Cavity quantum electro-optics. II. Input-output relations between traveling optical and microwave fields

In a previous paper [Phys. Rev. A 81, 063837 (2010)], I proposed a quantum model of the cavity electro-optic modulator, which can coherently couple an optical cavity mode to a microwave resonator mode and enable quantum operations on the two modes, including laser cooling of the microwave resonator, electro-optic entanglement, and backaction-evading optical measurement of a microwave quadrature. In this sequel, I focus on the quantum input-output relations between traveling optical and microwave fields coupled to the cavity electro-optic modulator. With red-sideband optical pumping, the relations are shown to resemble those of a beam splitter for the traveling fields, so that in the ideal case of zero parasitic loss and critical coupling, microwave photons can be coherently up converted to flying optical photons with unit efficiency, and vice versa. With blue-sideband pumping, the modulator acts as a nondegenerate parametric amplifier, which can generate two-mode squeezing and hybrid entangled photon pairs at optical and microwave frequencies. These fundamental operations provide a potential bridge between circuit quantum electrodynamics and quantum optics.