Transmon probe for quantum characteristics of magnons in antiferromagnets

The detection of magnons and their quantum properties, especially in antiferromagnetic (AFM) materials, is a substantial step to realize many ambitious advances in the study of nanomagnetism and the development of energy efficient quantum technologies. The recent development of hybrid systems based on superconducting circuits provides the possibility of engineering quantum sensors that exploit different degrees of freedom. Here, we examine the magnon-photon-transmon hybridisation based on bipartite AFM materials, which gives rise to an effective coupling between a transmon qubit and magnons in a bipartite AFM. We demonstrate how magnetically invisible magnon modes, their chiralities and quantum properties such as nonlocality and two-mode magnon entanglement in bipartite AFMs can be characterized through the Rabi frequency of the superconducting transmon qubit.


I. INTRODUCTION
During the last decade, there have been considerable advancements in the use of magnons for storing, transmitting, and processing information.This rapid progress has turned the emerging research field of magnonics into a promising candidate for innovating information processing technologies [1].The combination of magnonics with quantum information processing provides a highly interdisciplinary physical platform for studying various quantum phenomena in spintronics, quantum electrodynamics, and quantum information science.Indeed, the quantum magnonics exhibits distinct quantum properties, which can be utilized for multi-purpose quantum tasks [2][3][4][5][6].
Here, we examine the possibility to combine the advantageous features of transmon and AFM materials.To this end, we demonstrate effective coupling between a superconducting transmon qubit and a bipartite AFM material.We show how the polarized (chiral) magnons and bipartite magnon-magnon entanglement in the AFM can be detected through the measurement of Rabi frequency of the transmon qubit.The proposed setup is suitable for the experimental study of the quantum properties of magnons in a wide range of crystalline and synthetic AFM materials, such as NiO and MnO, MnF 2 and FeF 2 , two-dimensional Ising systems like MnPSe 3 , YIG-based synthetic AFMs, and perovskite manganites [21][22][23][24][25][26][27][28][29].
The outline of the paper is as follows : In sec.II we describe magnon-photon-transmon hybridization and derive the interacting Hamiltonian.In sec.III, we discuss two-mode magnon entanglement in AFM materials.In sec.IV, we obtain an effective magnon-transmon coupling and show how this effective coupling mechanism allows to experimentally study quantun charachteristics of magnons in antiferromagnetic materials.The paper ends with a conclusion in sec.V.

II. MAGNON-PHOTON-TRANSMON HYBRIDIZATION
In this section, we describe a photon-mediated coupling mechanism between a superconducting transmon qubit and polarized magnons in a bipartite AFM.We assume a hybrid system composed of a single crystal or synthetic AFM, a transmon-type superconducting qubit, and a microwave cavity, as illustrated in Fig. 1.The FIG. 1: (Color online) Schematic illustration of magnonphoton-transmon hybridization.A circularly polarized microwave cavity electromagnetic field, which is described by the vector potential A R;k (r, t), can interact with magnons in an antiferromagnetic material and a superconducting transmon qubit.The cavity walls are illustrated with yellow segments in the left panel.An antiferromagnetic material hosts two chiral magnons, which are shown with three-color balls in the cubic lattice inside the cavity.Two magnos are degenerate in the absence of magnetic field and a small external magnetic field B in the z direction breaks this degenerecy (see also Fig. 2).While the coupling between magnon and cavity filed is achieved through magnetic-dipole interaction, an electricdipole interaction describe coupling between cavity filed and transmon (right panel).
