Jean Michel Coron
email: coron@ann.jussieu.fr

Andreea Grigoriu
email: grigoriu@ceremade.dauphine.fr

Catalin Lefter
email: catalin.lefter@uaic.ro

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling

Keywords: numbers: 32.80.Qk, 03.65.Yz, 78.20.Bh

We analyse in this paper the Lyapunov trajectory tracking of the Schrödinger equation for a coupling control operator containing both a linear (dipole) and a quadratic (polarizability) term. We show numerically that the contribution of the quadratic part cannot be exploited by standard trajectory tracking tools and propose two improvements: discontinuous feedback and periodic (time-dependent) feedback. For both cases we present theoretical results and support them by numerical illustrations.

Introduction

We consider in this work the evolution of a quantum system with wavefunction Ψ(t) under the external influence of a laser field; the system satisfies the Time Dependent Schrödinger equation (TDSE)

i d dt Ψ(t) = H(t)Ψ(t), (1)
with H(t) a Hermitian operator; the control is realized by selecting a convenient laser intensity u(t). When the laser is shut off H(t) is the internal Hamiltonian of the system, denoted H 0 ; when the laser is present H(t) is the sum of H 0 and additional terms that describe the interaction of the system with the laser field. The first order term is the dipole coupling [START_REF] Shi | Optimal control of selective vibrational excitation in harmonic linear chain molecules[END_REF] of the form u(t)H 1 ; in the limit of small laser intensities this term may be enough to adequately describe the interaction. However, situations exist where the dipole coupling does not have enough influence on the system to reach the control goal; the goal may become accessible only after adding a polarizability term u 2 (t)H 2 in the expansion of H(t) (see e.g. [START_REF] Dion | Two-frequency IR laser orientation of polar molecules. numerical simulations for HCN[END_REF][START_REF] Dion | Laser-induced alignment dynamics of HCN: Roles of the permanent dipole moment and the polarizability[END_REF] and related works); to make effective use of this term one needs higher laser intensities u(t).

The focus of the paper is on practical procedures to find suitable control fields u(t) for the Hamiltonian H(t) = H 0 +u(t)H 1 +u 2 (t)H 2 by adapting feedback tracking control procedures to this setting. Here and in the following H 0 , H 1 and H 2 are n×n Hermitian matrices with complex coefficients and the control is the laser intensity u(t) ∈ R.

In what concerns the mere possibility to find a control, we recall that the controllability of the finite dimensional quantum system evolving with equation

i d dt Ψ(t) = (H 0 + u(t)H 1 + u 2 (t)H 2 )Ψ(t), (2)
can be studied via the general accessibility criteria [START_REF] Brockett | Lie theory and control systems defined on spheres[END_REF][START_REF] Sussmann | Controllability of nonlinear systems[END_REF] based on Lie brackets; more specific results can be found in [START_REF] Turinici | Beyond bilinear controllability: applications to quantum control[END_REF].

Let us consider for a moment the system with Hamiltonian H 0 + u(t)H 1 + v(t)H 2 , v(t) being a second control independent of u(t). It can be shown [START_REF] Turinici | Beyond bilinear controllability: applications to quantum control[END_REF] that this system is controllable under the same circumstances as H 0 + u(t)H 1 + u 2 (t)H 2 i.e. all target states that can be reached with Hamiltonian H 0 + u(t)H 1 + v(t)H 2 can also be reached by H 0 + u(t)H 1 + u 2 (t)H 2 (although obviously the second Hamiltonian is a particular case of the first for v(t) = u 2 (t)). This rather counter-intuitive result suggests that u 2 (t) can be considered, for the purpose of theoretical controllability, as independent of u(t); however, u 2 (t) having a particular functional dependence on u(t) will play a role at the level of the numerical procedure to find the control: in general finding the control for

H 0 + u(t)H 1 + u 2 (t)H 2 is more difficult than for H 0 + u(t)H 1 + v(t)H 2 .
The characterization of the controllability does not provide in general a simple and efficient way for open-loop trajectory generation. Optimal control techniques (cf., [START_REF] Maday | New formulations of monotonically convergent quantum control algorithms[END_REF] and [START_REF] Shi | Optimal control of selective vibrational excitation in harmonic linear chain molecules[END_REF] and the references herein) provide a first set of methods. A different approach consists in using feedback to generate trajectories and open-loop steering control [START_REF] Chen | Competitive tracking of molecular objectives described by quantum mechanics[END_REF][START_REF] Gross | Inverse quantum-mechanical control: A means for design and a test of intuition[END_REF][START_REF] Kosloff | Wavepacket dancing: Achieving chemical selectivity by shaping light pulses[END_REF]. More recent results can be found in [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] for decoupling techniques, in [START_REF] Beauchard | Implicit Lyapunov control of finite dimensional Schrödinger equations[END_REF][START_REF] Ferrante | Control of quantum systems using model-based feedback strategies[END_REF][START_REF] Grivopoulos | Lyapunov-based control of quantum systems[END_REF][START_REF] Maday | New formulations of monotonically convergent quantum control algorithms[END_REF][START_REF] Sugawara | General formulation of locally designed coherent control theory for quantum systems[END_REF][START_REF] Vaidya | Control of Heisenberg spin systems; lie algebraic decompositions and action-angle variables[END_REF][START_REF] Vettori | On the convergence of a feedback control strategy for multilevel quantum systems[END_REF] for Lyapunov-based techniques and in [START_REF] Altafini | Controllability of quantum mechanical systems by root space decomposition of su(n)[END_REF][START_REF] Constantinescu | Parametrizing quantum states and channels[END_REF][START_REF] Ramakrishna | Control of a coupled two-spin system without hard pulses[END_REF] for factorizations techniques of the unitary group.

