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Controllability of the bilinear Schr\"odinger equation with several controls and application to a 3D molecule

We show the approximate rotational controllability of a polar linear molecule by means of three nonresonant linear polarized laser fields. The result is based on a general approximate controllability result for the bilinear Schr\"odinger equation, with wavefunction varying in the unit sphere of an infinite-dimensional Hilbert space and with several control potentials, under the assumption that the internal Hamiltonian has discrete spectrum.


I. INTRODUCTION
Rotational molecular dynamics is one of the most important examples of quantum systems with an infinitedimensional Hilbert space and a discrete spectrum. Molecular orientation and alignment are well-established topics in the quantum control of molecular dynamics both from the experimental and theoretical points of view (see [19], [20] and references therein). For linear molecules driven by linearly polarized laser fields in gas phase, alignment means an increased probability direction along the polarization axis whereas orientation requires in addition the same (or opposite) direction as the polarization vector. Such controls have a variety of applications extending from chemical reaction dynamics to surface processing, catalysis and nanoscale design. A large amount of numerical simulations have been done in this domain but the mathematical part is not yet fully understood. From this perspective, the controllability problem is a necessary step towards comprehension.
We focus in this paper on the control by laser fields of the rotation of a rigid linear molecule in R 3 . This control problem corresponds to the control of the Schrödinger equation on the unit sphere S 2 . We show that the system driven by three fields along the three axes is approximately controllable for arbitrarily small controls. This means, in particular, that there exist control strategies which bring the initial state arbitrarily close to states maximizing the molecular orientation [21].

A. The model
We consider a polar linear molecule in its ground vibronic state subject to three nonresonant (with respect to the vibronic frequencies) linearly polarized laser fields. The control is given by the electric fields E = (u 1 , u 2 , u 3 ) depending on time and constant in space. We neglect in this model the polarizability tensor term which corresponds to the field-induced dipole moment. This approximation is correct if the intensity of the laser field is sufficiently weak. Despite its simplicity, this equation reproduces very well the experimental data on the rotational dynamics of rigid molecules (see [20]).
Up to normalization of physical constants (in particular, in units such that = 1), the dynamics is ruled by the equation where θ, ϕ are the spherical coordinates, which are related to the Euclidean coordinates by the identities x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, while ∆ is the Laplace-Beltrami operator on the sphere (called in this context the angular momentum operator), i.e., The wavefunction ψ(·, ·, t) evolves in the unit sphere S of H = L 2 (S 2 , C).

B. The main results
In the following we denote by ψ(T ; ψ 0 , u) the solution at time T of equation (1), corresponding to control u and with initial condition ψ(0; ψ 0 , u) = ψ 0 , belonging to S.
Our main result says that (1) is approximately controllable with arbitrarily small controls. Theorem 1.1: For every ψ 0 , ψ 1 belonging to S and every ε, The proof of the result is based on arguments inspired by those developed in [11], [7]. There are two main difficulties preventing us to apply those results to the case under consideration: firstly, we deal here with several control parameters, while those general results were specifically conceived for the single-input case. Notice that, because of symmetry obstructions, equation (1) is not controllable with only one of the three controls u 1 , u 2 , u 3 . Secondly, the general theory developed in [11], [7] is based on nonresonance conditions on the spectrum of the drift Schrödinger operator (the internal Hamiltonian). The Laplace-Belatrami operator on S 2 , however, has a severely degenerate spectrum. It is known, indeed, that the ℓ-th eigenvalue −iℓ(ℓ + 1) has multiplicity 2ℓ + 1. In [11] we proposed a perturbation technique in order to overcome resonance relations in the spectrum of the drift. This technique was applied in [8] to the case of the orientation of a molecule confined in a plane driven by one control. The planar case is already technically challenging and a generalization to the case of three controls in the space will hardly provide an apophantic proof of the approximate controllability result. We therefore provide a general multiinput result which can be applied to the control problem defined in (1), up to the computation of certain Lie algebras associated with its Galerkin approximations.
The structure of the paper is the following: in the next section we present the general multi-input abstract framework and we recall some previously known controllability and non-controllability results. In Section III we prove our main sufficient condition for approximate controllability. Finally, in Section IV we prove that the abstract result applies to system (1).

