Dipole–dipole-interaction-induced entanglement between two-dimensional ferromagnets

Wuhrer, D.; Rohling, N.; Belzig, W.

doi:10.1063/5.0189665

We investigate the viability of dipole–dipole interaction as a means of entangling two distant ferromagnets. To this end, we make use of the Bogoliubov transformation as a symplectic transformation. We show that the coupling of the uniform magnon modes can be expressed using four squeezing parameters, which we interpret in terms of hybridization, one-mode, and two-mode squeezing. We utilize the expansion in terms of the squeezing parameters to obtain an analytic formula for the entanglement in the magnon ground state using the logarithmic negativity as entanglement measure. Our investigation predicts that for infinitely large two-dimensional ferromagnets, the dipole–dipole interaction does not lead to significant long-range entanglement. However, in the case of finite ferromagnets, finite entanglement can be expected.

Magnetic materials exhibit a wide range of magnetic structures,^1–4 ranging from simple collinear magnetic structures including ferromagnets (FMs)¹ and antiferromagnets (AFMs)² to non-collinear structures like skyrmion lattices.^3,5–7 These magnetically ordered materials host collective excitations of coupled magnetic moments. These excitations can be described classically using spin waves, but are interpreted as particles—known as magnons—when treated quantum mechanically.^1,8,9

The interest in magnons rose in recent years^10,11 as they provide a platform for easy-to-manipulate long-distance transport with low dissipation.^12,13 They may provide an alternative to the Joule-heating-plagued electrons in information transport and processing,¹⁴ as a key computational device, transistors, can be realized by magnetic structures.¹⁵ Moreover, it is possible to link magnetic materials to conventional electronic devices using the (inverse-)spin-Hall effect,^12,13 enabling the integration of magnonic devices with state of the art technology. This is highly significant due to the reduction in computational technology size and the emergence of two-dimensional (2D) materials,^16–22 notably Van der Waals (VdW) materials.^23–26 These materials provide the potential to produce engineered synthetic systems with desired attributes by layering different materials together. Two-dimensional (2D) magnetic layers^27–36 represent just one of many potential building blocks. Therefore, it is necessary to enhance our understanding of the excitations of 2D magnetic materials, along with the impact of coupling between distinct magnetic materials, considering the potential for squeezing and entanglement within magnetic systems.

Similar to photons,^37,38 quantum squeezing and entanglement for magnons have been predicted theoretically^39–42 and observed in experiments.^43,44 However, in contrast to photons, squeezing is an inherent property of magnetic materials. Squeezing for magnons refers to quadrature squeezing. This means decreasing the variance of one observable while increasing the variance of a canonically conjugate observable, such as the spin components in the x- and y-direction in a squeezed FM with the z-axis as easy axis. This is because squeezed magnon states are a superposition of non-squeezed magnon Fock-states. Consequently, the system already holds a substantial number of non-squeezed magnons even in the ground state. This superposition has been proposed for exciting multiple quantum dots simultaneously and in the process entangle them⁴⁵ or being probed by quantum dots adjacent to the magnet.⁴⁶ It was also discussed that the entanglement due to magnons in VdW materials is switchable by electronic and magnetic means leading to electrical controllable entanglement of distant qubits.⁴⁷ As a result, magnons make an interesting tool for quantum computing.^48–51

While electric and magnetic dipole–dipole interactions were discussed in the context of entanglement previously for other platforms,^52–57 our work explores the feasibility of long-distance magnon entanglement through dipole–dipole interaction in two separated FMs, facilitating the entanglement of remote systems or by magnetic materials involved in the formation of VdW materials. By “long range,” we denote a distance greater than the decay length of the Heisenberg exchange, which experiences an exponential decay.

As a model system, we consider two 2D square lattice FMs separated by a distance l, see Fig. 1(a). The spins inside each FM are subjected to ferromagnetic Heisenberg interaction and two uniaxial anisotropies: one perpendicular to the plane of the 2D FMs (z-direction) and the other along one of the principle axes of the FMs (x-direction). Moreover, we consider the long-range interaction between the spins to be transmitted by dipole–dipole interaction. The Hamiltonian describing the interactions of spins inside FM A is given by

{\hat{H}}_{A} = {\hat{H}}_{ex}^{A} + {\hat{H}}_{an}^{A} + {\hat{H}}_{D}^{A},

(1)

with a similar expression for FM B. The Heisenberg exchange is given by

{\hat{H}}_{ex}^{A} = - \sum_{\begin{matrix} r_{i}, r_{j} \in A \\ r_{i} \neq r_{j} \end{matrix}} J_{i j} {\hat{S}}_{i} \cdot {\hat{S}}_{j}

(2)

with

J_{i j} = J_{j i} > 0

being the ferromagnetic exchange strength between spins

{\hat{S}}_{i}

and

{\hat{S}}_{j}

of spin length S at sites

r_{i}

and

r_{j}

in FM A preferring a parallel configuration of spins.

FIG. 1.

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(a) System of two FMs parallel to the x–y-plane separated by a distance l along the z-axis. The magnified area depicts the interactions between the spins of the different FMs. (b) Phase diagram showing the orientation of the spins in FM A and B depending on the distance l/a and the dipole–dipole interaction strength $D / K_{z}$ ⁠. For small distances or small dipole–dipole interactions, spins in both FM point out of plane forming a ferromagnetic ordering between the FMs (green, OOP FM). For large distances and large dipole–dipole interactions, spins in both FM lie in-plane forming an antiferromagnetic ordering between the FMs (yellow, IP AFM). The border between both phases (black solid line) is given by Eq. (7). The black dashed line shows the phases depending on the distance for $D = 0.5 K_{z}$ ⁠. The orientation of the spins inside each FM is depicted by the arrows with the figure-plane being the x–z-plane.

The anisotropy term in the Hamiltonian, Eq. , is given by

{\hat{H}}_{an}^{A} = - K_{z} \sum_{r_{i} \in A} {({\hat{S}}_{i}^{z})}^{2} - K_{x} \sum_{r_{i} \in A} {({\hat{S}}_{i}^{x})}^{2},

(3)

where K_z (K_x) is the strength of the uniaxial anisotropy along the z-axis (x-axis). We assume

K_{z} ≫ K_{x} > 0

such that the spins prefer to align parallel to the z-direction.

Finally, we consider dipole–dipole interaction between magnetic moments given by

{\hat{H}}_{D}^{A} = \sum_{\begin{matrix} r_{i}, r_{j} \in A \\ r_{i} \neq r_{j} \end{matrix}} D_{i j}^{A} [{\hat{S}}_{i} \cdot {\hat{S}}_{j} - 3 (e_{i j} \cdot {\hat{S}}_{i}) (e_{i j} \cdot {\hat{S}}_{j})] .

(4)

Here,

e_{i j}

is the unit vector along the connecting line of spins at

r_{i}

and

r_{j}

⁠.

D_{i j}^{A}

is the strength of dipolar coupling between two spins at

r_{i}

and

r_{j}

⁠, which is of the following form:

D_{i j}^{A} = \frac{a^{3} D^{A}}{{| r_{i} - r_{j} |}^{3}} = \frac{μ_{0} g_{1}^{A} g_{2}^{A} μ_{B}^{2}}{ℏ^{2}} \frac{1}{| r_{i} - r_{j} |^{3}}

(5)

with μ₀ being the vacuum magnetic permeability,

g_{i}^{A}

the gyromagnetic ratio of the interacting magnetic moments, μ_B the Bohr magneton,

ℏ

the reduced Planck constant, and a the lattice constant. Throughout this work, we will assume that

ℏ, a = 1

⁠, which means that all the interaction parameters (J_ij, K_x, K_z, and D^A) will be given in the unit of energy.

