Entanglement and asymmetric steering over two octaves of frequency difference

The development of quantum technologies which use quantum states of the light field interacting with other systems creates a demand for entangled states spanning wide frequency ranges. In this work we analyze a parametric scheme of cascaded harmonic generation which promises to deliver bipartite entangled states in which the two modes are separated by two octaves in frequency. This scheme is potentially very useful for applications in quantum communication and computation networks as well as providing for quantum interfaces between a wider range of light and atomic ensembles than is presently practicable. It doubles the frequency range over which entanglement is presently available.

Figure 1

Schematic of the system showing the nonlinear media inside an optical cavity and the input and output modes.

Figure 2

Positive-P and classical solutions for intensities in the traveling-wave configuration, with ${\kappa }_1=0.005$, ${\kappa }_2=4{\kappa }_1$, and $N_1\left(0\right)={10}^6$. The quantum solutions are the lines which begin oscillations at $\xi \approx 5$. The dimensionless interaction time, $\xi$, is equal to ${\kappa }_1\left|\langle {\alpha }_1\left(0\right)\rangle \right|t$. The equations were averaged over $1.4\times {10}^6$ stochastic trajectories. All quantities in this and subsequent figures are dimensionless.

Figure 3

Positive-P solutions for $\widehat{X}$ quadrature variances in the traveling-wave configuration, with parameters as in Fig. 2. The line at 1 is a guide to the eye.

Figure 4

Positive-P solutions for EPR steering between ${\omega }_2$ and ${\omega }_3$ in the traveling-wave configuration, with parameters as in Fig. 2. The line at 1 is a guide to the eye.

Figure 5

Positive-P and classical solutions for $N_1$ in the self-pulsing regime, for ${\kappa }_1=0.005$, ${\kappa }_2=4{\kappa }_1$, ${\gamma }_1=1$, ${\gamma }_2={\gamma }_3={\gamma }_1/2$, and $\epsilon =400$. The classical solution has the larger oscillations.

Figure 6

$\widehat{X}$ quadrature variances, for ${\kappa }_1=0.005$, ${\kappa }_2=4{\kappa }_1$, ${\gamma }_1=1$, ${\gamma }_2={\gamma }_3={\gamma }_1/2$, and $\epsilon =105$. The frequency axis is in units of the line width of the fundamental, ${\gamma }_1$.

Figure 7

The Reid criteria for steering between ${\omega }_2$ and ${\omega }_3$, for the same parameters as Fig. 6.

Figure 8

The Reid criteria for steering between modes 1 and 2, and 1 and 3, for the same parameters as Fig. 6. The solid line is $EP{R}_{12}$, the dash-dotted line is $EP{R}_{21}$, the dashed line is $EP{R}_{13}$, and the dotted line is $EP{R}_{31}$.