Abstract
We theoretically propose a scheme for generating nonreciprocal macroscopic bipartite and tripartite entanglement using the Barnett effect in cavity-magnon optomechanics. The system consists of an optomechanical cavity and a rotatable yttrium iron garnet (YIG) sphere. Our results indicate that under appropriate parameter conditions, both bipartite entanglement and genuine tripartite entanglement can be generated between the cavity mode, mechanical mode, and magnon mode. Moreover, when the YIG sphere rotates, adjusting the magnetic field direction can induce a positive or negative Barnett shift, which leads to the nonreciprocity of entanglement, where entanglement exists in one chosen magnetic field direction and disappears in the other. Meanwhile, the macroscopic tripartite entanglement in the system is robust against thermal noise. Our work provides a possible avenue for quantum information processing, quantum chiral device integration, and multi-node quantum networks construction.
Subject terms: Quantum physics, Quantum information, Quantum optics
Introduction
In recent years, ferromagnetic materials exemplified by yttrium iron garnet (YIG) have been extensively studied due to their high spin density and low damping rate. This has established cavity magnomechanical systems as excellent platforms for exploring various quantum phenomena, quantum information processing, and quantum network construction1–3. With further research, many interesting quantum phenomena have been demonstrated in cavity magnomechanical systems, such as ground state cooling4, magnon blockade5–7, quadrature squeezing8,9, macroscopic entanglement10, and Einstein–Podolsky–Rosen (EPR) steering11,12.
Quantum entanglement is one of the cornerstones of quantum mechanics, which describing the inseparability between particles. This nonclassical correlation is one of the essential differences between quantum mechanics and classical mechanics. This special property makes entanglement widely used in quantum communication13, quantum computation14, quantum teleportation15, and many other fields. Recently, many schemes have been proposed to generate steady-state entanglement in various quantum systems, such as reservoir engineering16–19, coherent feedback20–22, dark mode engineering23,24, and optical parametric amplifiers25,26. Moreover, magnon-based microwave-optical entanglement27,28 and tripartite entanglement between phonon-magnon-photon29 have also been implemented in cavity magnomechanical systems.
On the other hand, nonreciprocity describes the unidirectional invisibility of light in its propagation, that is, light can pass in one direction while being blocked in the other direction. This makes nonreciprocal physics have important significance in quantum communication, complex quantum network construction, and so on30,31. In recent years, the Sagnac effect induced by spin resonators has been extensively employed to study various nonreciprocal physical phenomena, including nonreciprocal entanglement and steering32–37, nonreciprocal squeezing38–40, and nonreciprocal photon blockade41,42. At the same time, experimental research on the Sagnac effect has also achieved significant success43. In addition, the Kerr effect of the magnon has also been shown to be able to generate nonreciprocity by changing the direction of the bias magnetic field44–46, and nonreciprocal tripartite entanglement has been successfully achieved47. Interestingly, recent studies have proposed a scheme for generating magnon blockade through the Barnett effect48. This represents a significant advance in the study of nonreciprocal physics and could be used to further explore nonreciprocal entanglement between macroscopic objects.
Inspired by previous studies, here we present a theoretical scheme for generating nonreciprocal macroscopic tripartite entanglement in a cavity magnomechanical system via the Barnett effect. The system consists of an optomechanical cavity and a YIG sphere placed in the cavity, where the photon-magnon and optomechanical couplings are induced by magnetic dipole interactions and radiation pressure interactions, respectively. It has been shown that by choosing the appropriate system parameters, photon, magnon, and phonon can become entangled with each other and generate genuine tripartite entanglement. Moreover, when the YIG sphere rotates, we can generate a positive or negative Barnett shift by changing the direction of the magnetic field. This leads directly to the nonreciprocal entanglement in the system, where entanglement exists in one magnetic field direction but disappears in the other. Finally, we present a more nuanced discussion of the nonreciprocity of tripartite entanglement through the chiral factor, and investigate the robustness of bipartite and tripartite entanglement against thermal noise, as well as the effect of mechanical damping rates on entanglement.
The remainder of this paper is organized as follows. Firstly, we introduce the model of this paper and its corresponding Hamiltonian. Secondly, we study the quantum Langevin equations and linearized dynamics of the system. Furthermore, we study bipartite and tripartite entanglement in the system and explore the nonreciprocity of entanglement. Finally, we present a brief conclusion.
