To study the influence of an anisotropic parabolic potential （APP） on the properties of a quantum dot （QD） qubit, we obtain the eigenenergies and eigenfunctions of the ground and first excited state of an electron, which is strongly coupled to the bulk longitudinal optical （LO） phonons, in a QD under the influence of an APP by the celebrated Lee-Low-Pines （LLP） unitary transformation and the Pekar type variational （PTV） methods. Then, this kind of two-level quantum system can be excogitated to constitute a single qubit. When the electron locates at the superposition state of its related eigenfunctions, we get the time evolution of the electron＇s probability density. Finally, the influence of an APP on the QD qubit is investigated. The numerical calculations indicate that the probability density will oscillate periodically and it is a decreasing function of the effective confinement lengths of the APP in different directions. Whereas its oscillatory period is an increasing one and will diminish with enhancing the electron-phonon （EP） coupling strength.
The Hamiltonian of a quantum rod with an ellipsoidal boundary is given by using a coordinate transformation in which the ellipsoidal boundary is changed into a spherical one.Under the condition of strong electron-longitudinal optical phonon coupling in the rod,we obtain both the electron eigenfunctions and the eigenenergies of the ground and first-excited state by using the Pekar-type variational method.This quantum rod system may be used as a two-level qubit.When the electron is in the superposition state of the ground and first-excited states,the probability density of the electron oscillates in the rod with a certain period.It is found that the oscillation period is an increasing function of the ellipsoid aspect ratio and the transverse and longitudinal effective confinement lengths of the quantum rod,whereas it is a decreasing function of the electron-phonon coupling strength.
