# Quantum Antenna as an Open System: Strong Antenna Coupling with Photonic Reservoir
## 2 Model of Quantum Antenna as an Open System  
The general model of the antenna is shown in Figure 1. We consider the antenna as a quantum dot placed inside a quantum wire which behaves as a waveguide (Figure 1a). Such structures were implemented experimentally in the form of tapered InP nanowire waveguide containing a single InAsP quantum dot [34; 35; 36]. The quantum dot is perfectly positioned on the axis of InP nanowire waveguides, where the emission is efficiently coupled to the fundamental waveguide mode. Figure 1: General illustration of the system used as a model of quantum antenna in the strong coupling regime. (**a**) The antenna configuration. The semiconductor InAsP quantum dot (red colored) positioned inside the semiconductor InP quantum wire (green colored). The quantum dot is coupled with quantum wire via quantum tunneling through the potential barriers. (**b**) The energy spectrum of the shown system in the case of EM field absence. It consists of two zones, valence and conduction, which are separated by the energy gap. The dominant support to the energy of quantum transitions is given by the small region near the band minimum at the zone center and is approximately equal to \(\hbar\omega_{0}\). From a theoretical point of view, this system may be considered as a defect of a crystal lattice. Therefore, the general model of the bulk crystal defects, which is based on Wannier functions [37], may be used for the antenna analysis. Similarly to [37], we define Wannier functions \(a_{n}\big{(}x-x_{j}\big{)}\) mutually coupled to Bloch functions \(\psi_{nh}(x)\) via relations  
\[a_{n}\big{(}x-x_{j}\big{)} = \frac{1}{\sqrt{N}}\underset{h}{\sum}e^{-ihx_{j}}\psi_{nh}(x) \tag{1}\] \[\psi_{nh}(x) = \frac{1}{\sqrt{N}}\underset{j}{\sum}e^{ihx_{j}}a_{n}\big{(}x-x_{j} \big{)} \tag{2}\]  
where \(n\) is a number of zone and \(h\) is a quasi-momentum of the charge. The Wannier function \(a_{n}\big{(}x-x_{j}\big{)}\) is strongly localized in the vicinity of the atom at the point \(x=x_{j}\). An arbitrary wave-function is given by \(\psi(x)=N^{-1/2}\underset{n,j}{\sum}f_{n}\big{(}x_{j}\big{)}a_{n}\big{(}x-x_{ j}\big{)}\), where the envelope \(f_{n}(x)\) in the continuous limit satisfies the Schrodinger equation  
\[\left[-\left(\frac{\hbar^{2}}{2m_{eff}}\right)^{2}\frac{\partial^{2}}{ \partial x^{2}}+U(x)\right]f(x)=[\varepsilon-\varepsilon_{c}(0)]f(x) \tag{3}\]  
where \(U(x)\) is the potential, averaged with respect to the crystal lattice, and \(m_{eff}\) and \(\varepsilon\) are the effective mass and energy of charge carrier, respectively. The Bloch function may be presented as \(\psi_{h}(x)=e^{ihx}u_{h}(x)\), where \(u_{h}(x)\) is a periodic function such that \(u_{h}(x+a)=u_{h}(x)\). The Wannier and Bloch states satisfy the uncertainty principle [37] in the form \(\Delta h\cdot\Delta x\cong 1\). This means that, for rather long antenna (large \(\Delta x\)), the main support to the wave-function is defined by a small region near the band minimum at the zone center. As a result, we obtain the approximate presentation for the wave function at the given zone  
\[ \begin{array}{ll}\psi(x)=&\frac{1}{N}\underset{j}{\sum}f\big{(}x_{j} \big{)}\underset{h}{\sum}e^{ih(x-x_{j})}u_{h}(x)\approx\\ &\frac{1}{N}\underset{j}{\sum}u_{0}(x)f\big{(}x_{j}\big{)}\underset{h}{ \sum}e^{ih(x-x_{j})}=\\ &\frac{1}{N}\underset{j}{\sum}u_{0}(x)f\big{(}x_{j}\big{)}\delta(x-x_{i})=u_{ 0}(x)f(x)\end{array}  \tag{4}\]  
Equation (4) shows that the quantum properties of antenna may be modeled by a two-level artificial atom with Fermionic quantum states \(|a\rangle\), \(|b\rangle\) separated by the energy \(\hbar\omega_{0}\) with the same spatial envelope \(f(x)\) (Figure 1b). We assume the length of antenna to be comparable to the wavelength. The area of quantum confinement is also comparable to a wavelength. It makes the retardation of EM field in its interaction with quantum emitter to be essential. The antenna is driven by the classical external field in the regime of arbitrary coupling.