The dipole–dipole interaction decays with the inverse of the third power of the distance between the spins and thus is classified as the long range.

The coupling between both FMs is also conveyed through dipolar interaction,

{\hat{H}}_{int} = \sum_{\begin{matrix} r_{i} \in A \\ r_{j} \in B \end{matrix}} D_{i j}^{int} [{\hat{S}}_{i} \cdot {\hat{S}}_{j} - 3 (e_{i j} \cdot {\hat{S}}_{i}) (e_{i j} \cdot {\hat{S}}_{j})] .

(6)

We assume identical FMs. Thus, Eq. yields similar dipole constants for the interaction inside each FM and between both FMs, and we assume

D^{A / B} = D^{int} = D

⁠.

The linear spin wave theory explains the excitation within magnetic systems as harmonic excitations, spin waves, by expanding the spin operators around the classical ground state using magnonic creation and annihilation operators. The classical configuration for each isolated FM is ferromagnetic, because we assume that the exchange interaction is dominant inside each FM. The anisotropies restrict the direction of magnetization to lie within the x–z-plane. The exact orientation is determined by the strength of the dipole–dipole interaction inside each and between both FMs. Thus, the classical ground state depends on the dipole interaction strength D and the distance l between both FMs. Only two orientations are taken by the FMs, see Fig. 1(b).

For short distances, a ferromagnetic out-of-plane configuration (OOP FM) between both FMs is taken due to the dominance of the dipole interaction between the systems, which favors a parallel, ferromagnetic alignment along the connecting line (⁠ $e_{z}$ ⁠). For large separations and weak dipole–dipole interaction also, the OOP FM configuration is taken. This is due to the fact that with D not only the interaction between the FMs is varied, but also the dipole–dipole interaction between spins inside each FM. Therefore, for weak dipole–dipole interaction, the spins prefer to align with K_z.

For significant distances, the spins align ferromagnetically in-plane due to the dominance of the dipole–dipole interaction within each FM over the anisotropy. Minimizing the interaction term when the spins are perpendicular to the connecting line results in an AFM ordering between the two ferromagnets. This is referred to as in-plane AFM (IP AFM) ordering.

In this work, we investigate square lattice FMs, where the phase boundary between OOP FM and IP AFM phase, seen as black solid line in Fig. 1(b), is given by

\tilde{A} = K_{z} - (K_{x} + 3 D_{x | 0}) = D_{0}^{int} - \frac{3}{2} (D_{x | 0}^{int} + D_{z | 0}^{int}) = Σ D_{0} .

(7)

The coefficient(s)

D_{α | 0} (D_{0}^{int}, D_{α | 0}^{int})

is (are) the sum(s) of the dipole interaction inside (between) the FMs, with α specifying the direction of the connection vector. The exact definition is given in the supplementary material.

The classical configuration is represented by the right handed eigensystems,

\begin{matrix} e_{i, 1} = (\begin{matrix} cos (ϑ_{i}) \\ 0 \\ - sin (ϑ_{i}) \end{matrix}), e_{i, 2} = (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), \\ e_{i, 3} = (\begin{matrix} sin (ϑ_{i}) \\ 0 \\ cos (ϑ_{i}), \end{matrix}) \end{matrix}

(8)

for

i \in {A, B}

⁠, where

ϑ_{A}

(⁠

ϑ_{B}

⁠) is the angle between the classical spin configuration of FM A (B) and the z-direction. Applying the linear Holstein–Primakoff transformation⁵⁸ yields for spins in FM A,

\begin{matrix} {\hat{S}}_{i, 1} \approx ℏ \sqrt{\frac{S}{2}} ({\hat{a}}_{i} + {\hat{a}}_{i}^{†}), {\hat{S}}_{i, 2} \approx - i ℏ \sqrt{\frac{S}{2}} ({\hat{a}}_{i} - {\hat{a}}_{i}^{†}), \\ {\hat{S}}_{i, 3} \approx ℏ (S - {\hat{a}}_{i}^{†} {\hat{a}}_{i}), \end{matrix}

(9)

and in FM B,

\begin{matrix} {\hat{S}}_{i, 1} \approx ℏ \sqrt{\frac{S}{2}} ({\hat{b}}_{i} + {\hat{b}}_{i}^{†}), {\hat{S}}_{i, 2} \approx - i ℏ \sqrt{\frac{S}{2}} ({\hat{b}}_{i} - {\hat{b}}_{i}^{†}), \\ {\hat{S}}_{i, 3} \approx ℏ (S - {\hat{b}}_{i}^{†} {\hat{b}}_{i}) \end{matrix}

(10)

with

{\hat{S}}_{i, α} = e_{A / B, α} \cdot {\hat{S}}_{i}

⁠.

We perform the Fourier transformation to account for the periodic structure of the system

{\hat{a}}_{k} = \frac{1}{\sqrt{N}} \sum_{r_{i} \in A} e^{- i r_{i} \cdot k} {\hat{a}}_{i}, {\hat{b}}_{k} = \frac{1}{\sqrt{N}} \sum_{r_{i} \in B} e^{- i r_{i} \cdot k} {\hat{b}}_{i},

(11)

where we assumed

N_{A} = N_{B} = N

⁠. The linear Holstein–Primakoff transformation, Eqs. , is an approximation of the spin operators in terms of bosonic creation and annihilation operators and is only valid for a small number of bosons

⟨ {\hat{a}}_{i}^{†} {\hat{a}}_{i} ⟩ ≪ 2 S

or

\sum_{k} ⟨ {\hat{a}}_{k}^{†} {\hat{a}}_{k} ⟩ ≪ 2 N S

⁠.

Inserting Eqs. into Eqs. and , the Hamiltonian takes the following form:

\hat{H} = \sum_{k \in {BZ}^{+}} {\hat{a}}_{k}^{†} \underset{H_{k}}{\underset{︸}{(\begin{matrix} E_{A, k} & μ_{1, k}^{*} & 2 ξ_{A, k}^{*} & μ_{2, k}^{*} \\ μ_{1, k} & E_{B, k} & μ_{2, k}^{*} & 2 ξ_{B, k}^{*} \\ 2 ξ_{A, k} & μ_{2, k} & E_{A, k} & μ_{1, k} \\ μ_{2, k} & 2 ξ_{B, k} & μ_{1, k}^{*} & E_{B, k} \end{matrix})}} {\hat{a}}_{k},

(12)

where we defined

{\hat{a}}_{k} = {({\hat{a}}_{k}, {\hat{b}}_{k}, {\hat{a}}_{- k}^{†}, {\hat{b}}_{- k}^{†})}^{⊤}

and

{BZ}^{+}

labels all

k

in the Brillouin zone with a positive (or zero) k_x component. The exact definition of each parameter contained by the Hamiltonian is given in the supplementary material.

The meaning of the different parameters, if only they together with the diagonal elements $E_{i, k}$ would appear in the Hamiltonian, is outlined below. The energy of a bare magnon mode ${\hat{a}}_{k}$ (⁠ ${\hat{b}}_{k}$ ⁠) inside FM A (B) is given by $E_{A, k} (E_{B, k})$ ⁠, while $μ_{1, k}$ results in a hybridization of the magnon modes of the different FMs. This does not change the vacuum state and corresponds to a rotation in the phase space spanned by the A and B magnons.

