Group theory for defects

Defects in solid-state materials have seen wide applicability in quantum information science, especially as quantum memories, because they have the potential to combine favorable coherence and non-classical emission properties of isolated atoms with the scalability and stability of solid-state technologies. These defects often introduce localized electronic states that can be effectively decoupled from the host lattice, thereby creating an artificial atom''. Such artificial atom systems can include simple substitutional or vacancy defects, as well as more complicated defect complexes that are comprised of multiple imperfections but effectively act as a single defect. To harness the electronic and optical properties of these defects, defects can be coupled to external fields, including electric, magnetic, and also strain, as well as to waveguides and cavity environments. Despite these advances, demands to the properties of defect systems are ever-increasing in complexity, such as specific level structures for emission of entangled photonic states or implementation of multi-qubit photonic gates. Therefore, the need for defect centers with engineered electronic, optical, and spin properties has motivated the search for novel defect centers and new knobs to tune their properties. In [this paper](https://pubs.acs.org/doi/10.1021/acsnano.0c10601), we add the chemical degree of freedom by demonstrating how these artificial atoms'' become artificial molecules'' when placed within a few angstroms of each other and investigate the properties of such coupled defects. In particular, we observe features analogous to molecules, such as the formation of delocalized, bonding and anti-bonding molecular'' orbitals arising from the hybridization of the localized, ``atomic" orbitals of the single defects. In addition, we show that, as expected, the degree of wave function overlap determines the magnitude of the bonding-antibonding energy gap. Because the degree of wave function overlap is determined by the defect and host lattice symmetries, the logical next step is to apply principles of group theory, enabling us to design defects with customized properties without having to run computationally expensive calculations from first principles. I suspect that the main challenge will be adapting group theory originally constructed for molecules directly to defects because the host lattice also imposes some symmetrical constraints. Some predictions we hope to make with group theory may include whether a defect will have energy levels within the band gap or within the electronic continuum, the excitation spectrum (specifically, the energy ordering and transition dipole moments) of defect complexes using the excited state structure of single defects, whether group symmetry rules change in 2D vs. 3D host materials, and the types (localized vs. delocalized) and magnitude of defect-phonon coupling. With these learnings, we may be able to predict, for instance, the chemical structure of an ultra-bright defect with an electronic structure analogous to the conjugated backbone of dye molecules, or defects that effectively decouple from the host lattice, enabling long phonon decoherence-limited lifetimes.