  The quantum harmonic oscillator (QHO), one of the most important and
ubiquitous model systems in quantum mechanics, features equally spaced energy
levels or eigenstates. Here we present a new class of nearly ideal QHOs formed
by hydrogenic substitutional dopants in an AlGaAs/GaAs heterostructure. On the
basis of model calculations, we demonstrate that, when a deta-doping Si donor
substitutes the Ga/Al lattice site close to AlGaAs/GaAs heterointerface, a
hydrogenic Si QHO, characterized by a restoring Coulomb force producing square
law harmonic potential, is formed. This gives rise to QHO states with energy
spacing of about ~8-9 meV. We experimentally confirm this proposal by utilizing
gate tuning and measuring QHO states using an aluminum single-electron
transistor (SET). A sharp and fast oscillation with period of ~7-8 mV appears
in addition to the regular Coulomb blockade (CB) oscillation with much larger
period, for positive gate biases above 0.5 V. The observation of fast
oscillation and its behavior is quantitatively consistent with our theoretical
result, manifesting the harmonic motion of electrons from the QHO. Our results
might establish a general principle to design, construct and manipulate QHOs in
semiconductor heterostructures, opening future possibilities for their quantum
applications.
