1. Introduction

Magnetism is a quantum phenomenon, the dynamics of which with a weak perturbation is usually discussed in the classical approximation. In this case, the quantum transition between the atomic levels is described in terms of precession of averaged magnetization. However, there are phenomena that stand behind the framework of the classical paradigm. This is the temperature dependence of magnetization, which is described by the density of quantum excitations - magnons. Another such phenomenon is the Bose - Einstein condensation of magnons (mBEC), which takes place only at their sufficient concentration. The critical density of magnons can be achieved by pumping of non-equilibrium magnons. A necessary density of magnons for yttrium iron garnet (YIG) films was calculated in [1] and corresponds to the deviation of the precessing magnetization at an angle of about 3 degrees.

The use of magnon degrees of freedom for building various macroscopic quantum systems caused an explosion of interest in our days [2–6]. The mBEC and related phenomena of spin superfluidity open new perspectives for quantum physics basic research as well as some modern technological applications for magnonics, quantum communications and quantum computing. The first observations of these phenomena were performed in antiferromagnetic superfluid $^{3}$He due to an extremely long life time of magnon quasiparticles [7–9]. Current growing interest in these phenomena is associated with their observation in antiferromagnetics at liquid helium temperatures [10] and yttrium iron garnet (YIG) film at room temperature [11–13], which opens up opportunities for their use in various quantum applications [14,15].

The mBEC is a quantum phenomenon, when due to the Bose statistics the macroscopic number of particles occupies the same quantum state. It is determined by the ratio between the temperature and the number of particles. Magnons are quasiparticles and its density at equilibrium conditions is determined by temperature, and is always below the density required for mBEC. However, their density can be increased by radio frequency pumping of magnetic resonance to the value required for mBEC.

There is a fundamental difference between the state of magnon in a YIG film magnetized in-plane and out-of-plane. In the first case, magnons are attracted to each other and in the second one – they are repealed. It follows from the dependence of resonance frequency on the angle of magnetization deflection, that is a density of magnons. In the first case the frequency decreases, while at the second case it increases in according with equations:

The repulsive interaction between magnons leads to a significant nonlinearity of the ferromagnetic resonance. The resonant frequency increases with an increase in the amplitude of the magnetization precession, the so-called foldover magnetic resonance. Recently, its properties were comprehensively explained in the framework of mBEC formation and magnon superflow [13]. Usually, the strip line radiofrequency (RF) pumping is used for excitation of magnetization precession. The energy absorbed by magnons gives information about the magnetization precession. This method makes it possible to record the value of the deflected magnetization averaged over the sample. Another RF method is measuring the signal on an additional receiving strip. In this case the magnetization can be measured locally in a certain region of the film [17]. However, there is a problem of the RF signal pickup on the receiving strip. Therefore, to measure the local value of the magnetization of the film as well as to study the spatial distribution of magnetization over the film it is advisable to use the optical method. This work is dedicated to the development of the optical method of registration of the magnetization precession in the conditions of mBEC formation and magnon superfluidity. We should also draw attention to the fact that photons directly interact with magnons and thus can be used to read quantum data, which should be taken into account in the further development of magnon quantum computers.

2. Optical setup

Here we developed a method of the mBEC investigation, based on the registration of optical radiation modulated by reflection from a magnetic film. Before proceeding to the discussion of the experimental setup, let us discuss the geometry of the interaction of light with the magnetization of the film. The deflected magnetization precesses around the direction of magnetic field as shown in Fig. 1(a). This type of motion corresponds to both the mBEC regime and linear ferromagnetic resonance. However, in the case of mBEC the precession phase is synchronized over the entire film even in the case of inhomogeneous magnetic field. This is a fundamental property of mBEC, carefully tested in experiments with superfluid $^{3}$He [9] and described in [18,19]. We have chosen a configuration shown in Fig. 1(a) in which the variable component of the magnetization $\textbf {m}_{\textbf {rot}}$ rotates in the plane of the film. In this configuration it is most easy to observe the effect of spin superfluidity which makes this geometry promising for the development of magnon qubits. However, this geometry is less convenient for the optical measurements, since the measurement of the in-plane component of the magnetization $\textbf {m}_{\textbf {rot}}$ requires oblique propagation of light relative to the film. In this case, the angle of incidence of light is limited by a high reflection coefficient of light from the film and substrate for oblique incidence. Therefore, we used a prism to illuminate the sample as shown in Fig. 1(b).

 

Fig. 1. Configuration of the magnetic field and film magnetization (a) and optical scheme of the experiment (b).

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The sample was glued to the prism by the substrate side through an immersion liquid. As a result, light from the prism entered the substrate first and then the film. After reflection from the air-YIG interface the light once again passed through the YIG film and the substrate. After passing through the output face of the prism it was directed to the receiving optical system of the setup. Due to the presence of a light wavevector component which is parallel to the YIG film the Faraday effect takes place and the reflected light acquires a rotation of polarization $\alpha$ proportional to the projection $\textbf {m}_{\textbf {rot}}$ onto the light wavevector into the prism. The phase of oscillations of the Faraday angle $\alpha$ is proportional to the angle $\varphi$ at the frequency of magnetization precession. At the same time, the amplitude of $\alpha$ is proportional to the angle $\theta$ (see Fig. 1) . Since $\textbf {m}_{\textbf {rot}}$ rotates in the film plane, the light polarization is modulated at the magnetization precession frequency.

