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arXiv:1310.2271 · Physical Review A

Steering the optimization pathway in the control landscape using constraints

José P. Palao, Daniel M. Reich, Christiane P. Koch

Published Nov 11, 2013• 41 citations
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Abstract

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.

DOI
10.1103/physreva.88.053409

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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.
Shaped pulses obtained by optimal control theory often possess unphysically broad spectra. In principle, the spectral width of a pulse can be restricted by an additional constraint in the optimization functional. However, it has so far been impossible to impose spectral constraints while strictly guaranteeing monotonic convergence. Here, we show that Krotov’s method allows for simultaneously imposing temporal and spectral constraints without perturbing monotonic convergence, provided the constraints can be expressed as positive semi-definite quadratic forms. The optimized field is given by an integral equation which can be solved efficiently using the method of degenerate kernels. We demonstrate that Gaussian filters suppress undesired frequency components in the control of non-resonant two-photon absorption.
We present a monotonic convergent quantum optimal control method that can be utilized to optimize the control field while exactly enforcing multiple equality constraints for steering quantum systems from an initial state towards desired quantum states. For illustration, special consideration is given to finding optimal control fields with (i) exact zero area and (ii) exact zero area along with constant pulse fluence. The method combined with these two types of constraints is successfully employed to maximize the state-to-state transition probability in a model vibrating diatomic molecule.
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.
Optimal control of quantum systems with complex constrained external fields is one of the longstanding theoretical and numerical challenges at the frontier of quantum control research. Here, we present a theoretical method that can be utilized to optimize the control fields subject to multiple constraints while guaranteeing monotonic convergence towards desired physical objectives. This optimization method is formulated in the frequency domain in line with the current ultrafast pulse shaping technique, providing the possibility for performing quantum optimal control simulations and experiments in a unified fashion. For illustrations, this method is successfully employed to perform multiple constraint spectral-phase-only optimization for maximizing resonant multiphoton transitions with desired pulses.
Fundamental selectivity limits of quantum control are pushed by introducing laser driven optimal dynamic discrimination to create distinguishing excitations on two nearly identical flavin molecules. Even with modest spectral resources, significant specificity is achieved with optimal pulse shapes, which amplify small molecular differences to create distinct, identifying signals. Rather than being a hindrance, system complexity appears to aid the control process and augments control field capability, which bodes well for implementation of quantum control in a variety of demanding applications.