Steering entanglement through exceptional points in non-hermitian cavity magnomechanics

Abstract

Cavity magnomechanical systems, which coherently couple magnons, photons, and phonons, offer a powerful platform for exploring quantum phenomena and developing hybrid quantum technologies. We study a non-Hermitian cavity magnomechanical system featuring a yttrium iron garnet (YIG) sphere driven by an external magnetic field, where magnons interact with cavity photons via magnetic dipole coupling and with phonons via magnetostrictive forces. Non-Hermiticity is introduced through a traveling optical field that directly excites the YIG sphere, enabling precise control over parity-time (PT) symmetry. Analysis of the eigenfrequency spectrum reveals a third-order exceptional point (EP) that demarcates distinct PT-symmetric phases: a unique configuration with coexisting broken and protected PT-symmetry regimes, tunable via the interplay of the traveling field strength and magnon-photon coupling. We also verify this by measuring the divergence around EP with Petermann Factor. Crucially, the system exhibits stable PT-symmetry only at specific field incidence angles, with instability or broken symmetry dominating elsewhere. Beyond PT-symmetry, we demonstrate robust quantum entanglement among the magnon, photon, and phonon subsystems. The exceptional point serves as a critical boundary, with unbroken and broken PT symmetry phases enabling dynamic entanglement swapping between subsystem pairs. By modulating the non-Hermitian parameters, we achieve controlled entanglement transfer and suppression of fluctuations, highlighting the system’s potential for quantum information processing. These results establish a direct connection between non-Hermitian topology, dynamical stability, and quantum correlations, providing a framework for leveraging PT symmetry in cavity magnomechanics for quantum technologies.

Introduction

Cavity magnomechanics, which integrates coherent magnon-photon and magnon-phonon interactions, has emerged as a powerful platform for exploring quantum electrodynamics and macroscopic quantum phenomena. Utilizing high-quality yttrium iron garnet (YIG) spheres, these systems achieve strong to ultra-strong coupling regimes due to their high spin density and robust spin-spin exchange interactions^1,2,3. The magnetostrictive interaction enables direct coupling between magnons and phonons, while magnetic dipole coupling mediates magnon-photon interactions, creating a versatile hybrid system for quantum transduction^4,5. Recent experiments have demonstrated groundbreaking capabilities, tripartite entanglement among photons, magnons, and phonons⁶, and enhanced sensing via cavity-enhanced coherent scattering⁷. These advances are further supported by theoretical proposals for quantum information protocols, such as deterministic entanglement generation between macroscopic ferrite samples⁸ and magnon-based quantum memories⁹. The unique interplay of spin, mechanical, and photonic degrees of freedom in cavity magnomechanics offers unprecedented opportunities for studying quantum-to-classical transitions and developing hybrid quantum technologies.

The integration of non-Hermitian physics and parity-time (PT) symmetry into cavity magnomechanics has opened new avenues for controlling quantum systems with engineered gain and loss. While traditional quantum systems rely on Hermitian Hamiltonians, non-Hermitian systems with PT symmetry can exhibit real energy spectra and unconventional dynamics, as first proposed by Bender and Boettcher¹⁰ and further enhanced by many notable studies,^{11,12,13,14,15}. In cavity magnomechanics, this framework enables the study of exceptional points (EPs)—degeneracies where eigenvalues and eigenvectors coalesce—leading to enhanced sensitivity in magnetometry¹⁶ and non-reciprocal phonon lasing¹⁷. Recent experiments have realized EPs in cavity magnon-polaritons¹⁸ and higher-order configurations¹⁹, revealing their role in manipulating quantum correlations and nonlinear responses. Theoretical studies further predict PT-symmetry-enhanced entanglement in magnomechanical systems²⁰ and non-Hermitian topological phases in coupled magnon-photon systems²¹. However, the interplay of PT symmetry with the tripartite interactions (magnon-photon-phonon) in cavity magnomechanics remains underexplored, particularly in regimes where gain-loss balance and nonlinearities compete to stabilize quantum states^22,23. This gap motivates our investigation into the PT-symmetric properties of non-Hermitian cavity magnomechanical systems.

In this work, we study a non-Hermitian cavity magnomechanical system comprising a YIG sphere coupled to a microwave cavity and driven by an external magnetic field orthogonally interacting with YIG. The non-Hermitian behavior is induced via a traveling optical field that introduces controlled gain and loss, leading to a third-order exceptional point (EP) in the system’s eigenfrequency spectrum. We first establish the conditions for PT symmetry, showing that the system exhibits a protected PT phase when the non-Hermitian strength matches the magnon-photon coupling rate, while deviations result in a bi-broken PT phase. Next, we perform a stability analysis, deriving the minimal conditions required to maintain dynamical stability across different PT phases. Finally, we investigate the entanglement dynamics between photons, magnons, and phonons, demonstrating how PT symmetry and EPs enable entanglement swapping and controlling quantum correlations. Our results provide a framework for exploiting non-Hermitian effects in quantum magnomechanics, with applications in sensing, information processing, and the study of macroscopic quantum phenomena.

Results

System description and Hamiltonian

We consider a single-crysta YIG microsphere, with diameter 10² μm to 1mm, placed inside a Fabry-Pérot cavity having length L ≈ 12.5 × 10⁻⁴m, Fig. 1,²³. 10G bias magnetic field B₀ orthogonality directly drives YIG exciting magnons $\hat{m}$. The coupling between B₀ and $\hat{m}$ can be defined as $\eta =(\sqrt{5}/4){\gamma }_{B}\sqrt{N}{B}_{0}$, having power P₀ ≈ 0.0164mW at frequency ω₀ ≈ 3.8 × 2π × 10⁸ Hz and γ_B/2π = 28G Hz/T. The total number of spins N = ρV with volume V and spin density ρ = 2 × 10²² cm⁻³. The magnons, oscillating with frequency ω_m ≈ 2π × 2 MHz and approximated decay rate κ_m = 2π × 3 MHz, interacts with previously existing microwave cavity photons, yielding in their mutual coupling due to dipole interactions. It should be mentioned here a microwave source also interacts with the system, exciting cavity photons $\hat{a}$, with wavelength higher than the diameter of YIG resulting in negligible effects of radiation pressure. That is the reason it is ignored and not discussed^5,18,24,25.

Fig. 1: Schematic diagram of the non-hermitian cavity magnomechanical system.