The Fig. 2 shows a scheme of the experimental setup. To detect the signal corresponding to the amplitude of the magnetization precession, we used radio frequency rather than optical heterodyning, as used in the Brillouin scattering setup. A laser beam was modulated at a frequency close to the microwave pump frequency. The beam was directed through a polarizer on a BK-7 glass prism of refractive index of 1.52 and base angles of 45°parallel to the prism base. It was incident on the sample substrate at 73°. Since the refractive index of the YIG film is 2.2, light hit the YIG-air interface at an angle of 41°and totally reflected back. In the experiment we used a laser system based on the Thorlabs diode L520P50 with wavelength 520 nm and optical power 50 mW. After passing through the prism and reflecting from the bottom film surface, the light entered the receiving system. The sample was placed in a uniform magnetic field generated by an electromagnet. A gradient of the field did not exceed 0.2 Oe/mm. Scanning was carried out by gradual sweeping of the electromagnet current. After reflecting from the film, the light was modulated in intensity due to modulation of the laser current and in polarization due to the Farady effect in the YIG film.

 

Fig. 2. Scheme of the experimental setup.

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The receiving system consisted of a telescope focusing light through a Wollaston prism onto a balanced photodetector. Due to the presence of the Wollaston prism, the photodetector current was modulated at a frequency corresponding to the difference between the laser modulation frequencies and the magnetization precession frequency. Thus, the optical signal is heterodyning with the separation of the intermediate difference frequency on the photodetector. The modulation depth of the laser by radio frequency in the experiment reached 20 % and the magneto-optical modulation in the magnetic film due to the Faraday effect was significantly less than 1 % due to a small Faraday effect in the YIG film. Thus, the radio frequency signal on the laser can be considered as a local oscillator. The frequency of the magnetization precession was equal to 1.9 GHz which was difficult to optically registrate without heterodyning. However, the difference frequency in the experiment was relatively low, about 9 kHz, which corresponded to the minimum laser and photodetector noise. Therefore, the experimental setup made it possible to use a low-frequency low-noise photodetector with a high sensitivity.

To pump magnons by a RF stimulus a microstrip waveguide was used. The waveguide was fabricated by photolithography on Rogers material and was gold plated. The waveguide width was 4 mm, which guaranteed uniform pumping of the sample over the entire surface. To avoid any optical distortion, the sample together with the prism was placed on top of the waveguide without using any gluing. An air gap of about 1 $\mu$m, which appeared due to the lack of gluing optically isolated the metal strip from the YIG sample and didn’t disturb optical measurements. At the same time, such gap was small enough not to affect the RF pumping.

In the inset in Fig. 2 the spectrum of the optical signal noise (red curve) and position of the magneto-optical signal at the difference frequency (blue dot curve) are presented. Also, in the inset one can see a spectrum of the radio signal at the difference frequency which was used for measuring the phase of the mBEC (black line). The investigated samples had shape of the elongated ellipse. Since the light falls on the samples obliquely at a small angle a part of the optical radiation could pass by the YIG film. To avoid this parasitic illumination an image of the film was illuminated by a laser was formed on a photodetector using a telescope and then the parasitic light was blocked by a diaphragm. The optical part of the setup could move relative to the sample for scanning.

3. Results of experiments

We used a 6 $\mu$m thick elliptically shaped monocrystalline YIG film with dimensions of 4 $\times$ 1.5 mm grown by the liquid phase epitaxy on a 500 $\mu$m thick gadolinium gallium garnet substrate. The high-frequency section of the setup was made in such a way that it was possible to carry out measurements simultaneously using the reflected microwave signal and with the optical reading of the mBEC. The magnetic resonanse was excited by a wide strip line, oriented along the main axis of the sample, so the RF field was rather homogeneous. Measurements were performed at a constant pump frequency about 1.9 kHz. An Agilent NN9917A vector network analyzer was used to pump magnons and measure the amplitude of the adsorbed energy.

The optical signal was processed by a data acquisition board. Using a fast Fourier transform, a harmonic at the difference frequency was extracted from the signal. For optical recording of the phase, the phase of the photodetector signal at the difference frequency was compared with the phase of the electronic signal obtained as a result of separating the difference frequency of the BEC pumping signals and laser modulation. For this purpose, via RF couplers the microwave pump and modulation signals of the laser attenuated by 10 dB were mixed and directed to a nonlinear detector. Then, the separated signal was sent to the data acquisition board, as an optical signal.

