A portion of a hypothetical Pareto front (solid line) for ODD in the objective space. The abscissa is the level of discrimination and the ordinate is the similarity of the systems. The Pareto front divides the objective space into a feasible and infeasible region (shaded) as a result of there being limited control resources. Starting from an arbitrary location, such as the point denoted with a star, a point on the Pareto front can be reached by following many paths. Sections II and III present specific algorithms for traversing the objective space horizontally and vertically along paths and , respectively. A procedure for taking an arbitrary path can be generated from sequential application of the latter two algorithms. Section IV presents an algorithm for moving along the front on paths or from an arbitrary starting point (bold dot).
Pareto fronts for different pulse lengths garnered from a pair of three-level quantum systems. The family of fronts illustrate the general principle that varying the available control resources can significantly influence the Pareto trade-offs. The numbered and lettered points refer to specific locations upon exploration of the similarity-discrimination plane (see the text for a detailed discussion). For example, with two physical systems and starting at point and moving horizontally to point , increasing the control resources (through in this case) can dramatically improve the level of discrimination. In contrast, with a fixed level of discrimination, , changing the physical systems and while also changing the control resources has little impact in traversing . In practice, approaching the Pareto front is an asymptotic process with extreme sensitivities in the limits (e.g., for a level of discrimination near 1.0). The Pareto fronts illustrated here span a level of discrimination between 0.005 and 0.995 within which the algorithm is numerically stable.
Percent differences for (a) energy levels and (b) dipole matrix elements where the point indices correspond to the identified points on the Pareto front in Fig. 2. Percent difference for an arbitrary Hamiltonian matrix element from system and from system is defined by the formula: . A general increase in Hamiltonian similarity may be accompanied by a decreased level of similarity for some individual matrix elements. For example, in the case of , the overall Hamiltonians are more similar but the energy levels are less similar while being compensated for by becoming more similar.
(a) Control field corresponding to point 4 in Figs. 2 and 3. (b) Populations of state (i.e., ) for systems and driven by the same control field in (a). The field is able to manipulate constructive interference in system and destructive interference in to yield dramatically different final values of allowing for a high level of discrimination.
Complex plane plots of for (a) system and (b) system where is the order contribution to the mechanism and . Each amplitude in (a) and (b) is a vector in the complex plane labeled by . The radial distance from the origin in the complex plane labels the rings in each plot. The magnitude of each contribution for either system or is an indication of its relative importance in the overall mechanism of discrimination. The transition amplitude is given by the sum over of all significant orders of contributions: . (c) The transition amplitude for systems and (labeled A and B) exhibit a large degree of constructive and destructive interference, respectively.
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