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.

Figure 1

Plots of the transmission spectra $s_{x_1}(\omega ,{\omega }_0)$ illustrating the dispersion relations of the magnon-polariton systems. The color scale is in logarithmic arbitrary units, where blue corresponds to low transmission and light green corresponds to high transmission. The dash between the labels of two modes indicates that the two are coupled. (a) Normal anticrossing curve and (b) coupling of the Kittel mode with the two cavity modes, $c$ and $d$. (c) $c$ mode coupled to two magnetostatic modes, $m$ and $n$, with $b_n\ne 0$, and (d) $c$ mode coupled to $m$ coupled to $n$, with $b_n=0$ and $g_{mn}=25$ MHz. One can obtain the same plot with $g_{mn}=0$ and $b_n\sim 0$. See the text for further details.

Figure 2

Section of the cavity body geometrical shape and profile of the $c$ and $d$ modes. The sample lies on a maximum of the rf magnetic field for both modes.

Figure 3

Anticrossing curve of spheres with different diameters. (a) and (b) show the dispersion relation of single spheres of different diameters coupled to the same cavity mode; the coupling strength is obtained by the fit (dashed lines) and scales as expected. (c) and (d) are the anticrossings of the same sphere coupled to the TM110 mode of two different cavities, and the dashed line is the lower-frequency hybrid mode. In (d) one can see a discrepancy between the theory and the result, possibly due to the fact that the sphere diameter is larger than one tenth of the microwave field wavelength.

Figure 4

Relation between the sphere's diameter and the full hybridization field $B_{\mathrm{fh}}$, measured with the 14-GHz cavity. The positive offset follows from the zero-field splitting of the YIG.

Figure 5

Coherent coupling of two spheres and the effect of their relative distance. On the left side of the plots four schematic drawings show the YIG (black sphere) position inside the pipe (blue). The pipe is placed on the cavity axis, parallel to the static magnetic field. (a) and (b) show the single-sphere configuration without and with magnetic field, which produces the hybrid modes indicated by the small vertical arrows. In (c) the second sphere is placed close to the first one, with the effect of increasing the coupling, i.e., the hybrid mode separation, and also of generating a number of dark modes in the intermediate-frequency interval. It is evident from (d) that the strength of such modes is reduced by tens of decibels thanks to the distance between the two YIGs, in agreement with what is expected from the model.

Figure 6

Ten-sphere HS: simulated (left; blue to light green corresponds to low to high transmission) and measured (right; bright colors correspond to higher transmission) anticrossing dispersion relations, obtained with the model based on ${\mathcal{H}}_4$ and a room temperature transmission measurement, respectively. The parameters used to obtain the left anticrossing curve ${\omega }_{-}$ are detailed in the text.

Figure 7

(a) Picture (left) and scheme (right) of the HS tested at a temperature of 90 mK and used in a FH [74]. See the text and Appendix pp3 for a detailed description of the apparatus. (b) Anticrossing curve obtained at 90 mK with the ${\omega }_{-}$ frequency evidenced by a dashed line and (c) tuning of the hybrid mode frequency ${\omega }_{-}$ with a field variation of 3 mT.

Figure 8

The instrument used to produce the spheres (left). Improvement of linewidth with subsequent steps for a batch of six spheres (right).