Hierarchy of magnon entanglement in antiferromagnets

Continuous variable entanglement between magnon modes in Heisenberg antiferromagnet with Dzyaloshinskii-Moryia (DM) interaction is examined. Different bosonic modes are identified, which allows to establish a hierarchy of magnon entanglement in the ground state. We argue that entanglement between magnon modes is determined by a simple lattice specific factor, together with the ratio of the strengths of the DM and Heisenberg exchange interactions, and that magnon entanglement can be detected by means of quantum homodyne techniques. As an illustration of the relevance of our findings for possible entanglement experiments in the solid state, a typical antiferromagnet with the perovskite crystal structure is considered, and it is shown that long wave length magnon modes have the highest degree of entanglement.

Quantum entanglement allows particles to act as a single non-separable entity, no matter how far apart they are. This is the feature that was initially used in the Einstein-Podolsky-Rosen (EPR) argument against completeness of quantum mechanics [1]. The original form of the EPR argument is closely related to continuous variable (CV) entanglement [2][3][4], which describes entanglement between bosonic modes. Such systems are characterized by an infinite number of allowed states, which makes them very different from the finite-dimensional Hilbert spaces associated with discrete variable (e.g., qubit) systems. Nevertheless, just as discrete variable entanglement, CV entanglement provides an essential resource for quantum technologies allowing for universal quantum information processing [5], the realization of quantum teleportation [5][6][7], quantum memories [5,8], and quantum enhanced measurement resolution [9].
It is natural to expect that quantum systems, in which information is carried in a wave-like form, in general can show entanglement. However, the question is how clear the entanglement can be demonstrated and what quantum systems might be appropriate for potential applications. In the solid state, there are several collective modes that could be suitable hosts of entanglement. Here, we focus on magnons, collective wave-like excitations of a magnet with a well-established quantum nature [10]. Typically magnons can be found in energy range of up to ∼ 500 meV, and with wave lengths spanning a range of hundreds of lattice constants to just a few. Low energy magnon excitations can be observed in different classes of magnetic materials, such as ferromagnets, antiferromagnets, and ferrimagnets, and each class has a vast space of materials to choose from. One of the broadest classes of antiferromagnets can be found in oxide compounds, in particular transition metal oxides [11], where for long wave lengths the dispersion relation is essentially linear.
In this paper, we focus on magnon CV entanglement in antiferromagnets, in which both the Heisenberg exchange and the Dzyaloshinskii-Moriya (DM) interactions may be relevant [10]. To begin with, we consider the quantum antiferromagnetic Heisenberg Hamiltonian on a bipartite lattice where S j is the spin-S operator on site j and J is the strength of the exchange interaction. Using the Holstein-Primakoff (HP) transformation at low temperatures (k B T J) followed by the Fourier transformation, one can express the spin Hamiltonian (trivial terms and zero-point energies are neglected from now on) in terms of bosonic operators as with the lattice specific parameter γ k = 1 z δ e ik·δ , z being the coordination number of the lattice and the sum over δ is carried out over nearest neighbors. Here, a † k (a k ) and b † k (b k ) are bosonic creation (annihilation) operators representing two magnon modes with wave vector k that are associated with the two sublattices (see Supplemental Material).
By employing the Bogoliubov transformation arXiv:2006.03479v1 [quant-ph] 5 Jun 2020 with u k and v k given by where |γ k | < 1, we obtain the Hamiltonian in diagonal form in terms of the new bosonic operators (α, β). For the antiferromagnetic magnon dispersion relation, we find The ground state of H 0 in the (α, β) modes reads |ψ 0 = k |0; α k |0; β k , which is a separable vacuum state with vanishing entropy of entanglement [12], i.e., E (α,β) 0 = 0. By making the inverse Bogoliubov transformation, we may express the ground state as with the two-mode generalized coherent state in the (a, b) occupation number basis |n; a k and |n; b k (see Supplemental Material for derivation). Here, the parameter r k and the phase φ k are given by e iφ k tanh evaluates CV entanglement between two magnon modes a k and b k . This expression indicates that the magnon CV entanglement in the (a, b) modes is, in the low temperature regime, solely determined by the lattice geometry encoded in the γ k parameter. Fig. 1 shows how the entropy of entanglement for the two-mode generalized coherent state varies with |γ k |, and the inset shows its dependence on k. The analysis presented here is appropriate for many classes of compounds. A concrete example that is known to exhibit only nearest neighbor Heisenberg exchange is SrMnO 3 with J = 17.1 meV [13]. The magnon dispersion of this magnetic insulator is known, both from experiments and theory, and it is shown in Fig. 1 (inset), where the band width illustrates the entropy of entanglement as a function of k. As is clear from the figure, when |γ k | approaches 1, the two-mode magnon entanglement becomes stronger and the entropy of entanglement formally diverges. The fact that the entanglement is largest close to the zone center is important since magnons typically are more distinct and long-lived in this regime, in comparison to the more short wave length magnons that have higher damping [10]. for the two-mode generalized coherent state |r k , φ k (red curve) and for the corresponding first and second excited states as a function of |γ k |. The magnon CV entanglement in the first excited states (α † k |r k , φ k and β † k |r k , φ k ) is shown in black. Blue and orange curves illustrate E The inset depicts magnon dispersion of SrMnO3 for a selected path of k along high-symmetry directions of the BZ. The width of the bands depicts the entropy of entanglement.