Methods
Model and Hamiltonian
In this paper, we present a hybrid quantum system consisting of a YIG sphere and an optomechanical cavity. As shown in Fig. 1, a rotatable YIG sphere is placed in the optomechanical cavity and is fully magnetizable by a static magnetic field 
along the z direction. This allows the frequency of the magnon to be regulated by the static magnetic field 
, which can be expressed as 
with the gyromagnetic ratio 
. When a ferromagnetic object rotates around a fixed axis, the spin and orbital magnetic moments of the electrons inside it show a significant orientation preference along the direction of the rotation axis, thus inducing magnetization of the ferromagnetic object. This physical phenomenon is known as the Barnett effect. Based on this mechanism, when YIG sphere rotates around the z-axis with an angular velocity 
, it generates an emergent magnetic field in the direction of the z-axis, which can be expressed as: 
. This magnetic field is known as the Barnett field and causes a shift in the magnon frequency49–53. In this case, the magnon frequency can be expressed as 
. Experimentally, we can fix the rotation direction of the YIG sphere and adjust the frequency shift either positively or negatively by changing the direction of the static magnetic field 
, which corresponds to 
(
) when the static magnetic field is positive (negative) along the z-axis. Different magnon frequency shifts will affect the generation of entanglement in the system, which is the core mechanism of nonreciprocal entanglement. In the rotating frame defined by exp
, the Hamiltonian of the cavity magnomechanical system can be written as
 |
1 |
where a (
), m (
), and b (
) are the annihilation (creation) operators of the cavity mode, magnon mode, and mechanical mode, respectively. 
and 
are the drive detuning of the cavity mode and magnon mode, respectively. The cavity and magnon modes are coupled via the magnetic dipole force with coupling strength 
. And 
denotes the single optomechanical coupling strength. The Rabi frequency 
represents the coupling strength of the magnon mode with the driving field.
Fig. 1.
Schematic of the cavity magnomechanical system, a YIG sphere that can be rotated about the z-axis is placed in the cavity and driven by a microwave driving field of frequency 
. The cavity mode, with frequency 
, is coupled to the magnon mode (frequency 
) through magnetic dipole interaction, and to the mechanical mode (frequency 
) via radiation pressure interaction.
Quantum Langevin equations and linearized dynamics
In this section, we study the quantum Langevin equations and linearized dynamics of the system. Based on the Eq. (1), the quantum Langevin equations of the system can be read as
 |
2 |
where 
and 
are the decay rates of the cavity mode and magnon mode respectively, and 
is the damping rate of the mechanical mode. The operators 
, 
, and 
are input noise operators characterized by the following correlation functions:
 |
3 |
where 
represents the average thermal phonons number associated with mechanical mode. Next, we will linearize the Langevin equation by rewriting the operator. We set all operators as 
, 
is the steady-state average value, and 
is the fluctuation operator. In this case, we can rewrite Eq. (2) as
 |
4 |
where 
is the normalized driving detuning of the cavity mode, and 
is the effective optomechanical coupling strength between the cavity mode and mechanical mode. In order to further study the bipartite and tripartite entanglement in the system, we introduce the orthogonal operator: 
and 
, and the corresponding quadrature operators of input quantum noise 
and 
. Then we can rewrite Eq. (4) as
 |
5 |
where
 |
6 |
and
 |
7 |
The coefficient matrix A can be written as
 |
8 |
It is important to note that the stability of the linearized optomechanical system can be measured by the Routh-Hurwitz criterion. To ensure that the system can evolve to a steady state, the chosen parameters are in the stability region. Due to the linearized dynamics and Gaussian nature of the input quantum noise, the steady state of the system is zero-mean Gaussian state. It can be represented by the 6 
6 covariance matrix 
, defined as
 |
9 |
And the covariance matrix 
satisfies the Lyapunov equation54
 |
10 |
where 
is the diffusion matrix, it can be written as 
.
An arbitrary target two-mode Gaussian state in the system can be described by a covariance matrix in the form of a 4 × 4 block matrix, it could be written as
 |
11 |
On this basis, we can quantify bipartite entanglement by introducing logarithmic negativity 
55. It can be expressed as follows
 |
12 |
where
 |
13 |
with
 |
14 |
Moreover, we quantify the tripartite entanglement among cavity mode, magnon mode, and mechanical mode in the system via minimal residual contangle 
56. It can be represented as
 |
15 |
When the nonzero minimum residual contangle 
, it means that genuine tripartite entanglement is generated in the system.