A YIG sphere is placed inside a microwave cavity driven by a strong magnetic field η exciting magnons $\hat{m}$ oscillating at ω_m, in coupling with the cavity photons $\hat{a}$. Magnomechanical interactions also excite phonon $\hat{b}$ from YIG oscillating with frequency ω_b. κ_m, κ_a and γ_b correspond to the dissipation associated with magnon, photon, and phonons, respectively. To make system non-Hermitain, a traveling field interacts with YIG at angle θ and strength Γ.

However, the microwave photonic mode oscillating with ω_a ≈ 2π × 5 GHz and Δ_a = ω₀ − ω_a, due to the high-quality Q factor of the cavity, can generate strong-coupling under condition G_ma > > κ_a, where G_ma is the magnon-photon coupling with cumulative cavity leakage κ_a ≈ 2π × 1 MHz^{5,26,27,28,29,30,31,32,33,34,35}. ${G}_{ma}=({g}_{s}{\mu }_{B}/2\hslash )\sqrt{{\mu }_{0}\hslash {\omega }_{a}{V}_{s}/V}$ is the relation for magnon-photon coupling, where g_s is the electron G-factor and μ_B is Bohr magneton. V_s/V is scaled spin saturation volume ratio as defined earlier, which V/V_s = 1 can define G_ma over the order of 10 MHz. Further, the magnetostrictive interactions between magnetic field and magnons generate phonon $\hat{b}$ around YIG, resulting in magnon-phonon coupling g_mb ∝ 1/D, which is defined over the inverse proportionality with YIG diameter⁵. We consider g_mb ≈ 2π × 9.88 mHz oscillating with frequency ω_b ≈ 2π × 9.88MHz having dissipation κ_a ≈ 2π × 300 Hz. However, magnomechanical coupling can be enhance by reducing the size of YIG. These values are selected by following the well performed experiments^2,5.

The total Hamiltonian of the system reads as,

$$\begin{array}{ll}\hat{H}=\hslash {\omega }_{a}{\hat{a}}^{\dagger }\hat{a}+\hslash {\omega }_{m}{\hat{m}}^{\dagger }\hat{m}+\hslash {\omega }_{b}\hat{b}{\hat{b}}^{\dagger }+{g}_{mb}{\hat{m}}^{\dagger }\hat{m}(\hat{b}+{\hat{b}}^{\dagger })\\\qquad+\,{G}_{ma}(\hat{a}+{\hat{a}}^{\dagger })(\hat{m}+{\hat{m}}^{\dagger })+i\eta ({\hat{m}}^{\dagger }{e}^{-i{\omega }_{0}t}-\hat{m}{e}^{i{\omega }_{0}t})\\\qquad-\,i\Gamma {e}^{i(\delta t+\theta )}(\hat{a}+{\hat{a}}^{\dagger })(\hat{m}+{\hat{m}}^{\dagger }),\end{array}$$

(1)

where $\hat{a}({\hat{a}}^{\dagger }),\hat{m}({\hat{m}}^{\dagger })$ and $\hat{b}({\hat{b}}^{\dagger })$ are the annihilation (creation) operator of the cavity, magnon and phonon, respectively, with $[\hat{O},{\hat{O}}^{\dagger }]=1(O=a,m,b)$. $\Gamma ={\mathscr{\alpha }}\sqrt{(\hslash /{\omega }_{m}{m}_{m})}$ represents the coupling strength of traveling field with the magnonic mode, where ${\mathscr{\alpha }}$ is the amplitude of the traveling field. δ and θ accounts for the frequency and incident angle of traveling field with cavity axis.

By applying the rotating-wave approximation which $(\hat{a}+{\hat{a}}^{\dagger })(\hat{m}+{\hat{m}}^{\dagger })\to ({\hat{a}}^{\dagger }\hat{m}+\hat{a}{\hat{m}}^{\dagger })$ and under the frame rotating at the drive frequency ω₀, we drive quantum Langevin equations (QLEs) to incorporate the associated dissipation and quantum noises with subsystems, for details the Methods. While driving QLEs, we also consider the traveling field time independent yielding in e^δt+iθ → e^iθ. After this, we linearized QLEs and, from Linearized QLEs, we computed matrix representation of the effective PT-symmetric Hamiltonian under condition ${\hat{H}}_{eff}^{PT}=({\mathcal{PT}}){\hat{H}}_{eff}{({\mathcal{PT}})}^{-1}$, where ${\mathcal{P}}$ corresponds to the parity operator while ${\mathcal{T}}$ is the time-reversal operator^36,37,38,

$${\hat{H}}_{eff}^{PT}\approx \left(\begin{array}{lll}{\Delta }_{a}+i{\kappa }_{a}&{G}_{ma}+i\Gamma {e}^{i\theta }&0\\ {G}_{ma}+i\Gamma {e}^{i\theta }&{\Delta }_{m}+i{\kappa }_{m}&{G}_{mb}\\ 0&{G}_{mb}&{\Delta }_{b}+i{\gamma }_{b}\end{array}\right),$$

(2)

where G_mb = g_mb〈m〉 and Δ_a,m,b = ω_a,m,b − ω₀, respectively.

Bi-broken P T-symmetric magnomechanics

From the effective Hamiltonian, one can easily guess the occurrence of three eigenvalues for the hybrid magnomechanical system, as illustrated in Fig. 2(a) and (b). It can be observed that at G_ma/Δ_a = 1, the eigenvalue spectrum overlaps in such particular way that results in singularity intersection or third-order EP. Along this intersection, the system is operating in a protected PT symmetry domain and all three eigenvalues coexist in real space having no imaginary part. On the other hand, on both sides of the EP axis, the system is under broken PT symmetry region, which means that the eigenvalues are appeared to be complex and have both real and imaginary parts, as illustrated in Fig. 2(a) and (b). One can also observe at G_ma = 0, all eigenvalues also appear to have zero imaginary parts. But, as G_ma = 0, it also means that the magnomechanical system is disintegrated and have no interaction resulting in trivial Hermitian state.

Fig. 2: Eigenfrequency spectrum as a function of normalized magnon-photon coupling G_ma/Δ_a.

The blue solid line is the real part Re[λ] while the red dashed line is for the imaginary part Im[λ] of the eigenvalues at Γ/ω_b = 1. a, b are for the incident angles θ = π/2 and θ = π of traveling field, respectively. The other parameters that we choose are κ_a/Δ_a = 0.08, κ_m/Δ_m = 0.08, G_mb/Δ_m ≈ 0.7 and γ_b/Δ_a = 1/1000.

