Quantum-state engineering in cavity magnomechanics formed by two-dimensional magnetic materials

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Quantum-state engineering in cavity

magnomechanics formed by two-dimensional

magnetic materials

Chun-Jie Yang1, QingJun Tong2,∗, and Jun-Hong An3,∗

1School of Physics, Henan Normal University, Xinxiang 453007, China

2School of Physical Science and Electronics, Hunan University, Changsha 410082,

China

3Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University,

Lanzhou 730000, China

E-mail: tongqj@hnu.edu.cn and anjhong@lzu.edu.cn

Abstract. Cavity magnomechanics has become an ideal platform to explore

macroscopic quantum eﬀects. Bringing together magnons, phonons, and photons in a

system, it opens many opportunities for quantum technologies. It was conventionally

realized by an yttrium iron garnet, which exhibits a parametric magnon-phonon

coupling ˆm†ˆm(ˆ

b†+ˆ

b), with ˆmand ˆ

bbeing the magnon and phonon modes. Inspired

by the recent realization of two-dimensional (2D) magnets, we propose a cavity

magnomechanical system using a 2D magnetic material with both optical and magnetic

drivings. It features the coexisting photon-phonon radiation-pressure coupling and

quadratic magnon-phonon coupling ˆm†ˆm(ˆ

b†+ˆ

b)2induced by the magnetostrictive

interaction. A stable squeezing of the phonon and bi- and tri-partite entanglements

among the three modes are generated in the regimes with a suppressed phonon number.

Compared with previous schemes, ours does not require any extra nonlinear interaction

and reservoir engineering and is robust against the thermal ﬂuctuation. Enriching the

realization of cavity magnomechanics, our system exhibits its superiority in quantum-

state engineering due to the versatile interactions enabled by its 2D feature.

Keywords: cavity magnomechanics, two-dimensional magnet, quantum-state engineer-

ing

arXiv:2210.15519v2 [quant-ph] 17 Feb 2024

1. Introduction

Hybrid quantum systems with multiple degrees of freedom are widely used in exploring

fundamental physics and building novel functional quantum devices [1,2,3,4]. The

heart of these applications is the designing of coherent couplings between diﬀerent

degrees of freedom in these hybrid systems. Cavity magnomechanics has emerged as

an ideal platform to study the coherent interactions between photons, phonons, and

magnons [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Fascinating applications

have been envisioned, such as low-temperature thermometer [21], quantum memory

for photonic quantum information [22], and building block for long-distance quantum

network [23].

Conventional cavity magnomechanics is realized by yttrium iron garnets (YIGs),

where the phonon linearly couples to the magnon as a result of its isotropic

magnetostrictive interaction. However, the emergence of many quantum phenomena,

e.g., the mechanical bistability [24], squeezing generation [25], and nonreciprocal

magnetic transmission [26], require carefully engineered nonlinearity. Thus, current

cavity magnomechanical systems dramatically resort to complex “self-Kerr” nonlinearity

[27] or the squeezed reservoir injection [28]. Developing new platforms to realize

cavity magnomechanics with versatile magnon-phonon couplings is greatly desired for

their practical applications in quantum technologies [9]. Recently, atomically thin

two-dimensional (2D) materials have become an exciting platform for exploring low-

dimensional physics and functional devices [29,30]. With the advent of 2D magnets

[31,32,33,34,35,36,37,38,39], it is now possible to add the magnon degree of

freedom in these atomically thin mechanical systems [40,41,42]. Building such a 2D

hybrid optical, magnonic, and mechanical system has immediate advantages over the

existing cavity magnomechanical systems based on the YIG. First, a 2D magnet has an

out-of-plane ﬂexural phonon mode that may exhibit possible high-order coupling to the

magnon due to its highly anisotropic magnetostrictive interactions [43,44,45], which

is key to the quantum-state engineering based on cavity magnomechanics. Second, a

mechanical oscillator made of a 2D magnet is sensitive to external forces due to its low

mass [46,47,48,49,50], which induces a photon-phonon radiation pressure absent in

the existing cavity magnomechanics [51,52]. Therefore, 2D magnetic materials may

open another avenue to realize cavity magnomechanics.

