Generic families of 2-level Hamiltonians

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Table of contents

1 Introduction
1.1 Préliminaires de physique quantique, théorème adiabatique
1.2 Thèmes d’études
1.2.1 Contrôle d’ensemble avec deux contrôles réels
1.2.2 Classification des singularités du champ non-mixant
1.2.3 Vers le contrôle d’ensemble avec un seul contrôle
2 Introduction 
2.1 Quantum physics preliminaries, adiabatic theorem
2.2 Topics of investigation
2.2.1 Ensemble control with two real controls
2.2.2 Classification of the singularities of the non-mixing field
2.2.3 Towards ensemble control with a single input
3 Ensemble control of quantum systems with two controls: conical case 
3.1 Introduction
3.2 Basic definitions and statement of the main results in the finite-dimensional case
3.3 Proof of the ensemble controllability result in the finite-dimensional case
3.4 Example 1: Two-level system driven by a chirped pulse
3.5 Permutations
3.6 Genericity
3.7 Multidimensional set of parameters
3.7.1 Chirped pulses for two-level systems with two parameters
3.7.2 Example 2: STIRAP
3.8 Extension to the infinite-dimensional case
3.8.1 Example 3: Eberly–Law-like models
3.9 Appendix
4 Ensemble control of quantum systems with two controls: non-conical case 
4.1 Introduction
4.2 Basic facts and normal forms
4.2.1 Generic families of 2-level Hamiltonians
4.2.2 Admissible transformations
4.2.3 Normal forms for the non parametric case
4.2.4 Normal forms for the parametric case
4.3 Generic global properties of the singular locus
4.3.1 Proof of Lemma 4.1.1 and Theorem 4.1.2
4.3.2 Generic self-intersections of ⇡(f)
4.4 Adiabatic control through a semi-conical intersection of eigenvalues
4.4.1 Adiabatic dynamics
4.4.2 Regularity of the eigenpairs along smooth control paths
4.4.3 Dynamical properties at semi-conical intersections of eigenvalues
4.5 Control of a continuum of systems
4.5.1 Ensemble adiabatic dynamics
4.5.2 Controllability properties between the eigenstates for the normal forms
4.5.3 The control path (u, v) exits from ⇡(f)
4.5.4 Proof of Theorem 4.1.3
4.6 Extension to n-level systems
4.6.1 Generic assumptions on n-level Hamiltonians and adiabatic decoupling
4.6.2 Adiabatic decoupling
4.6.3 Semi-conical intersections for n-level quantum systems
4.6.4 Controllability result
4.7 Appendix
4.7.1 Averaging theorems and estimates of oscillatory integrals
4.7.2 Two useful lemmas
5 Classification of the singularities of the non-mixing field 
5.1 General definition of the non-mixing field
5.2 The non-mixing field for two level systems
5.2.1 Classification of the singularities of the non-mixing field of a generic two-level system
5.2.2 Bifurcations of the non-mixing field for two-level systems and the avoided crossing problem
5.2.3 Parametric families of real Hamiltonians
5.3 The non-mixing field for general quantum systems
5.3.1 Useful results about line fields
5.3.2 Non-mixing field
5.3.3 Singularities of « j−1,j at intersections (j, j + 1)
6 Control of quantum systems with a single input 
6.1 General framework and main results
6.1.1 Problem formulation
6.1.2 Main results
6.2 Approximation results
6.2.1 Variation formula
6.2.2 Regularity of the eigenstates
6.2.3 Averaging of quantum systems
6.2.4 Perturbation of an adiabatic trajectory
6.2.5 Parametric case
6.3 Control of two-level systems
6.3.1 Control strategy for two-level systems and simulations
6.3.2 Robustness of the control strategy with respect to amplitude of control inhomogeneities
6.4 Control of STIRAP Process
Références