system hosts four modes including two magnon modes in an AFM compound, a transmon qubit, and a microwave cavity electromagnetic mode.The dynamics of the hybridized magnon-photon-transmon system can be described by the Hamiltonian where the term H m describes the magnon subsystem, H ph describes the microwave photon, H m-ph describes the magnon-photon interaction, H q describes the transmon and H ph-q describes the photon-transmon interaction.They are described in detail as follows: Two-mode magnon system: H m represents a twomode magnon Hamiltonian in a bipartite treatment of an AFM material.Consider an AFM spin Hamiltonian i,j S i I ij S j + i B • S i , where S i is the spin vector operator at lattice site i, I ij is the bi-linear interaction tensor matrix between sites i and j, and B is an external field.By applying the Holstein-Primakoff transformation at low temperature followed by the Fourier transformation to the AFM spin Hamiltonian, H m can be described in terms of a pair of interacting collective bosonic modes in the lattice momentum k-space as [8,9] (we assume = 1 throughout the paper) The a † k (a k ) and b † −k (b −k ) are bosonic creation (annihilation) operators on the two sublattices A and B with opposite magnetizations in the bipartite AFM.Bosonic operators on opposite sublattices commute and define a pair of interacting magnons in the Kittel (a, b) modes.The Kittel modes can be hybridized into the diagonal magnon modes (α, β) through the SU(1,1) Bogoliubov transformation where u k = cosh(r k ) and v k = sinh(r k )e iφ k with In terms of the (α, β) modes, the magnon Hamiltonian H k m takes the diagonal form The bosonic diagonal modes α and β describe two right and left circularly polarized (chiral) magnons [1,30], which are degenerate in the absence of an external magnetic field.As shown in Fig. 2, for a system with only diagonal components of J ij (=J), a magnetic field in the z direction, i.e., parallel to the magnetization of the two sublattices, breaks the degeneracy [8,9].Microwave photon: For the second term of the hybrid Hamiltonian in Eq. ( 1), we assume a right circularly polarized microwave cavity electromagnetic field with the single cavity mode frequency ω c k [9,18,19,30].This is described by the vector potential The vector k is the propagation direction of the electromagnetic wave, A 0 is the amplitude of the vector potential, and c k (c † k ) is the annihilation (creation) operator of the right circularly polarized photon with unit vector e R = 1 √ 2 (1, −i, 0).Both ω c k and A 0 can be tuned by changing the volume of the cavity and the separation distance between the two conductor plates in the cavity.Here, we focus on the lowest energy cavity mode and disregard contributions from the higher energy cavity modes.In the rotating frame, the photon contribution to the full Hamiltonian in Eq. ( 1) is for a given k.
Magnon-Photon interaction: By turning on the electromagnetic field, the magnon modes start to interact with the cavity mode through the magnetic-dipole coupling.Explicitly, the electromagnetic field induces a magnetic field B ph , which interacts with the total spin S of the AFM material through the Zeeman interaction term [9,18,19,30] In the rotating frame, the photon-induced magnetic field is given by B ph = ∇ × A k (r, 0).Following the bosonization procedure used to derive the Hamiltonian H k m , we obtain the bosonized resonant magnon-photon interaction Hamiltonian The off-resonant interaction (−g k m-ph c k β −k + H.c.) is neglected due to energy conservation.Here, the magnonphoton exchange coupling is with λ k = A 0 k √ S and we choose to study the case when k = (0, 0, k).
Transmon qubit: The third subsystem consists of a superconducting qubit described by the Hamiltonian [31] where the first term corresponds to the kinetic energy contribution from a capacitor and the second term is the potential energy contribution by a Josephson junction.At a sufficiently large E J /E C , the superconducting system enters the transmon qubit regime.Following the ladder operator approach, one may represent the momentum, n, and position, φ, operators in terms of bosonic annihilation (creation) operator η (η † ) as By using the ladder representation, one can write the Hamiltonian in Eq. ( 11) in the form of the following anharmonic oscillator Hamiltonian This follows from a Taylor expansion of the potential energy term in Eq. ( 11) and a rotating wave approximation.
