Periodic excitations of bilinear quantum systels - Semantic Scholar
arXiv:1103.1130v2 [math.OC] 31 Mar 2011
Periodic excitations of a bilinear quantum system Thomas Chambrion∗ April 1, 2011
Abstract A well known method to transfer the population of a quantum system from an eigenspace of the free Hamiltonian to another is to use a periodic control law with angular frequency equal to the difference of the eigenvalues. This paper gives a theoretical proof of this experimental result. We introduce a notion of efficiency and demonstrate its interest for the design of controls on the example of the rotation of a planar molecule.
Contents 1 Introduction 1.1 Effective control of quantum systems 1.2 Framework and notations . . . . . . 1.3 Main result . . . . . . . . . . . . . . 1.4 Content of the paper . . . . . . . . .
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2 Proof of the convergence result 2.1 Good Galerkyn approximation 2.2 Time reparametrization . . . . 2.3 Proof of Proposition 3 . . . . . 2.4 Efficiency of the transfer . . . . 2.5 Multiple resonant transitions .
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3 Rotation of a planar molecule 12 3.1 Presentation of the model . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Uniquely resonant case . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Not uniquely resonant case . . . . . . . . . . . . . . . . . . . . . 14 4 Conclusion
15
∗ The
author is with IECN UMR 7502, Nancy University, CNRS, INRIA, BP 70239, 54506 Vandœuvre-l` es-Nancy, France and INRIA Nancy Grand Est, projet CORIDA. [email protected]
1
1 1.1
Introduction Effective control of quantum systems
The state of a quantum system evolving on a finite dimensional Riemannian manifold Ω, with associated measure µ, is described by its wave function, that is, a point in the unit sphere of L2 (Ω,ZC). A system with wave function ψ is in a subset ω of Ω with the probability
|ψ|2 dµ.
ω
In the absence of interaction with the environment, the time evolution of the wave function is given by the Schr¨odinger equation i
1 ∂ψ = − ∆ψ + V (x)ψ(x, t), ∂t 2
where ∆ is the Laplace-Beltrami operator on Ω (with suitable boundary conditions) and V : Ω → R is a real function (usually called potential) accounting for the physical properties of the system. When submitted to an excitation by an external electric field (e.g. a laser), the Schr¨odinger equation reads i
∂ψ 1 = − ∆ψ + V (x)ψ(x, t) + u(t)W (x)ψ(x, t), ∂t 2
(1)
where W : Ω → R is a real function accounting for the physical properties of the laser and u is a real function of the time accounting for the intensity of the laser. A natural question, with many practical implications, is whether there exists a control u that steers the quantum system from a given initial wave function to a given target wave function (controllability issue) and, more important, how to build this control law (effective design of controls). Considerable efforts have been made by different communities to study the controllability of (1). We refer to [BCCS11, BL10, Ner10, Mir09] and references therein for a description of the known theoretical results concerning the existence of controls steering a given source to a given target. As proved in [Ner10, MS10, PS10], approximate controllability is a generic property for systems of the type of (1). Effective control algorithms have been obtained [WRD93, OKF98, BST08]. Most of the controls used in practice exhibit a remarkable pattern of periodic shape, with frequency corresponding to a resonance in the quantum system. They appear to work remarkably well, with little influence of the shape [BFS86]. However, to our knowledge, no theoretical proof of this effectiveness is available. The aim of the present paper is to provide a mathematically rigorous explanation of the surprising robustness of these control algorithms.
1.2
Framework and notations
We first reformulate the problem (1) in a more abstract framework. In a separable Hilbert space H endowed with norm k · k and Hilbert product h·, ·i, we 2
consider the following evolution problem: dψ = (A + uB)ψ dt
(2)
where (A, B) satisfies Assumption 1. Assumption 1. (A, B) is a pair of linear operators such that 1. A is skew adjoint with purely discrete spectrum (iλn )n∈N ; 2. the sequence (λn )n∈N takes value in (0, +∞), is non-decreasing and its only accumulation point is +∞; 3. there exists an Hilbert basis (φk )k∈N of H such that Aφk = λk φk for every k in N; 4. for every ψ in D(A), ψ belongs to D(B) and there exists sA,B < 1/2 such that kBψk ≤ k(iA)sA,B ψk; 5. for every u in R, A+uB is skew-adjoint, D(A+uB) = D(A) and D((A+ uB)2 ) = D(A2 ); 6. For every interval I containing 0, for every Radon measure u on I, t 7→ A(t) := eu([0,t))B Ae−u([0,t))B is a family of skew-adjoint operators with common domain D = D(A) and A is continuous with bounded variation from I to B(D, H); 7. For every interval I containing 0, for every Radon measure u on I, A :
t 7→ eu([0,t))B Ae−u([0,t))B is such that supt∈I A(t)−1 B(H,D(A)) < +∞ ; 8. there exists CA,B > 0 such that |=hAψ, Bψi| ≤ CA,B |hAψ, ψi| for every ψ in D(A). The Assumption 1.5 ensures that, for every constant u in R, A+uB generates a group of unitary propagators. Hence, for every initial condition ψ0 in H, for every piecewise constant control u, we can define the solution of (2) that we will note t 7→ Υut ψ0 . The map u 7→ ΥuT ψ0 can be extended to the closure of the set of piecewise constant functions for any norm such that u 7→ ΥuT ψ0 is locally uniformly continuous. Proposition 2 (Well-posedness). For every T in R+ , for every ψ0 in H, the map u 7→ ΥuT ψ0 admits a unique continuous extension to the set of Radon measures on [0, T ]. Moreover, the mapping Υ : (T, ψ0 , u) 7→ ΥuT ψ0 is continuous for the product of the standard distance of R, the Hilbert norm of H and the total variation in the set of Radon measures on R+ . The result of Proposotion 2 is classical under Assumption 1 for controls with bounded variations (see [Kat95]). It has recently been extended (see [BCC11]) to the set of Radon measures endowed with the distance of total variation. This framework includes L1loc functions and countable sums of Dirac masses. 3
1.3
Main result
Definition 1. A point (j, k) of N2 is said to be a uniquely resonant transition of (A, B) if (i) j 6= k, (ii) hφj , Bφk i = 6 0 and (iii) for every l, m, p in N, p|λj − λk | = |λl − λm | implies {j, k} = {l, m} and p = 1, or hφl , Bφm i = 0. A point (j, k) of N2 is said to be a multiple resonant transition of (A, B) if (i) j 6= k, (ii) hφj , Bφk i 6= 0 and (iii) there exist p, l, m ∈ N such that {l, m} 6= {j, k}, hφl , Bφm i = 6 0 and p|λj − λk | = |λl − λm |. Proposition 3. Let u∗ : R+ → R be a locally integrable function. Assume that u∗ is periodic with smallest period T = |λj2π −λk | for some uniquely resonant transition (j, k) of (A, B). If Z
T
u∗ (τ )ei(λj −λk )τ dτ 6= 0,
0
u∗ n then there exists T > 0 such that the sequence hφk , ΥnT ∗ (φj )i tends n∈N to 1 as n tends to infinity. ∗
This results provides a rigorous formulation of a well-known fact: to induce a transition between levels j and k of a quantum system, one can use a periodic excitation of frequency exactly equal to the difference of the corresponding eigenvalues. For almost every shape of the control, the trajectory eventually reaches any neighborhood of the target, provided the control is small enough and has the correct frequency. As it is well known from experiments, the situation is more intricate with multiple resonant transitions.
1.4
Content of the paper
The paper splits in two parts. The first one is a theoretical proof of Proposition 3. Some technical tools are introduced in Section 2.1. A time reparametrization (Section 2.2) allows to prove Proposition 3 (Section 2.3). In a second part, the theoretical results are tested on numerical simulations of the rotation of a planar molecule.
2
Proof of the convergence result
The strategy of the proof is inspired by [CMSB09, BCCS11] and relies upon the approximation of the original infinite dimensional system by its finite dimensional approximations.