In order to study feedback control of systems with Hamiltonian H(t) = H 0 + u(t)H 1 + u 2 (t)H 2 we adapt the analysis [START_REF] Jurdjevic | Controllability and stability[END_REF][START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF], initially proposed for bilinear quantum systems H 0 + u(t)H 1 . In the previous work it has been shown that the success of the feedback control depends on whether there exists (non-zero) direct coupling, through H 1 , between the target state and all other eigenstates. When H 1 has the same property for H(t) = H 0 + u(t)H 1 + u 2 (t)H 2 we show that same feedback formulas hold. However we argued that the polarizability term u 2 (t)H 2 is added when dipole u(t)H 1 is not enough to control the system; consequently the most interesting question is what happens when some of the (direct) coupling is realized by H 2 instead of H 1 . We show that the previous feedback formulas do not hold any more and we propose two alternatives. Our method is valid to track any eigenstate trajectory of a Schrödinger equation ( 2) when the Hamiltonian includes a second order coupling operator.

The balance of the paper is as follows: in Section 2 we introduce the main notations and the Lyapunov tracking feedback for a particular case. Section 3 contains the presentation of two types of feedback: discontinuous and time-dependent (periodic) forcing, that can be applied for all types of second order coupling operators. Both sections present theoretical results on the convergence illustrated by numerical simulations. Concluding remarks are presented in Section 4.

Tracking feedback design

Dynamics and global phase

We consider a n-level quantum system evolving under the equation [START_REF] Ancona | Patchy vector fields and asymptotic stabilization[END_REF]. The wave function Ψ = (Ψ j ) n j=1 is a vector in C n , verifying n j=1 |Ψ j | 2 = 1, thus it lives on the unit sphere S 2n-1 of C n . Physically, Ψ and e iθ(t) Ψ describe the same physical state for any global phase θ(t) ∈ R, i.e. Ψ 1 and Ψ 2 are identified when exists θ(t) ∈ R such that Ψ 1 = exp(iθ(t))Ψ 2 . To take into account such non trivial geometry we add a second control ω corresponding to θ (see also [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF]). Thus we consider the following control system

i d dt Ψ(t) = (H 0 + u(t)H 1 + u 2 (t)H 2 + ω(t))Ψ(t), (3)
where ω ∈ R is a new control playing the role of a gauge degree of freedom. We can choose it arbitrarily without changing the physical quantities attached to Ψ. With such additional fictitious control ω, we will assume in the sequel that the state space is S 2n-1 and the dynamics given by (3) admits two independent controls u and ω.

Lyapunov control design

Take a reference trajectory t → (Ψ r (t), u r (t), ω r (t)), i.e., a smooth solution of (3):

i d dt Ψ r = (H 0 + u r H 1 + u 2 r H 2 + ω r )Ψ r .
We introduce the following time varying function V (Ψ, t):

V (Ψ) = Ψ -Ψ r |Ψ -Ψ r = Ψ -Ψ r 2 (4)
where .|. denotes the Hermitian product. The fonction V is nonnegative for all t > 0 and all Ψ ∈ S 2n-1 and vanishes when Ψ = Ψ r , such a V is called a Lyapunov function.

The derivative of V is given by:

dV dt = 2(u -u r )Im( H 1 Ψ(t)|Ψ r ) + 2(u 2 -u 2 r )Im( H 2 Ψ(t)|Ψ r ) (5) + 2(ω -ω r )Im( Ψ(t) | Ψ r )
where Im denotes the imaginary part.

For convenience we denote:

I 1 = Im( H 1 Ψ(t)|Ψ r ) and I 2 = Im( H 2 Ψ(t)|Ψ r ). Note that if, for example, one takes u = u r (t) -k(I 1 + 2u r I 2 )/(1 + kI 2 ) ω = ω r (t) -cIm( Ψ(t) | Ψ r ), (6)
with k and c strictly positive parameters, one gets

dV /dt = - 2 k (u -u r ) 2 -2c(Im( Ψ(t) | Ψ r )) 2 ≤ 0,
and thus V is nonincreasing.

Remark 2.1 In order for the denominator 1 + kI 2 in Eqn. [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF] to be non-zero one notes that

|I 2 | ≤ | H 2 Ψ(t)|Ψ r | ≤ H 2 ; therefore 1 + kI 2 > 0 as soon as k < 1 H 2 .
From now on, unless specified otherwise, this condition will be supposed satisfied.