Here, ω q = √ 8E C E J −E C defines the Rabi transition frequency between the ground state |g and the first excited state |e , ξ = E C is the anharmonicity.In the transmon regime, the anharmonicity is negative and large enough that allows one to focus on the two lowest energy levels of the anharmonic oscillator as a transmon qubit, the Hamiltonian of which can be conveniently reduced to Photon-transmon interaction: The large electric dipole of the superconducting qubit, d = dη † +d * η, can strongly couple to the induced electric field of the microwave photon through electric-dipole coupling [31] H where dt determines the photon-induced electric field.If we assume d||e R , then, under the rotating wave approximation, the photon-qubit interaction is described by the Hamiltonian where the photon-qubit exchange coupling is given by with d = |d| being the strength of electric dipole of the superconducting transmon qubit.
Having specified each term in the Hamiltonian of Eq. ( 1), we conclude that the magnon-photon-transmon hybrid system is explicitly described by the bosonized Hamiltonian for a momentum k vector in the z-direction, the in-plane parallel photon polarization vector e R , and the superconducting dipole d||e R .
It is important to note that only the hybridized magnon in the α mode interacts with the photon and transmon modes in the Hamiltonian in Eq. (18).In other words, the β magnon mode is effectively decoupled from the other modes in the system.This is due to the fact that we use the right circularly polarized microwave cavity electromagnetic field, which only couples to the magnon with the same polarization, the α mode.On the one hand, if we use a left circularly polarized cavity field, it couples the β magnon mode with the photon and the transmon modes, and instead leaves the α magnon mode decoupled from the rest of the system.The hybrid quantum system described by Eq. ( 18) provides a promising platform to observe and verify quantum effects in quantum magnonics and exploit them for new quantum applications.Below we employ this hybrid platform to propose a new experimental setup for observing polarized twin magnon modes as well as intrinsic two-mode magnon entanglement in bipartite AFM materials via a transmon qubit.In the next section we briefly describe the basic concepts of two-mode entanglement in AFMs.

III. MAGNON-MAGNON ENTANGLEMENT
Let us focus on the two-mode magnon Hamiltonian described by H k m above.The coupling parameter g k m-m in Eq. ( 2), which is mainly given by the AFM coupling between the two opposite sublattices A and B, introduce a strong squeezing and entanglement between bosonic magnon modes in a way that all the eigenstates of H k m become entangled in the Kittel (a, b) modes [8,9].Explicitly, the complete energy eigenbasis of the Hamiltonian H k m can be expressed in the following form for positive integers x and y, and the two-mode squeezed vacuum ground state given in the Kittel (a, b) magnon basis.Here, x and y represent the number of magnons in the hybridized magnon modes α k and β −k , respectively.Note that the hybridized magnon modes (α, β) are related to the Kittel magnon modes (a, b) through Eq. (3).
Fig. 3 illustrates the entropies of entanglement of the energy eigenbasis in Eq. ( 19) for selected pairs of magnon numbers (x, y) as functions of the squeezing parameter r k .The squeezing parameter r k , which is given in Eq. ( 4) by the ratio of the magnon-magnon coupling g k m-m to the average single magnon energies in the Kittel modes, is actually the only parameter that determines the entropies of entanglement of the complete energy eigenbasis.This follows from the fact that the states in Eq. ( 19) are determined by (r k , φ k ) and φ k contributes only to the phase factors of the Schmidt coefficients in the Schmidt decompositions of these states.
We remind the reader that the entropy of entanglement for a bipartite state |ψ ∈ H A ⊗ H B is given by with χ n 's being the Schmidt coefficients in |ψ = n χ n |i n ; A |j n ; B , where |i n ; A and |j n ; B are orthonormal states in subsystem A and subsystem B, respectively [32].
