Controllability of the bilinear Schrödinger 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 Schrödinger 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.
Rotational molecular dynamics is one of the most important examples of quantum systems with an infinite-dimensional 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 [seideman, stapelfeldt] 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 . This control problem corresponds to the control of the Schrödinger equation on the unit sphere . 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 [sugny].
I-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 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 [stapelfeldt]).
Up to normalization of physical constants (in particular, in units such that ), the dynamics is ruled by the equation
where are the spherical coordinates, which are related to the Euclidean coordinates by the identities
while is the Laplace–Beltrami operator on the sphere (called in this context the angular momentum operator), i.e.,
The wavefunction evolves in the unit sphere of .
I-B The main results
In the following we denote by the solution at time of equation (1), corresponding to control and with initial condition , belonging to .
Our main result says that (1) is approximately controllable with arbitrarily small controls.
For every , belonging to and every , there exist and such that .
The proof of the result is based on arguments inspired by those developed in [noi, ancoranoi]. 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 , , . Secondly, the general theory developed in [noi, ancoranoi] is based on nonresonance conditions on the spectrum of the drift Schrödinger operator (the internal Hamiltonian). The Laplace–Belatrami operator on , however, has a severely degenerate spectrum. It is known, indeed, that the -th eigenvalue has multiplicity . In [noi] we proposed a perturbation technique in order to overcome resonance relations in the spectrum of the drift. This technique was applied in [noicdc] 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 multi-input 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
Let be an infinite-dimensional Hilbert space with scalar product and be (possibly unbounded) linear operators on , with domains . Let be a subset of . Let us introduce the controlled equation
We say that satisfies if the following assumptions are verified:
is an Hilbert basis of made of eigenvectors of associated with the family of eigenvalues ;
for every ;
is essentially skew-adjoint for every ;
if and then for every .
If satisfies then, for every , generates a unitary group . It is therefore possible to define the propagator at time of system (1) associated with a -uple of piecewise constant controls by concatenation. If, moreover, the potentials are bounded operators then the definition can be extended by continuity to every control law.
Let satisfy . We say that (2) is approximately controllable if for every in the unit sphere of and every there exist a piecewise constant control function such that
Let satisfy . We say that (2) is approximately simultaneously controllable if for every in , in , in , and there exists a piecewise constant control such that
Ii-a Short review of controllability results
The controllability of system (2) is a well-established topic when the state space is finite-dimensional (see for instance [dalessandro-book] and reference therein), thanks to general controllability methods for left-invariant control systems on compact Lie groups ([brock, jur]).
When is infinite-dimensional, it is known that the bilinear Schrödinger equation is not controllable (see [BMS, turinici]). 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 [Beauchard1, beauchard-coron, camillo]. 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, [beauchard-nersesyan, ancoranoi, noi, mirrahimi-aihp, Nersy, fratelli-nersesyan, nersesyan]), except for some approximate controllability result for specific systems ([ervedoza_puel]) and some general approximate controllability result between eigenfunctions based on adiabatic methods [adiabatiko].
Set , . For every in , define the orthogonal projection
Given a linear operator on we identify the linear operator preserving with its complex matrix representation with respect to the basis .
Iii Main abstract controllability result in the multi-input case
Let us introduce the set of spectral gaps associated with the -dimensional Galerkin approximation as
For every , let
The matrix corresponds to the choice of the controls and to the “activation” of the spectral gap . Define
The set represents “compatible dynamics” for the -dimensional Galerkin approximation (compatible, that is, with higher dimensional Galerkin approximations).
Theorem III.1 (Abstract multi-input controllability result)
Let for some . If for every there exist such that
then the system
is approximately simultaneously controllable.
The following technical result, which we shall use in the proof of Theorem III.1, has been proved in [ancoranoi].
Let be a positive integer and be such that for Let
Then, for every , we have
where Moreover, for every and there exists a sequence such that and
Iii-B Time reparametrization
For every piecewise constant function such that , for every , and with , we consider the system
System (4) can be seen as a time-reparametrisation of system (2). Let be the solution of (2) with initial condition associated with the piecewise constant control with components , . If , , for every , , then the solution of (4) with the initial condition associated with the controls satisfies
Controllability issues for system (2) and (4) are equivalent. Indeed, consider piecewise constant controls , and , with , achieving controllability (steering system (4) from to , in a time ) . Then the controls , defined by and , steer system (2) from to , in a time .
