Magnon Tunneling Between Two Magnon Bose-Einstein Condensates

The explosive development of quantum magnonics is explained by its potential of use in quantum computers, processing information and the formation of hybrid quantum systems. The processes of spatial correlation of quantum systems are fundamental and lead to such phenomena as the Josephson effect and superconductivity. In particular, they determine the phenomenon of the magnon superﬂuidity and magnon Bose condensation which were ﬁrst discovered in antiferromagnetic superﬂuid 3 He. In this article, we consider the features of magnon Bose condensation in yttrium iron garnet ﬁlm. We simulate the processes of magnon BEC coherency and magnon tunneling through the gap.


Introduction
Magnetism is a quantum phenomenon whose properties are usually considered in the classical approximation. In this approximation the quantum transitions between the atomic levels are described in terms of the precession of locally averaged magnetization. We should not forget that the basis of spin dynamics lies in the field of quantum mechanics, and their consideration in the form of classical physics is approximate. For example, the well known temperature dependence of magnetization stands out beyond the classical paradigm and is described by the density of magnons. Magnons are a quantum quasiparticles with integer spin which obey Bose statistics. According to the Holstein-Primakoff formalism 1 , they correspond to the quantum transition of a single spin, the energy of which is distributed over the ensemble of magnetic moments at a distance of exchange interaction. The density of equilibrium magnons is determined by the temperature. However, we can excite a large number of non-equilibrium magnons by magnetic resonance methods -the deflection of precessing magnetization at radio-frequency (RF) pumping. A great interest attracts the phenomenon of magnon Bose -Einstein condensation (mBEC), which occurs at a sufficient concentration of magnons. The question arose, as a relatively short time living non-equilibrium magnons can form a coherent state? Experimental investigations show that there is a sufficient time of its thermalization among themselves in case of impulse excitation. Moreover, under continuous excitation, a steady mBEC is formed.
It is easy to estimate the relation between the density of non-equilibrium magnons and the angle of magnetization deflection. The number of excited magnonsN related to the deviation of spinŜ z from its equilibrium value S 1 : whereâ † 0 andâ 0 are magnons creation and annihilation operators. As the magnon density increases, it can reach a critical value at which a Bose condensed state formed. The corresponding critical density of magnons for yttrium iron garnet (YIG) was considered in 2 and corresponds to the deviation of precessing magnetization at an angle of about 3 • . These phenomena, as well as the quantum transport of magnetization by magnon supercurrent are quantum phenomena and can be considered in the classical approximation with great care. However, according to a widespread concept that for N → ∞ the spin dynamics is approaching classical variables. Consequently, the calculations within the framework of the classical paradigm can be considered as a good approximation, of quantum processes. Possible contradictions between the results of classical calculations and experimental results are of paramount importance for the verification of the quantum properties of magnons. In particular, the existence of quantum entanglement for mBEC is of great interest, since it is a central source in many quantum information protocols, which naturally arise in any study of quantum technologies [3][4][5] .

MBEC properties
BEC states can be described by a macroscopic wave function 6 : where is the chemical potential and ψ 0 ( r) is real and normalized to the total number of particles N 0 , The mBEC wave function for time variables reads 7 : where ϕ and θ are the phase of precession and the angle of magnetization deflection, ∆ω is the difference between the Larmor frequency at a small excitation and frequency of mBEC precession. It plays a role of magnon chemical potential of magnon-magnon interaction. This wave function satisfies a Gross-Pitaevskii equation similarly to other BEC systems 8 ih where the first term describes the magnetic gradient energy, the second one -potential energy in magnetic field and the third one -the chemical potential of magnon-magnon interaction energy. This nonlinear term leads to frequency shift from the Larmor frequency, which plays a very important role for stability of mBEC and magnon superfluidity. The repulsive interaction between magnons responsible for formation of mexican hat energy potential 7 , typical for superconductivity and determines the value of critical supercurrent. In the case of attractive interaction the mBEC state and supercurrent are unstable.
In this article, we consider the out of plane magnetized YIG film, in which the interaction is repulsive, and the frequency increases with magnons density (deviation of magnetization), which reads: where M S is a saturated magnetization. At a small excitation the angle θ is near zero and the frequency of precession corresponds to the Larmor frequency ω 0 in the effective field (H − 4πM S ). But at the high excitation the frequency of precession increases on ∆ω = ω 0 (1 − cos θ ) with the magnetization deflection due to magnon interaction. At a permanent RF pumping the chemical potential of mBEC is determined by the RF frequency, which determines the density of nonequilibrium magnons, and therefore the angle of magnetization deflection. In other words, magnon density is determined by the frequency of excitation, and not on its intensity, while RF power is enough to compensate for magnons relaxation, as shown in 9 . Of course, only a part of non-equilibrium magnons condenses. Indeed, non-condensed magnons must follow mBEC due to interaction between them. This is similar to the case of superfluid 4 He, where only a few percent of atoms are Bose condensed, but the density of superfluid component corresponds to 100% of atoms at zero temperature. As a result the magnetic moment of mBEC state is