II. ABSTRACT FRAMEWORK
is essentially skew-adjoint for every u ∈ U ; (A4) if j = k and λ j = λ k then φ j , B l φ k = 0 for every l = 1, . . . , p. If (A, B 1 , . . . , B p , U, Φ) satisfies (A) then, for every (u 1 , . . . , u p ) ∈ R p , A + u 1 B 1 + · · · + u p B p generates a unitary group e t(A+u1B1+···+upBp) . It is therefore possible to define the propagator Υ u T at time T of system (1) associated with a p-uple of piecewise constant controls u(t) = (u 1 (t), . . . , u p (t)) by concatenation. If, moreover, the potentials B 1 , . . . , B p are bounded operators then the definition can be extended by continuity to every L ∞ control law.
Definition 2.2: Let (A, B, U, Φ) satisfy (A). We say that (2) is approximately controllable if for every ψ 0 , ψ 1 in the unit sphere of H and every ε > 0 there exist a piecewise constant control function u : . We say that (2) is approximately simultaneously controllable if for every r in N, ψ 1 , . . . , ψ r in H,Υ in U(H), and ε > 0 there exists a piecewise constant control u :

A. Short review of controllability results
The controllability of system (2) is a well-established topic when the state space H is finite-dimensional (see for instance [12] and reference therein), thanks to general controllability methods for left-invariant control systems on compact Lie groups ( [10], [14]).
When H is infinite-dimensional, it is known that the bilinear Schrödinger equation is not controllable (see [2], [22]). Hence, one has to look for weaker controllability properties as, for instance, approximate controllability or controllability between eigenstates of the Schödinger operator (which are the most relevant physical states). In certain cases where the dimension of the domain where the controlled PDE is defined is equal to one a description of the reachable set has been provided [3], [4], [5]. For dimension larger than one or for more general situations, the exact description of the reachable set appears to be more difficult and at the moment only approximate controllability results are available. Most of them are for the single-input case (see, in particular, [6], [7], [11], [15], [16], [18], [17]), except for some approximate controllability result for specific systems ( [13]) and some general approximate controllability result between eigenfunctions based on adiabatic methods [9].

B. Notation
Set b (l) jk = φ j , B l φ k , l = 1, . . . , p. For every n in N, define the orthogonal projection Given a linear operator Q on H we identify the linear operator π n Qπ n preserving span{φ 1 , . . . , φ n } with its n × n complex matrix representation with respect to the basis (φ 1 , . . . , φ n ).

MULTI-INPUT CASE
Let us introduce the set Σ N of spectral gaps associated with the N -dimensional Galerkin approximation as For every σ ∈ Σ N , let . . , v p ) corresponds to the choice of the controls v 1 , . . . , v p and to the "activation" of the spectral gap σ. Define The set M n 0 represents "compatible dynamics" for the ndimensional Galerkin approximation (compatible, that is, with higher dimensional Galerkin approximations).
Theorem 3.1 (Abstract multi-input controllability result): Let U = [0, δ] p for some δ > 0. If for every n 0 ∈ N there exist n > n 0 such that then the systeṁ is approximately simultaneously controllable.

A. Preliminaries
The following technical result, which we shall use in the proof of Theorem 3.1, has been proved in [7].
Lemma 3.2: Let κ be a positive integer and Then, for every τ 0 ∈ R, we have Moreover, for every R > 0 and ξ ∈ S 1 there exists a sequence (t k ) k∈N such that t k+1 − t k > R and

D. Galerkin approximation Definition 3.3:
Let N ∈ N. The Galerkin approximation of (5) of order N is the system in Ḣ

E. First step: choice of the order of the Galerkin approximation
In order to prove approximate simultaneous controllability, we should take r in N, ψ 1 , . . . , ψ r in H,Υ in U(H), and ε > 0 and prove the existence of a piecewise constant control u : [0, T ] → U such that Υ ψ k − Υ u T ψ k < ε, k = 1, . . . , r.
Notice that for n 0 large enough there exists U ∈ SU (n 0 ) such that | φ j ,Υψ k − π n0 φ j , U π n0 ψ k | < ε for every 1 ≤ k ≤ r and j ∈ N. This simple fact suggest to prove approximate simultaneous controllability by studying the controllability of (6) in the Lie group SU (n 0 ).