We propose such a cavity magnomechanical system using a 2D magnetic

material with both optical and magnetic drivings. A quantized description reveals

that this hybrid system has a combined parametric optomechanical and quadratic

magnomechanical interactions. The unique photon-phonon-magnon interaction endows

our system with the distinguished role in quantum-state engineering. We ﬁnd that a

stable phonon squeezing, and bi- and tri-partite entanglement among the three modes

are generated in the regimes with a suppressed phonon number. Steming from the unique

magnomechanical coupling in 2D magnets, the generation of these quantum eﬀects

requires neither the “self-Kerr” nonlinearity nor the squeezed-reservoir engineering.

laser

(a)

𝛾𝑐

𝜔𝑐

𝛾𝑚

𝜔𝑚

𝛾𝑏

𝜔𝑏

𝜒

(b)

Ω𝑐/𝜔𝑏

Δ𝑐/𝜔𝑏

|𝐺𝑏𝑚|

|𝐺𝑏𝑐|

Ω𝑐Ω𝑚

(c)

Figure 1. (a) Schematics of a cavity magnomechanical system: An optically driven

cavity interacted with a magnetic membrane under a magnetic driving. (b) Interactions

among the photons, phonons, and magnons with frequencies ωoand damping rates γo

(o∈ {c, b, m}). A parametric ampliﬁcation to the phonon mode with strength χis

induced. The photon-phonon coupling Gbc and the magnon-phonon coupling Gbm are

enhanced by the optical and magnetic drivings.(c) |Gbm/Gbc |as a function of ∆cand

Ωc. The parameters are from Fig. 2(b).

More importantly, due to the accompanied suppressed phonon number, our scheme

is robust against thermal noise, which exhibits a superiority over the conventional ones.

2. Model Hamiltonian

We consider a hybrid system of cavity optomechanics and 2D magnetic material. The

system consists of an optically driven cavity interacted with a 2D magnetic membrane

[see Fig. 1(a)]. Its spin interactions induce a collective wave, which couples to

the mechanical deformation of the magnet. The induced magnetoelastic energy is

E=σ

SRdV Pαβ BαβMαMβUαβ(r), where σ=N/V is the number density of the

magnetic atoms with Nand Vbeing the number of magnetic atoms and the volume

of the magnet, MSis the saturation magnetization, Mα(α=x, y, z) are the local

magnetization, Bαβ are magnetoelastic coupling constants, and Uαβ (r)=[∂βUα(r) +

∂αUβ(r)+Pγ∂αUγ(r)∂βUγ(r)]/2 are the strain tensors for the lattice displacement U(r)

[53,54]. The second-order term in Uαβ(r) is negligible in 3D systems (like YIG), which

leads to a radiation-pressure-like linear magnon-phonon coupling [5]. However, the 2D

nature of a magnetic membrane creates a unique ﬂexural phonon mode, which makes the

second-order term important [24]. Assuming Uxx(r) = Uyy (r) and Fourier transforming

Uαβ(r), the energy regarding ﬂexural magnon-phonon coupling reads (see Appendix A)

E=σB1

2M2

(M2

S− M2

z)k2

x|˜

Uz(k)|2

+σB2

2M2

Sh(MxMy+MyMx)kxky|˜

Uz(k)|2,(1)

where B1=Bαα and B2=Bαβ (α̸=β). Introducing the phonon ˜

Uz(k) =

[ℏ/(2m0ωb)]1/2(ˆ

b+ˆ

b†) and the magnon Mα= 2µBσˆ

Sα, with ˆ

Sx= 2−1/2S1/2( ˆm+ ˆm†),

Sy=−i2−1/2S1/2( ˆm−ˆm†) and ˆ

Sz=S−ˆm†ˆm, the ﬁrst term regarding to the dispersive

interaction reduces to

Hbm/ℏ=gbm ˆm†ˆm(ˆ

b†+ˆ

b)2,(2)

with the coupling strength gbm =B1

2m0ωb, where m0is the ion mass, ωband kxare

the resonance frequency and wave vector of the mechanical mode. The second term of

