(2021) Einstein-Podolsky-Rosen Entanglement and Asymmetric

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PHYSICAL REVIEW RESEARCH 3, 013192 (2021)

Einstein-Podolsky-Rosen entanglement and asymmetric steering between distant macroscopic mechanical and magnonic systems

Huatang Tan1,* and Jie Li2,3,† 1Department of Physics, Huazhong Normal University, Wuhan 430079, China 2Zhejiang Province Key Laboratory of Quantum Technology and Device, Department of Physics and State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China 3Kavli Institute of Nanoscience, Department of Quantum Nanoscience, Delft University of Technology, 2628CJ Delft, The Netherlands

(Received 15 October 2020; accepted 7 February 2021; published 26 February 2021)

We propose a deterministic scheme for establishing hybrid Einstein-Podolsky-Rosen (EPR) entanglement channel between a macroscopic mechanical oscillator and a magnon mode in a distant yttrium-iron-garnet (YIG) sphere across about 10 GHz of frequency difference. The system consists of a driven electromechanical cavity which is unidirectionally coupled to a distant electromagnonical cavity inside which a YIG sphere is placed. We ﬁnd that far beyond the sideband-resolved regime in the electromechanical subsystem, stationary phonon-magnon EPR entanglement can be achieved. This is realized by utilizing the output ﬁeld of the electromechanical cavity being an intermediary which distributes the electromechanical entanglement to the magnons, thus establishing the remote phonon-magnon entanglement. The EPR entanglement is strong enough such that phonon-magnon quantum steering can be attainable in an asymmetric manner. This long-distance macroscopic hybrid EPR entanglement and steering enable potential applications not only in fundamental tests of quantum mechanics at the macroscale, but also in quantum networking and one-sided device-independent quantum cryptography based on magnonics and electromechanics.

DOI: 10.1103/PhysRevResearch.3.013192

I. INTRODUCTION [1], due to their great use for, e.g., testing the validity of quantum mechanics [10,11] and probing decoherence theo- Long-distance entanglement has attracted extensive atten- ries [12,13] at large mass scales. In cavity optomechanics tion owing to its potential applications to the fundamental and electromechanics, which involve the hybrid coupling test of quantum mechanics [1], quantum networking [2,3], of massive mechanical resonators to a electromagnetic ﬁeld and quantum-enhanced metrology [4]. In such quantum [14,15], recent experiments have succeeded in preparing a tasks, hybrid quantum systems, composed of distinct physi- variety of quantum states of macroscopic mechanical oscilla- cal components with complementary functionalities, possess tors [16–23], including quantum squeezing and entanglement multitasking capabilities and thus may be better suited than of mechanical oscillators [16–19], nonclassical correlations others for speciﬁc tasks [5]. It has been proved that light between photons and phonons [20,21], single-phonon Fock mediation is an effective approach to achieving the entangle- states [22], optomechanical Bell nonlocality [23], etc. These ment between two remote systems that never interact directly macroscopic quantum states can also be useful in the pro- [6–8]. For example, hybrid Einstein-Podolsky-Rosen (EPR) cessing and communication of quantum information [24] and entanglement between distant macroscopic mechanical and ultrahigh precision measurement beyond the standard quan- atomic systems has been realized very recently via unidirec- tum limit [25]. In addition, hybrid interfaces of mechanical tional light coupling [8]. In addition, strong coherent coupling oscillators with other systems such as atomic ensembles [8,9], between a mechanical membrane and atomic spins has been nitrogen vacancy centers [26], and superconducting devices demonstrated using two cascade light-mediated coupling pro- [27] have already been realized [28]. cesses [9]. Apart from optomechanical and electromechanical sys- On the other hand, the realization of nonclassical effects of tems, hybrid systems based on magnons in macroscopic macroscopic objects is an ongoing effort in quantum science magnetic materials have increasingly become a new and promising platform for studying macroscopic quantum effects, attributed to magnons’ great frequency tunability, *[email protected] very low damping loss, and excellent coupling capabil- †[email protected] ity to microwave or optical photons, phonons, and qubits [29]. Experiments have realized strong cavity-magnon cou- Published by the American Physical Society under the terms of the pling [30–33] in yttrium-iron-garnet (YIG) spheres and other Creative Commons Attribution 4.0 International license. Further related interesting phenomena, such as magnon gradient distribution of this work must maintain attribution to the author(s) memory [34], exceptional point [35], bistability [36], and non- and the published article’s title, journal citation, and DOI. reciprocity and unidirectional invisibility in cavity magnonics

