Comparative Study of Quantum Features of Radiation Generated by Nondegenerate Down-Conversion

Abstract

A comparative study of quantum properties associated with coherent correlations between the modes of light generated by a nondegenerate parametric down-conversion process is presented. Particularly, the degree of nonclassicality that can be quantified by logarithmic negativity and quantum discord is compared, and a striking similar tendency is observed. In the same manner, a comparison of the quadrature variance, the smallest eigenvalue of the symplectic matrix and Einstein-Podolsky-Rosen (EPR) steering is made whereby a significant similarity is witnessed. The highest degree of quantum features of the radiation is in general manifested near threshold, whereas the quantifying measure related to the smallest eigenvalue of the symplectic matrix is found to capture a better degree of quantumness. Since each criterion renders only sufficient condition, it is hoped that witnessing consistent outcome across different criteria enhances the chance for experimental verification and practical application.

Appendix A: Correlations

In the following, the time evolution of the cavity mode variables and their correlations are obtain in Heisenberg’s picture to be

$$ \frac{d}{dt}\langle\hat{a}\rangle=-\frac{\kappa}{2}\langle\hat{a}\rangle+\mu\langle\hat{b}^{\dagger}\rangle, $$

(A1)

$$ \frac{d}{dt}\langle\hat{b}\rangle=-\frac{\kappa}{2}\langle\hat{b}\rangle+\mu\langle\hat{a}^{\dagger}\rangle, $$

(A2)

$$ \frac{d}{dt}\langle\hat{a}^{2}\rangle=-\kappa\langle\hat{a}^{2}\rangle+2\mu\langle\hat{a}\hat{b}^{\dagger}\rangle, $$

(A3)

$$ \frac{d}{dt}\langle\hat{b}^{2}\rangle=-\kappa\langle\hat{b}^{2}\rangle+2\mu\langle\hat{b}\hat{a}^{\dagger}\rangle, $$

(A4)

$$ \frac{d}{dt}\langle\hat{a}^{\dagger}\hat{a}\rangle= -\kappa\langle\hat{a}^{\dagger}\hat{a}\rangle-2\mu\langle\hat{a}^{\dagger}\hat{b}^{\dagger}\rangle, $$

(A5)

$$ \frac{d}{dt}\langle\hat{b}^{\dagger}\hat{b}\rangle= -\kappa\langle\hat{b}^{\dagger}\hat{b}\rangle-2\mu\langle\hat{b}^{\dagger}\hat{a}^{\dagger}\rangle, $$

(A6)

$$ \frac{d}{dt}\langle\hat{a}\hat{b}\rangle=-\kappa\langle\hat{a}\hat{b}\rangle-\mu\left( \langle\hat{a}^{\dagger}\hat{a}\rangle+\langle\hat{b}^{\dagger}\hat{b}\rangle\right)-\mu, $$

(A7)

$$ \frac{d}{dt}\langle\hat{a}\hat{b}^{\dagger}\rangle= -\kappa\langle\hat{a}\hat{b}^{\dagger}\rangle+\mu\left( \langle\hat{a}^{2}\rangle+\langle\hat{b}^{\dagger2}\rangle\right). $$

(A8)

On the other hand, with the aid of (A5), (A6) and (A7), one readily finds at steady state

$$ \begin{array}{@{}rcl@{}} \mu \langle\hat{a}^{\dagger}\hat{a}\rangle+\mu \langle\hat{b}^{\dagger}\hat{b}\rangle+k\langle\hat{a}\hat{b}\rangle&=&-\mu,\\ \kappa \langle\hat{a}^{\dagger}\hat{a}\rangle+2\mu \langle\hat{a}\hat{b}\rangle&=&0,\\ \kappa \langle\hat{b}^{\dagger}\hat{b}\rangle+2\mu \langle\hat{a}\hat{b}\rangle&=&0. \end{array} $$

(A9)

Simultaneously solving these equations thus leads to

$$ \langle\hat{a}^{\dagger}\hat{a}\rangle=\langle\hat{b}^{\dagger}\hat{b}\rangle=\frac{2\mu^{2}}{\kappa^{2}-4\mu^{2}}, $$

(A10)

$$ \langle\hat{a}\hat{b}\rangle=-\frac{\kappa\mu}{\kappa^{2}-4\mu^{2}}. $$

(A11)