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arXiv:0905.4867 · Physical Review A

Monotonically convergent optimal control theory of quantum systems under a nonlinear interaction with the control field

M. Lapert, Ronald Tehini, Gabriel Turinici, Dominique Sugny

Published Aug 18, 2008• 54 citations
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Abstract

We consider the optimal control of quantum systems interacting nonlinearly with an electromagnetic field. We propose monotonically convergent algorithms to solve the optimal equations. The monotonic behavior of the algorithm is ensured by a nonstandard choice of the cost, which is not quadratic in the field. These algorithms can be constructed for pure- and mixed-state quantum systems. The efficiency of the method is shown numerically for molecular orientation with a nonlinearity of order 3 in the field. Discretizing the amplitude and the phase of the Fourier transform of the optimal field, we show that the optimal solution can be well approximated by pulses that could be implemented experimentally.

DOI
10.1103/physreva.78.023408

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We consider the optimal control of quantum systems interacting nonlinearly with an electromagnetic field. We propose monotonically convergent algorithms to solve the optimal equations. The monotonic behavior of the algorithm is ensured by a nonstandard choice of the cost, which is not quadratic in the field. These algorithms can be constructed for pure- and mixed-state quantum systems. The efficiency of the method is shown numerically for molecular orientation with a nonlinearity of order 3 in the field. Discretizing the amplitude and the phase of the Fourier transform of the optimal field, we show that the optimal solution can be well approximated by pulses that could be implemented experimentally.
We develop a family of monotonically convergent iterative algorithms for solving a wide class of optimal control problems in which a dynamical system interacts nonlinearly with a control. The key idea is to divide a control into identical components and to introduce auxiliary steps to update each component at every iteration step. The algorithms are proved to exhibit monotonic convergence, which is also numerically confirmed through a case study of the control of molecular orientation. The numerical results show that high-quality solutions can be obtained by using the present algorithms.
We propose a monotonically convergent algorithm which can enforce spectral constraints on the control field (and extends to arbitrary filters). The procedure differs from standard algorithms in that at each iteration, the control field is taken as a linear combination of the control field (computed by the standard algorithm) and the filtered field. The parameter of the linear combination is chosen to respect the monotonic behavior of the algorithm and to be as close to the filtered field as possible. We test the efficiency of this method on molecular alignment. Using bandpass filters, we show how to select particular rotational transitions to reach high alignment efficiency. We also consider spectral constraints corresponding to experimental conditions using pulse-shaping techniques. We determine an optimal solution that could be implemented experimentally with this technique.
The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.
A wide range of cost functionals that describe the criteria for designing optimal pulses can be reduced to two basic functionals by the introduction of product spaces. We extend previous monotonically convergent algorithms to solve the generalized pulse design equations derived from those basic functionals. The new algorithms are proved to exhibit monotonic convergence. Numerical tests are implemented in four-level model systems employing stationary and/or nonstationary targets in the absence and/or presence of relaxation. Trajectory plots that conveniently present the global nature of the convergence behavior show that slow convergence may often be attributed to "trapping" and that relaxation processes may remove such unfavorable behavior.
The control of quantum dynamics is often concerned with finding time-dependent optimal control fields that can take a system from an initial state to a final state to attain the desired value of an observable. This paper presents a general method for formulating monotonically convergent algorithms to iteratively improve control fields. The formulation is based on a two-point boundary-value quantum control paradigm (TBQCP) expressed as a nonlinear integral equation of the first kind arising from dynamical invariant tracking control. TBQCP is shown to be related to various existing techniques, including local control theory, the Krotov method, and optimal control theory. Several accelerated monotonic convergence schemes for iteratively computing control fields are derived based on TBQCP. Numerical simulations are compared with the Krotov method showing that the new TBQCP schemes are efficient and remain monotonically convergent over a wide range of the iteration step parameters and the control pulse lengths, which is attributable to the trap-free character of the transition probability quantum dynamics control landscape.
This paper extends a monotonically convergent algorithm for quantum optimal control to treat systems with dissipation. The algorithm working with the density matrix is proved to exhibit quadratic and monotonic convergence. Several numerical tests are implemented in three-level model systems. The algorithm is exploited to control various targets, including the expectation value of a Hermitian operator, the modulus square of the expectation value of a non-Hermitian operator, and off-diagonal elements of the density matrix.