(a) Map of the energy splittings of the delocalized oxygen two-dimensional model. The and axes represent aluminium pair positions with . (b) The difference between the absolute dipole moment (in and directions) over the same range. We see either (red, upper and lower domains) or (green, left and right domains) dominated behavior in all defect locations, except where the oxygen is tightly confined (e.g., ). (c)–(f) The first excited state wave function of the oxygen atom and the acting potential of four configurations indicated on (a). For comparison with existing qubit experiments we plot contour lines corresponding (red to yellow, outer to inner) overlayed on (a) and (b). The same resonant frequencies are discussed in Fig. 3; hence (c)–(f) all represent configurations of ; whereas points i–vi are the locations selected for the strain study in Fig. 4. Black, dashed contour lines represent a minimum resolvable energy splitting of 10 kHz, and black, solid contour lines indicate where the aluminium atoms are so close that the oxygen confinement region no longer exists.
Coupling strength to a fictitious phase-qubit as a function of in the domains where is dominant [see Eq. (2)] for a set of constant splitting frequencies. For comparison with experimental results, and are expressed in frequency units.
(a) Depiction of a number of deformations which were applied to the aluminium atoms in the plane (see text). The response of one of the strain types is several orders of magnitude larger than the others (highlighted), which is indicative of an optical phonon mode, generating frequency splittings of for picometer deformations. A deformation range of 10 fm was applied to the points i–vi from Fig. 2, yielding responses that are both hyperbolic and symmetric (b). Over the range , the response is linear with a strain gradient, shown in (c) for both the - and -type defects in the direction.