1. Introduction

Bose condensate is a macroscopic quantum state of matter. The explosive growth of interest to these systems is associated with the hypothetical possibility of using them for new types of quantum computations involving a macroscopic ensemble of particles. In particular, the interest to Bose condensates of magnetic excitations — magnons — is connected with the variety of different states of the mBEC. In the frequently used films of yttrium iron garnet (YIG), the properties of magnons depend on the orientation of the magnetic field. When the film is magnetized in a plane, the minimum magnon energy corresponds to spin waves with a relatively large wave vector $k$. In this case, their state of the Bose condensate can be considered by analogy with the condensation of photons in matter. On the contrary, at a out of plane magnetized film, the minimum energy of magnons corresponds to zero $k$, which makes their Bose condensate similar to the atomic one. Of particular interest to these systems is due to the fact that quantum phenomena can be observed even at room temperature, when their properties go beyond the scope of classical physics. In this article, we will consider the magnon system in a YIG film under conditions when its properties cannot be described by the semiclassical theory.

Magnetism is, in principle, a quantum phenomenon, which is usually described in the quasi–classical approximation. However, there are a number of phenomena to which the quasi–classical consideration is not applicable. And above all, this is the Bose–Einstein condensation of magnons — elementary excitations of the ground magnetically ordered state. It follows from quantum statistics that magnons should form a coherent quantum state (Bose-Einstein condensed state, mBEC) at a concentration above the critical value $N_c$ at a given temperature. Magnons density is determined by the temperature and under stationary conditions it is always below $N_c$. However, the density of magnons can be significantly increased by exciting them by radio–frequency (RF) photons. This process corresponds to the magnetic resonance — deflection and precession of magnetization in the quasi–classical model of magnetism. In this article, we study the properties of magnons in an out–of–plane magnetized film of yttrium iron garnet (YIG) at room temperature. Under these conditions, the uniform precession of the magnetization is described by coherent magnons with $k = 0$, and correspond to the energy minimum of the system. Therefore, they are resistant to decay into spin waves [1]. The critical concentration of magnons $N_c$ for this case was calculated in [2] and corresponds to the deviation of the precession magnetization by approximately 2.5$^\circ$ in the quasi–classical approximation. The properties of a magnon BEC in this geometry are similar to those of an atomic Bose condensate. A fundamentally different experimental geometry was used for a number of experiments with magnon BEC, when the sample is magnetized in the plane of the film. In this case, the minimum energy corresponds to magnons with nonzero $k$. A coherent state of magnons was discovered in which the magnon phase is spatially inhomogeneous. The term Bose state should be used with some caution in this case. In this article we will not consider experiments in this geometry.

Bose condensation of stationary magnons was previously discovered in antiferromagnetic superfluid $^3$He-B [3]. It leads to the formation of a long-lived induction signal, which decayed orders of magnitude slower than it should be due to the inhomogeneity of the magnetic field. Spontaneous recovery of coherence after the decay of homogeneous precession [4], as well as a thousandfold narrowing of the resonance line [5], clearly indicated the formation of mBEC state in these experiments. The phenomenon of magnon supercurrent [6] was also discovered in these experiments. Despite the fact that antiferromagnetic $^3$He is a superfluid liquid, its mass superfluidity does not play any role in the formation of mBEC and magnon supercurrent. Superfluid properties are not included in any of the mBEC parameters. Thus, mBEC similar to that obtained in $^3$He can also be observed in solid magnets. In particular, the properties of magnons in out–of–plane magnetized YIG film are in many respects similar to the properties of magnons in $^3$He -B [7]. They are characterized by repulsion, as in $^3$He -B, which leads to an upward frequency shift as the magnetization deflects. Therefore, we can assume that the magnon BEC in YIG film can be observed analogous to the magnon BEC in $^3$He -B. The main advantage of antiferromagnetic superfluid $^3$He is the extremely long lifetime of magnons. The Gilbert damping constant is of the order of 10$^{-8}$, which makes it possible to observe Bose magnon condensation after turning off the RF excitation. For YIG film the Gilbert constant is by 3 orders of magnitude larger, which makes it problematic to observe the formation of mBEC after turning off the RF excitation.

