Nonreciprocal macroscopic tripartite entanglement in atom-optomagnomechanical system

We investigate how to generate the nonreciprocal macroscopic tripartite entanglement among the atomic ensemble, ferrimagnetic magnon and mechanical oscillator in a hybrid atom-optomagnomechanical system, where an ensemble of two-level atoms and a yttrium iron garnet micro-bridge supporting the magnon and mechanical modes are placed in a spinning optical resonator driven by a laser ﬁeld. The phonon being the quantum of the mechanical mode interacts with the magnon and the optical photon via magnetostriction and radiation pressure, respectively, and meanwhile the photon couples to the atomic ensemble. The results show that not only all bipartite entanglements but also the genuine tripartite entanglement among the atomic ensemble, magnon and phonon could be generated at the steady state. Moreover, the nonreciprocity of atom-magnon-phonon entanglement can be obtained with the aid of the optical Sagnac eﬀect by spinning the resonator, in which the entanglement is present in a chosen driving direction but disappears in the other direction. The nonreciprocal macroscopic tripartite entanglement is robust against temperature and could be ﬂexibly controlled by choosing the system parameters. Our work enriches the study of macroscopic multipartite quantum states, which may have potential applications in the development of quantum information storage and the construction of multi-node chiral


Introduction
In recent years, a variety of quantum systems including atomic ensembles [1], superconducting qubits [2], trapped ions [3], semiconductor quantum dots [4], and spins in magnetic materials or diamonds [5,6] have become promising candidates for studying the light-matter interactions, which are significantly applied in quantum information processing and quantum computing.Especially, magnons, quanta of collective spin wave excitation in yttrium iron garnet (YIG) as the representative magnetic material, provide a wide platform to build hybrid quantum networks and realize multi-functional quantum tasks via combining various quantum systems with different advantages.The strong and even super-strong couplings between magnons and microwave cavity photons have been reported due to the high spin density of YIG [7,8], and magnons can also interact with optical photons, superconducting qubits, and phonons [9,10].It is noted that the cavity optomagnomechanical system is proposed by means of the magnetostrictive and radiationpressure interactions mediated by the mechanical motion [11].In such a hybrid system, some fascinating quantum phenomena have been demonstrated, such as microwaveoptical conversion and entanglement [11,12], atom-magnon entanglement [13], Fanotype optical response and four-wave mixing process [14], and phonon-magnon-photon entanglement [15].
Entanglement is an essential quantum effect in hybrid quantum systems, particularly the multipartite continuous variable entanglement plays a key role in multi-party quantum communication and network [16,17].And the nonlocal correlation of macroscopic objects can be widely used in large-scale fundamental testing of quantum mechanics, quantum to classical transitions, and building more powerful sensing, communication, processing, and storage devices [18][19][20].In cavity optomechanics, significant progress has been made in the study of macroscopic entanglement via the optomechanical radiation pressure [21], where the entanglements of two distant macroscopic mechanical resonators and between the massive mechanical oscillator and the electromagnetic field or collective atomic spin oscillator have been reported [22][23][24][25][26]. Apart from the optomechanical system, cavity magnonics has gradually become an advantageous platform for studying macroscopic quantum entanglement [9,10].For example, two long-distance ferrimagnetic spheres can be entangled by exploiting the magnetostrictive interaction [27], squeezed-reservoir engineering [28], nonlinear Kerr effect of magnons [29], etc. Tan et al. [30] proposed a scheme to entangle a macroscopic mechanical motion and a YIG sphere mediated by the electromagnetic cavity far beyond the sideband-resolved regime.And Zhong et al. [31] recently presented that the macroscopic tripartite entanglement and steering of three YIG spheres can be achieved by a squeezed reservoir.Moreover, the multipartite quantum correlations among the magnons, photons, phonons, and atoms have been studied [32][33][34][35][36][37].