In non-Hermitian optical systems, PT-symmetry breaking typically involves the imbalance between system’s gain and loss configuration^21,39, therefore, it is crucial to observe the influence of Γ and its correspondence with G_ma. It should be mentioned here that, in following, by gain broken, we means G_ma/Γ < 1, and by lossy broken, we means G_maΓ > 1. It should not be confused with the gain that involves non-Hermiticity. The value of Γ should be equal to the G_ma, i.e. G_ma = Γ = 1 at the axis belonging to the EP (or at EP). In other case, if G_ma ≠ Γ, then the system will possess broken PT, because eigenvalues will have both real and imaginary parts. At G_ma/Γ > 1, the magnomechanical system appears to have lossy broken PT symmetry as G_ma is higher than the gain induced by Γ. While on other hand, at G_ma/Γ < 1, the system will operate with, sort-of, gain broken configuration because the gain excited by Γ is higher than G_ma. Physically, the system will be PT protected when there is symmetry of equality between G_ma and Γ. Whenever there is asymmetric equality then the system will possess broken PT. It will be further explained and verified by the Fig. 3(e) and (f).

Fig. 3: Effects of non-Hermitian strength and incident angle.

a–d Represent eigenfrequency spectrum as a function of G_ma/Δ_a. The incident angle θ = π/2 for (a) and (b), while θ = π for (c, d), respectively. The solid line is for Γ/ω_b = 1, the dashed line is for Γ/ω_b = 2. e, f are 3D plot of the eigenvalue with respect to non-hermitian strength Γ/ω_b and magnon-photon coupling G_ma/Δ_a, at angle θ = π/2 and θ = π, respectively. The other parameters are the same as in Fig. 2.

θ plays a crucial role in the non-Hermitian configuration of the system. From the Hamiltonian perspective, and according to Euler’s formula, only if the angle of incidence is π/2, π, 3π/2, 2π, the system can exist EPs. In our investigation, we found that the system only exhibits EPs when θ = π/2 or θ = π, because at only these angles, the traveling field acts as a gain to the system. When this gain matches the magnon’s loss, the system exhibits PT symmetry, naturally leading to the emergence of EPs. Because, at θ = π/2 and θ = π, the non-Hermitian term in the Hamiltonian differs by i, yielding in opposite real and imaginary parts of the eigenvalue spectrum. This can be clearly observed by comparing Fig. 2(a) and (b). On the other hand, when θ = 3π/2 or θ = 2π the traveling field acts as a loss, providing photonic energy to the system, similar to the effect of a pump laser, see ref.⁴⁰ for details. In such configuration, the system does not exhibit EP because of the lossy environment. For angles other than these four specific values, the real and imaginary parts of the eigenvalues can merge under certain parameters configuration, but it will not contain EPs and G_ma corresponding to these real and imaginary parts will be different. Thus, EPs do not exist at θ other than π/2 and π. One can note another interesting phenomenon that the real part Re[λ] at θ = π/2 is almost equal to the imaginary part at θ = π and vise-versa. It can be easily understood by imagining complex variables getting orthogonality shifted over the complex plan.

Figure 3 (a–d) further delves into the impact of non-Hermitian strength on the system. As previously discussed, when θ is π/2 and π, system loss is positively correlated with non-Hermitian strength. When we increase the non-Hermitian strength Γ/ω_b from 1 to 2, the EP also shifts from G_ma/Δ_a = 1 to G_ma/Δ_a = 2, maintaining the balance between system loss and gain. Therefore, we conclude that in this system, the PT-symmetry condition is achieved when the non-Hermitian strength equals the magnon-photon coupling rate, i.e.,Γ/ω_b = G_ma/Δ_a. Alternatively saying, the increase in Γ/ω_b shifts the interface between gain broken and lossy broken to higher G_ma regions, providing more effective control over the position of EPs.

Further, we plotted the eigenvalue spectrum versus these parameters, as illustrated in Fig. 3(e, f), where we plots the imaginary parts of the eigenvalues as functions of Γ/ω_b and G_ma/Δ_a. One can note that the values or the parametric position illustrating the occurrence of EPs move diagonally between G_ma and Γ. Means, all of the eigenvalues coexist diagonally having zero value for all or the eigenvalues. Thus, the EPs exist at points where both G_ma and Γ are equal with each other. It further proves our argument about the tunability of the non-Hermitian EPs. Both adjusting the traveling field to control the non-Hermitian strength and modifying the magnetic field to control G_ma offer significant operational flexibility.

It is worth noting that, unlike the canonical two-mode PT-symmetric system where eigenvalues are either purely real or occur strictly as complex-conjugate pairs, the three-mode hybrid system studied here exhibits a richer spectral structure. In particular, beyond the exceptional point one eigenvalue remains real, while the other two form a complex-conjugate pair, consistent with the generalized pseudo-Hermitian behavior of higher-dimensional non-Hermitian systems. This feature is clearly visible in Figs. 2, 3, where one eigenfrequency branch persists on the real axis while the remaining branches bifurcate into the complex plane. The apparent asymmetry between the complex branches arises from the effective Hamiltonian approximations, but does not alter the essential pseudo-Hermitian character of the spectrum.

Further, the order of the exceptional point (EP) depends critically on the coupling structure of the system. In the absence of magnon–phonon coupling (G_mb = 0), the phonon mode decouples and the EP₃ reduces to a second-order EP arising solely from the magnon–photon subspace. In contrast, setting the magnon–photon coupling to zero (G_ma = 0) renders the system uncoupled, where apparent level crossings are simple degeneracies rather than true EPs, since no non-diagonalizable structure emerges. Thus, nonzero coupling is essential for the realization of higher-order EPs in our hybrid system.

Eigenvectors and Petermann factor near the EP₃

Having discussed the eigenvalue behavior around the third-order exceptional point EPs (EP₃), we now turn to the properties of the corresponding eigenvectors and the associated Petermann factor⁴¹. In the vicinity of an EPs, the eigenvalues of the non-Hermitian Hamiltonian H(G_ma) exhibit a characteristic cubic-root splitting. Let $\epsilon ={G}_{ma}-{G}_{ma}^{{\rm{EP}}}$ denote the small deviation of the system parameter G_ma from its EPs value. Then, the eigenvalues scale as

$${\lambda }_{j}(\epsilon )\,\approx \,{\lambda }_{{\rm{EP}}}+\alpha \,{\epsilon }^{1/3}{e}^{i2\pi j/3},\quad j=0,1,2,$$

(3)

where α is a complex constant determined by the local expansion of $\det [H(g)-\lambda I]$. This fractional power-law dependence ~ ϵ^1/3 is a hallmark of cubic-root branch point singularities.