For the energy eigenstates in Eq. ( 19), we obtain the following normalized Schmidt decompositions where δ = |x − y|.Here, the Schmidt coefficients are given by p (x,y) for m = min{x, y}, with and f (m,δ) n;k that satisfies the following recursive relations with f (0,0) n;k = 1 for each n.From Eqs. ( 23)- (25), it is clear that the absolute value of the Schmidt coefficients |p (x,y) n;k |, and thus the entanglement entropies of all energy eigenbasis states in the Kittel magnon modes (a, b), namely, are single variable functions of the squeezing parameter r k .In other words, the squeezing parameter r k is the only entanglement parameter that determines two-mode magnon entanglement in the AFM system described by In the following we show how a superconducting transmon qubit can be used to observe different magnons and the squeezing/entanglement parameter r k .The latter allows us to quantify quantum characteristics such as twomode squeezing and entanglement in AFM materials.

IV. SENSING MAGNONS AND THIER QUANTUM CHARACTERISTICS WITH TRANSMONS A. Magnon-transmon effective coupling
The Hamiltonian in Eq. ( 18), that allows for magnonphoton-transmon hybrid states, provides an effective photon mediated magnon-transmon coupling.To determine this effective coupling rate one may use the Schrieffer-Wolff unitary transformation [33], to effectively decouple the photon mode from magnon and transmon modes in the hybrid Hamiltonian up to first order.Consider the following decomposition of the hybrid Hamiltonian in Eq. ( 18) where we neglect the magnon β mode as it is decoupled from the rest of the Hamiltonian H k .By using the Baker-Campbell-Haussdorf formula, the transformation in Eq. ( 27) can be expanded as This three-mode Schrieffer-Wolff Hamiltonian can be made block diagonal turning the system into a two-mode magnon-transmon subsystem decoupled from a one-mode photon subsystem by choosing the generator W k such that By substituting the solution of Eq. ( 30) into Eq.( 29), one can obtain the standard form of the Schrieffer-Wolff Hamiltonian up to first order in the interaction term V k .Equation ( 30) always has a definite solution as the perturbative component V k is off-diagonal in the eigenbasis of H k;0 .By solving Eq. ( 30), we obtain the generator of the Schrieffer-Wolff transformation that leads to the following block diagonal hybrid Hamiltonian where As the photon mode is effectively decoupled from the rest of the Hamiltonian in Eq. ( 33), the effective magnontransmon interacting Hamiltonian reads

B. Transmon-qubit to probe magnons and their quantum characteristics in AFMs
The computational basis of the transmon qubit consits of the ground and first excited states |0 ≡ |g and |1 ≡ |e , respectively, of the anharmonic oscillator in the transmon regime.In this case, the raising and lowering operators of the transmon qubit can be represented as η † = |1 0| and η = |0 1|.The eigenstates of the number operator are {|0, 0 ; |1, 0 , |0, 1 ; ...; |n, 0 , |n − 1, 1 ; ...}, where the first entry counts the number of magnons in the hybridized mode α and the second entry labels the qubit state.These eigenstates span the magnon-qubit Hilbert space.The number operator commutes with the effective Hamiltonian in Eq. ( 35), i.e., This implies that the effective Hamiltonian takes the block diagonal form: where n is the eigenvalue of the number operator N k , i.e., counts the total number of magnon and transmon excitations.Except for the case n = 0 that the Hamiltonian submatrix is a 1D block, for each n > 0 the block Hamiltonians H k;n m−q are 2 × 2 matrix of the form with ∆ k = ω α k − ω q /2 being the detuning between magnon and qubit frequencies.By shifting the qubit energy levels |0 and |1 with the amount of ∆ k , we may rewrite the Hamiltonian in Eq. ( 39) as a effective single transmon qubit Hamiltonian for each n.Here, Ω k = Ω x k +iΩ y k = g k m-q characterizes the Rabi frequency of the qubit, I is the 2 × 2 identity matrix and σ l , l = x, y, z, are the Pauli matrices in the ordered effective qubit basis {|n, 0 , |n − 1, 1 }.This Hamiltonian results in the following energy eigensystem: with Suppose the transmon qubit is initialized in the state |0 at time t = 0 for a fixed n, for instance n = 1, that is |ψ(0) = |1, 0 .Governed by the effective qubit Hamiltonian in Eq. ( 40), the initial state evolves to after time