Geometric property of energetic cost in transitionless quantum driving
K. Z. Li, D. M. Tong · Physical Review A · 2022
Transitionless quantum driving is a useful approach to mimic the adiabatic evolution of a quantum system so that the system can rapidly evolve from its initial state to the target state without transitions between instantaneous eigenstates of the reference Hamiltonian. To quantify the energetic cost in the transitionless quantum driving process, the instantaneous cost is defined as the Frobenius norm of the driving Hamiltonian and the integral of instantaneous cost is used to describe the total cost in the entire evolution. In this paper, we find that the minimal integral of instantaneous cost has a geometric property being equal to the length of the evolution path on the Riemannian manifold spanned by the control parameters with a positive definite metric, but independent of the evolution details such as the changing rate of the parameters. Based on this property, we further show that the optimal transitionless quantum driving with the minimum total cost corresponds to the geodesic path on the Riemannian manifold, which provides a method for optimizing transitionless quantum driving.