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.

Figure 1

Schematic of cavity electro-optic modulator coupled to traveling fields. The physics remains essentially the same regardless of the actual types of the optical and microwave resonators.

Figure 2

Red-sideband optical pumping schemes. (a) A doubly resonant configuration with an off-resonant pump. (b) A triply resonant configuration with a resonant pump in a different polarization mode [8]. Both schemes suppress interactions with the off-resonant field at ${\omega }_a-2{\omega }_b$.

Figure 3

Root-locus plot for increasing red-sideband pump strength $\left|g\alpha \right|$.

Figure 4

Conversion efficiency $R\left(0\right)/\eta$ at zero detuning versus the cooperativity parameter $G_0$.

Figure 5

The color plot shows the conversion efficiency $R\left(\omega \right)/\eta$ at ${\Gamma }_a={\Gamma }_b$ and a fixed $\eta$ with respect to $G_0$ on the horizontal axis and $\Omega \equiv 2\omega /\sqrt{{\Gamma }_a{\Gamma }_b}$ on the vertical axis. The solid lines are the imaginary parts of the poles.

Figure 6

The color plot shows the conversion efficiency $R\left(\omega \right)/\eta$ at critical coupling ($G_0=1$) and a fixed $\eta$ with respect to $\ln ({\Gamma }_b/{\Gamma }_a)$ on the horizontal axis and $\Omega \equiv 2\omega /\sqrt{{\Gamma }_a{\Gamma }_b}$ on the vertical axis. The solid lines are the imaginary parts of the poles. The bandwidth is maximum when ${\Gamma }_a={\Gamma }_b$ and the imaginary parts of the poles are the farthest apart.

Figure 7

Blue-sideband optical pumping scheme in (a) a doubly resonant configuration with an off-resonance pump and (b) a triply resonant configuration with a resonant pump.

Figure 8

Root-locus plot for increasing blue-sideband pump strength $\left|g\alpha \right|$.

Figure 9

Idler gain $\mathcal{R}\left(0\right)/\eta$ in dB at zero detuning versus $G_0$.

Figure 10

Nonclassicality parameter $\Lambda \left(0\right)$ at zero detuning versus $G_0$.