Results
Nonreciprocal macroscopic entanglement
In this section, we will examine in detail nonreciprocal bipartite and tripartite entanglement in the system. In Fig. 2, we discuss the effects of coupling strength 
and magnon detuning 
on the bipartite and tripartite entanglement assuming that the YIG sphere is not rotating (
). Since in this scheme, we assume that the system is working in the resolved sideband condition and the strong coupling condition is satisfied between the cavity mode and the mechanical mode. Therefore, we can select the following experimentally feasible parameters while satisfying the stability conditions 57: 
, 
, 
, 
, 
, and 
. Meanwhile, for better generation and distribution of entanglement, we choose: 
. From Fig. 2, we can found that 
changes with 
in a completely opposite trend to 
and 
. This phenomenon indicates that optomechanical entanglement, as a quantum resource, is partially distributed to other subsystems, enabling genuine tripartite entanglement among cavity mode, magnon mode, and mechanical mode, as demonstrated in Fig. 2d. This phenomenon can be explained by the effective Hamiltonian, under strong coupling conditions, it can be written as:
 |
16 |
In Eq. (16), the first term is the parametric downconversion type Hamiltonian, which generates entanglement between the cavity and mechanical modes. The latter two terms are the beam-splitter type Hamiltonian, and the optomechanical entanglement generated via the parametric downconversion interaction can be partially distributed to the other two subsystems through this state-exchange interaction. This transfer of entanglement enables genuine tripartite entanglement to occur in the system. Additionally, the beam-splitter type interaction between cavity mode and mechanical mode can also significantly cool the mechanical mode, which helps to improve the robustness of the system. Interestingly, we note that all entanglements can be maximized near 
. The optomechanical entanglement 
, photon-magnon entanglement 
, magnon-phonon entanglement 
, and tripartite entanglement 
. This phenomenon can be attributed to the decisive role played by the magnon in the process of entanglement assignment in the system when it is in resonance with the Stokes sideband. Similar mechanisms have been demonstrated and used in previous studies to realize atom-photon-phonon entanglement58.
Fig. 2.
Logarithmic negativity (a) 
, (b) 
, (c) 
, and (d) minimum residual contangle 
as functions of coupling strength 
and magnon detuning 
. The selected system parameters are visible in the main text.
In a previous study, we discussed bipartite and tripartite entanglement in the system when the YIG sphere is not rotating. In this case, the system is reciprocal. However, when the YIG sphere rotates, the resulting Barnett field causes the magnon frequency to shift. By changing the direction of the static magnetic field 
, we are able to determine whether the frequency shift is positive or negative. This effect is the source of the nonreciprocity of entanglement in the system. Next, we will investigate the nonreciprocal entanglement in the system through numerical simulations. In Fig. 3, we study the influence of magnon detuning on bipartite and tripartite entanglement in the system under three different cases of 
, 
, and 
, respectively. These three cases correspond to the blue, black, and red dotted lines in Fig. 3. As shown in Fig. 3a–c, when the static magnetic field is reversed along the z-axis direction (i.e., 
), the optomechanical entanglement 
, photon-magnon entanglement 
, and magnon-phonon entanglement 
all peak near 
, with maxima of 0.17, 0.18, and 0.16, respectively. It is worth noting that the entanglement in the other magnetic field direction disappears completely under this condition. In addition, the tripartite entanglement in Fig. 3d shows similar properties: when 
, the tripartite entanglement peaks at 
with a maximum value of 
, and it disappears completely in the opposite magnetic field direction. Thus, we demonstrate that ideal nonreciprocal entanglement can be generated through this system.
Fig. 3.
When the YIG sphere rotates, logarithmic negativity (a) 
, (b) 
, (c) 
, and (d) minimum residual contangle 
as functions of magnon detuning 
. Here we choose 
, and the other parameters remain the same as those given in Fig. 2.
In Fig. 4, we further study the effect of the Barnett shift 
on the nonreciprocity of tripartite entanglement. Moreover, we also discuss the robustness of the system and the influence of mechanical damping rate on entanglement. In Fig. 4a, we see that as 
increases, if 
, the entanglement will then increase monotonically, while for 
, the entanglement will decay rapidly to 0. To quantify the nonreciprocity of tripartite entanglement in a system, we introduce chiral factor here, which can be written as:
 |
17 |
The chiral factor is a number greater than 0 and less than or equal to 1. In this interval, a larger value of 
represents a stronger nonreciprocity. When 
, the system can achieve complete nonreciprocity. In Fig. 4b, we study the effect of magnon detuning 
and Barnett shift 
on the nonreciprocity of tripartite entanglement using the chiral factor 
. We can find that the chiral factor 
increases gradually as the Barnett shift 
increases. When 
takes the proper value and 
, the chiral factor 
can reach the maximum value (
). This proves that nonreciprocal macroscopic tripartite entanglement can be realized in this system. We next investigate the robustness of entanglement to thermal noise in the system and the effect of the mechanical damping rate 
on entanglement. The blue, black, red, and green dotted lines in Fig. 4c, d represent 
, 
, 
, and 
, respectively. In Fig. 4c we can find that all entanglement will weaken as the average number of thermal phonons 
increases. We see that optomechanical entanglement 
, which is most sensitive to thermal noise, can still exist in a place for 
. Moreover, 
and 
with better robustness can still exist when 
and 
, respectively. At the same time, the macroscopic tripartite entanglement between the three modes can also be maintained at 
. This means that both bipartite and tripartite entanglement in the system have excellent robustness to thermal noise, which is due to the cooling effect of cavity mode on mechanical mode. We also plot entanglement as a function of the mechanical damping rate 
in Fig. 4d. We can obviously observe that when 
, the entanglement will not change obviously, but when the damping rate 
continues to increase, the entanglement will experience significant decay. However, the macroscopic tripartite entanglement can still exist under a high damping rate, even if 
. Furthermore, all bipartite entanglements exist within the range of 
.