Nonreciprocity representing the unidirectional invisibility of light in propagation has been an indispensable tool to construct the optical functional components and complex information processing network [38].Recently, the Fizeau light-dragging effect induced by spinning the optical resonator leads to a positive or negative frequency shift of optical frequency, which has become an experimentally feasible approach to explore various quantum nonreciprocal phenomena including nonreciprocal photon blockade [39,40], quantum entanglement and steering [31,37,41,42], phonon laser [43], and sideband responses [44].Interestingly, Wang et al. [45] theoretically show the nonreciprocal magnon blockade could be generated based on the microwave cavity-mediated qubit-magnon coherent and dissipative couplings at the quantum limit level.And the magnon Kerr effect also can be used to create the nonreciprocal quantum entanglement via tuning the bias magnetic field direction [46].Exploring the nonreciprocal multipartite entanglement of macroscopic objects is of great significance to the storage and transmission of information in complex quantum chiral network, but the study is still few and needs to be further supplemented.
Inspired by the above, we propose a scheme to generate the nonreciprocal macroscopic tripartite entanglement among the atomic ensemble, ferrimagnetic magnon and mechan-ical oscillator in a magnon-based hybrid system.In this system, the optomechanical and magnomechanical couplings are mediated by the mechanical oscillator via the magnetostrictive and radiation pressure interactions, and the optical photon in a spinning resonator couples to an ensemble of two-level atoms in the low-excitation limit.By driving the magnon with a sufficiently strong red-detuned microwave field, the magnon-phonon entanglement is initially prepared, which is partially distributed to the atom-magnon and atom-phonon subsystems when the optical photon and the atom are respectively scattered onto the two sidebands of the phonon.Specifically, the genuine tripartite entanglement among the atom, magnon, and phonon modes could be achieved, and the stationary entanglement is robust against thermal fluctuations.Moreover, the nonreciprocal bipartite and tripartite entanglements are induced by the Fizeau light-dragging effect by spinning the resonator, and the ideal nonreciprocal tripartite entanglement could be obtained by properly choosing the system parameters.
In Sect.2, we introduce a hybrid atom-optomagnomechanical system composed of a YIG micro-bridge supporting the magnon and phonon, an spinning optical resonator, and an atomic ensemble, provide its Hamiltonian and the corresponding quantum Langevin equations (QLEs), and study the linearized dynamics of this system.In Sect.3, we show the generation of macroscopic atom-magnon-phonon entanglement and discuss the nonreciprocity of multipartite entanglement under the influence of various system parameters.In Sect.4, a summary is given.

Model and equation
We consider a hybrid atom-optomagnomechanical system consisting of an atomic ensemble of two-level atoms, a magnon, a phonon, and an optical photon, as shown in Fig. 1(a) and 1(b).A YIG micro-bridge attached to a small high-reflectivity mirror pad on the surface and an atomic ensemble are placed in a spinning optical resonator driven by a laser field.The YIG crystal is driven by a microwave field and simultaneously in a uniform bias magnetic field along the z direction, where the magnetostriction-induced phononmode interacts with the magnon.The optical photon couples to the phonon and the atomic ensemble via the radiation-pressure [12,13] and collective Tavis-Cummings-type interactions [47][48][49], respectively.The Hamiltonian of the system is given by the sum of the free term and the interaction term where m(m † ) and c(c † ) are annihilation (creation) operators of the magnon and the optical photon, respectively, satisfying [j, j † ] = 1(j = m, c).S z,± = N a i=1 σ i z,± are collective spin operators of an ensemble of N a two-level atoms with Pauli matrices σ z,± and satisfy the commutation relations [S + , S -] = S z and [S z , S ± ] = ±2S ± .q and p are the dimensionless position and momentum of the phonon ([q, p] = i ).ω m , ω c , ω a ,and ω b are resonant frequencies of the magnon, optical photon, atomic ensemble and phonon modes, respectively.The the atom is in the low-excitation limit.In this case, all atoms are initially prepared in the ground state (S z S z -N a ), and the dynamics of the atomic polarization can be described in terms of bosonic operators.The atomic annihilation operator could be defined On the other hand, due to the optical Sagnac effect, the counterpropagating light would travel different effective optical path lengths by spinning the resonator.In this case, the optical photon experiences an opposite Fizeau frequency shift s (ω c → ω c + s ), degenerating into the clockwise (CW) and counterclockwise (CCW) modes [39,41].In a rotating frame with respect to ω L (a † a + c † c) + ω 0 m † m, the bosonized Hamiltonian can be written as where a = ω aω L , m = ω mω 0 , c = ω cω L + s , and g N = g a √ N describes the effective atom-photon coupling strength.If the resonator is rotated in CW direction (see Fig. 1(a))) with an angular velocity , the Sagnac-Fizeau shifts are [51] where n(R) is the refractive index (radius) of the resonator, c(λ) is the speed (wavelength) of light in vacuum, and the dispersion term dn/dλ is relatively small and can be ignored.
s > 0 ( s < 0) denotes that the optical CCW (CW) mode is driven from the left-hand (right-hand) side.
