key: cord-0179039-k43j2yg9
authors: Yu, Mei; Shen, Heng; Li, Jie
title: Magnetostrictively induced stationary entanglement between two microwave fields
date: 2019-09-12
journal: nan
DOI: nan
sha: d4ba828f0d2db8b4df66e22cd03baea1b0deed32
doc_id: 179039
cord_uid: k43j2yg9

We present a scheme to entangle two microwave fields by using the nonlinear magnetostrictive interaction in a ferrimagnet. The magnetostrictive interaction enables the coupling between a magnon mode (spin wave) and a mechanical mode in the ferrimagnet, and the magnon mode simultaneously couples to two microwave cavity fields via the magnetic dipole interaction. The magnon-phonon coupling is enhanced by directly driving the ferrimagnet with a strong red-detuned microwave field, and the driving photons are scattered onto two sidebands induced by the mechanical motion. We show that two cavity fields can be prepared in a stationary entangled state if they are respectively resonant with two mechanical sidebands. The present scheme illustrates a new mechanism for creating entangled states of optical fields, and enables potential applications in quantum information science and quantum tasks that require entangled microwave fields.

We present a scheme to entangle two microwave fields by using the nonlinear magnetostrictive interaction in a ferrimagnet. The magnetostrictive interaction enables the coupling between a magnon mode (spin wave) and a mechanical mode in the ferrimagnet, and the magnon mode simultaneously couples to two microwave cavity fields via the magnetic dipole interaction. The magnon-phonon coupling is enhanced by directly driving the ferrimagnet with a strong red-detuned microwave field, and the driving photons are scattered onto two sidebands induced by the mechanical motion. We show that two cavity fields can be prepared in a stationary entangled state if they are respectively resonant with two mechanical sidebands. The present scheme illustrates a new mechanism for creating entangled states of optical fields, and enables potential applications in quantum information science and quantum tasks that require entangled microwave fields.

Many quantum information tasks, e.g., quantum teleportation [1] , quantum metrology [2] , and fundamental tests of quantum mechanics [3] , require optical entangled states. Conventionally, they have been generated via parameteric downconversion with nonlinear crystals [4] . Alternative efficient approaches have been adopted by utilizing, e.g., four-wave mixing in optical fibers [5] and atomic vapors [6] , quantum dots [7] , and periodically poled lithium niobate waveguide [8] , to name but a few. In the microwave (MW) domain, the entangled fields are typically produced by using the nonlinearity in Josephson parametric amplifiers [9] , or by injecting a squeezed vacuum through a linear MW beamsplitter [10] . In the field of optomechanics, two MW fields can get entangled by coupling to a common mechanical resonator via radiation pressure [11] . Despite their different forms, most of the mechanisms utilize the nonlinearity of the physical processes.

In this article, we present a mechanism, distinguished from all previous approaches, for creating continuous-variable (CV) entanglement of MW fields by using the nonlinear magnetostrictive interaction in a ferrimagnet. Specifically, two MW cavity fields couple to a magnon mode in a ferrimagnetic yttrium-iron-garnet (YIG) sphere [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] , and simultaneously the magnon mode couples to a phonon mode embodied by the vibrations of the sphere induced by the magnetostrictive force [22] . Due to the intrinsic low frequency of the phonon mode, it owns a large thermal occupation at typical cryogenic temperatures. We thus drive the magnon mode with a red-detuned MW field (with the detuning equal to the mechanical frequency), which leads to the stimulation of the anti-Stokes process, i.e., a MW photon interacts with a phonon and converts into a magnon of a higher frequency [23, 24] . This process corresponds to the cooling of the phonon mode, a prerequisite for observing quantum effects in the system [25] . The strong magnon drive also enhances the effective magnon-phonon coupling, and when this coupling is sufficiently strong, magnomechanical entanglement is created, similar to the mechanism of creating optome-chanical entanglement with a strong red-detuned drive [26] . The entanglement originates from the nonlinear magnetostrictive coupling, and could be distributed to two MW fields due to the linear magnon-photon coupling. We prove its validity, and two MW fields get maximumly entangled when they are respectively resonant with the two mechanical sidebands (we assume the resolved sidebands [14] [15] [16] [17] [18] [19] [20] [21] [22] ) on both sides of the driving field, which are generated by the scattering of phonons. We verify the entanglement in both the quantitative and qualitative ways, i.e., by calculating the logarithmic negativity and by using the Duan criterion for CV systems. Both can be implemented straightforwardly by homodyning the outputs of the two cavity fields.

We first present a general model of the scheme, then solve the system dynamics by means of the standard Langevin formalism and the linearization treatment, and study the entanglement in the stationary state. Finally, we show strategies to measure/verify the optical entanglement, and provide possible configurations for experimental realizations.

