
Sagnac effect, we can obtain nonreciprocal photon-phonon and photon-magnon
super-Rabi oscillations under conditions of optically driving resonance
transitions. Opening dissipative channels for such super-Rabi oscillations
enables the realization of directional bundle emissions of entangled photon-
phonon pairs and photon-magnon pairs by transferring pure multiquanta state to
bundled multiquanta outside of the system. This nonreciprocal emission is a
flexible switch that can be controlled with precision, and simultaneous
emissions of different entangled pairs (such as photon-phonon or photon-magnon
pairs) can even emerge but in opposite directions by driving the resonator
from different directions. This ability to flexibly manipulate the system
allows us to achieve directional entangled multiquanta emitters, and has also
potential applications for building hybrid quantum networks and on-chip
quantum communications.

Nonreciprocity plays a key role in various devices such as isolators and
circulators Potton2004 ; Lodahl2017MS ; Sounas2017Alu . Nonreciprocal devices,
allowing the flow of signal from one side but not the other, have important
applications for the realizations of noise-free information processing and
amplifiers Kamal2011CD ; Liu2021ZZ ; Sliwa2015HN ; Kerckhoff2015LC ;
Metelmann2015Clerk ; Malz2018TB ; Shi2015YF , one-way communications
Feng2011AH ; Scheucher2016HW , unidirectional invisibility Lin2011RE ;
Wang2019RY , and directional lasing Bahari2017NV ; Jiang2018MC ; Xu2021LL ;
Xu2021Song . Many nonreciprocal devices have been achieved based on magnetic
biasing Saleh2007Teich ; Jalas2013PE , nonlinear optics Fan2012WV ; Cao2017WD
; Graf2022RS ; Yang2019XH , atomic gases Wang2013ZG ; Ramezani2018JW ,
optomechanical resonators Peterson2017LC ; Bernier2017TK ; Shen2016ZC ;
Lepinay2019DOK ; Lepinay2020OKM ; Manipatruni2009RL ; Shen2023ZC , spinning
resonators Maayani2018DK , non-Hermitian optics Bender2013FB ; Peng2014OL ;
Chang2014JH , and acoustic materials Fleury2014SS ; Zangeneh-Nejad2018Fleury ;
Wiederhold2019SA ; Coulais2017SA ; Cui2018YC . However, previous works have
mainly focused on classical regimes rather than quantum regimes. Recently,
many nonreciprocal quantum phenomena have been predicted, including single-
photon insolation and circulation Tang2022NT ; Xia2018NX ; Shen2011BS ;
Dong2021XZ ; Ren2022WF , one-way flow of thermal noise Barzanjeh2018AX , and
nonreciprocal quantum blockade Wang2022XX ; Huang2018ML ; Li2019HX ;
Jing2021SX ; Shen2020WW ; Wang2019WY  and quantum entanglement Ren2022 ;
Jiao2022LL ; Jiao2022ZZ .

Multiquanta physics is an increasing popular research line in the field of
quanta state manipulation, with important applications in many fields such as
quantum metrology and lithography Giovannetti2006LM ; DAngelo2001CY ,
biodetection Denk1990SW ; Horton2013WK ; Chu2011HZ , and lasing engineering
Gauthier1992WM . Current methods generating multiquanta states mainly rely on
post-selection Pan2012CL , Rydberg atoms Bienias2014CF ; Maghrebi2015GB ;
Jachymski2016BB ; Firstenberg2016AH ; Liang2018VC , and
waveguide/cavity/circuit quantum electrodynamics (QED) systems
GarciaMaraver2004CE ; Koshino2013IY ; Douglas2016CC ; Sanchez-Burillo2016MMGR
; Gonzalez-Tudela2017PK ; Ota2011IK ; Munoz2014VT ; Sanchez-Munoz2015LT ;
Hargart2016MRC ; Chang2016GTM ; Munoz2018LV ; Satzinger2018ZC ; Qin2019MM ;
Bin2020LL ; Cosacchi2022MPS ; Zou2022LL ; Gou2022HW ; Bin2021WL ; Ma2021LR ;
Liu2023HT ; Dong2019Li ; Ren2021DX . In particular, the continuous generation
of multiquanta state has been proposed under atom-driven systems Munoz2014VT ;
Sanchez-Munoz2015LT ; Munoz2018LV ; Bin2020LL ; Zou2022LL ; Gou2022HW ;
Bin2021WL ; Ma2021LR ; Liu2023HT ; Deng2021SY ; Yuan2023XD , where the unit of
emission is replaced by a bundle of strongly correlated $n$ quanta. This
multiquanta bundle emission means that the quantum emitter releases energy in
the group of $n$ quanta, which is different from the lasing Bahari2017NV ;
Jiang2018MC ; Xu2021LL ; Xu2021Song  that releases energy in the form of
single-quanta emission and investigates the amplification of phonons or
magnons. This bundle emission is a powerful source for achieving Heisenberg-
limited quantum metrology Deng2023LC ; Chu2018KY . However, as far, the
realization of bundle emission has not reach exquisite controllability such as
strong nonreciprocity. Previously suggested bundle emissions were all
reciprocal, and overcoming this limitation to achieve controllable directional
bundle emission has remained challenging. Exquisite controllability for bundle
emission is highly desirable and may have an important role in the design of
directional multiquanta emitters.

Here, we propose an experimentally feasible scheme to achieve switchable
nonreciprocal bundle emissions of quantum entangled pairs in a setup
consisting of a strongly driven atom coupled to an optically pumped opto-
magnetic resonator. Spinning the resonator introduces a frequency split in the
countercirculating modes via the Sagnac effect Maayani2018DK . The Jaynes-
Cummings (JC) interaction combined with linear interaction in the resonator
can jointly induce the resonant excitation of the dressed atom together with a
single photon and a single phonon (or magnon), when the resonator is driven in
a chosen direction. The combination of resonator spinning and optically driven
resonance transition leads to photon-phonon (or photon-magnon) super-Rabi
oscillations Strekalov2014  that occur in one direction but not in the other.
Combined with system dissipations, this super-Rabi oscillation can transfer
the pure multiquanta state to bundles of two strongly correlated quanta of
completely different natures outside of the system, achieving nonreciprocal
bundle emission of quantum pairs.

Figure 1:  (a,b) Schematic of the model. Fixing the resonator spinning along
the counterclockwise direction, the photon-phonon (photon-magnon) pairs arise
by driving the resonator from the (a) left ($\mathcal{LD}$) or (b) right
($\mathcal{RD}$). (c-d) Excitation spectra $S_{o}(\epsilon)$ [$o=a,b,m$
correspond to photon, phonon, and magnon modes] versus $\Delta_{ad}/\epsilon$
for (c) $\mathcal{LD}$ and (d) $\mathcal{RD}$, where $\Delta_{\sigma
a}/\omega_{b}=-3.1$, $\omega_{m}/\omega_{b}=1.05$,
$|\Delta_{F}|/\omega_{b}=0.025$, $\lambda_{a\sigma}/\omega_{b}=0.3$,
$\lambda_{ab}/\omega_{b}=\lambda_{am}/\omega_{b}=0.022$, $\xi/\omega_{b}=0.8$,
$\gamma/\omega_{b}=0.001$, and $\kappa/\omega_{b}=0.005$. (e) Photon-phonon
and (f) photon-magnon transitions in the Mollow ladder from $|+\rangle$ to
$|-\rangle$, where the states $|000-\rangle_{2}$ and $|110\rangle_{1}$
($|101+\rangle_{1}$) are degenerate in (e) [(f)]. After a subsequent emission,
the system goes back to the state $|000+\rangle_{0}$.

Accompanied by this emission, the hybrid system also enables the occurrence of
photon-phonon (or photon-magnon) bipartite entanglement and photon-phonon-atom
(or photon-magnon-atom) tripartite entanglement. Thus, our proposal
continuously generates nonreciprocal entangled photon-phonon (or photon-
magnon) pairs. Additionally, entangled photon-phonon and photon-magnon pairs
can be emitted simultaneously in opposite directions when the resonator is
driven from different directions. These directional hybrid quantum effects,
unattainable in purely optical or reciprocal systems, offer new resources for
one-way quantum communications, such as directionally transferring quantum
information with quantum entangled pairs in future quantum networks Kimble2008
; Dong2015WW ; Riedinger2016 ; Wehner2018EH . This unidirectional flow can
prevent backscattering in quantum processing, as transmission signals are
strictly directed along a predefined pathway Wang2009CJ ; Chen2011HD . It also
offers potential application for nonreciprocal quantum metrology, e.g.,
utilizing nonreciprocality to eliminate the interference typically caused by
back-reflections, and applying quantum entangled states to enhance quantum
sensing, overcoming the standard quantum limit and approaching the Heisenberg
limit Xia2023AP ; Gessner2018PS ; Duarte2021Taylor .

