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arXiv:0801.3935 · Physical Review Letters

Monotonic Convergent Optimal Control Theory with Strict Limitations on the Spectrum of Optimized Laser Fields

C. Gollub, Markus Kowalewski, Regina de Vivie‐Riedle

Published Aug 13, 2008• 76 citations
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Abstract

We present a modified optimal control scheme based on the Krotov method, which allows for strict limitations on the spectrum of the optimized laser fields. A frequency constraint is introduced and derived mathematically correct, without losing monotonic convergence of the algorithm. The method guarantees a close link to learning loop control experiments and is demonstrated for the challenging control of nonresonant Raman transitions, which are used to implement a set of global quantum gates for molecular vibrational qubits.

DOI
10.1103/physrevlett.101.073002

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We present a modified optimal control scheme based on the Krotov method, which allows for strict limitations on the spectrum of the optimized laser fields. A frequency constraint is introduced and derived mathematically correct, without losing monotonic convergence of the algorithm. The method guarantees a close link to learning loop control experiments and is demonstrated for the challenging control of nonresonant Raman transitions, which are used to implement a set of global quantum gates for molecular vibrational qubits.
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.
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We show how additional constraints, restricting the spectrum of the optimized pulse or confining the system dynamics, can be used to steer optimization in quantum control towards distinct solutions. Our examples are multiphoton excitation in atoms and vibrational population transfer in molecules. We show that a spectral constraint is most effective in enforcing nonresonant two-photon absorption pathways in atoms and avoiding unnecessarily broad spectra in Raman transitions in molecules. While a constraint restricting the system to stay in an allowed subspace is also capable of identifying nonresonant excitation pathways, it does not avoid spurious peaks in the pulse spectrum. Both constraints are compatible with monotonic convergence but imply different additional numerical costs.
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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.