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We propose a device architecture capable of direct quantum electro-opticalconversion of microwave to optical photons. The hybrid system consists of aplanar superconducting microwave circuit coupled to an integratedwhispering-gallery-mode microresonator made from an electro-optical material.We show that electro-optical (vacuum) coupling rates $g_0$ as large as$\sim2\pi \, \mathcal{O}(10-100)$ kHz are achievable with currently availabletechnology, due to the small mode volume of the planar microwave resonator.Operating at millikelvin temperatures, such a converter would enablehigh-efficiency conversion of microwave to optical photons. We analyze theadded noise, and show that maximum conversion efficiency is achieved for amulti-photon cooperativity of unity which can be reached with optical power aslow as $ \mathcal{O}(1)\,\mathrm{mW} $.

On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator