# Quantum Antenna as an Open System: Strong Antenna Coupling with Photonic Reservoir
## 4 Radiation Properties of Quantum Antenna
For simplifying the integration in (27), we will use the conventional assumptions from macroscopic antenna theory [14]  
\[\left|\mathbf{r}-\mathbf{r}^{\prime}\big{(}x^{\prime}\big{)}\right|\approx R \tag{36}\]  
\[\left|\mathbf{r}-\mathbf{r}^{\prime}\big{(}x^{\prime}\big{)}\right|\approx R- x^{\prime}\cos\theta \tag{37}\]  
for amplitude and phase factors, respectively, where \(R,\theta\) are coordinates for the spherical system with origin at the antenna center (exponentially attenuated factors in (30) are related to the amplitude and thus approximated according to (36). As a result, the intensity of radiation may be presented in terms of radiation pattern from conventional antenna theory [14]. It reads  
\[\left\langle\mathbf{\hat{E}}^{(-)}(\mathbf{r},t)\cdot\mathbf{\hat{E}}^{(+)}( \mathbf{r},t)\right\rangle_{SS}\approx\frac{\omega_{0}^{4}d_{ab}^{2}}{\left(4 \pi\varepsilon_{0}\right)^{2}c^{4}}\frac{1}{R^{2}}\sin^{2}\theta\cdot\xi(\theta) \tag{38}\]  
where  
\[\xi(\theta)=2\text{Re}\int\limits_{-l/2}^{l/2}dx\cdot f(x)\int\limits_{x}^{l/2 }f^{*}\big{(}x^{\prime}\big{)}\left\langle\hat{\sigma}^{+}(0)\hat{\sigma}^{-} \bigg{(}\frac{x^{\prime}-x}{c}\cos\theta\bigg{)}\right\rangle_{SS}dx^{\prime} \tag{39}\]  
(\(\theta<|\pi/2|\)). For illustrating the qualitative properties of radiation pattern we consider the simplest model of perfect linear antenna [14], quantum envelope of which has the constant spatial amplitude and the linear phase distribution \(f(x)=e^{ik\varphi x}\), where \(\varphi\) is a normalized phase shift per unit length. The integration in (33) gives  
\[\xi(\theta)=\frac{1}{2}\Bigg{\{}\bigg{(}\frac{\sin\Psi}{\Psi}\bigg{)}^{2}+ \frac{1}{2}\bigg{(}\frac{\sin\Psi_{+}}{\Psi_{+}}\bigg{)}^{2}+\frac{1}{2}\bigg{(} \frac{\sin\Psi_{-}}{\Psi_{-}}\bigg{)}^{2}\Bigg{\}} \tag{40}\]  
with \(\Psi=kl(\cos\theta-\varphi)/2\), \(\Psi_{\pm}=\Psi\pm(\Omega_{R}/\omega_{0})\cos\theta\). the adiabatic field variation. For example, the right-hand line may be removed to the invisible region while the two others are left in the visible one. This mechanism manifests itself in the strong decrease of the emitted power at the line \(\omega=\omega_{0}-\Omega_{R}\) compared to two others for rather large Rabi-frequency (Figures 3a and 4a). The conventional antenna parameters--such as directivity, gain, etc. [14]--may be defined by the ordinary way for the quantum antenna considered in this paper. However, it is more natural to consider separately each component of Mollow triplet. For the side lobes (the two last terms in Equation (37)) these values will depend on the driving electromagnetic field. Figure 3: The elementary radiation patterns of the lines \(\omega=\omega_{0}\) (green), \(\omega=\omega_{0}+\Omega_{R}\) (red) and \(\omega=\omega_{0}-\Omega_{R}\) (blue) in logarithmic scale for different values of phase shift. \(kl/2=4\pi\), \(\Omega_{R}=0.2\omega_{0}\). (**a**) \(\varphi=1.0\); (**b**) \(\varphi=1.2\). \[K\big{(}x,x^{\prime};t\big{)}=\left\langle\phi^{+}\bigg{(}t-\frac{|\mathbf{r}- \mathbf{r}^{\prime}(x)|}{c}\bigg{)}\cdot\phi^{-}\bigg{(}t-\frac{|\mathbf{r}- \mathbf{r}^{\prime}(x^{\prime})|}{c}\bigg{)}\right\rangle \tag{33}\]0 The elementary radiation patterns of the lines \(\omega=\omega_{0}\) (green), \(\omega=\omega_{0}+\Omega_{R}\) (red), and \(\omega=\omega_{0}-\Omega_{R}\) (blue) for different values of Rabi-frequency. The patterns are in the ratio 2:1:1 according to the ratio of integrated intensities in the peaks. The patterns are presented in logarithmic scale. \(kl/2=2\pi\), \(\varphi=0.8\). (**a**) \(\Omega_{R}=0.001\omega_{0}\); (**b**) \(\Omega_{R}=0.2\omega_{0}\).