The model. The system consists of two MW cavity modes, a magnon mode, and a mechanical oscillator, as shown in Fig. 1 (a). The magnons, as quantized spin wave, are the collective excitations of a large number of spins inside a massive YIG sphere. The magnon mode couples to two MW cavity modes via magnetic dipole interaction, and, simultaneously, to a mechanical vibrational mode via the magnetostrictive force [22] [23] [24] . We consider the size of the YIG sphere to be much smaller than the MW wavelengths, hence neglecting any radiation pressure on the sphere induced by the MW fields. The Hamiltonian of the system reads

where a j (m) and a † j (m † ) are, respectively, the annihilation and creation operators of the cavity mode j (magnon mode), arXiv:1909.05936v1 [quant-ph] 12 Sep 2019 FIG. 1: (a) General model of the scheme. A magnon mode m in a YIG sphere couples to two MW fields a 1 and a 2 via magnetic dipole interaction, and to a phonon mode b via magnetostrictive interaction. (b) Mode frequencies and linewidths. The magnon mode with frequency ω m is driven by a strong MW field at frequency ω 0 , and the mechanical motion of frequency ω b scatters the driving photons onto the two sidebands at ω 0 ± ω b . If the magnon mode is resonant with the blue (anti-Stokes) sideband, and the two cavity fields with frequencies ω 1,2 are respectively resonant with the two sidebands, the two cavity fields get entangled.

, and q and p are the dimensionless position and momentum quadratures of the mechanical mode, thus [q, p] = i. ω j , ω m , and ω b are the resonance frequencies of the cavity mode j, the magnon mode, and the mechanical mode, respectively, and the magnon frequency can be adjusted in a large range by altering the external bias magnetic field H via ω m = γ 0 H, where the gyromagnetic ratio γ 0 /2π = 28 GHz/T. G 0 is the single-magnon magnomechanical coupling rate, and g j denotes the coupling rate between the magnon mode with the cavity mode j, which can be (much) larger than the dissipation rates κ j and κ m of the cavity and magnon modes, g j > κ j , κ m , leading to cavity-magnon polaritons [14] [15] [16] [17] [18] [19] [20] [21] . The Rabi frequency Ω = For convenience, we switch to the rotating frame with respect to the drive frequency ω 0 , and by including input noises and dissipations of the system, we obtain the following quantum Langevin equations (QLEs)

where ∆ j = ω j − ω 0 , ∆ m = ω m − ω 0 , γ is the mechanical damping rate, and a in j , m in are input noise operators with zero mean value acting on the cavity and magnon modes, respectively, which are characterized by the following correlation functions [27] : a in

. The Langevin force operator ξ, accounting for the Brownian motion of the mechanical oscillator, is autocorrelated as

, where a Markovian approximation has been taken valid for a large mechanical quality factor Q m = ω b /γ 1 [28] . The equilibrium mean thermal photon, magnon, and phonon numbers

, with k B the Boltzmann constant and T the environmental temperature.

Because the magnon mode is strongly driven, it has a large amplitude | m | 1, and further owing to the cavity-magnon beamsplitter interactions the two cavity fields are also of large amplitudes. This allows us to linearize the system dynamics around the steady-state averages by writing any mode operator as a c-number plus its fluctuation operator O = O + δO, (O = a j , m, q, p) , and neglecting small second-order fluctuation terms. Substituting those linearized mode operators into Eq. (2), the equations are then separated into two sets of equations, respectively, for steady-state averages and for quantum fluctuations. The solutions of the averages are obtained, which are

i∆ j +κ j m , and m is given by

(i∆ m +κ m )(i∆ 1 +κ 1 )(i∆ 2 +κ 2 ) + g 2 1 (i∆ 2 +κ 2 ) + g 2 2 (i∆ 1 +κ 1 )

,

with∆ m = ∆ m + G 0 q the effective detuning of the magnon mode including the frequency shift caused by the magnetostrictive interaction. It takes a simpler form

when |∆ j |, |∆ m | κ j , κ m . Let us introduce the quadratures of the quantum fluctuations (δX 1 , δY 1 , δX 2 , δY 2 , δx, δy, δq, δp), 

where u(t)= δX 1 (t),δY 1 (t),δX 2 (t),δY 2 (t),δx(t),δy(t),δq(t),δp(t) T ,

2κ m y in (t), 0, ξ(t) T is the vector of noises entering the system, and the drift matrix A is given by

where G = i √ 2G 0 m is the effective magnomechanical coupling rate. By using the result of Eq. (4), we obtain

which shows that the coupling can be significantly enhanced with a strong magnon drive. We are interested in the quantum correlation of two MW fields in the steady state. The steady state of the system is a four-mode Gaussian state due to the linearized dynamics and the Gaussian nature of input noises. Such a state is fully characterized by an 8×8 covariance matrix (CM) C with its entries defined as C i j (t) = 1 2 u i (t)u j (t ) + u j (t )u i (t) (i, j = 1, 2, ..., 8). It can be obtained straightforwardly by solving the Lyapunov equation [29] AC

where D = diag κ 1 (2N 1 +1) ,

We adopt the logarithmic negativity [30] to quantify the entanglement between the two MW cavity fields. It is a full entanglement monotone under local operations and classical communication [31] and an upper bound for the distillable entanglement [30] . The logarithmic negativity for Gaussian states is defined as [32] 

whereν − = min eig|iΩ 2Cmw | (with the symplectic matrix Ω 2 = ⊕ 2 j=1 iσ y and the y-Pauli matrix σ y ) is the minimum symplectic eigenvalue of the CMC mw = PC mw P, with C mw the CM of the two MW fields, which is obtained by removing in C the rows and columns associated with the magnon and mechanical modes, and P = diag(1, −1, 1, 1) is the matrix that performs partial transposition on CMs [33] .