t, which give rise to the following Rabi oscillation This indicates that the probability of finding the transmon qubit in the state |1 after time t oscillates with the frequency and intensity Note that the maximum intensity I k = 1 occurs at the zero detuning ∆ k = 0, which is equivalent to the following qubit parameter tuning The detuning can be achieved, for instance, by appropriate adjustments of photon frequency and amplitude of vector potential as well as an applied magnetic field in the z direction, as depicted in Fig. 1.As a result of zero detuning, the angular frequency of the Rabi oscillation becomes where is the the Einstein-Podolsky-Rosen (EPR) function for the two-mode ground state |ψ 00 (r k , φ k ) given by Eq. ( 20) [8,9] (see appendix for details about EPR).The EPR function, which characterizes the Bell-type nonlocal correlations known as EPR nonlocality, is a highly relevant concept in the study of continuous variable entanglement [34,35].We can always assume the parameter Γ k in Eq. ( 4) to be real-valued, in which case φ k = 0 or π and thus Since the ground state EPR function and the magnonmagnon entanglement entropies all depend on the same entanglement (squeezing) parameter, one may observe the magnon-magnon entanglement through the EPR function ∆ [ψ 00 (r k , φ k )] and in fact through the qubit angular frequency in Eq. (47) of the Rabi oscillation.For instance, we obtain the entanglement entropy for the twomode ground state as a function of the qubit angular frequency through for φ k = 0, π.Eq. ( 51) follows from Eqs. ( 47) and (49).The entanglement entropies of all magnon eigenbasis states given by Eq. ( 26) are actually functions of the qubit angular frequency through the relation in Eq. ( 51).
In practice the entanglement entropy, Eq. ( 50), is a function of the parameter r k , which can be identified by Eq. ( 51) once the qubit angular frequency f k has been determined experimentally.Figure 4 illustrates, as an example, the two-mode magnon entanglement in the ground (vacuum) state and number of excited states against the EPR function ∆ [ψ 00 (r k , φ k )] ∝ f k , for AFM spin lattices.Two distinct regions, the non-local bipartite entangled state, φ k = π, and the local bipartite entangled state, φ k = 0, with transition point at ∆ [ψ 00 (r k , φ k )] = 1 can be distinguished in Fig. 4. The region of stronger magnon-magnon entanglement for non-local two-mode magnon state is observed by the EPR uncertainty relation ∆ [ψ 00 (r k , φ k )] < 1.The clear relation between the EPR function and the two-mode magnon entanglement entropy allows for experimental quantification of magnon-magnon entanglement through the EPR function ∆ [ψ 00 (r k , φ k )] and indeed the frequency, f k , of Rabi oscillation of the transmon qubit.It is worth mentioning that the EPR nonlocality has been used for verification of entanglement in optical and atomic systems based on homodyne detection and types of interferometry setups [36][37][38][39][40][41][42].However, these types of measurement setups are not realistic for magnon systems, since these technologies are mainly based on beam splitters that have limitations for characterizing magnon entanglement.We propose as a solution, a mechanism and measurement setup that rely on qubit-light-matter interaction as a probe to observe the EPR function and thus EPR nonlocality and the degree of magnon-magnon entanglement.Moreover, Eq. ( 46) shows that at the zero detuning, the magnon frequency in the hybridized α mode can also be observed through qubit frequency.
A similar procedure and formulation as above hold if we couple the transmon qubit to a bipartite AFM material instead through the magnon β mode, for instance, by using oppositely (left) circularly polarized light.Using different polarization for the photon would allow one to detect the twin chiral magnon modes in bipartite AFM materials.Fig. 5 shows that the angular frequency f k of the Rabi oscillation of a transmon qubit can observe and distinguish the two hybridized magnong modes in the system provided that appropriate polarized light is used.The figure also shows the correlation between indistinguishablity of the two hybridized magnong modes, EPR nonlocality, and the entanglement between Kittel magnon modes.The higher the indistinguishability (around the zone center), the higher the non-locality and entanglement.

