We report on the selective excitation of single impurity-bound exciton states in a GaAs double quantum well (DQW). The structure consists of two quantum wells (QWs) coupled by a thin tunnel barrier. The DQW is subject to a transverse electric field to create spatially indirect inter-QW excitons with electrons and holes located in different QWs. We show that the presence of intra-QW charged excitons (trions) blocks carrier tunneling across the barrier to form indirect excitons, thus opening a gap in their emission spectrum. This behavior is attributed to the low binding energy of the trions. Within the tunneling blockade regime, emission becomes dominated by processes involving excitons bound to single shallow impurities, which behave as two-level centers activated by resonant tunneling. The quantum nature of the emission is confirmed by the anti-bunched photon emission statistics. The narrow distribution of emission energies ($sim 10$~meV) and the electrical connection to the QWs make these single-exciton centers interesting candidates for applications in single-photon sources.
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