2643-1564/2021/3(1)/013192(8) 013192-1 Published by the American Physical Society HUATANG TAN AND JIE LI PHYSICAL REVIEW RESEARCH 3, 013192 (2021)

electromagnonical cavity, a ferrimagnetic YIG sphere with a diameter about hundreds of micrometers is placed. The YIG sphere is also put in a uniform bias magnetic field, giving rise to magnetostatic modes of spin waves in the sphere. The cavities pb,m, with their output fields sent to measurement apparatuses are used to read out the phonon and magnon states, respectively, and the two cavities (a2, pm) can be a cross-shaped cavity coupled to the YIG sphere glued on the end of a cantilever [52]. The cavity electromechanical system can be considered as an aluminum membrane micromechanical oscillator coupled to two microwave cavity modes [53]. The two cavities (a1,2) can be connected via two microwave FIG. 1. Schematic diagram. A driven electromechanical cav- circulators and coaxial cable to enable directional transfer of ity (a1) is unidirectionally coupled to an electromagnonical cavity information from cavity a1 to cavity a2 [6]. The magnons, (a2) where a YIG sphere in a uniform magnetic field is placed. which characterize quanta of the uniform magnetostatic mode The probing cavities pb and pm are used to read out and detect (i.e., Kittel mode [54]) in the YIG sphere, are coupled to the phonon-magnon entangled state. The two-cavity configuration the cavity mode via magnetic dipole interaction. Including , (a2 pm) of the microwave optomagnonic system can be a cross- the driving of the first cavity, we can write the interaction shaped cavity [52], and the YIG sphere is glued on the end of a Hamiltonian of the whole system, given by (¯h = 1) cantilever. The cavity electromechanical subsystem can be considered as an aluminum membrane micromechanical oscillator coupled Hˆ = ω aˆ†aˆ + ω bˆ†bˆ +g˜ aˆ†aˆ (bˆ+bˆ†) to two microwave cavity modes [53]. ab 1 1 1 b ab 1 1 ∗ ω − ω − E i d t −E i d t † , i( d e aˆ1 d e aˆ1 ) (1a) [37]. Moreover, entanglement-based single-shot detection of a ˆ = ω † +ω † + † + + † , Ham 2aˆ2aˆ2 mmˆ mˆ gam(â2 aˆ2 )(m ˆ mˆ ) (1b) single magnon with a superconducting qubit has been realized

