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

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 ten gigahertz 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 a 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 macro scale, but also in quantum networking and one-sided device-independent quantum cryptography based on magnonics and electromechanics.

Introduction.-Long-distance entanglement has attracted extensive attention owing to its potential applications to the fundamental test of quantum mechanics [1], quantum networking [2,3], and quantum-enhanced metrology [4]. In such quantum tasks, hybrid quantum systems, composed of distinct physical components with complementary functionalities, possess multitasking capabilities and thus may be better suited than others for specific tasks [5]. It has been proved that light mediation is an effective approach to achieving the entanglement between two remote systems that never interact directly [6][7][8]. For example, hybrid EPR entanglement between distant macroscopic mechanical and atomic systems has been realized very recently via unidirectional light coupling [8]. In addition, strong coherent coupling between a mechanical membrane and atomic spins has been demonstrated by two cascade light-mediated coupling processes [9].
Here we consider a microwave-mediated phononmagnon interface and focus on how to deterministi- A driven electromechanical cavity (a1) is unidirectionally coupled to an electromagnonical cavity (a2) where a YIG sphere in a uniform magnetic field is placed. The probing cavities p b and pm are used to read out and detect the phonon-magnon entangled state. The twocavity configuration (a2, pm) of the microwave optomagnonic system can be a cross-shape cavity [49], and the YIG sphere is glued on the end of a cantilever.
cally establish hybrid EPR entanglement channel between a macroscopic mechanical oscillator and a distant YIG sphere across about ten gigahertz of frequency difference. The system consists of two unidirectionallycoupled electromechanical and electromagnonical cavities inside which a mechanical oscillator and a YIG sphere are placed, respectively. We find that far beyond the electromechanical sideband-resolved regime, strong stationary phonon-magnon EPR entanglement and steering can be achieved, as a result of electromechanical output photon-phonon entanglement distributed via the unidirectional cavity coupling. Further, the one-way steering from phonons to magnons is established and adjustable over a wide range of feasible parameters. The entanglement and steering are robust against the frequency dismatch between the two cavities, unidirectional cavitycoupling loss, and environmental temperature.
Model.-We consider a driven electromechanical cavity that is unidirectionally coupled to an electromagnonical cavity, as shown in Fig.1. For the electromechanical cavity, the cavity resonance is modulated by the motion of the mechanical oscillator, giving rise to the electromechanical coupling. Inside the 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 p b,m , with their output fields sent to measurement apparatuses, are used to read out the phonon and magnon states, respectively, and the two cavities (a 2 , p m ) can be a cross-shape cavity coupled to the YIG sphere glued on the end of a cantilever [49]. The magnons, which characterize quanta of the uniform magnetostatic mode (i.e., Kittel mode [50]) in the YIG sphere, are coupled to the cavity mode via magnetic dipole interaction. In the rotating frame with respect to the frequency ω d of the drive with amplitude E d , the system's HamiltonianĤ =Ĥ ab +Ĥ am reads ( = 1) where the bosonic annihilation operatorsâ j ,m andb denote the cavity, magnon, and mechanical modes with frequencies ω j , ω m and ω b , respectively. The detunings where β is the gyromagnetic ratio and H B is the strength of the uniform bias magnetic field.g ab represents the single-photon electromechanical coupling, and g am denotes the magnetic-dipole coupling between the cavity and magnons, g am ∝ √ N with N being the number of spins. In the recent experiment [33], a strong coupling, g am ∼ 47 MHz, much larger than the cavity and magnon linewidths about 2.7 MHz and 1.1 MHz, has been achieved, whereas the electromechanical couplingg ab is typically weak, but it can be enhanced by using a strong cavity drive. Ultrastrong electromechanical coupling in the linear regime has been reported [51]. Under a strong drive, Eq.(1) can be linearized by replacing the operators byô → ô ss +ô (o = a j , b, m), where ô ss denotes the steady-state amplitudes of the modes, leading to the linearized Hamiltonian where ∆ 1 = δ 1 + 2g ab Re[ b ss ] and g ab =g ab â 1 ss , with and κ 1 and γ b are the damping rates of the cavityâ 1 and mechanical mode, respectively. The unidirectional coupling between the two cavity fields can be described as follows: the output fieldâ out 1 (t) is used as the input fieldâ in 2 (t) to drive the cavityâ 2 but not vice versa, i.e., where the transmission loss is taken into account, with the coupling efficiency η, and the output fieldâ out k B T 2 − 1) −1 is the equilibrium mean thermal photon number at environmental temperature T 1 (T 2 ) and k B the Boltzmann constant. By using Eqs. (2) and including the dissipations and input noises of the system, the equations of motion are derived as where κ 2 and γ m are the damping rates of the cavitŷ a 2 and magnon mode, respectively. The noise operatorŝ b in (t) andm in (t) are independent and satisfy the same correlations asâ in 1,η (t), with the mean thermal excitation numbersn th b,m at temperature T b,m . When starting from Gaussian states, the system governed by the linearized Eq.(5) evolves still in Gaussian, whose state is completely determined by the covariance 1 , a 2 , b, m). The covariance matrixσ satisfieṡ where the drift matrix Here with D 1 = κ 1 (n th 1 + 1/2)I, D 2 = κ 2 η(n th 1 + 1/2) + (1 − η)(n th η + 1/2) I, D 12 = √ ηκ 1 κ 2 I/2, and D s = γ s (n th s + 1/2)I (s = b, m).