Let us focus on the important case when the reference trajectory corresponds to an equilibrium: u r = 0, ω r = -λ and Ψ r = φ where φ is an eigenvector of H 0 associated to the eigenvalue λ ∈ R (H 0 φ = λφ, φ = 1). We obtain:

I 1 = Im( H 1 Ψ(t)|φ), I 2 = Im( H 2 Ψ(t)|φ). (7)
Then (6) becomes a static-state feedback:

u = -kI 1 /(1 + kI 2 ) ω = -λ -cIm( Ψ(t) | φ ). (8)
Denote by λ j , with j = 1, ..., N the eigenvalues of the matrix H 0 . Let φ 1 , . . . , φ n be an orthogonal system of corresponding eigenvectors.

We say that H 0 has non-degenerate spectrum if for all j = l, j, l = 1, ..., n, λ j = λ l . Although V being nonincreasing is a very important property, this is not enough to ensure that the target state φ is reached asymptotically. The following theoretical result for the feedback (8) explains when convergence to target state holds: Theorem 2.1 Consider (3) with Ψ ∈ S 2n-1 and an eigenstate φ ∈ S 2n-1 of H 0 associated to the eigenvalue λ. Take the static feedback [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]

with c > 0, k < 1
H 2 and suppose that the spectrum of H 0 is non-degenerate. Then the two following propositions are true:

(i) The limit set of the closed loop system (3) is in the intersection of S 2n-1 with the vector space E = Rφ α Cφ α where φ α is any eigenvector of H 0 not co-linear to φ such that φ α |H 1 φ = 0.

(ii) If E = Rφ, the limit set is a subset of {φ, -φ}.

The proof follows the same ideas as in [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] and it can be found in [START_REF] Grigoriu | Lyapounov control of Schodinger equations: beyond the dipole approximation[END_REF].

Remark 2.2

The theorem above shows that tracking to φ works when all eigenstates of H 0 , φ 2 , ....φ n , other than φ are coupled to φ by H 1 . However we do not know what happens when some of the coupling are realized by H 2 instead (the theorem does not apply but the system is still controllable cf. [START_REF] Turinici | Beyond bilinear controllability: applications to quantum control[END_REF]). We analyze such a case in Section 3. Note that, as pointed out in the Introduction, one uses the model

H 0 + u(t)H 1 + u 2 (

Examples and simulations

In order to solve (3), we use the following numerical scheme:

ψ((m + 1)∆t) = e -i∆t(H 0 +u(m∆t)H 1 +u 2 (m∆t)H 2 )+ω(m∆t)) ψ(m∆t), (9)
where m is the index of the time step, ∆t = T /M is the discretization time step, and M is the total number of time steps. Numerical simulations have been performed for a three-dimensional test system with H 0 , H 1 and H 2 given by:

H 0 =    0 0 0 0 1 0 0 0 3 2    , H 1 =    0 1 1 1 0 0 1 0 0    , H 2 =    0 0 1 0 0 0 1 0 0    . (10)
In this case the wave function is Ψ = (Ψ 1 , Ψ 2 , Ψ 3 ) T . We use the Lyapunov control (8) in order to reach the first eigenstate φ = (1, 0, 0) of energy λ = 0, at the final time T .

Remark 2.3

We note that the conditions of Theorem 2.1 are fulfilled since:

φ 2 |H 1 φ = 0 and φ 3 |H 1 φ = 0. In addition H 2 = 1.
In Figure 1 we plot the evolution of V = V (Ψ) = Ψ -φ|Ψ -φ and u, corresponding to system defined by [START_REF] Coron | Control and nonlinearity[END_REF] with feedback [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. We can remark a fast convergence of the Lyapunov function V towards zero, that implies the convergence of Ψ towards φ = (1, 0, 0). The target goal is achieved with high accuracy at T = 100.

(t = 0) = (0, 1/ √ 2, 1/ √ 2)
; system defined by (10) with feedback [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. We take k = 0.2, c = 0.8.

We consider another three-dimensional test system with H 0 , H 1 and H 2 given by:

H 0 =    0 0 0 0 1 0 0 0 3 2    , H 1 =    0 1 0 1 0 0 0 0 0    , H 2 =    0 0 1 0 0 0 1 0 0    . (11)
In this case the wave function is Ψ = (Ψ 1 , Ψ 2 , Ψ 3 ) T . We use the previous Lyapunov control in order to reach the first eigenstate φ = (1, 0, 0) of energy λ = 0, at the final time T .

Simulations of Figure 2 start with (0, 1/ √ 2, 1/ √ 2) as initial condition for Ψ. Such a feedback reduces the distance to the first state but does not ensure its convergence to φ = (1, 0, 0) (the Lyapunov function V (Ψ), does not reach the value zero, but stalls at V = 10 -0.1 .) This is not due to a lack of controllability. This system is controllable since the Lie algebra spanned by H 0 /i, H 1 /i and H 2 /i coincides with u(3) (see [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF]). As explained in Remark 2.2, such convergence deficiency comes from the fact that operator H 1 couples φ only with the state φ 2 . We plot the evolution of V (Ψ), u, I 1 and I 2 , corresponding to system defined by [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback (8), in Figure 2 and Figure 3.

