Discovery of the Classical Bose–Einstein Condensation of Magnons in Solid Antiferromagnets Yu

ISSN 00213640, JETP Letters, 2011, Vol. 94, No. 1, pp. 68–72. © Pleiades Publishing, Inc., 2011. Original Russian Text © Yu.M. Bunkov, E.M. Alakshin, R.R. Gazizulin, A.V. Klochkov, V.V. Kuzmin, T.R. Safin, M.S. Tagirov, 2011, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 94, No. 1, pp. 68–72. Discovery of the Classical Bose–Einstein Condensation of Magnons in Solid Antiferromagnets Yu. M. Bunkova, E. M. Alakshinb, R. R. Gazizulinb, A. V. Klochkovb, V. V. Kuzminb, T. R. Safinb, and M. S. Tagirovb a Institut Néel, CNRS, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France b Kazan (Volga region) Federal University, ul. Kremlevskaya 18, Kazan, 420008 Russia Received May 19, 2011 Results of experiments in which the Bose–Einstein condensate of magnons is created in the CsMnF3 easy plane antiferromagnet in a system with coupled nuclear–electron precession with dynamical frequency shift are presented. This condensate is similar to the Bose–Einstein condensate of magnons in superfluid 3HeA in aerogel. DOI: 10.1134/S0021364011130066 Bose–Einstein condensation (BEC) is one of the important difference. In case of the classical BEC pre most striking macroscopic quantum phenomena. A dicted by Einstein, the quantum distribution of parti macroscopic number of particles form a coherent cles changes rather than the vacuum in which they quantum state, which is described by a common wave exist, whereas a phase transition is accompanied by a function. The formation of such a state under certain change in the ground state of the system (vacuum). conditions was predicted by Einstein in 1925 (see The validity of using the term BEC to describe such review [1]). A nearly ideal BEC state has been discov states is controversial. The magnon condensate stud ered in supercooled atomic gases. The BEC state is ied in this work is analogous to the atomic condensate. strongly distorted in Bose liquids by intrinsic interac In order to distinguish it from a phase transition, we tions, but it remains the main formation mechanism of refer to it as a classical BEC. a macroscopic quantum state. The main dynamical The classical BEC of magnons and spin superfluid property of the BEC is superfluidity, which can be 3 described as a quantum transport phenomenon ity were experimentally discovered in superfluid He B [5]. In subsequent studies, six different superfluid emerging in the presence of BEC wavefunction gradi 3 ents. He states in which BEC appears have been discov ered. Various experiments in which the BEC was Strictly speaking, BEC theory is applicable to sta observed are reviewed in [6–8]. In all cases, the BEC ble particles. However, it can also be applied to quasis forms in a system of excited nonequilibrium magnons. table particles if their lifetime is much longer than Pulsed or continuous pumping is used for their excita characteristic Bose condensate formation times. In tion at the NMR frequency. The magnetization particular, although atomic gases consist of stable par becomes tilted by a certain angle β, which corresponds ticles, they evaporate from a trap in about a second, to the generation of magnons with the density but this is enough time to form the BEC. Magnons in superfluid 3He are quasiparticles, but their BEC can SS– χH N ==z 1()– cosβ , (1) exist for longer times [2] and it can even be maintained ប បγ continuously [3]. Before we turn to the description of magnon BEC, where Sz is the equilibrium magnetization and Sz is its we should clarify our terminology. A phase transition projection on the direction of the external magnetic to a magnetically ordered state has been experimen field. Excited magnons usually precess with local pre tally discovered in certain magnetic systems. This cession frequencies, so that the induction signal transition was described in terms of a change in the becomes out of phase during the time of inhomoge number of equilibrium magnons (see review [4]). neous broadening of the magnetic resonance line. The Their density reached macroscopic values during the BEC phenomenon consists in the formation of a transition, indicating the formation of a new phase. coherent state via magnons, where they precess with This phenomenon was called the BEC of magnons. the same phase and frequency despite the inhomoge Magnons obey Bose–Einstein statistics, and appro neous external conditions. In the case of continuous priately, their density is described by the same formu NMR, the pump frequency determines the chemical las as those for the BEC of atoms. However, there is an potential of the system. The condensate is formed 68 DISCOVERY OF THE CLASSICAL BOSE–EINSTEIN CONDENSATION 69 within a certain region or at a low magnon density, to observe