In the standard fluctuation-dissipation theorem, the QLEs of the system can be obtained where γ a , κ m , and γ b are dissipation rates of the atomic excited level, the magnon, and the phonon, respectively.a in , c in , and m in are the corresponding input noise operators with zero mean, which satisfy the following correlation functions: j(t)j † (t ) = [N j (ω j ) + 1]δ(tt ), and j(t) † j(t ) = N j (ω j )δ(tt )(j = a, c, m).ξ (t) is the Hermitian Brownian noise operator, which is essentially non-Markovian.But for a large mechanical quality factor , the Markovian approximation can be taken, so it can be considered as a The equilibrium mean thermal number of each mode is defined as where κ B and T are the Boltzmann constant and the environmental temperature, respectively.
By continuously driving the magnon and the optical photon with strong pumping fields, the magnomechanical and optomechanical couplings can be enhanced, which leads to The linearized QLEs describing the quantum fluctuations of the system can be expressed as where T and the drift matrix A is given by detuning including the frequency shift due to the radiation pressure (magnetostriction), where the mechanical displacement is q are the corresponding effective coupling strengths, where The average value of the atomic mode is a = -ig N c /(γ a + i a ).Under the optimal conditions , the average values can be simplified as m -iE 1 / ˜ m and c iE 2 a /(g 2 N -˜ c a ), which are the pure imaginary numbers.Correspondingly, the effective coupling strengths G m and G c are approximately real numbers.If the real parts of all eigenvalues of A are negative, the system is stable and reaches a steady state when t → ∞.The steady-state conditions of system can be derived by the Routh-Hurwitz criterion [53] (i.e., |A -λI| = 0).In this work, the characteristic equation is in the form of eighth-order polynomial for the 8 × 8 drift matrix A, which is difficult to solve the stability conditions analytically.Here we numerically calculate and guarantee that all parameters satisfy the stability conditions.Due to the linearized dynamics and Gaussian input noise, the system is a continuous variable (CV) four-mode Gaussian state, which can be characterized by a 8 × 8 covariance matrix (CM) V defined as ).The steady-state CM V can be obtained by solving the Lyapunov equation [54] AV where ] is the diffusion matrix defined as n i (t)n j (t ) + n j (t )n i (t) /2 = D ij δ(tt ).The minimal residual contangle R min τ [55] is taken to quantify the genuine tripartite entanglement among the atomic, the magnon and the phonon, which is given by where ) is the residual contangle, which is a CV analogue of tangle for discrete-variable tripartite entanglement, C u|v is defined

Results and discussion
We numerically calculate the bipartite and tripartite entanglements among three of modes atom ensemble, magnon and phonon versus detunings ˜ m and a for the case = 0 at the steady state, which are shown in Fig. 2.Here the static system has reciprocity regardless of the driving direction.The experimentally feasible parameters are taken as [12,13,32] ω m /2π = 10 GHz, ω b /2π = 25 MHz, λ c = 1550 nm, γ a /2π = κ m /2π = 1 MHz, γ b /2π = 100 Hz, κ c /2π = 1.5 MHz, g N /2π = 16 MHz, G m /2π = 8 MHz, G c /2π = 3 MHz, g c /2π = 1 kHz, T = 10 mK, and the atom ensemble is tuned to off-resonant coupling with the optical photon ( a = -˜ c ).The effective magnomechanical coupling G m /2π = 8 MHz corresponds to a microwave drive power P 0 1.44 mW for a 8 × 3 × 1 μm 3 YIG micro-bridge with g m /2π = 25 Hz [14].From Fig. 2(a), 2(b) and 2(c), it can be found that all bipartite entanglements among the atom, magnon and phonon modes can coexist in the parameters regime around ˜ m ω b and a -ω b .Interestingly, the genuine tripartite entanglement of the three modes could be achieved at steady state, which is demonstrated by the minimum residual contangle R min τ greater than zero in Fig. 2 (d).The similar mechanism has been demonstrated to achieve the genuine atom-light-mirror entanglement in cavity op-tomechanics and magnon-photon-phonon entanglement in cavity magnomechanics, respectively [32,50].Different from that, our work combines the respective advantages of the two systems, and further prepares the macroscopic tripartite entanglement among the atomic ensemble, massive ferrimagnetic magnon and mechanical oscillator based on the cooperative effect of magnetostrictive and radiation-pressure interactions.