[38]. Recent schemes for achieving quantum phenomena of where the bosonic annihilation operatorsa ˆ j,ˆm, and bˆ denote magnons, including squeezing and entanglement, quantum the cavity, magnon, and mechanical modes with frequencies steering, magnon blockade, and magnon-mediated microwave ω j, ωm, and ωb, respectively. The magnon frequency ωm = entanglement, have already been proposed [39–51]. βHB, where β is the gyromagnetic ratio and HB is the strength Here we consider a microwave-mediated phonon-magnon of the uniform bias magnetic field. The magnon frequency is interface and focus on how to deterministically establish thus adjustable in a large range by changing the strength HB. hybrid EPR entanglement channel between a macroscopic gãb represents the single-photon electromechanical coupling, distant mechanical oscillator and a YIG sphere across about and gam denotes the magnetic-dipole√ coupling between the 10 GHz of frequency difference. The system consists of cavity and magnons, gam ∝ N with N being the number of two unidirectionally coupled electromechanical and electro- spins. In the recent experiment [31], a strong coupling, gam ∼ magnonical cavities inside which a mechanical oscillator and 47 MHz, much larger than the cavity and magnon line widths a YIG sphere are placed, respectively. We find that far beyond about 2.7 MHz and 1.1 MHz, has been achieved. Typically, the electromechanical sideband-resolved regime, strong sta- the coupling gam {ω2,ωm}∼ GHz, and the microwave tionary phonon-magnon EPR entanglement and steering can † + + † photon-magnon coupling term gam(â2 aˆ2 )(m ˆ mˆ ) can be be achieved, as a result of electromechanical output photon- † † approximated into gam(â mˆ + aˆ mˆ ) by discarding the fast os- phonon entanglement distributed via the unidirectional cavity 2 2 cillating termsa ˆ mˆ + aˆ†mˆ †. The electromechanical coupling coupling. Further, the one-way steering from phonons to 2 2 g˜ is typically weak, but it can be enhanced by the strong magnons is established and adjustable over a wide range ab cavity drive with the amplitude E and frequency ω . Ultra- of feasible parameters. The entanglement and steering are d d strong electromechanical coupling in the linear regime has robust against the frequency dismatch between the two cav- been reported [55]. In the rotating frame with respect to the ities, unidirectional cavity-coupling loss, and environmental frequency ω , the Hamiltonian becomes temperature. d This paper is organized as follows: In Sec. II the system Hˆ = δ aˆ†aˆ + ω bˆ†bˆ +g˜ aˆ†aˆ (bˆ+bˆ†)−i(E∗aˆ −E aˆ†), (2a) and working equations are presented. In Sec. III the measures ab 1 1 1 b ab 1 1 d 1 d 1 of entanglement and steering are briefly reviewed. In Sec. IV ˆ = † + † + † + † , Ham 2aˆ2aˆ2 mmˆ mˆ gam(â2mˆ aˆ2mˆ ) (2b) the results are presented in detail. In Sec. V some discussion and the conclusion are given. where the detunings δ1 = ω1 − ωd , 2 = ω2 − ωd , and m = ωm − ωd . Under a strong drive, Eq. (2) can be linearized by re- → + = , , II. MODEL placing the operators byo ˆ oˆ ss oˆ (o a j b m), where oˆ ss denotes the steady-state amplitudes of the modes, leading We consider a driven electromechanical cavity that is to the linearized Hamiltonian unidirectionally coupled to an electromagnonical cavity, as shown in Fig. 1. For the electromechanical cavity, the cavity ˆ lin = † + ω ˆ† ˆ + + † ˆ + ˆ† , Hab 1aˆ1aˆ1 bb b gab(â1 aˆ1 )(b b ) (3a) resonance is modulated by the motion of the mechanical oscil- ˆ lin = † + † + † + † , lator, giving rise to the electromechanical coupling. Inside the Ham 2aˆ2aˆ2 mmˆ mˆ gam(â2mˆ aˆ2mˆ ) (3b)

