1. Introduction

Magnetism is, in principle, a quantum phenomenon, which is usually described in the semiclassical approximation. However, there are a number of phenomena to which the semiclassical consideration is not applicable. The most important among them is Bose-Einstein condensation of magnons — elementary excitations of the ground magnets 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 $N_c$. The density of magnons 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 phenomenological model of magnetism. We can estimate the relationship between the density of non-equilibrium magnons and the angle of magnetization deflection. According to the Holstein–Primakof transformation the number of excited magnons $\hat {N}$, associated with the deviation of the spin $\hat {S}_z$ from its equilibrium value of $S$, can be written as [1]:

The magnon density can reach the critical value of $N_c$, with a sufficient amplitude of the excitation, which inevitably should lead to the formation of Bose - Einstein condensate. This effect was shown for the first time in experiments with superfluid $^3$He [2–5]. A spontaneous 1000-fold shrinking of the resonance line due to mBEC and superfluid transport of magnons was observed. Later, similar properties were observed in solid antiferromagnets [6–8].

Fundamental progress in this area was achieved after the discovery of magnon BEC in a film of yttrium iron garnet (YIG). The main advantage of YIG is that magnetic ordering exists even at room temperature. This makes mBEC in YIG very promising for possible applications for quantum computing without cooling. There are two different states of magnons in a YIG film. In the case of out of plane magnetization of YIG film, magnons are characterized by a repulsive interaction. This means that the coherent magnetization precession with wave vector $\vec k = 0$ is stable and corresponds to the energy minimum [9], similar to the atomic BEC. The observation of magnon BEC at this geometry was descibed in [10–12]. In this case, mBEC can be viewed as a macroscopic quantum object suitable for quantum calculations [13]. In contrast, in the case of in plane magnetization of YIG film, magnons with $\vec k = 0$ are characterized by an attractive interaction and the homogeneous precession is unstable. In this case, the energy minimum corresponds to a spin wave with $\vec k$ equal to about $10^5 \mathrm{~cm}^{-1}$ and directed along the magnetic field [9]. In this geometry, the formation of high-amplitude interference of spin waves was observed [14]. Traditionally, this pattern of spin waves is interpreted as mBEC with non-zero $\vec k$. The possibility of mBEC formation by traveling magnons with a quasi-one-dimensional direction differs greatly from the original concept of Bose-Einstein condensation and requires theoretical discussion. The formation of this state can also be considered analogous to the well-known formation of a photon resonance in an optical resonator during pumping. Regardless of the interpretation, this state can be used for vector computing [15,16].

Magnon BEC is usually formed in the region of RF excitation. But it can cross the boundary of the excitation area due to the phenomenon of magnon supercurrent, as it was shown in the experiments with superfluid $^3$He-B [17,18]. The Gilbert relaxation coefficient $\alpha$ in $^3$He-B is very small, only about $10^{-8}$, what allows mBEC to spread over relatively long distances. However, in YIG, the rate of magnon relaxation is 3 orders of magnitude larger and is about $10^{-5}$ at the best. Therefore, the spread of the mBEC over the YIG sample at large distances is limited. In the current experiments, we have found that mBEC can spread over a macroscopic distance, more than several mm from the excitation zone, and fill the entire sample. This result opens up the possibilities for the use of the magnon supercurrent state similar to the one with superconducting quantum devices.

There are several ways to implement optical measurements of magnetization dynamics in a YIG film. The most commonly used method is based on the Brillouin scattering of light on traveling magnons [14,19]. However, certain difficulties arise for its application to stationary magnons. Recently, we developed a magneto-optical method of the mBEC investigation based on the Faraday effect [20]. This method has demonstrated a high sensitivity in measuring the amplitude and phase of the magnetization precession in the region of magnon excitation. In this article, we apply this method for the first time to the detection of spin dynamics outside the region of direct magnon excitation.