At an EP₃, all three eigenvectors collapse into a single defective state, forming a Jordan block of order three. Denoting the right and left eigenvectors by $\left\vert {\psi }_{R,j}\right\rangle$ and $\left\langle {\psi }_{L,j}\right\vert$, one finds

$$\left\vert {\psi }_{R,0}\right\rangle \,\simeq \,\left\vert {\psi }_{R,1}\right\rangle \,\simeq \,\left\vert {\psi }_{R,2}\right\rangle \,\to \,\left\vert {\psi }_{{\rm{EP}}}\right\rangle ,$$

(4)

with an analogous relation for $\left\langle {\psi }_{L,j}\right\vert$. This collapse signals a complete loss of modal orthogonality.

The non-orthogonality of left and right eigenvectors is quantified by the Petermann factor⁴¹,

$${K}_{j}\,=\,\frac{\langle {\psi }_{L,j}| {\psi }_{L,j}\rangle \langle {\psi }_{R,j}| {\psi }_{R,j}\rangle }{{\left\vert \langle {\psi }_{L,j}| {\psi }_{R,j}\rangle \right\vert }^{2}}.$$

(5)

For Hermitian systems, one has K_j = 1, reflecting orthogonality. In contrast, near an EP_N the Petermann factor diverges as K_j ~ ϵ^{− (N−1)}, so that for EP₃ one obtains K_j ~ ϵ⁻², 7D17D1ϵ → 0. Thus, for EP₃ one obtains K_j ~ ϵ⁻², with all three eigenmodes contributing to the divergence.

The theoretical framework described above applies directly to our PT-symmetric magnon-photon-phonon system. The calculated eigenvalue spectra in Figs. 2, 3 clearly display the expected coalescence of both the real and imaginary components of the eigenvalues at the critical relation between the magnon-photon coupling G_ma and the non-Hermitian parameter Γ, marking the onset of higher-order exceptional points (EPs). This spectral behavior is a hallmark of non-Hermitian degeneracies, where multiple eigenmodes merge simultaneously, leading to drastic modifications in the system’s dynamical response.

Consistently, the Petermann factor ${\mathcal{K}}$, shown in Fig. 4, exhibits sharp resonant peaks at the same parametric values where the EPs emerge. These peaks signify the divergence of eigenvectors and the concomitant breakdown of bi-orthogonality near the EPs. In particular, Fig. 4(a) demonstrates that for Γ/ω_b = 1.0, the EP appears at G_ma = 1Δ_a, in full agreement with the eigenfrequency spectrum. The presence of three distinct peaks is especially noteworthy: it indicates that all three hybridized modes of the system simultaneously participate in the EP₃ coalescence, confirming the higher-order non-Hermitian degeneracy.

Fig. 4: Petermann factor ${\mathcal{K}}$ corresponding to three system eigenfrequency λ₁ (black), λ₂ (blue), and λ₃ (red) versus G_ma/Δ_a.

a, b are for non-hermitian strength Γ/ω_b = 1.0 and Γ/ω_b = 2.0, respectively, at G_mb = 0.09Δ_m. While c corresponds to the effects of G_mb = 1.0Δ_m, at Γ/ω_b = 1.0. The remaining numerical parameters are the same as in Fig. 2.

A further increase in the non-Hermitian parameter to Γ/ω_b = 2.0 shifts the location of these divergences, as illustrated in Fig. 4(b). The peaks of ${\mathscr{K}}$ are displaced toward larger G_ma, reflecting the shift of the exceptional point in parameter space. This demonstrates the tunability of the EP₃ through dissipation engineering, where stronger non-Hermiticity drives the system into new regions of coalescence. The divergence in ${\mathscr{K}}$, therefore, directly tracks the motion of the eigenvalues in the complex plane, providing a powerful diagnostic tool for mapping EP dynamics.

Finally, Fig. 4(c) highlights the role of the magnon-phonon coupling strength G_mb. When G_mb is increased to 1.0Δ_a, comparable to the magnon-photon coupling, the divergence of ${\mathscr{K}}$ near the EPs is significantly reduced. This suppression implies that stronger magnon-phonon hybridization counteracts the EP-enhanced non-orthogonality, effectively stabilizing the system against extreme sensitivity associated with the higher-order degeneracy. Hence, large G_mb values act as a control knob for moderating the influence of non-Hermitian effects on the hybrid modes. Taken together, the simultaneous observation of eigenvalue bifurcation and Petermann factor divergence provides robust and complementary evidence for the realization of an EP₃ in our hybrid magnon-photon-phonon platform. These results confirm the intrinsically non-Hermitian character of the effective Hamiltonian ${\hat{H}}_{{\rm{eff}}}^{PT}$ and open up avenues for exploiting EP-enhanced sensitivity, mode control, and dissipative engineering in hybrid quantum systems.

Stability criterion with P T-symmetry

To defined a particular set of parameters where system operates in a stable configuration, we use the Routh-Hurwitz stability criterion to develop stability conditions, see Methods for details. We strictly follow these conditions while choosing parameters in our numerical calculations. To further understand the stability, we drive a stability parameter from the stability conditions, reading as,

$$\begin{array}{rcl}{\mathcal{S}} &=& 1+\frac{{\left({G}_{ma}-i\Gamma {e}^{i\theta }\right)}^{2}\left(2{\kappa }_{a}{\kappa }_{m}{\omega }_{b}+{\Delta }_{a}\left({G}_{mb}^{2}-2{\omega }_{b}{\Delta }_{m}\right)\right.}{\left({\kappa }_{a}^{2}+{\Delta }_{a}^{2}\right)\left(-{G}_{mb}^{2}{\Delta }_{m}\right.}....\\ &&....\frac{+\left.{\left({G}_{ma}-i\Gamma {e}^{i\theta }\right)}^{2}{\omega }_{b}\right)}{+{\omega }_{b}\left.\left({\kappa }_{m}^{2}+{\Delta }_{m}^{2}\right)\right)},\end{array}$$

(6)

where Δ_m > 0 and Δ_a > 0 while κ_a + κ_m > γ_b/2 in order to fulfill the stability conditions.

From ${\mathscr{S}}$, one can graphically analysis the stability of the system, especially with respect to θ, as illustrated in Fig. 5. ${\mathscr{S}}$ is plotted versus G_ma/Δ_a and G_mb/Δ_m, with θ set at π/2 and in comparison with the PT-symmetric eigenvalues of the system. Here colored regions correspond to the stable regimes while white areas illustrate the instability of the system. It should be noted that the reason behind specifically choosing θ = π/2 is that the system appears to be unstable at θ = π and non-PT-symmetric on other angles. The reason is at when θ = π, the traveling field is parallel to the intracavity optical field and perpendicular to the magnetic field. This configuration functions similarly to adding a pump optical field along the x-axis, supplying photons and energy to the system, thereby driving it into an unstable state.