_Model and hybrid super-Rabi oscillations_.—We consider theoretically a hybrid
system, with a coherently driven two-level atom coupled to a spinning yttrium
iron garnet (YIG) microresonator with coupling strength $\lambda_{a\sigma}$,
as shown in Fig. 1(a). The optical modes in the microcavity can be modulated
by the mechanical breathing mode Aspelmeyer2014KM  and magnetization
procession Chai2022SZ , when an optical pump with frequency $\omega_{d}$
drives at the optical modes. The resonator spins at an angular velocity
$\Omega$, causing the resonance frequencies of the countercirculating optical
modes to undergo an opposite Fizeau shift, i.e.,
$\omega_{a}\to\omega_{a}+\Delta_{F}$, with $\Delta_{F}=\pm
rn_{r}\omega_{a}\Omega[1-1/n_{r}^{2}-(\lambda/n_{r})(dn_{r}/d\lambda)]/c$.
Here $\omega_{a}$ is the frequency of the optical modes of a stationary
resonator, $r$ is the resonator radius, $n_{r}$ is the refractive index of the
material, and $\lambda$ ($c$) is the wavelength (speed) of light in vacuum.
The dispersion term $(\lambda/n_{r})(dn_{r}/d\lambda)$ characterizes the
relativistic origin of the Sagnac effect and is typically small, and
$\Delta_{F}>0$ and $\Delta_{F}<0$ correspond to driving the resonator from the
left and right, respectively. In the frame rotating at $\omega_{d}$, the
system effective Hamiltonian derived by a standard linearization procedure is
($\hbar=1$) supp

$\displaystyle H\\!=$
$\displaystyle(\Delta_{ad}\\!+\\!\Delta_{F})a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m+\Delta_{\sigma
d}\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(a^{\dagger}+a)(b^{\dagger}+b)+\lambda_{am}(a^{\dagger}+a)(m^{\dagger}+m)$
$\displaystyle+\lambda_{a\sigma}(a\sigma^{\dagger}+a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma),$
(1)

where $a$ ($b$, $m$) is the annihilation operator of the optical (mechanical,
magnon) mode, $\sigma=|g\rangle\langle e|$ is the lowering operator of the
atom, $\Delta_{ad}$ ($\Delta_{\sigma d}$) is the laser detuning with respect
to the optical mode (atom), with $\Delta_{\sigma a}=\Delta_{\sigma
d}-\Delta_{ad}$. Moreover, $\omega_{b}$ ($\omega_{m}$) is the frequency of the
mechanical (magnon) mode, $\lambda_{ab}$ ($\lambda_{am}$) is the effective
linear coupling strength that is proportional to the classical cavity
amplitude, and $\xi$ is the effective driving strength. Strong driving on the
atom can dress the atomic levels, forming a Mollow ladder of manifolds. As
shown in Figs. 1(e,f), each manifold includes many dressed states
$|n_{a}n_{b}n_{m}\pm\rangle$, where $n_{a}(n_{b},n_{m})$ represents the photon
(phonon, magnon) number. This is different from the well-known Mollow ladder
in the single atom and cavity QED systems Mollow1969 ; Cohen-
Tannoudji1977Reynaud ; Zakrzewski1991LM ; Ulhaq2012We ; Carreno2017VL ;
Nienhuis1993 ; Valle2012GTL ; Munoz2014VT ; Sanchez-Munoz2015LT ; Munoz2018LV
; Bin2020LL . The dressed states of the atom $|\pm\rangle=c_{\pm}|g\rangle\pm
c_{\mp}|e\rangle$ have corresponding eigenvalues
$E_{|\pm\rangle}=\Delta_{\sigma d}/2\pm\sqrt{\Delta_{\sigma
d}^{2}+4\xi^{2}}/2$, where $c_{\pm}=\sqrt{2}\xi/[\Delta_{\sigma
d}^{2}+4\xi^{2}\pm\Delta_{\sigma d}\sqrt{\Delta_{\sigma
d}^{2}+4\xi^{2}}]^{1/2}$ supp .

First, when the spinning resonator is driven from the left, and the total
energy of a single clockwise photon and a single magnon matches with the
transition between the $|+\rangle$ and $|-\rangle$, i.e.,
$(\Delta_{ad}+|\Delta_{F}|)+\omega_{m}-\sqrt{\Delta_{\sigma
d}^{2}+4\xi^{2}}\approx 0$ supp , the photon-magnon resonance transition from
the state $|000+\rangle$ to the photon-magnon state $|101-\rangle$ can be
induced by combining the JC interaction and linear optomagnetic interaction.
This is verified by the super-Rabi oscillation
$|000+\rangle\leftrightarrow|101-\rangle$ shown in Fig. 2(a). In contrast,
when the spinning resonator is driven from the right, there is no transition
to multiquata states due to detuning supp , as shown in the inset of Fig.
2(a). This nonreciprocal generation of photon-magnon states is also shown by
the excitation spectra in Figs. 1(c,d). The blue area in Fig. 1(c) corresponds
to the excitation of the photon-magnon state, but this excitation in the same
parameter regime cannot be seen in Fig. 1(d). Second, by varying the driving
frequency detuning at the pink area of Fig. 1(d), the transition from
$|000+\rangle$ to $|110-\rangle$ can be induced by the combination of JC
interaction and linear optomechnical interaction. Figure 2(b) shows that
photon-phonon super-Rabi oscillations occur when driving the resonator from
the right but not from the left. Lastly, when the detuning is varied at the
orange areas of Figs. 1(c,d), the photon-phonon resonance transition can be
induced when the resonator is driven from the left, while the photon-magnon
resonance transition can also be induced when the resonator is driven from the
right supp . Figure 2(c) shows the nonreciprocal photon-phonon and photon-
magnon super-Rabi oscillations. Note that the occurrence of super-Rabi
oscillations is a crucial step for multiquanta emissions.

Figure 2:  (a–c) The population dynamics of the system state at the photon-
phonon (photon-magnon) bundle resonances. (d–g) A quantum trajectory during
photon-magnon emission in the same regime as in (a), but in the presence of
dissipation. Panels (a), (b), and (c) correspond to the blue, pink, and orange
areas in Figs. 1(c,d), and $P_{n_{a}n_{b}n_{m}l}=|\langle
n_{a}n_{b}n_{m}l|\psi(t)\rangle|^{2}$ ($n_{a},n_{b},n_{m}=0,1$ and
$l=+,-,e/g$).
Figure 3:  Cross correlation $g_{1,oo^{\prime}}^{(2)}(0)$ and bundle
correlation $g_{2,oo^{\prime}}^{(2)}(0)$ versus $\kappa/\omega_{b}$ when
driving the resonator from the (a-c) left and (d–f) right. Panels (a,d),
(b,e), and (f,g) correspond to the resonance regimes indicated in blue, pink,
and orange areas of Figs. 1(c,d). The shaded areas correspond to the regime of
$g_{1,ab}^{(2)}(0)>1$ ($g_{1,am}^{(2)}(0)>1$) and $g_{2,ab}^{(2)}(0)<1$
($g_{2,am}^{(2)}(0)<1$). Other system parameters are the same as in Figs.
1(c,d).

_Nonreciprocal emission of entangled photon-phonon and photon-magnon
pairs_.—System dissipations need to be considered here for triggering quantum
emission. The dynamical behavior of the dissipative system can be described by
a quantum master equation

$\displaystyle\frac{d\rho}{dt}=$
$\displaystyle-i[H,\rho]+\sum_{o=a,b,m}\kappa\mathcal{L}[o]\rho+\gamma\mathcal{L}[\sigma]\rho,$
(2)

where $\mathcal{L}[o]\rho=(2o\rho o^{\dagger}-\rho
o^{\dagger}o-o^{\dagger}o\rho)/2$ is a Lindblad term, $\gamma$ is the atomic
decay rate, and $\kappa$ is the decay rate of photon (phonon, magnon). System
dissipations enable the above photon-phonon (photon-magnon) states to be the
bundles of strongly correlated single photons and phonons (magnons) outside of
the system. In Figs. 2(d-g), we depict a brief quantum trajectory to
demonstrate the bundle emissions by calculating the populations of states
using a quantum Monte Carlo method. Initially, the system is mainly in a
superposition state of the vacuum state and photon-magnon state. Triggered by
dissipation, a quantum collapse is likely to occur into the photon-magnon
state, resulting in the emission of a single magnon and leaving the system in
the single-photon state. Subsequently, a single cavity photon is emitted
within a very short temporal window, completing the photon-magnon bundle
emission. Consequently, the system emits two strongly correlated single photon
and single magnon, leaving it in a vacuum state. Note that there is randomness
in the emission sequence of the photon and magnon. Following a direct photon
emission from the atom decay, a photon-magnon state is again prepared for the
next bundle emission. In the subsequent cycle, the system undergoes the same
cascade emission of pair of photon-magnon, each accompanied by a single photon
emission that is at another frequency and does not disrupt the bundle. The
processes of the photon-phonon and photon-magnon bundle emissions are similar.