Although the main focus of our study is on the ground state, we also consider magnon CV entanglement for the two-fold degenerate first excited states α † k |r k , φ k and β † k |r k , φ k , as well as for the three-fold degenerate second excited states (α † k ) 2 |r k , φ k , (β † k ) 2 |r k , φ k , and α † k β † k |r k , φ k . For these states, we find that the entropy of entanglement behaves in a similar way as for the ground state, see Fig. 1, though being slightly larger.
The arithmetic mean of the squared quadrature variances is directly related to the geometry of the spin lattice with Re[γ k ] being the real part of γ k . Here, X A k (X B k ) and P A k (P B k ) are the dimensionless position and momentum quadratures of the a k (b k ) mode [14], and Var r k ,φ k (V ) is the variance of a given Hermitian operator V with respect to the state |r k , φ k .
For ∆(r k , φ k ) < 1, which corresponds to Re[γ k ] < 1 − |γ k | 2 − 1, the two-mode generalized coherent state |r k , φ k is a two-mode squeezed state with mean variance being the associated EPR-uncertainty [3]. For ∆(r k , φ k ) 1, on the other hand, the EPR-uncertainty is constant and equal to 1. In the latter case, the amount of nonlocal correlations vanishes [3] although the magnon CV entanglement is nontrivial; thus, the magnon CV entanglement can only be related to the EPR-uncertainty in the squeezing domain.
The relation in Eq. (9) allows one to evaluate the CV entanglement in terms of ∆(r k , φ k ). For real γ k , which correspond to φ k = 0 or π, we find The parameter ∆(r k , φ k ), which depends on the lattice geometry and the choice of k, can be accessed experimentally by detecting coherences of quantum fields, the quadratures, with homodyne detection techniques [15,16] adapted to a possible magnon-photon coupling [17,18]. This may be an avenue forward for experimental detection of the magnon CV entanglement.
In order to further explore the material-specific features of magnon entanglement, we consider a more general spin Hamiltonian, that also has Dzyaloshinskii-Moriya interaction, with H DM = ij D ij · (S i × S j ) being the DM term with D ij = −D ji pointing along the same fixed direction D for all nearest neighbor spin pairs. By assuming D = |D|, H takes the form which is not diagonal anymore in the (α, β) modes. Without loss of generality, we assume real-valued u k in the Bogoliubov transformation of Eq. (3). The second summation of off-diagonal terms on the right hand side implies that there is mixing between α and β modes in the presence of the DM interaction. This may cause extra magnon CV entanglement in the ground state of the system. To see this, we diagonalize H by applying another Bogoliubov transformation where ζ k and η k are given by provided Γ k = iDγ k J √ 1−|γ k | 2 with |Γ k | < 1. In the (α k ,β k ) modes, the Hamiltonian H takes the diagonal form with the dispersion relation The ground state of the diagonal Hamiltonian is a product state |ψ = k |0;α k |0;β k , where |0;α k and |0;β k are vacuum states ofα k andβ k , respectively. In this basis, the magnon entanglement is absent. Using the inverse transformation back into the (α, β) modes, we express the ground state as with the entangled two-mode generalized coherent state where |n; α k and |n; β k are the nth excitation of α k and β k , respectively. Here,r k andφ k are specified by e iφ k tanhr k = ζ k η * k withr k ≡r k (γ k , D J ) 0, and In the case of D = 0, the only relevant term is n = 0, i.e., |r k ,φ k = |0; α k |0; β k and thus |ψ = |ψ 0 . The entropy of entanglement in the (α, β) modes is a function of |γ k | and the relative coupling strength In the (α, β) modes, the two-mode magnon entanglement, Eq. (18), is non-trivial, while, as shown above, the symmetric Heisenberg interaction H 0 on its own does not generate any magnon entanglement in these modes, i.e., E (α,β) 0 = 0. Thus, the antisymmetric DM interaction is mainly responsible for the entanglement contribution in Eq. (18), and we identify E (α,β) DM = E (α,β) as the DMinduced entanglement.