Fig. 5: Stability parameter ${\mathcal{S}}$ versus normalized G_ma/Δ_a and G_mb/Δ_m.

a, b are for non-hermitian strength Γ/ω_b = 1.0, while c, d are for Γ/ω_b = 2.0, respectively. e, f illustrate ${\mathscr{S}}$ versus κ_a/Δ_a and κ_m/Δ_m at fixed G_ma/Δ_a = Γ/ω_b = 1 and θ = π/2. The other parameters used in calculation are same as in Fig. 2.

Centering around the EP, the system exhibits two symmetric stability regions on either side of G_mb/Δ_m = 1, displaying strength vise opposite stability trends with increase in G_ma/Δ_a, as can be seen in Fig. 5(a, d). In regions with a lower magnon-phonon coupling rate, the system is more stable. When moving away from the EP, the system reaches maximum stability in regions of weak magnon-phonon coupling rate and minimum stability in regions of strong magnon-phonon coupling rate. The comparison here between stability and eigenvalues is only versus the G_ma and other axis is independent. One can note that the system only appears to be unstable at higher values of G_ma/Δ_a on both higher and lower regions of G_mb/Δ_m.

Further, the stability of the system crucially depends on the position of EP (or in other words, depends on the relation between Γ and G_ma), which is saturated between the center of stability regions. Therefore, any change in Γ results in a shift in entire stability region similarly as they appear in eigenvalue spectrum, where EP shifts towards higher values of G_ma/Δ_a, as can be seen in Fig. 5(c, d). But it is interesting to note that similar unstable region, which was appearing at higher coupling rates, is now also appearing at lower coupling rates. It means that the collective system is now possessing bi-unstable parametric characteristics. Such tunability stability through both G_ma and G_mb can offers valuable insights for designing stable non-Hermitian systems and exploring their potential applications in quantum information science.

In addition, we explored the impact of dissipations on system stability. The results reveal two clear boundaries between stability regions corresponding to higher and lower values, as shown in Fig. 5(e, f). The system tends to be more stable with higher magnon dissipation κ_m, while the dissipation of the optical field significantly determines the threshold for system stability. In conditions of low dissipation, it is challenging for the system to remain stable. However, non-Hermitian strength can partially compensate for this. Comparing Fig. 5(e) and (f), a stronger non-Hermitian strength allows the system to achieve stability under lower optical field dissipation conditions.

PT-symmetry mediated entanglement

In this section, we will discuss the entanglement between associated subsystems, photon, magnon and phonon. To ensure the realization of entanglement in the system to a last degree, we have to select the optimal effective interactions among the three modes, by keeping them in study-state via quantum correlations. Later, we measure the entanglement by adopting bipartite approach means by considering two subsystem at one time and tracing-out other. Further, we apply Routh-Hurwitz stability criterion to generate stability conditions and strictly follow these condition while choosing the numerical parameters, as discussed earlier during stability analysis. The entanglement generation and stability criterion procedures are discussed with details in Method Sections.

In Fig. 3, we show three bipartite entanglements versus normalized detuning Δ_a/κ_a(κ_a is fixed) and cavity dissipation κ_a/γ_b(γ_b is fixed). E_am, E_mb, and E_ab denote the photon-magnon, magnon-phonon, and photon-phonon entanglement, respectively. The graphical analysis reveals conspicuous alterations in the entanglement dynamics between subsystems of the system due to the non-Hermitian effects. Based on our previous research, we selected entanglement values in the PT-unbroken region (dashed line: G_ma = 1Δ_a) and the PT-broken region (solid line: G_ma = 0.5Δ_a and dotted line: G_ma = 1.5Δ_a) to investigate the role of PT symmetry. In Fig. 6(a), around Δ_a/κ_a ≈ G_ma + G_mb, bipartite entanglement exhibits significant fluctuations, which increase with increase in magnon-photon coupling, especially when G_ma > Γ. At Δ_a/κ_a ≈ G_ma + G_mb, the photon, magnon, and phonon modes become strongly hybridized, pushing the system close to the stability boundary and thereby suppressing steady-state entanglement. It reveals the fact that we can use Γ, to some extent, suppress these fluctuations, leading towards the application of EPs. But for higher values of Δ_a, the entanglement is appeared to increasing. As, the cavity magnomechanical PT-symmetric systems have been realized in numerous experiments^42,43. Therefore, current observation is insightful for achieving relatively stable or more pronounced entanglement fluctuations through a control coupling rate. The influence of a strong magnon pump on cavity field detuning directly affects the system, leading to nonlinear effects caused by Kerr nonlinearity, which is inversely proportional to the volume of the sphere^6,44,45.

Fig. 6: Steady-state entanglement through EPs.

The bipartite entanglement E_n (where n → (am, mb, ab)) as a function of normalized microwave cavity mode detuning Δ_a/κ_a (a) and cavity dissipation κ_a/γ_b (b). The blue, red, and black curves represent E_am(photon−magnon), E_mb(magnon−phonon), E_ab (photon-phonon), respectively. The solid, dashed, and dotted lines indicate that the values of photon-magnon coupling G_ma/Δ_a are 0.5, 1 and 1.5, respectively, which makes the system under the PT symmetry protection and PT broken regimes. Here Γ/ω_b = 1.0, while the other parameters used are the same as in Fig. 2.

While the effect of detuning on magnon-phonon and photon-phonon entanglement is weak, it is significant for photon-magnon entanglement, indicating a direct relationship with the coupling interactions that generate entanglement. The variation trend of entanglement between photons and magnons induced by magnetic dipole interaction is opposite to that of entanglement between photons and phonons, as well as between magnons and phonons, generated by magnetostrictive interaction, similar to radiation pressure-like interaction, concerning changes in cavity detuning Δ_a. Fig. 6(b) further confirms that under PT symmetry-breaking conditions, the bipartite entanglement increases significantly with parameter variations. With the enhancement of cavity field dissipation, entanglement transitions from within the photon-magnon and magnon-phonon subsystems towards the photon-phonon subsystem and rapidly approaches zero.