2.1
Good Galerkyn approximation
In this Section, we explain how to construct a good Galerkyn approximation of the original system. The term “good” refers to the fact that the error made when replacing the original system by its Galerkyn approximation is bounded 4
uniformly with respect to the control u. What follows is a very simplified version of a much more general construction (valid also for operators with continuous spectrum) presented in [BCC11]. Under Assumption 1.2, −iA is a self-adjoint, bounded from below operator. For every α in R, the translation from A to A + αiIdH has no effect on the dynamics of (2) but a physically irrelevant shift in the phases. Up to a suitable translation, one may assume that iA is a self-adjoint operator of H with positive P eigenvalues. For every ψ in D(A), −iAψ = j∈N λP j hφj , ψiφj . For every s > 0, the linear operator (−iA)s is defined by (−iA)s ψ = j∈N λsj hφj , ψiφj , for every ψ in D((−iA)s ). Proposition 4. For every ψ0 ∈ D(A2 ) and K > 0, there exists CK such that for every T ≥ 0 and for every control u for which kukL1 < K, one has |hAΥuT (ψ0 ), ΥuT (ψ0 )i| < CK . Proof. Let x(t) be a solution of x˙ = (A + u(t)B) x with initial condition ψ0 . By Assumption 1.5, x(t) belongs to D(A2 ) for every t. Consider the real mapping f : t 7→ |hAx(t), x(t)i|. Since x(0) belongs to D(A), f is differentiable and d d f (t) = i hAx(t), x(t)i dt dt = ihAx(t), (A + u(t)B)x(t)i + ihA(A + u(t)B)x(t), x(t)i = 2i2 =hAx(t), (A + u(t)B)x(t)i = −2=hAx(t), u(t)Bx(t)i, so that, thanks to Assumption 1.4, |f 0 (t)| ≤ 2|u(t)||hAx(t), Bx(t)i| ≤ CA,B |u(t)|f (t). Gronwall lemma implies that |hAx(t), x(t)i| ≤ eCA,B
Rt
The result follows by taking CK = eCA,B
Rt
0
|u|(τ )dτ
0
|hAψ0 , ψ0 i|.
|u|(τ )dτ
(3)
|hAψ0 , ψ0 i|.
For every N in N, we define the orthogonal projection πN
:H ψ
→ 7 →
H P
j≤N hφj , ψiφj
Proposition 5. For every > 0, K ≥ 0, n ∈ N, there exists N ∈ N such that for every control u, for every 0 ≤ s < 1, kukL1 ≤ K =⇒ |h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj )i| <
(4)
for every t ≥ 0, and j = 1, . . . , n. As a consequence, there exists N ∈ N such that for every control u, kukL1 ≤ K =⇒ kB(Id − πN )Υut (φj )k < , for every t ≥ 0, and j = 1, . . . , n. 5
(5)
Proof. Fix j ∈ {1, . . . , n}. For every N > 1, one has ∞ X
|h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj i| =
λsn |hφn , Υut (φj )i|2
n=N +1 ∞ X
≤
λs−1 N
≤
u u λs−1 N |hAΥt (φj ), Υt (φj )i|.
λn |hφn , Υut (φj )i|2
n=N +1
By Proposition 4, there exists CK such that |hAΥut (φj ), Υut (φj )i| ≤ CK for every t in R+ and u in the K ball of L1 . Since s < 1 then |h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj )i| tends to 0, uniformly with respect to u, as N tends to infinity. By Assumption 1.4, there exists sA,B in (0, 1/2) such that p kBxk ≤ k(iA)sA,B xk = |h(iA)2sA,B x, xi| for every x ∈ D(A) and (5) is a direct consequence of (4) applied with s ∈ (2sA,B , 1). Definition 2. Let N ∈ N. Denote LN = span(φ1 , . . . , φN ). The Galerkyn approximates of A and B of order N are the operators A(N ) : H → H and B (N ) : H → H defined by A(N ) = πN ALN
and
B (N ) = πN BLN .
We define the system (ΣN ) as x˙ = (A(N ) + uB (N ) )x (N ),u
and call Xt
(ΣN )
the propagator of (ΣN ).
With an obvious abuse of notation, we will sometimes identify the operators A(N ) and B (N ) with their restrictions to the invariant space span1≤l≤N {φl } and with their matrices in the basis (φl )1≤l≤N . Entries of B (N ) are denoted (bl1 ,l2 = hφl1 , Bφl2 i)l1 ,l2 . With these identifications, (ΣN ) turns into a finite dimensional system in CN . Proposition 6 (Good Galerkin Approximation). For every > 0, K ≥ 0, n ∈ N, there exists N ∈ N such that for every u ∈ L1 (0, ∞) (N ),u
kukL1 ≤ K =⇒ kΥut (φj ) − Xt
φj k < ,
(6)
for every t ≥ 0 and i = 1, . . . , n. Proof. Let > 0, K ≥ 0, n ∈ N be given. By Proposition 5, there exists N ∈ N such that, for every u ∈ L1 (0, ∞) for which kukL1 ≤ K, we have kB(Id − πN )Υut (φj )k < 6
, K
for every t ≥ 0 and j = 1, . . . , n. Fix j in 1, . . . , n and consider yj : t 7→ πN Υut (φj ). The mapping t 7→ yj (t) is absolutely continuous and, for almost every t ≥ 0, y˙ j (t) = (A(N ) + uB (N ) )yj (t) + u(t)πN B(Id − πN )Υut (φj ). Hence, for every t ≥ 0, (N ),u
yj (t) = Xt
Z φj +
t
(N ),u
Xt−s πN B(Id − πN )Υus (φj )u(τ )dτ
(7)
0
The norm of t 7→ B(Id − πN )Υut (φj ) is less than or equal to /K for every t ≥ 0 (N ),u (N ),u and, since Xt is unitary, kyj (t) − Xt (φj )k ≤ kukL1 /K ≤ . .
2.2
Time reparametrization
We note P C the set of the piecewise constant functions for which there exist two sequences (uj )1≤j≤p and (tj )1≤j≤p with value in (0, +∞) such that X u= uj χ[τj ,τj +tj ) , 1≤j≤p−1
where χ is the characteristic function and the sequence (τj ) is defined by induction: τ1 = 0, τj+1 = τj + tj . An element u of P C will be denoted (uj , tj )1≤j≤p . The involutive mapping P : PC (uj , tj )1≤j≤p
→ 7→
PC
1 uj , uj tj
1≤j≤p
performs a time reparametrization of the system (2). Indeed, introduce the control system dψ = (u(t)A + B)ψ(t), (8) dt ˇ ut ψ0 . whose solution with initial condition ψ0 will be denoted by t 7→ Υ ˇ Pu Rt Proposition 7. For every ψ0 in H, for every control u in P C, Υ ψ = u(τ )dτ 0 0 u Υt ψ0 . Proof. This follows from the equality et(A+uB) = etu( u A+B ) , valid on every interval where u is constant. 1
7
2.3
Proof of Proposition 3
Let > 0, u∗ a non vanishing piecewise constant function, T -periodic, j < k two integers be given asR in the hypothesis of Proposition 3. t The function t 7→ 0 |u∗ (τ )|dτ is non-decreasing. We denote with v ∗ its RT reciprocal function. We define also I = 0 |u∗ (τ )|dτ or, equivalently, v ∗ (I) = T . RT If 0 u∗ (τ ) exp [i(λj − λk )τ ] dτ 6= 0, let us introduce K=
πI R . T∗ i(λ 2 0 u (τ )e j −λk )τ dτ |bj,k |
By Proposition 6, there exists N in N such that for every l < k, for every u, for every t in R, (N ),u
kukL1 < K =⇒ kΥut (φl ) − Xt
φl k < ,
where X (N ),u is the propagator associated with the control system (ΣN ). We introduce the sequence un = n1 u∗ . For every n in N, un is a nonR nT RT vanishing T -periodic function and 0 |un (τ )|dτ = 0 |u∗ (τ )|dτ . For every n, for every t, P|un |(t) = nP|u∗ | (nt) . In particular, P|un | is a
I n
periodic function and
I n
Z
Z
P|un |(τ )dτ = 0
I
P|u∗ |(τ )dτ = T.
0
The primitive vn of P|un | taking value 0 at 0, Z vn : t 7→
t
P|un |(τ )dτ = v ∗ (nt),
0 +
satisfies, for every t in R , vn
I t+ n
Z
I/n
P|un |(τ )dτ = vn (t) + T.
= vn (t) + 0
Equivalently, one can define vn as the reciprocal of the increasing function t 7→ Rt |u (τ )|dτ . n 0 Let us note t 7→ xn (t) the solution of x˙ = (A(N ) + un (t)B (N ) )x(t) with initial condition φj . The set [0, T ] can be written as a finite union of disjoint intervals [0, T ] = J1+ ∪ J1− ∪ . . . ∪ Jp+ ∪ Jp− , 8
(9)
such that u∗ takes positive (resp. negative) values on J + = ∪pl=1 Jl+ (resp. J − = ∪pl=1 Jl− ). For t in J − , (ΣN ) writes dx = (A(N ) + (−un (t))(−B (N ) ))x(t). dt We apply the P reparametrization to the positive function |u∗ | separately on [−1] every intervals in J + and J − . Defining the sets G+ (J + ) = {l ∈ n = (vn ) Rs [−1] + + − − R |∃s ∈ J , 0 |un (τ )|dτ = l} and Gn = (vn ) (J ) = {l ∈ R+ |∃s ∈ R s J − , 0 |un (τ )|dτ = l}, we obtain the dynamics of yn = xn ◦ vn :
dyn = dt
(P(|un |)(t)A(N ) + B (N ) )yn (t) (P(|un |)(t)A(N ) − B (N ) )yn (t)
for almost every t ∈ G+ , for almost every t ∈ G−
(10)
For every t, we define zn (t) as Z t zn (t) = exp − P|un |(τ )dτ A(N ) yn (t) = e−vn (t)A yn (t). 0
Notice that, for every t, for every l in N, |hφl , zn (t)i| = |hφl , yn (t)i|. From (10), one deduces the dynamics of zn , valid for almost every t in − G+ n ∪ Gn : (N ) (N ) dzn = sg(un ◦ vn )e−vn (t)A B (N ) evn (t)A zn (t). dt Notice that, for every k, the mapping t 7→ hφk , zn i is Lipschitz continuous with Lipschitz constant kBφk k. Finally, we define the time varying N × N matrix Mn (t) equal to (N )
sg(un ◦ vn )e−vn (t)A
(N )
B (N ) evn (t)A
.
