Under the influence of an applied magnetic field(MF), the eigenenergies and the eigenfunctions of the ground and the first excited states(GFES) are obtained by using a variational method of the Pekar type(VMPT) in a strong electron-LO-phonon coupling asymmetrical Gaussian potential quantum well(AGPQW). This AGPQW system may be employed as a two-level qubit. The numerical results have indicated(i) that when the electron situates in the superposition state of the GFES, we obtain the time evolution and the coordinate change of the electron probability density in the AGPQW,(ii) that due to the presence of the asymmetrical potential in the growth direction of the AGPQW, the probability density shows double-peak configuration, whereas there is only one peak if the confinement is a two dimensional symmetric one in the xy plane of the AGPQW,(iii) that the oscillatory period is a decreasing function of the cyclotron frequency of the MF, the height of the AGPQW and the polaron radius,(iv) and that as the range of the confinement potential(RCP) decreases the oscillatory period will decrease firstly and then increase and it will take a minimum when R =-0.234 nm. Under the influence of an applied magnetic field (MF), the eigenenergies and the eigenfunctions of the ground and the first excited states (GFES) are obtained by using a variational method of the Pekar type (VMPT) in a strong electron-LO-phonon The asymmetric Gaussian potential quantum well (AGPQW). This numerical result has indicated (i) that when the electron situates in the superposition state of the GFES, we obtain the time evolution and the coordinate change of the electron probability density in the AGPQW, (ii) that due to the presence of the asymmetrical potential in the growth direction of the AGPQW, the probability density shows double-peak configuration, there there only only one peak if the confinement is a two dimensional symmetric one in the xy plane of the AGPQW, (iii) that the oscillatory period is a decreasing function of the cyclotron frequency of the MF, the height of the AGPQW and the polaron radius, (iv) and th at as the range of the confinement potential (RCP) decreases the oscillatory period will decrease first and then increase and it will take take a minimum when R = -0.234 nm.