Entanglement in non degenerate optical parametric oscillator with partially coherent driving

At the moment quantum properties of light in optical parametric oscillators are actively studied. As a model of OPO, we consider three modes of a resonator with frequencies ωp (pump) and ωs=ωi=ωp/2 (signal an idler),... [ view full abstract ]

At the moment quantum properties of light in optical parametric oscillators are actively studied. As a model of OPO, we consider three modes of a resonator with frequencies ω_p (pump) and ω_s=ω_i=ω_p/2 (signal an idler), interacting nonlinearly due to the presence of quadratic nonlinearity. Pumping of the oscillator is performed by external continuous laser radiation with central frequency ω_p. The main approach to study the dynamics of the discussed system that present in literature is to use Fokker-Planck equation for Wigner or positive P representation. In such papers to determine the entanglement in the system criteria based on quadratic quantities and valid only for Gaussian quantum states are used. In our work we perform direct numerical calculation of master equation in Lindblad form using quantum trajectories method. This allow us to calculate the full value of logarithmic negativity E_N without any assumptions. Besides, our calculations show that quantum correlations between pump and signal(idler) modes are negligibly small and in this assumption we perform analytic solution of Heisenberg-Langevin equations for the considered system. Comparison of our two approaches can be seen in fig.1(a) where the results of calculation of mean photon number in signal mode are presented. There points are the result of numeric simulation, solid curve is analytic solution and dashed line is the result of classical consideration of the system. Our calculations show that E_N goes to the stationary value and its dependence on pump amplitude is present on fig.1(b). For analytic calculations we use formula for E_N derived for Gaussian states. The difference between two approaches at high pump amplitudes is the evidence that photon statistics in the OPO above threshold is not Gaussian. In our previous work we have shown that phase noise in the pump may dramatically influence the entangled state formation in parametric systems. Here we again see that taking into account such noise, inevitably presented in all laser sources, leads to finite lifetime of the entanglement in the system. And the lifetime rapidly decreases with the increase of the pump amplitude, see fig.2.

This work was supported by the Ministry of Education and Science of the Russian Federation under contract No.14.W03.31.0032.