Lemma 8. Let M † be the constant N × N matrix whose entries are, for 1 ≤ RI l1 , l2 ≤ N , m†l1 ,l2 = (bl1 ,l2 /I) 0 exp (i(λl1 − λl2 )v ∗ (τ )) dτ if T (λl1 − λl2 ) ∈ 2πZ (N ),un
and zero else. Then, for every t in R+ , the sequence (Xt exp(tM † ) as n tends to infinity.
)n tends to
Remark 1. The matrix M † is clearly related to the differential of u 7→ ΥuT at u = 0 (see in particular [BL10, Eq. (34)]). Proof of Lemma 8. For every t, define the integer s = b tn I c. For every l1 , l2 such that T (λl1 − λl2 ) ∈ / 2πZ, Z
t
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ =
0
Z
sI/n
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ +
0
Z
t
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ
sI/n
9
Z t sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ 0 Z sI/n i(λl1 −λl2 )vn (τ ) dτ + I/n ≤ sg(un ◦ vn )e 0 s Z X I/n sg(un ◦ vn )ei(λl1 −λl2 )(vn (τ )+mT ) dτ + I/n ≤ 0 m=1 s Z I/n X i(λl1 −λl2 )vn (τ ) i(λl1 −λl2 )mT e dτ + I/n ≤ sg(un ◦ vn )e 0 m=1 2 I ≤ +1 . |1 − exp(iT (λl1 − λl2 ))| n This last quantity obviously tends to zero as n tends to infinity. If T (λl1 − λl2 ) ∈ 2πZ, then s Z X m=1
I/n
sg(un◦ vn )ei(λl1 −λl2 )(vn (τ )+mT ) dτ
I/n
Z
sg(un◦ vn )ei(λl1 −λl2 )vn (τ ) dτ
= s
0
0
=
j n
Z
Z
T
I
sg(u∗◦ v ∗ )ei(λl1 −λl2 )v
∗
(τ )
dτ.
0
This last quantity tends to t I
Z
I
sg(u∗ ◦ v ∗ )ei(λl1 −λl2 )v
∗
(τ )
dτ =
0
t I
u∗ (τ )ei(λl1 −λl2 )τ dτ,
0
as n tends to infinity, uniformly with respect to t in any compact set. From [AS04, Lemma 8.10], the solutions of x˙ = Mn (t)x with initial condition x0 tend uniformly with respect to t in any compact set to the solutions of x˙ = M † x with initial condition x0 . As a consequence of Lemma 8, for every t, zn (t) tends to exp(tM † )φj as n tends to infinity. Hence, if (j, k) is a uniquely resonant transition of (A, B), then |hφk , zn (t)i| = |hφk , xn (vn (t))i| tends to Z ! t|bj,k | T ∗ u (τ )ei(λl1 −λl2 )τ dτ sin I 0 as n tends to infinity. Choosing Tn∗ = v ∗
nπI
R , T 2|bj,k | 0 u∗ (τ )ei(λl1 −λl2 )τ dτ
one gets that |hφk , xn (Tn∗ )i| tends to 1 as n tends to infinity.
10
By definition of v ∗ , for every n in N, Z
Tn∗
|un (τ )|dτ ≤ 0
πI = K. R T ∗ 2|bj,k | 0 u (τ )ei(λl1 −λl2 )τ dτ u
By definition of N , from Proposition 6, hφk , xn (Tn∗ )i is -close to hφk , ΥTnn∗ φj i u for every n in N, hence the sequence |hφk , ΥTnn∗ φj i| accumulates in an n∈N
neighborhood of 1. The fact that the sequence (Tn∗ )n∈N does not depend upon gives the desired convergence. Knowing that v ∗ is non-decreasing and v ∗ (lI) = lT for every l in N, one deduces the asymptotic behavior of Tn∗ as n tends to infinity, nT ∗ ≤ Tn∗ ≤ (n + 1)T ∗ where πT R . T∗ = T ∗ 2|bj,k | 0 u (τ )ei(λl1 −λl2 )τ dτ Notice finally that the mapping t 7→ |hφk , xn (t)i| = |hφk , zn ◦ (vn∗ )[−1] (t)i| has Lipschitz constant less than kBφk k sup |u∗ |/n. This completes the proof of Proposition 3 in the case where u∗ is a non vanishing piecewise constant function. If u∗ is a locally integrable T -periodic function, let (u∗,l )l∈N be a sequence of non-vanishing piecewise constant T -periodic functions converging to u∗ in L1loc ∗,l with ku∗,l kL1 ([0,T ]) ≤ kukL1 ([0,T ]) for every l. We define, for every n, un,l = un , Rt vn,l : t 7→ 0 P|un,l |(τ )dτ and Mn,l (t) = sg(un,l ◦ vn,l (t))e−vn,l (t)A B (N ) evn,l (t)A . Rt For every t, the matrix 0 Ml,l (τ )dτ tends to tM † , uniformly with respect to t in a compact set, as l tends to infinity. Hence the solutions of x˙ = Ml,l (t)x tend to the solutions of x˙ = M † x, uniformly with respect to the time in a compact interval, as l tends to infinity. That concludes the proof of Proposition 3.
2.4
Efficiency of the transfer
For every non constant T-periodic u∗ , we define the efficiency of u∗ with respect to the transition (j, k) as the real quantity: Z I 1 Z T ∗ 1 E (j,k) (u∗ ) = sg(u∗ ◦ v ∗ )e(λj −λk )v (τ ) dτ = u∗ (τ )ei(λj −λk )τ dτ . I 0 I 0 For every u, 0 ≤ E (j,k) (u) ≤ 1. In the case where a transition (j, k) is uniquely resonant, supu E (j,k) (u) = 1 (consider a sequence of controls that tends to a periodic sum of Dirac functions). An example of u∗ with zero efficiency is presented in Section 3. An intuitive explanation of the efficiency could be the following: the L1 norm of the control (considered as a cost, closely related to the time) needed to induce the transition between levels j and k using periodic controls of the form un is equal to π/(2|bj,k |E (j,k) (u∗ )).
11
2.5
Multiple resonant transitions
The procedure we have used in the present paper (namely, use any periodic control with non zero efficiency with respect to the transition (j, k)) cannot be applied if the transition (j, k) is multiple resonant. To generate anyway a transfer from level j to level k, one should chose a control u such that E (j,k) (u) be as large as possible and E (l1 ,l2 ) (u) be zero (or arbitrarily close to zero) for every l1 , l2 such that λl1 − λl2 ∈ (λj − λk )Z. The algorithm we have described in [BCCS11] allows to build u such that E (j,k) (u) > 0.43, with E (l1 ,l2 ) (u) arbitrarily small for every l1 , l2 satisfying {l1 , l2 } = 6 {j, k} and |λl1 − λl2 | = 6 |λj − λk |.
3
Rotation of a planar molecule
In this Section, we apply our results to the well studied example of the rotation of a planar molecule (see [Bou99, BCM+ 09, BCCS11, ST05]).
3.1
Presentation of the model
We consider a linear molecule with fixed length and center of mass. We assume that the molecule is constrained to stay in a fixed plane and that its only degree of freedom is the rotation, in the plane, about its center of mass. The state of the system at time t is described by a point θ 7→ ψ(t, θ) of L2 (Ω, C) where Ω = R/2πZ is the one dimensional torus. The Schr¨odinger equation writes i
∂ψ (t, θ) = −∆ψ(t, θ) + u(t) cos(θ)ψ(t, θ), ∂t
(11)
where ∆ is the Laplace-Beltrami operator on Ω. The self-adjoint operator −∆ has purely discrete spectrum {k 2 , k ∈ N}. All its eigenvalues are double but zero which is simple. The eigenvalue zero is associated to the constant functions. The eigenvalue k 2 for k > 0 is associated to the two eigenfunctions θ 7→ √1π cos(kθ) and θ 7→ √1π sin(kθ). The Hilbert space H = L2 (Ω, C) splits in two subspaces He and Ho , respectively the spaces of even and odd functions of H. The spaces He and Ho are stable under the dynamics of (11), hence no global controllability is to be expected in H.
