Microwave to optical photon conversion via fully concentrated rare-earth-ion crystals

Most investigations of rare-earth ions in solids for quantum information have used crystals where the rare-earth ion is a dopant. Here, we analyze the conversion of quantum information from microwave photons to optical frequencies using crystals where the rare-earth ions, rather than being dopants, are part of the host crystal. These concentrated crystals are attractive for frequency conversion because of their large ion densities and small linewidths. We show that conversion with both high efficiency and large bandwidth is possible in these crystals. In fact, the collective coupling between the rare-earth ions and the optical and microwave cavities is large enough that the limitation on the bandwidth of the devices will instead be the spacing between magnon modes in the crystal.

Figure 1

The device used to convert microwave photons into optical photons. A rare-earth crystal (doped rare-earth crystal for previous device [30], fully concentrated for proposed device) is placed within a microwave resonator and an optical cavity. A static magnetic field is then applied in the $\widehat{z}$ direction. This controls the frequency of the spin resonance (magnon resonance in current device).

Figure 2

(a) Energy-level diagram used to describe the dynamics of the previous conversion device [30]. An input microwave photon, in mode $\widehat{a}$, excites a spin excitation in the $k\mathrm{th}$ rare-earth ion, state ${|1\rangle }_k$. A coherent driving field with Rabi frequency $\mathrm{\Omega }$ then drives the spin excitation to an optical excitation, state ${|2\rangle }_k$. From state ${|2\rangle }_k$, the device decays into the ground state ${|g\rangle }_k$, releasing an optical photon in mode $\widehat{b}$ in the process. Each state transition is driven off resonantly, indicated by the detunings ${\mathrm{\Delta }}_{o,k}$ and ${\mathrm{\Delta }}_{\mu ,k}$. (b) Energy-level diagram used to describe the dynamics of our proposed conversion device. Strong interactions between the spins means that using the single atom picture becomes problematic. The conversion occurs via a similar off-resonant Raman-like process to that of the previous conversion device. The state $|1\rangle$ has one excitation in the Kittel mode and is driven off resonance (an amount ${\mathrm{\Delta }}_M$) by both a microwave input field (coupling strength $G_{\mu }$) and a two optical photon Raman process (coupling strength $G_{o,\mathrm{\Omega }}$). Each optical state ${|2\rangle }_k$ is shifted due to the spin interactions. However, as the optical detuning ${\mathrm{\Delta }}_{o,k}$ is much larger than this energy shift, the effect is negligible to our calculations.

Figure 3

Diagram showing proposed geometry (not to scale) for our (a) microwave resonator and (b) optical resonator. A convex lens is used in the optical resonator to focus the beam into the center of the sphere to minimize spherical aberration. (c), (d) show the modeled field amplitudes of the microwave and optical resonators. The solid white lines in (c) outline the loop gap, and in (d) outline the crystal of diameter 2 mm. The dashed line in (d) indicates the Gaussian beam width.