Adiabatic Control of the Schrödinger Equation via Conical Intersections of the Eigenvalues
TL;DR: This paper presents a constructive method to control the bilinear Schrödinger equation via two controls based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, and if the latter are conical.
Abstract: In this paper, we present a constructive method to control the bilinear Schrodinger equation via two controls. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits eigenvalue intersections, and if the latter are conical (as it happens generically). In this framework, we are able to spread on several levels connected by conical intersections a state initially concentrated in a single energy level. We provide sharp estimates on the dependence of the error with respect to the controllability time. Moreover, we identify some special curves in the space of controls that improve the precision of the adiabatic approximation, when passing through conical intersections, with respect to classical adiabatic theory.
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TL;DR: 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.
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 quantum-enhanced 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
572 citations
TL;DR: Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry as mentioned in this paper, and they refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal controller problem.
146 citations
TL;DR: In this paper, the controllability of a closed control-affine quantum system driven by two or more external fields is studied in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters.
Abstract: We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties.
58 citations
TL;DR: In this article, a sampling-based learning control (SLC) method is proposed for robust control of quantum systems with uncertainties, which includes two steps of training and testing, where an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution.
Abstract: Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with uncertainties. The SLC method includes two steps of training and testing. In the training step, an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution. A gradient flow-based learning algorithm is developed to find the control for the augmented system. In the process of testing, a number of additional samples are tested to evaluate the control performance, where these samples are obtained through sampling the uncertainty parameters according to a possible distribution. The SLC method is applied to three significant examples of quantum robust control, including state preparation in a three-level quantum system, robust entanglement generation in a two-qubit superconducting circuit, and quantum entanglement control in a two-atom system interacting with a quantized field in a cavity. Numerical results demonstrate the effectiveness of the SLC approach even when uncertainties are quite large, and show its potential for robust control design of quantum systems.
55 citations
TL;DR: In this article, the authors present a sufficient condition for approximate controllability of the bilinear discrete-spectrum Schrodinger equation in the multi-input case and apply it to a rotating polar linear molecule driven by three orthogonal external fields.
51 citations
References
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Book•
01 Jan 1966
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Abstract: "The monograph by T Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4) Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8) The fundamentals of semigroup theory are given in chapter 9 The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10
The first edition is now 30 years old The revised edition is 20 years old Nevertheless it is a standard textbook for the theory of linear operators It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field
Zentralblatt MATH, 836
19,846 citations
TL;DR: In this article, the Adiabatensatz in der neuen Quantenmechanik wird fur den Fall des Punktspektrums in mathematisch strenger Weise bewiesen, wobei er sich bei einer vorubergehenden Entartung des mechanischen Systems als gultig erweist.
Abstract: Der Adiabatensatz in der neuen Quantenmechanik wird fur den Fall des Punktspektrums in mathematisch strenger Weise bewiesen, wobei er sich auch bei einer vorubergehenden Entartung des mechanischen Systems als gultig erweist.
1,156 citations
"Adiabatic Control of the Schrödinge..." refers background or methods in this paper
...nding on εand γ. In particular, (4) is approximately spread controllable on Σ. III. SURVEY OF BASIC RESULTS A. The adiabatic theorem One of the main tools used in this paper is the adiabatic theorem ([8], [15], [21], [24]); here we recall its formulation, adapting it to our framework. For a general overview see the monograph [27]. We remark that we refer here exclusively to the time-adiabatic theorem...
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...l intersections between the eigenvalues. Conical intersections constitute a well-known notion in molecularphysics. They have an important role in the Born–Oppenheimer approximations (see for instance [8], [18], [27], where they appear for finite dimensional operators). In the finite dimensional case they have been classified by Hagedorn [14]. A unified characterization of conical intersections seems t o ...
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819 citations
"Adiabatic Control of the Schrödinge..." refers background or methods in this paper
...2:If there are more than two parts of the spectrum which are sepa rated by a gap, then it is possible to generalize the adiabatic Hamiltonian in the following way ([21]):...
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...One of the main tools used in this paper is the adiabatic theor em ([8], [15], [21], [24]); here we recall its formulation, adapting it to our framework....
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