3.2
Uniquely resonant case
We first concentrate on the space Ho . The restriction A of i∆ to Ho is skew adjoint, with simple eigenvalues (ik 2 )k∈N associated to the eigenvectors 1 . φk : θ 7→ √ sin(kθ) π k∈N The restriction B of ψ 7→ cos(θ)ψ to Ho is skew-symmetric. The couple (A, B) satisfies Assumption 1.
12
The Galerkyn approximation of A and B at order 0 i 0 ··· 0 1/2 .. 0 4i . . . . (N ) (N ) A = . and B = i 0 . . . . . . . . . 0 .. 2 0 ··· 0 N i 0
N are 1/2
0
0 .. .
1/2
..
. ···
0 1/2 0
··· .. . .. .
0 .. .
0 0 1/2 1/2 0
Our aim is to transfer the wave function from the first eigenspace to the second one. The numerical simulation will be done on some finite dimensional space CN . The constant CA,B of Assumption 1.4 is 1. The controls we will use in the following have L1 norm less than 13/3 and, from Proposition 4, the A norm of Υut (φ1 ) will remain less than exp(13/3) ≈ 76 for every time. From Proposition 6, the error done p when replacing the original system by its Galerkyn approximation of order 76/10−4 ≈ 872 is smaller than = 10−4 . This estimate is indeed very conservative and it can be improved using the regularity of the operator B. Proposition 9. Let k be an integer. If B is bounded, then for every t in R+ , for every locally integrable control u (not necessarily periodic), Z t l k k−1 X |hφl , B l φ1 | Z t kB k φk k |hφk , Υut φ1 i| ≤ |u(τ )|dτ + |u(τ )|dτ . l! k! 0 0 l=0
Proof. We do the proof in the case where u (and hence Pu) is a finite sum of Rt Dirac functions (i.e., t 7→ 0 Pu(τ )dτ is piecewise constant). Since the set of finite sums of Dirac functions is dense (for the metric of total variation) in the set of Radon measures, the result can be extended to locally integrable controls by continuity (Proposition 2). ˙ For every t ≥ 0, consider t 7→ ψ(t), the solution of ψ(t) = (P|u|(t)A + sg(u)B)ψ(t) with Rinitial condition φ1 . t Set z(t) = e− 0 P|u|(s)dsA ψ(t) and define zk (t) = hφk , q(t)i. Because of the boundedness of B, for every 1 ≤ l ≤ k, the function t 7→ qk (t) is piecewise C l for every l ≤ k and, for almost every t ≥ 0, l d
qk (t) = | φk , M1 (t)l q(t) | = | φk , B l q(t) | = | B l φk , q(t) |. dtl In particular, the lth derivative of qk admits everywhere in R a left and a right limit. The moduli of these two limits is the same, only the phase may be (l) discontinuous. Notice that thanks to the Cauchy–Schwarz inequality, |qk (t)| ≤ kB l φk k. After l integrations on [0, t], one finds |hφk , q(t)i| ≤
l−1 X |hφk , B j ψ0 i| j=0
j!
The result follows from Proposition 7. 13
tj +
kB l φk k l t . l!
Applying Proposition 9, the particular tri-diagonal form of B yields |hφk , Υut φ1 i| ≤
kukkL1 . k!
If kukL1 ≤ 13/3, then kπ22 B(Id − π22 )Υut (φ1 )k ≤ 2.10−8 for every t in R+ . Plugging this inequality in (7), one gets that the error done when replacing the original system by its Galerkyn approximation of order 22 is smaller than = 10−7 when kukL1 ≤ 13/3. The transition (1, 2) is not uniquely resonant, since, for instance, λ4 = 42 , λ5 = 52 and 52 − 42 = 9 = 3(22 − 12 ). However, for every l1 , l2 such that λl1 − λl2 ∈ 3Z and hφl1 , Bφl2 i = 6 0, one has l1 > 2 and l2 > 2. Hence the limit matrix M † (Lemma 8) lets invariant the subspace generated by φ1 and φ2 and the result of Proposition 3 applies. We illustrate the notion of efficiency on some examples of control, namely u∗ : t 7→ cosl (3t) for l ∈ {1, 2, 3, 4, 5}. The efficiency is zero when l is even. In numerical simulations, the quantity (22),u∗ |hφ2 , Xt φ1 i| is less than 2.10−5 for every t < 500 (see Figure below for l = 2). When l is odd, the efficiency is not zero. To estimate numerically the efficiency, one considers, for n = 10 and n = 30, the first maximum p† of (N ),u∗ /n t 7→ |hφ2 , Xt φ1 i|, reached at time t† , and computes (1 − p† )nπ . R t† 2|hφ1 , Bφ2 i| 0 |u∗ (τ )|dτ The Scilab source codes used for the simulation are available on the web page http://www.iecn.u-nancy.fr/∼chambrio/PreprintUK.html. We sum up the results in the following tabular. Control u∗ n Time t† Precision Numerical (Theoretical Efficiency) 1 − p† Efficiency n=1 6.8 2.10−2 73% t 7→ cos(3t) n = 10 63 4.10−4 78% π/4 ≈ 79% n = 30 189 3.10−5 78% n=1 8.9 2.10−2 83% t 7→ cos(3t)3 n = 10 84 2.10−4 88% 9π/32 ≈ 88% n = 30 252 2.10−5 88% n=1 10 7.10−3 93% t 7→ cos(3t)5 n = 10 101 2.10−4 92% 75π/256 ≈ 92% n = 30 302 2.10−5 92%
3.3
Not uniquely resonant case
We concentrate on the space He . The restriction A of i∆ to He is skew adjoint, with simple eigenvalues (ik 2 )k∈N∪{0} associated to the eigenvectors (φk )k∈N∪{0} , with φk : θ √1π 7→ cos(kθ) for k in N and φ0 : θ 7→ √12π . The restriction 14
B of ψ 7→ cos(θ)ψ to He is skew-symmetric. The couple (A + i, B) satisfies Assumption 1. The translation from A to A + i just induces a phase shift and will be neglected in the following. The Galerkyn approximation of A and B at order N are √i 0 0 · · · 0 2 0 0 ··· 0 . .. √i i . .. 0 . . 2 .. .. 2 0 (N ) i . ,B .. .. A(N ) = = i . . . . 0 0 0 .. ... .. 0 .. .. i i 2 . 0 . 0 ··· 0 (N − 1) i 0
···
2
0
i 2
2
0
Our aim is to transfer the population from the first eigenspace, associated with eigenvalue 0, to the second one, associated with eigenvalue i. The transition (1, 2) is not uniquely resonant, and in contrary to what happens on the space of odd eigenfunctions, the limit matrix M † does not stabilize the space spanned by φ1 and φ2 . Note however that B only connects level 2 to levels 1 and 3. In other words, it is enough to find a 2π-periodic function u∗ such that E (2,3) (u∗ ) is zero and E (1,2) (u∗ ) is not zero (and as large as possible) P to induce the desired transfer. This is achieved, for instance, with u∗ = n 2δ2πn + δ2πn−π/3 + δ2πn+π/3 (this control is the limit of the sequence of piecewise constant controls build in [BCCS11]), for which E (1,2) (u∗ ) = 1+cos(π/3) = 34 and E (2,3) (u∗ ) = 0. 2 Another example is presented on Figure 2.
4
Conclusion
This main result of this paper will not surprise anyone already familiar with quantum control. Its interest lies rather, first in the theoretical rigorous proof of the convergence and second, in the interpretation of the notion of efficiency, seen as a measure of the wasted L1 norm of the control to achieve a given transition. Despite some recent advances in the field of bilinear control of skew-adjoint operators, many questions remain open. Among other topics, future works may concentrate on the design of time-efficient controls or on the control of systems where operators have continuous spectrum.
Acknowledgments This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, by the ANR project GCM, program “Blanche”, project number NT09-504490, and by the Inria NancyGrand Est “COLOR” project. It is a pleasure for the author to thank Pierre Rouchon who suggested him this question, Ugo Boscain, Marco Caponigro, Julien Salomon, Mario Sigalotti 15
and Dominique Sugny for many discussions and advices, Nabile Boussa¨ıd for his most valuable help on the functional analysis topics and Denzil Millichap for many corrections. The author wishes also to thank the Institut Henri Poincar´e (Paris, France) for providing research facilities and a stimulating environment during the “Control of Partial and Differential Equations and Applications” program in the Fall 2010.
