High-Cooperativity Cavity QED with Magnons at Microwave Frequencies

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.

Figure 1

Two reentrant-type cavities with two resonance posts. Cavity $\uparrow \uparrow$ demonstrates operation in the “dark” mode, and cavity $\uparrow \downarrow$ demonstrates operation in the “bright” mode. (a) shows a 3D cross-sectional view of the cavities with $\overrightarrow{E}$ and $\overrightarrow{B}$ field directions, and (b) shows a top view of the field modeling results computed in the equatorial plane of the YIG sphere.

Figure 2

Cavity-resonance frequencies and YIG-sphere filling factors as a function of (a) the distance between the two posts and (b) the cavity height, for both the dark and bright modes. The distance is given in terms of the sphere diameter $d$.

Figure 3

Cavity-resonance frequencies and the YIG-sphere filling factor as a function the post gap size, for both the dark and bright modes.

Figure 4

Schematic of the experiment. For clarity, not all antiradiation shields are shown.

Figure 5

Transmission through the cavity as a function of frequency and applied dc magnetic field. The inset demonstrates the region of interaction between the dark mode and the magnon resonance. The plot labeled $\uparrow \downarrow$ shows the frequency response of the interaction between the $f_{\uparrow \downarrow }$ cavity mode and the magnon mode $M_1$ at $B=0.743\mathrm{T}$, and the bare cavity mode outside resonance. The plot labeled $\uparrow \uparrow$ shows the interaction between $f_{\uparrow \uparrow }$ and the magnon mode $M_2$ at $B=0.471\mathrm{T}$, as well as the bare cavity mode outside the resonance. Dashed curves represent Lorentzian fits to the data.

Figure 6

(a) Magnon modes observed near the dark cavity mode $\uparrow \uparrow$ at $f_{\uparrow \uparrow }=13.9\mathrm{GHz}$. Red crosses and shaded areas are theoretical predictions for magnon modes of order $(n,m)$ [46]; (b) interaction between the magnon doublet $M_3$ and the dark cavity mode. Avoided crossings between photon and magnon modes are observed as transmission dips. The dashed curves show the three-mode interaction model fit.

Figure 7

Origins of the different couplings of the dark cavity mode to two orthogonal realizations [$c_R^{†}{c}_R$ (a) and $c_L^{†}{c}_L$ (b) in Eq. (3)] of the $M_3$ mode which is (2,2) magnon-sphere mode. The plot shows the top cavity view with two black circles denoting the post orientation with respect to the sphere.

Figure 8

Predictions of the optimized system response with the cavity tuned to the resonant frequency observed in the experiment.