The strong correlation between the single photon and single phonon (magnons)
in a bundle can be determined by the correlation
$g_{1,oo^{\prime}}^{(2)}(0)=\langle
o^{\dagger}oo^{\prime{\dagger}}o^{\prime}\rangle/(\langle
o^{\dagger}o\rangle\langle o^{\prime{\dagger}}o^{\prime}\rangle)$ Glauber1963
; Scully and Zubairy1997 ; Rebic2002PT , as shown in Fig. 3. However, the
correlations between bundles themselves here need to be quantified by the
generalized second-order correlation of the bundle Munoz2014VT

$\displaystyle
g_{2,oo^{\prime}}^{(2)}(\tau)=\frac{\langle(oo^{\prime})^{{\dagger}}(0)(oo^{\prime})^{{\dagger}}(\tau)(oo^{\prime})(\tau)(oo^{\prime})(0)\rangle}{\langle(oo^{\prime})^{{\dagger}}(0)(oo^{\prime})(0)\rangle\langle(oo^{\prime})^{{\dagger}}(\tau)(oo^{\prime})(\tau)\rangle},$
(3)

where $\tau$ is time-delay. Figures 3(a,c,e,f) show the evolution of photon-
phonon (photon-magnon) bundle from antibunching, through coherence, to
bunching. However, as shown in the shaded areas in Figs. 3 (a,d) [Figs. 3
(b,e)], the photon-magnon (photon-phonon) pairs only occur when the resonator
is driven in one direction, due to the absence of the excitation of
multiquanta states for driving in the other; i.e., there is a nonreciprocal
generation of antibunched photon-magnon (photon-phonon) pairs. In Figs. 3
(c,f), we present that in this parameter regime shown by the orange areas of
Figs. 1(c,d), the nonreciprocal emissions of antibunched photon-phonon and
photon-magnon pairs simultaneously emerge in opposite directions when driving
the resonator from different directions. The strongly antibunched bundles
actually correspond to single photon-phonon (photon-magnon) pairs with
$g_{2,ab}^{(2)}(0)\ll 1$ ($g_{2,am}^{(2)}(0)\ll 1$), analogous to single
photon with $g_{1,aa}^{(2)}(0)\ll 1$. Moreover, it is also possible to realize
nonreciprocal emissions of entangled photon-phonon and photon-magnon pairs in
our proposal, since the system involves multiple degrees of freedom of
completely different nature.

To verify the above emitted quantum pairs are entangled, we calculate the
entanglement between different particles Amico2008FO ; Walter2016GE ;
Fabre2020Treps ; Coffman2000KW ; Adesso2006lluminati ; Rebic2010MM  based on
quantum Fisher information (QFI) and covariance. For an arbitrary separable
quantum states of $N$ particles, the QFI $F_{Q}$ and covariance ${\rm
Var}(A_{j})_{\rho}$ satisfy Gessner2016PS ; Tian2002XS

$\displaystyle F_{Q}[\rho_{\rm sep},\sum_{j=1}^{N}A_{j}]\leq
4\sum_{j=1}^{N}{\rm Var}(A_{j})_{\rho_{\rm sep}}\equiv B_{n},$ (4)

where $A_{j}$ is a local observable for the $j$th particle. The violation of
Eq. (4) for any choice of $A_{j}$ is a sufficient criterion for entanglement.
To find the optimal local operator, we write $A_{j}={\bf c}_{j}\cdot{\bf
A}_{j}$, where ${\bf A}_{j}=(A_{j}^{(1)},A_{j}^{(2)},\dots)^{T}$ and ${\bf
c}_{j}=(c_{j}^{(1)},c_{j}^{(2)},\dots)$ are the set of operators for the $j$th
particle and corresponding coefficients. The full operator is then $A({\bf
c})={\bf c}\cdot{\bf A}$, with the combined vectors ${\bf c}=({\bf c}_{1},{\bf
c}_{2},\dots,{\bf c}_{N})$ and the operators ${\bf A}=[{\bf A}_{1},\dots,{\bf
A}_{N}]^{T}$. In our model, we select ${\bf
A}_{j}=(\sigma_{j}^{x},\sigma_{j}^{y},\sigma_{j}^{z})^{T}$ for the atom and
${\bf A}_{j}=(x_{j},p_{j},x_{j}^{2},p_{j}^{2},(x_{j}p_{j}+p_{j}x_{j})/2)^{T}$
for the mode $o$, where $x=o+o^{\dagger}$ and $p=-i(o-o^{\dagger})$ Tian2002XS
. The QFI and covariance can be written as $F_{Q}[\rho,A({\bf c})]={\bf
c}Q_{\rho}^{\mathcal{A}}{\bf c}^{T}$ and $\sum_{j=1}^{N}{\rm Var}({\bf
c}_{j}\cdot{\bf A}_{j})={\bf c}\Gamma_{\rho}^{\mathcal{A}}{\bf c}^{T}$,
respectively Hyllus2010GS . Here, $Q_{\rho}^{\mathcal{A}}$ has elements
$(Q_{\rho}^{\mathcal{A}})_{jj^{\prime}}^{mm^{\prime}}=2\sum_{k,k^{\prime}}\frac{(p_{k}-p_{k^{\prime}})^{2}}{p_{k}+p_{k^{\prime}}}\langle\psi_{k}|A_{j}^{(m)}|\psi_{k^{\prime}}\rangle\langle\psi_{k^{\prime}}|A_{j^{\prime}}^{(m^{\prime})}|\psi_{k}\rangle$
using the spectral decomposition
$\rho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$ Braunstein1994Caves , and
$\Gamma_{\rho}^{\mathcal{A}}$ has elements
$(\Gamma_{\rho}^{\mathcal{A}})_{jj^{\prime}}^{mm^{\prime}}={\rm
Cov}(A_{j}^{(m)},A_{j^{\prime}}^{(m^{\prime})})_{\rho}$ and
$(\Gamma_{\rho}^{\mathcal{A}})_{jj}^{mm^{\prime}}=0$. The optimal ${\bf c}$
can be found by calculating the optimal eigenvalue and corresponding
eigenstate of the matrix $Q_{\rho}^{\mathcal{A}}-\Gamma_{\rho}^{\mathcal{A}}$
supp .

Figure 4:  (a-b) The Quantities $D_{1}^{ab(m)\sigma}$ and $D_{1}^{ab(m)}$
characterizing the photon-phonon-atom (photon-magnon-atom) entanglement and
photon-phonon (photon-magnon) entanglement versus $\kappa/\omega_{b}$. The
shaded areas correspond to the antibundle bundle regime as shown in Figs.
3(c,f). System parameters are the same as in Figs. 3(c,f).

A photon-phonon separable quantum state and a photon-phonon-atom fully
separable quantum state can be given by
$\rho_{ab}=\sum_{k}p_{k}\rho_{a}^{k}\otimes\rho_{b}^{k}$ and
$\rho_{ab\sigma}=\sum_{k^{\prime}}p_{k^{\prime}}\rho_{a}^{k^{\prime}}\otimes\rho_{b}^{k^{\prime}}\otimes\rho_{\sigma}^{k^{\prime}}$
($\rho_{j}$ is the reduced density operator), respectively. If the quantity
$W^{ab}_{1}=F_{Q}[\rho_{ab},A({\bf c})]-B_{1}(\rho_{ab})>0$, the photon and
phonon modes are entangled. Similarly, if
$W^{ab\sigma}_{1}=F_{Q}[\rho_{ab\sigma},A({\bf c})]-B_{1}(\rho_{ab\sigma})>0$,
there is entanglement between the atom, photon mode, and phonon mode supp ;
Tian2002XS . Here it is only necessary to replace mode $b$ with mode $m$ to
characterize the entanglement between the magnon mode and others. Corresponded
to the parameter regime in Figs. 3 (c,f), in Fig.4 we show the quantities
$D_{1}^{ab(m)}={\rm max}\\{0,W^{ab(m)}_{1}\\}$ and $D_{1}^{ab(m)\sigma}={\rm
max}\\{0,W^{ab(m)\sigma}_{1}\\}$ versus $\kappa/\omega_{b}$. Comparing Fig.
3(c,f) and Fig. 4, it is clear that there is the photon-phonon bipartite
entanglement with $D^{ab}_{1}>0$ and photon-phonon-atom tripartite
entanglement with $D^{ab\sigma}_{1}>0$ in the antibunched bundle regime when
driving the resonator from the left, while there is the photon-magnon
bipartite entanglement with $D^{am}_{1}>0$ and photon-magnon-atom tripartite
entanglement with $D^{am\sigma}_{1}>0$ in the antibunched bundle regime when
driving the resonator from the right. This confirms that nonreciprocal
generation of antibunched photon-phonon (photon-magnon) entangled pairs can be
achieved simultaneously but in opposite directions when we drive the resonator
from different directions. The proposed hybrid system could serve as a
nonreciprocal quantum emitter of entangled photon-phonon (photon-magnon)
pairs.

