Bright bichromatic entanglement and quantum dynamics of sum frequency generation

We investigate the quantum properties of the well-known process of sum frequency generation, showing that it is potentially a very useful source of nonclassical states of the electromagnetic field, some of which are not possible with the more common techniques. We show that it can produce quadrature squeezed light, bright bichromatic entangled states, and symmetric and asymmetric demonstrations of the Einstein-Podolsky-Rosen paradox. We also show that the semiclassical equations totally fail to describe the mean-field dynamics when the cavity is strongly pumped.

Figure 1

Schematic of the intracavity scheme, showing the ${\chi }^{\left(2\right)}$ crystal with nonlinear interaction strength $\kappa$, the two mirrors, the two pump fields, and the three optical modes at frequencies ${\omega }_j$, where ${\omega }_3={\omega }_1+{\omega }_2$.

Figure 2

(Color online) Mean intensities from an average over $2.67\times {10}^6$ stochastic trajectories of the positive-$P$ equations (6). The horizontal axis is a scaled dimensionless time, $\zeta =\kappa \left|{\alpha }_1\left(0\right)\right|t$. All quantities plotted in this and subsequent graphs are dimensionless.

Figure 3

(Color online) The $X$ quadrature variance for the field at ${\omega }_3$ and the Fano factor for the intensity sum of the two low-frequency fields. A value of less than one signifies squeezing or sub-Poissonian statistics, respectively. The two low-frequency fields show excessive noise in both $X$ and $Y$ quadratures and exhibit super-Poissonian intensity fluctuations.

Figure 4

(Color online) The Duan-Simon correlation $V\left(X_1+X_2\right)+V\left(Y_1-Y_2\right)$ and the Reid EPR correlation $V_{inf}\left(X_j\right)V_{inf}\left(Y_j\right)$ for the two low-frequency fields. With symmetric inputs, the EPR correlation is equal for $j=1$ and 2. Note that the Duan-Simon value has been divided by four so that both entanglement and the EPR paradox are demonstrated for values less than 1.

Figure 5

(Color online) The regions of stability and instability as a function of $ϵ$ and ${\gamma }_3/\gamma$ for $\kappa ={10}^{-2}$ and $\gamma ={\gamma }_1={\gamma }_2=1$. In the region to the right of the line there is always at least one eigenvalue with a negative real part and the linearization process for calculating spectra is not valid.

Figure 6

(Color online) The spectral variance of the high-frequency quadrature ${\widehat{X}}_3$ for $\kappa ={10}^{-2}$, ${\gamma }_1={\gamma }_2=\gamma =1$, and ${\gamma }_3=10$. The solid line is for ${ϵ}_1={ϵ}_2=200$, giving ${\alpha }_1={\alpha }_2=0.25{\alpha }_c$, the dash-dotted line is for ${ϵ}_1={ϵ}_2=400$, giving ${\alpha }_1={\alpha }_2=0.61{\alpha }_c$, and the dashed line is for ${ϵ}_1={ϵ}_2=600$, giving ${\alpha }_1={\alpha }_2=0.95{\alpha }_c$.

Figure 7

(Color online) The Duan-Simon entanglement correlations for the low-frequency modes 1 and 2 and the same parameters as Fig. 6. The two quadratures which violate the inequality (less than 4) and hence signify entanglement are $X_1+X_2$ and $Y_1-Y_2$. The solid line is for ${ϵ}_1={ϵ}_2=200$, giving ${\alpha }_1={\alpha }_2=0.25{\alpha }_c$, the dash-dotted line is for ${ϵ}_1={ϵ}_2=400$, giving ${\alpha }_1={\alpha }_2=0.61{\alpha }_c$, and the dashed line is for ${ϵ}_1={ϵ}_2=600$, giving ${\alpha }_1={\alpha }_2=0.95{\alpha }_c$.

Figure 8

(Color online) The spectral EPR product of the inferred variances for modes 1 and 2, and the same parameters as Fig. 6. The solid line is for ${ϵ}_1={ϵ}_2=200$, giving ${\alpha }_1={\alpha }_2=0.25{\alpha }_c$, the dash-dotted line is for ${ϵ}_1={ϵ}_2=400$, giving ${\alpha }_1={\alpha }_2=0.61{\alpha }_c$, and the dashed line is for ${ϵ}_1={ϵ}_2=600$, giving ${\alpha }_1={\alpha }_2=0.95{\alpha }_c$.

Figure 9

(Color online) Demonstration of asymmetric steering for the parameters ${\gamma }_1=1$, ${\gamma }_2=40$, ${\gamma }_3=2$, ${ϵ}_1=400$, ${ϵ}_2=2400$, and $\kappa =0.01$. The correlation EPR12 shows that 2 can steer 1, while EPR21 shows that 1 cannot steer 2.

Figure 10

(Color online) Intracavity intensities predicted by averaging $2.2\times {10}^5$ stochastic trajectories of the full positive-$P$ representation equations, for ${\gamma }_1={\gamma }_2=1$, ${\gamma }_3=10$, ${ϵ}_1={ϵ}_2={10}^3$, and $\kappa =0.01$. The straight lines are the semiclassical steady-state predictions for the same parameters.