Under the influence of the radio-frequency pumping field, the magnetization deviates by a certain angle $\theta$ from the direction of the vector H and precesses around it at a frequency of the pump signal $\omega$. In this case, the out-of-plane component of magnetization remains constant while the in-plane component, m$_{rot}(t)$, rotates along the film normal at frequency $\omega$. Its direction is determined by the angle $\varphi = \varphi _{rf}+\Delta \varphi + \omega t$. There $\varphi _{rf}$ represents phase of RF pump and $\Delta \varphi$ is a phase shift due to the equilibrium relation between magnon pumping and magnon relaxation (see [11]).

2. Optical setup

In the experiment we used a 6 $\mu$m thick elliptically shaped YIG film with dimensions of 5 $\times$ 1.5 mm. The ferromagnetic resonance was excited with a narrow strip line 0.2 mm wide, placed at a distance of 0.6 mm from the endpoint of the sample ellipse, and oriented perpendicular to its major axis, as shown in Fig. 1. We used a BK7 glass prism in order to obtain large angle of light incidence values. The prism was glued with immersion liquid on the gadolinium gallium garnet substrate of the YIG sample. The prism allows directing a light beam into the film at an angle enough to provide total internal reflection from the YIG-air interface. The dimensions of the light spot on the sample are about 30 $\times$ 500 $\mu$m. This creates the possibility to measuring m$_{rot}(t)$. After reflection, the polarization of the light is rotated by an angle $\alpha (t)$ due to the Faraday effect. The amplitude of angle $\alpha$ is determined on the projection of m$_{rot}(t)$ on the light beam inside the magnetic film and the angle $\varphi$. In this case, the amplitude of the $\alpha (t)$ is determined by the former, while its phase is related to the angle $\varphi$. Thus, we can obtain the magnitude and phase distribution of the mBEC over the whole YIG sample by scanning the laser beam along the garnet film. The principle of the measuring scheme is explained in more detail in Ref. [20].

 

Fig. 1. Scheme of the experimental setup.

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The light transmitted through the sample was directed to the Wollaston prism oriented at 45$^{\circ }$ with respect to the initial polarization of light. The orthogonally polarized components of light were detected by a balance photo detector. For optical registration of signals at the frequencies of resonance, about 1.9 GHz, we used the optical heterodyne scheme. The radiation of a diode laser at the wavelength of 520 nm was modulated at the frequency shifted by several kHz from the BEC pump frequency $\omega$. As a result, the optical signal modulation appears on the difference frequency, which was measured by photo detectors. The resulting signal contains information about the phase and amplitude of magnetization precession.

The aim of this experiment was to investigate the density of magnons outside the region of their excitation. The set up was equipped with a system for moving the optical beam relative to the sample. For precise positioning of the laser beam, we used an imaging system consisting of a telescope and a diaphragm. This system made it possible to visualize the position of the laser beam on the sample.

The excitation of magnons was controlled by the magnitude of the external magnetic field and the RF power. At low excitation, a linear magnetic resonance was observed. We have used this field as a reference point (marked as zero). With an increase in the RF power, the resonance region expanded towards a lower field. This effect is known as foldover magnetic resonance. It is related to the fact that when the precessing magnetization deflects, the demagnetizing field decreases and, accordingly, the resonance shifts to lower fields at a fixed excitation frequency. From this shift of the resonance field, we calibrate the scale of the magnetization deflection angle. This effect in our samples was studied in detail in Ref. [11,20].

To explore the distribution of magnons in the sample, we scan with the light beam along the main axis of the sample ellipse and at each point sweep the field down starting from the field a bit larger than the field of the linear ferromagnetic resonance. We characterize the observed point on the sample by the coordinate $x$ measured from one side of the ellipse (inset in Fig. 2). As for the magnetic field sweep, it is given by the difference between the magnetic field and the magnetic field of the linear ferromagnetic resonance (FMR) — "FMR mismatch". The optical signal from the region of the sample covered by the strip line ($4.1<x<4.3$ mm) shows clearly the magnon excitation for a wide field range (see Fig. 2).

