Spin-photon interaction in a cavity with time-reversal symmetry breaking

Employing a sapphire whispering gallery mode resonator, we demonstrate features of the spin-photon interaction in cavities with broken time-reflection symmetry. The broken symmetry leads to a lifting of the degeneracy between left-handed and right-handed polarized cavity photons, which results in an observable gyrotropic effect. In the high-$Q$ cavity limit, such a situation requires a modification of the Tavis-Cummings Hamiltonian to take into account conservation of spin angular momentum and the corresponding selection rules. As a result, the system is represented by a system of two linearly coupled bosonic modes, with each one coupled to its own subensemble of two-level systems with different energy splittings. In the experimental example, these subensembles originate from ${\text{Fe}}^{3+}$ impurity ions effectively seen as a two-level system at the interaction frequency. The temperature dependence of the population of each subensemble (in terms of effective susceptibility of the medium) is determined experimentally in accordance with the theoretical predictions revealing various paramagnetic impurity types in the solid. The regimes of backscatterer and spin ensemble domination are discussed and compared.

Figure 1

Energy diagram of the ${\text{Fe}}^{3+}$ ion interaction with a double resonance in a cavity with the broken T-symmetry. (a) The case of a spin-increasing ion transition where ${\omega }_{+}$ is tuned to the doublet, and (b) the case when ${\omega }_{-}$ is tuned to the doublet.

Figure 2

Experimental setup: The incident signal is attenuated at various stages of the dilution refrigerator, the transmitted signal is amplified both at 4 K and at room temperature. The backaction noise of the cold amplifier is damped by a milli-Kelvin isolator. Antiradiation shields are not shown.

Figure 3

Interaction of the two subensembles of ${\text{Fe}}^{3+}$ ions with two different WGM doublets at 36 mK and $B\approx 44$ mT. (a) Spin-increasing ion transition ${\omega }_{+}$ tuned to the doublet $2\pi f_A$ (13.259 GHz), where $g_{+}\gg g_{RL}$ and $g_{-}\sim 0$. (Bb) Spin-decreasing ion transition ${\omega }_{-}$ tuned to the doublet $2\pi f_B$ (10.810 GHz), where $g_{-}\gg g_{RL}$ and $g_{+}\sim 0$.

Figure 4

Interactions of the ${\text{Fe}}^{3+}$ ion ensemble with two WGM doublets. Solid curves are theoretical predictions. Squares denote spin-increasing transitions [situation (a) in Figs. 1 and 3], and circles denote spin-decreasing transitions [situation (b) in Figs. 1 and 3]. Dashed lines are particular transitions discussed in the text.

Figure 5

Weak interaction of the ensemble of ${\text{Fe}}^{3+}$ ions with two WGM doublets in the transitional case $g_{\pm }\to g_{RL}$ ($T\sim 5$ K). (a) The case of a spin-increasing ion transition ${\omega }_{+}$, and (b) the case of a spin-decreasing transition.

Figure 6

Comparison of the spin-dominated regime ($T=36$ mK, $g_{\pm }\gg g_{RL}$) and the spin regime with significant backscatter (5 K, $g_{\pm }\to g_{RL}$). The former interaction can be modeled by a two oscillator model (one WGM and one spin ensemble). The latter requires a three oscillator model (two for WGMs, and one for the spin ensemble).

Figure 7

Measured and predicted susceptibilities of two WG modes $f_A$ and $f_B$ coupled to two transitions ${\omega }_{+}$ and ${\omega }_{-}$.