_Experimental feasibility and conclusions._ —Our proposal could be implemented
in a hybrid setup Aspelmeyer2014KM ; Chai2022SZ , where a coherently driven
two-level atom (e.g., rubidium or cesium atom) is coupled to a spinning YIG
microsphere fixed on a rotating platform Maayani2018DK . The microsphere
supports two countercirculating optical whispering gallery modes (WGMs), which
are strongly driven by the input light through a high-index prism
Aspelmeyer2014KM . The WGMs can be modulated by the mechanical breathing mode
Aspelmeyer2014KM  and magnetization procession Chai2022SZ  when the input
light pumps at the WGM. The microsphere resonator can also be replaced with
the toroidal or bottle microresonator Zhang2016ZZ . The atom is cooled and
trapped near the microresonator using the magneto-optical trap and optical
dipole trap OShea2013JV ; Shomroni2014RL ; Will2021MR ; Aoki2006DW ;
Alton2011SA ; Dayan2008PA ; Junge2013OSV ; Thompson2013TdL . The resonator’s
evanescent field, which decays exponentially outside its surface, interacts
with the atom. This interaction strength depends on the resonator’s properties
and the distance of the atom to its surface. Placing the atom closer to the
resonator Alton2011SA ; Will2021MR , scaling down the size of the
microresonator to reduce the mode volume Buck2003Kimble ; Spillane2005KV , or
employing other physical methods such as quantum squeezing Lu2015WJ ;
Qin2018ML ; Burd2021SK ; Burd2024KS ; Qin2024Nori  can increase the coupling
strength (more details in the Supplemental Material). Additionally, the
phonons (magnons) counts and other types of statistical processing of photons
and phonons (magnons) can be measured by applying an auxiliary system to
convert the mechanical (magnon) signals Meenehan2015MM ; Bender2019KB . The
multipartite entanglement can be characterized by a class of nonlinear
squeezing parameters Tian2002XS ; Gessner2019SP . With this special design, we
theoretically predict that the nonreciprocal entangled photon-phonon pairs and
photon-magnon pairs with corresponding antibunching $g^{(2)}_{2,ab}=0.49$ and
$g^{(2)}_{2,am}=0.29$ could be achieved ($\omega_{b}/2\pi=2\,{\rm GHz}$,
$\omega_{m}/2\pi=2.1\,{\rm GHz}$, $\Delta_{\sigma a}/2\pi=-6.2\,{\rm GHz}$,
$|\Delta_{F}|/2\pi=50\,{\rm MHz}$, $\lambda_{a\sigma}/2\pi=600\,{\rm MHz}$,
$\lambda_{ab}/2\pi=\lambda_{am}/2\pi=44\,{\rm MHz}$, $\xi/2\pi=1.6\,{\rm
GHz}$, $\gamma/2\pi=2\,{\rm MHz}$, and $\kappa/2\pi=16\,{\rm MHz}$)
Aspelmeyer2014KM ; Chai2022SZ ; Zhang2016ZZ ; OShea2013JV ; Shomroni2014RL ;
Will2021MR ; Aoki2006DW ; Alton2011SA ; Dayan2008PA ; Junge2013OSV ;
Buck2003Kimble ; Spillane2005KV ; Thompson2013TdL .

In conclusion, we have shown how to achieve and exquisitely manipulate
nonreciprocal bundle emission of entangled photon-phonon pairs and photon-
magnon pairs. The system enables the simultaneous emission of entangled
photon-phonon and entangled photon-magnon pairs, but in opposite directions,
by driving the resonator from different directions. This work establishes a
connection between nonreciprocal physics and multiquanta emission, providing a
way to manipulate multiquanta states using nonreciprocal devices in optics
Fan2012WV ; Cao2017WD ; Yang2019XH , atomic gases Wang2013ZG ; Ramezani2018JW
, and acoustic materials Fleury2014SS ; Zangeneh-Nejad2018Fleury ;
Wiederhold2019SA . The proposed hybrid system can serve as a directional
quantum emitter of entangled pairs, continuously preparing entangled sources.
This nonreciprocal entangled source may provides unconventional resources for
quantum communication Kimble2008 ; Dong2015WW ; Riedinger2016 ; Wehner2018EH
and metrology Xia2023AP ; Gessner2018PS ; Duarte2021Taylor .

X.L. is supported by the National Key Research and Development Program of
China grant 2021YFA1400700 and the National Natural Science Foundation of
China (Grant No. 11974125). Q.B. is supported by the National Natural Science
Foundation of China (Grant No. 12205109). H.J. is supported by the National
Natural Science Foundation of China (Grants No. 11935006 and No. 11774086) and
the Science and Technology Innovation Program of Hunan Province (Grant No.
2020RC4047). 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), and the Moonshot R $\&$ D Grant
Number JPMJMS2061], the Asian Office of Aerospace Research and Development
(AOARD) (via Grant No. FA2386-20-1-4069), and the office of naval research
(ONR). The computation iscompleted in the HPC Platform of Huazhong University
of Science and Technology.

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Supplemental Material for “Nonreciprocal Bundle Emissions of Quantum Entangled
Pairs”

Supplemental Material for “Nonreciprocal Bundle Emissions of Quantum Entangled
Pairs”

Supplemental Material for “Nonreciprocal Bundle Emissions of Quantum Entangled
Pairs” Qian Bin

Hui Jing

Ying Wu

Franco Nori

Xin-You Lü

In this Supplemental Material, we present the technical details of the
effective Hamiltonian and multiquanta resonances discussed in the main text.
In Sec. I, we derive the system’s effective Hamiltonian using the standard
linearization procedure. In Sec. II, we provide a detailed derivation of
multiquanta resonance conditions within the Mollow regime. In Sec. III, we
discuss the entanglement witness based on quantum Fisher information and
covariance. In Sec. IV, we discuss the experimental feasibility of our
theoretical model in detail.

##  I I. Derivation of the effective Hamiltonian via the quantum master
equation

The Hamiltonian of the system we consider is

$\displaystyle H_{t}=$
$\displaystyle(\omega_{a}+\Delta_{F})a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m+\omega_{\sigma}\sigma^{\dagger}\sigma+\lambda_{ab}^{\prime}a^{\dagger}a(b^{\dagger}+b)+\lambda_{am}^{\prime}a^{\dagger}a(m^{\dagger}+m)+\lambda_{a\sigma}^{\prime}(a^{\dagger}\sigma+a\sigma^{\dagger})$
$\displaystyle+\xi_{d}(a^{\dagger}e^{-i\omega_{d}t}+ae^{i\omega_{d}t})+\xi_{p}(\sigma^{\dagger}e^{-i\omega_{d}t}+\sigma
e^{i\omega_{d}t}),$ (S1)

where $a$ ($b$, $m$) is the annihilation operator of the optical (mechanical,
magnon) mode with corresponding frequency $\omega_{a}$ ($\omega_{b}$,
$\omega_{m}$), $\sigma=|g\rangle\langle e|$ is the lowering operator of the
atom with frequency $\omega_{\sigma}$, $\Delta_{F}$ is the Fizeau shift,
$\lambda_{a\sigma}^{\prime}$ ($\lambda_{ab}^{\prime}$,
$\lambda_{am}^{\prime}$) is the coupling strength between the optical mode and
atom (mechanical mode, magnon mode), $\xi_{d}$ ($\xi_{p}$) and $\omega_{d}$
are the amplitude and frequency of the driving at the optical mode (atom),
respectively. In the frame rotating with frequency $\omega_{d}$, Eq. (I)
becomes

$\displaystyle H_{t}=$
$\displaystyle(\omega_{a}-\omega_{d}+\Delta_{F})a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m+(\omega_{\sigma}-\omega_{d})\sigma^{\dagger}\sigma+\lambda_{ab}^{\prime}a^{\dagger}a(b^{\dagger}+b)+\lambda_{am}^{\prime}a^{\dagger}a(m^{\dagger}+m)+\lambda_{a\sigma}^{\prime}(a^{\dagger}\sigma+a\sigma^{\dagger})$
$\displaystyle+\xi_{d}(a^{\dagger}+a)+\xi_{p}(\sigma^{\dagger}+\sigma).$ (S2)

Taking dissipations into consideration, the dissipative dynamics of the system
can be described by the quantum master equation

$\displaystyle\frac{d}{dt}\rho=$
$\displaystyle-i[H_{t},\rho]+\frac{\kappa_{a}}{2}(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a)+\frac{\kappa_{b}}{2}(2b\rho
b^{\dagger}-b^{\dagger}b\rho-\rho b^{\dagger}b)$
$\displaystyle+\frac{\kappa_{m}}{2}(2m\rho m^{\dagger}-m^{\dagger}m\rho-\rho
m^{\dagger}m)+\frac{\gamma}{2}(2\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho-\rho\sigma^{\dagger}\sigma).$
(S3)