Squeezing inside FM A (B) is mediated by $ξ_{A, k}$ (⁠ $ξ_{B, k}$ ⁠) entangling modes with wave vectors $\pm k$ inside each FM, while squeezing of $\pm k$ modes of the different FMs is a result of $μ_{2, k}$ ⁠. Squeezing alters the vacuum state as the squeezed magnon states are superpositions of populated A and B magnon states.⁴¹ If we combine different of these parameters, a clear distinction between squeezing and hybridization for each parameter cannot be given anymore.

In order to diagonalize the Hamiltonian, we perform a four-dimensional (4D) Bogoliubov transformation,^59,60 which is a symplectic transformation converting our bare magnon operators

{\hat{a}}_{k}

and

{\hat{b}}_{k}

into squeezed magnon operators

{\hat{α}}_{k}

and

{\hat{β}}_{k}

⁠. The Bogoliubov transformation can be written as

{\hat{a}}_{k} = (\begin{matrix} {\hat{a}}_{k} \\ {\hat{b}}_{k} \\ {\hat{a}}_{- k}^{†} \\ {\hat{b}}_{- k}^{†} \end{matrix}) = (\begin{matrix} U_{k} & V_{k} \\ V_{k}^{*} & U_{k}^{*} \end{matrix}) (\begin{matrix} {\hat{α}}_{k} \\ {\hat{β}}_{k} \\ {\hat{α}}_{- k}^{†} \\ {\hat{β}}_{- k}^{†} \end{matrix}) = U_{k} {\hat{α}}_{k}

(13)

with

U_{k} U_{k}^{†} - V_{k} V_{k}^{†} = 1, U_{k} V_{k}^{⊤} - V_{k} U_{k}^{⊤} = 0 .

(14)

The elements of

U_{k}

(⁠

V_{k}

⁠) are responsible for hybridization (squeezing), with

U_{k} = (\begin{matrix} u_{1, k} & u_{2, k} \\ u_{3, k} & u_{4, k} \end{matrix}), V_{k} = (\begin{matrix} v_{1, k} & v_{2, k} \\ v_{3, k} & v_{4, k} \end{matrix}) .

(15)

Solving the eigenvalue equation of the Hamiltonian yields the elements of the Bogoliubov matrix

U_{k}

and the eigenenergies

ε_{α / β, k}

of the squeezed magnon modes,

ε_{α / β, k} = \sqrt{{| E_{k} \pm μ_{1, k} |}^{2} - {| μ_{2, k} \pm 2 ξ_{k} |}^{2}},

(16)

where we used that we deal with similar FMs yielding

E_{A, k} = E_{B, k} = E_{k}

and

ξ_{A, k} = ξ_{B, k} = ξ_{k}

and that we regard either the OOP FM or the IP AFM phase. Figure 2 shows the energies of magnons

{\hat{α}}_{0}

and

{\hat{β}}_{0}

depending on the distance. In all figures, we regarded a system with nearest neighbor Heisenberg interaction of strength J. We see that for large distances, both converge toward the energy

{\tilde{ε}}_{0} = \sqrt{ε_{0}^{2} - 4 | ξ_{0} |^{2}}

of the squeezed magnons in the case of non-interacting FMs. At the phase transition,

ε_{α, 0}

shows a jump, while

ε_{β, 0}

vanishes.

FIG. 2.

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Energy of the $k = 0$ squeezed magnon modes depending on the distance l/a between both FMs in units of the energy of the eigenmodes in non-interacting FMs, ${\tilde{ε}}_{0}$ ⁠. At the phase transition, $l / a \approx 0.33$ ⁠, the energy $ε_{β, 0}$ vanishes. For large distances, $ε_{α, 0}$ and $ε_{β, 0}$ approach ${\tilde{ε}}_{0}$ (black, dashed line). Parameters are chosen as $D = 0.5 \times 10^{- 4} | J |, K_{z} = 10^{- 4} | J |$ ⁠, and $K_{x} = 10^{- 6} | J |$ ⁠.

From two-mode squeezed cases for an AFM,⁴¹ a harmonic spin spiral⁶¹ and most important for a FM with dipole interaction, we expect the largest squeezing, which is proportional to the entanglement, for the uniform $k = 0$ mode. An investigation of the system at hand shows the maximal squeezing for the $k = 0$ mode as discussed in the supplementary material. Thus, in the following, we will limit our considerations to the $k = 0$ mode and will drop the wave vector index for clarity.

For the uniform mode, all parameters in Eq. are real yielding real Bogoliubov parameters of the following form:

u_{1} = \sqrt{\frac{E + μ_{1} + ε_{α}}{4 ε_{α}}}, u_{2} = - \sqrt{\frac{E - μ_{1} - ε_{β}}{4 ε_{β}}}

(17)

with the remaining matrix elements given in the supplementary material. Figure 3 shows the dependence of the Bogoliubov matrix elements on the distance of both FMs. For distances

l / a < 0.33

⁠, in the OOP FM phase, we have non-zero values for all elements of U and close to zero value for all elements of V. Thus, the ferromagnetic structure is dominated by hybridization (U) with little to no squeezing (V). Due to the vanishing magnon energy of the

\hat{β}

magnons at the phase transition

l / a \approx 0.33

⁠, we see divergences in u₂, u₄, v₂, and v₄. These divergences lead to a breakdown of the linear Holstein–Primakoff approximation, Eqs. and , in a narrow region around the phase transition as the number of

{\hat{a}}_{k}

and

{\hat{b}}_{k}

magnons is connected to

| v_{i, k} |^{2}

⁠. Thus, in this region, the results should be treated with caution.

FIG. 3.

Dependence of the Bogoliubov matrix elements on the distance l for the k = 0 mode with D = 0.5 × 10 − 4 | J | , K z = 10 − 4 | J |, and K x = 10 − 6 | J |. Divergences and jumps occur at the phase transition l / a ≈ 0.33. (a) Elements of U 0. (b) Elements of V 0.

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Dependence of the Bogoliubov matrix elements on the distance l for the $k = 0$ mode with $D = 0.5 \times 10^{- 4} | J |, K_{z} = 10^{- 4} | J |$ ⁠, and $K_{x} = 10^{- 6} | J |$ ⁠. Divergences and jumps occur at the phase transition $l / a \approx 0.33$ ⁠. (a) Elements of $U_{0}$ ⁠. (b) Elements of $V_{0}$ ⁠.

In the IP AFM configuration (⁠ $l / a > 0.33$ ⁠), we see an increase in all elements of $U$ compared to the OOP FM configuration with the elements responsible for squeezing (V) being comparable in magnitude. This is due to the large inherent degree of squeezing of AFMs.⁴¹

The Bogoliubov transformation for the system at hand is an element of the ten-dimensional group of 4D symplectic matrices

Sp (4, R)

⁠.⁶² The limitation to the

k = 0

mode leaves us with a real Bogoliubov matrix, which can be parameterized by four parameters. We connect the Bogoliubov matrix to the generators of the group and introduce the four-mode squeezing operator,

{\hat{S}}_{4} (θ) = exp (\sum_{i = 1}^{4} θ_{i} {\hat{Φ}}_{i})

(18)

with θ_i being the real parameter corresponding to the generator

{\hat{Φ}}_{i}

⁠, which in quantum representation are given by

{\hat{Φ}}_{1 / 2} = {\hat{a}}^{†} {\hat{a}}^{†} - \hat{a} \hat{a} \mp ({\hat{b}}^{†} {\hat{b}}^{†} - \hat{b} \hat{b}),

(19)

{\hat{Φ}}_{3} = {\hat{a}}^{†} \hat{b} - \hat{a} {\hat{b}}^{†} + {\hat{a}}^{†} \hat{b} - \hat{a} {\hat{b}}^{†},

(20)

{\hat{Φ}}_{4} = {\hat{a}}^{†} {\hat{b}}^{†} - \hat{a} \hat{b} + {\hat{a}}^{†} {\hat{b}}^{†} - \hat{a} \hat{b} .

