Emergence of phase coherence in a magnon Bose–Einstein condensate

MainThe spontaneous emergence of phase coherence1,2 is a fundamental property of Bose–Einstein condensates (BECs)3,4,5,6,7,8, including quasiparticle condensates1,9,10,11,12,13. So far, the BEC phase has only been revealed through phenomena dependent on spatial phase differences, such as interference14,15, second-order coherence16 and macroscopic BEC motions—supercurrents17, superfluidity18,19,20,21 and Josephson oscillations20,22,23. In the case of magnon condensates2,24,25,26, coherence is manifested in the observation of phenomena such as quantized vorticity15, supercurrents17,27,28, Bogoliubov waves29 or Josephson oscillations23. Even though these experiments provide conclusive evidence of coherence, so far it has been impossible to observe its spontaneous emergence because the employed detection methods, such as Brillouin light scattering spectroscopy30, rely on ensemble averaging over many excitation cycles and therefore do not provide access to the instantaneously selected condensate phase. These previous observations of coherence could thus not rule out the possibility of coherent properties being introduced by external variables, such as the coherent pumping of quasiparticles into the spin system11,31,32.However, the formation of a random phase in concurrence with an increase of coherence is an inherent and necessary property of a BEC, and provides evidence of the condensation process. Because the quantum phase is not directly observable, magnonics provides a special opportunity for these measurements2,33. The magnon BEC can also be represented as a semi-classical macroscopic object, and, therefore, it is possible to determine the phase of its coherent state with respect to an external reference. The concept of quantum-classical correspondence allows to draw conclusions about the coherence of the quantum mechanical phase from the coherence of the classical phase. By utilizing this correspondence, we are able to observe the spontaneous establishment of BEC phases. These phases are independent of each other as well as the external reference signal and pump phases. The BEC phase is randomly formed in every repetition of the experiment. In our work, we present an electronic approach to both excite and measure a magnon BEC in a single set-up and furthermore have the ability to directly measure the evolution of the coherent state phase of the condensate as well as its amplitude. In turn, this enables us to directly observe the emergence of coherence as well as the randomness of the phase in the BEC.Magnon BEC propertiesMagnons can be described as the delocalized quantum excitations of an ordered ensemble of spins. In its simplest form, this means the flipping of a single spin in an otherwise magnetically perfectly ordered material. Because a single flipped spin is not an eigenstate of the system, it is delocalized in the form of a Bloch state34. For high occupation numbers, this leads to a macroscopic precession of the magnetization. As quasiparticles, magnons are bosonic35 and thus can undergo Bose–Einstein condensation.Magnons in magnetically ordered materials possess several defining properties, one of which is their strong anisotropy. In films, the magnon dispersion relation strongly depends on the orientation of the bias magnetic field, relative to the wave vector k of a magnon. This leads to considerable deviations in the behaviour for different sample orientations. In contrast to most previous works on magnonic BEC, which utilized an in-plane magnetized film24,36 and showed Bose–Einstein condensation in the backward-volume geometry at wavenumber k ≠ 0, we choose an out-of-plane magnetization (that is, forward-volume) geometry, and thus we are observing condensation at k ≈ 0. This enables us to efficiently use inductive detection of the BEC-signal with a 100-μm-wide microstrip antenna. Therefore, we are able to conduct phase-resolved single-shot measurements of our BEC. For further information on the experimental geometry, see the Methods. The condition for the formation of a BEC can be derived from the occupation of the Bose–Einstein distribution of a gas of bosonic particles or quasiparticles37$${N}_{{\rm{g}}}(T,\mu )=\mathop{\int}\nolimits_{0}^{\infty }\frac{D(\varepsilon )}{{e}^{(\varepsilon -\mu )/{k}_{{\rm{B}}}T}-1}{\rm{d}}\varepsilon ,$$ (1) where Ng denotes the total number of particles in the magnon gas, D(ε) is the density of states at energy ε, μ is the chemical potential, T is the temperature and kB is the Boltzmann constant. For quasiparticles, it is possible to increase the number of particles in the system by external pumping16,18,38, so that, whenever the number of particles in the system exceeds the maximum of Ng(T0, μ = ϵ0) at a finite temperature T0, the excess magnons accumulate in the lowest available energy state, that is, in the ground state3,25, resulting in a thermodynamic phase transition. In contrast to BECs in atoms, the quasiparticle (for example, magnon) density can be largely increased, and a BEC can be observed even at room temperature24,39.In the BEC state, the ground state forms a coherent ensemble of bosons, whose behaviour is described by a single collective wave function. A crucial property of this state is the spontaneous