F. Second step: control in SU (n)
Let n ≥ n 0 satisfy hypothesis (3). It follows from standard controllability results on compact Lie groups (see [14]) that for every U ∈ SU (n) there exists a path M : [0, where the chronological notation −→ exp t 0 V s ds is used for the flow from time 0 to t of the time-varying equationq = V s (q) (see [1]). More precisely, there exists a finite partition in intervals (I k ) k of [0, T v ] such that for every t ∈ I k either there exist v 1 , . . . , v p ∈ [0, 1] and σ ∈ Σ N such that In particular, G. Third step: control of M N Lemma 3.4: For every N ∈ N, δ > 0, and for ev- v 1 (s), . . . , v p (s))ds → 0 uniformly with respect to t ∈ [0, T v ] as h tends to infinity.
Following the smoothing procedure of [7, Proposition 5.5] one can construct the desired sequence of control z h . The idea is to approximate ω h (t) by suitable piecewise linear functions with slope greater than 1/δ. Then z h can be constructed from the derivatives of these functions.
As a consequence of last proposition by [1,Lemma 8 uniformly with respect to t ∈ [0, T v ] as h tends to infinity.

H. Fourth step: control of the infinite-dimensional system
Next proposition states that, roughly speaking, we can pass to the limit as N tends to infinity without losing the controllability property proved for the finite-dimensional case. Its proof can be found in [7,Proposition 5.6]. It is based on the particular form (7) of the operators involved, since the fact that the operator has several zero elements guarantees that the difference between the dynamics of the infinite-dimensional system and the dynamics of the Galerkin approximations is small. Proposition 3.5: For every ε > 0, for every δ > 0, and for every trajectory U ∈ SU (n) there exist piecewise constant controls u j : [0, T u ] → [0, δ], j = 1, . . . , p such that the associated propagator Υ u of (2) satisfies | π n φ j , U π n φ | − | φ j , Υ u Tu φ | < ε for every φ ∈ span{φ 1 , . . . , φ n } with φ = 1 and for every j in N.
We recall now a controllability result for the phases (see [7, Proposition 6.1 and Remark 6.3]). This property, stated in the proposition below, together with the controllability up to phases proved in the previous section, is sufficient to conclude the proof of Theorem 3.1.
Thanks to this notation we can conveniently represent the matrices corresponding to the controlled vector field (projected on L). A computation shows that the control potential in the z direction, −i cos θ, projected on L, has a matrix representation with respect to the chosen basis .

B. Useful bracket relations
From the identity [e j,k , e n,m ] = δ kn e j,m − δ jm e n,k we get the relations [E j,k , E k,n ] = E j,n , [F j,k , F k,n ] = −E j,n , and [E j,k , F k,n ] = F j,n and The relations above can be interpreted following a "triangle rule": the bracket between an operator coupling the states Y m ℓ and Y n k and an operator coupling the states Y m ℓ and Y n ′ k ′ couples the states Y n k and Y n ′ k ′ . On the other hand, the bracket is zero if two operators couple no common states. Moreover,

C. Controllability result
We prove the following result, which allows us to apply the abstract controllability criterium obtained in the previsous section. We obtain then Theorem 1.1 as a corollary of Theorem 3.1. Notice that the conclusions of Theorem 3.1 allow us to claim more than the required approximately controllability, since simultaneous controllability is obtained as well.
Proposition 4.1: The Lie algebra L generated by A, B 1 , B 2 , B 3 is the whole algebra su(4ℓ + 4).