Pre-selection of optical transitions in rare-earth ions in crystals perspective for quantum information processing

A systematic analysis of decoherence rates due to electron–phonon interactions for optical transitions of rare-earth dopant ions in crystals is presented in the frame of the point charge model. For this model, the large value of any one of the matrix elements of the unit tensor operator U ^{( k )} of rank k for transitions within the 4f-electronic configuration, viz. U2, U4 or U6, is enough to ensure the strong optical transition between different levels, while the Stark–Stark transitions within the multiplet can be characterized by the matrix element U2 alone, the influence of elements U4, U6 being of much smaller order of magnitude and neglected. The circumstance that exactly such Stark–Stark transitions within the multiplet define the efficiency of electron–phonon interaction and, consequently, the decoherence rate (except for the case of lowest, less than approximately 2–4?K, temperatures), enables selection of optical transitions which are strong enough and at the same time are characterized by relatively small decoherence rates. Correspondingly, these optical transitions, provided that they lie in an appropriate spectral range and the gap to the nearest neighboring energy level is large enough (>500?cm^?1) to prevent eventual fast phonon-assisted relaxation, should be considered as prospective for subsequent use in quantum informatics processing and communication. The list of such pre-selected transitions is given; the applicability area and limitations of our approach are discussed.