We shift the optical, phonon, and mechanical modes with their mean values
$\alpha$, $\beta$, and $\mu$, i.e., $a=\alpha+\delta a$, $b=\beta+\delta b$,
and $m=\mu+\delta m$, Eq. (I) becomes

$\displaystyle\frac{d}{dt}\rho=$
$\displaystyle-i[H_{t},\rho]+\frac{\kappa_{a}}{2}[2(\alpha+\delta
a)\rho(\alpha^{\ast}+\delta a^{\dagger})-(\alpha^{\ast}+\delta
a^{\dagger})(\alpha+\delta a)\rho-\rho(\alpha^{\ast}+\delta
a^{\dagger})(\alpha+\delta a)]$
$\displaystyle+\frac{\kappa_{b}}{2}[2(\beta+\delta b)\rho(\beta^{\ast}+\delta
b^{\dagger})-(\beta^{\ast}+\delta b^{\dagger})(\beta+\delta
b)\rho-\rho(\beta^{\ast}+\delta b^{\dagger})(\beta+\delta b)]$
$\displaystyle+\frac{\kappa_{m}}{2}[2(\mu+\delta m)\rho(\mu^{\ast}+\delta
m^{\dagger})-(\mu^{\ast}+\delta m^{\dagger})(\mu+\delta
m)\rho-\rho(\mu^{\ast}+\delta m^{\dagger})(\mu+\delta m)]$
$\displaystyle+\frac{\gamma}{2}[2\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho-\rho\sigma^{\dagger}\sigma]$
$\displaystyle=$
$\displaystyle-i[H_{t},\rho]+\frac{\kappa_{a}}{2}(\alpha^{\ast}\delta
a\rho-\alpha^{\ast}\rho\delta a-\alpha\delta a^{\dagger}\rho+\alpha\rho\delta
a^{\dagger})$ $\displaystyle+\frac{\kappa_{b}}{2}(\beta^{\ast}\delta
b\rho-\beta^{\ast}\rho\delta b-\beta\delta b^{\dagger}\rho+\beta\rho\delta
b^{\dagger})+\frac{\kappa_{m}}{2}(\mu^{\ast}\delta m\rho-\mu^{\ast}\rho\delta
m-\mu\delta m^{\dagger}\rho+\mu\rho\delta m^{\dagger})$
$\displaystyle+\frac{\kappa_{a}}{2}(2\delta a\rho\delta a^{\dagger}-\delta
a^{\dagger}\delta a\rho-\rho\delta a^{\dagger}\delta
a)+\frac{\kappa_{b}}{2}(2\delta b\rho\delta b^{\dagger}-\delta
b^{\dagger}\delta b\rho-\rho\delta b^{\dagger}\delta b)$
$\displaystyle+\frac{\kappa_{m}}{2}(2\delta m\rho\delta m^{\dagger}-\delta
m^{\dagger}\delta m\rho-\rho\delta m^{\dagger}\delta
m)+\frac{\gamma}{2}(2\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho-\rho\sigma^{\dagger}\sigma)$
$\displaystyle=$
$\displaystyle-i[H_{1}+H_{2},\rho]+\frac{\kappa_{a}}{2}(2\delta a\rho\delta
a^{\dagger}-\delta a^{\dagger}\delta a\rho-\rho\delta a^{\dagger}\delta
a)+\frac{\kappa_{b}}{2}(2\delta b\rho\delta b^{\dagger}-\delta
b^{\dagger}\delta b\rho-\rho\delta b^{\dagger}\delta b)$
$\displaystyle+\frac{\kappa_{m}}{2}(2\delta m\rho\delta m^{\dagger}-\delta
m^{\dagger}\delta m\rho-\rho\delta m^{\dagger}\delta
m)+\frac{\gamma}{2}(2\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho-\rho\sigma^{\dagger}\sigma),$
(S4)

where the Hamiltonians $H_{1}$ and $H_{2}$ are, respectively,

$\displaystyle H_{1}=$
$\displaystyle(\omega_{a}-\omega_{d}+\Delta_{F})|\alpha|^{2}+\omega_{b}|\beta|^{2}+\omega_{m}|\mu|^{2}+\lambda_{ab}^{\prime}|\alpha|^{2}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}|\alpha|^{2}(\mu^{\ast}+\mu)+\xi_{d}(\alpha^{\ast}+\alpha)$
$\displaystyle+[(\omega_{a}-\omega_{d}+\Delta_{F})\alpha^{\ast}+\lambda_{ab}^{\prime}\alpha^{\ast}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}\alpha^{\ast}(\mu^{\ast}+\mu)+\xi_{d}+\frac{i}{2}\alpha^{\ast}\kappa_{a}]\delta
a$
$\displaystyle+[(\omega_{a}-\omega_{d}+\Delta_{F})\alpha+\lambda_{ab}^{\prime}\alpha(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}\alpha(\mu^{\ast}+\mu)+\xi_{d}-\frac{i}{2}\alpha\kappa_{a}]\delta
a^{\dagger}$
$\displaystyle+(\omega_{b}\beta^{\ast}+\lambda_{ab}^{\prime}|\alpha|^{2}+\frac{i}{2}\beta^{\ast}\kappa_{b})\delta
b+(\omega_{b}\beta+\lambda_{ab}^{\prime}|\alpha|^{2}-\frac{i}{2}\beta\kappa_{b})\delta
b^{\dagger}$
$\displaystyle+(\omega_{m}\mu^{\ast}+\lambda_{am}^{\prime}|\alpha|^{2}+\frac{i}{2}\mu^{\ast}\kappa_{m})\delta
m+(\omega_{m}\mu+\lambda_{am}^{\prime}|\alpha|^{2}-\frac{i}{2}\mu\kappa_{m})\delta
m^{\dagger},$ (S5)

and

$\displaystyle H_{2}=$
$\displaystyle[\omega_{a}-\omega_{d}+\Delta_{F}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]\delta
a^{\dagger}\delta a+\omega_{b}\delta b^{\dagger}\delta b+\omega_{m}\delta
m^{\dagger}\delta m+(\omega_{\sigma}-\omega_{d})\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}^{\prime}(\alpha\delta
a^{\dagger}+\alpha^{\ast}\delta a)(\delta b^{\dagger}+\delta
b)+\lambda_{am}^{\prime}(\alpha\delta a^{\dagger}+\alpha^{\ast}\delta
a)(\delta m^{\dagger}+\delta m)+\lambda_{a\sigma}^{\prime}(\delta
a\sigma^{\dagger}+\delta
a^{\dagger}\sigma)+\lambda_{a\sigma}^{\prime}(\alpha\sigma^{\dagger}+\alpha^{\ast}\sigma)$
$\displaystyle+\lambda_{ab}^{\prime}\delta a^{\dagger}\delta a(\delta
b^{\dagger}+\delta b)+\lambda_{am}^{\prime}\delta a^{\dagger}\delta a(\delta
m^{\dagger}+\delta m)+\xi_{p}(\sigma^{\dagger}+\sigma).$ (S6)

Setting $\lambda_{ab}=\lambda_{ab}^{\prime}|\alpha|$,
$\lambda_{am}=\lambda_{am}^{\prime}|\alpha|$,
$\lambda_{a\sigma}=\lambda_{a\sigma}^{\prime}$, and
$\xi=\lambda_{a\sigma}^{\prime}|\alpha|+\xi_{p}$, the above equation Eq. (I)
becomes

$\displaystyle H_{2}=$
$\displaystyle[\omega_{a}-\omega_{d}+\Delta_{F}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]\delta
a^{\dagger}\delta a+\omega_{b}\delta b^{\dagger}\delta b+\omega_{m}\delta
m^{\dagger}\delta m+(\omega_{\sigma}-\omega_{d})\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(\delta a^{\dagger}+\delta a)(\delta
b^{\dagger}+\delta b)+\lambda_{am}(\delta a^{\dagger}+\delta a)(\delta
m^{\dagger}+\delta m)+\lambda_{a\sigma}(\delta a\sigma^{\dagger}+\delta
a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma)$
$\displaystyle+\lambda_{ab}^{\prime}\delta a^{\dagger}\delta a(\delta
b^{\dagger}+\delta b)+\lambda_{am}^{\prime}\delta a^{\dagger}\delta a(\delta
m^{\dagger}+\delta m).$ (S7)

