Spectral analysis of the linearized triple-quantum-dot shuttle in the adiabatic regime

Abstract

In this paper we seek to perform a spectral analysis of the linearized Hamiltonian of the triple-quantum-dot shuttle in the adiabatic regime. This system consists of a linear array of three quantum dots where a central quantum dot oscillates between two quantum dots that remain fixed, the linearization of the Hamiltonian is through tunneling rates. For our analysis we visualize the problem from the point of view of the displaced oscillator picture, which allows us to build a convenient composite base to rewrite HˆTQDS and by approximating the evolution of the system to an adiabatic evolution it is possible to find the energies of the system through a simple analytic diagonalization process and a search for the roots of a cubic polynomial, as well as the corresponding eigenvectors. The Hamiltonian of the shuttle can be interpreted as a particular case of a more general Hamiltonian in a perturbative scheme, but the mathematical treatment does not proceed in the formalism of perturbation theory. The analytical results obtained were compared with the exact results from numerical diagonalization for different levels of the adiabatic regime, showing that the more strictly the adiabatic condition is fulfilled, the analytical results tend to be exact quickly. In addition, we were able to reproduce some isolated exact solutions from the analytical solution model developed by Kus et al. in 1986 and taken up by Peralta et al. in 2022 to apply it to the shuttle.