emergence of coherence, which means that the initial phase state of the condensate must be independent of the external pumping1,2. As a consequence, in a sequence of individual single-shot experiments, the BEC generated in each of the realizations establishes a random phase.Phase evolution of the BECIn our experiments, we observe the time evolution of a parametrically pumped gas of magnons. Both pumping and detection of the magnetic response are achieved by a microstrip antenna, inductively coupled to a yttrium–iron–garnet film (see the Methods for details). The sample is pumped for 1 μs with varying microwave powers, and the response of the system is recorded throughout the pumping duration, as well as during the free evolution of the magnonic system up to 2.5 μs after the pumping is switched off. For a complete sketch of the set-up, see Extended Data Fig. 1. It is important to note that this approach enables us to perform single-shot measurements without the need for integration over consecutive measurements.The inductive antenna primarily detects long-wavelength magnetization dynamics, acting as a spatial low-pass filter whose sensitivity gradually decreases with increasing magnon wavenumber. Via the magnon dispersion relation ω(k), this spatial selectivity translates into an effective frequency detection window with a width of several hundred megahertz above the Kittel frequency, which is much broader than the observed linewidth of the condensate mode.For the parameters of the present experiment, the parametrically excited magnons of the fundamental thickness mode correspond to wavelengths of approximately 18 μm. Although this wavelength is smaller than the antenna width (100 μm), the spatial Fourier spectrum of the antenna does not impose a sharp cut-off. Consequently, coherent spin waves with such wavelengths would still produce a detectable signal, albeit with an amplitude reduced by an estimated factor of 18 relative to the k ≈ 0 mode. Nevertheless, no coherent signal at the parametric frequency is observed. This indicates that the parametrically excited magnons do not form a coherent oscillation, consistent with strong phase randomization in the ensemble of parametrically generated modes.The time evolution of a measurement can be divided into three sections: (1) The first is the pumping, during which new magnons are introduced into the system by parametric pumping. Parametric pumping excites pairs of magnons with opposite wave vectors (k, −k) satisfying ω±k = ωp/2. In the perpendicularly magnetized geometry used here, all in-plane wave vectors experience an identical parametric coupling and excitation threshold, so that a continuum of parametrically excited modes is generated. While the microwave drive fixes the phase sum within each pair (k, −k), the phases of different mode pairs remain mutually uncorrelated (ref. 40, p. 270). As a result, the parametrically excited magnon population forms a random-phase ensemble with large occupation numbers but without a global phase. Nonlinear four-magnon scattering redistributes magnons across the spectrum via cascade-like processes and kinetic instability, in which two parametrically excited magnons generate one excitation near the spectral minimum and another at a higher frequency. Because the phases of different parametrically excited modes are mutually uncorrelated, these processes redistribute populations without introducing a deterministic phase reference for the k ≈ 0 mode. As the magnon density reaches the critical threshold, (2) the thermalization section begins, and the excess magnon gas thermalizes into the coherent condensate state. Finally, after the pump is switched off and the condensate is fully formed, we can observe (3) the free evolution of the magnon BEC. These sections are schematically illustrated in Fig. 1. As an additional illustration of these distinct sections and the abrupt onset of coherence, a representative waveform was converted into an audio file, enabling the emergence of a well-defined tone to be perceived directly. Further details are provided in Supplementary Section 3.Fig. 1: Sketch of the measurement set-up and the magnon system dynamics.Full size imagea, A microstrip antenna on an out-of-plane magnetized YIG film is connected to a microwave pump source. The alternating magnetic field of the microstrip is coupled to the ordered YIG spin system and excites magnons parametrically at half of the pumping frequency. The pumped magnons thermalize and undergo a transition to the BEC state. The resulting precession of the magnetization induces a signal in the antenna, which is then further amplified and detected by a phase-sensitive measurement set-up. b, The magnonic dispersion relation of an out-of-plane magnetized YIG film and a schematic representation of the parametric pumping scheme. The bold black line indicates the fundamental mode with the minimum at k = 0, where grey lines denote higher-order thickness modes. The microwave photons couple to the magnetic system and parametrically induce magnons at half the pumping frequency over a broad range of wavenumbers. After further incoherent scattering, the magnons accumulate at the bottom of the spectrum, where Bose--Einstein condensation occurs. c, Schematic view of the spatiotemporal behaviour