Implicit Lyapunov Control for the Quantum Liouville Equation

TL;DR

This paper introduces an implicit Lyapunov control method with perturbations for degenerate quantum systems, ensuring convergence to target states even when traditional conditions are not met.

Contribution

It proposes a novel implicit Lyapunov control approach with perturbations for degenerate quantum systems, expanding controllability beyond standard assumptions.

Findings

Proves convergence of the control system under the new method

Designs control laws for diagonal and non-diagonal target states

Numerical simulations confirm effectiveness of the approach

Abstract

A quantum system whose internal Hamiltonian is not strongly regular or/and control Hamiltonians are not full connected, are thought to be in the degenerate cases. In this paper, convergence problems of the multi-control Hamiltonians closed quantum systems in the degenerate cases are solved by introducing implicit function perturbations and choosing an implicit Lyapunov function based on the average value of an imaginary mechanical quantity. For the diagonal and non-diagonal tar-get states, respectively, control laws are designed. The convergence of the control system is proved, and an explicit design principle of the imaginary mechanical quantity is proposed. By using the proposed method, the multi-control Hamiltonians closed quantum systems in the degenerate cases can converge from any initial state to an arbitrary target state unitarily equivalent to the initial state. Finally, numerical simulations are studied to verify the effectiveness of the proposed control method.