Vibrational Effects on the Formation of Quantum $W$ States

We theoretically investigate the formation of $W$ states in a tripartite system composed of three charge qubits coupled to vibrational modes. The electromechanical coupling is responsable for second order virtual processes that result in an effective electron-electron interaction between neighbor qubits, which yields to the formation of $W$ states. Based on the Lang-Firsov transformation and perturbation theory, we analytically solve the quantum dynamics, providing a mathematical expression for the maximally entangled $W$ state. Dephasing is also taken into accout, paying particular attention on the robustness of bipartite entanglement against local dephasing processes.