013192-2 EINSTEIN-PODOLSKY-ROSEN ENTANGLEMENT AND … PHYSICAL REVIEW RESEARCH 3, 013192 (2021) where = δ + 2˜g Re[bˆ ] and g = g˜ aˆ , with where the drift matrix 1 1 ab ss ab ab 1 ss ⎛ ⎞ 2 A1 0 Aab 0 2Ed 2igãb|aˆ1 ss| aˆ = , bˆ =− . (4) ⎜A12 A2 0 Aam⎟ 1 ss κ + i ss γ + iω A =⎝ ⎠. (8) 1 2 1 b 2 b Aab 0 Ab 0 0 Aam 0 Am Here κ1 ≡ κ1e + κ1i denotes the total dissipation rate of the κ κ cavity modea ˆ1, with 1e and 1i being the cavity external κx −2x Here A ={ , } =−( κ )/2, A ={ , } = γ x 1 2 2 x x y b m and internal loss rates, respectively, and b is the mechanical γ − − y 2 y / ≡ ω =− 00 damping rate. (2y γy ) 2 with b b, Aab (02G ), √ ab The unidirectional coupling between the two cavity fields 0 gam Aam = ( ), and A12 =− ηκ1κ2I. The diffusion out −gam 0 can be described as follows: the output fielda ˆ1 (t )isusedas in matrix the input fielda ˆ2e(t ) to drive the cavitya ˆ2 but not vice versa, i.e., D D D 0 D = 1 12 ⊕ b , (9) √ D D 0 D in = η out + − η in , 12 2 m aˆ2e(t ) aˆ1 (t ) 1 aˆη (t ) (5) th th with D1 = κ1(¯n + 1/2)I, D2 = κ2[η(¯n + 1/2) + (1 − where the transmission loss is taken into account, with 1 √ 1 η)(¯nth + 1/2)]I, D = ηκ κ I/2, and D = γ (¯nth + the coupling efficiency η, and the output fielda ôut (t ) = η 12 1 2 s s s √ 1 1/2)I (s = b, m). κ aˆ (t ) + aîn (t ). The thermal noise operatorsa în(t )(l = 1e 1 1e l We are interested in the quantum correlations in the steady ,η in in† = 1e ) satisfy the nonzero correlations aˆl (t )âl (t ) states which can be solved by setting the left-hand side of in† th + δ δ − in = thδ δ − (¯nl 1) ll (t t ) and aˆl (t )âl (t ) n¯l ll (t t ), Eq. (7) to be zero. Note that the stability of the present master- ω ω h¯ 1 − h¯ 2 − th = kBT − 1 th = kBT − 1 slave cascade system is merely determined by the stability of wheren ¯1e (e 1 1) [¯nη (e 2 1) ]isthe equilibrium mean thermal photon number at environmental the electromechanical subsystem, since the master subsystem is not influenced by the salve subsystem and the latter in- temperature T1 (T2) and kB the Boltzmann constant. By using Eqs. (3) and including the dissipations and input noises of the volves only linear mixing of the cavity and magnon modes. system, the equations of motion are derived as The stability is therefore guaranteed when all the eigenvalues of the drift matrix of the electromechanical subsystem κ √ d 1 † in ˜ ≡ A1 Aab aˆ =− + i aˆ − ig (bˆ + bˆ ) − κ aˆ (t ) Aab (A A ) have negative real parts. Applying the Routh- dt 1 2 1 1 ab 1e 1e ab b √ Hurwitz criterion [56], the stability condition is found to be − κ in , 1iaˆ1i (t ) (6a) γ 2 κ2 d κ √ S = b + ω2 1 + 2 − 4ω g2 > 0, (10a) aˆ =− 2 + i aˆ − ig mˆ − ηκ κ aˆ 1 4 b 4 1 b 1 ab dt 2 2 2 2 am 1e 2e 1 √ κ γ κ γ − ηκ in − − η κ in = κ 1 b + ω2 + γ 1 b + 2 2eaˆ1e(t ) (1 ) 2eaˆη (t ) S2 1 b b 1 √ 4 4 − κ in , 2iaˆ2i(t ) (6b) κ2 × κ 1 + κ γ + 2 d γ √ 1 1 b 1 bˆ =− b + iω bˆ − ig a + a† − γ bˆ t , 4 b ab(ˆ1 ˆ1 ) b in( ) (6c) dt 2 γ 2 γ √ + γ 1 + κ γ + ω2 d m b 1 b b mˆ =− + im mˆ − igamaˆ2 − γmmˆ in(t ), (6d) 4 dt 2 κ2 γ 2 κ ≡ κ + κ γ − κ + γ 2 1 + 2 b + ω2 where 2 2e 2i and m are the damping rates of the ( 1 b) 1 1 κ κ 4 4 cavitya ˆ2 and magnon mode, respectively, with 2e and 2i being the external and internal loss rates of the second cavity. − ω 2 > . in in 4 b 1gab 0 (10b) The internal noisesa ˆ1i (t ) anda ˆ2i(t ) are independent of the in in noisesa ˆ1e(t ) anda ˆη (t ) but have the same correlations as the ˆ two latter respectively. The noise operators bin(t ) andm ˆ in(t ) III. PHONON-MAGNON ENTANGLEMENT in satisfy the same correlations asa ˆ1e,η (t ), but with the mean AND STEERING th thermal excitation numbersn ¯ at temperature T , and with b,m b m The phonon-magnon entanglement can be witnessed when frequencies ω , . b m [57] When starting from Gaussian states, the system gov- θ θb θ θb erned by the linearized Eq. (6) evolves still in Gaussian, 4V xˆ m + fxxˆ V pˆ m − fy pˆ E = m b m b < 1, (11) whose state is completely determined by the covariance bm 2 (1 + fx fy ) matrix σ jj =μ jμ j + μ j μ j /2 −μ j μ j , where μ = (ˆx1, pˆ1, xˆ2, pˆ2, xˆb, pˆb, xˆm, p√ˆm ) with the quadrature operators√ where V (ô) denotes the variance of the operatoro ˆ, and the = + † / =− − † / = defined byx ˆ (ô oˆ ) 2 andp ˆ i(ô oˆ ) 2(o angles θb,m and fx,y are used to minimize the variances, with , , , σ a1 a2 b m). The covariance matrix ˆ satisfies fx fy > 0. A tighter criterion is [58] θ θ σ = σ + σ T + , = m m < , ˙ A A D (7) Sm|b 4Vinf xˆm Vinf pˆm 1 (12)

