Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

Beauchard, Karine; Laurent, Camille

doi:10.1016/j.matpur.2010.04.001

Cited by 136 publications

(229 citation statements)

References 60 publications

(124 reference statements)

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“…An important result is a general obstruction property to exact controllability [33,36,37]. This was recently amended by positive results about exact [38][39][40] and approximate controllability [32,[41][42][43][44], based on Galerkin techniques. For the specific case of a generalized Jaynes-Cummings model, i.e., several two-level systems coupled to a harmonic oscillator, symmetry methods were used to assess controllability [45,46].…”

Section: State Of the Artmentioning

confidence: 99%

Training Schrödinger’s cat: quantum optimal control

et al. 2015

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Abstract. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments. ForewordThe authors of this paper represent the QUAINT consortium, a European Coordination Action on Optimal Control of Quantum Systems, funded by the European Commission Framework Programme 7, Future Emerging Technologies FET-OPEN programme and the Virtual Facility for Quantum Control (VF-QC). This consortium has considerable expertise in optimal control theory and its applications to quantum systems, both in existing areas, such as spectroscopy and imaging, and in emerging quantum technologies, such as quantum information processing, quantum communication, quantum simulation a e-mail: fwm@lusi.uni-sb.de and quantum sensing. The list of challenges for quantum control has been gathered by a broad poll of leading researchers across the communities of general and mathematical control theory, atomic, molecular-, and chemical physics, electron and nuclear magnetic resonance spectroscopy, as well as medical imaging, quantum information, communication and simulation. 144 experts in these fields have provided feedback and specific input on the state of the art, mid-term and long-term goals. Those have been summarized in this document, which can be viewed as a perspectives paper, providing a roadmap for the future development of quantum control. Because such an endeavour can hardly ever be complete (there are many additional areas of quantum control applications, such as spintronics, nano-optomechanical technologies etc.), this roadmap

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Section: State Of the Artmentioning

confidence: 99%

Training Schrödinger’s cat: quantum optimal control

et al. 2015

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“…The reference [5], by the authors of this paper, improves this result and establishes the exact controllability of Equation (7), locally around the ground state in H 3 , with controls u ∈ L 2 ((0, T ), R), in arbitrary time T > 0, and with generic functions µ when N = 1, Ω = (0, 1). This result, proved with V = 0 in [5], can be extended to an arbitrary potential V , as explained in [29].…”

Section: Local Exact Results In 1dmentioning

confidence: 57%

“…This result, proved with V = 0 in [5], can be extended to an arbitrary potential V , as explained in [29]. The proof relies on a smoothing effect, that allows one to conclude with the inverse mapping theorem (instead of Nash-Moser's one).…”

Section: Local Exact Results In 1dmentioning

confidence: 88%

Local exact controllability of the 2D-Schrödinger-Poisson system

Beauchard¹,

Laurent²

2017

Journal De L’École Polytechnique — Mathématiques

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International audienceIn this article, we investigate the exact controllability of the 2D-Schrödinger-Poisson system, which couples a Schrödinger equation on a bounded domain of R 2 with a Poisson equation for the electrical potential. The control acts on the system through a Neumann boundary condition on the potential, locally distributed on the boundary of the space domain. We prove several results, with or without nonlinearity and with dierent boundary conditions on the wave function, of Dirichlet type or of Neumann type

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“…The linear test [2,3,4], Lyapunov approach [17,5,20,19,6] were used to give a precise description of the attainable set of toy models of (1.1) where x belongs to a compact set (and −∆ + V thus has a discrete spectrum). Very few works have considered the actual form of (1.1), including the continuous part of the spectrum of −∆ + V .…”

Section: Physical Motivationmentioning

confidence: 99%

An approximate controllability result with continuous spectrum: the Morse potential with dipolar interaction

Boussaïd¹,

Caponigro²,

Chambrion³

2015

2015 Proceedings of the Conference on Control and Its Applications

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This note presents a positive approximate controllability result for a bilinear quantum system modeling a chemical bond. The main difficulties are due to the presence of a continuous spectrum part for the uncontrolled Hamiltonian (modeled by a Morse potential) and the unboundedness of the interaction potential (dipolar interaction). The proof is based on averaging technique and spectral analysis.1 Problem Specification. Physical motivationIn this note, we consider the dynamics of a molecular bond, such as a H-H bond in a dihydrogen molecule (Figure 1), modeled by a bilinear Schrödinger equationwhere ℏ = h 2π is the reduced Planck constant, m is the reduced mass of the system, x > 0 is the length of the bond between the atomic nuclei, ∆ is the Laplacian (with Dirichlet boundary conditions) on (0, +∞), V : (0, +∞) → R is the potential of the free dynamics (without external interaction) and W : (0, +∞) → R is the interaction potential, modeling the influence of an electric field (e.g., a laser) on the molecule. The function u : [0, +∞) → R is the intensity of the external field. The wave function ψ ∈ L 2 ((0, +∞), C) the quantum state of the system. For instance, the probability for the molecular bond to have length in|ψ(x, t)| 2 dx. In this note, we focus on the case where V : x → D e (1 − e −a(x−xe) ) 2 is the celebrated Morse potential [18]. With an abuse of notation we identify the function V with the multiplicative operator ψ → V ψ in L 2 ((0, +∞), C). The constants D e and a represent, respectively, the depth and the width of the potential well, the length x e is the equilibrium length of the bond. For a dihydrogen molecule in standard conditions, D e ≈38292 cm −1 , a≈1.4426 and x e ≈7.741410−11 m (see [13]). The Morse potential is nowadays essentially used for academic purposes and it has been replaced in practical applications by more realistic potentials, see for instance, [13, page 65] and [15]. It has the advantage of a complete explicit analytic description of the eigenstates of the free Hamiltonian. Indeed, the operator −∆ + V admits a finite number of simple eigenvalues λ j = hν 0 j + 1 2 − hν 0 j +

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Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control

Cited by 136 publications

References 60 publications

Training Schrödinger’s cat: quantum optimal control

Training Schrödinger’s cat: quantum optimal control

Local exact controllability of the 2D-Schrödinger-Poisson system

An approximate controllability result with continuous spectrum: the Morse potential with dipolar interaction

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