Continuous time dissipation-assisted quantum walks on a finite lattice

We consider a possible dissipative quantum state transport scheme which makes use of a system which is moving on an $N$-site 1-dimensional lattice, coupled to an environment. The time-evolved interaction Hamiltonian for this system is similar in form to the Hamiltonian for a system undergoing a quantum walk, so the system is said to be undergoing a dissipation-assisted quantum walk. We then derive the master equation describing the dynamics of the system, making use of the Redfield equation in doing so. Numerical evaluation of the resulting master equation shows that it is possible for this quantum state transport scheme to be used to transport excited states to the end of the lattice, so long as the coupling between the system and the environment is weak. Furthermore, the resulting state will be a pure state, making it ideal as well for dissipative preparation of pure quantum states.