013192-3 HUATANG TAN AND JIE LI PHYSICAL REVIEW RESEARCH 3, 013192 (2021)

mode, conditioned on the measurements of the mechanical position and momentum, with the optimal gains fo = θ θ θ θ m − m b / b = , V (ôm ) oˆm oˆb V (ôb )(o x y). Equation (12) shows that the Heisenberg uncertainty is seemingly violated, embodying the original EPR paradox [59,60]. Moreover, the conditional magnon squeezed states can be generated when Eq. (12)is held, and it therefore reflects that the magnonic states can be steered by mechanics via the EPR entanglement and local measurements, characterizing quantum steering [61], a type of quantum nonlocality [62] and originally named by Schrödinger in response to the EPR paradox [63]. Similarly, the reverse steering from the magnon to the phonon exists if θ θ = ˆ b ˆ b < . Sb|m 4Vinf Xb Vinf Yb 1 (13) It should be emphasized that the criteria (12) and (13) are sufficient and necessary for Gaussian states and Gaus- sian measurements (e.g., homodyne detection) [62]. One-way steering is present when either of Eqs. (12) and (13) is held. One-way property of quantum steering makes it intrinsically distinct from entanglement, and it is useful for, e.g., one-sided FIG. 2. Density plots of the mechanical-magnon entanglement device-independent quantum cryptography [64]. Note that the κ = κ Ebm and steering Sm|b vs the cavity dissipation rates 1,2,the smaller values of Ebm, Sm|b, and Sb|m mean stronger entangle- drive-magnon detuning m, and the drive-cavity detuning 1,2.In ment and steering. (a) and (c), 1 =−2 = ωb,andgab = 0.5gam = 0.5ωb;in(b)and (d), κ = 10ω , =−ω ,andg = 0.5g = 0.5ω .Wetakeη = b m b ab am b IV. RESULTS 1, and the other parameters are provided in the text. In the plots, the steering Sb|m is absent and not plotted, and the blank areas mean the In Figs. 2 and 3 the dependence of the steady-state en- absence of the entanglement and steering (similarly hereinafter). tanglement Ebm and steering Sm|b and Sb|m on some key parameters is plotted. We adopt experimentally feasible pa- θ θ θ θ θ m ≡ m + b m ≡ m − where Vinf (ˆxm ) V (ˆxm fxxˆb ) and Vinf (ˆpm ) V (ˆpm rameters ωb/2π = 10 MHz, ωm/2π = 10 GHz, γb/2π = θ b γ / π = . κ / π = = fy pˆb ) represent the inferred variances of the magnon 100 Hz, m 2 1 5MHz, 1i,2i 2 1 MHz, and T

FIG. 3. Density plots of the entanglement Ebm and steering Sm|b and Sb|m vs the cavity dissipation rates k1 and k2 (a)–(c) and couplings gam and gab (d)–(f). We take gab = 0.5gam = ωm in (a)–(c), κ1 = κ2 = 10ωb in (d)–(f), m =−ωb, 1 =−2 = ωb, and the other parameters are the same as in Fig. 2.