(21)

Each of these (pair of) generators can be identified as a one- or two-mode operation.

θ_{1} {\hat{Φ}}_{1} + θ_{2} {\hat{Φ}}_{2}

results in one-mode squeezing of modes inside FM A and B with squeezing parameters

r^{A / B} = θ_{2} \pm θ_{1}

for FM A (B).

{\hat{Φ}}_{3}

(⁠

{\hat{Φ}}_{4}

⁠) results in hybridization (two-mode squeezing) of the magnon modes of the different FMs. Thus, each isolated parameters θ_i can be identified with either one-mode, two-mode squeezing, or hybridization. However, this clear separation cannot be made if multiple parameters are involved, as the different generators do not commute. Ongoing, we will refer to all θ_i as “squeezing parameters.”

The analytic treatment in terms of a manageable number of squeezing parameters [ $N (2 N + 1)$ ] strongly depends on the dimension of the Bogoliubov transformation (2N), Eq. (13). This number could be increased by coupling of different $k$ modes, for example in finite magnets, making translational invariance crucial for the analytic treatment of the problem at hand.

We use the properties of the symplectic group to connect the elements of the Bogoliubov matrix to the squeezing parameters,

\hat{α} = U^{- 1} \hat{a} = {\hat{S}}_{4} (θ) \hat{a} {\hat{S}}_{4}^{- 1} (θ),

(22)

which yields

u_{1} = cosh (θ_{2}) cos (θ) + \frac{θ_{1}}{θ} sin (θ) sinh (θ_{2}),

(23)

u_{2} = \frac{θ_{4}}{θ} sin (θ) sinh (θ_{2}) + \frac{θ_{3}}{θ} sin (θ) cosh (θ_{2})

(24)

with

θ = \sqrt{{(θ_{3})}^{2} - {(θ_{1})}^{2} - {(θ_{4})}^{2}}

⁠. The remaining matrix elements expressed in terms of the squeezing parameters are given in the supplementary material.

Figure 4 shows the dependence of the squeezing parameters on the distance. They show a similar behavior as the elements of $U$ ⁠. The parameters identified with intra FM (⁠ $θ_{1}, θ_{2}$ ⁠) and inter FM (θ₄) squeezing are close to zero in the OOP FM phase and finite in the IP AFM phase, while the parameter identified with inter FM hybridization (θ₃) is of a finite value for the whole parameter range. This behavior supports our identification based on the generators $Φ_{i}$ ⁠. The divergence of the squeezing parameters is a result of vanishing energy of the ${\hat{β}}_{0}$ magnon modes at the phase transition.

FIG. 4.

Dependence of the squeezing parameters θ , θ 1 , θ 2 , θ 3 , θ 4 on the distance l for the k = 0 mode. Divergences occur at the phase transition l / a ≈ 0.33. Parameters are D = 0.5 × 10 − 4 | J | , K z = 10 − 4 | J |, and K x = 10 − 6 | J |.

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Dependence of the squeezing parameters $θ, θ_{1}, θ_{2}, θ_{3}, θ_{4}$ on the distance l for the $k = 0$ mode. Divergences occur at the phase transition $l / a \approx 0.33$ ⁠. Parameters are $D = 0.5 \times 10^{- 4} | J |, K_{z} = 10^{- 4} | J |$ ⁠, and $K_{x} = 10^{- 6} | J |$ ⁠.

To determine the entanglement between the

k = 0

magnon modes

\hat{a}

and

\hat{b}

in the squeezed vacuum state,

{| 0 ⟩}_{Sq} = {\hat{S}}_{4} (θ) ({| 0 ⟩}_{A} \otimes {| 0 ⟩}_{B}),

(25)

of the system, we use the logarithmic negativity,⁶³

E_{N} = max {0, - ln (2 η^{-})}

(26)

as entanglement measure, with

η^{-}

being the lowest symplectic eigenvalue of the partially transposed symmetric covariance matrix

γ^{T_{B}}

⁠. The symmetric covariance matrix is given by

γ_{i j} = \frac{1}{2} ⟨ {{\hat{R}}_{i}, {\hat{R}}_{j}} ⟩,

(27)

where we used that we only deal with

⟨ {\hat{R}}_{i} ⟩ = 0

⁠, which can always be achieved by local unitary operations.

The anticommutator is represented by

{\dots}

⁠, and

\hat{R}

is the phase space vector with

\hat{R} = (\begin{matrix} {\hat{q}}_{A} \\ {\hat{q}}_{B} \\ {\hat{p}}_{A} \\ {\hat{p}}_{B} \end{matrix}) = \underset{M}{\underset{︸}{\frac{1}{\sqrt{2}} (\begin{matrix} 1_{2 \times 2} & 1_{2 \times 2} \\ - i 1_{2 \times 2} & i 1_{2 \times 2} \end{matrix})}} \hat{a} = M U \hat{α},

(28)

where we used Eq. . The partial transposition of a state is defined as the transposition of a density matrix applied to only one subsystem, where the chosen subsystem is irrelevant for the computation of the symplectic eigenvalues.

The lowest symplectic eigenvalue of the partially transposed covariance matrix in terms of the squeezing parameters is given by

η^{-} = \frac{1}{2} | 2 | \frac{θ_{4}}{2 θ} sin (2 θ) - \frac{θ_{3} θ_{1}}{θ^{2}} {sin}^{2} (θ) | - \sqrt{1 + 4 {| \frac{θ_{4}}{2 θ} sin (2 θ) - \frac{θ_{3} θ_{1}}{θ^{2}} {sin}^{2} (θ) |}^{2}} |,

(29)

which is the central result of this work.

From Eq. (29), we see that all squeezing parameters influence the entanglement except for θ₂, because $Φ_{2}$ commutes with all other generators. To further investigate the entanglement, we concentrate on limiting cases. If only squeezing inside each FM is present (⁠ $θ_{3}, θ_{4} = 0, θ = i | θ_{1} |$ ⁠), entanglement between modes of the different FM vanishes, which is expected as there is no interaction between the FMs.

Only hybridization between the FMs (⁠ $θ_{1}, θ_{2}, θ_{4} = 0, θ = | θ_{3} |$ ⁠) results also in a vanishing entanglement as the vacuum for hybridized modes is equal to the vacuum of the bare modes $\hat{a}$ and $\hat{b}$ and thus is a separable state.

Two-mode squeezing between both FMs (⁠ $θ_{1}, θ_{2}, θ_{3} = 0, θ = i | θ_{4} |$ ⁠) results in entanglement linear in the squeezing parameter $E_{N} = 2 | θ_{4} |$ similar to the entanglement in the case of simple two-mode squeezing.¹⁰ Therefore, known results are reproduced by the analytic formula given in Eq. (29). Further, we see that squeezing, be it intra- or intermagnet squeezing, is necessary to establish entanglement, which is in agreement with earlier results.^64,65

The distance dependence of the entanglement can be seen in Fig. 5 where we regard finite and infinite 2D FMs. Figure 5 shows similar behavior of the entanglement for all lattice sizes for $l / a \leq 1$ ⁠. The phase transition from an OOP FM configuration to an IP AFM configuration appears around $l / a \approx 0.33$ ⁠, in the case of the infinite lattice, and entails a divergence of the entanglement. This is shifted to larger distances for finite lattices. The divergence once again arises due to the vanishing magnon energy at the phase transition.

