A perturbative analytical solution for 3-level systems: The 3-State Rabi model and the linearized Triple-Quantum-Dot Shuttle

In this paper we present an analytical solution for the 3-State Rabi model (3SRM) and linearized Triple-Quantum-Dot Shuttle (TQDS) from the perspective of perturbation theory. The 3SRM is an immediate three-state extension of the Rabi Hamiltonian, where the Pauli matrices ${\hat{\sigma }}_x$ and ${\hat{\sigma }}_z$ are replaced by the spin-1 matrices ${\hat{J}}_x$ and ${\hat{J}}_z$, respectively. The TQDS is a nanoelectromechanical system consisting of a linear array of three quantum dots (QDs), with one energy level in each dot participating in the dynamics, and where the central QD is modeled by a simple quantum harmonic oscillator that oscillates between the outer QDs, which are kept fixed. When a gate voltage is applied to the outer QDs, electronic transport is induced by tunneling from a source to a drain in a controlled manner. The Hamiltonian of both models is built from three important energetic contributions: one coming from the oscillation, another corresponding to the site energy (atomic Hamiltonian) and a last one that refers to the interaction between the electronic states and the oscillation modes. In the perturbation theory scheme, we analyze two cases for each model, where either the interaction Hamiltonian or the atomic Hamiltonian is used as the perturbation. We calculate the energies corrected to second order at most, and the corresponding eigenstates. The analytical results obtained were compared with the numerically exact solution and an analytical solution previously calculated in the adiabatic approximation regime. For the 3SRM we find that the perturbative approach gives results limited to small values of the perturbing parameter; however, for the TQDS, incorporating degenerate perturbation theory, we are able to reproduce anticrossings in the energy spectrum, thus complementing the results obtained using the adiabatic approximation.