Collective behavior of ${\mathrm{Cr}}^{3+}$ ions in ruby revealed by whispering gallery modes

We present evidence for the collective action of ${\mathrm{Cr}}^{3+}$ ion impurities in a highly concentrated ruby crystal coupled to microwave whispering gallery modes (WGMs). The cylindrical geometry of the crystal allows the creation of superradiant or “spin-mode” doublets, with spatial structures similar to that of WGMs. The formation of these spin patterns allows us to observe directly different selection rules, namely, wave number and azimuthal phase matching. The demonstration is made via an avoided level crossing between spin and photon mode doublets as well as absence of coupling between spin modes of different wave numbers. The effect is observable due to strong spin-photon coupling (67 MHz) exceeding both spin ensemble and cavity losses as well as the photon doublet splitting. We demonstrate that a four harmonic oscillator model not only with coupling between photon resonances (0.43 MHz) but also with a spin doublet (73 MHz) is necessary to accurately describe these results.

Figure 1

Four harmonic oscillator model depiction.

Figure 2

Experimental setup. The crystal and superconducting magnet are cooled to 4 K in a cryogenic refrigerator, while the microwave coupling probes are aligned to excite WGH modes ($E_z, H_r, H_{\varphi }$ components). Two probes are used to view the relevant modes in transmission (S21) on a vector network analyzer (VNA) from which $Q$ factors can be determined as well as the frequency shift caused by coupling to spins.

Figure 3

Spectroscopy results showing the interaction of the generated and pumped WGMs with the $\left|+3/2\right.〉\to \left|+1/2\right.〉{\mathrm{Cr}}^{3+}$ electron spin transition as the magnetic field is swept.

Figure 4

Interaction of the 9.545 GHz WGM (${\mathrm{WGH}}_{7,1,1}$) and the ${\mathrm{Cr}}^{3+}\left|+3/2\right.〉\to \left|+1/2\right.〉$ transition at $B=0.68$ T. Asymptotes (dashed) and four SHO model fit (solid).

Figure 5

Interaction of the 12.74 GHz WGM and the ${\mathrm{Cr}}^{3+}\left|-3/2\right.〉\to \left|-1/2\right.〉$ transition.

Figure 6

$Q$ factor of the hybrid $s$ and $c$ modes at 9.54 GHz as B field is swept.