FIG. 5.

Entanglement of the k = 0 magnon modes in FM A and FM B for infinite (blue, solid), 100 × 100 (orange, dotted), and 500 × 500 (green, dashed) sites FMs. We used D = 0.5 × 10 − 4 | J | , K z = 10 − 4 | J |, and K x = 10 − 6 | J |.

View large Download slide

Entanglement of the $k = 0$ magnon modes in FM A and FM B for infinite (blue, solid), 100 × 100 (orange, dotted), and 500 × 500 (green, dashed) sites FMs. We used $D = 0.5 \times 10^{- 4} | J |, K_{z} = 10^{- 4} | J |$ ⁠, and $K_{x} = 10^{- 6} | J |$ ⁠.

Aside from the divergence, the entanglement takes it maximum value at approximately $l / a \approx 0.5$ ⁠, rapidly decreases afterward, and, in the infinite case, vanishes already at $l / a \approx 2$ ⁠. The origin of the maximum is still up to debate and subject to future work. The finite cases with 100 × 100 and 500 × 500 lattices do show a plateau in the entanglement around $l / a \approx 2$ ⁠, which is the higher, the smaller the lattice is. Increasing the distance up to the order of the system diameters (100a or 500a, respectively) results in a decrease in entanglement that is proportional to $1 / l^{3}$ ⁠. This behavior is derived from the source of interaction: the dipole–dipole interaction, Eq. (6).

We infer that no ground-state long-range entanglement between infinite large magnets is present due to dipole interaction, while for finite magnets, the entanglement can take a finite value. Thus, for large magnetic structures, dipole interaction alone will not be enough to establish ground-state long-range entanglement between magnetic materials and other interactions or structures need to be considered. We remark that we always assumed periodic boundary conditions for magnons in finite systems. Also, for small distances (⁠ $l / a < 1$ ⁠), other interactions between the magnetic moments may be of significance, which require investigation in future work.

At this point, we would like to emphasize that we have studied an experimentally easily realizable system of two distant FMs. While the exact structure may differ from a simple cubic lattice structure, the main point, the interaction between the FMs via the dipole–dipole interaction, is present in any magnetic structure. The importance of this work is further emphasized by the possibility of using the entangled magnons in finite size magnets to establish long-range entanglement between spin-qubits, as proposed by Skogvoll et al.⁴⁵ or Yuan et al.⁴⁷ This is important for the future experimental realization of quantum computing.^48–51

We investigated the entanglement of magnon modes in two 2D ferromagnets coupled by dipole–dipole interaction by representing the Bogoliubov matrix as an element of the symplectic group to represent its matrix elements in terms of four squeezing parameters, each identified with either two-mode squeezing or hybridization.

Investigating the Bogoliubov matrix elements and the squeezing parameters shows a clear dominance of hybridization over squeezing in the OOP FM phase and a significant contribution of squeezing in the IP AFM phase, which is in agreement with already known results. The system shows a finite hybridization for all distances due to the long-range character of the dipole–dipole interaction, while the squeezing vanishes for large distances.

Using the logarithmic negativity, we derived an analytic expression for the entanglement in terms of the squeezing parameters and were able to reproduce known limiting cases and showed that besides squeezing between both FMs also, squeezing inside one FM paired with hybridization between both FMs is enough to ensure entanglement between both systems, while hybridization alone is not enough. This clearly shows the importance of squeezing for the entanglement of magnon modes. For infinite systems, the entanglement already becomes insignificant at $l / a = 2$ implying no long-range entanglement. However, for finite systems, we see a finite plateau in the entanglement before it tends toward zero for distances of the order of the system diameter. The value of this plateau decreases with increasing system size.

Concerning potential applications, we note that the current high interest in 2D van der Waals magnets²⁷ is explored experimentally worldwide. These systems are usually fabricated in small flakes, and our prediction might be relevant to discuss the entanglement of two (finite-size) flakes of two-dimensional ferromagnetic materials separated by an insulating layer, thus avoiding a direct electronic contact. These entanglement properties may have applications in quantum computing.⁴⁵ Future investigations could concentrate on the possible tuning of entanglement between two close FM structures by an applied magnetic field, which changes the configuration taken by the system or the effect of different orientations and lattice types of the two FMs.

SUPPLEMENTARY MATERIAL

See the supplementary material for extended calculations regarding the classical ground state, quantum ground state, and squeezing parameters.

This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via the Collaborative Research Center SFB 1432 Project Nos. 425217212 and 417034116.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

D. Wuhrer: Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). N. Rohling: Formal analysis (supporting); Software (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (equal). W. Belzig: Conceptualization (lead); Funding acquisition (lead); Methodology (equal); Project administration (lead); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

REFERENCES

1.

F. J.

Dyson

, “

General theory of spin-wave interactions

,”

Phys. Rev.

102

,

1217

–

1230

(

1956

).

https://doi.org/10.1103/PhysRev.102.1217

Google Scholar

Crossref

2.

M. L.

Néel

, “

Propriétés magnétiques des ferrites; ferrimagnétisme et antiferromagnétisme

,”

Ann. Phys.

12

,

137

–

198

(

1948

).

https://doi.org/10.1051/anphys/194812030137

Google Scholar

Crossref

3.

A. N.

Bogdanov

and

D. A.

Yablonskii

, “

Thermodynamically stable ‘vortices’ in magnetically ordered crystals. The mixed state of magnets

,”

Sov. Phys. JETP

68

,

101

–

103

(

1989

).

Google Scholar

4.

M.

Bode

,

M.

Heide

,

K.

von Bergmann

,

P.

Ferriani

,

S.

Heinze

,

G.

Bihlmayer

,

A.

Kubetzka

,

O.

Pietzsch

,

S.

Blügel

, and

R.

Wiesendanger

, “

Chiral magnetic order at surfaces driven by inversion asymmetry

,”

Nature

447

,

190

–

193

(

2007

).

https://doi.org/10.1038/nature05802

Google Scholar

Crossref

PubMed

5.

M.

Garst

,

J.

Waizner

, and

D.

Grundler

, “

Collective spin excitations of helices and magnetic skyrmions: Review and perspectives of magnonics in non-centrosymmetric magnets

,”

J. Phys. D

50

,

293002

(

2017

).

https://doi.org/10.1088/1361-6463/aa7573

Google Scholar

Crossref

6.

S. A.

Díaz

,

J.

Klinovaja

, and

D.

Loss

, “

Topological magnons and edge states in antiferromagnetic skyrmion crystals

,”

Phys. Rev. Lett.

122

,

187203

(

2019

).

https://doi.org/10.1103/PhysRevLett.122.187203

Google Scholar

Crossref

PubMed

7.

S. A.

Díaz

,

T.

Hirosawa

,

J.

Klinovaja

, and

D.

Loss

, “

Chiral magnonic edge states in ferromagnetic skyrmion crystals controlled by magnetic fields

,”

Phys. Rev. Res.

2

,

013231

(

2020

).

https://doi.org/10.1103/PhysRevResearch.2.013231

Google Scholar

Crossref

8.

C.

Kittel

,

Introduction to Solid State Physics

(

Wiley John and Sons

,

2004

).

Google Scholar

9.

W.

Nolting

and

A.

Ramakanth

,

Quantum Theory of Magnetism

(

Springer

,

Berlin/Heidelberg

,

2009

).

Google Scholar

Crossref

10.

H.

Yuan

,

Y.

Cao

,

A.

Kamra

,

R. A.

Duine

, and

P.