Figure 3 shows the influence of the effective optical detuning, thermal fluctuations and mechanical damping on the bipartite and tripartite entanglements among the atom, magnon and phonon modes in the static system ( = 0).In Fig. 3(a), three bipartite entanglements interplay with each other and the degree of entanglements could be controlled by detuning ˜ c .It can be seen that the variation trend of magnon-phonon (blue dashed) entanglement with detuning ˜ c is opposite to that of the atom-magnon (black dot-dashed) and atom-phonon (red solid) entanglements.This indicates that the magnon-phonon entanglement originally prepared in the magnomechanical system is used as a quantum resource, which is partially distributed to the atom-phonon and atom-magnon subsystems, thus yielding a genuinely macroscopic tripartite entangled state in the hybrid system.The tripartite entanglement in terms of the minimum residual contangle R min τ versus ˜ c is illustrated in Fig. 3(b).It can be found that the atom-magnon-phonon entanglement is sensitive to detuning ˜ c , and the maximum tripartite entanglement can reach 0.037 at ˜ c = 0.5ω b .Figure 3(c) shows the bipartite and tripartite entanglements under the effect of ambient temperature with optimized ˜ c .We can see that all entanglements are degraded by the increase of thermal fluctuations.The macroscopic tripartite entanglement can survive up to T 210 mK, and the atom-magnon and atom-phonon entanglement even exist for bath temperature over 350 mK.Due to the attachment of a mirror on the surface of YIG bridge, an additional mechanical damping of the phonon is inevitably introduced.The steady-state entanglements versus the mechanical damping γ b with optimal ˜ c are shown in Fig. 3(d).It is obvious that the entanglement remains almost constant with the increase of damping rate to γ b /2π ∼ 5 × 10 3 Hz, and the macroscopic tripartite entanglement is still present with a large damping rate γ b /2π ∼ 5 × 10 4 Hz.
It is essential to elucidate the mechanism for generating macroscopic tripartite entanglement in such quadripartite compound hybrid system with large frequency mismatch.For this purpose, we proceed via the linearized Hamiltonian of the static system ( = 0) for quantum fluctuations where When the magnon is driven by a sufficiently strong red-detuned microwave field ( ˜ m ω b ), the magnomechanical anti-Stokes scattering is activated and the strong magnomechanical coupling (G m /2π = 8 MHz) that breaks the weak coupling condition (G m ω b ) of the rotating-wave (RW) approximation.In this case, the magnomechanical RW term ∝ δm † δb + δmδb † and the optomechanical coupling δc † δb + δcδb † (the weak optomechanical coupling strength satisfies the RW approximation condition for the optical photon driven by a red-detuned laser field) are linear beam-splitter interactions, which can significantly cool the low-frequency mechanical mode (∼MHz) in the resolved sideband limit (ω b κ m , κ c ).The heat of phonon thermal excitation can be eventually dissipated to the environment through where ˜ c is optimized for each entanglement, respectively.The other parameters are the same as in Fig. 2 magnon and photon decay.The counter-RW term ∝ δm † δb † + δmδb corresponding to the parametric down-conversion interaction, which plays an important role in generating the magnon-phonon entanglement.When the optical and atom modes are scattered by the mechanical vibration onto the two sidebands ( a = -˜ c , see Fig. 1(c)), the magnonphonon entanglement is partially distributed to the atom-phonon and atom-magnon subsystems through the atom-photon and optomechanical state exchange interactions, realizing a genuinely tripartite entangled state in the system.Then we show that the classical nonreciprocity of mean optical photon number and quantum nonreciprocity of macroscopic entanglements can be observed by spinning the resonator.Here we adopt the angular velocity = 20 KHz and the