Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Integrable Case

Bonnard, Bernard; Sugny, Dominique

doi:10.1137/080717043

Cited by 53 publications

(53 citation statements)

References 13 publications

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“…We then search for the time-minimum control field, which drives the system from S to M . The general solution of the time-optimal control problem has been given in a series of papers, both for the bounded [5,19,20] and the unbounded cases [21]. It can be shown that the structure of the time-optimal control law can be mainly described by two geometric objects of the Bloch ball which play a central role in the present analysis: the magic plane of equation…”

Section: The Model Systemmentioning

confidence: 99%

Optimal control of the signal-to-noise ratio per unit time for a spin-1/2 particle

et al. 2014

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We investigate the maximum signal to noise ratio per unit time that can be achieved for a spin 1/2 particle subjected to a periodic pulse sequence. Optimal control techniques are applied to design the control field and the position of the steady state, leading to the best signal to noise performance. A complete geometric description of the optimal control problem is given in the unbounded case. We show the optimality of the well-known Ernst angle solution, which is widely used in spectroscopic and medical imaging applications, over a large control space allowing use of shaped pulses.

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Section: The Model Systemmentioning

confidence: 99%

Optimal control of the signal-to-noise ratio per unit time for a spin-1/2 particle

et al. 2014

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“…In this case the purity ρ of the system cannot be controlled and the study of the extremals reduces to the analysis of the geodesics of the Grushin metric on the two sphere of revolution, which was done in [6]. The geodesics are a) meridian circles or b) periodic trajectories in the plane (φ, p φ ) withθ periodic.…”

Section: Classification Of the Asymptotics Of The Extremal Solutionsmentioning

confidence: 99%

“…There are several reasons to consider that for the time-minimal control problem, the smooth continuation method is well adapted to analyze the system. Indeed, from our previous study [6,12] we know that when Γ = γ + and γ − = 0, the time-minimizing control problem can be reduced to analyze the quasi-Riemannian metric…”

Section: Introductionmentioning

confidence: 99%

The smooth continuation method in optimal control with an application to quantum systems

Bonnard

Shcherbakova

Sugny³

2010

ESAIM: COCV

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Abstract. The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the timeminimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.Mathematics Subject Classification. 49K15, 65K10, 81V55.

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“…The control problem consists of finding a trajectory of the state variables solving (5), starting at the completely mixed state (i.e., r = 0) and ending at the apogee. It is important to note, however, that the dynamics (5) has a singularity at the origin since…”

Section: Introductionmentioning

confidence: 99%

Time-minimum control of quantum purity for 2-level Lindblad equations

Clark¹,

Bloch²,

Colombo³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - S

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We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra-and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity.The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

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Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Integrable Case

Cited by 53 publications

References 13 publications

Optimal control of the signal-to-noise ratio per unit time for a spin-1/2 particle

Optimal control of the signal-to-noise ratio per unit time for a spin-1/2 particle

The smooth continuation method in optimal control with an application to quantum systems

Time-minimum control of quantum purity for 2-level Lindblad equations

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