of the macrospin system. The different stages of the condensation process are indicated. During the pumping, incoherent high-energy magnons are introduced into a narrow frequency band of the spectrum. As the density of magnons increases, thermalization starts and the system evolves to an incoherent magnon gas state, which then transitions towards the ordered BEC state. Afterwards, only the BEC is detected, at which point the magnon signal becomes fully coherent.Using the directly measured signal, we are able to determine the phase of the magnetization precession at arbitrary points in time during and after pumping. The evolution of the phase is a clear measure of coherence. A non-coherent signal exhibits a random-phase trace, whereas a coherent signal manifests itself with a linear phase evolution. The time dependencies of the signal before and after condensation for three independent measurements at 23 dBm are shown in Fig. 2. In our set-up, we record a signal that is in phase to an external reference local osicllator, as well as a 90° shifted one, which will in turn be interpreted as the real and imaginary parts of a complex function, that is, in-phase I(t) and quadrature Q(t) components (Suppementary Section 1). Thus, the phase of the oscillation can be determined as the complex angle between the real and imaginary parts, as shown in Fig. 2b. The signal and its phase reach a high degree of coherence only after condensation. Further analysis of the phase shows uniform phase development, indicating a coherent condensate, and also the spontaneity of the initial phase in every experiment. This analysis is presented in the ‘Data Analysis’ section in the Methods. The inlays in Fig. 2 show histograms of the phase measured at the beginning of the indicated time intervals for 1,000 independent experimental runs. At the beginning of the pumping sequence, the signal amplitude, which is proportional to the square root of the magnon density, is low, and no coherent behaviour is apparent. As more and more magnons are injected into the system, the signal starts to show some periodic and apparently coherent traits, indicating the onset of Bose–Einstein condensation. After the pump is switched off, condensation develops further in the absence of perturbation of the magnon system by the pumping field, and an increase in the population of the ground state is observed. At the same time, the coherence of the signal increases. Because all generation and detection equipment in the experimental set-up is phase-locked, and the signal generator is primed to start pumping with a constant phase-offset throughout the whole measurement, all externally induced factors are stable within a few degrees of phase shift, and the uniform phase distribution must be an intrinsic property of the BEC (see the distribution of the pumping phase indicated by the grey sectors in the first and second insets in Fig. 2a). This leads to the conclusion that the emergence of a stable phase evolution is indeed spontaneous and not induced by external factors.Fig. 2: Time evolution of the signal.Full size imagea, The real component and phase of the measured complex signal at different time slices at a pumping power of 23 dBm. For each time slice, three randomly selected measurements are depicted. Lines are a guide to the eye. The system is pumped from ton = 0 μs to toff = 1 μs. The pump signal is filtered out, and only oscillations near the bottom of the spectrum are measured. The inset shows angular histograms of the wave-function phase distribution over 1,000 measurements at the start time of each panel. Amplitude denotes the percentage of measurements in this cluster. The grey bars denote the phase distribution of the pump, whereas the red bars denote the phase distribution of the magnon signal. The distribution of the magnon signal shows no apparent clustering and is uniform, indicating randomness of the formed BEC phase in each realization of the experiment. This can be seen as a strong indication that the observed coherence is a result of true BEC and not an induced coherence by external factors1. b, The complex phase of the signal is analysed at the same time slices as in a. No coherent phase evolution can be observed at the beginning of the pumping pulse. As the system undergoes condensation, the phase starts to show more and more coherent behaviour. In the BEC state, a uniform phase evolution of the signal is observed.Source dataEmergence of coherenceAs a measure of coherence, we used the integral of the normalized autocorrelation of the direct signal over a 100-ns boxcar window, shifted over the entire measurement interval. Comparing this with the theoretically ideal autocorrelation (that is, the autocorrelation of a perfect f(t) = eiωt pulse with a boxcar envelope) allows us to generate a dimensionless measure of coherence—the coherence value, where 0 denotes perfect white noise and 1 denotes a perfectly coherent signal. The boxcar window is shifted continuously to obtain the same time resolution in the coherence value as the original signal.Figure 3 shows the evolution of frequency and coherence throughout the measurement. At the beginning of the pumping, the signal is not registered at all, as the injected parametric magnons are not yet thermalized and are located