Yan

, “

Quantum magnonics: When magnon spintronics meets quantum information science

,”

Phys. Rep.

965

,

1

–

74

(

2022

).

https://doi.org/10.1016/j.physrep.2022.03.002

Google Scholar

Crossref

11.

A.

Barman

,

G.

Gubbiotti

,

S.

Ladak

et al, “

The 2021 magnonics roadmap

,”

J. Phys.: Condens. Matter

33

,

413001

(

2021

).

https://doi.org/10.1088/1361-648X/abec1a

Google Scholar

Crossref

12.

L. J.

Cornelissen

,

J.

Liu

,

R. A.

Duine

,

J. B.

Youssef

, and

B. J.

van Wees

, “

Long-distance transport of magnon spin information in a magnetic insulator at room temperature

,”

Nat. Phys.

11

,

1022

–

1026

(

2015

).

https://doi.org/10.1038/nphys3465

Google Scholar

Crossref

13.

R.

Lebrun

,

A.

Ross

,

S. A.

Bender

,

A.

Qaiumzadeh

,

L.

Baldrati

,

J.

Cramer

,

A.

Brataas

,

R. A.

Duine

, and

M.

Kläui

, “

Tunable long-distance spin transport in a crystalline antiferromagnetic iron oxide

,”

Nature

561

,

222

–

225

(

2018

).

https://doi.org/10.1038/s41586-018-0490-7

Google Scholar

Crossref

PubMed

14.

A. V.

Chumak

,

V.

Vasyuchka

,

A.

Serga

, and

B.

Hillebrands

, “

Magnon spintronics

,”

Nat. Phys.

11

,

453

–

461

(

2015

).

https://doi.org/10.1038/nphys3347

Google Scholar

Crossref

15.

A. V.

Chumak

,

A. A.

Serga

, and

B.

Hillebrands

, “

Magnon transistor for all-magnon data processing

,”

Nat. Commun.

5

,

45

–

53

(

2014

).

https://doi.org/10.1038/ncomms5700

Google Scholar

Crossref

16.

K. S.

Novoselov

,

D.

Jiang

,

F.

Schedin

,

T. J.

Booth

,

V. V.

Khotkevich

,

S. V.

Morozov

, and

A. K.

Geim

, “

Two-dimensional atomic crystals

,”

Proc. Natl. Acad. Sci. U. S. A.

102

,

10451

–

10453

(

2005

).

https://doi.org/10.1073/pnas.0502848102

Google Scholar

Crossref

PubMed

17.

K. S.

Novoselov

,

A.

Mishchenko

,

A.

Carvalho

, and

A. H.

Castro Neto

, “

2D materials and van der Waals heterostructures

,”

Science

353

(

6298

),

aac9439

(

2016

).

https://doi.org/10.1126/science.aac9439

Google Scholar

Crossref

PubMed

18.

K. F.

Mak

,

C.

Lee

,

J.

Hone

,

J.

Shan

, and

T. F.

Heinz

, “

Atomically thin MoS₂: A new direct-gap semiconductor

,”

Phys. Rev. Lett.

105

,

136805

(

2010

).

https://doi.org/10.1103/PhysRevLett.105.136805

Google Scholar

Crossref

PubMed

19.

B.

Radisavljevic

,

A.

Radenovic

,

J.

Brivio

,

V.

Giacometti

, and

A.

Kis

, “

Single-layer MoS₂ transistors

,”

Nat. Nanotechnol.

6

,

147

–

150

(

2011

).

https://doi.org/10.1038/nnano.2010.279

Google Scholar

Crossref

PubMed

20.

C.

Elias

,

P.

Valvin

,

T.

Pelini

,

A.

Summerfield

,

C. J.

Mellor

,

T. S.

Cheng

,

L.

Eaves

,

C. T.

Foxon

,

P. H.

Beton

,

S. V.

Novikov

,

B.

Gil

, and

G.

Cassabois

, “

Direct band-gap crossover in epitaxial monolayer boron nitride

,”

Nat. Commun.

10

,

2639

(

2019

).

https://doi.org/10.1038/s41467-019-10610-5

Google Scholar

Crossref

PubMed

21.

K. S.

Novoselov

,

A. K.

Geim

,

S. V.

Morozov

,

D.

Jiang

,

Y.

Zhang

,

S. V.

Dubonos

,

I. V.

Grigorieva

, and

A. A.

Firsov

, “

Electric field effect in atomically thin carbon films

,”

Science

306

,

666

–

669

(

2004

).

https://doi.org/10.1126/science.1102896

Google Scholar

Crossref

PubMed

22.

A. S.

Mayorov

,

R. V.

Gorbachev

,

S. V.

Morozov

,

L.

Britnell

,

R.

Jalil

,

L. A.

Ponomarenko

,

P.

Blake

,

K. S.

Novoselov

,

K.

Watanabe

,

T.

Taniguchi

, and

A. K.

Geim

, “

Micrometer-scale ballistic transport in encapsulated graphene at room temperature

,”

Nano Lett.

11

,

2396

–

2399

(

2011

).

https://doi.org/10.1021/nl200758b

Google Scholar

Crossref

PubMed

23.

S. J.

Haigh

,

A.

Gholinia

,

R.

Jalil

,

S.

Romani

,

L.

Britnell

,

D. C.

Elias

,

K. S.

Novoselov

,

L. A.

Ponomarenko

,

A. K.

Geim

, and

R.

Gorbachev

, “

Cross-sectional imaging of individual layers and buried interfaces of graphene-based heterostructures and superlattices

,”

Nat. Mater.

11

,

764

–

767

(

2012

).

https://doi.org/10.1038/nmat3386

Google Scholar

Crossref

PubMed

24.

A. K.

Geim

and

I. V.

Grigorieva

, “

Van der Waals heterostructures

,”

Nature

499

,

419

–

425

(

2013

).

https://doi.org/10.1038/nature12385

Google Scholar

Crossref

PubMed

25.

A.

Castellanos-Gomez

,

X.

Duan

,

Z.

Fei

,

H. R.

Gutierrez

,

Y.

Huang

,

X.

Huang

,

J.

Quereda

,

Q.

Qian

,

E.

Sutter

, and

P.

Sutter

, “

Van der Waals heterostructures

,”

Nat. Rev. Methods Primers

2

,

58

(

2022

).

https://doi.org/10.1038/s43586-022-00139-1

Google Scholar

Crossref

26.

H.

Yang

,

S. O.

Valenzuela

,

M.

Chshiev

,

S.

Couet

,

B.

Dieny

,

B.

Dlubak

,

A.

Fert

,

K.

Garello

,

M.

Jamet

,

D.-E.

Jeong

,

K.

Lee

,

T.

Lee

,

M.-B.

Martin

,

G. S.

Kar

,

P.

Sénéor

,

H.-J.

Shin

, and

S.

Roche

, “

Two-dimensional materials prospects for non-volatile spintronic memories

,”

Nature

606

,

663

–

673

(

2022

).

https://doi.org/10.1038/s41586-022-04768-0

Google Scholar

Crossref

PubMed

27.

M.

Gibertini

,

M.

Koperski

,

A.

Morpurgo

, and

K.

Novoselov

, “

Magnetic 2D materials and heterostructures

,”

Nat. Nanotechnol.

14

,

408

–

419

(

2019

).

https://doi.org/10.1038/s41565-019-0438-6

Google Scholar

Crossref

PubMed

28.

K. S.

Burch

,

D.

Mandrus

, and

J.-G.