 

Fig. 2. Spatial distribution of the magnetization precession angle (a) and magnetization precession phase (b) as a function of the observation coordinate when sweeping the magnetic field down. On the left, the position of the strip line (SL) is schematically shown, as well as the region of the light beam (LB), which was swept along the main axis of the sample.

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3. Results of experiments

On sweeping the magnetic field down, the chemical potential of the non-equilibrium magnon gas increases as well as its density and the mBEC forms [11,20]. At some values of magnetic field magnons spread from the region of excitation and fill the entire sample.

The distribution of the magnon density along the sample for three values of magnetic field are shown in Fig. 3 (a). The graphs represent slices of the color plot in Fig. 2 (a) along the dashed lines. The whole sample gets filled by magnons at the FMR mismatch fields of around -11 and -28 Oe (curves 2 and 3 in Fig. 3 (a)), while for a smaller field, mBEC remains only in the region of excitation. The point is that the processes of magnetic relaxation are quadratic in the angle of magnetization deflection. It can be assumed that the pump power is not enough to compensate for the relaxation processes in the entire volume of the mBEC at high magnon densities.

 

Fig. 3. Spatial dependences of the amplitude (a) and phase (b) of the precessing magnetization at a fixed field shift from the resonant one at -48 Oe (1), -28 Oe (2) and -11 Oe (3). Dependences are obtained from Fig. 2. It is clearly seen that at a high concentration of magnons (2, 3) the magnons precess coherently with a phase different from that of the RF pumping.

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The method of optical registration of magnetization precession allows us to measure the phase shift of precession from the phase of RF pumping. The measured distributions of phase shift $\Delta \varphi$ as a function of position and field are shown in Fig. 2 (b). The phase dependence is shown in Fig. 3 (b). The phase gradient between the region of magnon excitation and the rest of the sample is clearly seen. This phase gradient excites the magnon superflow from the region of magnon excitation to the rest of the sample, which is observed for the first time. It follows from our results that the phase of mBEC precession is homogeneous. This clearly signals the mBEC formation. Any gradient of phase should excite the magnon supercurrent, which redistributes the magnon density and restores coherence. Some attention should be paid to the areas near boundaries of the sample. At $x=1$ mm, it can be seen that the gradient increases along with the density of magnons. At lower $x$ the accuracy of the phase measurement decreases due to the small optical signal amplitude. The same applies to the position regions with $x>4.7$ mm.

It is important to note that the distribution of magnons has a number of features (see Fig. 3). For example, there are dips in the magnon density in the regions $x=1.8, 2.4$ and $3.2$ mm. Due to their regularity, it can be assumed that they are associated with the standing Goldstone wave (the wave of the second sound of magnons). The magnon Goldstone wave was observed earlier in superfluid $^3$He [21] and in antiferromagnets [22].

4. Conclusions

To conclude, we have developed an optical method for studying the precession of magnetization, which makes it possible to visualize the formation of mBECs, as well as their spatial distribution. It follows from this experiment that the critical density of magnon BEC formation in yttrium iron garnet corresponds to a magnetization deflection angle of about 3$^\circ$, which is in good agreement with the prediction made in Ref. [23]. In addition, the formation of a superfluid magnon state at a macroscopic distance from the region of magnon excitation is shown. This state can be used for various types of qubits by analogy with superconducting qubits. The setup and technique described in this article can be used to develop and test qubits based on the circulation of a magnon superfluid liquid in a ring with a Josephson junction [24]. The experimental method described in this research can be used to check the quality of YIG samples, which is very important for the design and manufacture of magnonic devices.

There has been an exploding interest in the use of coherent magnons for a variety of macroscopic quantum systems during the last few years [25–29]. The mBEC and phenomena of magnon superfluidity open new perspectives for quantum physics research as well as some modern technological applications for magnonics, quantum communications, and quantum computing. We have developed magneto-optical visualization methods that are demanded for studying the quantum dynamics of magnons, as well as for the development of quantum devices based on magnons.