References [AS04]
Andrei A. Agrachev and Yuri L. Sachkov. Control theory from the geometric viewpoint, volume 87 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II.
[BCC11]
Nabile Boussa¨ıd, Marco Caponigro, and Thomas Chambrion. Approximate controllability of perturbations of the quantum harmonic oscillators. not published, 2011.
[BCCS11] Ugo Boscain, Marco Caponigro, Thomas Chambrion, and Mario Sigalotti. A weak spectral condition for the controllability of the bilinear Schr¨ odinger equation with application to the control of a rotating planar molecule. arXiv:1101.4313v1, 2011. [BCM+ 09] Ugo Boscain, Thomas Chambrion, Paolo Mason, Mario Sigalotti, and Dominique Sugny. Controllability of the rotation of a quantum planar molecule. In Proceedings of the 48th IEEE Conference on Decision and Control, pages 369–374, 2009. [BFS86]
R. Bl¨ umel, S. Fishman, and U. Smilansky. Excitation of molecular rotation by periodic microwave pulses. a testing ground for anderson localization. The Journal of Chemical Physics, 84(5):2604–2614, 1986.
[BL10]
Karine Beauchard and Camille Laurent. Local controllability of 1D linear and nonlinear Schr¨odinger equations with bilinear control. J. Math. Pures Appl., 94(5):520–554, 2010.
[Bou99]
J. Bourgain. On growth of Sobolev norms in linear Schr¨odinger equations with smooth time dependent potential. J. Anal. Math., 77:315–348, 1999.
[BST08]
M. Belhadj, J. Salomon, and G. Turinici. A stable toolkit method in quantum control. J. Phys. A, 41(36):362001, 10, 2008.
[CMSB09] Thomas Chambrion, Paolo Mason, Mario Sigalotti, and Ugo Boscain. Controllability of the discrete-spectrum Schr¨odinger equation driven by an external field. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 26(1):329–349, 2009. 16
[Kat95]
Tosio Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
[Mir09]
Mazyar Mirrahimi. Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 26(5):1743–1765, 2009.
[MS10]
Paolo Mason and Mario Sigalotti. Generic controllability properties for the bilinear Schr¨odinger equation. Communications in Partial Differential Equations, 35:685–706, 2010.
[Ner10]
Vahagn Nersesyan. Global approximate controllability for Schr¨ odinger equation in higher Sobolev norms and applications. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 27(3):901–915, 2010.
[OKF98]
Y. Ohtsuki, H. Kono, and Y. Fujiyama. Qantum control of nuclear wave packets by locally designed optimal pulses. Journal of chemical physics, 109(21):9318–9331, 1998.
[PS10]
Yannick Privat and Mario Sigalotti. The squares of the Laplacian–Dirichlet eigenfunctions are generically linearly independent. ESAIM: COCV, 16:794–807, 2010.
[ST05]
Julien Salomon and Gabriel Turinici. Control of molecular orientation and alignment by monotonic schemes. In Proceedings of the 24-th IASTED International Conference on modelling, identification and control, pages 64–68, February 16-18, 2005.
[WRD93] Warren S. Warren, Herschel Rabitz, and Mohammed Dahleh. Coherent Control of Quantum Dynamics: the Dream Is Alive. Sciences, 259:1581–1589, 1993.
17
1.8e−05
1.6e−05
1.4e−05
1.2e−05
1.0e−05
8.0e−06
6.0e−06
4.0e−06
2.0e−06
0.0e+00 0
50
100
150
200
250
300
0
50
100
150
200
250
300
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Figure 1: Evolution of the modulus of the second coordinate when applying the controls : t 7→ cos2 (3t)/30 (top) and t 7→ cos3 (3t)/30 (bottom) on the planar molecule (odd subspace) with initial condition φ1 . The simulations have been done on a Galerkyn approximation of size N = 22.
18
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
Figure 2: Evolution of the modulus of coordinate when applying the the second 1 2 t+π/3 control 20 2 cos2 2t + cos2 t−π/3 + cos on the planar molecule 2 2 (even subspace) with initial condition φ1 . The simulation has been done on a Galerkyn approximation of size N = 22. Precision 1 − p† is equal to 2.10−3 . Numerical efficiencies are 38% (theoretical: 3/8) for the transition (1, 2) and less than 5.10−4 for the transition (2, 3) (theoretical: 0).
19
Periodic excitations of a bilinear quantum system Thomas Chambrion∗ April 1, 2011
Abstract A well known method to transfer the population of a quantum system from an eigenspace of the free Hamiltonian to another is to use a periodic control law with angular frequency equal to the difference of the eigenvalues. This paper gives a theoretical proof of this experimental result. We introduce a notion of efficiency and demonstrate its interest for the design of controls on the example of the rotation of a planar molecule.
Contents 1 Introduction 1.1 Effective control of quantum systems 1.2 Framework and notations . . . . . . 1.3 Main result . . . . . . . . . . . . . . 1.4 Content of the paper . . . . . . . . .
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2 2 2 4 4
2 Proof of the convergence result 2.1 Good Galerkyn approximation 2.2 Time reparametrization . . . . 2.3 Proof of Proposition 3 . . . . . 2.4 Efficiency of the transfer . . . . 2.5 Multiple resonant transitions .
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4 4 7 8 11 12
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3 Rotation of a planar molecule 12 3.1 Presentation of the model . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Uniquely resonant case . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Not uniquely resonant case . . . . . . . . . . . . . . . . . . . . . 14 4 Conclusion
15
∗ The
author is with IECN UMR 7502, Nancy University, CNRS, INRIA, BP 70239, 54506 Vandœuvre-l` es-Nancy, France and INRIA Nancy Grand Est, projet CORIDA. [email protected]
1
1 1.1
Introduction Effective control of quantum systems
The state of a quantum system evolving on a finite dimensional Riemannian manifold Ω, with associated measure µ, is described by its wave function, that is, a point in the unit sphere of L2 (Ω,ZC). A system with wave function ψ is in a subset ω of Ω with the probability
|ψ|2 dµ.
ω
In the absence of interaction with the environment, the time evolution of the wave function is given by the Schr¨odinger equation i
1 ∂ψ = − ∆ψ + V (x)ψ(x, t), ∂t 2
where ∆ is the Laplace-Beltrami operator on Ω (with suitable boundary conditions) and V : Ω → R is a real function (usually called potential) accounting for the physical properties of the system. When submitted to an excitation by an external electric field (e.g. a laser), the Schr¨odinger equation reads i
∂ψ 1 = − ∆ψ + V (x)ψ(x, t) + u(t)W (x)ψ(x, t), ∂t 2
(1)
where W : Ω → R is a real function accounting for the physical properties of the laser and u is a real function of the time accounting for the intensity of the laser. A natural question, with many practical implications, is whether there exists a control u that steers the quantum system from a given initial wave function to a given target wave function (controllability issue) and, more important, how to build this control law (effective design of controls). Considerable efforts have been made by different communities to study the controllability of (1). We refer to [BCCS11, BL10, Ner10, Mir09] and references therein for a description of the known theoretical results concerning the existence of controls steering a given source to a given target. As proved in [Ner10, MS10, PS10], approximate controllability is a generic property for systems of the type of (1). Effective control algorithms have been obtained [WRD93, OKF98, BST08]. Most of the controls used in practice exhibit a remarkable pattern of periodic shape, with frequency corresponding to a resonance in the quantum system. They appear to work remarkably well, with little influence of the shape [BFS86]. However, to our knowledge, no theoretical proof of this effectiveness is available. The aim of the present paper is to provide a mathematically rigorous explanation of the surprising robustness of these control algorithms.