(t = 0) = (0, 1/ √ 2, 1/ √ 2)
; system defined by [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. The feedback (8) fails to reach the target, V stalls at V = 10 -0.1 . We take k = 0.2, c = 0.8. . Time evolution of I 1 and I 2 ; system defined by [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. We note that I 1 converges to zero. Contrary to I 1 , I 2 does not converge to zero.

Discontinuous and periodic feedback

In order to stabilize the system when formulas [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] are ineffective, we propose two methods. The first one is to use a special discontinuous feedback ([2, 6, 12, 26], as well as [START_REF] Coron | Control and nonlinearity[END_REF]Section 11.4] and the references therein). The second approach is through periodic time dependent feedback ( [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF][START_REF] Coron | On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback law[END_REF] as well as [10, Sections 11.2 and 12.4] and the references therein).

Discontinuous feedback

For the case of discontinuous feedback we consider the regions:

A = {|I 1 | < δ and I 2 < - √ δ}, B = {|I 1 | < δ and I 2 > √ δ}, C = {|I 1 | > δ/2 or |I 2 | < 2 √ δ}. Note that A, B
, C are open sets; the regions A, C, respectively B, C are overlapping. For k 1 , k 2 , c, δ > 0 we define the control as follows:

u(I 1 , I 2 ) =          k 1 I 2 , in A \ C 0, in B \ C -k 2 I 1 /(1 + k 2 I 2 ), in C \ (A ∪ B) f or A ∩ C and A ∩ B see below, ω = -λ -cIm( Ψ(t) | φ ). ( 12
)
The definition of u(I 1 , I 2 ) on A ∩ C is either u(I 1 , I 2 ) = k 1 I 2 (i.e. formula for set A) or u(I 1 , I 2 ) = -k 2 I 1 /(1 + k 2 I 2 ) (i.e. formula for set C) is such a way to ensure continuity (with respect to time) of the feedback u(I 1 , I 2 ). Remark that a discontinuity may appear when reaching the border ∂C of C situated in the interior of A (here the formula used is always u(I 1 , I 2 ) = k 1 I 2 ). Same considerations apply for A ∩ B (continuity).

We define the propagator S 1 Ψ 0 by solving the feedback equation such that: if the initial state Ψ 0 ∈ A ∩ C, we initiate with the feedback corresponding to C and if the initial state Ψ 0 ∈ B ∩ C, we initiate also with the feedback corresponding to C. We define also the propagator S 2 Ψ 0 by solving the feedback equation such that : if the initial state Ψ 0 ∈ A ∩ C, we initiate with the feedback corresponding to A and if the initial state Ψ 0 ∈ B ∩ C, we initiate also with the feedback corresponding to B. We continue with the feedback for the given region until reaching the boundary of this one, then we switch to the feedback corresponding to the next overlapping region. Observe that neither S 1 nor S 2 define a classical dynamical system, that is the semigroup property is lost. One has instead:

S 1 (t + s)Ψ 0 = S 1 (t)S 1 (s)Ψ 0 or S 1 (t + s)Ψ 0 = S 2 (t)S 1 (s)Ψ 0 . ( 13
)
The propagators S 1 (t)Ψ 0 and S 2 (t)Ψ 0 are solutions in the sense of Carathéodory for the feedback controlled system and depend continuously on the initial data. These solutions are well defined locally and they are globally defined on [0, ∞[; this follows from the fact that intervals of time between switching moments are bounded from bellow by a strictly positive constant.

We now turn to the question of the negativity of dV /dt.

/dt = uI 1 + u 2 I 2 = u(I 1 + uI 2 ) = k 1 I 2 (I 1 + k 1 (I 2 ) 2
). On the set A the term I 2 is less than -√ δ and thus k 1 I 2 is strictly negative. From the definition of A the term I 1 + k 1 (I 2 ) 2 is lower bounded by -δ + k 1 δ; thus for k 1 > 1 the term I 1 + k 1 (I 2 ) 2 is strictly positive thus dV /dt < 0.

Therefore k 1 > 1 and k 2 < 1 H 2 imply dV /dt ≤ 0 with dV /dt = 0 only when u(I 1 , I 2 ) = 0. Moreover one has the following convergence result for the feedback [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF] (see also [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF]). C of H 0 associated to the eigenvalue λ. Take the feedback [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF]

with k 1 > 1, k 2 < 1
H 2 and c, δ > 0. If H 0 is not degenerate and for every k with φ k = φ either φ k |H 1 φ = 0 or φ k |H 2 φ = 0 then the limit set of Ψ(t) reduces to a solution of the uncontrolled system, with

|I 1 | < δ, |I 2 | ≤ C √ δ with a constant C depending only on H 0 .
Proof of Theorem 3.1. Up to a shift on ω and H 0 , we can assume that λ = 0. Trajectories corresponding to the propagator S 1 are relatively compact so the limit points at infinity form a limit set Ω δ which is compact and connected.