We are interested in the quantum correlations in the steady states which can be solved by setting the lefthand side of Eq.(6) to be zero. Note that the stability of the present master-slave cascade system is merely determined by the stability of the electromechanical subsystem, since the master subsystem is not influenced by the salve subsystem and the latter only involves linear mixing of the cavity and magnon modes. The stability is therefore guaranteed when all the eigenvalues of the drift matrix of the electromechanical subsystemÃ ab ≡ A1 A ab We take η=1 and the other parameters are provided in the text. In the plots, the steering S b|m is absent and not plotted, and the blank areas mean the absence of the entanglement and steering (similarly hereinafter).

when [52]
where V (ô) denotes the variance of the operatorô, and the angles θ b,m and f x,y are used to minimize the variances, with f x f y > 0. A tighter criterion is [53] where ) represent the inferred variances of the magnon mode, conditioned on the measurements of the mechanical position and momentum, with the optimal Eq.(10) shows that the Heisenberg uncertainty is seemingly violated, embodying the original EPR paradox [54,55]. Moreover, the conditional magnon squeezed states can be generated when Eq.(10) is hold, and it therefore reflects that the magnonic states can be steered by mechanics via the EPR entanglement and local measurements, characterizing quantum steering [56], a type of quantum nonlocality [57] and originally termed by Schödinger in response to the EPR paradox [58]. Similarly, the reverse steering from the magnon to the phonon exists if One-way steering is present when either of Eqs. (10) and (11) is hold. One-way property of quantum steering makes it intrinsically distinct from entanglement and it is useful for, e.g., one-sided device-independent quantum cryptography [59]. Note that the smaller values of E bm , S m|b , and S b|m mean stronger entanglement and steering.
Results.-In Figs.2-3, the dependence of the steadystate entanglement E bm and steering S m|b and S b|m on some key parameters are plotted. We adopt experimentally feasible parameters ω b /2π = 10 MHz, ω m /2π = 10 GHz, γ b /2π = 100 Hz, γ m /2π = 1.5 MHz, and T = T 1,2 = T m,b = 30 mK [33,38,51]. We take the detuning ∆ 1 = ω b for cooling the mechanical mode, except for Figs.2 (b) and (d). We see that the EPR entanglement and steerings between the mechanical oscillator and the YIG sphere can be achieved in the steady-state regime. The phonon-to-magnon steering S m|b shows similar properties to the entanglement. By contrast, the reverse steering S b|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., γ m ≫ γ b , and it is merely present for unbalanced cavity dissipation rates κ 1 and κ 2 or relatively large couplings g ab and g am , as shown in Fig.3 (c) and (f). Moreover, the steering S m|b is stronger than the reverse S b|m when both of them are present.