Iii-C Interaction framework
Let , and for . Let be the solution of (4) with initial condition associated with the controls and set
For set , then satisfies
Notice that for every and for every -uple of piecewise constant controls , .
Iii-D Galerkin approximation
Let . The Galerkin approximation of (5) of order is the system in
Iii-E First step: choice of the order of the Galerkin approximation
In order to prove approximate simultaneous controllability, we should take in , in , in , and and prove the existence of a piecewise constant control such that
Notice that for large enough there exists such that
for every and . This simple fact suggest to prove approximate simultaneous controllability by studying the controllability of (6) in the Lie group .
Iii-F Second step: control in
Let satisfy hypothesis (3). It follows from standard controllability results on compact Lie groups (see [jur]) that for every there exists a path such that
where the chronological notation is used for the flow from time to of the time-varying equation (see [book2]). More precisely, there exists a finite partition in intervals of such that for every either there exist and such that
Iii-G Third step: control of
For every , , and for every piecewise constant and there exists a sequence of piecewise constant functions from to , such that
uniformly with respect to as tends to infinity.
In other words, every piecewise constant path in can be approximately tracked by system (6).
Proof. Fix . We are going to construct the control by applying recursively Lemma III.2. Consider an interval in which , , and are constantly equal to , and respectively. Apply Lemma III.2 with , , and to be fixed later depending on . Then, for every , there exist and a sequence such that , , and such that
Set , , and define the piecewise constant function
Note that by choosing for and we have that is non-decreasing.
Following the smoothing procedure of [ancoranoi, Proposition 5.5] one can construct the desired sequence of control . The idea is to approximate by suitable piecewise linear functions with slope greater than . Then can be constructed from the derivatives of these functions.
As a consequence of last proposition by [book2, Lemma 8.2] we have that
uniformly with respect to as tends to infinity.
Iii-H Fourth step: control of the infinite-dimensional system
Next proposition states that, roughly speaking, we can pass to the limit as tends to infinity without losing the controllability property proved for the finite-dimensional case. Its proof can be found in [ancoranoi, 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.
For every , for every , and for every trajectory there exist piecewise constant controls such that the associated propagator of (2) satisfies
for every with and for every in .
We recall now a controllability result for the phases (see [ancoranoi, 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 III.1.
Iv D molecule
Let us go back to the system presented in the introduction for the orientation of a linear molecule,
A basis of eigenvectors of the Laplace–Beltrami operator is given by the spherical harmonics , which sastisfy
We are first going to prove that for every the system projected on the -dimensional linear space
is controllable. More precisely, chosen a reordering of the spherical harmonics in such a way that
we are going to prove that
Iv-a Matrix representations
Denote by the set of integer pairs . Consider an ordering . Let be the -square matrix whose entries are all zero, but the one at line and column which is equal to . Define
By a slight abuse of language, also set . The analogous identification can be used to define .
Thanks to this notation we can conveniently represent the matrices corresponding to the controlled vector field (projected on ). A computation shows that the control potential in the direction, , projected on , has a matrix representation with respect to the chosen basis
Similarly, we associate with the control potentials in the and directions, and respectively, the matrix representations
The matrix representation of the Schrödinger operator is the diagonal matrix
Now consider , in such a way that . Hence, where
Iv-B Useful bracket relations
From the identity
we get the relations , , and and
The relations above can be interpreted following a “triangle rule”: the bracket between an operator coupling the states and and an operator coupling the states and couples the states and . On the other hand, the bracket is zero if two operators couple no common states.
From (10) we find also that
if and only if or , with
Iv-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 I.1 as a corollary of Theorem III.1. Notice that the conclusions of Theorem III.1 allow us to claim more than the required approximately controllability, since simultaneous controllability is obtained as well.
The Lie algebra generated by is the whole algebra .
Thanks to the matrix relations obtained in Section IV-B, the proof of the proposition can be easily reduced to the proof of the following lemma.
The Lie algebra contains the elementary matrices
Proof of Lemma IV.2. First, we want to prove that
We use the fact that
and, by induction, for ,