Quantum dot lattice as nano-antenna for collective spontaneous emission

We present a theory for the collective spontaneous emission of timed Dicke states in a periodic 2D-array of quantum dots (QDs) coupled by dipoledipole (d-d) interactions. The master equation is first reformulated with respect to the timed Dicke basis. As a result, we obtain simple analytical relations for the spontaneous decay rate, collective Lamb shift and radiative pattern. The collective spontaneous emission in QD-array manifests itself in strong directivity, whereby the radiative pattern consists of a set of strong radiative lobes. The direction of the first lobe is dictated by the pumping direction, while the other lobes correspond to diffractive rays due to the periodicity. The influence of d-d interactions on the radiation decay of timed Dicke states in QD arrays is identical to the influence of an environment to single-particle excited states similar to the action of a structured photonic reservoir. For a rectangular 2D-array, the equivalent structured photonic reservoir has a form of a hollow rectangular waveguide with perfectly conductive walls. For lattice periods comparable to the radiation wavelength the decay rate shows sharp peaks due to Van-Hove singularities in the photonic density of states (PDOS) similar to the Purcell effect in photonic crystals. The optical nanoantenna under study allows tuning of the radiation pattern by varying the timing.