The engineered non-Hermiticity in our cavity magnomechanical system manifests through the traveling field term $-i\Gamma {e}^{i\theta }({\hat{a}}^{\dagger }\hat{m}+\hat{a}{\hat{m}}^{\dagger })$ in Eq. (1), creating a tunable gain-loss landscape that directly modulates quantum entanglement dynamics. This synthetic non-Hermitian parameter Γ serves as a powerful control knob, enabling precise manipulation of the system’s quantum correlations while preserving their coherence properties. As shown in Fig. 7, the bipartite entanglement measures E_am, E_mb, and E_ab exhibit distinct responses to cavity detuning Δ_a/κ_a and dissipation κ_a/γ_b that are strongly mediated by the non-Hermitian strength.

Fig. 7: Non-Hermitian control of entanglement.

The bipartite entanglement E_n as a function of normalized microwave cavity mode detuning Δ_a/κ_a (a, b) and cavity dissipation κ_a/γ_b (c, d). The blue, red, and black curves represent E_am, E_mb, E_ab, respectively. The solid and dashed lines indicate that the values of non-Hermitian strength Γ/ω_b are 1 and 0.5, respectively. The other parameters are the same with 2.

Our results reveal several key quantum control features. First, reducing Γ induces a systematic blue shift in the entanglement spectra (Fig. 7(a-b)), demonstrating how non-Hermitian strength effectively renormalizes the system’s energy landscape. The PT-symmetric regime (Γ = G_ma) maintains remarkable entanglement stability with fluctuations ΔE_am < 0.1, while the PT-broken phase shows enhanced but more volatile correlations (ΔE_am ≈ 0.3) that could be harnessed for sensitive quantum sensing. Notably, cavity dissipation affects entanglement redistribution rather than simple decay - quantum correlations transfer between subsystems before vanishing, suggesting a novel dissipation-assisted entanglement engineering protocol.

The entanglement phase diagrams in Fig. 8 provide deeper insight into the system’s quantum many-body physics. The magnon-phonon coupling G_mb, typically limited to weak values in conventional magnomechanics⁴⁴, can be effectively enhanced through non-Hermitian engineering, as evidenced by the expanded high-entanglement regions in Fig. 8(a–c). This enhancement occurs without increasing pump power, avoiding the usual trade-off between coupling strength and nonlinear decoherence.

Fig. 8: Steady-state entanglement dependance on magnon-photon coupling and non-Hermitian strength.

Entanglement phase diagrams showing (a–c) E_am, E_mb, and E_ab versus Δ_a/κ_a and G_mb/Δ_m (top row), and (d–f) versus Δ_a/κ_a and Γ/ω_b (bottom row). The color scheme used in density plot illustrates the strength of entanglement, as illustrated by the plot-legend, while white regions account for zero entanglement. Parameters match Fig. 2.

Three distinct entanglement regimes emerge: For G_mb/Δ_m < 1.5, the system exhibits magnon-phonon dominated entanglement (E_mb ≈ 0.5) that is remarkably robust against detuning variations. In the transition region (1.5 < G_mb/Δ_m < 2.0), quantum correlations gradually transfer from E_mb to E_am, while above G_mb/Δ_m > 2.0, photon-magnon entanglement becomes dominant (E_am > 0.6) as the hybrid system enters a strongly coupled regime.

The non-Hermitian control demonstrated in Fig. 8(d–f) reveals even more sophisticated manipulation possibilities. The equivalence between Γ and G_ma in Eq. (8) allows the traveling field angle θ to effectively implement negative coupling interactions - a feature impossible in conventional Hermitian systems. This capability enables on-demand switching between different entanglement configurations, suggesting new protocols for quantum state engineering in hybrid magnonic systems.

Discussion

Cavity magnomechanics, which integrates magnons, photons, and phonons through magnetic dipole and magnetostrictive interactions, provides a rich platform for exploring non-Hermitian quantum phenomena and their applications in quantum technologies. In this work, we investigated a non-Hermitian cavity magnomechanical system comprising a YIG sphere driven by a magnetic field, where magnon-photon coupling is mediated by magnetic dipole interactions and magnon-phonon coupling arises from magnetostrictive forces. Non-Hermiticity was engineered via a traveling optical field, enabling precise control over parity-time (PT) symmetry and exceptional points (EPs).

Our analysis revealed a third-order exceptional point in the eigenfrequency spectrum, which separates unique PT-symmetric phases: a protected PT-symmetric regime occurring only when the non-Hermitian strength matches the magnon-photon coupling, and bi-broken PT phases elsewhere. We also measure the Petermann Factor to observe the divergence in eigen vector spectrum around EPs. Stability conditions demonstrated that the system remains stable near the EP but becomes unstable at the EP itself, with stability tunable via the traveling field’s incidence angle and the magnon-phonon coupling rate. Notably, excessive cavity dissipation can destabilize the system, but this can be mitigated by adjusting the non-Hermitian parameters.

Beyond PT symmetry, we demonstrated that quantum entanglement among the subsystems is profoundly influenced by non-Hermitian dynamics. The PT-symmetric regimes, especially with the broken phases, enable controlled entanglement swapping between magnon-photon and magnon-phonon pairs. This dynamic control, achieved by tuning the non-Hermitian strength and coupling rates, suppresses fluctuations and optimizes entanglement for quantum information tasks.

Looking ahead, our findings pave the way for experimental realizations of non-Hermitian magnomechanics, with potential applications in quantum sensing, information processing, and error-resilient quantum memory. Future work could explore higher-order EPs in multi-mode systems or integrate nonlinearities to further enhance entanglement control. By bridging non-Hermitian physics with quantum magnomechanics, this work offers a framework for developing robust quantum technologies leveraging engineered gain and loss.

Methods

Heisenberg equations of motion

By considering the rotating-wave approximation $(\hat{a}+{\hat{a}}^{\dagger })(\hat{m}+{\hat{m}}^{\dagger })\to ({\hat{a}}^{\dagger }\hat{m}+\hat{a}{\hat{m}}^{\dagger })$ for the total Hamiltonian mentioned in main text and applying the frame rotating at the drive frequency ω₀, the total system Hamiltonian will read as,

$$\begin{array}{rcl}\hat{H}&=&{\Delta }_{c}{\hat{a}}^{\dagger }\hat{a}+{\Delta }_{m}{\hat{m}}^{\dagger }\hat{m}+{\Delta }_{b}{\hat{b}}^{\dagger }\hat{b}\\ &&+({G}_{ma}-{\rm{i}}\Gamma {{\rm{e}}}^{{\rm{i}}\theta })(\hat{a}{\hat{m}}^{\dagger }+{\hat{a}}^{\dagger }\hat{m})+{g}_{mb}{\hat{m}}^{\dagger }\hat{m}({\hat{b}}^{\dagger }+\hat{b})\\ &&+{\rm{i}}\eta ({\hat{m}}^{\dagger }-\hat{m}),\end{array}$$