To have a clearer picture of the hierarchy of the magnon CV entanglement, we transform the ground state |ψ back into the original (a, b) modes with the two-mode generalized coherent state The total entropy of entanglement in the (a, b) modes for this state is Since E (a,b) | D=0 = E (a,b) 0 , we may write where E (a,b) DM vanishes in the absence of DM interaction, and we refer to it as the DM-induced entanglement in the (a, b) modes. Unlike the (α, β) modes, in the (a, b) modes both the Heisenberg and the DM interactions induce nonzero contributions to the magnon entanglement in the ground state.
The magnon CV entanglement is an intrinsic property of antiferromagnets that depends on the geometry of the spin lattice as encoded in γ k and on the relative coupling strength D J . Both parameters are material-dependent and can vary strongly from system to system. This opens an interesting route to search for suitable entanglement hosts among the existing thousands of magnetic compounds, and poses a natural question of how magnon entanglement can be detected in an experiment. Similar to what was discussed above in the pure Heisenberg case, the magnon CV entanglement in the presence of DM interaction can be measured experimentally by detecting quadratures corresponding to the arithmetic mean variance ∆(r k ,φ k ) in a homodyne detection setup [15,16] adapted to a possible magnon-photon coupling [17,18]. In the (a, b) modes, one may extract the parameterr k from ∆(r k ,φ k ) = cosh 2r k − sinh 2r k cosφ k (23) to evaluate the total entanglement E (a,b) in Eq. (21).
Here, we used Eq. (9) replacing the state |r k , φ k by the state |r k ,φ k .
We conclude with some final remarks. In the analysis of different bosonic modes, we notice different types of two-mode magnon entanglement residing in the ground state. In Fig. 4, we compare entropies of entanglement for an antiferromagnet with a simple cubic crystal structure, where the Hamiltonian is effectively described by nearest neighbor Heisenberg exchange as well as DM interaction with typical ratio D/J ≈ 30% [19][20][21]. It can be seen that from (a, b) modes to (α, β) modes the Heisenberg contribution to entanglement decreases while the DM-induced magnon CV entanglement increases. This is due to the fact that different bosonic modes represent different tensor product structures of the Hilbert space [22]. While the (a, b) modes describe naturally identifiable magnon modes, being associated with each sublattice, the (α, β) and (α,β) modes are hybridized and their number states are described by superpositions of excitations in the (a, b) modes. Although the stronger entanglement, a feature particularly useful in quantum information science and technology, is available in the (a, b) modes, the usefulness of these modes in an actual experiment must be determined by a suitable tensor product decomposition.
The condition for diagonalizing the Hamiltonians in terms of bosonic operators is that |γ k | < 1 in the pure Heisenberg case and |γ k | 2 < J 2 J 2 +D 2 in the presence of DM interaction. Since D is typically less than a few tenths of J for most materials [19][20][21], this condition is not satisfied only for a very small part of the BZ, e.g., the region around zone center. We would like to remark that the entire BZ may be included in this analysis by considering a Hamiltonian that possesses single ion uniaxial anisotropy, −K(n · S) 2 , e.g., with easy-axis n along the direction of the DM vector, as long as |1+ 2K zJ | > 1 + D 2 J 2 . Note that any change of symmetry in the Hamiltonian introduces new magnon modes and hence new levels of entanglement contribution in the hierarchy of two-mode magnon entanglement. Our general message is not changed by this, although the technical level of calculations may become more intricate. It is interesting to note that already very mild uniaxial anisotropy of 0.001% of J along z axis, when included in a pure Heisenberg Hamiltonian (D = 0), allows to regularize the magnon CV entanglement dependence on |γ k |. At |γ k | = 1, we obtain E (a,b) 0 ≈ 9.04. In a more generic case of Heisenberg-DMI with D/J = 0.1, z = 6 and uniaxial anisotropy of K/J =0.015 along D, we find E (a,b) ≈ 8.094 and E (α,β) ≈ 4.766 at |γ k | = 1.
In contrast to Ref. [23], where photoinduced spin dynamics was employed to trigger entanglement between a pair of magnon modes, our analysis shows that ground state two-mode magnon entanglement in antiferromagnets is an intrinsic property of the magnetic structure that is already given by the geometry of the spin lattice and exchange couplings, which should be accessible to experimental detection. We have examined the magnon entanglement in quantum magnetic structures with nearest neighbor antiferromagnetic Heisenberg exchange and DM interaction. The analysis is appropriate for many classes of compounds, but we would like to mention in particular the transition metal oxides that have a vast crystallographic phase space, which allows both for tunability of D J as well as γ k . A concrete example that is known to exhibit only nearest neighbor Heisenberg exchange is SrMnO 3 [13]. We also note that materials like La 2 CuO 4 [24], FeBO 3 , and CoCO 3 [25] are well studied antiferromagnets that are known to have DM interaction in the here studied range of D J .