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SIAM Journal on Control and Optimization

Controllability of Quantum Systems Having Weakly Conically Connected Spectrum

Ruikang Liang, Ugo Boscain, Mario Sigalotti

Published Jan 15, 2025• 0 citations
Publisher

Abstract

Abstract. We explore the controllability of a closed multi-input control-affine quantum system. Previous studies have demonstrated that a spectrum connected by conical intersections which do not pile up yields exact controllability in finite dimension and approximate controllability in infinite dimension. Actually, the property that intersections between eigenvalues are conical and that they do not pile up is generic. However, in physical situations, due to symmetry of the system, the spectrum can exhibit intersections that are not conical and possibly pile up. We extend the controllability result to cover these types of situations under the hypothesis that the intersections have at least one conical direction and the piled-up intersections have “rationally unrelated germs.” Finally, we provide a testable first-order sufficient condition for controllability. Physically relevant examples are provided.

DOI
10.1137/23m1597356

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Abstract. We explore the controllability of a closed multi-input control-affine quantum system. Previous studies have demonstrated that a spectrum connected by conical intersections which do not pile up yields exact controllability in finite dimension and approximate controllability in infinite dimension. Actually, the property that intersections between eigenvalues are conical and that they do not pile up is generic. However, in physical situations, due to symmetry of the system, the spectrum can exhibit intersections that are not conical and possibly pile up. We extend the controllability result to cover these types of situations under the hypothesis that the intersections have at least one conical direction and the piled-up intersections have “rationally unrelated germs.” Finally, we provide a testable first-order sufficient condition for controllability. Physically relevant examples are provided.
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The property of controllability of quantum systems is revisited. In the case of a system having this property, couplings to external agents are available whose adjustment can guide the state to any chosen target on a suitably defined manifold (or arbitrarily close to any such target) at any chosen time. With due consideration to unbounded operators corresponding to physical observables possessing continuous spectra, sufficient conditions for controllability based on Lie-algebraic arguments are obtained. The results are not limited to finite-dimensional systems, nor to infinite-dimensional systems with discrete spectra. The applicability of the results to systems with both bound and scattering states is demonstrated for the case of the one-dimensional Poschl-Teller potential. Attention is also directed to transitivity of the pertinent Lie algebra of a system without drift as a necessary and sufficient condition for controllability.