outside of the detection frequency band. As the thermalization process evolves, an increase in coherence can be observed. After the pumping is completed, the frequency of the ground state drifts towards lower frequencies, causing a temporary reduction of coherence. This phenomenon can be attributed to the high density of magnons during the pumping section and the resulting nonlinear shift due to magnon–magnon interactions41. Once the magnon gas has completely thermalized and the BEC has absorbed the excess of the hot, thermal magnons, the coherence value reaches its final value. Then, a plateau of high coherence is observed, indicating the formation of a stable BEC. The oscillations in coherence present in this plateau are still a subject of investigation. A possible explanation stems from incoherent magnons that travelled out of the probing area during pumping and were then reflected from the edge of the sample. When crossing the antenna, these magnons would decrease the overall coherence of our signal, before being partially absorbed by the BEC or leaving the detection range of the antenna again. This speculation is supported by the evolution of amplitude presented in Fig. 4. It is apparent that the oscillations in amplitude and coherence coincide over the whole ensemble. The gradual decrease in coherence, late after the pump is turned off, can be understood by the decrease in amplitude and thus the deterioration of the signal-to-noise ratio. A comparison of the coherence value with the amplitude of the signal also shows that the BEC is phase-stable until the signal vanishes due to the finite lifetime of the magnons. In other words, the BEC does not undergo dephasing, but the magnonic damping lowers the amplitude of the oscillation until the signal becomes indistinguishable from noise. At this point, the signal itself is no longer detectable, although the correlation analysis (that is, the coherence value) still shows a clearly detectable coherence. This can be seen in Fig. 4. At t = 3.5 μs, the amplitude of the signal has vanished below the noise threshold, but a clear remanence in coherence can be observed.Fig. 3: Emergence of coherence at a pumping power of 23 dBm.Full size imagea, Spectrogram of the signal during the measurement. The condensation process can be observed after the density of the magnon gas reaches the critical value for condensation, as indicated by the increase in signal amplitude. The downward shift of the frequency is an effect of the decrease in the overall number of magnons after the pump is switched off and the corresponding increase in net magnetization (Supplementary Section 2). b, The evolution of coherence as a measure of the quotient of the integrals of the autocorrelation of the signal and the ideal monochromatic signal. A value of 1 indicates a perfectly coherent signal, and zero denotes perfect white noise. The given measurements are an average of 1,000 measurements, with the shaded area denoting the standard deviation. The emergence of coherence can clearly be observed. After the pump is switched off and thermalization is finished, the BEC is fully formed and peak coherence is reached. c, Indicated are the absolute values of the autocorrelation function of the measured signal at different time windows, compared with those of a fully coherent \(f(t)={e}^{i{\omega }_{0}t}\) signal of the same frequency. The three slices are 100 ns wide and are located at ti = 0.1 μs, tii = 0.9 μs and tiii = 1.25 μs. The ratio of the integral of the measured autocorrelation is used to calculate coherence values seen in b.Source dataFig. 4: Pump-power dependence of coherence and signal amplitude.Full size imagea,b, Comparison of temporal dependencies of the coherence value (a) and amplitude (b) of the signal collected from the antenna, for different pumping powers. The shaded areas denote the standard derivation of the ensemble. By changing the pump power, a threshold for condensation can be obtained. At 20 dBm, there is no increase in coherence. At 21 dBm, an increase in coherence is observed, around t = 1 μs. Due to the low amplitude and the resulting low signal-to-noise ratio, the increase of the coherence value is not very pronounced. At 22 dBm, a strong increase in coherence can be observed, although the amplitude of the signal is still very low. This indicates a condensation threshold of 21 dBm in this particular set-up. Increasing the pump power above 23 dBm does not lead to higher coherence, although a higher amplitude can be observed. Remarkably, the coherence of the BEC does not depend on the absolute amplitude of the signal. At higher powers, high levels of coherence can be seen even after the amplitude drastically decreases. This is a strong indication that the BEC does not lose its coherent properties during decay.Source dataDiscussionWhile the existence of BECs in a variety of systems was demonstrated by former studies (see, for example, ref. 21), no direct proof of randomness in the established condensate phase has yet been provided. Our system provides access to the coherent state phase, which is in our semi-classical system equivalent to the quantum mechanical phase, allowing one to directly probe the coherence of the BEC state. In this work, we demonstrate the emergence of phase coherence during the