(7)

here Δ_a = ω_a − ω₀, Δ_m = ω_m − ω₀, Δ_b = ω_b − ω₀ being the detuning of the microwave cavity mode, magnon mode, and phononic mode, respectively. After this, we drive quantum Langevin equations to incorporate the associated dissipation with subsystems and govern the time dynamics in the form of equations of motion, reading as,

$$\dot{\hat{a}}=-(i{\Delta }_{a}+{\kappa }_{a})\hat{a}-(i{G}_{ma}+\Gamma {e}^{i\theta })\hat{m}+\sqrt{2{\kappa }_{a}}{\hat{a}}_{in},$$

(8)

$$\begin{array}{rcl}\dot{\hat{m}}&=&-(i{\Delta }_{m}+{\kappa }_{m})\hat{m}-(i{G}_{ma}+\Gamma {e}^{i\theta })\hat{a}-i\hat{m}\hat{q}\\ &&+\eta +\sqrt{2{\kappa }_{m}}{\hat{m}}_{in},\end{array}$$

(9)

$$\dot{\hat{q}}={\omega }_{b}p,$$

(10)

$$\dot{\hat{p}}=-{\omega }_{b}\hat{q}-{\gamma }_{b}\hat{p}-{g}_{mb}{\hat{m}}^{\dagger }\hat{m}+\xi ,$$

(11)

here $\hat{p}$ and $\hat{q}$ are the dimensionless position and momentum quadrature of the mechanical modes, with [q, p] = i, defined conventionally from bosonic field operator of phonons $\hat{b},{\hat{b}}^{\dagger }$. ${\hat{a}}_{in},{\hat{m}}_{in},\xi$ are the Markovian input noise of the optical cavity, magnon and mechanical modes, respectively, the damping of mechanical is described by γ_b. The quantum noise correlation functions⁴⁶ are considered as,

$$\begin{array}{rcl}\langle {a}_{in}(t){a}_{in}^{\dagger }\left({t}^{{\prime} }\right)\rangle &=&\left[{N}_{a}\left({\omega }_{a}\right)+1\right]\delta (t-{t}^{{\prime} }),\\ \langle {a}_{in}^{\dagger }(t){a}_{in}\left({t}^{{\prime} }\right)\rangle &=&{N}_{a}\left({\omega }_{a}\right)\delta \left(t-{t}^{{\prime} }\right),\\ \langle {m}_{in}(t){m}_{in}^{\dagger }\left({t}^{{\prime} }\right)\rangle &=&\left[{N}_{m}\left({\omega }_{m}\right)+1\right]\delta \left(t-{t}^{{\prime} }\right),\\ \langle {m}_{in}^{\dagger }(t){m}_{in}\left({t}^{{\prime} }\right)\rangle &=&{N}_{m}\left({\omega }_{m}\right)\delta \left(t-{t}^{{\prime} }\right),\\ \langle \xi (t)\xi \left({t}^{{\prime} }\right)+\xi \left({t}^{{\prime} }\right)\xi (t)\rangle /2&\simeq &{\gamma }_{b}\left[\left(2{N}_{b}({\omega }_{b})+1\right)\delta \left(t-{t}^{{\prime} }\right)\right..\end{array}$$

(12)

The steady-state mean values can be obtain from above equation by simply putting time-derivative equal to zero and solving them for individual subsystems, given as,

$$\begin{array}{rcl}\langle a\rangle &=&-\frac{i{G}_{ma}+\Gamma {e}^{i\theta }}{i{\Delta }_{a}+{\kappa }_{a}}\langle m\rangle ,\\ \langle m\rangle &=&\frac{\eta -(i{G}_{ma}+\Gamma {e}^{i\theta })\langle a\rangle }{i{\Delta }_{m}+{\kappa }_{m}+{g}_{mb}({b}_{s}+{b}_{s}^{* })},\\ \langle b\rangle &=&-\frac{i{g}_{mb}| \langle m\rangle {| }^{2}}{i{\Delta }_{b}+{\gamma }_{b}}+\xi ,\end{array}$$

(13)

Since the magnon and cavity modes are strongly driven, resulting in large amplitude ∣〈m〉∣ ≫ 1, ∣〈a〉∣ ≫ 1, therefore, each Heisenberg operator can rewritten as a sum of its steady-state mean value and its corresponding quantum fluctuation $\hat{O}=\langle O\rangle +\delta \hat{O}(O=a,m,b)$, reading as,

$$\begin{array}{rcl}\dot{\delta }\hat{a}&=&-(i{\Delta }_{a}+{\kappa }_{a})\delta \hat{a}-(i{G}_{ma}+\Gamma {e}^{i\theta })\delta \hat{m}\\ &&+\sqrt{2{\kappa }_{a}}{\hat{a}}_{in},\end{array}$$

(14)

$$\begin{array}{rcl}\dot{\delta }\hat{m}&=&-(i{\Delta }_{m}+{\kappa }_{m})\delta \hat{m}-(i{G}_{ma}+\Gamma {e}^{i\theta })\delta \hat{a}\\ &&-i{G}_{mb}\hat{m}\delta \hat{b}+\sqrt{2{\kappa }_{m}}{\hat{m}}_{in},\end{array}$$

(15)

$$\dot{\delta }\hat{b}=-(i{\Delta }_{b}+{\gamma }_{b})\delta \hat{b}-i{G}_{mb}\delta \hat{m},$$

(16)

where Δ_m ≈ Δ_m + g_mb〈q〉 is the effective magnon-drive detuning including the frequency shift due to the magnomechanical interaction, ${G}_{mb}=\sqrt{2}{g}_{mb}{m}_{s}$ is the effective magnomechanical coupling rate. One can approximate the effective Hamiltonian matrix representation as following,

$${\hat{H}}_{eff}\approx \left(\begin{array}{lll}{\Delta }_{a}-i{\kappa }_{a}&{G}_{ma}-i\Gamma {e}^{i\theta }&0\\ {G}_{ma}-i\Gamma {e}^{i\theta }&{\Delta }_{m}-i{\kappa }_{m}&{G}_{mb}\\ 0&{G}_{mb}&{\Delta }_{b}-i{\gamma }_{b}\end{array}\right).$$

(17)

Here it should be noted that we are omitting vectors for bosonic field creation and annihilation operators for simplicity. Further, by inducing parity operator ${\mathscr{P}}$, which will lead to the transformation $\hat{a}\leftrightarrow -\hat{m}$, ${\hat{a}}^{\dagger }\leftrightarrow -{\hat{m}}^{\dagger }$, and time-reversal operator ${\mathscr{T}}$, causing the transformation $\hat{a},{\hat{a}}^{\dagger }\leftrightarrow \hat{a},{\hat{a}}^{\dagger }$, $\hat{m},{\hat{m}}^{\dagger }\leftrightarrow -\hat{m},{\hat{m}}^{\dagger }$ and i ↔ − i, which appears as the main transformation regarding our Hamiltonian. Furthermore, by using the mentioned transformation, which will lead to the satisfaction of the condition ${\hat{H}}_{eff}^{PT}=({\mathscr{PT}}){\hat{H}}_{eff}{({\mathscr{PT}})}^{-1}$, one can reach to the effective PT-symmetric Hamiltonian mentioned in Eq.(2)^36,37,38.