013192-4 EINSTEIN-PODOLSKY-ROSEN ENTANGLEMENT AND … PHYSICAL REVIEW RESEARCH 3, 013192 (2021)

T1,2 = Tm,b = 30 mK [31,36,55]. We take the detuning 1 = ωb for cooling the mechanical mode, except for Figs. 2(b) and 2(d). We see that the EPR entanglement and steering between the mechanical oscillator and the YIG sphere can be achieved in the steady-state regime. The phonon-to-magnon steering Sm|b shows similar properties to the entanglement. By contrast, the reverse steering Sb|m from the magnons to the phonons is absent, mainly due to a much larger magnon damping rate than that of the mechanics, i.e., γm γb, and it is merely present for unbalanced cavity dissipation rates κ1 κ and 2 or relatively large couplings gab and gam,asshownin FIG. 4. Density plots of the entanglement Ebm and steering Sm|b 2 /κ η κ = Figs. 3(c) and 3(f), altering the effective damping rates gab 1 vs temperature T and the cavity coupling efficiency , with 1 2 /κ κ = ω =− =− = ω = . = . ω and gam 2 of the mechanical and magnon modes. Moreover, 2 10 b, 1 2 m b, gab 0 5gam 0 5 b,and the steering Sm|b is stronger than the reverse Sb|m when both the other parameters are the same as in Fig. 2. The steering Sb|m is of them are present. This asymmetric steering means that the absent and not plotted. ability of an experimenter Alice, for example, to distantly control magnon states (e.g. achieving conditional squeezed magnon states) via her local Gaussian measurements on the Figure 4 reveals that the entanglement and steering are ro- mechanical mode is different from the ability of another ex- bust against the inefficiency of the unidirectional coupling and perimenter Bob to distantly control mechanical states by his thermal fluctuations. They can survive up to T > 10 mK for local Gaussian measurements on the magnon mode. Further, a realistic coupling efficiency η = 0.5, and the entanglement the one-way steering indicates that Alice can obtain squeezed is more robust than the steering, since the latter embodies magnon states, but Bob cannot achieve the squeezing of me- a stronger quantum correlation. We note that in the current chanics by local homodyne detection. experiments in microwave domain [6,7], the cascade-cavity- Specifically, as shown in Fig. 2 the entanglement and steer- coupling efficiency up to η ≈ 0.75 can be achieved. For the ing become maximal under strong cavity dissipation, κ1,2 YIG sphere, the cooling to 10 mK to 1 K by using a di- {gab, gam,ωb}, far beyond the sideband-resolved regime of lution refrigerator, merely with small line broadening, has the electromechanical system, and they are also optimized been achieved in the experiment [31]. In addition, around the = = = when the detuning m =−ωb. This can be understood as temperature T1,2 Tb Tm 30 mK by using a cryostat in follows: the phonon-magnon entanglement in fact originates the experiments [31,36,55], the mean thermal magnon number th ≈ th ≈ from the photon-phonon entanglement built up by the elec- n¯m 0, while the mean thermal phonon numbern ¯b 60. tromechanical coupling. As studied by one of us in Ref. [65], on the bad-cavity condition κ1 ωb, the stationary entanglement between the mechanical oscillator and the cavity V. DISCUSSION AND CONCLUSION ω − ω output photons at frequency d b becomes maximal and For the parameters considered ωb/2π = 10 MHz, much stronger than the intracavity-photon-phonon entangle- ωm/2π = 10 GHz, 1 =−2 =−m = ωb, and ment, which does not yet exhibit quantum steering; via the κ1 = 10ωb, to achieve the electromechanical coupling unidirectional photon-photon coupling and photon-magnon √ g ≡ g˜ P κ / h¯ω (2 + κ2/4) = 0.5ω , the pumping coupling, the output photon-phonon entanglement is then ab ab d 1e c1 1 1 b P = μ partially transferred into the phonon-magnon entanglement. power d 1 W is required, given the single-photon / π ≈ Since the transfer efficiency depends on the product κ κ electromechanical couplingg ãb 2 150 Hz [55]. When a 1 2 μ and resonates also at frequency ω − ω for the photons and 400- m-diameter YIG sphere is considered, the number of d b ≈ × 16 magnons in the second cavity, the phonon-magnon entangle- spins in the sphere N 7 10 , with the net spin density of ρ = . × 27 −3 ment is therefore maximal for the bad cavities and at the the sphere s 2 1 √10 m , and the electromagnonical ≡ = ω detuning =−ω . coupling gam gm0 N b can be obtained for the m b / π ≈ Figure 2 also shows that under the bad-cavity condition, single-spin coupling gm0 2 38 mHz [31]. In addition, for the steady-state entanglement and steering are attainable in the given couplings gab and gam, the mean magnon number | |2 ≈ . × 8 = 5 relatively wide ranges of the detunings , . They are present mˆ 3 4 10 2Ns for the spin number s 2 of the 1 2 + in the red-detuned regime of the electromechanical subsystem ground-state ferrum ion Fe3 , which ensures the condition of (1 > 0) for the stability consideration and become maxi- low-lying excitation of the spin ensemble necessary for the ≈ mum at the instability threshold 1 0.√ We also see that present model. exactly due to the strong cavity coupling κ1κ2, the entan- As for the detection of the EPR entanglement and steering, glement and steering can still be achieved even for largely we adopt the well-established method [66] by coupling the different two cavity frequencies (i.e., 1 = 2), demonstrat- mechanical and magnon modes to the separate probing fields ing their robustness against the frequency mismatch between pˆb andp ˆm (see Fig. 1). When the probing cavityp ˆb is driven the two cavities. In addition, as depicted in Fig. 3, the optimal by a weak red-detuned pulse and the cavityp ˆm is resonant with entanglement and steering are obtained for the cooperativity the magnon mode, the beam-splitter-like Hamiltonian Hˆbp = C = 2 /κ γ ≈ × 3 C = 2 /κ γ ≈ ˆ † + ˆ† ˆ = † + † parameters b gab 1 b 6 10 and m gam 2 m gbp(bpˆb b pˆb) and Hmp gmp(ˆmpˆm mˆ pˆm ) are thus ac- 1, which can readily be realized with current state-of-the-art tivated. Therefore, the states of the mechanical and magnon experimental technology [36,55]. modes can be transferred onto the probes. By homodyning

