Strong coupling between whispering gallery modes and chromium ions in ruby

We report the study of interactions between cavity photons and paramagnetic ${\mathrm{Cr}}^{3+}$ spins in a ruby ${(\mathrm{Cr}}^{3+}:{\mathrm{Al}}_2{\mathrm{O}}_3)$ whispering gallery mode (WGM) resonator. Examining the system at microwave frequencies and millikelvin temperatures, spin-photon couplings up to 610 MHz or about 5% of photon energy are observed between the impurity spins and high quality factor $\left(Q>{10}^5\right)$ WGM. Large tunability and spin-spin interaction allows operation in the strong coupling regime. The system exhibits behavior not predicted by the usual Tavis-Cummings model because of interactions within the two-level spin bath, and the existence of numerous photonic modes.

Figure 1

(a) Energy level splitting of the ${\mathrm{Cr}}^{3+}$ ion. The ground state of this system is the $|-\frac{3}{2}\rangle$ state. (b) Relative susceptibility of the transitions as a function of temperature, with an applied dc magnetic field of 0.08 T.

Figure 2

Composite image of a series of measurements of the power transmission $\left(S{}_{21}\right)$ of the ruby resonator, at frequencies around 13.9 GHz at a minimum temperature approximately 20 mK. The plot shows a number of WGMs experiencing different susceptibilities due to the ${\mathrm{Cr}}^{3+}$ ESR. The line representing the bare ESR transition is also shown.

Figure 3

Model of the coupling between a WGM at 13.9 GHz [Eq. (8)], and ${\mathrm{Cr}}^{3+}$ ESR. The case of low and high coupling are shown in blue (dark gray) and red (light gray) respectively. In the low coupling case the two modes are symmetric about the two axes around the interaction. In the high coupling case, the two modes are not symmetric due to the influence of the antiparallel ESR transition. The shaded area in the subfigure labeled I is rotated and presented as subfigure II. The dashed line shows the expected result in the case of high coupling which is symmetric about both axes, rather than the nonsymmetric result observed (solid line).

Figure 4

Series of measurements around 13.9 GHz showing avoided crossings between different WGMs with different susceptibility to ${\mathrm{Cr}}^{3+}$. The difference in susceptibility causes the modes to tune across each other. Plots A and C are a subset of the measurements of Fig. 2 at 20 mK; plots B and D were measured at a slightly higher temperature of 40 mK, which may result a marginally weaker coupling to the ESR.

Figure 5

Power transmission of the strong coupling between the hybridized ${\alpha }_3$ and ${\eta }_3$ WGMs in Fig. 4. The trace is calculated from the model, Eq. (10).