resonator with n = 1.48 and R = 1.1 mm, which are feasible in the current experimental technology [41,51].Due to the optical Sagnac effect, the frequency of the optical photon driven from the left-hand (right-hand) side experiences a blue (red) shift s > 0 ( s < 0) (see Fig. 1(d)) in the spinning resonator.Accordingly, the effective optomechanical detuning ( ˜ c = ω cω L + sg c q ) are different for the CW and CCW modes.And the mean optical photon number exhibits nonreciprocity (| c( s > 0) | 2 = | c( s < 0) | 2 ) for the same driving field pump power in opposite directions, which is demonstrated in Fig. 4(a).Furthermore, the optomechanical coupling strength (G c = √ 2g c c ) and the average value /ω b are also affected, which leads to the irreversibility of entanglement when the optical photon is driven in opposite directions.The stationary nonreciprocal bipartite entanglements versus detuning ˜ c are respectively shown in Fig. 4(b), 4(c), and 4(d).Taking the atommagnon entanglement in Fig. 4(d) as an example, it can be seen that the atomic ensemble and magnon are strongly entangled at ˜ c = ω b when the optical photon is driven in CW ( s < 0), yet fully separated in the opposite driving direction ( s > 0).
The macroscopic tripartite entanglement among the atomic ensemble, ferrimagnetic magnon and mechanical oscillator also inherits the nonreciprocal property, which is depicted in Fig. 5(a).It is evident that the tripartite entanglement is the strongest at ˜ c = 0.5ω b without spinning (black dashed).When the resonator is selectively rotated,  rection s > 0 or s < 0 is illustrated in Fig. 5(c).Obviously, once the optical resonator is rotated, the nonreciprocity of the atom-magnon-phonon entanglement would appear.As the rotation speed increases, the tripartite entanglement is gradually weakened for s > 0, but the entanglement could be significantly enhanced for s < 0.
In order to further study the nonreciprocity of macroscopic atom-magnon-phonon entanglement, the chiral factor is chosen to quantify, which is defined as The chiral factor χ > 0 (χ ≤ 1) denotes that the nonreciprocity of tripartite entanglement is present.Figure 5(d) shows the chiral factor versus detuning ˜ c and rotation speed .It can be found that the tripartite is reciprocal (χ = 0) without the spinning of the resonator = 0.However, the nonreciprocal tripartite entanglement appears by spinning the resonator, and the nonreciprocity of entanglement can be flexibly implemented by the optical detuning and rotating speed of the resonator.The nearly ideal macroscopic nonreciprocal tripartite entanglement could be obtained (χ = 1) in the steady-state system.Finally, we discuss the validity of the system and the detection of nonreciprocal entanglement.The above results are valid only when atom and magnon are in low-lying excitations.When the single-atom excitation probability ) is much smaller than 1 and a † a N a , the atomic polarization can be described by boson operators.In this paper, we obtain a N a ∼ 10 7 with g a /2π = 2 × 10 3 Hz [50] and | a | ω b , which satisfies P a ∼ 0.006 1.For the magnon, m † m = 5.1× 10 10 5N = 5.1× 10 11 , which meets the low-excitation condition.The steady-state entanglement properties can be verified by experimentally measuring the corresponding CMs using homodyne techniques (see Ref. [22,54]).A weak microwave (optical) probe field is sent to resonantly couple to the magnon (atomic ensemble), thus the magnon (atomic polarization) quadratures can be read out by homodyning the microwave (optical) cavity output.The mechanical quadratures can be measured by adjusting the detuning and bandwidth of an adjacent optical cavity, both the position and momentum of the harmonic oscillator can be measured by homodyning the output of this second cavity.Once the quadratures of atomic ensemble, magnon and mechanical phonon are obtained, the nonreciprocal bipartite and tripartite entanglements could be calculated.

