A theory of bipolaron states in quantum wires with a parabolic potential well is developed applying the Feynman variational principle. The basic parameters of the bipolaron ground state (the binding energy, the number of phonons in the bipolaron cloud, the effective mass, and the bipolaron radius) are studied as a function of sizes of the potential well. Two cases are considered in detail: a cylindrical quantum wire and a planar quantum wire. Analytical expressions for the bipolaron parameters are obtained at large and small sizes of the quantum well. It is shown that at $R\ensuremath{\gg}1$ [where
R means the radius (half width) of a cylindrical (planar) quantum wire, expressed in Feynman units], the influence of confinement on the bipolaron binding energy is described by the function $\ensuremath{\sim}{1/R}^{2}$ for both cases, while at small sizes this influence is different in each case. In quantum wires, the bipolaron binding energy
W(R) increases logarithmically with decreasing radius. The shapes and the sizes of a nanostructure, which are favorable for observation of stable bipolaron states, are determined.
The stability of large Fr\"ohlich bipolarons in the presence of a static magnetic field is investigated with the path integral formalism. We find that the application of a magnetic field (characterized by the cyclotron frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$) favors bipolaron formation: (i) The critical electron-phonon coupling parameter ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$ (above which the bipolaron is stable) decreases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ and (ii) the critical Coulomb repulsion strength
UC (below which the bipolaron is stable) increases with increasing ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The binding energy and the corresponding variational parameters are calculated as a function of \ensuremath{\alpha}, U, and ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. Analytical results are obtained in various limiting cases. In the limit of strong electron-phonon coupling (\ensuremath{\alpha}\ensuremath{\gg}1) we obtain for ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\ll}1 that
Eestim\ensuremath{\approxeq}
Eestim(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0)+c(u)${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{4}$ with c(u) an explicitly calculated constant, dependent on the ratio u=U/\ensuremath{\alpha}, where U is the strength of the Coulomb repulsion. This relation applies both in two dimensions (2D) and in 3D, but with a different expression for c(u). For ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\gg}${\mathrm{\ensuremath{\alpha}}}^{2}$\ensuremath{\gg}1 we find in 3D
Eestim\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-${\mathrm{\ensuremath{\alpha}}}^{2}$A(u)
ln2(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathrm{\ensuremath{\alpha}}}^{2}$) [also with an explicit analytical expression for A(u)] whereas in 2D
Eestim2D\ensuremath{\approxeq}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$-\ensuremath{\alpha}\ensuremath{\surd}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$\ensuremath{\pi}(u-2-\ensuremath{\surd}2)/2. The validity region of the Feynman-Jensen inequality for the present problem, bipolarons in a magnetic field, remains to be examined. \textcopyright{} 1996 The American Physical Society.
On the basis of Lee-Low-Pines( LLP) unitary transformation,the eigenenergy and eigenfunction of the ground-state and the first excited state of the strong-coupling bipolaron in two-dimensional quantum dot( QD) were obtained by using the variational method of Pekar type. A qubit was formed by overlaying both the ground state and the first excited state of the bipolaron system. Numerical calculations indicate that the oscillating period T0 of qubits decreases with the increasing the electron-phonon coupling strength α,the confinement strength ω0of the quantum dot,and the dielectric constant ratio η; the distribution of the probability density Q of the electrons in quantum dot oscillates periodically with time t,angle coordinate φ2,and the dielectric constant ratio η,and there is a maximum at near the center and zero away from the center of quantum dot.
Based on the variational method of Pekar type, we study the energies and the wave-functions of the ground and the first-excited states of magneto-bipolaron, which is strongly coupled to the LO phonon in a parabolic potential quantum dot under an applied magnetic field, thus built up a quantum dot magneto-bipolaron qubit. The results show that the oscillation period of the probability density of the two electrons in the qubit decreases with increasing electron–phonon coupling strength α , resonant frequency of the magnetic field ω c , confinement strength of the quantum dot ω 0 , and dielectric constant ratio of the medium η ; the probability density of the two electrons in the qubit oscillates periodically with increasing time t , angular coordinate φ 2 , and dielectric constant ratio of the medium η ; the probability of electron appearing near the center of the quantum dot is larger, and the probability of electron appearing away from the center of the quantum dot is much smaller.