Routh-Hurwitz stability conditions

The linearized quantum Langevin equations describing the quadrature fluctuations ($\delta \hat{X},\delta \hat{Y},\delta \hat{x},\delta \hat{y},\delta \hat{q},\delta \hat{p}$) with $\delta \hat{X}=(\delta \hat{a}+\delta {\hat{a}}^{\dagger })/\sqrt{2}$, $\delta \hat{Y}=i(\delta {\hat{a}}^{\dagger }-\delta \hat{a})/\sqrt{2}$, $\delta \hat{x}=(\delta \hat{m}+\delta {\hat{m}}^{\dagger })/\sqrt{2}$, $\delta \hat{y}=i(\delta {\hat{m}}^{\dagger }-\delta \hat{m})/\sqrt{2}$, can be written as

$$\dot{u}(t)=Au(t)+n(t),$$

(18)

where $u(t)={[\delta \hat{X}(t),\delta \hat{Y}(t),\delta \hat{x}(t),\delta \hat{y}(t),\delta \hat{q}(t),\delta \hat{p}(t)]}^{{\rm{T}}}$,$n(t)={[\sqrt{2{\kappa }_{a}}{\hat{X}}_{in}(t),\sqrt{2{\kappa }_{a}}{\hat{Y}}_{in}(t),\sqrt{2{\kappa }_{m}}{\hat{x}}_{in}(t),\sqrt{2{\kappa }_{m}}{\hat{y}}_{in}(t),0,\xi (t)]}^{T}$ is the vector of input noises, and the drift matrix A is given by

$$A=\left(\begin{array}{llllll}-{\kappa }_{a}&{\Delta }_{a}&0&{G}_{ma}-i\Gamma {e}^{i\theta }&0&0\\ {\Delta }_{a}&-{\kappa }_{a}&-({G}_{ma}-i\Gamma {e}^{i\theta })&0&0&0\\ 0&{G}_{ma}-i\Gamma {e}^{i\theta }&-{\kappa }_{m}&{\Delta }_{m}&0&0\\ -({G}_{ma}-i\Gamma {e}^{i\theta })&0&-{\Delta }_{m}&-{\kappa }_{m}&-{G}_{mb}&0\\ 0&0&0&0&0&{\omega }_{b}\\ 0&0&-{G}_{mb}&0&-{\omega }_{b}&-{\gamma }_{b}\end{array}\right).$$

(19)

Later, we apply Routh-Hurwitz stability criteria on the above matrix ${\mathcal{A}}$ in order to govern the stability conditions for the magnomechanical system. Routh-Hurwitz stability criteria states that if any root of the characteristic polynomial of matrix ${\mathcal{A}}$ is on the left-half plan then the system will be unstable. By adopting and following that mechanism, we developed certain parametric stability conditions, given as,

$$\begin{array}{l}{\left({G}_{ma}-i\Gamma {e}^{i\theta }\right)}^{2}\left(2{\kappa }_{a}{\kappa }_{m}{\omega }_{b}+{\Delta }_{a}\left({G}_{mb}^{2}-2{\omega }_{b}{\Delta }_{m}\right)\right.\\ +\left.{\left({G}_{ma}-i\Gamma {e}^{i\theta }\right)}^{2}{\omega }_{b}\right)+\left({\kappa }_{a}^{2}+{\Delta }_{a}^{2}\right)\left(-{G}_{mb}^{2}{\Delta }_{m}\right.\\ +\left.{\omega }_{b}\left({\kappa }_{m}^{2}+{\Delta }_{m}^{2}\right)\right) > 0,\\ {\kappa }_{a}+{\kappa }_{m} > \frac{{\gamma }_{b}}{2},\\ {\Delta }_{m} > 0,\\ {\Delta }_{a} > 0.\end{array}$$

(20)

We strictly follow these conditions while computing results presented and discussed in main text.

Entanglement generation

Due to the linearized of the Langevin equation and the Gaussian nature of the quantum noise, the system will decay to a stable Gaussian state which can be completely characterized by a 6 × 6 covariance matrix in the phase space as follows ${\mu }_{ij}=\frac{1}{2}\langle {u}_{i}(t){u}_{j}(t^{\prime} )+{u}_{j}(t^{\prime} ){u}_{i}(t)\rangle (i,j=1,2,...,6)$. The steady-state covariance matrix μ can be obtained directly by solving the Lyapunov equation⁴⁷

$$A\mu +\mu {A}^{{\rm{T}}}=-D,$$

(21)

where D = diag[κ_a(2N_a + 1), κ_a(2Na + a), κ_m(2N_m + 1), κ_m(2N_m + 1), 0, γ_b(2N_b + a)], is the diffusion matrix which is defined by ${D}_{ij}\delta (t-t^{\prime} )=\langle {n}_{i}(t){n}_{j}(t^{\prime} )+{n}_{j}(t^{\prime} ){n}_{i}(t)\rangle /2$. To investigate bipartite and tripartite entanglement of the system, we use the logarithmic negativity E_n to quantify the Gaussian bipartite entanglement

$${E}_{n}=\max [0,ln(2{\eta }^{-})],$$

(22)

where ${\eta }^{-}\equiv {2}^{-1/2}{[\Sigma (V)-{(\Sigma {(V)}^{2}-4\det V)}^{1/2}]}^{1/2}$ with $\Sigma (V)\equiv \det {V}_{1}+\det {V}_{4}-2\det V2$ and here, V is the associated 4 × 4 covariance matrix of any two modes, which can be written as

$$V=\left(\begin{array}{ll}{V}_{1}&{V}_{2}\\ {V}_{2}^{{\rm{T}}}&{V}_{4}\end{array}\right).$$

(23)