1.2
Framework and notations
We first reformulate the problem (1) in a more abstract framework. In a separable Hilbert space H endowed with norm k · k and Hilbert product h·, ·i, we 2
consider the following evolution problem: dψ = (A + uB)ψ dt
(2)
where (A, B) satisfies Assumption 1. Assumption 1. (A, B) is a pair of linear operators such that 1. A is skew adjoint with purely discrete spectrum (iλn )n∈N ; 2. the sequence (λn )n∈N takes value in (0, +∞), is non-decreasing and its only accumulation point is +∞; 3. there exists an Hilbert basis (φk )k∈N of H such that Aφk = λk φk for every k in N; 4. for every ψ in D(A), ψ belongs to D(B) and there exists sA,B < 1/2 such that kBψk ≤ k(iA)sA,B ψk; 5. for every u in R, A+uB is skew-adjoint, D(A+uB) = D(A) and D((A+ uB)2 ) = D(A2 ); 6. For every interval I containing 0, for every Radon measure u on I, t 7→ A(t) := eu([0,t))B Ae−u([0,t))B is a family of skew-adjoint operators with common domain D = D(A) and A is continuous with bounded variation from I to B(D, H); 7. For every interval I containing 0, for every Radon measure u on I, A :
t 7→ eu([0,t))B Ae−u([0,t))B is such that supt∈I A(t)−1 B(H,D(A)) < +∞ ; 8. there exists CA,B > 0 such that |=hAψ, Bψi| ≤ CA,B |hAψ, ψi| for every ψ in D(A). The Assumption 1.5 ensures that, for every constant u in R, A+uB generates a group of unitary propagators. Hence, for every initial condition ψ0 in H, for every piecewise constant control u, we can define the solution of (2) that we will note t 7→ Υut ψ0 . The map u 7→ ΥuT ψ0 can be extended to the closure of the set of piecewise constant functions for any norm such that u 7→ ΥuT ψ0 is locally uniformly continuous. Proposition 2 (Well-posedness). For every T in R+ , for every ψ0 in H, the map u 7→ ΥuT ψ0 admits a unique continuous extension to the set of Radon measures on [0, T ]. Moreover, the mapping Υ : (T, ψ0 , u) 7→ ΥuT ψ0 is continuous for the product of the standard distance of R, the Hilbert norm of H and the total variation in the set of Radon measures on R+ . The result of Proposotion 2 is classical under Assumption 1 for controls with bounded variations (see [Kat95]). It has recently been extended (see [BCC11]) to the set of Radon measures endowed with the distance of total variation. This framework includes L1loc functions and countable sums of Dirac masses. 3
1.3
Main result
Definition 1. A point (j, k) of N2 is said to be a uniquely resonant transition of (A, B) if (i) j 6= k, (ii) hφj , Bφk i = 6 0 and (iii) for every l, m, p in N, p|λj − λk | = |λl − λm | implies {j, k} = {l, m} and p = 1, or hφl , Bφm i = 0. A point (j, k) of N2 is said to be a multiple resonant transition of (A, B) if (i) j 6= k, (ii) hφj , Bφk i 6= 0 and (iii) there exist p, l, m ∈ N such that {l, m} 6= {j, k}, hφl , Bφm i = 6 0 and p|λj − λk | = |λl − λm |. Proposition 3. Let u∗ : R+ → R be a locally integrable function. Assume that u∗ is periodic with smallest period T = |λj2π −λk | for some uniquely resonant transition (j, k) of (A, B). If Z
T
u∗ (τ )ei(λj −λk )τ dτ 6= 0,
0
u∗ n then there exists T > 0 such that the sequence hφk , ΥnT ∗ (φj )i tends n∈N to 1 as n tends to infinity. ∗
This results provides a rigorous formulation of a well-known fact: to induce a transition between levels j and k of a quantum system, one can use a periodic excitation of frequency exactly equal to the difference of the corresponding eigenvalues. For almost every shape of the control, the trajectory eventually reaches any neighborhood of the target, provided the control is small enough and has the correct frequency. As it is well known from experiments, the situation is more intricate with multiple resonant transitions.
1.4
Content of the paper
The paper splits in two parts. The first one is a theoretical proof of Proposition 3. Some technical tools are introduced in Section 2.1. A time reparametrization (Section 2.2) allows to prove Proposition 3 (Section 2.3). In a second part, the theoretical results are tested on numerical simulations of the rotation of a planar molecule.
2
Proof of the convergence result
The strategy of the proof is inspired by [CMSB09, BCCS11] and relies upon the approximation of the original infinite dimensional system by its finite dimensional approximations.
2.1
Good Galerkyn approximation
In this Section, we explain how to construct a good Galerkyn approximation of the original system. The term “good” refers to the fact that the error made when replacing the original system by its Galerkyn approximation is bounded 4
uniformly with respect to the control u. What follows is a very simplified version of a much more general construction (valid also for operators with continuous spectrum) presented in [BCC11]. Under Assumption 1.2, −iA is a self-adjoint, bounded from below operator. For every α in R, the translation from A to A + αiIdH has no effect on the dynamics of (2) but a physically irrelevant shift in the phases. Up to a suitable translation, one may assume that iA is a self-adjoint operator of H with positive P eigenvalues. For every ψ in D(A), −iAψ = j∈N λP j hφj , ψiφj . For every s > 0, the linear operator (−iA)s is defined by (−iA)s ψ = j∈N λsj hφj , ψiφj , for every ψ in D((−iA)s ). Proposition 4. For every ψ0 ∈ D(A2 ) and K > 0, there exists CK such that for every T ≥ 0 and for every control u for which kukL1 < K, one has |hAΥuT (ψ0 ), ΥuT (ψ0 )i| < CK . Proof. Let x(t) be a solution of x˙ = (A + u(t)B) x with initial condition ψ0 . By Assumption 1.5, x(t) belongs to D(A2 ) for every t. Consider the real mapping f : t 7→ |hAx(t), x(t)i|. Since x(0) belongs to D(A), f is differentiable and d d f (t) = i hAx(t), x(t)i dt dt = ihAx(t), (A + u(t)B)x(t)i + ihA(A + u(t)B)x(t), x(t)i = 2i2 =hAx(t), (A + u(t)B)x(t)i = −2=hAx(t), u(t)Bx(t)i, so that, thanks to Assumption 1.4, |f 0 (t)| ≤ 2|u(t)||hAx(t), Bx(t)i| ≤ CA,B |u(t)|f (t). Gronwall lemma implies that |hAx(t), x(t)i| ≤ eCA,B
Rt
The result follows by taking CK = eCA,B
Rt
0
|u|(τ )dτ
0
|hAψ0 , ψ0 i|.
|u|(τ )dτ
(3)
|hAψ0 , ψ0 i|.
For every N in N, we define the orthogonal projection πN
:H ψ
→ 7 →
H P
j≤N hφj , ψiφj
Proposition 5. For every > 0, K ≥ 0, n ∈ N, there exists N ∈ N such that for every control u, for every 0 ≤ s < 1, kukL1 ≤ K =⇒ |h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj )i| <
(4)
for every t ≥ 0, and j = 1, . . . , n. As a consequence, there exists N ∈ N such that for every control u, kukL1 ≤ K =⇒ kB(Id − πN )Υut (φj )k < , for every t ≥ 0, and j = 1, . . . , n. 5
(5)
Proof. Fix j ∈ {1, . . . , n}. For every N > 1, one has ∞ X
|h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj i| =
λsn |hφn , Υut (φj )i|2
n=N +1 ∞ X
≤
λs−1 N
≤
u u λs−1 N |hAΥt (φj ), Υt (φj )i|.
λn |hφn , Υut (φj )i|2
n=N +1
By Proposition 4, there exists CK such that |hAΥut (φj ), Υut (φj )i| ≤ CK for every t in R+ and u in the K ball of L1 . Since s < 1 then |h(−iA)s (Id − πN )Υut (φj ), (Id − πN )Υut (φj )i| tends to 0, uniformly with respect to u, as N tends to infinity. By Assumption 1.4, there exists sA,B in (0, 1/2) such that p kBxk ≤ k(iA)sA,B xk = |h(iA)2sA,B x, xi| for every x ∈ D(A) and (5) is a direct consequence of (4) applied with s ∈ (2sA,B , 1). Definition 2. Let N ∈ N. Denote LN = span(φ1 , . . . , φN ). The Galerkyn approximates of A and B of order N are the operators A(N ) : H → H and B (N ) : H → H defined by A(N ) = πN ALN
and
B (N ) = πN BLN .
We define the system (ΣN ) as x˙ = (A(N ) + uB (N ) )x (N ),u
and call Xt
(ΣN )
the propagator of (ΣN ).
With an obvious abuse of notation, we will sometimes identify the operators A(N ) and B (N ) with their restrictions to the invariant space span1≤l≤N {φl } and with their matrices in the basis (φl )1≤l≤N . Entries of B (N ) are denoted (bl1 ,l2 = hφl1 , Bφl2 i)l1 ,l2 . With these identifications, (ΣN ) turns into a finite dimensional system in CN . Proposition 6 (Good Galerkin Approximation). For every > 0, K ≥ 0, n ∈ N, there exists N ∈ N such that for every u ∈ L1 (0, ∞) (N ),u
kukL1 ≤ K =⇒ kΥut (φj ) − Xt
φj k < ,
(6)
for every t ≥ 0 and i = 1, . . . , n. Proof. Let > 0, K ≥ 0, n ∈ N be given. By Proposition 5, there exists N ∈ N such that, for every u ∈ L1 (0, ∞) for which kukL1 ≤ K, we have kB(Id − πN )Υut (φj )k < 6
, K
for every t ≥ 0 and j = 1, . . . , n. Fix j in 1, . . . , n and consider yj : t 7→ πN Υut (φj ). The mapping t 7→ yj (t) is absolutely continuous and, for almost every t ≥ 0, y˙ j (t) = (A(N ) + uB (N ) )yj (t) + u(t)πN B(Id − πN )Υut (φj ). Hence, for every t ≥ 0, (N ),u
yj (t) = Xt
Z φj +
t
(N ),u
Xt−s πN B(Id − πN )Υus (φj )u(τ )dτ
(7)
0
The norm of t 7→ B(Id − πN )Υut (φj ) is less than or equal to /K for every t ≥ 0 (N ),u (N ),u and, since Xt is unitary, kyj (t) − Xt (φj )k ≤ kukL1 /K ≤ . .
