  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 report on the design, demonstration and
operation of nearly perfect QHOs in AlGaAs/GaAs heterostructure. On the basis
of model calculations, we demonstrate that, when a substitutional 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 ~8-9 meV. We experimentally confirm this proposal by utilizing Stark
effect 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 qualitatively 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.
