Nonclassicality criteria: Quasiprobability distributions and correlation functions

We use the exact calculation of the quantum mechanical, temporal characteristic function $\chi \left(\eta \right)$ and the degree of second-order coherence $g^{\left(2\right)}\left(\tau \right)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P\left(\beta \right)$ of the Glauber-Sudarshan coherent or P representation of the density of state and the Mandel $Q_M\left(\tau \right)$ parameter, versus the criteria associated with the two-time correlation function $g^{\left(2\right)}\left(\tau \right)$.

Figure 1

Temporal second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for $\overline{n}=0.1,r=0.1$, and $\left|\alpha \right|=0$. One has $g^{\left(2\right)}\left(0\right)=3.1625$ and ${lim}_{\tau \to \infty }{g}^{\left(2\right)}\left(\tau \right)=1.6603$. Both nonclassical inequalities in (21) are violated and the statistics is super-Poissonian.

Figure 2

Plot of $\left[\right|g^{\left(2\right)}\left(0\right)-1|-|g^{\left(2\right)}\left(\tau \right)-1\left|\right]$ for $\overline{n}=0.1,r=0.1$, and $\left|\alpha \right|=0$, which asymptotically approaches 1.5022. The nonclassical inequality, (22) is violated and $g^{\left(2\right)}\left(\tau \right)$ is strictly classical.

Figure 3

Temporal second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for $\overline{n}=0.1,r=0.1$, and $\left|\alpha \right|=2$. One has $g^{\left(2\right)}\left(0\right)=0.9975$ and ${lim}_{\tau \to \infty }{g}^{\left(2\right)}\left(\tau \right)=1.0180$. The correlation is strictly nonclassical since both inequalities in (21) are satisfied. The statistics is sub-Poissonian.

Figure 4

Plot of $\left(\right|g^{\left(2\right)}\left(0\right)-1|-|g^{\left(2\right)}\left(\tau \right)-1\left|\right)$ for $\overline{n}=0.1,r=0.1$, and $\left|\alpha \right|=2$, which asymptotically approaches $-0.0155$. The behavior is classical for $\mathrm{\Omega }\tau \le 2.5793$ and nonclassical for $\mathrm{\Omega }\tau >2.5793$ according to (22).

Figure 5

Temporal second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for $\overline{n}=1.0,r=0.1$, and $\left|\alpha \right|=0$, where $g^{\left(2\right)}\left(0\right)=2.0859$ and ${lim}_{\tau \to \infty }{g}^{\left(2\right)}\left(\tau \right)=1.9402$. The behavior is nonclassical for $0<\mathrm{\Omega }\tau <0.5605$ and classical for $\mathrm{\Omega }\tau \ge 0.5605$.

Figure 6

Plot of $\left(\right|g^{\left(2\right)}\left(0\right)-1|-|g^{\left(2\right)}\left(\tau \right)-1\left|\right)$ for $\overline{n}=1.0,r=0.1$, and $\left|\alpha \right|=0$, which asymptotically approaches 0.1457. The behavior is nonclassical for $0<\mathrm{\Omega }\tau <0.5605$ and classical for $\mathrm{\Omega }\tau \ge 0.5605$.

Figure 7

Plot of the Mandel parameter $Q_M\left(\tau \right)$ for $\overline{n}=0.1,r=0.1$, and $\left|\alpha \right|=2$. The behavior is nonclassical for $0\le \mathrm{\Omega }\tau <1.7704$ and classical for $\mathrm{\Omega }\tau \ge 1.7704$ since $Q_M\left(0\right)=-0.0104$.