On the limit set Ω δ , V is constant and from the relation [START_REF] Dion | Two-frequency IR laser orientation of polar molecules. numerical simulations for HCN[END_REF],

Ω δ is invariant either to S 1 or to S 2 , that is if Ψ 1 ∈ Ω δ then S 1 (t)Ψ 1 ∈ Ω δ , t > 0 or S 2 (t)Ψ 1 ∈ Ω δ , t > 0.
The limit set Ω δ is a union of trajectories of equation ( 3) corresponding either to the propagator S 1 or to the propagator S 2 , along this trajectories V is constant, so dV /dt = 0 where V is defined by [START_REF] Brockett | Lie theory and control systems defined on spheres[END_REF]. The equation dV /dt = 0 means that:

u(I 1 + uI 2 ) = 0, ( 14
)
Im Ψ, φ = 0. ( 15
)
Since u is defined by (12) it follows that the limit set, Ω δ , consists in fact of trajectories of the uncontrolled system:

i d dt Ψ = H 0 Ψ. ( 16
)
with the solutions of the form:

Ψ = n j=1 b j e -iλ j t φ j . (17)
For the same reasons as above we obtain that the limit set Ω δ is characterized by:

Ω δ ⊂ {I 1 = 0 and |I 2 | < 2 √ δ} ∪ {|I 1 | < δ and I 2 > √ δ}. ( 18
)
From relation [START_REF] Gross | Teaching lasers to control molecules in the presence of laboratory field uncertainty and measurement imprecision[END_REF] we have that on the limit set |I 1 | < δ. We substitute [START_REF] Grivopoulos | Lyapunov-based control of quantum systems[END_REF] in [START_REF] Ferrante | Control of quantum systems using model-based feedback strategies[END_REF] and we have:

Im Ψ, φ = Im(b 1 ) φ, φ + n j=2 Im(b j φ j , φ e -iλ j t ) = 0. ( 19
)
We obtain Im(b 1 ) = 0. We denote by

J 1 = {j|j = 1, H 1 φ j |φ ) = 0} and J 2 = {j|j = 1, H 2 φ j |φ ) = 0}.
We have by the hypothesis that J 1 ∪ J 2 = {2, 3, . . . , n}.

We substitute ( 17) in ( 7), and we obtain:

I 1 = Im(b 1 ) H 1 φ, φ + j∈J 1
Im(b j H 1 φ j , φ e -iλ j t ), ( 20)

I 2 = Im(b 1 ) H 2 φ, φ + j∈J 2 Im(b j H 2 φ j , φ e -iλ j t ). ( 21
)
Since Im(b 1 ) = 0 we have:

I 2 = j∈J 2 Im(b j H 2 φ j , φ e -iλ j t ) = j∈J 2 B j sin(λ j t + θ j ). ( 22
)
where the coefficients B j = 0 if and only if b j = 0, j ∈ J 2 . We define M = sup(I 2 ) and m = inf(I 2 ). There exists C > 0 independent of B j and θ j such that M -Cm. Since on the limit set Ω δ ,

I 2 -2 √ δ it is easy to verify that |I 2 | C √ δ.
Remark 3.1 In order to make the conclusion of the theorem more precise note that if Ψ δ is a trajectory of ( 16) belonging to Ω δ , then when δ converges to zero, Ψ δ → φ , if the initial state is different of -φ. Accordingly, when I 1 , I 2 are small V (Ψ) will also be small and the system is close to the target state. The practical question is then how small should one choose δ. A way to circumvent this question is to consider not a constant value δ but one that decreases over time; this way the problem will take itself care of finding the good value of δ for a given precision. Numerical results (not shown here) confirm the interest of this approach. Remark 3.2 An important ingredient of the proof is finding the limit sets of the evolution, which itself depends very much on the choice of the sets A, B, C and of the controls u. The general rationale behind these choice are to modify formula [START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF] minimally in order to have good properties of Ω δ .

3.1.1.

Examples for non-degenerate cases. We take the system [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] and apply the discontinuous feedback [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF]. Simulations of Figure 4 describe the evolution of the Lyapunov function V (Ψ) and control u, for the initial state Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2). In this case:

k 1 = 1.1, k 2 = c = 0.8 and δ = 1.e -4.
It appears that this feedback is quite efficient for system [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF]. We present the evolution of I 1 and I 2 corresponding to system defined by [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF], with feedback (12), in Figure 5. We consider next the five-dimensional system (see [START_REF] Tersigni | On using shaped light pulses to control the selectivity of product formation in a chemical reaction: An application to a multiple level system[END_REF]) defined by

H 0 =   1.0 0 0 0 0 0 1.2 0 0 0 0 0 1.3 0 0 0 0 0 1.4 0 0 0 0 0 2.15   , H 1 =   0 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0   , H 2 =   0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   . (23)
We use the previous Lyapunov control in order to reach the first eigenstate φ = (1, 0, 0, 0, 0, 0) of energy λ = 1, at the final time T . Note that here H 2 = 1. Simulations of Figure 6 describe the evolution of the Lyapunov function V (Ψ) and control u, for the initial state

Ψ(t = 0) = (0, 1/ √ 4, 1/ √ 4, 1/ √ 4, 1/ √ 4). We take k 1 = 1.1, k 2 = c = 0.8 and δ = 1.e -4.
We present the evolution of I 1 and I 2 corresponding to system defined by [START_REF] Maday | New formulations of monotonically convergent quantum control algorithms[END_REF], with feedback [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF], in Figure 7.