Park

, “

Magnetism in two-dimensional van der Waals materials

,”

Nature

563

,

47

–

52

(

2018

).

https://doi.org/10.1038/s41586-018-0631-z

Google Scholar

Crossref

PubMed

29.

C.

Gong

,

L.

Li

,

Z.

Li

,

H.

Ji

,

A.

Stern

,

Y.

Xia

,

T.

Cao

,

W.

Bao

,

C.

Wang

,

Y.

Wang

,

Z. Q.

Qiu

,

R. J.

Cava

,

S. G.

Louie

,

J.

Xia

, and

X.

Zhang

, “

Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals

,”

Nature

546

,

265

–

269

(

2017

).

https://doi.org/10.1038/nature22060

Google Scholar

Crossref

PubMed

30.

B.

Huang

,

G.

Clark

,

E.

Navarro-Moratalla

,

D. R.

Klein

,

R.

Cheng

,

K. L.

Seyler

,

D.

Zhong

,

E.

Schmidgall

,

M. A.

McGuire

,

D. H.

Cobden

,

W.

Yao

,

D.

Xiao

,

P.

Jarillo-Herrero

, and

X.

Xu

, “

Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit

,”

Nature

546

,

270

–

273

(

2017

).

https://doi.org/10.1038/nature22391

Google Scholar

Crossref

PubMed

31.

F.

Schulz

,

K.

Litzius

,

L.

Powalla

,

M. T.

Birch

,

R. A.

Gallardo

,

S.

Satheesh

,

M.

Weigand

,

T.

Scholz

,

B. V.

Lotsch

,

G.

Schütz

,

M.

Burghard

, and

S.

Wintz

, “

Direct observation of propagating spin waves in the 2D van der Waals ferromagnet Fe₅GeTe₂

,”

Nano Lett.

23

,

10126

–

10131

(

2023

).

https://doi.org/10.1021/acs.nanolett.3c02212

Google Scholar

Crossref

PubMed

32.

X.

Ma

,

R.

Li

,

B.

Zheng

,

L.

Huang

,

Y.

Zhang

,

S.

Wang

,

C.

Wang

,

H.

Tan

,

Y.

Lu

, and

B.

Xiang

, “

Ferromagnetism above room temperature in two intrinsic van der Waals magnets with large coercivity

,”

Nano Lett.

23

,

11226

–

11232

(

2023

).

https://doi.org/10.1021/acs.nanolett.3c03716

Google Scholar

Crossref

PubMed

33.

C.

Gong

,

P.

Zhang

,

T.

Norden

,

Q.

Li

,

Z.

Guo

,

A.

Chaturvedi

,

A.

Najafi

,

S.

Lan

,

X.

Liu

,

Y.

Wang

,

S.-J.

Gong

,

H.

Zeng

,

H.

Zhang

,

A.

Petrou

, and

X.

Zhang

, “

Ferromagnetism emerged from non-ferromagnetic atomic crystals

,”

Nat. Commun.

14

,

3839

(

2023

).

https://doi.org/10.1038/s41467-023-39002-6

Google Scholar

Crossref

PubMed

34.

Y.

Xu

,

A.

Ray

,

Y.-T.

Shao

,

S.

Jiang

,

K.

Lee

,

D.

Weber

,

J. E.

Goldberger

,

K.

Watanabe

,

T.

Taniguchi

,

D. A.

Muller

,

K. F.

Mak

, and

J.

Shan

, “

Coexisting ferromagnetic–antiferromagnetic state in twisted bilayer CrI₃

,”

Nat. Nanotechnol.

17

,

143

–

147

(

2022

).

https://doi.org/10.1038/s41565-021-01014-y

Google Scholar

Crossref

PubMed

35.

Y.

Zhong

,

C.

Peng

,

H.

Huang

,

D.

Guan

,

J.

Hwang

,

K. H.

Hsu

,

Y.

Hu

,

C.

Jia

,

B.

Moritz

,

D.

Lu

,

J.-S.

Lee

,

J.-F.

Jia

,

T. P.

Devereaux

,

S.-K.

Mo

, and

Z.-X.

Shen

, “

From Stoner to local moment magnetism in atomically thin Cr₂Te₃

,”

Nat. Commun.

14

,

5340

(

2023

).

https://doi.org/10.1038/s41467-023-40997-1

Google Scholar

Crossref

PubMed

36.

R.

Cheng

,

L.

Yin

,

Y.

Wen

,

B.

Zhai

,

Y.

Guo

,

Z.

Zhang

,

W.

Liao

,

W.

Xiong

,

H.

Wang

,

S.

Yuan

,

J.

Jiang

,

C.

Liu

, and

J.

He

, “

Ultrathin ferrite nanosheets for room-temperature two-dimensional magnetic semiconductors

,”

Nat. Commun.

13

,

5241

(

2022

).

https://doi.org/10.1038/s41467-022-33017-1

Google Scholar

Crossref

PubMed

37.

D. F.

Walls

, “

Squeezed states of light

,”

Nature

306

,

141

–

146

(

1983

).

https://doi.org/10.1038/306141a0

Google Scholar

Crossref

38.

L.-A.

Wu

,

H. J.

Kimble

,

J. L.

Hall

, and

H.

Wu

, “

Generation of squeezed states by parametric down conversion

,”

Phys. Rev. Lett.

57

,

2520

–

2523

(

1986

).

https://doi.org/10.1103/PhysRevLett.57.2520

Google Scholar

Crossref

PubMed

39.

A.

Kamra

,

W.

Belzig

, and

A.

Brataas

, “

Magnon-squeezing as a niche of quantum magnonics

,”

Appl. Phys. Lett.

117

,

090501

(

2020

).

https://doi.org/10.1063/5.0021099

Google Scholar

Crossref

40.

A.

Kamra

,

E.

Thingstad

,

G.

Rastelli

,

R. A.

Duine

,

A.

Brataas

,

W.

Belzig

, and

A.

Sudbø

, “

Antiferromagnetic magnons as highly squeezed Fock states underlying quantum correlations

,”

Phys. Rev. B

100

,

174407

(

2019

).

https://doi.org/10.1103/PhysRevB.100.174407

Google Scholar

Crossref

41.

D.

Wuhrer

,

N.

Rohling

, and

W.

Belzig

, “

Theory of quantum entanglement and structure of the two-mode squeezed antiferromagnetic magnon vacuum

,”

Phys. Rev. B

105

,

054406

(

2022

).

https://doi.org/10.1103/PhysRevB.105.054406

Google Scholar

Crossref

42.

A.

Kamra

and

W.

Belzig

, “

Super-Poissonian shot noise of squeezed-magnon mediated spin transport

,”

Phys. Rev. Lett.

116

,

146601

(

2016

).

https://doi.org/10.1103/PhysRevLett.116.146601

Google Scholar

Crossref

PubMed

43.

J.

Zhao

,

A. V.

Bragas

,

D. J.

Lockwood

, and

R.

Merlin

, “

Magnon squeezing in an antiferromagnet: Reducing the spin noise below the standard quantum limit

,”

Phys. Rev. Lett.

93

,

107203

(

2004

).

https://doi.org/10.1103/PhysRevLett.93.107203

Google Scholar

Crossref

PubMed

44.

J.

Zhao

,

A. V.

Bragas

,

R.

Merlin

, and

D. J.

Lockwood

, “

Magnon squeezing in antiferromagnetic

{MnF}_{2}

and

{FeF}_{2}

,”

Phys. Rev. B

73

,

184434

(

2006

).

https://doi.org/10.1103/PhysRevB.73.184434

Google Scholar

Crossref

45.

I. C.

Skogvoll

,

J.

