Acceleration of quantum optimal control theory algorithms with mixing strategies

arXiv:0903.5106v1 [physics.comp-ph] 30 Mar 2009 Acceleration of quantum optimal control theory algorithms with mixing strategies Alberto Castro∗and E. K. U. Gross Institut f¨ur Theoretische Physik and European Theoretical Spectroscopy Facility, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany (Dated: October 31, 2018) Abstract We propose the use of mixing strategies to accelerate the convergence of the common iterative algorithms utilized in Quantum Optimal Control Theory (QOCT). We show how the non-linear equations of QOCT can be viewed as a “fixed-point” non-linear problem. The iterative algorithms for this class of problems may benefit from mixing strategies, as it happens, e.g. in the quest for the ground-state density in Kohn-Sham density functional theory. We demonstrate, with some numerical examples, how the same mixing schemes utilized in this latter non-linear problem, may significantly accelerate the QOCT iterative procedures. ∗Electronic address: alberto@physik.fu-berlin.de 1 I. INTRODUCTION Quantum optimal control theory[1, 2, 3, 4] (QOCT) answers the following question: A sys- tem can be driven, during some time interval, by one or various external fields whose temporal dependence is determined by a set of “control” functions. Given an objective (e.g., to maximize the transition probability to a prescribed final state, the so-called target state), what are the control functions that best achieve this objective? In the more general context of dynamical systems, optimal control theory is widely used for engineering problems, and its modern formulation was established in the 1950’s.[5] The trans- lation of these ideas to Quantum Mechanics was initiated in the 1980’s.[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] Recently, the field has received increasing attention due to the parallel advances in experimental control techniques: femto – and atto – second laser sources with pulse shaping,[19, 20, 21] and learning loop algorithms.[22] These new developments call for corre- sponding theoretical efforts. The computational solution of the QOCT equations may impose an enormous burden. Any algorithm requires multiple forward and backward propagations of the quantum system under study. This can be very cumbersome, depending on the level of theory employed to model the process. The development of efficient algorithms is therefore essential. And, in fact, rather ef- ficient schemes already exist.[23, 24, 25, 26] The most effective choices are closely related and can be grouped in a unified framework.[27] The equations to be solved are non-linearly coupled initial-value partial differential equations, and must be solved iteratively. These iterative proce- dures can be described in the following way: one input field is passed to an “iteration functional” that tests its performances and produces an improved “output” field. This output field can then be used as input for the iteration functional. Upon solution, output and input fields coincide at the “fixed-point” of the iteration functional. We must therefore search for the fixed-point of some non-linear functional. One prominent example of this kind of fixed-point problems is the Kohn-Sham (KS) formulation of density- functional theory (DFT).[28] In this field, it was soon realized that the naive use of the output produced in one iteration as input for the next one leads to poor (or no) convergence, and this observation suggested the use and development of “mixing” techniques:[29, 30, 31] the input for each iteration is a smart combination of the output of the previous iteration and several inputs or outputs of former iterations. The result is typically a very significant acceleration in the conver- 2 gence – and even the possibility of finding a solution in cases where no mixing (or trivial “linear” mixing) is unable of finding one. In this work, we propose the use of those mixing strategies to accelerate the convergence of the iterative algorithms used in QOCT. We demonstrate how they can significantly reduce the iteration count – yet the performance and degree of gain, of course, depends on the details of each particular model. The procedure should be viewed as a scheme to accelerate (and not substitute) the existent iterative algorithms; in particular, it will be made evident that the mixing should be switched on after a couple iterations have been made and the control function is not too far away from the solution – fortunately, it is precisely the regime where the existent algorithms behave better. The description of the proposed methodology is provided in Section II. Some numerical evi- dence supporting the advantages of its use is shown in Section III. Atomic units are used through- out. II. METHODOLOGY We recall the essential equations of QOCT, making no attempt to state them in full generality – the basic ideas can be generalized in different ways suitable for a broad class of situations; however the reader should find no difficulties to translate our

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