013192-5 HUATANG TAN AND JIE LI PHYSICAL REVIEW RESEARCH 3, 013192 (2021) the outputs of the probes and measuring the variances of the our proposal is promising to be realized in the near future. quadratures, one can verify the entanglement and steering. This distant hybrid EPR entanglement and steering between In conclusion, we present a deterministic scheme for two truly massive objects may ﬁnd its applications in the fun- creating a hybrid EPR entanglement channel between a damental study of macroscopic quantum phenomena, as well macroscopic mechanical oscillator and a magnon mode in a as in quantum networking and one-sided device-independent distant macroscopic YIG sphere. The entanglement is created quantum cryptography. Further investigations would conclude in the electromechanical cavity and distributed remotely to quantum-state exchange between a mechanical oscillator and the electromagnonical cavity by coupling the output ﬁeld of a distant YIG sphere. the former to the latter. Strong stationary phonon-magnon EPR entanglement and steering can be achieved under re- ACKNOWLEDGMENTS alistic parameters far beyond the sideband-resolved regime. These features clearly distinguish from the existing proposals This work is supported by the National Natural Sci- [39,46] where the entanglement is generated locally under the ence Foundation of China (No. 11674120), the Funda- condition of resolved sidebands, and thus the present proposal mental Research Funds for the Central Universities (No. manifests its unique advantages. With the rapid progress in CCNU20TD003), and the European Research Council Project realizing the coupling between hybrid massive systems [8,9], (Grant No. ERC StG Strong-Q, 676842).

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