Second order correlation function

The correlator $\langle\ud{a}(t)\ud{a}(t+\tau)a(t+\tau)a(t)\rangle$ needed to compute $g^{(2)}(\tau)$ , requires to apply the quantum regression formula in Eq. (2.114) for the general set of operators $C_{\{m,n,\mu,\nu\}}=\ud{a}^ma^n\ud{b}^\mu b^\nu$ (that includes $\ud{a}a$ ) with $\Omega_1=\ud{a}$ and $\Omega_2=a$ . As we noted in Sec. 2.7, the quantum regression formula is the same as that used to compute the mean values of Sec. 3.2. The set of correlators $C_{\{m,n,\mu,\nu\}}$ is again reduced to $\tilde{\mathcal{N}}_1$ (see Fig. 3.2). We write explicitly these four correlators, linked to $G^{(2)}(\tau)$ , in the form $\langle\ud{a}(t)C_{\{m,n,\mu,\nu\}}(t+\tau)a(t)\rangle$ and construct the vector

$\displaystyle \mathbf{w}(t,t+\tau)=\left( \begin{array}{c} \langle\ud{a}(t)(\ud... ...(t)\rangle\\ \langle\ud{a}(t)(a\ud{b})(t+\tau)a(t)\rangle \end{array}\right)\,.$

(3.73)

The equation we must solve reads

$\displaystyle \frac{d\mathbf{w}(t,t+\tau)}{d\tau}=-\mathbf{M}_0\mathbf{w}(t,t+\tau)+\mathbf{p}n_a(t)\,,$

(3.74)

with the same matrix $\mathbf{M}_0$ and pumping vector $\mathbf{p}$ [Eqs. (3.7a)] that already appeared in Eq. (3.6) for the mean values. If we concentrate on the SS case, the initial conditions $\mathbf{w}(t,t)$ are given by the SS values,

$\displaystyle \mathbf{w}^\mathrm{SS}=\left( \begin{array}{c} \langle\ud{a}\ud{a... ...le^\mathrm{SS}\\ \langle\ud{a}a\ud{b}a\rangle^\mathrm{SS} \end{array}\right)\,,$

(3.75)

and $n_{a}(t)\rightarrow n_a^\mathrm{SS}$ . These new four average quantities can be obtained by applying once more the quantum regression formula, in the same way we did to obtain $\mathbf{v}(t,t)$ , having in mind the schema on the right of Fig. 3.2. However, the quantities in $\mathbf{w}^\mathrm{SS}$ correspond to $C_{\{\tilde\eta\}}$ in the second manifold, $\{\tilde\eta\}\in\tilde{\mathcal{N}}_2$ (not plotted in Fig. 3.2). We need to obtain all correlators in $\tilde{\mathcal{N}}_2$ :

$\begin{subequations}\begin{align}&\{1,1,1,1\}\,,\\ &\{2,2,0,0\}\,,\,\{2,1,0,1\}\... ...\,\{1,0,1,2\}\,,\\ &\{2,0,0,2\}\,,\,\{0,2,2,0\}\,. \end{align}\end{subequations}$

The new set of equations link these elements among themselves and to those in the first manifold, that we already computed. Now that we know how to obtain $\mathbf{w}^\mathrm{SS}$ , we can write the solution of Eq. (3.77) as:

$\displaystyle \mathbf{w}(\tau)=e^{-\mathbf{M}_0\tau}\mathbf{w}^\mathrm{SS}-\mathbf{M}_0^{-1}(e^{-\mathbf{M}_0\tau}-1)\mathbf{p}n_a^\mathrm{SS}\,.$

(3.77)

Remembering from Sec. 3.2 that $\mathbf{M}_0^{-1}\mathbf{p}=\mathbf{u}^\mathrm{SS}$ , the solution can be further simplified into:

$\displaystyle \mathbf{w}(\tau)=\mathbf{u}^\mathrm{SS}n_a^\mathrm{SS}+ e^{-\math... ...0\tau}\Big(\mathbf{w}^\mathrm{SS}-\mathbf{u}^\mathrm{SS}n_a^\mathrm{SS}\Big)\,.$

(3.78)

If we define the SS fluctuation vector,

$\begin{displaymath}\mathbf{f}^\mathrm{SS}=\mathbf{w}^\mathrm{SS}-\mathbf{u}^\mat... ...a^\mathrm{SS})^2}-D^\mathrm{SS}\Big)^*\\ \end{array}\right)\,,\end{displaymath}$

(3.79)

we can write a general expression for $g^{(2)}(\tau)$ (with $\tau>0$ ):

$\displaystyle g^{(2)}(\tau)=\frac{[\mathbf{w}(\tau)]_1}{(n_a^\mathrm{SS})^2}=1+\frac{[e^{-\mathbf{M}_0\tau}\,\mathbf{f}^\mathrm{SS}]_1}{(n_a^\mathrm{SS})^2}\,,$

(3.80)

where $[\mathbf{x}]_1$ means that we take the first element of the vector $\mathbf{x}$ . Note that for the present system, $(g^{(2)})^{\mathrm{SS}}=2$ and

$\displaystyle \langle\ud{a}a\ud{b}b\rangle^\mathrm{SS}=2n_a^\mathrm{SS}n_b^\mathrm{SS}-\frac{n_a^\mathrm{SS}P_b+n_b^\mathrm{SS}P_a}{\Gamma_a+\Gamma_b}\,.$

(3.81)

A more explicit expression of $g^{(2)}(\tau)$ in terms of the system parameters, can be obtained by analogy with the solution for $\mathbf{u}^\mathrm{SE}(t)$ in Eq. (3.9), by exchanging

for $\tau$ and the elements in $\mathbf{u}(0)$ for those in $\mathbf{f}^\mathrm{SS}/(n_a^\mathrm{SS})^2$ . However, $g^{(2)}(\tau)$ can be more straightforwardly obtained from the relation $g^{(2)}(\tau)=1+\vert g^{(1)}(\tau)\vert^2$ , that applies for thermal photons (Eq. 2.115). Two emission events are independent from each other in this case, where we also have $S^{(2)}_\mathrm{corr}(\omega)=0$ . For the sake of completeness, we can write the explicit expressions in the SS for $g^{(2)}(\tau)$ and $S^{(2)}(\omega)$ :

$\begin{subequations}\begin{align}&g^{(2)}(\tau)=1+e^{-2\Gamma_+\tau}\Big\{ \frac... ...2R_\mathrm{i})^2+(\omega-2R_\mathrm{r})^2}\Big]\,, \end{align}\end{subequations}$

with the definitions of Eq. (3.39) and (3.12). As it corresponds to bosons, the emission presents bunching, that is, the second photon prefers to be emitted together with the first one, $g^{(2)}(0)>g^{(2)}(\tau)$ . The SS $g^{(2)}(\tau)$ goes from its zero delay value,

, towards the infinite delay value,

, with damped oscillations in SC [see Fig. 3.18(a) in solid blue] and exponential decay in the WC. In all cases, the transient in $\tau$ happens at twice the speed of $g^{(1)}(\tau)$ , given also in terms of

and $\Gamma_+$ . The noise spectrum in SC consists of three peaks, one centered at zero and the other two at $\pm2R_\mathrm{r}$ [see Fig. 3.18(b)].