MW entanglement and its detection. In Fig. 2 we present the main results of the entanglement between two MW cavity fields. The stationary entanglement is guaranteed by the negative eigenvalues (real parts) of the drift matrix A. Figure 2 (a) shows clearly that the maximum entanglement is achieved when the two cavity fields are respectively resonant with the two mechanical sidebands [see Fig. 1 (b) ], i.e., ∆ 1 = −∆ 2 ± ω b , where "±" sign is taken due to the symmetry of the two cavity fields. And the magnon mode resonant with the blue sideband∆ m ω b corresponds to the anti-Stokes process, which significantly cools the phonon mode, thus eliminating the main obstacle for observing entanglement [23] . We have employed experimentally feasible parameters [22] : ω m /2π = 10 GHz, ω b /2π = 10 MHz, γ/2π = 10 2 Hz, κ m /2π = κ 1 /2π = 1 MHz, g 1 /2π = 3.8 MHz, G/2π = 4.5 MHz, and T = 20 mK. We use a strong magnon-phonon coupling G > κ m to create magnomechanical entanglement. This means that a strong magnon driving field should be used, and in order to avoid unwanted magnon Kerr effect [34, 35] the bare coupling rate G 0 should not be too small [23, 24] . For the optimal case ∆ 1 = −∆ 2 ± ω b in Fig. 2 (a) , a driving power of 6.3 mW (0.57 mW) should be used to yield G/2π = 4.5 MHz for G 0 /2π = 0.3 Hz (1 Hz), while keeping the Kerr effect negligible. In Fig. 2 (b) , we analyse the optimal coupling 2: (a) Density plot of the entanglement E N between two MW cavity fields vs (a) ∆ 1 and ∆ 2 , (b) κ 2 /κ 1 and g 2 /g 1 (κ 1 , g 1 are fixed). We take∆ m = 0.9ω b , κ 2 = κ 1 , g 2 = g 1 in (a), and ∆ 1 = −∆ 2 = ω b in (b). See text for the other parameters. rates g 1,2 and decay rates κ 1,2 , and find that in both situations ∆ 1 = −∆ 2 ω b and ∆ 1 = −∆ 2 −ω b , close coupling rates should be used, and the cavity that is resonant with the red (blue) mechanical sideband should have a smaller (larger) decay rate than the other. The generated optical entanglement can be detected by measuring the CM of the two cavity fields, which can be realized by homodyning the outputs of the two cavity fields [26, 36] .

Alternatively, one can also verify the entanglement by using the Duan criterion [37] , which requires simpler experimental operations, i.e., one does not have to measure all the entries of the 4 × 4 CM, but measure only two collective quadratures [38] . Specifically, a sufficient condition for entanglement is that the two collective quadratures satisfy the inequality

where X + = X 1 + X 2 , and Y − = Y 1 − Y 2 . Figure 3 (a) shows that in two areas around ∆ 1 = −∆ 2 ±ω b the inequality is fulfilled, indicating that the two cavity fields are entangled.

Experimental implementations. We now discuss possible configurations that could realize the proposal. Two MW cavities and each cavity containing a cavity mode are preferred. In this situation, the frequencies of the cavity fields can be adjusted flexibly to match the two mechanical sidebands. The two cavities could be placed vertically in the horizontal plane with the YIG sphere located in the intersection (near the maximum magnetic fields) of the cavity fields. This can be realized in a cross-shape 3D MW Fabry-Perot cavity or coplanar waveguide [39, 40] . Such a cross configuration may reduce the Q factor of the cavities induced by the damage to boundary conditions. We thus study the Duan criterion for taking larger cavity decay rates in Fig. 3 (b) . It shows that with much larger decay rates, the two cavity fields are still entangled. As a secondary choice, the two MW fields can also be two cavity modes in a single cavity. In this case, the free spectrum range of the cavity should match twice the mechanical frequency. This requires a high-frequency mechanical mode with a high Q factor that strongly couples to the magnon mode, in order to reduce the cavity length. Given mechanical frequency of Fig. 2 (a) , and we take optimal detunings ∆ 1 = 0.9ω b and ∆ 2 = −1.1ω b in (b). ∼100 (250) MHz , a cavity with the length of about 75 (30) cm is needed. Although silica can be inserted into the cavity to reduce its length, it still calls for a dilution fridge with large volume to provide cryogenic temperature environment.

Conclusions. We present a new mechanism for creating MW entangled states based on magnetostrictive interaction in a ferrimagnetic YIG sphere. The mechanism makes use of the nonlinearity of such a magnomechanical interaction. The entanglement is in the steady state and robust against the cavity dissipations. We show strategies to detect the entanglement and possible configurations that are promising to realize the proposal. This work may find applications in quantum information science, quantum metrology, and quantum tasks that require entangled CV MW fields.

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