2.2
Time reparametrization
We note P C the set of the piecewise constant functions for which there exist two sequences (uj )1≤j≤p and (tj )1≤j≤p with value in (0, +∞) such that X u= uj χ[τj ,τj +tj ) , 1≤j≤p−1
where χ is the characteristic function and the sequence (τj ) is defined by induction: τ1 = 0, τj+1 = τj + tj . An element u of P C will be denoted (uj , tj )1≤j≤p . The involutive mapping P : PC (uj , tj )1≤j≤p
→ 7→
PC
1 uj , uj tj
1≤j≤p
performs a time reparametrization of the system (2). Indeed, introduce the control system dψ = (u(t)A + B)ψ(t), (8) dt ˇ ut ψ0 . whose solution with initial condition ψ0 will be denoted by t 7→ Υ ˇ Pu Rt Proposition 7. For every ψ0 in H, for every control u in P C, Υ ψ = u(τ )dτ 0 0 u Υt ψ0 . Proof. This follows from the equality et(A+uB) = etu( u A+B ) , valid on every interval where u is constant. 1
7
2.3
Proof of Proposition 3
Let > 0, u∗ a non vanishing piecewise constant function, T -periodic, j < k two integers be given asR in the hypothesis of Proposition 3. t The function t 7→ 0 |u∗ (τ )|dτ is non-decreasing. We denote with v ∗ its RT reciprocal function. We define also I = 0 |u∗ (τ )|dτ or, equivalently, v ∗ (I) = T . RT If 0 u∗ (τ ) exp [i(λj − λk )τ ] dτ 6= 0, let us introduce K=
πI R . T∗ i(λ 2 0 u (τ )e j −λk )τ dτ |bj,k |
By Proposition 6, there exists N in N such that for every l < k, for every u, for every t in R, (N ),u
kukL1 < K =⇒ kΥut (φl ) − Xt
φl k < ,
where X (N ),u is the propagator associated with the control system (ΣN ). We introduce the sequence un = n1 u∗ . For every n in N, un is a nonR nT RT vanishing T -periodic function and 0 |un (τ )|dτ = 0 |u∗ (τ )|dτ . For every n, for every t, P|un |(t) = nP|u∗ | (nt) . In particular, P|un | is a
I n
periodic function and
I n
Z
Z
P|un |(τ )dτ = 0
I
P|u∗ |(τ )dτ = T.
0
The primitive vn of P|un | taking value 0 at 0, Z vn : t 7→
t
P|un |(τ )dτ = v ∗ (nt),
0 +
satisfies, for every t in R , vn
I t+ n
Z
I/n
P|un |(τ )dτ = vn (t) + T.
= vn (t) + 0
Equivalently, one can define vn as the reciprocal of the increasing function t 7→ Rt |u (τ )|dτ . n 0 Let us note t 7→ xn (t) the solution of x˙ = (A(N ) + un (t)B (N ) )x(t) with initial condition φj . The set [0, T ] can be written as a finite union of disjoint intervals [0, T ] = J1+ ∪ J1− ∪ . . . ∪ Jp+ ∪ Jp− , 8
(9)
such that u∗ takes positive (resp. negative) values on J + = ∪pl=1 Jl+ (resp. J − = ∪pl=1 Jl− ). For t in J − , (ΣN ) writes dx = (A(N ) + (−un (t))(−B (N ) ))x(t). dt We apply the P reparametrization to the positive function |u∗ | separately on [−1] every intervals in J + and J − . Defining the sets G+ (J + ) = {l ∈ n = (vn ) Rs [−1] + + − − R |∃s ∈ J , 0 |un (τ )|dτ = l} and Gn = (vn ) (J ) = {l ∈ R+ |∃s ∈ R s J − , 0 |un (τ )|dτ = l}, we obtain the dynamics of yn = xn ◦ vn :
dyn = dt
(P(|un |)(t)A(N ) + B (N ) )yn (t) (P(|un |)(t)A(N ) − B (N ) )yn (t)
for almost every t ∈ G+ , for almost every t ∈ G−
(10)
For every t, we define zn (t) as Z t zn (t) = exp − P|un |(τ )dτ A(N ) yn (t) = e−vn (t)A yn (t). 0
Notice that, for every t, for every l in N, |hφl , zn (t)i| = |hφl , yn (t)i|. From (10), one deduces the dynamics of zn , valid for almost every t in − G+ n ∪ Gn : (N ) (N ) dzn = sg(un ◦ vn )e−vn (t)A B (N ) evn (t)A zn (t). dt Notice that, for every k, the mapping t 7→ hφk , zn i is Lipschitz continuous with Lipschitz constant kBφk k. Finally, we define the time varying N × N matrix Mn (t) equal to (N )
sg(un ◦ vn )e−vn (t)A
(N )
B (N ) evn (t)A
.
Lemma 8. Let M † be the constant N × N matrix whose entries are, for 1 ≤ RI l1 , l2 ≤ N , m†l1 ,l2 = (bl1 ,l2 /I) 0 exp (i(λl1 − λl2 )v ∗ (τ )) dτ if T (λl1 − λl2 ) ∈ 2πZ (N ),un
and zero else. Then, for every t in R+ , the sequence (Xt exp(tM † ) as n tends to infinity.
)n tends to
Remark 1. The matrix M † is clearly related to the differential of u 7→ ΥuT at u = 0 (see in particular [BL10, Eq. (34)]). Proof of Lemma 8. For every t, define the integer s = b tn I c. For every l1 , l2 such that T (λl1 − λl2 ) ∈ / 2πZ, Z
t
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ =
0
Z
sI/n
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ +
0
Z
t
sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ
sI/n
9
Z t sg(un ◦ vn )ei(λl1 −λl2 )vn (τ ) dτ 0 Z sI/n i(λl1 −λl2 )vn (τ ) dτ + I/n ≤ sg(un ◦ vn )e 0 s Z X I/n sg(un ◦ vn )ei(λl1 −λl2 )(vn (τ )+mT ) dτ + I/n ≤ 0 m=1 s Z I/n X i(λl1 −λl2 )vn (τ ) i(λl1 −λl2 )mT e dτ + I/n ≤ sg(un ◦ vn )e 0 m=1 2 I ≤ +1 . |1 − exp(iT (λl1 − λl2 ))| n This last quantity obviously tends to zero as n tends to infinity. If T (λl1 − λl2 ) ∈ 2πZ, then s Z X m=1
I/n
sg(un◦ vn )ei(λl1 −λl2 )(vn (τ )+mT ) dτ
I/n
Z
sg(un◦ vn )ei(λl1 −λl2 )vn (τ ) dτ
= s
0
0
=
j n
Z
Z
T
I
sg(u∗◦ v ∗ )ei(λl1 −λl2 )v
∗
(τ )
dτ.
0
This last quantity tends to t I
Z
I
sg(u∗ ◦ v ∗ )ei(λl1 −λl2 )v
∗
(τ )
dτ =
0
t I
u∗ (τ )ei(λl1 −λl2 )τ dτ,
0
as n tends to infinity, uniformly with respect to t in any compact set. From [AS04, Lemma 8.10], the solutions of x˙ = Mn (t)x with initial condition x0 tend uniformly with respect to t in any compact set to the solutions of x˙ = M † x with initial condition x0 . As a consequence of Lemma 8, for every t, zn (t) tends to exp(tM † )φj as n tends to infinity. Hence, if (j, k) is a uniquely resonant transition of (A, B), then |hφk , zn (t)i| = |hφk , xn (vn (t))i| tends to Z ! t|bj,k | T ∗ u (τ )ei(λl1 −λl2 )τ dτ sin I 0 as n tends to infinity. Choosing Tn∗ = v ∗
nπI
R , T 2|bj,k | 0 u∗ (τ )ei(λl1 −λl2 )τ dτ
one gets that |hφk , xn (Tn∗ )i| tends to 1 as n tends to infinity.
