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The Journal of Physical Chemistry A

Reference Trajectory Tracking for Locally Designed Coherent Quantum Controls

Mazyar Mirrahimi, Gabriel Turinici, Pierre Rouchon

Published Feb 26, 2005• 60 citations
Publisher

Abstract

Local time control methods are used in the simulation of quantum control phenomena because they conveniently ensure an increase of a predefined performance index and also avoid singularities associated with tracking procedures. However, the drawback of the existing implementations is that they only take into account one-photon, direct transitions and may stop at nonoptimal values of the index. We propose in this paper a modification of the currently used algorithms that addresses this issue and explain how the convergence is improved. Furthermore, when iterations are required, we show that this approach can be inserted into a monotonically convergent algorithm.

DOI
10.1021/jp0472461

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Local time control methods are used in the simulation of quantum control phenomena because they conveniently ensure an increase of a predefined performance index and also avoid singularities associated with tracking procedures. However, the drawback of the existing implementations is that they only take into account one-photon, direct transitions and may stop at nonoptimal values of the index. We propose in this paper a modification of the currently used algorithms that addresses this issue and explain how the convergence is improved. Furthermore, when iterations are required, we show that this approach can be inserted into a monotonically convergent algorithm.
Numerical simulations of (bilinear) quantum control often rely on either monotonically convergent algorithms or tracking schemes. However, despite their mathematical simplicity, very limited intuitive understanding exists at this time to explain the former type of algorithms. Departing from the usual mathematical formalization, we present in this paper an interpretation of the monotonic algorithms as finite horizon, local in time, tracking schemes. Our purpose is not to present a new class of procedures but rather to introduce the necessary rigorous framework that supports this interpretation. As a by-product we show that at each instant, estimates of the future quality of the current control field are available and used in the optimization. When the target is expressed as reaching a prescribed final state, we also present an intuitive geometrical interpretation as the minimization of the distance between two correlated trajectories: one starting from the given initial state and the other backward in time from the target state. As an illustration, a stochastic monotonic algorithm is introduced. Numerical discretizations of the two procedures are also presented.
The computer simulations of quantum control use several approaches including local tracking procedures that prescribe the controlling field through the requirement that a certain functional be decreasing and monotonic algorithms that solve the Euler-Lagrange equations for a predefined cost functional. While different in implementation, recent works [1] hinted that these two classes share some common characteristics. We propose in this contribution a rigorous ground for such conclusions and discuss the precise formulation that allows to construct this equivalence.
The problem of finding the optimal control in numerical computer simulations of quantum control phenomena is usually addressed through the introduction of monotonically convergent algorithms that are guaranteed to improve the cost functional at each step. A recent extension of these algorithms implements a search for a control with given bounds. Within this context, this paper will present a generalisation of this algorithm to a more general class of local in time constraints. Numerical results that illustrate some of the properties of the algorithms introduced are given. KEY WORDS quantum control, bilinear control, monotonic algorithms, numerical methods, non-linear control 1
A general local control theory for manipulating quantum system dynamics is developed. Basic concept of the present theory is lying in the realization of monotonous increasing condition of the performance index, which is locally (in time domain) defined to major how the present quantum state satisfies the current objective. The local control field is designed to satisfy the above condition taking into account the equation of motion of the system. It is found, through the formulation, that the monotonous increasing condition can be achieved as long as the performance index is given as a function of expectation values of time-dependent observable operators, whose equation of motion is governed by the field-free system Hamiltonian or Liouvillian. It is also shown that the present theory is a generalization of the local optimization approach which has been successfully applied to many of molecular dynamics control problems. As for the special cases, performance indices for “transition path control,” “population distribution control,” and “wave packet shaping” are proposed. The theory is applied to vibrational control problems of the one-dimensional model system of hydrogen fluoride. The results show that the present method works effectively for the population dynamics control as well as the wave packet shaping.
This paper explores an adaption of the coordinate ascent approach to quantum control problems. It was motivated by the observation that several of the existing monotone-converging schemes for quantum control may be viewed as approximations of the well-known coordinate ascent method. Our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional, without invoking its derivatives. It is based on recasting the performance functional as a local tracking function which gives the „future” quality of a control at each moment in time. Back propagation of a basis of the target space enables these performance functional (i.e., tracking function) evaluations to be done efficiently. The performance functional may include, or not include, regularization terms as needed. Convergence of the resulting algorithm and its relation to previous schemes are discussed, and numerical examples are provided. In our tests, the coordinate ascent algorithm exhibited rapid convergence to high yield controls. Moreover, it is readily adapted to different restrictions on the controls imposed by the physics of the system under study, e.g., controls with spectrum restrictions in laser control problems, and discrete controls on a specified time grid in electronically controlled nanodevices, or in NMR.
A control strategy for tracking the trajectory of a time-variant objective function in closed quantum systems is proposed in this paper. The Lyapunov stability theorem is used to design the control law. Meanwhile, the issue of singularity is discussed and the problem of large value of control field occurred during the process of tracking are solved by using an adaptive regulation algorithm of objective function. Simulation results demonstrate the effectiveness and feasibility of the method proposed. We also explore the relationship of effect between control parameters and the error of the control system.