The steady-state mean values $\alpha$, $\beta$, and $\mu$ satisfy

$\displaystyle(\omega_{a}-\omega_{d}+\Delta_{F})\alpha+\lambda_{ab}^{\prime}\alpha(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}\alpha(\mu^{\ast}+\mu)+\xi_{d}-\frac{i}{2}\alpha\kappa=0,$
$\displaystyle\omega_{b}\beta+\lambda_{ab}^{\prime}|\alpha|^{2}-\frac{i}{2}\beta\kappa_{b}=0,$
$\displaystyle\omega_{m}\mu+\lambda_{am}^{\prime}|\alpha|^{2}-\frac{i}{2}\mu\kappa_{m}=0,$
(S8)

the system dynamics is then decided by

$\displaystyle\frac{d}{dt}\rho=$
$\displaystyle-i[H_{2},\rho]+\frac{\kappa_{a}}{2}(2\delta a\rho\delta
a^{\dagger}-\delta a^{\dagger}\delta a\rho-\rho\delta a^{\dagger}\delta
a)+\frac{\kappa_{b}}{2}(2\delta b\rho\delta b^{\dagger}-\delta
b^{\dagger}\delta b\rho-\rho\delta b^{\dagger}\delta b)$
$\displaystyle+\frac{\kappa_{m}}{2}(2\delta m\rho\delta m^{\dagger}-\delta
m^{\dagger}\delta m\rho-\rho\delta m^{\dagger}\delta
m)+\frac{\gamma}{2}(2\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho-\rho\sigma^{\dagger}\sigma).$
(S9)

Under the condition of strong driving at the optical mode, the higher-order
terms $\lambda_{ab}^{\prime}\delta a^{\dagger}\delta a(\delta
b^{\dagger}+\delta b)$ and $\lambda_{am}^{\prime}\delta a^{\dagger}\delta
a(\delta m^{\dagger}+\delta m)$ can be neglected safely because of
$|\alpha|\gg 1$ and
$\lambda_{ab}^{\prime},\lambda_{am}^{\prime}\ll\lambda_{ab},\lambda_{am}$.
Then the system Hamiltonian $H_{2}$ can be further simplified as

$\displaystyle H_{\rm eff}=$
$\displaystyle[\omega_{a}-\omega_{d}+\Delta_{F}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]\delta
a^{\dagger}\delta a+\omega_{b}\delta b^{\dagger}\delta b+\omega_{m}\delta
m^{\dagger}\delta m+(\omega_{\sigma}-\omega_{d})\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(\delta a^{\dagger}+\delta a)(\delta
b^{\dagger}+\delta b)+\lambda_{am}(\delta a^{\dagger}+\delta a)(\delta
m^{\dagger}+\delta m)+\lambda_{a\sigma}(\delta a\sigma^{\dagger}+\delta
a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma).$ (S10)

Assuming $\delta a\to a$, $\delta b\to b$, and $\delta m\to m$, the above
equation becomes

$\displaystyle H=$
$\displaystyle[\omega_{a}-\omega_{d}+\Delta_{F}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m+(\omega_{\sigma}-\omega_{d})\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(a^{\dagger}+a)(b^{\dagger}+b)+\lambda_{am}(a^{\dagger}+a)(m^{\dagger}+m)+\lambda_{a\sigma}(a\sigma^{\dagger}+a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma)$
$\displaystyle=$
$\displaystyle[\omega_{a}-\omega_{d}+\Delta_{F}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m$
$\displaystyle+[\omega_{\sigma}-\omega_{a}+\omega_{a}-\omega_{d}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)-\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)-\lambda_{am}^{\prime}(\mu^{\ast}+\mu)]\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(a^{\dagger}+a)(b^{\dagger}+b)+\lambda_{am}(a^{\dagger}+a)(m^{\dagger}+m)+\lambda_{a\sigma}(a\sigma^{\dagger}+a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma)$
$\displaystyle=$
$\displaystyle(\Delta_{ad}+\Delta_{F})a^{\dagger}a+\omega_{b}b^{\dagger}b+\omega_{m}m^{\dagger}m+(\Delta_{ad}+\Delta_{\sigma
a})\sigma^{\dagger}\sigma$
$\displaystyle+\lambda_{ab}(a^{\dagger}+a)(b^{\dagger}+b)+\lambda_{am}(a^{\dagger}+a)(m^{\dagger}+m)+\lambda_{a\sigma}(a\sigma^{\dagger}+a^{\dagger}\sigma)+\xi(\sigma^{\dagger}+\sigma),$
(S11)

where
$\Delta_{ad}=\omega_{a}-\omega_{d}+\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)+\lambda_{am}^{\prime}(\mu^{\ast}+\mu)$
and $\Delta_{\sigma
a}=\omega_{\sigma}-\omega_{a}-\lambda_{ab}^{\prime}(\beta^{\ast}+\beta)-\lambda_{am}^{\prime}(\mu^{\ast}+\mu)$.

##  II II. Derivation of multiquanta resonances conditions

In the large pumping regime, the strong driving laser can dress the atom, and
the system forms new eigenstates that are a quantum superposition of the bare
states $\\{|e\rangle,|g\rangle\\}$. The subsystem Hamiltonian for the strongly
driven atom is

$\displaystyle H_{\sigma}=$ $\displaystyle(\Delta_{\sigma
a}+\Delta_{ad})\sigma^{\dagger}\sigma+\xi(\sigma^{\dagger}+\sigma),$ (S12)

with the eigenvalues

$\displaystyle E_{|\pm\rangle}=\frac{\Delta_{\sigma
a}+\Delta_{ad}\pm\sqrt{(\Delta_{\sigma a}+\Delta_{ad})^{2}+4\xi^{2}}}{2},$
(S13)

and corresponding eigenstates

$\displaystyle|+\rangle=c_{+}|g\rangle+c_{-}|e\rangle,~{}~{}~{}|-\rangle=c_{-}|g\rangle-
c_{+}|e\rangle,$ (S14)

where

$\displaystyle c_{\pm}=\sqrt{\frac{2\xi^{2}}{(\Delta_{\sigma
a}+\Delta_{ad})^{2}+4\xi^{2}\pm(\Delta_{\sigma
a}+\Delta_{ad})\sqrt{(\Delta_{\sigma a}+\Delta_{ad})^{2}+4\xi^{2}}}}$ (S15)

and $c_{+}^{2}+c_{-}^{2}=1$. Together with the photon, phonon, and magnon
modes, and ignoring the influences of the JC interaction and the linear
interactions in the optomagnetic system on the energy structure, the
eigenvalues of the system Hamiltonian become

$\displaystyle
E_{|n_{a}n_{b}n_{m}\pm\rangle}=n_{a}(\Delta_{ad}+\Delta_{F})+n_{b}\omega_{b}+n_{m}\omega_{m}+\frac{\Delta_{\sigma
a}+\Delta_{ad}\pm\sqrt{(\Delta_{\sigma a}+\Delta_{ad})^{2}+4\xi^{2}}}{2}.$
(S16)

In the following, we discuss the resonance conditions considered in the main
text. For convenience, we always fix the resonator spinning along the
counterclockwise direction. The Fizeau shifts $\Delta_{F}>0$ and
$\Delta_{F}<0$ correspond to the situations of driving the resonator from the
left and right, respectively.

First, when the spinning resonator is driven from the left, and the total
energy of a single clockwise photon and a single magnon matches the transition
between the states $|+\rangle$ and $|-\rangle$, i.e.,
$E_{|000+\rangle}=E_{|101-\rangle}$, we have

$\displaystyle(\Delta_{ad}+|\Delta_{F}|)+\omega_{m}-\sqrt{(\Delta_{\sigma
a}+\Delta_{ad})^{2}+4\xi^{2}}=0,$ (S17)

and

$\displaystyle\Delta_{ad}=\frac{\Delta_{\sigma
a}^{2}+4\xi^{2}-(\omega_{m}+|\Delta_{F}|)^{2}}{2(\omega_{m}+|\Delta_{F}|-\Delta_{\sigma
a})}.$ (S18)

In other words, these two states $|000+\rangle$ and $|101-\rangle$ are
degenerate when the atom is driven at the photon-magnon resonance. When the
spinning resonator is driven from the right, and the total energy of a single
counterclockwise photon and a single magnon matches with the transition
between the states $|+\rangle$ and $|-\rangle$, the photon-magnon resonance
condition becomes

$\displaystyle\Delta_{ad}=\frac{\Delta_{\sigma
a}^{2}+4\xi^{2}-(\omega_{m}-|\Delta_{F}|)^{2}}{2(\omega_{m}-|\Delta_{F}|-\Delta_{\sigma
a})}.$ (S19)

The above results correspond to the excitation of the photon-magnon state
$|101-\rangle$. The parameter conditions are in agreement with the results
shown in the blue area of Fig.1(c) and the orange area of Fig.1(c) in the main
text, respectively.

