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IEEE Conference on Decision and Control (CDC)

Motion planning in quantum control via intersection of eigenvalues

Ugo Boscain, Francesca Carlotta Chittaro, Paolo Mason, Rémi Pacqueau, Mario Sigalotti

Published Dec 1, 2010• 6 citations
Publisher

Abstract

In this paper we consider the problem of inducing a transition in a controlled quantum mechanical system whose spectrum loses simplicity for some values of the control. We study the situation in which the Hamiltonian of the system is real, and we are in presence of two controls. In this case, eigenvalue crossings are generically conical. Adiabatic approximation is used to decouple a finite dimensional sub system from the original one (usually infinite dimensional). The main advantage of this method is that as a byproduct of the controllability result it permits to get an explicit expression of the controls. Moreover it may be used in the case in which the dependence of the Hamiltonian from the controls is non-linear, for which at the moment, no other method works. In this paper we study the basic block of this controllability method, that is a two by two system whose spectrum presents a conical intersection. We show how to control exactly this system with a control strategy that can be slowed down. The possibility of slowing down the control law is essential to obtain an adiabatic decoupling from the rest of the system with an arbitrary precision.

DOI
10.1109/cdc.2010.5717409

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In this paper we consider the problem of inducing a transition in a controlled quantum mechanical system whose spectrum loses simplicity for some values of the control. We study the situation in which the Hamiltonian of the system is real, and we are in presence of two controls. In this case, eigenvalue crossings are generically conical. Adiabatic approximation is used to decouple a finite dimensional sub system from the original one (usually infinite dimensional). The main advantage of this method is that as a byproduct of the controllability result it permits to get an explicit expression of the controls. Moreover it may be used in the case in which the dependence of the Hamiltonian from the controls is non-linear, for which at the moment, no other method works. In this paper we study the basic block of this controllability method, that is a two by two system whose spectrum presents a conical intersection. We show how to control exactly this system with a control strategy that can be slowed down. The possibility of slowing down the control law is essential to obtain an adiabatic decoupling from the rest of the system with an arbitrary precision.
The main purpose of the thesis is to study the links between the singularities of the spectrum of a controlled quantum Hamiltonian and the controllability issues of the associated Schr"odinger equation.The principal issue that is developed is how to control a parameter-dependent family of quantum systems with a common control input. This problem of ensemble controllability is linked to the design of a robust control strategy when a parameter (a resonance frequency or a control field inhomogeneity for instance) is unknown, and is an important issue for experimentalists.Thanks to the study one-parametric families of Hamiltonians and their generic singularities, we give an explicit control strategy for the ensemble controllability problem when geometric conditions on the spectrum of the Hamiltonian are satisfied. The result is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. Then we study the problem of ensemble controllability under less restrictive hypotheses on the spectrum, namely the presence of non-conical singularities. Under generic conditions such non-conical singularities are not present for single systems, but appear for one-parametric families of systems.For the study of a single system, we focus on a class of curves in the space of controls, called the non-mixing curves (defined in cite{Bos}), that can optimize the adiabatic dynamics near conical and non-conical intersections. They are linked to the geometry of the eigenspaces of the controlled Hamiltonian and the adiabatic approximation holds with higher precision along them.We propose to study the compatibility of the adiabatic approximation with the rotating wave approximation. Such approximations are usually done in cascade by physicists. My work shows that this is justified for finite dimensional quantum systems only under certain conditions on the time scales. We also study ensemble control issues in this case.
In this article we discuss how to control a parameter-dependent family of quantum systems. Our technique is based on adiabatic approximation theory and on the presence of curves of conical eigenvalue intersections of the controlled Hamiltonian. As particular cases, we recover chirped pulses for two-level quantum systems and counterintuitive solutions for three-level stimulated Raman adiabatic passages. The proposed technique works for systems evolving both in finite-dimensional and infinite-dimensional Hilbert spaces. We show that the assumptions guaranteeing ensemble controllability are structurally stable with respect to perturbations of the parameterized family of systems.
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.
We demonstrate coherent control of motional dynamics in trapped Rydberg ions engineered to exhibit a conical intersection between adiabatic potential-energy surfaces. Using quantum optimal control, an optimally shaped electric field drives the motional wave packet between prescribed spatial configurations on microsecond timescales. Localized nonadiabatic coupling breaks the symmetry of the dynamics and produces a directed trajectory: after only a few early passages through the conical intersection region, the packet proceeds toward the target with high fidelity. In contrast, in the Born-Oppenheimer limit, where such coupling is absent, the optimized control yields symmetric, multicycle oscillations rather than a comparably directed displacement. While both approaches reach the target at the chosen final time, the underlying trajectories are qualitatively different. This work demonstrates that engineered conical intersections can serve as a potential control resource for directional motional dynamics, complementing pulse-shaping methods in trapped-ion systems. This directional motion of the ions, enabled by the conical intersection, may provide a route toward improved control of motional states in trapped-ion quantum information processing.
The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrödinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space—and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonian have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrödinger control problem. The applicability of the resulting theorem is illustrated with selected examples.