We demonstrate large normal-mode splitting between a magnetostatic mode (the Kittel mode) in a ferromagnetic sphere of yttrium iron garnet and a microwave cavity mode. Strong coupling is achieved in the quantum regime where the average number of thermally or externally excited magnons and photons is less than one. We also confirm that the coupling strength is proportional to the square root of the number of spins. A nonmonotonic temperature dependence of the Kittel-mode linewidth is observed below 1 K and is attributed to the dissipation due to the coupling with a bath of two-level systems.

Rigidity of an ordered phase in condensed matter results in collective excitation modes spatially extending in macroscopic dimensions 1 . Magnon is a quantum of an elementary excitation in the ordered spin system, such as ferromagnet. Being low dissipative, dynamics of magnons in ferromagnetic insulators has been extensively studied and widely applied for decades in the contexts of ferromagnetic resonance 2,3 , and more recently of Bose-Einstein condensation 4 as well as spintronics 5,6 . Moreover, towards hybrid systems for quantum memories and transducers, coupling of magnons and microwave photons in a resonator have been investigated 7-10 . However, quantumstate manipulation at the single-magnon level has remained elusive because of the lack of anharmonic element in the system. Here we demonstrate coherent coupling between a magnon excitation in a millimetre-sized ferromagnetic sphere and a superconducting qubit, where the interaction is mediated by the virtual photon excitation in a microwave cavity. We obtain the coupling strength far exceeding the damping rates, thus bringing the hybrid system into the strong coupling regime. Furthermore, we find a tunable magnon-qubit coupling scheme utilising a parametric drive with a microwave. Our approach provides a versatile tool for quantum control and measurement of the magnon excitations and thus opens a new discipline of quantum magnonics.Single electron spins, being a natural and genuine twolevel system, play crucial roles in numerous applications in quantum information processing. The intrinsic drawbacks, however, are its small magnetic moment µ B , the Bohr magneton, and the limited spatial extension of the electron wavefunction, making coherent coupling with an electromagnetic field rather weak. To circumvent the problems, paramagnetic spin ensembles have been actively studied using atoms 11 , NV centres 12,13 , and rareearth ions in a crystal 14,15 . The coupling strength is largely enhanced by the square-root of the number of spins involved. At the same time, a collective spin excitation mode, which matches the input electromagnetic-field mode, is spanned in the spatially and spectrally extended ensemble. However, with an increased spin density for stronger coupling, the spin-spin interactions among the ensemble drastically degrade the coherence of the system and thus make a trade-off.We move one-step further by introducing ferromagnets. Even though they typically have a spin density several orders of magnitude higher, the strong exchange and dipolar interactions among the spins dominate their dynamics and form narrow-linewidth magnetostatic modes. The simplest mode has the uniform spin precessions of the rigid spins in the whole volume, called the Kittel mode. Coherent coupling between the Kittel-mode magnons and microwave photons in a cavity was recently demonstrated in the quantum regime 8 .Superconducting qubits are also an excellent example of quantized collective excitations in macroscopic-scale electrical circuits, where the nonlinearity of Josephs...

Quanta of collective spin excitations are observed in a ferromagnet by measurements of a superconducting quantum bit.

The techniques of microwave quantum optics are applied to collective spin excitations in a macroscopic sphere of ferromagnetic insulator. We demonstrate, in the single-magnon limit, strong coupling between a magnetostatic mode in the sphere and a microwave cavity mode. Moreover, we introduce a superconducting qubit in the cavity and couple the qubit with the magnon excitation via the virtual photon excitation. We observe the magnon-vacuum-induced Rabi splitting. The hybrid quantum system enables generation and characterization of non-classical quantum states of magnons.

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We propose a novel descriptor of materials, named 'cation fingerprints', based on the chemical formula or concentrations of raw materials and their respective properties. To test its performance, this method was used to predict the viscosity of glass materials using the experimental database INTERGLAD. Using artificial neural network models, we succeeded in predicting the temperature required for glass to have a specific viscosity within a root-mean-square error of 33.0°C. We were also able to evaluate the effect of particular target raw materials using a model trained without including the specific target raw material. The results show that cation fingerprints with a neural network model can predict some unseen combinations of raw materials. In addition, we propose a method for estimating the prediction accuracy by calculating cosine similarity of the input features of the material which we want to predict.

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