Coherent coupling between multiple ferrimagnetic spheres and a microwave cavity at millikelvin temperatures

The spin resonance of electrons can be coupled to a microwave cavity mode to obtain a photon-magnon hybrid system. These quantum systems are widely studied for both fundamental physics and technological quantum applications. In this paper, the behavior of a large number of ferrimagnetic spheres coupled to a single cavity is tested. We use second-quantization modeling of harmonic oscillators to theoretically describe our experimental setup and understand the influence of several parameters. The magnon-polariton dispersion relation is used to characterize the system, with a particular focus on the vacuum Rabi mode splitting due to multiple spheres. We combine the results obtained with simple hybrid systems to analyze the behavior of a more complex one and show that it can be devised in such a way to minimize the degrees of freedom needed to completely describe it. By studying single-sphere coupling two possible size effects related to the sample diameter have been identified, while multiple-sphere configurations reveal how to upscale the system. This characterization is useful for the implementation of an axion-to-electromagnetic field transducer in a ferromagnetic haloscope for dark matter searches. Our dedicated setup, consisting of ten 2-mm-diameter yttrium iron garnet spheres coupled to a copper microwave cavity, is used for this aim and studied at millikelvin temperatures. Moreover, we show that applications of optimally controlled hybrid systems can be foreseen for setups embedding a large number of samples.