formation of a BEC of quasiparticles, applying a phase-sensitive electromagnetic measurement technique to magnons—the quanta of spin waves. With this approach, we succeeded in exploiting the quantum-classical correspondence in magnonic systems to overcome the fundamental hurdles in the study of coherent states phase evolution, thus opening up new perspectives in the field of experimental research of condensates. We were able to observe the emergence of coherence during and after pumping, which strongly indicates the existence of a BEC even in a heavily disturbed environment. The demonstrated ability to enhance and control the magnon flux towards the spectral minimum by adjusting the pumping geometry has strong implications for processes that rely on coherent, low-energy magnons, such as magnon supercurrents and Josephson-like oscillations in magnon condensates. These phenomena are critically dependent on the condensate density and its phase stability. In this context, the increased population of bottom magnons achievable under transverse pumping provides a promising pathway towards strengthening phase-coherent magnon transport, which in turn facilitates robust Josephson dynamics. We are able to directly measure the emergence of coherence of magnon condensate in a solid-state system and directly show independence of the established condensate phase from any external sources, which has long been required as a necessary feature to prove the formation of a true BEC1.MethodsExperimental set-upThe coherence measurements were performed by pumping a 2.1-μm-thick film of yttrium–iron–garnet with lateral dimensions of 9.5 mm by 2.7 mm, at 7.8 GHz with varying powers through a 100-μm-wide microstrip antenna for 1 μs at an external magnetic field of 281 mT. The reflected signal was separated using a frequency diplexer, filtered by a 5-GHz low-pass filter to suppress all remains of the strong pumping, and amplified by a low-noise amplifier (Qotana DBLNA202000800B) before being fed into the RF (radio frequency) port of an IQ mixer (otherwise known as in-phase/quadrature mixer) (Miteq IRM0218LC1Q). The IQ mixer was supplied with a 3.2-GHz local oscillator frequency, and the I and Q outputs were routed to an oscilloscope (Teledyne LeCroy HDO6054). A more detailed description of the working mechanisms of the IQ mixer set-up can be found in Supplementary Section 1. The detailed set-up is depicted in Extended Data Fig. 1. For the acquisition of the displayed spectra, the IQ mixer and oscilloscope were replaced by a microwave switch (Miteq S138BDU1) and a spectrum analyser (Agilent E4446A). The consistency of the data obtained by the spectrum analyser and the IQ mixer has been validated, as shown in Extended Data Fig. 2.Data analysisThe method chosen to determine the coherence value was the autocorrelation of the normalized complex signal over a 100-ns-wide window, using the NumPy Python package42. Because the signal z(t) = I(t) + iQ(t) is complex, this leads to a complex autocorrelation function. However, only the absolute value of the autocorrelation function is relevant for determining coherence. To calculate the coherence value at a given time t, the absolute value of the autocorrelation function of z is integrated, and the resulting value is normalized by dividing it by the integral of the absolute value of the autocorrelation function of a perfectly coherent dummy signal \(f(t)={e}^{i{\omega }_{0}t}\). Thus, the normalized coherence value is obtained:$$\begin{array}{rcl}{{\mathcal{C}}}_{w}(t)&=&{\mathcal{N}}\mathop{\displaystyle\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\tau \left(\overline{z\cdot {{\rm{box}}}_{t,w}}\star \overline{z\cdot {{\rm{box}}}_{t,w}}\right)(\tau )\\ &=&{\mathcal{N}}\mathop{\displaystyle\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\tau \mathop{\displaystyle\int}\nolimits_{-\infty }^{\infty }{\rm{d}}{\tau }^{{\prime} }\overline{{z}^{* }({\tau }^{{\prime} }){{\rm{box}}}_{t,w}({\tau }^{{\prime} })}\,\overline{z({\tau }^{{\prime} }+\tau ){{\rm{box}}}_{t,w}({\tau }^{{\prime} }+\tau )}\,\,,\end{array}$$where z(t) is the complex signal, w = 100 ns is the width of the autocorrelation window, \({{\rm{box}}}_{t,w}{(\tau )}={\Theta (\tau -t-\frac{w}{2})}-{\Theta}{ (\tau -t+\frac{w}{2})}\), \({\mathcal{N}}=(\mathop{\int}\nolimits_{-\infty}^{\infty}{\rm{d}}\tau (\overline{{e}^{i{\omega}t}\cdot {\mathrm{box}}_{t,w}}\star \overline{{e}^{i{\omega}t}\cdot {\mathrm{box}}_{t,w}})(\tau))\!\!\scriptsize{-1\atop}\), the bar denotes vector normalization, ★ a convolution and * the complex conjugate. The width of the analysis window has a strong influence on the evaluated coherence value. A shorter window provides better temporal resolution and reduces temporal smearing. However, due to the finite sampling rate and measurement noise, fewer data points contribute to the evaluation, which reduces the statistical reliability of the resulting coherence value. Because the coherence is evaluated over a finite window and discrete time sampling, a small, non-zero baseline may remain even for purely thermal signals and therefore does not by itself indicate coherent magnetization dynamics. The window length of 100 ns was therefore chosen as a compromise between temporal resolution and statistical robustness of the evaluated coherence.