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...

Engineered quantum systems enabling novel capabilities for communication, computation, and sensing have blossomed in the last decade. Architectures benefiting from combining distinct and complementary physical quantum systems have emerged as promising platforms for developing quantum technologies. A new class of hybrid quantum systems based on collective spin excitations in ferromagnetic materials has led to the diverse set of experimental platforms which are outlined in this review article. The coherent interaction between microwave cavity modes and collective spin-wave modes is presented as the backbone of the development of more complex hybrid quantum systems. Indeed, quanta of excitation of the spin-wave modes, called magnons, can also interact coherently with optical photons, phonons, and superconducting qubits in the fields of cavity optomagnonics, cavity magnomechanics, and quantum magnonics, respectively. Notably, quantum magnonics provides a promising platform for performing quantum optics experiments in magnetically-ordered solid-state systems. Applications of hybrid quantum systems based on magnonics for quantum information processing and quantum sensing are also outlined briefly.

Coherent conversion of microwave and optical photons in the single-quantum level can significantly expand our ability to process signals in various fields. Efficient up-conversion of a feeble signal in the microwave domain to the optical domain will lead to quantum-noise-limited microwave amplifiers. Coherent exchange between optical photons and microwave photons will also be a stepping stone to realize long-distance quantum communication. Here we demonstrate bidirectional and coherent conversion between microwave and light using collective spin excitations in a ferromagnet. The converter consists of two harmonic oscillator modes, a microwave cavity mode and a magnetostatic mode called Kittel mode, where microwave photons and magnons in the respective modes are strongly coupled and hybridized. An itinerant microwave field and a traveling optical field can be coupled through the hybrid system, where the microwave field is coupled to the hybrid system through the cavity mode, while the optical field addresses the hybrid system through the Kittel mode via Faraday and inverse Faraday effects. The conversion efficiency is theoretically analyzed and experimentally evaluated. The possible schemes for improving the efficiency are also discussed.

We experimentally implement a system of cavity optomagnonics, where a sphere of ferromagnetic material supports whispering gallery modes (WGMs) for photons and the magnetostatic mode for magnons. We observe pronounced nonreciprocity and asymmetry in the sideband signals generated by the magnon-induced Brillouin scattering of light. The spin-orbit coupled nature of the WGM photons, their geometrical birefringence, and the time-reversal symmetry breaking in the magnon dynamics impose the angular-momentum selection rules in the scattering process and account for the observed phenomena. The unique features of the system may find interesting applications at the crossroad between quantum optics and spintronics.