Lidal

,

J.

Danon

, and

A.

Kamra

, “

Tunable anisotropic quantum Rabi model via a Magnon–spin-qubit ensemble

,”

Phys. Rev. Appl.

16

,

064008

(

2021

).

https://doi.org/10.1103/PhysRevApplied.16.064008

Google Scholar

Crossref

46.

A.-L. E.

Römling

,

A.

Vivas-Viaña

,

C. S.

Muñoz

, and

A.

Kamra

, “

Resolving nonclassical magnon composition of a magnetic ground state via a qubit

,”

Phys. Rev. Lett.

131

,

143602

(

2023

).

https://doi.org/10.1103/PhysRevLett.131.143602

Google Scholar

Crossref

PubMed

47.

H. Y.

Yuan

,

A.

Kamra

,

D. M. F.

Hartmann

, and

R. A.

Duine

, “

Electrically switchable entanglement channel in van der Waals magnets

,”

Phys. Rev. Appl.

16

,

024047

(

2021

).

https://doi.org/10.1103/PhysRevApplied.16.024047

Google Scholar

Crossref

48.

S.

Wehner

,

D.

Elkouss

, and

R.

Hanson

, “

Quantum internet: A vision for the road ahead

,”

Science

362

,

eaam9288

(

2018

).

https://doi.org/10.1126/science.aam9288

Google Scholar

Crossref

PubMed

49.

G. J.

Milburn

and

S. L.

Braunstein

, “

Quantum teleportation with squeezed vacuum states

,”

Phys. Rev. A

60

,

937

–

942

(

1999

).

https://doi.org/10.1103/PhysRevA.60.937

Google Scholar

Crossref

50.

C. H.

Bennett

,

G.

Brassard

, and

N. D.

Mermin

, “

Quantum cryptography without Bell's theorem

,”

Phys. Rev. Lett.

68

,

557

–

559

(

1992

).

https://doi.org/10.1103/PhysRevLett.68.557

Google Scholar

Crossref

PubMed

51.

J.

Yin

,

Y.-H.

Li

,

S.-K.

Liao

,

M.

Yang

,

Y.

Cao

,

L.

Zhang

,

J.-G.

Ren

,

W.-Q.

Cai

,

W.-Y.

Liu

,

S.-L.

Li

,

R.

Shu

,

Y.-M.

Huang

,

L.

Deng

,

L.

Li

,

Q.

Zhang

,

N.-L.

Liu

,

Y.-A.

Chen

,

C.-Y.

Lu

,

X.-B.

Wang

,

F.

Xu

,

J.-Y.

Wang

,

C.-Z.

Peng

,

A. K.

Ekert

, and

J.-W.

Pan

, “

Entanglement-based secure quantum cryptography over 1,120 kilometres

,”

Nature

582

,

501

–

505

(

2020

).

https://doi.org/10.1038/s41586-020-2401-y

Google Scholar

Crossref

PubMed

52.

L.

Chomaz

,

I.

Ferrier-Barbut

,

F.

Ferlaino

,

B.

Laburthe-Tolra

,

B. L.

Lev

, and

T.

Pfau

, “

Dipolar physics: A review of experiments with magnetic quantum gases

,”

Rep. Prog. Phys.

86

,

026401

(

2023

).

https://doi.org/10.1088/1361-6633/aca814

Google Scholar

Crossref

53.

T.

Comparin

,

F.

Mezzacapo

, and

T.

Roscilde

, “

Multipartite entangled states in dipolar quantum simulators

,”

Phys. Rev. Lett.

129

,

150503

(

2022

).

https://doi.org/10.1103/PhysRevLett.129.150503

Google Scholar

Crossref

PubMed

54.

T. V.

Tscherbul

,

J.

Ye

, and

A. M.

Rey

, “

Robust nuclear spin entanglement via dipolar interactions in polar molecules

,”

Phys. Rev. Lett.

130

,

143002

(

2023

).

https://doi.org/10.1103/PhysRevLett.130.143002

Google Scholar

Crossref

PubMed

55.

G.

Furman

,

V.

Meerovich

, and

V.

Sokolovsky

, “

Entanglement of dipolar coupling spins

,”

Quantum Inf. Process.

10

,

307

–

315

(

2011

).

https://doi.org/10.1007/s11128-010-0198-6

Google Scholar

Crossref

56.

G.

Furman

,

V.

Meerovich

, and

V.

Sokolovsky

, “

Entanglement in dipolar coupling spin system in equilibrium state

,”

Quantum Inf. Process.

11

,

1603

–

1617

(

2012

).

https://doi.org/10.1007/s11128-011-0320-4

Google Scholar

Crossref

57.

G.

Bornet

,

G.

Emperauger

,

C.

Chen

,

B.

Ye

,

M.

Block

,

M.

Bintz

,

J. A.

Boyd

,

D.

Barredo

,

T.

Comparin

,

F.

Mezzacapo

,

T.

Roscilde

,

T.

Lahaye

,

N. Y.

Yao

, and

A.

Browaeys

, “

Scalable spin squeezing in a dipolar Rydberg atom array

,”

Nature

621

,

728

–

733

(

2023

).

https://doi.org/10.1038/s41586-023-06414-9

Google Scholar

Crossref

PubMed

58.

T.

Holstein

and

H.

Primakoff

, “

Field dependence of the intrinsic domain magnetization of a ferromagnet

,”

Phys. Rev.

58

,

1098

–

1113

(

1940

).

https://doi.org/10.1103/PhysRev.58.1098

Google Scholar

Crossref

59.

N. N.

Bogoliubov

, “

On a new method in the theory of superconductivity

,”

Nuovo Cimento

7

,

794

–

805

(

1958

).

https://doi.org/10.1007/BF02745585

Google Scholar

Crossref

60.

J. G.

Valatin

, “

Comments on the theory of superconductivity

,”

Nuovo Cimento

7

,

843

–

857

(

1958

).

https://doi.org/10.1007/BF02745589

Google Scholar

Crossref

61.

D.

Wuhrer

,

L.

Rózsa

,

U.

Nowak

, and

W.

Belzig

, “

Magnon squeezing in conical spin spirals

,”

Phys. Rev. Res.

5

,

043124

(

2023

).

https://doi.org/10.1103/PhysRevResearch.5.043124

Google Scholar

Crossref

62.

T.

Colas

,

J.

Grain

, and

V.

Vennin

, “

Four-mode squeezed states: Two-field quantum systems and the symplectic group

Sp (4, R)

,”

Eur. Phys. J. C

82

,

6

–

82

(

2022

).

https://doi.org/10.1140/epjc/s10052-021-09922-y

Google Scholar

Crossref

63.

G.

Vidal

and

R. F.

Werner

, “

Computable measure of entanglement

,”

Phys. Rev. A

65

,

032314

(

2002

).

https://doi.org/10.1103/PhysRevA.65.032314

Google Scholar

Crossref

64.

M. G. A.

Paris

, “

Entanglement and visibility at the output of a Mach-Zehnder interferometer

,”

Phys. Rev. A

59

,

1615

–

1621

(

1999

).

https://doi.org/10.1103/PhysRevA.59.1615

Google Scholar

Crossref

65.

M. M.

Wolf

,

J.

Eisert

, and

M. B.

Plenio

, “

Entangling power of passive optical elements

,”

Phys. Rev. Lett.

90

,

047904

(

2003

).

https://doi.org/10.1103/PhysRevLett.90.047904

Google Scholar

Crossref

PubMed

Dipole–dipole-interaction-induced entanglement between two-dimensional ferromagnets

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