If one sweeps the magnetic field down, the chemical potential of the non-equilibrium magnon gas changes along with its density. As a result, the mBEC forms and magnon density and amplitude of the signal are increased, as was shown previously [13]. If the RF power is large enough then the angle of magnetization precession doesn’t depend on the excitation power. It becomes fully determined by the difference between the excitation RF frequency and the frequency of spin dynamics in the linear regime. It indicates that mBEC appears [13,17]. To ensure mBEC formation in the current experiments we have measured the RF signals at different RF powers and found that the mBEC regime appears starting from the RF power of 3 mW. The growth of the RF signal from the mBEC state is shown in Fig. 3. In this figure the dependences of the magnitude and phases of the optically registered magnetization precession as a function of the external magnetic field H are presented. Figure 3 shows optically measured amplitude and phase of the magnonic dynamics in a spatially narrow region of 25 $\mu$m in the center of the sample. This region is located in the position 2.5 mm in Fig. 5. It can be seen that the signals measured by the optical (blue circles) and radio frequency (black squares) methods almost coincide with each other. In addition, the optical method of registration allows to measure directly the difference between the phase of the magnetization precession and the phase of the RF field (red crosses). In Fig. 3 we showed the phase only in that band of the magnetic field where the optical amplitude is nonzero (for the magnetic field larger than 2645 Oe). For the magnetic field below 2645 Oe the phase is undefined.

 

Fig. 3. Optical and RF signal dependences on magnetic field H for the uniform RF pumping at 3 mW power ($\bullet$ – magnitude of precession measured by the optical method, $\square$ – magnitude of precession measured by the RF method, $\times$ $-$ phase of precession measured by the optical method.

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In contrast to a linear case, the density of magnons is determined not by the pump power, but rather by the field shift. This shift corresponds to the difference between Larmor frequency at a given field and RF frequency. It determines the chemical potential of mBEC and magnon density. The magnon annihilation is compensated by magnon creation due to the RF field. This process keeps magnon density constant. The phase of the mBEC precession relative to the RF pump phase is automatically adjusted so that the stationary conditions are carried out [13]. As a result, we can observe a delay of the precession phase with an increase of the precession amplitude as shown in Fig. 3.

The power absorbed by mBEC from RF field pumping is:

This relation was checked and approved in experiments with mBEC in $^{3}$He-B [9]. Figure 4 demonstrates that for different RF pump energies the magnitude of the signal increases proportionally to the phase difference up to some phase difference of around $30^{\circ }$. To compare signals of absorbed energy with different pump power, we calibrated the amplitude of the precession signal to the amplitude of the RF field according to Eq. (3). For a larger phase differences the proportionality gets broken and the signal magnitude increases slower than the phase. This behavior can be explained by the inhomogeneity of local conditions, as can be seen from the Fig. 5.

 

Fig. 4. The phase portrait of the magnitude of the mBEC optical signal with magnetic field sweep. Dashed line is the theoretical curve, triangles and squares represent experimental data at 3 mW and 16 mW of the RF pump power respectively.

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Fig. 5. Distribution of the mBEC magnitude (a) and phase (b) over the YIG film in the case of uniform RF pumping at 3 mW power.

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Theory predicts that the mBEC collapse should take place for the phase difference of $90^{\circ }$. This corresponds to the maximum of energy, pumped up to the spin system. The critical phase difference in the experiment is a bit lower, the collapse appears for the phase difference of about $77^{\circ }$. Apparently, the relaxation rate is not uniform and, respectively, the angle $90^{\circ }$ will be reached unevenly. In a region of greater relaxation, mBEC is destroyed what provokes its collapse throughout the sample. This is clearly visible in Fig. 5.

4. Spatial distribution of mBEC

The experimental setup allows to scan the distribution of the precessing magnetization along the sample by moving the optical observation point (Fig. 5). The following properties of the signal appear. Firstly, it can be seen that the external magnetic field is inhomogeneous. We suppose that the peculiarities in the distributions of the optical phase and amplitude are caused by the inhomogeneous magnetic field caused by the demagnetization field of the sample and some defects in the region near 1.5 mm. Resonance comes first in the area between 1.5 and 2 mm. In this area there is a minimum of the total magnetic field. Secondly, the edge effect is visible due to a sharp decrease in the demagnetization field. It is also clear that the distribution of the phase of the precession has heterogeneity. In the region of 1.5 mm, relaxation is significantly accelerated, which can be apparently associated with a local defect. It is in the area that the critical phase is achieved in $90^{\circ }$ and provokes the destruction of the mBEC in the sample. In addition, the emerging gradient of the phase of precession leads to the spatial current of magnon, which transfers magnons to the area with a larger relaxation. In the conditions of mBEC, this current has a character of the spin superfluid current.

5. Conclusions

We have developed the optical method of registration of the magnetization precession, which allows us to measure the amplitude and phase of mBEC, as well as their spatial distribution. It is not sensitive to the pumping signal crosstalk to the receiving RF probe. The parameters of the optical signals and the ratio between their phase and the amplitude are similar to those obtained in experiments with superfluid 3He-B [8], which confirms the formation of the magnon BEC in these experiments. Magneto-optical imaging methods are necessary to study the dynamics of superfluid magnon currents and also to adjust the devices made on their basis. They are needed to develop magnon qubits as well as reading quantum data.