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...

The Hubbard model, containing only the minimum ingredients of nearest neighbor hopping and on-site interaction for correlated electrons, has succeeded in accounting for diverse phenomena observed in solid-state materials. One of the interesting extensions is to enlarge its spin symmetry to SU(N > 2), which is closely related to systems with orbital degeneracy. Here we report a successful formation of the SU(6) symmetric Mott insulator state with an atomic Fermi gas of ytterbium ( 173 Yb) in a three-dimensional optical lattice. Besides the suppression of compressibility and the existence of charge excitation gap which characterize a Mott insulating phase, we reveal an important difference between the cases of SU(6) and SU(2) in the achievable temperature as the consequence of different entropy carried by an isolated spin. This is analogous to Pomeranchuk cooling in solid 3 He and will be helpful for investigating exotic quantum phases of SU(N ) Hubbard system at extremely low temperatures. * Electronic address: taie@scphys.kyoto-u.ac.jp 1

We report the realization of a novel degenerate Fermi mixture with an SU(2)×SU(6) symmetry in a cold atomic gas. We successfully cool the mixture of the two fermionic isotopes of ytterbium 171Yb with the nuclear spin I=1/2 and 173Yb with I=5/2 below the Fermi temperature T_{F} as 0.46TF for 171Yb and 0.54TF for 173Yb. The same scattering lengths for different spin components make this mixture featured with the novel SU(2)×SU(6) symmetry. The nuclear spin components are separately imaged by exploiting an optical Stern-Gerlach effect. In addition, the mixture is loaded into a 3D optical lattice to implement the SU(2)×SU(6) Hubbard model. This mixture will open the door to the study of novel quantum phases such as a spinor Bardeen-Cooper-Schrieffer-like fermionic superfluid.