# Quantum Antenna as an Open System: Strong Antenna Coupling with Photonic Reservoir
## 2 Model of Quantum Antenna as an Open System
Particularly, an essential role in the formation of radiation is played by the Rabi-oscillations produced by the antenna feeding. Therefore, its interaction with the EM field cannot be considered by different types of perturbation theories. The total Hamiltonian of antenna in the EM-field within the bounds of given model is \(\hat{H}=\hat{H}_{0}+\hat{V}\), where  
\[\hat{H}_{0}=\frac{\hbar\omega_{0}}{2}(|a\rangle\langle a|-|b\rangle\langle b|) \tag{5}\]  
\(\hat{H}_{0}\) is Hamiltonian of free antenna and  
\[\hat{V}=-\frac{\hbar\Omega_{R}}{2}\Big{(}|b\rangle\langle a|e^{i\omega t}+|a \rangle\langle b|e^{-i\omega t}\Big{)} \tag{6}\]is the interaction Hamiltonian with the external field. We will limit our consideration to the strong coupling regime and not consider the ultra-strong one [17], thus Hamiltonian (6) is written in rotating-wave approximation [1]. The states \(|a\rangle,|b\rangle\) are the excited and ground states of antenna without field, \(\omega_{0},\omega\) are the frequencies of optical transition and driven field, respectively and \(\Omega_{R}\) is the Rabi-frequency. The radiation of quantum antenna coupled to photonic reservoir is described in rotating-wave approximation [1] by Hamiltonian  
\[\hat{H}_{\text{int}}=\hbar\underset{\mathbf{k}}{\sum}\frac{g_{\mathbf{k}}}{l} \int\limits_{-l/2}^{l/2}e^{-i(\mathbf{k}\cdot\mathbf{e})x}\Big{[}\hat{b}_{ \mathbf{k}}\hat{\sigma}^{+}f(x)e^{-i\omega_{\mathbf{k}}t}+\text{H.c.}\Big{]}dx \tag{7}\]  
where \(g_{\mathbf{k}}=d_{ab}\cos(\mathbf{e}\cdot\mathbf{k}/|\mathbf{k}|)\sqrt{\omega _{\mathbf{k}}/2\hbar e_{0}V}\) is a coupling factor of \(k\)-th EM-field mode of reservoir with antenna, \(d_{ab}\) is the dipole moment, \(\hat{\sigma}^{+}=|b\rangle\langle a|\) is a Fermionic-type lowering operator of antenna excitation, \(\hat{b}_{\mathbf{k}}\) is a creation operator of photon in the \(k\)-th mode, \(\mathbf{e}\) is the unit vector along the antenna axis, \(\omega_{\mathbf{k}},\mathbf{k}\) are the frequency and wave-vector of \(k\)-th mode, respectively and \(V\) is the normalization volume. We will use the Eigenfunctions of Hamiltonian \(\hat{H}=\hat{H}_{0}+\hat{V}\) as a basis functions. For this, we transform all operators according to \(\hat{A}\Rightarrow\hat{U}\hat{A}\hat{U}^{+}\), where \(\hat{U}(t)=\left( \begin{array}{cc}e^{i\omega t/2}&0\\ 0&e^{-i\omega t/2}\end{array} \right)\) is the transformation operator. The transformed Hamiltonian components are \(\hat{H}_{0}\Rightarrow\hbar\Delta(|a\rangle\langle a|-|b\rangle\langle b|)/2\) and \(\hat{V}\Rightarrow-\hbar\Omega_{R}(|b\rangle\langle a|+|a\rangle\langle b|)/2\), where \(\Delta=\omega_{0}-\omega\) is the frequency detuning. The Eigenmodes are defined by relation \(\hat{H}|\Psi_{\alpha}\rangle=\
u_{\alpha}|\Psi_{\alpha}\rangle\) where \(\alpha=1,2\). The eigenvalues are given by \(\
u_{\alpha}=\pm\
u=\pm\sqrt{\Delta^{2}+\Omega_{R}^{2}}/2\), with corresponding Eigenmodes  
\[|\Psi_{1}\rangle=Ce^{i\
u t}(g|a\rangle+|b\rangle) \tag{8}\]  
\[|\Psi_{2}\rangle=Ce^{-i\
u t}(|a\rangle-g|b\rangle) \tag{9}\]  
where \(C=1/\sqrt{1+g^{2}}\) is a normalization factor and \(g=\Omega_{R}/(\Delta+2\
u)\). The Eigenmodes (8) and (9) are orthonormal: \(\left\langle\Psi_{\alpha}|\Psi_{\beta}\right\rangle=\delta_{\alpha\beta}\). These states describe the interaction of excited and ground states via Rabi-oscillations with frequency \(\
u\). The state of emitting antenna is characterized by a \(2\times 2\) density matrix, \(\rho\), which satisfies the master equation  
\[\dot{\rho}=-\frac{i}{\hbar}\big{[}\hat{H},\rho\big{]}+\hat{\Im}(t) \tag{10}\]  
The last term is the Lindblad super-operator which models antenna emission as its coupling to a photonic thermal reservoir. It reads in its conventional form  
\[