Conclusion
In conclusion, we present an approach to create and manipulate the nonreciprocal macroscopic atom-magnon-phonon entanglement in a hybrid atom-optomagnomechanical system composed of an ensemble of two-level atoms, a YIG micro-bridge, and a spinning optical resonator driven by a laser field.It is found that the atom, magnon, and phonon modes can be simultaneously entangled with each other by appropriately driving the magnon and photon, and the genuine tripartite entangled state of the three modes is generated with the feasible parameters.Furthermore, the atom-magnon-phonon entanglement depending on the optical detuning could be enhanced in a chosen direction but strongly suppressed in another direction by spinning the resonator.The stationary nonreciprocal macroscopic entanglement is robust against thermal fluctuations and could be flexibly manipulated by the spinning speed, size and structure of the resonator, optomechanical and magnomechanical coupling strength, dissipations and detunings of optical, microwave, and magnon modes.The nonreciprocal multipartite entanglement among macroscopic objects in the magnon-based hybrid system may have practical applications in quantum information processing, quantum network construction and quantum chiral device integration.

Figure 1 √
Figure 1 (a)-(b) Hybrid atom-optomagnomechanical system.An optical mode driven by a laser at frequency ω L in a spinning resonator couples to an ensemble of two-level atoms (a) and a magnon mode (m) in a YIG micro-bridge by the mediation of a phonon mode (b) induced by magnetostriction.The YIG micro-bridge is driven by a strong microwave field at frequency ω 0 , and simultaneously is in a uniform bias magnetic field along the z direction.(c) Frequencies and linewidths of the system.When the magnon is resonant with the blue sideband (anti-Stokes) of the driven field at frequencies ω 0 + ω b and the optical photon and atom modes are scatted by the phonon onto two sidebands, the genuine atom-magnon-phonon entanglement is generated at the steady state.(d) By fixing the CW rotation of the resonator, we have the frequency ω cs (ω c + s ) for the optical CW (CCW) mode when the light is driven from the right-hand (left-hand) large steady-state coherent amplitudes | m |, | c | 1.Therefore, the dynamics of the system can be linearized around the steady-state values by writing the operators as k = k +δk and neglecting the second-order fluctuation terms.The quadratures of the atomic ensemble, magnon, and optical photon are defined as X j = (j + j † )/ √ 2 and Y j = i(j †j)/ √ 2, and similarly for the quadrature components of input noise operators X in j = (j in + j † in )/ √ 2 and

Figure 2
Figure 2 Density plot of logarithmic negativity related to bipartite entanglement (a) E mb , (b) E ab , (c) E am , and (d) tripartite atom-magnon-phonon entanglement R min τ versus detunings ˜ m and a with ˜ c =a and = 0. See text for the other parameters.

Figure 3
Figure 3 (a) Bipartite entanglement E mb , E ab , and E am and (b) tripartite entanglement R min τ versus detuning ˜ c .Bipartite and tripartite entanglement versus (c) temperature T and (d) mechanical damping rate γ b

Figure 4
Figure 4 (a) Mean optical photon number | c | 2 versus detuning ˜ c with a = -0.5ωb in the presence of the spinning.Nonreciprocal bipartite entanglement (b) E mb , (c) E ab , and (d) E am versus detuning ˜ c with a = -˜ c .We take ˜ m = ω b and = 20 KHz in all plots.The other parameters are the same as in Fig. 2

Figure 5
Figure 5 Nonreciprocal tripartite entanglement R min τ (a) versus detuning ˜ c , the temperature T with ˜ c = 1.4ω b , and (c) the rotation speed with ˜ c = 0.95ω b . (d) The chiral factor χ of atom-magnon-phonon entanglement as a function of detuning ˜ c and rotation speed .The other parameters are the same as in Fig. 2 Guangling Cheng is supported by the National Natural Science Foundation of China (Grant No. 11905064) and the Natural Science Foundation China of Jiangxi Province (Grant No. 20232ACB201013), Wenxue Zhong is supported by the National Natural Science Foundation of China (Grant No. 12165007), Aixi Chen is supported by the National Natural Science Foundation of China (Grants No. 12175199), and Qianjun Zheng is supported by the Graduate Innovative Special Fund Projects of Jiangxi Province (Grant No. YC2022-s490).YIG, Yttrium iron garnet; QLEs, Quantum Langevin equations; CW, Clockwise; CCW, Counterclockwise; CV, Continuous variable; CM, Covariance matrix; RW, Rotating-wave.