Second, when the spinning resonator is driven from the left, and the total
energy of a single clockwise photon and a single phonon matches the transition
between the states $|+\rangle$ and $|-\rangle$, i.e.,
$E_{|000+\rangle}=E_{|110-\rangle}$ (the states $|000+\rangle$ and
$|110-\rangle$ are degenerate), we have

$\displaystyle(\Delta_{ad}+|\Delta_{F}|)+\omega_{b}-\sqrt{(\Delta_{\sigma
a}+\Delta_{ad})^{2}+4\xi^{2}}=0,$ (S20)

and

$\displaystyle\Delta_{ad}=\frac{\Delta_{\sigma
a}^{2}+4\xi^{2}-(\omega_{b}+|\Delta_{F}|)^{2}}{2(\omega_{b}+|\Delta_{F}|-\Delta_{\sigma
a})}.$ (S21)

When the spinning resonator is driven from the right, and the total energy of
a single counterclockwise photon and a single phonon matches with the
transition between the states $|+\rangle$ and $|-\rangle$, the photon-phonon
resonance condition becomes

$\displaystyle\Delta_{ad}=\frac{\Delta_{\sigma
a}^{2}+4\xi^{2}-(\omega_{b}-|\Delta_{F}|)^{2}}{2(\omega_{b}-|\Delta_{F}|-\Delta_{\sigma
a})}.$ (S22)

The above results correspond to the excitation of the photon-phonon state
$|110-\rangle$. The parameter conditions are in agreement with those displayed
in the orange area of Fig.1(c) and the pink area of Fig.1(d) in the main text,
respectively.

Lastly, because of the system parameters chosen in the main text, the results
from the right sides of Eq.(S19) and Eq.(S21) are the same, as shown in the
orange areas of Figs.1(c) and 1(d). This demonstrates that, in this parameter
regime, the photon-phonon state is resonantly excited when the resonator is
driven from the left, while the photon-magnon state is also resonantly excited
when the resonator is driven from the right.

##  III III. The optimal local operators in quantum Fisher information

Multipartite entanglement can be characterized based on quantum Fisher
information (QFI) and covariance. For an arbitrary separable quantum states of
$N$ particles, the QFI and covariance must satisfy the inequality
Gessner2016PS ; Tian2002XS

$\displaystyle F_{Q}[\rho_{\rm sep},\sum_{j=1}^{N}A_{j}]\,\leq
4\sum_{j=1}^{N}{\rm Var}(A_{j})_{\rho_{\rm sep}}\equiv B_{n},$ (S23)

where $A_{j}$ is a local observable for the $j$th particle, and ${\rm
Var}(A_{j})_{\rho}=\langle A_{j}^{2}\rangle_{\rho}-\langle
A_{j}\rangle_{\rho}^{2}$. The violation of Eq. (S23) for any choice of $A_{j}$
is a sufficient criterion for entanglement. However, certain choices of
operators $A_{j}$ may be better suited than others to detect the entanglement
of a given state $\rho$. The optimal local operator can be given by a
combination of accessible operators $A_{j}=\sum_{m}c_{j}^{(m)}A_{j}^{(m)}={\bf
c}_{j}\cdot{\bf A}_{j}$, where ${\bf
A}_{j}=(A_{j}^{(1)},A_{j}^{(2)},\dots)^{T}$ and ${\bf
c}_{j}=(c_{j}^{(1)},c_{j}^{(2)},\dots)$ are the set of operators for $j$th
particle and corresponding coefficients. The full operator is given by $A({\bf
c})=\sum_{j=1}^{N}{\bf c}_{j}\cdot{\bf A}_{j}={\bf c}\cdot{\bf A}$, with the
combined vectors ${\bf c}=({\bf c}_{1},{\bf c}_{2},\dots,{\bf c}_{N})$ and the
operators ${\bf A}=[{\bf A}_{1},\dots,{\bf A}_{N}]$. To detect the
multipartite entanglement in our model, we choose the family of accessible
operators ${\bf A}_{j}=(\sigma_{j}^{x},\sigma_{j}^{y},\sigma_{j}^{z})^{T}$ for
the atom and ${\bf
A}_{j}=(x_{j},p_{j},x_{j}^{2},p_{j}^{2},(x_{j}p_{j}+p_{j}x_{j})/2)^{T}$ for
the mode $o$ ($o=\\{a,b,m\\}$), where $x=o+o^{\dagger}$ and
$p=-i(o-o^{\dagger})$ Tian2002XS . Then the optimal local operators can be
given by finding the optimal combined vectors ${\bf c}$. According to Eq.
(S23), $F_{Q}[\rho,A({\bf c})]-4\sum_{j=1}^{N}{\rm Var}({\bf c}_{j}\cdot{\bf
A}_{j})$ must be nonpositive for arbitrary choices of ${\bf c}$ whenever
$\rho$ is separable. To find the optimal ${\bf c}$, we expand the QFI and
covariance in their matrix forms. The QFI can be written as $F_{Q}[\rho,A({\bf
c})]={\bf c}Q_{\rho}^{\mathcal{A}}{\bf c}^{T}$ Hyllus2010GS , where
$Q_{\rho}^{\mathcal{A}}$ is the QFI matrix with elements

$\displaystyle(Q_{\rho}^{\mathcal{A}})_{jj^{\prime}}^{mm^{\prime}}=2\sum_{k,k^{\prime}}\frac{(p_{k}-p_{k^{\prime}})^{2}}{p_{k}+p_{k^{\prime}}}\langle\psi_{k}|A_{j}^{(m)}|\psi_{k^{\prime}}\rangle\langle\psi_{k^{\prime}}|A_{j^{\prime}}^{(m^{\prime})}|\psi_{k}\rangle$
(S24)

on the basis of spectral decomposition
$\rho=\sum_{k}p_{k}|\psi_{k}\rangle\langle\psi_{k}|$ Braunstein1994Caves . The
covariance can be written as $\sum_{j=1}^{N}{\rm Var}({\bf c}_{j}\cdot{\bf
A}_{j})={\bf c}\Gamma_{\rho}^{\mathcal{A}}{\bf c}^{T}$, where the covariance
matrix $\Gamma_{\rho}^{\mathcal{A}}$ has elements
$(\Gamma_{\rho}^{\mathcal{A}})_{jj^{\prime}}^{mm^{\prime}}={\rm
Cov}(A_{j}^{(m)},A_{j^{\prime}}^{(m^{\prime})})_{\rho}$ and
$(\Gamma_{\rho}^{\mathcal{A}})_{jj}^{mm^{\prime}}=0$. Combining the quantum
Fisher matrix $Q_{\rho}^{\mathcal{A}}$ with the covariance matrix
$\Gamma_{\rho}^{\mathcal{A}}$, the separability criterion reads

$\displaystyle{\bf c}(Q_{\rho}^{\mathcal{A}}-\Gamma_{\rho}^{\mathcal{A}}){\bf
c}^{T}<0.$ (S25)

The optimal ${\bf c}$ can be given by finding the optimal eigenvalue and
corresponding eigenstate of the matrix
$Q_{\rho}^{\mathcal{A}}-\Gamma_{\rho}^{\mathcal{A}}$. Further, we obtain the
optimal local operators $A_{j}$ for the entanglement witness.

A photon-phonon separable state can be described as
$\rho_{ab}=\sum_{k}p_{k}\rho_{a}^{k}\otimes\rho_{b}^{k}$ ($\rho_{j}$ is the
reduced density operator). The right-hand side of of Eq. (S23) characterizes
the bounds of a given state and local operator $A_{j}$. In our case, we have

$\displaystyle B_{1}(\rho_{ab})=4[{\rm Var}(A_{a}){\rho_{a}}+{\rm
Var}(A_{b})_{\rho_{b}}],$ $\displaystyle B_{2}(\rho_{ab})=4[{\rm
Var}(A_{a}+A_{b})_{\rho_{ab}}].$ (S26)

If the quantity $W^{ab}_{1}=F_{Q}[\rho_{ab},A({\bf c})]-4\sum_{j=a,b}{\rm
Var}(A_{j})_{\rho_{j}}=F_{Q}[\rho_{ab},A({\bf c})]-B_{1}(\rho_{ab})>0$, the
photon-phonon state is not separable, i.e., there is entanglement between the
photon mode and phonon mode. The inequality $F_{Q}[\rho_{ab},A({\bf c})]\leq
B_{2}(\rho_{ab})$ is a bound valid for all physical states $\rho_{ab}$.

