Semiclassical Wigner distribution for a two-mode entangled state generated by an optical parametric oscillator

We derive the steady-state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. We assume that the pump mode is strongly damped, which permits its adiabatic elimination. When the elimination is correctly executed, the resulting stochastic equations contain multiplicative noise terms and do not admit a potential solution. However, we develop a heuristic scheme leading to a satisfactory steady-state solution. This provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the optical parametric oscillator. A non-Gaussian distribution is obtained for the above threshold solution.

Figure 1

Squeezed quadrature variance as a function of the pump level. We compare the results of the positive-$P$ (solid black line) and truncated-Wigner stochastic equations with [solid (red) line] and without (dotted black line) multiplicative noise terms. The first two curves are almost indistinguishable; the third one decreases monotonically. We chose $g^2=0.01$ in all calculations and ${\gamma }_0=10\gamma$ in the positive-$P$ simulation. All plotted quantities are dimensionless.

Figure 2

Conditional Wigner distribution for $y_1=0$ and $y_2=0$, (a) below ($\mu =0.5$), (b) at ($\mu =1.0$), and (c) above the threshold ($\mu =1.5$). In all cases $g^2=0.01$. All plotted quantities are dimensionless.

Figure 3

Marginal distributions (a) below threshold, $\mu =0.8$; (b) at threshold, $\mu =1.0$; and (c) above threshold, $\mu =1.2$. In all cases $g^2=0.01$. All plotted quantities are dimensionless.

Figure 4

Comparison between the values of the antisqueezed variable $\langle x_{+}^2\rangle$ obtained from the linearized theory (dashed line) with those obtained from the Wigner function of Ref. [7] (solid line) and the one given by Eq. (12) (crosses). In all cases we have set $g^2=0.01$. All plotted quantities are dimensionless.

Figure 5

Comparison between the values of the squeezed variable $\langle x_{-}^2\rangle$ obtained from the linearized theory (solid line) with those obtained from the Wigner function of Ref. [7] (open circles) and the one given by Eq. (12) (solid diamonds). In all cases we set $g^2=0.01$. All plotted quantities are dimensionless.