Towards an analytical solution for the triple quantum dot shuttle

Abstract

In this paper we look for analytical solutions of the triple-quantum-dot shuttle (TQDS) in the linear tunneling regime. This system consists of three quantum dots arranged in a straight line, where the end dots remain fixed while the center dot oscillates between them and its motion is modeled by a quantum harmonic oscillator. In the linear tunneling approximation for the Hamiltonian HˆTQDS, we consider an analytical method developed by Kuś and Lewenstein in 1986 that enables us to find exact isolated solutions for a Hamiltonian of the HˆTQDS-type using the Bargmann representation. This allows us to obtain some energy values and their respective eigenvectors with the condition that the parameters that describe the system comply with certain constraints called compatibility conditions. The latter give rise to the possibility of developing a criterion that leads us to establish under what conditions the Kuś method is applicable in the solution of systems represented by Hamiltonians of the HˆTQDS-type.