# Quantum Antenna as an Open System: Strong Antenna Coupling with Photonic Reservoir
## 4 Radiation Properties of Quantum Antenna  
In this section, we will consider the power radiation pattern of quantum antenna in the strong coupling regime. The far field emitted by antenna [1] presented as a superposition of the partial supports produced by elementary dipoles induced at the antenna surface. The positive-frequency part of field operator produced by the single dipole quantum emitter [1] in spherical system is given by \(\hat{\mathbf{E}}^{(+)}(\mathbf{r},t)=\mathbf{e}\cdot\omega_{0}^{2}d_{ab} \phi^{-}(t-r/c)/4\pi\varepsilon_{0}c^{2}r\), where \(r\) is the spherical radial coordinate (dipole assumed to be placed at the origin). The corresponding operator of the antenna field is given by  
\[\hat{\mathbf{E}}^{(+)}(\mathbf{r},t)=\frac{\omega_{0}^{2}d_{ab}}{4\pi \varepsilon_{0}c^{2}}\mathbf{e}\int\limits_{-l/2}^{l/2}\frac{1}{|\mathbf{r}- \mathbf{r}^{\prime}(x^{\prime})|}f\big{(}x^{\prime}\big{)}\hat{\sigma}^{-} \bigg{(}t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}(x^{\prime})|}{c}\bigg{)}dx^{\prime} \tag{30}\]  
The normally ordered operator of intensity in the far field zone is given by  
\[ \begin{array}{cc}\hat{\mathbf{E}}^{(-)}(\mathbf{r},t)\cdot&\hat{\mathbf{E}} ^{(+)}(\mathbf{r},t)=\frac{\omega_{0}^{4}d_{ab}^{2}}{(4\pi\varepsilon_{0})^{2 }c^{4}}\int\limits_{-l/2}^{l/2}\int\limits_{-l/2}^{l/2}\frac{f(x)}{|\mathbf{r} -\mathbf{r}^{\prime}(x)|}\frac{f^{*}(x^{\prime})}{|\mathbf{r}-\mathbf{r}^{ \prime}(x^{\prime})|}\hat{\sigma}^{+}\bigg{(}t-\frac{|\mathbf{r}-\mathbf{r}^{ \prime}(x)|}{c}\bigg{)}\\ &\hat{\sigma}^{-}\bigg{(}t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}(x^{\prime})|} {c}\bigg{)}dx^{\prime}dx\end{array}  \tag{31}\]The observable value of intensity is expressed through the two-time correlation function of polarization [1]. It reads  
\[\left\langle\mathbf{\hat{E}}^{(-)}(\mathbf{r},t)\cdot\mathbf{\hat{E}}^{(+)}( \mathbf{r},t)\right\rangle=\frac{\omega_{0}^{4}d_{ab}^{2}}{\left(4\pi\varepsilon _{0}\right)^{2}c^{4}}\int\limits_{-l/2}^{l/2}\int\limits_{-l/2}^{l/2}\frac{f( x)}{|\mathbf{r}-\mathbf{r}^{\prime}(x)|}\frac{f^{*}(x^{\prime})}{|\mathbf{r}- \mathbf{r}^{\prime}(x^{\prime})|}K(x,x^{\prime};t)dx^{\prime}dx \tag{32}\]  
where  
\[K\big{(}x,x^{\prime};t\big{)}=\left\langle\phi^{+}\bigg{(}t-\frac{|\mathbf{r}- \mathbf{r}^{\prime}(x)|}{c}\bigg{)}\cdot\phi^{-}\bigg{(}t-\frac{|\mathbf{r}- \mathbf{r}^{\prime}(x^{\prime})|}{c}\bigg{)}\right\rangle \tag{33}\]  
We will consider the steady state of antenna given by Equations (23) and (24). It is a stationary process for which the correlation function is time-independent. Thus, Equation (28) may be rewritten as  
\[K\big{(}x,x^{\prime}\big{)}=\left\langle\phi^{+}(0)\cdot\phi^{-}\bigg{(}\frac{ |\mathbf{r}-\mathbf{r}^{\prime}(x)|-|\mathbf{r}-\mathbf{r}^{\prime}(x^{\prime })|}{c}\bigg{)}\right\rangle \tag{34}\]  
The correlation function (29) is equal to the correlation function of resonance fluorescence considered in detail in [1]. An approximate relation for steady state is given by  
\[\left\langle\phi^{+}(0)\cdot\phi^{-}(\tau)\right\rangle_{SS}=\frac{1}{4}\bigg{(} e^{-\frac{\Gamma}{2}\tau}+\frac{1}{2}e^{-\frac{3\Gamma}{4}\tau}\cdot e^{-i \Omega_{R}\tau}+\frac{1}{2}e^{-\frac{3\Gamma}{4}\tau}\cdot e^{i\Omega_{R}\tau }\bigg{)}e^{-i\omega_{0}\tau} \tag{35}\]  
where \(\Gamma=\Gamma(\omega_{0})\). The time shift in antenna is stipulated by the phase shift of radiation from different points and is given by \(\tau=\left(|\mathbf{r}-\mathbf{r}^{\prime}(x)|-|\mathbf{r}-\mathbf{r}^{\prime }(x^{\prime})|\right)/c\).