In this article, we used another method to demonstrate the BEC state of magnons in YIG film. We have studied the destribution of magnon state outside the region of their excitation. For this purposes we have developed the original optical setup [8,9]. Our experiments showed the formation of mBEC on the dimensions of the entire sample in the region far away from the region of RF excitation. We carried out a computer simulation of the experiment within the framework of the quasi – classical Landau–Lifshitz–Gilbert theory and may conclude that the LLG theory well describes the emission of spin waves from the excitation region at a small excitation amplitude. However, within the framework of this theory, we are unable to obtain coherent precession of magnetization outside the excitation region. Consequently, in the experimental conditions we went beyond the applicability of the semiclassical theory.

2. Experimental setup

The experiments were carried out on an elliptical YIG film 6 $\mu$m thick and 4.5 $\times$ 1.5 mm in size. The geometry of the experiment is schematically shown in Fig. 1. The sample was grown by epitaxy on a gallium gadolinium garnet substrate. The elliptical shape of the sample was chosen to reduce the possibility of the secondary resonance modes formation. Magnetic resonance was excited by a narrow strip line 0.2 mm wide, oriented perpendicular to the main axis of the sample. It was located at a distance of 1 mm from one of the sides of the sample.

 

Fig. 1. Scheme of experimental geometry. A beam of linearly polarized light illuminates the sample through a prism. The reflected beam is directed to the detectors. The polarization of optical beam changes due to the Faraday rotation on the magnetization component M along the beam of light. The received signal contains information about the angle of magnetization deflection $\theta$ and phase of precession $\varphi$ in relation to the phase of RF pumping.

Download Full Size | PDF

The spatial distribution of the magnon density and their phase was recorded with an optical setup. The laser beam moved along the main axis of the sample, as shown in Fig. 1. The reflected beam contained information about the phase and amplitude of the precessing magnetization due to the Faraday rotation effect. A detailed description of the optical setup and the principle of reading out the parameters of the magnetization precession was presented in [8]. Modification of the optical setup for studying the magnetization precession outside the excitation region, as well as preliminary results, was presented in [9].

The states of magnons along the main axis of the sample were studied by using the beam of linearly polarized light, which was sent to the sample through a prism. The position of the laser beam was scanned along the main axis of the sample. The optical polarization of reflected beam changes due to the Faraday rotation when interacting with the component of the magnetization M along light beam. Therefore, the Faraday angle is sensitive to the magnetization dynamics, in particular to its deflection angle $\theta$ and phase $\varphi$. The reflected beam of light was directed to a balance photodetector through a Wollaston prism. To translate the results to lower frequencies, we used light modulation at a frequency shifted from the resonant frequency by about 12 kHz. This made it possible to record the parameters of the reflected signal using low–frequency detectors. After detecting the signals, we obtained the amplitude and phase of the magnon precession at a given point of the sample and for a given magnetic field. By scanning the magnetic field and the illumination point of the sample, we obtained the amplitude and phase distribution of magnons in the sample depending on the position and field.

3. Experimental results

In this article, we publish the results of an experimental comparison of the magnon distribution outside the excitation region at low and high RF pumping powers. The experimental results of spatial distribution of amplitude and phase of the magnetization precession as a function of the magnetic field at a low excitation level of 0.05 mW (Fig. 2(a, c)) and at a high excitation level of 6 mW (Fig. 2(b, d) are shown in Fig. 2. The deflection angle was calibrated by the field shift of a signal at a previous experiments at the region of RF excitation by taking into account the foldover effect [10]. At low power, the deviation of the magnetization corresponds approximately to 4$^\circ$ (Fig. 2(a)) in the region of the strip line (between 3.5 and 3.7 mm) and a resonance field of about 2621 Oe and slightly below. But it decreases greatly immediately outside the excitation region. At high RF power, the magnetization deviation reaches about 20$^\circ$ (Fig. 2(b)) in a resonant field and a further increase with a decrease in the magnetic field due to "Folded" resonance when the resonance field shifts due to decrease of demagnetization field at magnetization deflection. See [8,10] for details.

