Position-momentum-entangled photon pairs in nonlinear waveguides and transmission lines

We analyze the correlation properties of light in nonlinear waveguides and transmission lines, predict the position-momentum realization of the Einstein-Podolsky-Rosen paradox for photon pairs in Kerr-type nonlinear photonic circuits, and we show how two-photon entangled states can be generated and detected.

Figure 1

Entangled two-photon states in nonlinear waveguides. (a) Spectrum of a two-photon state, $\tilde{E}=(E-2{\omega }_{k_0})\left|\beta \right|/{\kappa }^2$, with total momentum $2k_0$ in a waveguide with quadratic dispersion (1) for $\beta <0, \kappa >0$ (left) and $\beta >0, \kappa <0$ (right). The solid line corresponds to the continuous spectrum, while the single eigenvalue corresponding to the entangled state is shown by a dashed line. (b) Wigner function of the two-photon entangled state. It takes negative values, which is a hallmark of non-Gaussian entangled states.

Figure 2

Entangled photon pairs in chains of coupled nonlinear resonators. (a) Schematic of the setup. (b) An example of a two-photon spectrum for $J_0\sim u$ and ${\omega }_c=0$. (c) Split of the bound-state eigenvalue from the spectrum edge $\mathrm{\Delta }E$ as a function of $J_0=4Jcos\left(k_{0}a\right)$. (d) Joint probability distribution $P$ for various ratios $J_0/u$.

Figure 3

Steady-state average number of photons $N_j$ (dashed line) and on-site zero-time-delay correlation function $g_{jj}^{\left(2\right)}\left(0\right)$ (solid line) in resonator $j$, as a function of pumping frequency ${\omega }_p$, for a coherently pumped short chain (three resonators) with pumping amplitude $F={10}^{-2}\left|u\right|$, photon decay rate $\gamma =0.1\left|u\right|$, and pair momentum $2k_0=0$. (a) $J_0=0.1\left|u\right|, u<0$; (b) $J_0=0.1\left|u\right|, u>0$; (c) $J_0=\left|u\right|, u>0$; (d) $J_0=4\left|u\right|, u>0$. Note that far from the resonances, $g_{jj}^{\left(2\right)}\left(0\right)\to 1$.