**Figure 3.18:** SS values of $g^{(2)}(\tau)$ (solid blue) $g_{ab}^{(2)}(\tau)$ (dashed purple) (a) and $S^{(2)}(\omega)$ (b) in SC given by Eq. (3.85). The spectra is decomposed in three peaks with Lorentzian (dotted purple lines) and dispersive parts (dashed green). Parameters: $\Delta=0$ , $\gamma_a=g$ , $\gamma_b=0.5g$ , and .
$\includegraphics[width=0.45\linewidth]{chap3/g2/g2.eps}$ $\includegraphics[width=0.45\linewidth]{chap3/g2/S2.eps}$

Another correlator of interest is the cross second order correlation function,

$\displaystyle g_{ab}^{(2)}(\tau)=\frac{\langle\ud{a}(t)(\ud{b}b)(t+\tau)a(t)\rangle}{n_a^{\mathrm{SS}}n_b^{\mathrm{SS}}}\,,$

(3.83)

a quantity related with the probability of emitting an exciton (direct emission) at $\tau$ after having emitted a photon and

. This is computed together with the $g^{(2)}(\tau)$ :

$\displaystyle g_{ab}^{(2)}(\tau)=\frac{[\mathbf{w}(\tau)]_2}{n_a^{\mathrm{SS}} ... ...M}_0\tau}\,\mathbf{f}^{\mathrm{SS}}]_2}{n_a^{\mathrm{SS}} n_b^{\mathrm{SS}}}\,.$

(3.84)

In this case, the correlator goes from $2-(P_a/n_a+P_b/n_b)/(\Gamma_a+\Gamma_b)$ to

[see Fig. 3.18(a) in dashed purple]. The zero delay value is not

in general. It is exactly

only when $P_a/n_a+P_b/n_b=\Gamma_a+\Gamma_b$ , and the modes behave as if they were independent with $\langle n_an_b\rangle ^{\mathrm{SS}}=n_a^{\mathrm{SS}}n_b^{\mathrm{SS}}$ . One case is the very WC, with the modes really independent, and the other one, is the very SC, where excitation is equally shared between the modes as $n_a^{\mathrm{SS}}=n_b^{\mathrm{SS}}=(P_a+P_b)/(\Gamma_a+\Gamma_b)$ . However these two cases are clearly distinguished in the rest of the dynamics. In the very SC, the oscillations of $g_{ab}^{(2)}(\tau)$ reach

alternating exactly with $g^{(2)}(\tau)$ . In the most general case, $g_{ab}^{(2)}(\tau)$ can have starting values between

and

, as in Fig. 3.19. In their SC experiments, Hennessy et al. (2007) computed this kind of cross two-photon counting for large detuning, where the cavity and excitonic modes can be clearly resolved separately. Contrary to our present case, they observed antibunching with $g_{ab}^{(2)}(\tau)>g_{ab}^{(2)}(0)$ , demonstrating that the modes are coupled and that the QD is not bosonic.

**Figure 3.19:** SS values of $g^{(2)}(\tau)$ (solid blue) $g_{ab}^{(2)}(\tau)$ (dashed purple) in SC at resonance. Parameters: $\Delta=0$ , $\gamma_a=2g$ , $\gamma_b=0.5g$ , and .
$\includegraphics[width=0.45\linewidth]{chap3/g2/g2c.eps}$

The full dynamics of the general one-time average values in $\mathbf{u}(t)$ , that we computed in Sec. 3.2 only for the SS or in the absence of pump, can be obtained now with no additional effort by simple analogy with Eq. (3.81):

$\displaystyle \mathbf{u}(t)=e^{-\mathbf{M}_0t}\mathbf{u}(0)+(1-e^{-\mathbf{M}_0t})\mathbf{u}^\mathrm{SS}\,.$

(3.85)

The transient part of the dynamics, $\mathbf{u}^\mathrm{TR}(t)=e^{-\mathbf{M}_0t}(\mathbf{u}(0)-\mathbf{u}^\mathrm{SS})$ , has the same mathematical expression as $\mathbf{u}^\mathrm{SE}(t)$ , making the substitutions ( $\gamma_{a,b}\rightarrow \Gamma_{a,b}$ ), ( $\gamma_{\pm}\rightarrow\Gamma_{\pm}$ ) and $\mathbf{u}(0)\rightarrow (\mathbf{u}(0)-\mathbf{u}^\mathrm{SS})$ . Therefore, the frequency/damping of the oscillations in the transient is also given by real/imaginary part of