Fig. 4.
(a) The relationship between nonreciprocal tripartite entanglement and the Barnett shift 
. (b) The chiral factor as a function of the magnon detuning and Barnett shift 
, and the effect of (c) the average thermal phonon number 
and (d) the mechanical damping rate 
on entanglement in the system.
Finally, we will briefly discuss the experimental feasibility of the present scheme and the potential applications of nonreciprocal entanglement. In view of recent research progress, strong optomechanical coupling within the parameter range 
as well as optomechanical cavities with higher quality factors have been experimentally verified in a variety of optomechanical systems59–61. Based on these results, we can choose the following experimentally feasible parameters: 
, 
, 
, and 
. In addition, the magnitude of the Barnett shift is directly related to the rotational angular velocity of the YIG sphere. Recent experimental studies have shown that the angular velocity of the YIG sphere can reach the order of 
by using the levitated crystal62,63. Meanwhile, theoretical studies have shown that, based on the Einstein-De Haas effect, the angular velocity of the YIG sphere is expected to be further advanced to 10 
49. However, the high rotation speed of the YIG sphere may lead to an increase in temperature52,64, which can be solved by placing the system in a low-temperature vacuum environment. Another point to note is that the inherent Kerr effect of the magnon may have some influence on the bipartite and tripartite entanglement in the system. This problem can be solved by choosing a suitable size of the YIG sphere. It has been referenced that the Kerr coefficient magnitude is extremely weak (
) and can be approximated to be negligible when the diameter of the YIG sphere is about 250 
65. Therefore, the experimental feasibility of the present scheme can be ensured by applying YIG spheres of this size. Further, the model proposed in this paper has significant potential for practical applications. In this scheme, nonreciprocal bipartite and tripartite quantum entanglement between the magnon, phonon and photon is successfully realized by exploiting the Barnett effect. Remarkably, this entanglement arises only in a specific direction and disappears completely in the opposite direction. This unique directional property makes nonreciprocal entanglement valuable for applications in quantum information processing tasks, such as quantum teleportation66,67, quantum transduction68 and quantum sensing69. In addition, since the nonreciprocal quantum correlation between macroscopic devices can be used to realize unidirectional quantum information transmission, this scheme also shows potential applications in the construction of multi-node chiral quantum networks70,71.
Discussion
In conclusion, we have presented a scheme to realize nonreciprocal macroscopic tripartite entanglement in cavity-magnon optomechanics via the Barnett effect. By selecting appropriate system parameters, it is possible to achieve all bipartite entanglements among the cavity mode, magnon mode, and mechanical mode, as well as genuine tripartite entanglement. However, when the YIG sphere rotates, we can control the direction of the magnetic field to produce a positive or negative Barnett shift. In this case, the nonreciprocal macroscopic tripartite entanglement between three modes can be generated. In addition, we find that all entanglement are robust against thermal noise. Our study not only provides a possible approach for multipartite entanglement between macroscopic devices, but also has potential applications in several aspects, such as quantum information processing and the preparation of thermal noise-resistant quantum resources.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants Nos. 62101479, 62475226, 62071412, 12074330, and 12375020; the Natural Science Foundation of Jilin Province under Grant No. 20240101013JC; and the Young Talents Support Project of Association of Science and Technology of Jilin Province under Grant No. QT202425.
Author contributions
This paper was co-authored by Ping-Chi Ge, Yikyung Yu, Hao-Tian Wu, Xue Han, Hong-Fu Wang, and Shou Zhang. Ping-Chi Ge was responsible for drafting the full text, Yikyung Yu conducted numerical modeling and computation. Hao-Tian Wu provided expertise in the theoretical analysis. Xue Han, Hong-Fu Wang, and Shou Zhang supervised the project. All authors actively discussed the results and contributed to the final version of the manuscript.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Xue Han, Email: xuehan@ybu.edu.cn.
Hong-Fu Wang, Email: hfwang@ybu.edu.cn.
Shou Zhang, Email: szhang@ybu.edu.cn.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.