Second order correlation function

In order to obtain the correlator $\langle\ud{\sigma_1}(t)\ud{\sigma_1}(t+\tau)\sigma_1(t+\tau)\sigma_1(t)\rangle$ , and $g^{(2)}(\tau)$ , but also the cross correlator $\langle\ud{\sigma_1}(t)\ud{\sigma_2}(t+\tau)\sigma_2(t+\tau)\sigma_1(t)\rangle$ to compute $g_{12}^{(2)}(\tau)$ , we can proceed as in Sec. 3.6. Once more, the quantum regression formula in Eq. (2.114), for the general set of operators $C_{\{m,n,\mu,\nu\}}=\ud{\sigma_1}^m\sigma_1^n\ud{\sigma_2}^\mu \sigma_2^\nu$ , is the same as that used to compute the mean values. We solve the same equations, the only difference as compared with the expressions for the LM, are the steady state vectors,

$\begin{displaymath}\mathbf{w}^\mathrm{SS}= \left( \begin{array}{c} 0\\ n_B\\ 0... ...\ n_1n_2-n_B\\ n_{12}n_1\\ n_{12}^*n_1 \end{array}\right)\,.\end{displaymath}$

(4.43)

$\tau>0$

$\begin{subequations}\begin{align}&g^{(2)}(\tau)=1+\frac{[e^{-\mathbf{M}_0\tau}\,... ...bf{M}_0\tau}\,\mathbf{f}^\mathrm{SS}]_2}{n_1n_2}\, \end{align}\end{subequations}$

where $[\mathbf{x}]_1$ means that we take the first element of the vector $\mathbf{x}$ . At zero delay we have, $g^{(2)}(\tau=0)=0$ and $g_{12}^{(2)}(\tau=0)=n_B/(n_1n_2)\leq 1$ , and at infinite delay, $g_1^{(2)},g_{12}^{(2)}(\tau\rightarrow\infty)\rightarrow 1$ . As it corresponds to fermions, the emission presents antibunching, that is, the second particle cannot be emitted at the same time if it is of the same nature as the first one, but in a posterior emission, $g^{(2)}(0)<g^{(2)}(\tau)$ (see Fig. 4.10). If they are different kinds, there is no antibunching. $g_{12}^{(2)}(\tau=0)=1$ when the two dots behave as independent. For the optimum coupling, in Fig. 4.11, $g_{12}^{(2)}(\tau=0)$ can be as small as

**Figure 4.10:** SS values of $g^{(2)}(\tau)$ (solid blue) $g_{12}^{(2)}(\tau)$ (dashed purple) (a) and $S^{(2)}(\omega)$ (b) in SC given by Eq. (4.44). The parameters are those in Fig. 3.18 for comparison: $\Delta=0$ , $\gamma_{E1}=g$ , $\gamma_{E2}=0.5g$ , $P_{E1}=0.5g$ and $P_{E2}=0.1g$ .
$\includegraphics[width=0.4\linewidth]{chap4/g2/g2F.eps}$ $\includegraphics[width=0.4\linewidth]{chap4/g2/S2F.eps}$

**Figure 4.11:** SS values of $g^{(2)}(\tau)$ (solid blue) $g_{12}^{(2)}(\tau)$ (dashed purple) in (MC) at resonance. Parameters are chosen so that the coupling is optimum $G=\sqrt{2}g$ : $\gamma_{E1}=P_{E2}=0$ and $P_{E1}=\gamma_{E2}=2g$ (see Fig. 4.8).
$\includegraphics[width=0.4\linewidth]{chap4/g2/g2c.eps}$