Examples for degenerate cases.

There are various situations where the condition of non degeneracy of the Hamiltonian H 0 , present in Theorem 2.1 and Theorem 3.1 is non fulfilled. One such example is given below (see [START_REF] Gross | Teaching lasers to control molecules in the presence of laboratory field uncertainty and measurement imprecision[END_REF][START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]):

H 0 = 0 0 0 0 0 0.04556 0 0 0 0 0.095683 0 0 0 0 0.095683 , H 1 = 0 1 1 -1 1 0 1 1 1 1 0 0 -1 1 0 0 , H 2 = 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 . ( 24
)
The internal Hamiltonian H 0 is degenerate since λ 3 = λ 4 = 0.095683, but it can be stabilized using the discontinuous feedback defined by [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF]. Here H 2 = 1.

Simulations of Figure 8 describe the evolution of the Lyapunov function V (Ψ) and control u, system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] This positive result for degenerate system shows that the theoretical results are sufficient but not necessary; however the approach may fail in some particular degenerate cases.This is consistent with the literature on quantum control that shows that degenerate cases have special structure (starting even with controllability criteria). . Time evolution of I 1 and I 2 ; system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback (12).

Periodic feedback

Although the discontinuous feedback [START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF] gives satisfactory results in terms of the control quality, the fact that it is discontinuous motivates trying to find additional procedures. To this end we introduce in this section a periodic, time dependent, feedback u = u(t, Ψ) stabilizing (3) to the ground state φ. The idea is to use a highly oscillatory field component whose linear contribution averages to zero while the quadratic part averages to a constant; then we compare the asymptotic behavior of the system with the behavior of the averaged system. We recall that we are in the case when the reference Theorem 3.2 Under the hypotheses:

(i) λ j = λ l for j = l, (ii) for any j = 2, .., n : H 1 φ j |φ = 0 or H 2 φ j |φ = 0, the averaged system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] is globally asymptotically stable on S 2n-1 \ {-φ} in the sense (recall [START_REF] Sugawara | General formulation of locally designed coherent control theory for quantum systems[END_REF]) that every solution Ψ av of [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] with an initial state other than -φ tends to φ as t tends to +∞.

Proof of Theorem 3.2 Up to a shift on ω and H 0 , we may assume that λ = 0. LaSalle's principle (see, e.g., [START_REF] Khalil | Nonlinear Systems[END_REF]Theorem 3.4,page 115]) says that the trajectories of the system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] converge to the largest invariant set contained in dV av /dt = 0. The equation dV /dt = 0 means that:

I av 1 = 0, (I av 2 ) -= 0, Im( Ψ av (t)|φ ) = 0, (32)
and therefore α = β = 0. On the Ω-limit set of a trajectory, V is constant. Since the Ω-limit set is invariant under the flow generated by [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] it follows that it consists in fact of trajectories of the uncontrolled system:

i d dt Ψ av = H 0 Ψ av . (33)
The solutions of (33) have the form:

Ψ av = n j=1 b j e -iλ j t φ j , (34)
We substitute [START_REF] Turinici | Beyond bilinear controllability: applications to quantum control[END_REF] in [START_REF] Sussmann | Controllability of nonlinear systems[END_REF] and we obtain:

Im( Ψ av (t)|φ ) = Im(b 1 ) φ, φ + n j=2
Im(b j φ j , φ e -iλ j t ),

I av 1 = Im(b 1 ) H 1 φ, φ + j∈J 1 Im(b j H 1 φ j , φ e -iλ j t ), (35)
I av 2 = Im(b 1 ) H 2 φ, φ + k∈J 2 Im(b j H 2 φ j , φ e -iλ j t ). ( (36)
) 37
From equation ( 32) and ( 35) we obtain that Im(b 1 ) = 0. Since along the trajectories in Ω, I av 1 ≡ 0 we obtain b j = 0, j ∈ J 1 . Using Im(b 1 ) = 0 we have:

I av 2 = j∈J 2 Im(b j H 2 φ j , φ e -iλ j t ) = j∈J 2 B j sin(λ j t + θ j ), (38)
where the coefficients B j = 0 if and only if b j = 0, j ∈ J 2 . Since I av 2 ≥ 0 ∀t, it follows that I 2 av ≡ 0. We have thus b j = 0, j = 1. We obtain that Ω ⊂ {φ, -φ}. This concludes the proof of Theorem 3.2.

Our next theorem shows that our time-varying feedback laws lead to some kind of "practical global asymptotic stability on S 2n-1 \ {-φ}" if ε > 0 is small enough and if the assumptions of Theorem 3.2 hold (see also [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF]).

Using (47) and applying (46) with τ + T for the new value of τ , one gets that

|Ψ(T + τ + t) -φ| < min{δ, δ 1 } ≤ δ ∀t ∈ [T, 2T ].
Keeping going, an easy induction argument on the integer m shows that, more generally, for every nonnegative integer m,

|Ψ(mT + τ + t) -φ| < min{δ, δ 1 } ≤ δ ∀t ∈ [T, 2T ],
which implies (39).