Eq. (1) is negligible when the resonance frequency of the phonon is much smaller than

that of the magnon [5]. The quadratic magnon-phonon coupling in Eq. (2) is absent

in 3D magnets, which is associated with the 2D nature of the membrane. The photon

also exerts a radiation pressure on the magnetic membrane, which triggers the photon-

phonon coupling ˆ

Hbc/ℏ=gbcˆc†ˆc(ˆ

b†+ˆ

b) (see Appendix B). Our system not only gives a

novel realization of the rapidly developing cavity magnomechanics, but also generalizes

the conventional systems into a quadratic magnon-phonon coupling regime. We note

that this non-linear interaction may lead to parametric instability or chaotic dynamics in

evolution of the coupled system [55,56], and can also be used to engineer non-Gaussian

states of the magnons and phonons [57].

In the rotating frame with ˆ

H0=ℏωdcˆc†ˆc+ℏωdm ˆm†ˆm, the total Hamiltonian in the

presence of both optical and magnetic drivings reads

H/ℏ= ∆cˆc†ˆc+ ∆mˆm†ˆm+ωbˆ

b†ˆ

b+gbcˆc†ˆc(ˆ

b+ˆ

b†)

+gbm ˆm†ˆm(ˆ

b†+ˆ

b)2+ (Ωcˆc†+ Ωmˆm†+ H.c.).(3)

Here ˆcis the annihilation operator of the cavity with frequency ωc,gbc is the photon-

phonon coupling strength, and ∆c/m =ωc/m −ωdc/dm are the photon and magnon

detunings to their driving frequencies ωdc/dm. The Rabi frequencies of the driving ﬁelds

on the cavity and magnet are Ωm=γB0√2NS/4 [58] and Ωc=pP γc/ℏωdc [59], where

γ/2π= 28 GHz/T is the gyromagnetic ratio, B0is the amplitude of the drive magnetic

ﬁeld, Pis the input laser power, and γcis the damping rate of the cavity mode. We

have eliminated the direct magnon-photon interaction in the model, as the coupling

strength is rather weak due to the signiﬁcantly reduced number of spins in 2D system.

The dynamics is governed by the master equation

W(t) = i

ℏ[W(t),ˆ

+ [γcˇ

Lˆc+γmˇ

Lˆm+γb¯n0ˇ

Lˆ

b†+γb(1 + ¯n0)ˇ

Lˆ

b]W(t),(4)

where W(t) is the density matrix of the three-mode system, ˇ

Lˆo·= 2ˆo·ˆo†− {ˆo†ˆo, ·} is

the Lindblad superoperator, γoare the damping rates of the three bosonic modes, and

¯n0= [exp(ℏωb/kBT)−1]−1is the mean thermal excitation number of the environment

felt by the phonon.

The strong optical and magnetic drivings make the steady-state occupations of

the three modes have large amplitudes. This allow us to linearize the Hamiltonian

on one hand and enhance both of the magnon-phonon and photon-phonon couplings

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摘要：

Quantum-stateengineeringincavitymagnomechanicsformedbytwo-dimensionalmagneticmaterialsChun-JieYang1,QingJunTong2,∗,andJun-HongAn3,∗1SchoolofPhysics,HenanNormalUniversity,Xinxiang453007,China2SchoolofPhysicalScienceandElectronics,HunanUniversity,Changsha410082,China3KeyLaboratoryofQuantumTheoryandApp...

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Quantum-state engineering in cavity magnomechanics formed by two-dimensional magnetic materials

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