Specifically, as shown in Fig.2 the entanglement and steering become maximal under strong cavity dissipation, κ 1,2 ≫ {g ab , g am , ω b }, far beyond the sideband-resolved regime of the electromechanical system, and they are also optimized when the detuning ∆ m = −ω b . This can be understood as follows: the phonon-magnon entanglement in fact originates from the photon-phonon entanglement built up by the electromechanical coupling. As studied by one of us in Ref. [60], on the bad-cavity condition κ 1 ≫ ω b , the stationary entanglement between the mechanical oscillator and the cavity output photons at frequency ω d − ω b becomes maximal and much stronger than the intracavity-photon-phonon entanglement which does not yet exhibit quantum steering; via the unidirectional photon-photon coupling and photon-magnon coupling, the output photon-phonon entanglement is then partially transferred into the phonon-magnon entanglement. Since the transfer efficiency depends on the product κ 1 κ 2 and resonates also at frequency ω d − ω b for the photons and magnons in the second cavity, the phononmagnon entanglement is therefore maximal for the bad cavities and at the detuning ∆ m = −ω b . Figure 2 also shows that under the bad-cavity condition, the steady-state entanglement and steering are attainable in relatively wide ranges of the detunings ∆ 1,2 . They are present in the red-detuned regime of the electromechanical subsystem (∆ 1 > 0) for the stability consideration and become maximum at the instability threshold ∆ 1 ≈ 0. We also see that exactly due to the strong cavity coupling √ κ 1 κ 2 , the entanglement and steering can still be achieved even for largely different two cavity frequencies (i.e., ∆ 1 = ∆ 2 ), demonstrating their robustness against the frequency dismatch between the two cavities. In addition, as depicted in Fig.3 the optimal entanglement and steering are obtained for the cooperativity parameters C b = g 2 ab /κ 1 γ b ≈ 6 × 10 3 and C m = g 2 am /κ 2 γ m ≈ 1, which can readily be realized with current state-of-the-art experimental technology [38,51]. Fig.4 reveals that the entanglement and steering are robust against the inefficiency of the unidirectional coupling and thermal fluctuations. They can survive up to T > 100 mK for a realistic coupling efficiency η = 0.5, and the entanglement is more robust than the steering, since the latter embodies a stronger quantum correlation. We note that in the current experiments in microwave domain [6,7], the cascade-cavity-coupling efficiency up to η ≈ 0.75 can be achieved. For the YIG sphere, the cooling to 10 mK ∼ 1 K by using a dilution refrigerator, merely with small line broadening, has been achieved in the experiment [33]. In addition, around the temperature T 1,2 = T b = T m = 30 mK by using cryostat in the experiments [33,38,51], the mean thermal magnon numbern th m ≈ 0, while the mean thermal phonon number n th b ≈ 60. Discussion and Conclusion.-For the parameters considered ω b /2π = 10 MHz, ω m /2π = 10 GHz, ∆ 1 = −∆ 2 = −∆ m = ω b , and κ 1 = 10ω b , to achieve the electromechanical coupling g ab ≡ g ab √ P d κ 1 / ω c1 (∆ 2 1 + κ 2 1 /4) = 0.5ω b , the pumping power P d = 1 µW is required, given the single-photon electromechanical couplingg ab /2π ≈ 150 Hz [51]. When a 400-µm-diameter YIG sphere is considered, the number of spins in the sphere N ≈ 7 × 10 16 , with the net spin density of the sphere ρ s = 2.1 × 10 27 m −3 , and the electromagnonical coupling g am ≡ g m0 √ N = ω b can be obtained for the single-spin coupling g m0 /2π ≈ 38 mHz [33]. In addition, for the given couplings g ab and g am , the mean magnon number | m | 2 ≈ 3.4 × 10 8 ≪ 2N s for the spin number s = 5 2 of the ground-state ferrum ion Fe 3+ , which ensures the condition of low-lying excitation of the spin ensemble necessary for the present model.
As for the detection of the EPR entanglement and steering, we adopt the well established method [61] by coupling the mechanical and magnon modes to the separate probing fieldsp b andp m (see Fig.1). When the probing cavityp b is driven by a weak red-detuned pulse and the cavityp m is resonant with the magnon mode, the beam-splitter-like HamiltonianĤ bp = g bp (bp † b +b †p b ) and H mp = g mp (mp † m +m †p m ) are thus activated. Therefore, the states of the mechanical and magnon modes can be transferred onto the probes. By homodyning the outputs of the probes and measuring the variances of the quadratures, one can verify the entanglement and steering.
In conclusion, we present a deterministic scheme for creating hybrid EPR entanglement channel between a macroscopic mechanical oscillator and a magnon mode in a distant macroscopic YIG sphere. The entanglement is created in the electromechanical cavity and distributed remotely to the electromagnonical cavity by coupling the output field of the former to the latter. Strong stationary phonon-magnon EPR entanglement and steering can be achieved under realistic parameters far beyond the sideband-resolved regime. These features clearly distinguish from the existing proposals [41,48] where the entanglement is generated locally under the condition of resolved sidebands, and thus the present proposal manifests its unique advantages. With the rapid progress in realizing the coupling between hybrid massive systems [8,9], our proposal is promising to be realized in the near future. This distant hybrid EPR entanglement and steering between two truly massive objects may find its applications in the fundamental study of macroscopic quantum phenomena, as well as in quantum networking and one-sided device-independent quantum cryptography. Further investigations would conclude quantumstate exchange between a mechanical oscillator and a distant YIG sphere.