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arXiv:0707.1883 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum optimal control theory

Jan Werschnik, Eberhard K. U. Gross

Published Sep 4, 2007• 470 citations
arXivHTMLPDF

Abstract

The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.

DOI
10.1088/0953-4075/40/18/r01

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The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.
In this work, we investigate how and to which extent a quantum system can be driven along a prescribed path in Hilbert space by a suitably shaped laser pulse. To calculate the optimal, i.e., the variationally best pulse, a properly defined functional is maximized. This leads to a monotonically convergent algorithm which is computationally not more expensive than the standard optimal-control techniques to push a system, without specifying the path, from a given initial to a given final state. The method is successfully applied to drive the time-dependent density along a given trajectory in real space and to control the time-dependent occupation numbers of a two-level system and of a one-dimensional model for the hydrogen atom.
Quantum control refers to our ability to manipulate quantum systems. This tutorial-style chapter focuses on the use of classical electromagnetic fields to steer the system dynamics. In this approach, the quantum nature of the control stems solely from the underlying dynamics, through the exploitation of destructive and constructive interference to reach the control target. We first discuss two basic control principles -- coherent control which uses manipulation in frequency or time to design these interferences, and adiabatic following where access to the control target is enabled by tracking the time-dependent ground state. For complex control targets and system dynamics that exceed the scope of these basic principles, optimal control theory provides a powerful suite of tools to design the necessary protocols. A key consideration for the successful application of optimal control theory is a proper choice of the optimization functional. All concepts are illustrated using recent work from my research group, with a focus on controlling atoms and superconducting qubits. The chapter concludes with an outlook on integrating coherent control with engineered dissipation and a discussion of open questions in the field.
The examples of quantum-state manipulation and coherent excitation discussed in this monograph present idealizations of actual quantum systems, simplifications that allow straightforward theoretical description. As one moves beyond the models of isolated atoms, few essential states, and transform-limited pulses to deal with more realistic models that can describe experimental reality, the basic tools described hitherto require elaboration and extension. Theoretical treatments of large molecules and chemical reactions involving laser-induced changes rely upon numerical simulation more than on analytic solutions. This final chapter discusses two of the themes applicable to that work: control theory and optimization. Control theory Classical control theory, as followed by mathematicians and engineers, deals with procedures for manipulating the input (the “controls”) of a dynamically changing system to obtain a desired output of the system. In a closed-loop control system some device measures the output and, using a feedback loop, alters the input (via a control element ) to bring the output closer to conformity with a goal. These techniques typically find application in control of experiments but they also work for theoretical modeling. An open-loop control system is one without such feedback; the controls are adjusted in accord with some established plan. Design of a suitable control mechanism (a control function ratioing output to input) must ensure that the system is stable (i.e. a finite input signal produces a finite output signal) and controllable (i.e. it is possible to obtain the desired output).
Quantum optimal control theory (QOCT) aims at finding an external field that drives a quantum system in such a way that optimally achieves some predefined target. In practice, this normally means optimizing the value of some observable, a so-called merit function. In consequence, a key part of the theory is a set of equations, which provides the gradient of the merit function with respect to parameters that control the shape of the driving field. We show that these equations can be straightforwardly derived using the standard linear response theory, only requiring a minor generalization: the unperturbed Hamiltonian is allowed to be time dependent. As a result, the aforementioned gradients are identified with certain response functions. This identification leads to a natural reformulation of QOCT in terms of the Keldysh contour formalism of the quantum many-body theory. In particular, the gradients of the merit function can be calculated using the diagrammatic technique for nonequilibrium Green's functions, which should be helpful in the application of QOCT to computationally difficult many-electron problems.
Abstract Quantum optimal control (QOC) is a set of methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. This tutorial paper introduces the basic elements of this theory based on the Pontryagin maximum principle, in a physicist-friendly way. An analogy with classical Lagrangian and Hamiltonian mechanics is proposed to present the main results used in this field. Emphasis is placed on the different numerical algorithms to solve a QOC problem. Several examples ranging from the control of two-level quantum systems to that of Bose–Einstein condensates (BECs) in a one-dimensional optical lattice are studied in detail, using both analytical and numerical methods. Codes based on shooting method and gradient-based algorithms are provided. The connection between optimal processes and the quantum speed limit is also discussed in two-level quantum systems. In the case of BEC, the experimental implementation of optimal control protocols is described, both for two-level and many-level cases, with the current constraints and limitations of such platforms. This presentation is illustrated by the corresponding experimental results.
We present a novel derivation of the Quantum Optimal Control Theory (QOCT). First, we identify the QOCT equations as the Euler-Lagrange equations of the functional associated to the control scheme considering a general window function that accounts for the measurement process. In this framework, we prove that the extremal functions found by Peirce et al. [Phys. Rev. A 37, 4950 (1988)], can be obtained using the well-established theory of variations with appropriately chosen boundary conditions. Second, we derive the QOCT canonical equations to maximize an observable at a given time T by taking the limit of an infinite narrow window function, that is, a delta distribution. We show that the costate is discontinuous and vanishes after the measurement time. Finally, we illustrate the convergence to the limiting case with numerical computations on a spin-chain Heisenberg model.