BECs in magnets, it is necessary to find a which is determined by the chemical potential. When system where the fourthorder term is positive. the NMR frequency is changed, the chemical poten 3 tial also changes, which corresponds to an increase Superfluid He is a liquid antiferromagnet. Its B either in the condensate region or in the magnon den phase is a unique threesublattice antiferromagnet, 2 sity. There is one basic advantage as compared to the where term FSO(Ψ ) is zero up to a tilt angle of 104°. case of the atomic BEC. We can generate additional At larger angles, it is positive. The BEC appears as magnons, which compensate the relaxation of the soon as the tilt angle becomes larger than 104°, i.e., coherent state, and thus the BEC can be maintained when the magnon density is high. It is qualitatively dif continuously. Thus, the pulse method corresponds to ferent from the atomic BEC. Another phase, 3HeA, the formation of the BEC under a fixed particle den has the structure of a twosublattice antiferromagnet. sity, while the continuous method corresponds to the Ψ 2 formation of the BEC under a fixed chemical poten The term FSO( ) is negative, and the spatial insta tial. bility of homogeneous precession is observed [9]. However, it was shown in [10] that when the orbital Theoretical description of the magnon BEC is part of the order parameter is oriented along the mag based on the Gross–Pitaevskii equation for the order netic field, this term becomes positive, which makes parameter: the BEC possible. It has been discovered in experi 3 Ψ()r, t ===〈〉 Ψˆ ()r, t , N Ψ 2, N d3r Ψ 2, (2) ments with superfluid HeA placed in aerogel (a ∫ porous silica structure with 98% porosity) that uniaxial where n is the local density of magnons. If the lifetime compression of aerogel leads to the orientation of the of magnons is long enough and their dissipation can be orbital angular momentum along the compression axis neglected, the Gross–Pitaevskii equation can be writ [11, 12]. When the magnetic field was oriented along ten in the standard form the compression axis, NMR signals, characteristic of the BEC, were detected [13, 14]. In this case, the mag ∂Ψ δᏲ –i = , (3) netization precession frequency depends on tilt angle ∂t δΨ* β as where Ᏺ{Ψ} is the free energy functional. In the case Ω2 of the coherent state of precession ωω–= L cosβ, (7) L 2ω ω Ψ()r, t = Ψ()r ei t (4) where Ω2 is the square of the frequency of the NMR and the Gross–Pitaevskii equation is transformed into L the Ginzburg–Landau equation with ω = μ: longitudinal vibration mode. δᏲ In the case of small NMR excitation amplitudes in – μΨ = 0. (5) δΨ the linear regime, an NMR signal is detected only at * the frequency corresponding to the condition cosβ = In the case of a low magnon concentration, the free 1. When the pumping amplitude is increased, the mag energy functional in the Ginzburg–Landau theory has non generation rate becomes larger than the relaxation the form rate. The equilibrium BEC is formed with a chemical potential corresponding to the pump frequency. In ⎧⎫2 order to satisfy this condition during an increase in the Ᏺ μᏺ 3 ∇Ψ []Ψω () ω 2 – = ∫d r⎨⎬ ++L r – FSO ,(6) NMR frequency (or a decrease in magnetic field), ⎩⎭2m angle β increases, so that the precession frequency corresponds to the pump frequency. The signal ampli ω γ where L(r) = H(r) is the local Larmor frequency, tude increases significantly, because the total magneti which plays the role of an external potential like zation precesses coherently at the pump frequency. Earth’s gravitation field in the case of the atomic con This BEC state is an eigenstate of the magnetic system. 2 The external pumping determines the chemical densate. The last, nonlinear, term FSO(Ψ ) appears due to spin–orbit coupling. It is analogous to the potential of the system and, correspondingly, the mag fourthorder term, which takes into account interac non density. A relatively longlived NMR signal is tion in the atomic BEC. This term determines the sta observed when NMR pumping is turned off. Its dura bility of the BEC. The particles are repelled from each tion is several times longer than the time derived from other when this term is positive. In the opposite case, the inhomogeneity of the external magnetic field [14]. the particles form clusters. In the case of magnetically Here, we described BEC formation in the superfluid ordered systems, they correspond to Suhl or other 3HeA phase, because the free energy and all experi types of instabilities of homogeneous precession of mental results of BEC formation in CsMnF3 given in magnetization. This instability is also observed in this paper are very similar to those obtained in the other types of magnetically ordered systems.

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