Implicit Lyapunov control of finite dimensional Schrödinger equations

Abstract

An implicit Lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system. The main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable. The controlled Lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem. The convergence analysis is done using LaSalle invariance principle. Closed-loop simulations illustrate the interest of such feedback laws for the open-loop control of a test case considered by chemists.

Optimal Lyapunov quantum control of two-level systems: Convergence and extended techniques

“…1, we observe that the control field changes at dV /dt = 0, or Im(ab * ) = 0 in Eq. (12). Without loss of generality, a can be set to be a real number, and φ and γ 0 determine the design of the control field.…”

“…It uses feedback design to construct control fields but applies the fields to a quantum system in an open-loop way. It provides us with a simple way to design control fields for the manipulation of quantum state transfer [7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…For an arbitrary initial state, the listed cases cover all situations encountered in the method of optimal Lyapunov control used in (12). We will discuss these cases in the next three subsections, where the global phase of the quantum state is neglected throughout the discussions.…”

Stability analysis of discontinuous quantum control systems with dipole and polarizability coupling

“…A solution is to present the problem as a minimization of a cost functional, that describes the goal to be achieved, and eventually some other constraints. This approach leaded to procedures such as stochastic iterative approaches (e.g., genetic algorithms) [16], iterative critical point methods (monotonic algorithms) [17,24,28], trajectory tracking or local control procedures ( [2,5,9,12,18,20,22] etc.). One advantage of this class of methods is that we obtain explicit control fields.…”

Analysis of Effectiveness of Lyapunov Control for Non-Generic Quantum States

“…Most papers have used the Hilbert-Schmidt (HS) distance as the most natural Lyapunov function and in this setting [4]- [6] showed that the target state is asymptotically stable under a sufficient condition equivalent to controllability of the linearized system for pure states represented by wavefunctions, although an additional control had to be added to fix the relative phase of the state. An alternative design based on an implicit Lyapunov function was proposed in [7] to render (pure) target states asymptotically stable when the linearized system is not controllable. The more general case of systems whose states must be represented by density operators was recently considered in [9], [10], but our analysis for generic quantum states [12], for instance, showed the dynamical landscape and convergence behavior to be more complicated than described in [10].…”