3.2.1.

Examples for non-degenerate cases. We take the system [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] and apply the periodic feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF] with α et β defined by [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]. Simulations of Figure 10 describe the evolution of the Lyapunov function V (Ψ) for the initial state Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2). It appears that the periodic feedback is quite efficient for system [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF]. See Figure 10.

(t = 0) = (0, 1/ √ 2, 1/ √ 2)
; system defined by [START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5. We take the system [START_REF] Maday | New formulations of monotonically convergent quantum control algorithms[END_REF] and apply the periodic feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF] with α et β defined by [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]. Simulations of Figure 11 describe the evolution of the Lyapunov function V (Ψ) and control u for the initial state Ψ(t = 0) = (0, 1/ √ 4, 1/ √ 4, 1/ √ 4, 1/ √ 4). Agreement with the theoretical results presented above is obtained. See Figure 11. 23) with feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5.

3.2.2.

Examples for degenerate cases. We take the system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] and we apply the periodic feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF] with α et β defined by [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF]. Simulations of Figure 12 describe the evolution of the Lyapunov function V (Ψ) and control u, system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] starting from the initial state Ψ(t = 0) = (0, 1/ √ 3, 1/ √ 3, 1/ √ 3). We present the evolution of I 1 and I 2 corresponding to system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF], with feedback (25), in Figure 13. ; system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5. . Time evolution of I 1 and I 2 ; system defined by [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback (25).

Conclusions

We focus in this paper on designing trajectory tracking (feedback) procedures for a control system with polarizability terms u 2 (t)H 2 present. We find that a straightforward application of the previous results only work for systems that are controllable without the polarizability term. To be able to find a control field that exploit the polarizability coupling we propose two different solutions: the first one is to use a discontinuous feedback with memory terms, the other is to use time-dependent (periodic) forcing. In both cases we present related theoretical results and numerically implement these techniques on prototypical examples. The time-dependent feedback is seen to generally produce smoother controls.

Figure 1 .

 1 Figure 1. Evolution of the Lyapunov function V (Ψ) (blue line) and control u (green line); initial condition Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2); system defined by (10) with feedback[START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. We take k = 0.2, c = 0.8.

Figure 2 .

 2 Figure 2. Evolution of the Lyapunov function V (Ψ) (blue line) and control u (green line); initial condition Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2); system defined by[START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback[START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. The feedback (8) fails to reach the target, V stalls at V = 10 -0.1 . We take k = 0.2, c = 0.8.

Figure 3

 3 Figure 3. Time evolution of I 1 and I 2 ; system defined by[START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback[START_REF] Coron | Global asymptotic stabilization for controllable systems without drift[END_REF]. We note that I 1 converges to zero. Contrary to I 1 , I 2 does not converge to zero.

(

  i) when u(I 1 , I 2 ) = -k 2 I 1 /(1 + k 2 I 2 ), i.e. in the region C \ (A ∪ B) and possibly A ∩ C and B ∩ C, we obtain the following constraint on k 2 (cf. also Remark 2.1): k 2 < 1 H 2 ; with this provision dV /dt < 0 in this region. (ii) when u = 0, i.e. in the region B \ C and possibly C ∩ B: dV /dt = 0 (iii) when u(I 1 , I 2 ) = k 1 I 2 , i.e. in the region A \ C and possibly A ∩ C: dV

Theorem 3 . 1 1 C

 311 Consider (3) with Ψ ∈ S 2n-and an eigenstate φ ∈ S 2n-1

Figure 4 .Figure 5 .

 45 Figure 4. Evolution of the Lyapunov function V (Ψ)(blue line) and control u (green line); initial condition: Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2); system defined by (11) with feedback (12)( k 1 = 1.1, k 2 = c = 0.8, δ = 1.e -4.).

Figure 6 .

 6 Figure 6. Evolution of the Lyapunov function V (Ψ)(blue line) and control u (green line); initial condition: Ψ(t = 0) = (0, 1/ √ 4, 1/ √ 4, 1/ √ 4, 1/ √ 4); system defined by (23) with feedback (12) ( k 1 = 1.1, k 2 = c = 0.8, δ = 10 -4 ).

Figure 7 .

 7 Figure 7. Time evolution of I 1 and I 2 ; system defined by (23) with feedback (12); |I 1 | < δ and |I 2 | Cδ.

5 VFigure 8 .

 58 Figure 8. Evolution of the Lyapunov function V (Ψ) (blue line) and control u (green line); initial condition Ψ(t = 0) = (0, 1/ √ 3, 1/ √ 3, 1/ √ 3); system defined by (24) with feedback (12) (k 1 = 1.1, k 2 = c = 0.8, δ = 1.e -4).

Figure 9

 9 Figure 9. Time evolution of I 1 and I 2 ; system defined by[START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback[START_REF] Coron | A relation between continuous time-varying and discontinuous feedback stabilization[END_REF].

Figure 10 .