V. CONCLUSION
In conclusion, we demonstrate microwave cavity mediated hybridization of superconducting transmon qubit and chiral magnons in bipartite AFM materials.We derive analytical expressions for the hybridized Hamiltonian and the coupling strengths.This coupling allows us not only to identify magnons in AFM materials, but also to verify their chirality and to characterize the nonlocality and bipartite entanglement between Kittel magnon modes in the system.These are all observed through measurement of the angular frequency of Rabi oscillation in the transmon qubit.We hope the present work opens up a new route to experimentally access rich quantum properties of magnons in AFM materials.The broad range of crystalline and synthetic AFM materials, such as the oxides NiO and MnO, the fluorides MnF 2 and FeF 2 , 2D Ising systems like MnPSe 3 , YIG-based synthetic AFMs, and perovskite manganites [21][22][23][24][25][26][27][28][29], provide a space for experimental observation of the present results.and entanglement (black solid curve) between Kittel magnon modes in the vacuum ground state for different values of lattice momentum k.Similar results can be obtained for excited states.We assume uniaxial AFM materials [9] with simple cubic lattice structure subjected to external magnetic field in the z direction.The lattice momentum k takes its values along (0, 0, 1) direction with the lattice constant set to unity.We consider the nearest neighbor Heisenberg interaction J and the easy-axis anisotropy Kz with model parameter values: J = 10meV , Kz = 0.01J, B = 2.5T for the amplitude of the magnetic field in z direction, and S = 1/2.For the microwave cavity photon we assume A0 = 1meV and ωc = 0.05meV .acknowledgments

FIG. 2 :
FIG. 2: (Color online) Magnon energy dispersions ωα k and ω β −k in the first Brillion zone of a square lattice with lattice constant a = 1 for an easy-axis AFM.As model parameters, we use |J| = 1meV for antiferromagnetic Heisenberg exchange, Kz = 0.01J for uniaxial anisotropy, and S = 1/2.Two magnons are degenerate in the absence of an external magnetic field µBB = 0 (left panel).A magnetic field µBB = 1meV in the z direction breaks the degeneracy (right panel).

FIG. 3 :
FIG. 3: (Color online) Entanglement of magnon eigenstates corresponding to pairs of magnon numbers (x, y) against the entanglement (squeezing) parameter r k .
FIG. 4: (Color online) Entanglement entropies of magnon eigenstates corresponding to selected pairs of magnon numbers (x, y) against the EPR function ∆ [ψ00(r k , φ k )] for AFM spin lattices.Stronger entanglement is observed for non-local states associated to φ k = π, whereas φ k = 0 represents a local state regime with weaker magnon-magnon entanglement.

FIG. 5 :
FIG.5:(Color online) Left panel: The angular frequency, f k , of the Rabi oscillation of transmon qubit depending on whether the transmon qubit is coupled to the magnon in α mode (red) through right circularly polarized photon or to the magnon in β mode (blue) through left circularly polarized photon.The inset shows the corresponding dispersion energies for the two hybridized magnon modes α (red) and β (blue).Right panel: EPR function (gray dashed curve) and entanglement (black solid curve) between Kittel magnon modes in the vacuum ground state for different values of lattice momentum k.Similar results can be obtained for excited states.We assume uniaxial AFM materials[9] with simple cubic lattice structure subjected to external magnetic field in the z direction.The lattice momentum k takes its values along (0, 0, 1) direction with the lattice constant set to unity.We consider the nearest neighbor Heisenberg interaction J and the easy-axis anisotropy Kz with model parameter values: J = 10meV , Kz = 0.01J, B = 2.5T for the amplitude of the magnetic field in z direction, and S = 1/2.For the microwave cavity photon we assume A0 = 1meV and ωc = 0.05meV .