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## Ensemble control with two real controls

For a general closed quantum system under the action of a control u 2 Rm and depending on a parameter z, the corresponding controlled equation is of the form i d dt (t) = H(u(t), z) (t), (t) 2 H, (2.7) with H(u, z) essentially self-adjoint on the separable complex Hilbert space H for every value of u and z. The parameter z can be used either to describe a family of physical systems on which acts a common field driven by u or a physical systems for which the value of one parameter is not known precisely.

The controllability properties of systems of this form has been studied both for discrete and continuous sets of parameters. The case of a finite set of systems is characterized in [14, 35]. In [29] the asymptotic ensemble stabilization is studied for countable sets of parameters. In [63], [13] a proof of a strong notion of ensemble controllability has been obtained for a two-level system. Numerical ensemble control in the case of a continuum of parameters has been throughly studied for two-level systems [79, 28, 75]. Our aim is to generalize the theorems 2.1.2 and 2.1.3 to the issue of Ensemble control. We restrict our study to the problem of ensemble approximate controllability between eigenstates. Excepted when it is explicitely mentioned, we will study finite dimensional real quantum systems, that is systems whose Hamiltonian belong to the set Sn(R) of real symmetric n-dimensional matrices. However the main results of controllability remain valid when H(u, v) = H0+uH1+vH2 where H0,H1,H2 are essentially self-ajoint operators on H with a common dense domain, and satisfying Condition (R), (u, v) 2 U, where U is a connected open set of R2, and an adaptation of this condition to the parametric case (see Chapter 2.2.1).

### Classification of the singularities of the non-mixing field

For a general Hamiltonian H depending on two real controls, the non-mixing curves between %j−1 and %j for j 2 2 have been defined in [22] as the curves & = (&(t))t2[0,1] of R2 along which ˙$j−1(&(t)) is orthogonal to $j(&(t)), for every t 2 [0, 1]. Then, by Theo- rem 2.1.1, the precision of the adiabatic approximation along such a curve is improved. In particular, the error of order p✏ in Equation (2.6) for a control path at a conical intersection is transformed into ✏ along a non-mixing curve. This property has been used in [22] for a precise control of the Schödinger Equation with real Hamiltonians and an extension has been presented in [31] for complex Hamiltonians with three real controls.

In this section, we study the singularities of the non-mixing curves for two level sys- tems, then for more general quantum system.

#### Towards ensemble control with a single input

An important issue of quantum control is to design explicit control laws for the problem of the single input bilinear Schrödinger equation, that is i d dt = (H0 + uH1) (2.12) where belongs to the unit sphere in a Hilbert space H, H0 is a self adjoint operator rep- resenting a drift term called free Hamiltonian, H1 is a self-adjoint operator representing the control coupling and u : [0, T] ! R, T > 0. Important theoretical results of controlla- bility have been proved with different techniques (see [7, 12, 17] and references therein).

For the problem with two or more inputs, adiabatic methods are a nowadays classical way to get an explicit expression of the controls and can be used under geometric conditions on the spectrum of the controlled Hamiltonian (see the articles [10, 22, 62] and references therein), and our results about these methods are presented in Section 2.2.1 of the Intro- duction and are developed in Chapter 3. However, these methods are effective for inputs of dimension at least 2. Our aim is then to extend a single-input bilinear Schrödinger equation into a two-inputs bilinear Schrödinger equation in the same spirit as the Lieextensions introduced by Sussmann and Liu in [66] and [77], then to apply the well-known adiabatic techniques to the extended system. The first step of this procedure is well known by physicists and it is called the rotating-wave approximation (RWA, for short). It is a decoupling approximation to get rid of highly oscillating terms when the system is driven by a real control. This approximation is based on a first-order averaging procedure (see [71, 77, 66, 23] for more informations about averaging of dynamical systems). This ap- proximation is known to work well for a small detuning from the resonance frequency and a small amplitude. For a review of the RWA and its limitations see [37] and [44, 45, 51]. In [24], the mathematical framework has been set for infinite-dimensional quantum systems, formalizing what physicists call Generalized Rabi oscillations and showing that the RWA is valid for a large class of quantum systems. The adiabatic and RWA involve different time scales, and it is natural to ask whether or not they can be used in cascade. The aim of Chapter 2.2.3 is to show the validity of such an approximation under a certain condition on the time scales involved in the dynamics, using an averaging procedure. Then the well-known results of adiabatic theory (see [22, 18, 78]) can be applied in order to get transitions between the eigenstates of the free Hamiltonian. It leads us to design control laws achieving the inversion of a Spin- 1 2 particule and population transfers in the STIRAP process that are robust with respect to inhomogeneities of the amplitude of the control input (see [81, 10] and Chapter 3). As a byproduct of the use of a control oscillating with a small frequency detuning, the proposed method is not expected to be robust with respect to inhomogeneities of the resonance frequencies, that is inhomogeneities of the drift term H0.

**Proof of the ensemble controllability result in the finite-dimensional case**

Theorem 3.3.1. Consider a C3 map [z0, z1] ⇥ U 3 (z, u) 7! Hz(u) 2 Herm(N). Let u : [0, 1] ! U be a C2 control. For every z 2 [z0, z1] and t 2 [0, 1], let !z 1(t), . . . ,!z N(t) be the eigenvalues of Hz(u(t)) repeated according to their multiplicities and denote by (« z 1(t), . . . , »z N(t)) an orthonormal basis of associated eigenvectors. Assume that for every z 2 [z0, z1] and every j 2 J1,NK, (!zj , »zj ) 2 C2([0, 1],R ⇥ CN).

**Extension to the infinite-dimensional case**

The results of the previous sections extend, under some suitable regularity assump- tions, to the case where CN is replaced by an infinite-dimensional complex separable Hilbert space H.

In order to avoid excessive technicalities, we present this extension in the case where the Hamiltonian H depends affinely on the controls and where the controlled Hamiltonians are bounded. (For the general nonlinear case, one could follow the approach in [30].) We then consider a Hamiltonian of the type Hz(u) = Hz 0 + Xd j=1 ujHz j .

**Table of contents :**

**1 Introduction– fr – **

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 ofquantumsystemswithtwo 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 Classificationof the singularities of thenon-mixingfield **

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 ofquantumsystemswitha 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 **