Quantum antenna as an open system: strong antenna coupling with photonic reservoir

We proposed the general concept of quantum antenna in the strong coupling regime. It is based on the theory of open quantum systems. The antenna emission to the space is considered as an interaction with the thermal photonic reservoir. For modeling of the antenna dynamics is formulated a master equation with the correspondent Lindblad super-operators as the radiation terms. It is shown that strong coupling dramatically changes the radiation pattern of antenna. The total power pattern splits to three partial components; each corresponds to the spectral line of Mollow triplet. We analyzed the dependence of splitting from the length of antenna, shift of the phase, and Rabi-frequency. The predicted effect opens a way for implementation of multi-beam electrically tunable antennas, potentially useful in different nano-devices.


2.The model of quantum antenna as an open system
The general model of the antenna is shown on Figure 1. We consider the antenna as a quantum dot placed inside the quantum wire, which manifests itself as a waveguide. (Figure 1a). Such type of devices was implemented experimentally at the form of tapered InP nanowire waveguide containing a single InAsP quantum dot [34][35][36]. Quantum dot is perfectly positioned on-axis of InP nanowire waveguides, where the emission is efficiently coupled to the fundamental waveguide mode.
From 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. Similar [37], we define Wannier functions . The Wannier and Bloch states are satisfy uncertainty principle [37] in the form 1 hx     . It means, that for rather long antenna (large x  ) the main support to the wave-function is defined by the small region near the band minimum at the zone center. As a result, we obtain the approximate presentation Equation (4) shows, that the quantum properties of antenna may be modeled by the two-level artificial atom with Fermionic quantum states a , b separated by the energy 0  with the same spatial envelope   fx (Fig 1b).
We assume the length of antenna to be comparable with the wavelength. The area of quantum confinement is comparable with the 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. In particular, an essential role in the formation of radiation is played by the Rabioscillations produced by the antenna feeding. Therefore, its interaction with EM-field cannot be considered by different types of perturbation theory.
The total Hamiltonian of antenna in the EM-field within the bounds of given model is H is Hamiltonian of free antenna and is the interaction Hamiltonian with the external field. We will limit our consideration by the strong coupling regime and not touch ultra-strong one [17], thus Hamiltonian (2) is written in rotating-wave approximation [1]. The states , ab are excited and ground states of antenna without field, 0 ,  are the frequencies of optical transition and driven field, respectively, R  is the Rabi-frequency.
The radiation of quantum antenna into photonic reservoir is described in rotating-wave approximation [1]  is the Fermionic-type lowering operator of antenna excitation,b k is a creation operator of photon in the k-th mode, e is the unit vector along the antenna axis, ,  k k are the frequency and wave-vector of k-th mode,V is the normalization volume.

 
The eigenvalues are given by The last term is the Lindblad super-operator, which models antenna emission as its coupling with photonic thermal reservoir. It reads in its conventional form The next step of simplification consists in the standard transformation from summation over k to the frequency integration [1] using the replacement where azimuthal integration have been carried out and the new variable of integration / kc   k have been used. As a result, we obtain The main support to the time integral is given by the narrow vicinity of frequency   k . It allow to use the approximation The relaxation parameter     is a spontaneous emission frequency. Its difference with Weisskopf -Wigner result for individual atom [1] consists in the special interference dictated by the relative phase shift of different EM-modes over antenna axis. As a result, its frequency dependence doesn't add up to   , where coefficients   are elements of matrix The relaxation parameters are obtained from rather long, but trivial calculations. They are given by The element 22  may be found from the probability conservation low 22 11 1   [1].

3.Dynamics of quantum antenna: qualitative analysis
The strong coupling regime corresponds to the condition R    , which is equal to 1 g  . We will analyze for simplicity antenna dynamics in the regime of exact resonance (zero detuning As it leads from (20), the antenna emission has not vanished, while the emission properties are independent on its initial state.

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] (25) The normally ordered operator of intensity in the far field zone is given by ll ab x ll E r e r r r r rr (26) The observable value of intensity expressed through the two-time correlation function of polarization [1].  (27) where We will consider the steady state of antenna given by equations (23), (24). It is stationary process, for which the correlation function is time-independent. Thus, equation (28) may be rewritten as The correlation function (29) is equal to the correlation function of resonance fluorescence in detail considered in [1].
The time shift in antenna is stipulated by the phase shift of radiation from different points and given by For simplifying the integration in (27), we will use the conventional assumptions to macroscopic antenna theory [14]: for amplitude and phase factors, respectively, where , R  are coordinates for the spherical system with origin at the antenna center (exponentially attenuated factors in (30) are related to the amplitude ones and approximated accordingly (31a)). As a result, the intensity of radiation may be presented in terms of radiation pattern, conventional for antenna theory [14]. It reads For illustrating the qualitative properties of radiation pattern we consider the simplest model of perfect linear antenna [14], which quantum envelope has the constant spatial amplitude and the linear phase distribution , where  is a phase shift per unit length. The integration in (33) Equation (37) allows analyzing the qualitative behavior of radiation pattern, some of which aspects is seen to be in agreement with theory of resonance fluorescence [1]. The total radiation pattern represents the sum of three elementary patterns   2 sin /  of the form of perfect wire antenna [14], The patterns become more and more jagged with the length increasing (Figures 3,4). Simultaneously, effect of pattern separation increases and even the absolute separation becomes reachable (the angle of maximum for the main lobe of one pattern coincides with the minimum for another one). The typical radiation patterns of wire antennas are highly dependent on the value of phase shift. The value cr   defines the regime of axial radiation. For the phase shift exceeding the critical value cr   , the main lobe removes to the invisible region [14] and becomes not related to the observable angles  .Therefore, the antenna emission is completely formed as a superposition of side lobes, which are incoherent. As a result, the emitted power decreases, which makes this regime to be not suitable for a lot of applications. For the classical wire antennas 1 cr   [14]. In our case we have the own critical shift for every partial diagram:

5.Conclusion
In summary, we developed the model of quantum antenna in the strong coupling regime basing on the general theory of open quantum systems [1,33]. The far-field zone of antenna radiation is considered as the thermal photonic reservoir. We formulated and solved the master equation with Lindblad-terms related to the energy losses via antenna emission. The general concept was applied to the wire antenna with Fermionic type of excitation. Spectral density of power in far-field zone was calculated. It is shown, that the strong coupling regime dramatically changes the radiation pattern compared with the macroscopic antennas and optical nanoantennas of different well-known types. The calculated radiation pattern consists of three components, each corresponding to the resonance line in the Mollow triplet and turned one with respect to another. The value of turn strongly depends on the geometric parameters of antenna, energy spectra of used materials and the value of coupling (Rabi-frequency). It opens a new ways for high-effective electric control of antennas characteristics for using in different nano-photonic applications. On the other hand, the influence of EM-field on the pattern should be accounted from the point of view of electromagnetic compatibility in nanoscale [37,38].
Author Contributions: Developments of the physical models, derivation of the basis equations, interpretation of the physical results and righting the paper have been done by Alexei Komarov and Gregory Slepyan jointly.

Conflicts of Interest:
The authors declare no conflict of interest.