Experimental observation of mBEC
MBEC was observed as the spontaneously self-organized phase-coherent precession of magnetization in an antiferromagnetic superfluid 3 He-B 10,11 . The formation of the state with coherent precession of magnetization even at a very inhomogeneous magnetic field was explained by magnons supercurrent due to the spatial gradient of phase of precession ∇ϕ 12 .

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There was observed a number of quantum phenomena like a phase-slippage at a critical magnon supercurrent in a long channel [13][14][15] , magnon Josephson effect 16,17 and magnon-current vortices 18,19 . The review of magnon supercurrent and mBEC investigations in 3 He can be found in 7,20,21 . The question of magnon entanglement was also investigated in 3 He-B. There was observed experimentally the Josephson oscillations between two mBEC states 22 . The existence of quantum magnetic phenomena in quantum superfluid 3 He do not cause surprise. However, these phenomena are not directly related to superfluidity of 3 He and can be found in other magnetic materials. Thus, mBEC and magnon superfluidity was observed in solid materials, including antiferromagnets with a coupling nuclear electron precession [23][24][25][26] and YIG film, magnetized in plane 27,28 and out of plane 9,29 .
Many experimental results of mBEC investigations does not contradict the classic model when the mBEC state is viewed in the framework of the macrospin model. Therefore, in this article we use a quasi-classical model assuming that its results can be considered a good approximation for the quantum behavior of mBEC 11,30 .
In this article we present the results of numerical modeling of magnon tunneling between the two mBEC, located in two samples of YIG separated by a thin gap. This configuration corresponds to tunneling experiments and Josephson effect in superconductors. The overlap of the wave functions of magnon condensates occurs due to the dipole-dipole interaction in this case. The considered model gives a realistic estimate of the magnon tunneling and its experimental verification in the planned experiments will shed light on the quantum properties of mBEC.