A photon-phonon-atom fully separable state can be described as
$\rho_{ab\sigma}=\sum_{k^{\prime}}p_{k^{\prime}}\rho_{a}^{k^{\prime}}\otimes\rho_{b}^{k^{\prime}}\otimes\rho_{\sigma}^{k^{\prime}}$.
In this case, the bounds become

$\displaystyle B_{1}(\rho_{ab\sigma})=4[{\rm Var}(A_{a}){\rho_{a}}+{\rm
Var}(A_{b})_{\rho_{b}}+{\rm Var}(A_{\sigma})_{\rho_{\sigma}}],$ $\displaystyle
B_{2}(\rho_{ab\sigma})=4\,{\rm max}\\{{\rm Var}(A_{a}+A_{b})_{\rho_{ab}}+{\rm
Var}(A_{\sigma})_{\rho_{\sigma}},{\rm
Var}(A_{a}+A_{\sigma})_{\rho_{a\sigma}}+{\rm Var}(A_{b})_{\rho_{b}},{\rm
Var}(A_{b}+A_{\sigma})_{\rho_{b\sigma}}+{\rm Var}(A_{a})_{\rho_{a}}\\},$
$\displaystyle B_{3}(\rho_{ab\sigma})=4[{\rm
Var}(A_{a}+A_{b}+A_{\sigma})\rho_{ab\sigma}].$ (S27)

If $W^{ab\sigma}_{1}=F_{Q}[\rho_{ab\sigma},A({\bf
c})]-4\sum_{j=a,b,\sigma}{\rm
Var}(A_{j})_{\rho_{j}}=F_{Q}[\rho_{ab\sigma},A({\bf
c})]-B_{1}(\rho_{ab\sigma})>0$, there is entanglement between the atom, photon
mode, and phonon mode. Furthermore, if
$W^{ab\sigma}_{2}=F_{Q}[\rho_{ab\sigma},A({\bf c})]-B_{2}(\rho_{ab\sigma})>0$,
the state is fully inseparable, i.e., there is a fully inseparable photon-
magnon-atom tripartite entanglement Tian2002XS . The inequality
$F_{Q}[\rho_{ab\sigma},A({\bf c})]\leq B_{3}(\rho_{ab\sigma})$ is a bound
valid for all physical states $\rho_{ab\sigma}$. Here it is only necessary to
replace mode $b$ with mode $m$ to characterize the entanglement between the
magnon mode and others.

##  IV IV. Discussion of Experimental feasibility

Regarding experimental implementations, our proposal could be implemented in a
hybrid setup Aspelmeyer2014KM ; Chai2022SZ , where a coherently driven two-
level atom is coupled to a spinning a spinning yttrium iron garnet (YIG)
microsphere fixed on a rotating platform Maayani2018DK . The microsphere
supports a mechanical breathing mode and two countercirculating optical
whispering gallery modes (WGMs), with the WGM being strongly driven by the
input light Aspelmeyer2014KM . Access to the input and output light fields of
the microresonator is provided by a tapered optical fiber coupler interfaced
with the resonator. The WGMs are modulated by the mechanical breathing mode
when the input light pumps at the WGM. Moreover, because of the spin-orbit
coupling of light, the WGMs have a spin along the $z$ direction. By applying a
magnetic field parallel to the equator to the microsphere and exciting the
magnon mode with an antenna coupling to a microwave field, the WGMs could also
be modulated by the dynamic magnetic field via the Faraday effect Chai2022SZ .
Here, scaling down the microsphere size to reduce the mode volumes of both
magnon and optical fields, or engineering the microresonator structure such as
microrings to increase the overlap between optical and magnon fields, can
enhance the coupling between the optical mode and magnon mode. The effective
linear coupling strength between the optical and magnon (mechanical) mode that
is proportional to the classical cavity amplitude can also be adjusting by the
strong driving on the optical mode.

Moreover, the selection of microresonator is not limited to a microsphere and
can also be replaced with a toroidal or bottle microresonator Zhang2016ZZ .
The two-level atom of choice can be a cesium atom (e.g., the $6S_{1/2},F=4\to
6P_{3/2},F^{\prime}=5$ transition in cesium atom) Aoki2006DW ; Dayan2008PA ;
Alton2011SA  or rubidium atom (e.g., the $5S_{1/2},F=3\to
5P_{3/2},F^{\prime}=4$ transition in rubidium atom) Junge2013OSV ; OShea2013JV
; Shomroni2014RL ; Will2021MR . The atom can be coupled with the
microresonator by trapping the atom in the vicinity of the microresonator.
However, trapping a single atom in the vicinity of a microresonator is indeed
a daunting task due to several technical and physical complexities. First,
precisely positioning and maintaining a single atom near a microresonator
requires extremely high stability and accuracy in spatial control, since the
optical fields around microresonators can vary significantly over nanometer
scales. The use of optical tweezers or other trapping methods to hold a single
atom in place involves complex manipulations of light fields. The intensity,
phase, and polarization of these light fields must be precisely controlled to
create stable trapping potentials without causing excessive heating or
ionization of the atom. Second, efficient coupling of the atom to the
microresonator’s evanescent field requires the atom to be placed within a
fraction of the wavelength of the light used, further complicating the spatial
control challenge. Lastly, atoms must typically be cooled to very low
temperatures to reduce thermal motion, which otherwise could lead to loss of
control over the atom’s position relative to the microsphere. Any
perturbations from the environment (like mechanical vibrations,
electromagnetic noise, etc.) can displace the atom from its optimal position,
making sustained interaction with the microresonator difficult.

In recent years, several experiments have successfully achieved coupling
between individual cesium atoms (or rubidium atoms) and various types of
microresonators, including microspheres, microtoroidal resonators, and bottle
microresonators OShea2013JV ; Shomroni2014RL ; Will2021MR ; Aoki2006DW ;
Alton2011SA ; Dayan2008PA ; Junge2013OSV . Initially, these experiments
utilize a magneto-optical trap to trap and cool atomic clouds in a ‘magneto-
optical trap chamber’. Subsequently, the trapping and cooling beams are
switched off, allowing the atoms to fall onto the microresonator, thus
facilitating the coupling of the individual atoms with the evanescent field of
the microresonator’s WGMs. Furthermore, the addition of an optical dipole
trap—created by the interference of the incoming laser beam with its
reflection from the microresonator surface—is more likely to trap the falling
atoms, thereby increasing the interaction time between the atom and the
microresonator Will2021MR ; Thompson2013TdL . Based on the experimental
groundwork, our model can also employ the techniques described in these
studies to achieve the trapping of a single atom and its coupling with a
microresonator. In our approach, the atom is initially cooled and trapped
within a chamber using a magneto-optical trap, then transported to the
vicinity of the microresonator. Here, the microresonator’s WGM evanescent
field, which decays exponentially outside the resonator’s surface, interacts
with the atom OShea2013JV ; Shomroni2014RL ; Will2021MR ; Aoki2006DW ;
Alton2011SA ; Dayan2008PA ; Junge2013OSV ; Thompson2013TdL . An optical dipole
trap may be introduced to enhance the atom’s trapping and prolong its
interaction time with the microresonator Will2021MR ; Thompson2013TdL . The
decay rate of the cesium or rubidium atom used in trapping near a
microresonator has linewidth of tens of ${\rm MHz}$. The coupling strength
between the selected atom and the microresonator depends on the properties of
the microresonator and the distance between the atom and the microresonator’s
surface. Placing the atom closer to the surface of the microresonator
Alton2011SA ; Will2021MR , or reducing the size of the microresonator to
reduce the mode volume Buck2003Kimble ; Spillane2005KV , can enhance this
coupling strength. H. J. Kimble and his collaborators, using finite element
models, have given the atom-microresonator coupling rate can exceed
$2\pi\times 700\,{\rm MHz}$ in microresonator structure given suitable
experimental parameters Buck2003Kimble ; Spillane2005KV . In addition to these
microresonator-specific techniques, the atom-microresonator coupling can also
be enhanced by other physical methods. For instance, both theoretical and
experimental evidence suggests that a quantum squeezing method can enhance
interactions between quantum systems, even without precise knowledge of the
system parameters Lu2015WJ ; Qin2018ML ; Burd2021SK ; Burd2024KS ; Qin2024Nori
. Additionally, the counting of phonons (magnons) and other types of
statistical processing of photons and phonons (magnons) can be measured by
applying an auxiliary system to convert the mechanical (magnon) signals
Meenehan2015MM ; Bender2019KB . The multipartite entanglement can be
characterized by a class of nonlinear squeezing parameters Tian2002XS ;
Gessner2019SP . For this special design, we theoretically predicts that the
nonreciprocal entangled photon-phonon pairs and photon-magnon pairs with
corresponding antibunching $g^{(2)}_{2,ab}=0.29$ and $g^{(2)}_{2,am}=0.49$
could be achieved with the parameters $\omega_{b}=2\pi\times 2\,{\rm GHz}$,
$\omega_{m}=2\pi\times 2.1\,{\rm GHz}$, $\Delta_{\sigma a}=-2\pi\times
6.2\,{\rm GHz}$, $|\Delta_{F}|=2\pi\times 50\,{\rm MHz}$,
$\lambda_{a\sigma}=2\pi\times 600\,{\rm MHz}$,
$\lambda_{ab}=\lambda_{am}=2\pi\times 44\,{\rm MHz}$, $\xi=2\pi\times
1.6\,{\rm GHz}$, $\gamma=2\pi\times 2\,{\rm MHz}$, and $\kappa=2\pi\times
16\,{\rm MHz}$ Aspelmeyer2014KM ; Chai2022SZ ; Zhang2016ZZ ; OShea2013JV ;
Shomroni2014RL ; Will2021MR ; Aoki2006DW ; Alton2011SA ; Dayan2008PA ;
Junge2013OSV ; Buck2003Kimble ; Spillane2005KV ; Thompson2013TdL .

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