Using a submillimeter-sized YIG (yttrium-iron-garnet) sphere mounted in a magnetic-field-focusing cavity, we demonstrate an ultrahigh cooperativity of 10 5 between magnon and photon modes at millikelvin temperatures and microwave frequencies. The cavity is designed to act as a magnetic dipole by using a novel multiple-post approach, effectively focusing the cavity magnetic field within the YIG crystal with a filling factor of 3%. Coupling strength (normal-mode splitting) of 2 GHz (equivalent to 76 cavity linewidths or 0.3 Hz per spin) is achieved for a bright cavity mode that constitutes about 10% of the photon energy and shows that ultrastrong coupling is possible in spin systems at microwave frequencies. With straightforward optimizations we demonstrate that this system has the potential to reach cooperativities of 10 7 , corresponding to a normal-mode splitting of 5.2 GHz and a coupling per spin approaching 1 Hz. We also observe a three-mode strong-coupling regime between a dark cavity mode and a magnon-mode doublet pair, where the photon-magnon and magnon-magnon couplings (normal-mode splittings) are 143 and 12.5 MHz, respectively, with a HWHM bandwidth of about 0.5 MHz.

Resonant photon modes of a 5mm diameter YIG sphere loaded in a cylindrical cavity in the 10-30GHz frequency range are characterised as a function of applied DC magnetic field at millikelvin temperatures. The photon modes are confined mainly to the sphere, and exhibited large mode filling factors in comparison to previous experiments, allowing ultrastrong coupling with the magnon spin wave resonances. The largest observed coupling between photons and magnons is 2g/2π = 7.11 GHz for a 15.5 GHz mode, corresponding to a cooperativity of C = 1.51±0.47×10 7 . Complex modifications beyond a simple multi-oscillator model, of the photon mode frequencies were observed between 0 and 0.1 Tesla. Between 0.4 to 1 Tesla, degenerate resonant photon modes were observed to interact with magnon spin wave resonances with different couplings strengths, indicating time reversal symmetry breaking due to the gyrotropic permeability of YIG. Bare dielectric resonator mode frequencies were determined by detuning magnon modes to significantly higher frequencies with strong magnetic fields. By comparing measured mode frequencies at 7 Tesla with Finite Element modelling, a bare dielectric permittivity of 15.96 ± 0.02 of the YIG crystal has been determined at about 20mK.

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