 

Fig. 2. (a)–(b) Experimental observation of magnon density spatial distribution in units of magnetization deflection angle as a function of sweeping down magnetic field at 0.05 mW (a), and at 6 mW (b) of RF pumping energy. (c)–(d) The spatial distribution of the magnon phase as function of sweeping down magnetic field at 0.05 mW (c) and at 6 mW (d) RF pumping energy.

Download Full Size | PDF

Of great interest is the spatial distribution of the phase of precession, shown in Fig. 2(c, d)). We see that at low excitation power (c) magnons propagate from the excitation region in the form of spin waves. The length of these waves changes with the shift of the magnetic field. Naturally, the magnetization precession frequency should coincide with the RF pumpining frequency. In the region where RF field is not applied, the corresponding frequency shift appears due to the gradient energy of spin waves. Therefore, as the magnetic field decreases, the length of the spin waves also decreases.

The experimental results obtained with relatively low excitation are well described by the LLG theory. To verify this, we performed a computer simulation of the magnetization precession under experimental conditions. We used the MuMax$^3$ micromagnetic simulation software package [11] and obtained good agreement with the experimental results (See Supplement 1 for supporting content). The results of computer simulation are shown in Fig. 3(a) and (c) for the experimental conditions presented on the corresponding frames in Fig. 2. It can be seen that the results of the experiment and simulation on frames (a) and (c) are in qualitative agreement with each other. This means that the LLG theory describes the experimental results well at a relatively low excitation.

 

Fig. 3. The results of computer calculations of experiments. (a)–(b) The spatial distribution of magnon density in units of magnetization deflection angle as a function of sweeping down magnetic field at 0.4 Oe (a), and 4 Oe (b) RF field. (c)–(d) The results of computer calculations of spatial distribution of the magnon phase as function of sweeping down magnetic field at 0.4 Oe (c) and 4 Oe (d) RF field.

Download Full Size | PDF

Experimental results on the spatial destribution of the precessing magnetization have a completely different form at large excitation amplitudes and, correspondingly, high densities of excited magnons. We see that in the range of fields from 2621 to 2598 Oe magnons fill the entire sample. Moreover, spin waves are not formed. The precession occurs spatially uniformly. That is, we experimentally observe the formation of a coherent precession, that is, the formation of a Bose magnon condensate. This result confirms the theoretical estimation of magnon BEC formation at the deviation angle above 3$^\circ$, as predicted in [2].

We attempted to simulate these results by solving the LLG equations. It should be noted that coherent precession of magnetization was previously obtained using computer simulations for an in–plane magnetized YIG film [12]. We carried out similar simulations for the in-plane magnetized case and confirmed the results: the sample is filled by standing spin waves formed by spin waves propagating in opposite directions and having $k$-vector corresponding to the positive and negative minima of magnon energy [12]. At the same time, in the out–of–plane magnetic field the magnon energy minimum takes place at zero $k$-vector. Therefore, one should expect uniform spin precession all over the sample. However, we were unable to obtain simulations the coherent precession outside the excitation region as is observed for our experimental out–of–plane magnetization geometry.

The results of our computer simulation for high excitation using LLG theory are shown in Fig. 3(b) and (d). The LLG theory predicts the formation of a wide region of spin waves at any excitation level. We have investigated this situation in detail and found out that LLG based modeling can give only a limited region of uniform precession near the excitation stripe line. For the parameters of the experimentally studied magnetic sample this region is about $20$ $\mu$m wide and the other part of the sample is filled by nonuniform precession corresponding to propagating spin waves. It hints, that in experimental studies we deal with a phenomenon which should be described beyond the quasi-classical theory.

As the deflection angle increases, the magnon relaxation rate increases. It is proportional to the square of the deflection angle and the distance from the excitation region to the edge of the sample [8,10]. The BEC state collapses when the flow of magnons from the excitation region cannot compensate for the relaxation. In the lower part of the sample, it collapses into a field of 2588 Oe. But it is preserved in the higher part of the sample, since the distance to the age is much smaller. So it collapses in a field of 2565 Oe.

