Spatially Separated Generalized Two-Mode Squeezed Vacuum States in Lossy Coupled Resonator Optical Waveguides

Abstract

Two-mode squeezed states have potential applications in quantum teleportation, quantum computation, and quantum information. Spontaneous parametric down conversion (SPDC) is one of the processes that can be used to generate two-mode squeezed optical states, both in bulk and integrated systems. In this work, we theoretically model SPDC in a vertically-pumped, lossy, coupled-resonator optical waveguide (CROW), shown in Fig. 1(a), to generate counterpropagating continuous variable (CV) entangled states. For a CROW in which the cavities are identical, using the nearest neighbour TB approximation, the CROW mode dispersion is given by $\tilde{\omega}_{Fk} \approx \tilde{\omega}_{F}[1+ \tilde{\beta}_1 \cos (kD)] \equiv \omega _{Fk}-i \gamma_{Fk},$ , where $\tilde{\omega}_{F}$ is the individual-cavity complex mode frequency, $\tilde{\beta}_{1}$ is the complex coupling parameter, D is the periodicity of the CROW, and k is the Bloch vector. We consider the case of a pump that is Gaussian in time and space, with spatial and temporal full width at half maxima of $\Delta r_{FMHM} = 2 \sqrt{l n2}/ \sigma_{+}$ and $\Delta t_{FMHM} = 2 \sqrt{l n2}/ \tau / (\sigma\_D)$ , respectively, where σ + and σ_ are the k-space width parameters (see below), and $\tau = Re \{1 / (\tilde{\omega}_{F} \tilde{\beta}_1)\}$ is the minimum time for the light to propagate one period.