 10 Figure 10. Evolution of the Lyapunov function V (Ψ)(blue line) and control u (green line); initial condition: Ψ(t = 0) = (0, 1/ √ 2, 1/ √ 2); system defined by[START_REF] Coron | Lyapunov control of Schrödinger equations:beyond the dipole coupling[END_REF] with feedback[START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5.

Figure 11 .

 11 Figure 11. Evolution of the Lyapunov function V (Ψ)(blue line) and control u (green line); initial condition: Ψ(t = 0) = (0, 1/ √ 3, 1/ √ 3, 1/ √ 3); system defined by (23) with feedback[START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5.

Figure 12 . 3

 123 Figure 12. Evolution of the Lyapunov function V (Ψ)(blue line) and control u (green line); initial condition Ψ(t = 0) = (0, 1/ √ 3, 1/ √ 3, 1/ √ 3); system defined by[START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback[START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF]. We take ε = 1.e -3, k = 0.8, c = 0.5.

Figure 13

 13 Figure 13. Time evolution of I 1 and I 2 ; system defined by[START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF] with feedback[START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF].

" (ANR), Projet Blanc C-QUID number BLAN-3-139579. § Financial support from INRIA Rocquencourt projet MicMac and OMQP, from "Agence Nationale de la Recherche" (ANR), Projet Blanc C-QUID number BLAN-3-139579 and from PICS NSF-CNRS. Financial support by Université Paris Dauphine and CNCSIS grant PNII ID-404/2007-2010 acknowledged

trajectory corresponds to an equilibrium.

We consider the following time dependent feedback: u(t, Ψ) = α(Ψ) + β(Ψ) sin(t/ε). [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF] We substitute [START_REF] Phan | Learning control of quantum-mechanical systems by laboratory identification of effective input-output maps[END_REF] in [START_REF] Beauchard | Implicit Lyapunov control of finite dimensional Schrödinger equations[END_REF] and we obtain the system:

The averaged system is given by (see [START_REF] Khalil | Nonlinear Systems[END_REF] pages 402-410):

We identify the coefficients α and β such that the averaged system is asymptotically stable. We use a Lyapunov technique to stabilize the averaged system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] around the ground state φ. We take again the function V defined by (4), which is nonnegative for all Ψ ∈ S 2n-1 and vanishes when Ψ = φ.

The derivative of V along a trajectory of the averaged system ( 26) is given by:

We denote:

When, for instance, we take:

we obtain:

and thus dV /dt ≤ 0, for c > 0 and k < 1 H 2 (cf. Remark 2.1), i.e. V is nonincreasing along the trajectories of the averaged system. In particular φ is a stable point for the averaged system, i.e. such that ∀δ > 0, ∃δ ′ > 0 such that (|Ψ av (0) -φ| < δ ′ ) ⇒ (|Ψ av (t) -φ| < δ, ∀ t ∈ [0, +∞)). [START_REF] Sugawara | General formulation of locally designed coherent control theory for quantum systems[END_REF] We have the following asymptotic stability result: Theorem 3.3 Let us assume that (i) and (ii) of Theorem 3.2 hold. Let V be a neighborhood of -φ and let δ ∈ (0, +∞). Then there exist T > 0 and ǫ 0 > 0 such that, for every τ > 0 and for every solution Ψ of (26) with ǫ ∈ (0, ǫ 0 ) and Ψ(τ ) ∈ S 2n-1 \ V, |Ψ(t) -φ| < δ for every t τ + T.

(39)

Proof of Theorem 3.3. The key ingredient is the following classical lemma (see, e.g., [21, pages 415-417] or [29, Section 3.2]).

Lemma 3.1 Let T > 0. There exists C and ε 0 > 0 such that, for every τ ∈ R and for every ε ∈ (0, ε 0 ), if Ψ : [τ, τ + T ] → S 2n-1 is a solution of (26) and Ψ av is the solution of the averaged system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF] such that Ψ av (τ ) = Ψ(τ ), then

Let δ 1 > 0 be such that

By [START_REF] Sugawara | General formulation of locally designed coherent control theory for quantum systems[END_REF], there exists δ 2 > 0 such that, for every solution Ψ av of the averaged system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF],

By Theorem 3.2, there exists T > 0 such that, for every solution Ψ av of the averaged system [START_REF] Rabitz | Quantum control design via adaptive tracking[END_REF],

By Lemma 3.1 and (42), there exists ǫ 1 > 0 such that, for every ε ∈ (0, ε 1 ), for every τ ∈ R and for every solution Ψ of ( 26),

By Lemma 3.1 and (41), there exists ε 2 > 0 such that, for every ε ∈ (0, ε 2 ), for every τ ′ ∈ R and for every solution Ψ of ( 26),

Let us check that the conclusion of Theorem 3.3 holds with ε 0 = min{ε 1 , ε 2 }. Let ε ∈ (0, min{ε 1 , ε 2 }), let τ > 0 and let Ψ be a solution of [START_REF] Prieur | Asymptotic controllability and robust asymptotic stabilizability[END_REF] such that Ψ(τ ) ∈ S 2n-1 \ V.