Methods and results
The magnetization dynamics in two identical YIG square films is calculated using the open-source finite-difference micromagnetic solver MuMax 3 (see Ref. 31 ). Simulations were done on the high performance platform "Zhores" based on SkolTech 32 . We simulate magnetization dynamics at zero temperature in square films with a side length a = 5 µm, thickness d = 10 nm separated by a gap with length l gap Computational cell size was chosen 10 × 10 × 10 nm 3 (see Fig. 1). The following magnetic parameters are used: µ 0 M S = 165 mT, A = 3.49 × 10 −12 J/m. Crystallographic anisotropy has been neglected. The external dc magnetic field with magnitude of B z = 0.25 T applied along z-axis and microwave field with an amplitude b rf = 100 µT and frequency f excited along x-axis only in the left film. Magnons in the right film were excited only by its tunneling from the left film.
We simulate the properties of our system for different parameters, such as distance between films l gap , Gilbert damping α G , and driving frequency f . The spatial distribution of the angle of precession (a) and the phase of precession (b) along the axis x are presented in Fig. 2. These results are obtained for the RF frequency of 2.45 GHz, which corresponds to the frequency shift in the central part of the sample ∆ω = 6 GHz. This shift decreases near the edges due to the decrease in the demagnetization field. Consequently, the angle of deviation of magnetization is also reduced. The magnon tunneling from the left sample leads to a formation of mBEC in the right sample except for the large gap l gap = 500 nm in which coherent coupling is broken and, consequently, the magnon tunneling stopped. The phase leap through the gap become larger with increasing of the distance between samples, which corresponds to a Josephson relation between the samples. The tunneling rate is proportional to the phase leap. Please take into account that the magnon BEC filling up completely the right side sample by spin current, which arise due to the gradient of phase of precession. However, for the gap of 500 nm, the magnetic tunneling speed is insufficient for mBEC formation and typical spin waves are formed. The phase rapidly rotates along the sample and is not very representitive. The profile of magon density (angle of deflection) is determined by the demagnetization factor near the gap. The local field increases and consequently,∆ω decreases. In these simulations α G = 2 × 10 − 5 We simulated mBEC properties at different relaxation rates. The results are shown in Fig 3. From this figure it is clear that higher damping leads to an increase in the phase gradient and the magnon flow. The magnon density and angle of deflection determined by the frequency shift ∆ω and, consequently, by the RF frequency. In Fig. 4 shown the precession amplitude and phase for four different pumping frequencies. We see that the tunneling speed decreases with a decrease in magnon density. In addition, the relaxation rate and the magnon relaxation also decrease with the magnon density. Pay attention to the fact that for 2.40 GHz pumping the magnon density is not enough to fill all the right samples. Small artifacts near the edges are explained by the huge increase of calculation time to achieve quasi-stationary conditions for a small density of magnons.

Conclusion
Magnon BEC is a unique coherent quantum state that exists at room temperature. Traditionally, the magnetic resonance is considered in the linear approximation when MBEC is not formed. The question araises, how the existence of mBEC can be shown experimentally? This is the direct result of quantum statistics when the macroscopic amount of bosonic quasiparticles occupy the lowest level 2 . In other words, the question of the reality of mBEC with a given density of nonequilibrium magnons is equivalent to the question of the reality of quantum physics. The non-linear properties of magnetic resonance are traditionally considered in a framework of a single non-linear oscillator 33 . This model is applicable only to very small samples with the dimensions of the order of exchange interaction or for the case of an absolutely homogeneous external field, which is not a case for general conditions. In this model, the amplitude of the resonance is proportional to the excitation power. Indeed, for a quantum model, the amplitude is proportional to the frequency shift, and does not depend on the RF power, if it is enough for mBEC formation. A complete verification of the quantum behavior of nonequilibrium magnon gas was performed in antiferromagnetic superfluid 3 He, the results of which were awarded by the prestigious F. London prize. An experimental verification for the quantum behavior of nonequilibrium magnon gas in YIG was described in 9 .
MBEC is a very promising system for quantum calculations. The use of mBEC allows for an increase in energy scales via bosonic enhancement 34 , resulting in gate operations that can be performed at a macroscopically large energy scale and can be considered as a perspective platform for quantum computing 35 . The spatial entanglement between Bose-Einstein condensates was recently considered in the review 36 . YIG samples chain with mBEC, interacting through magnon Josephson junctions, described in our article, can be viewed as a multi qubits system. The spatial distribution of mBEC at the edge of the sample is crucial for this type of qubits design. Owing to the equivalence of quantum and classical approaches for a large concentration of magnons, our calculations of the magnetic tunneling are applicable to quantum systems. Indeed, magnon transport properties for quantum and classical models are very different. In our calculations for a magnon transport in the right sample, we used a classical approach, which is equivalent to solid-body rotation. In reality, the transport phenomena should be described by supercurrent. It should result in a smoothing of phase gradient. Experimental verification of deviation from a solid body model is very important.
The other approach for magnon BEC and its application for quantum computing are demonstrated in in-plane magnetized YIG films ( 28 in references there).
A very important question of the thermal limit for the magnon qubit functionality since mBEC exists even at room temperature. Thermal fluctuations may possibly be averaged for a macroscopic number of identical magnons in the condensate. Therefore, the quantum computing based on mBEC may possibly be performed even at a room temperature.