Let’s consider the spatial distribution of magnon amplitude and phase at a fixed field in more detail. In Fig. 4(a) the spatial distribution of the signal amplitude is shown for an external magnetic field of 2607 Oe and 2615 Oe, marked by a dashed lines in Fig. 2(a), and at an RF pump power of 0.05 mW. The critical magnon concentration is reached only in the RF pump region, while outside this region the magnon concentration is much lower and a magnon gas can be described in the semiclassical approximation. The very different destribution is shown in Fig. 4(b) at an excitation power of 6 mW. In this case, the magnons filled the entire sample in approximately the same concentration, excluding the edge regions.

 

Fig. 4. (a)–(b) The spatial distribution of the magnon density in units of the magnetization deflection angle in the fields corresponding to the dashed lines in Fig. 2(a,b) at a pump energy of 0.05 mW (a) and 6 mW (b). (c)–(d) The spatial distribution of the magnon phase at fields corresponding to the dashed lines in Fig. 2(c,d) at a pump energy of 0.05 mW (c) and 6 mW (d).

Download Full Size | PDF

The spatial distribution of the magnon precession phase is shown in Fig. 4(c), (d). With a small excitation of 0.05 mW, a phase rotation is observed, which depends on the distance from the excitation region, which reaches 7$\pi$ per mm in a field of 2615 Oe and 17$\pi$ per mm in a field of 2607 Oe.

The phase distribution changes drastically at 6 mW excitation as shown in Fig. 4(d). In this case, a sharp turn of the precession phase near the excitation region corresponds to the flow of magnons from the excitation region. For small magnon relaxation, this turn corresponds to about 180$^\circ$. It decreases with increasing relaxation. This experimental result requires further theoretical study. It is important to note that micromagnetic modeling using the MuMax$^3$ program shows the formation of spin waves at any, even relatively large, excitation amplitude. Apparently the formation of a coherent state of precession lies beyond the applicability of the quasiclassical LLG theory.

4. Conclusion

Magnon Bose condensation and the associated magnon supercurrent in the antiferromagnetic superfluid $^{3}$He are well-known quantum phenomena that have received world recognition [13] and were awarded by F. London Memorial Prize. The existence of magnons Bose condensate state in solid-state magnets caused a lot of controversy. A number of experimental results previously obtained in an out–of–plane magnetized YIG film were considered as indirect evidence of magnon Bose condensation [10,14]. In this paper, we have demonstrated the Bose condensation of magnons with $k$ equal to zero, by direct optical observation of the coherent precession of magnetization at the nonuniform RF pumping. This result goes beyond the framework of the semiclassical LLG theory. This opens up new prospects for research in quantum physics, as well as for some modern technological applications of magnonics, quantum communication and quantum computing.

The Bose condensation of magnons considered in this article is formed by magnons with $k = 0$. It has direct analogies with the atomic Bose condensate. It should be noted that another type of magnon BEC was observed in in–plane magnetized YIG film [15]. In this case, it is formed by spin waves with a non-zero wave vector k and can be modeled using semiclassical computer simulations of LLG equations as it was confirmed in [16]. However, even in this geometry it is possible to obtain Bose condensation of magnons at $k = 0$ in the case of resonant excitation [17].

Under some conditions, many processes in the macroscopic quantum systems can be described by classical dynamical equations: Gross–Pitaevsky equation in cold gases; two–fluid hydrodynamic in superfluid $^4$He; Landau–Lifshitz equations in magnets; Leggett equations for spin dynamics in superfluid $^3$He, etc. But the applicability of these equations are restricted by different conditions on frequency range, temperature and even on geometry. For example, in superconductors the Ginzburg–Landau equations are applicable only close $T_c$, while the Gross–Pitaevsky equations practically have no applicability range. Instead, one must use the Gor’kov equations for quantum mechanical Green’s functions. In the experiment, considered in this article, the process of formation of coherent precession is not described by classical equations. This means that in this particular case the conditions of applicability of the LLG equations are violated. Apparently, it is necessary to expand this theory by including the dynamics of magnons as quasiparticles.