10
By definition of v ∗ , for every n in N, Z
Tn∗
|un (τ )|dτ ≤ 0
πI = K. R T ∗ 2|bj,k | 0 u (τ )ei(λl1 −λl2 )τ dτ u
By definition of N , from Proposition 6, hφk , xn (Tn∗ )i is -close to hφk , ΥTnn∗ φj i u for every n in N, hence the sequence |hφk , ΥTnn∗ φj i| accumulates in an n∈N
neighborhood of 1. The fact that the sequence (Tn∗ )n∈N does not depend upon gives the desired convergence. Knowing that v ∗ is non-decreasing and v ∗ (lI) = lT for every l in N, one deduces the asymptotic behavior of Tn∗ as n tends to infinity, nT ∗ ≤ Tn∗ ≤ (n + 1)T ∗ where πT R . T∗ = T ∗ 2|bj,k | 0 u (τ )ei(λl1 −λl2 )τ dτ Notice finally that the mapping t 7→ |hφk , xn (t)i| = |hφk , zn ◦ (vn∗ )[−1] (t)i| has Lipschitz constant less than kBφk k sup |u∗ |/n. This completes the proof of Proposition 3 in the case where u∗ is a non vanishing piecewise constant function. If u∗ is a locally integrable T -periodic function, let (u∗,l )l∈N be a sequence of non-vanishing piecewise constant T -periodic functions converging to u∗ in L1loc ∗,l with ku∗,l kL1 ([0,T ]) ≤ kukL1 ([0,T ]) for every l. We define, for every n, un,l = un , Rt vn,l : t 7→ 0 P|un,l |(τ )dτ and Mn,l (t) = sg(un,l ◦ vn,l (t))e−vn,l (t)A B (N ) evn,l (t)A . Rt For every t, the matrix 0 Ml,l (τ )dτ tends to tM † , uniformly with respect to t in a compact set, as l tends to infinity. Hence the solutions of x˙ = Ml,l (t)x tend to the solutions of x˙ = M † x, uniformly with respect to the time in a compact interval, as l tends to infinity. That concludes the proof of Proposition 3.
2.4
Efficiency of the transfer
For every non constant T-periodic u∗ , we define the efficiency of u∗ with respect to the transition (j, k) as the real quantity: Z I 1 Z T ∗ 1 E (j,k) (u∗ ) = sg(u∗ ◦ v ∗ )e(λj −λk )v (τ ) dτ = u∗ (τ )ei(λj −λk )τ dτ . I 0 I 0 For every u, 0 ≤ E (j,k) (u) ≤ 1. In the case where a transition (j, k) is uniquely resonant, supu E (j,k) (u) = 1 (consider a sequence of controls that tends to a periodic sum of Dirac functions). An example of u∗ with zero efficiency is presented in Section 3. An intuitive explanation of the efficiency could be the following: the L1 norm of the control (considered as a cost, closely related to the time) needed to induce the transition between levels j and k using periodic controls of the form un is equal to π/(2|bj,k |E (j,k) (u∗ )).
11
2.5
Multiple resonant transitions
The procedure we have used in the present paper (namely, use any periodic control with non zero efficiency with respect to the transition (j, k)) cannot be applied if the transition (j, k) is multiple resonant. To generate anyway a transfer from level j to level k, one should chose a control u such that E (j,k) (u) be as large as possible and E (l1 ,l2 ) (u) be zero (or arbitrarily close to zero) for every l1 , l2 such that λl1 − λl2 ∈ (λj − λk )Z. The algorithm we have described in [BCCS11] allows to build u such that E (j,k) (u) > 0.43, with E (l1 ,l2 ) (u) arbitrarily small for every l1 , l2 satisfying {l1 , l2 } = 6 {j, k} and |λl1 − λl2 | = 6 |λj − λk |.
3
Rotation of a planar molecule
In this Section, we apply our results to the well studied example of the rotation of a planar molecule (see [Bou99, BCM+ 09, BCCS11, ST05]).
3.1
Presentation of the model
We consider a linear molecule with fixed length and center of mass. We assume that the molecule is constrained to stay in a fixed plane and that its only degree of freedom is the rotation, in the plane, about its center of mass. The state of the system at time t is described by a point θ 7→ ψ(t, θ) of L2 (Ω, C) where Ω = R/2πZ is the one dimensional torus. The Schr¨odinger equation writes i
∂ψ (t, θ) = −∆ψ(t, θ) + u(t) cos(θ)ψ(t, θ), ∂t
(11)
where ∆ is the Laplace-Beltrami operator on Ω. The self-adjoint operator −∆ has purely discrete spectrum {k 2 , k ∈ N}. All its eigenvalues are double but zero which is simple. The eigenvalue zero is associated to the constant functions. The eigenvalue k 2 for k > 0 is associated to the two eigenfunctions θ 7→ √1π cos(kθ) and θ 7→ √1π sin(kθ). The Hilbert space H = L2 (Ω, C) splits in two subspaces He and Ho , respectively the spaces of even and odd functions of H. The spaces He and Ho are stable under the dynamics of (11), hence no global controllability is to be expected in H.
3.2
Uniquely resonant case
We first concentrate on the space Ho . The restriction A of i∆ to Ho is skew adjoint, with simple eigenvalues (ik 2 )k∈N associated to the eigenvectors 1 . φk : θ 7→ √ sin(kθ) π k∈N The restriction B of ψ 7→ cos(θ)ψ to Ho is skew-symmetric. The couple (A, B) satisfies Assumption 1.
12
The Galerkyn approximation of A and B at order 0 i 0 ··· 0 1/2 .. 0 4i . . . . (N ) (N ) A = . and B = i 0 . . . . . . . . . 0 .. 2 0 ··· 0 N i 0
N are 1/2
0
0 .. .
1/2
..
. ···
0 1/2 0
··· .. . .. .
0 .. .
0 0 1/2 1/2 0
Our aim is to transfer the wave function from the first eigenspace to the second one. The numerical simulation will be done on some finite dimensional space CN . The constant CA,B of Assumption 1.4 is 1. The controls we will use in the following have L1 norm less than 13/3 and, from Proposition 4, the A norm of Υut (φ1 ) will remain less than exp(13/3) ≈ 76 for every time. From Proposition 6, the error done p when replacing the original system by its Galerkyn approximation of order 76/10−4 ≈ 872 is smaller than = 10−4 . This estimate is indeed very conservative and it can be improved using the regularity of the operator B. Proposition 9. Let k be an integer. If B is bounded, then for every t in R+ , for every locally integrable control u (not necessarily periodic), Z t l k k−1 X |hφl , B l φ1 | Z t kB k φk k |hφk , Υut φ1 i| ≤ |u(τ )|dτ + |u(τ )|dτ . l! k! 0 0 l=0
Proof. We do the proof in the case where u (and hence Pu) is a finite sum of Rt Dirac functions (i.e., t 7→ 0 Pu(τ )dτ is piecewise constant). Since the set of finite sums of Dirac functions is dense (for the metric of total variation) in the set of Radon measures, the result can be extended to locally integrable controls by continuity (Proposition 2). ˙ For every t ≥ 0, consider t 7→ ψ(t), the solution of ψ(t) = (P|u|(t)A + sg(u)B)ψ(t) with Rinitial condition φ1 . t Set z(t) = e− 0 P|u|(s)dsA ψ(t) and define zk (t) = hφk , q(t)i. Because of the boundedness of B, for every 1 ≤ l ≤ k, the function t 7→ qk (t) is piecewise C l for every l ≤ k and, for almost every t ≥ 0, l d
qk (t) = | φk , M1 (t)l q(t) | = | φk , B l q(t) | = | B l φk , q(t) |. dtl In particular, the lth derivative of qk admits everywhere in R a left and a right limit. The moduli of these two limits is the same, only the phase may be (l) discontinuous. Notice that thanks to the Cauchy–Schwarz inequality, |qk (t)| ≤ kB l φk k. After l integrations on [0, t], one finds |hφk , q(t)i| ≤
l−1 X |hφk , B j ψ0 i| j=0
j!
The result follows from Proposition 7. 13
tj +
kB l φk k l t . l!
Applying Proposition 9, the particular tri-diagonal form of B yields |hφk , Υut φ1 i| ≤
kukkL1 . k!
If kukL1 ≤ 13/3, then kπ22 B(Id − π22 )Υut (φ1 )k ≤ 2.10−8 for every t in R+ . Plugging this inequality in (7), one gets that the error done when replacing the original system by its Galerkyn approximation of order 22 is smaller than = 10−7 when kukL1 ≤ 13/3. The transition (1, 2) is not uniquely resonant, since, for instance, λ4 = 42 , λ5 = 52 and 52 − 42 = 9 = 3(22 − 12 ). However, for every l1 , l2 such that λl1 − λl2 ∈ 3Z and hφl1 , Bφl2 i = 6 0, one has l1 > 2 and l2 > 2. Hence the limit matrix M † (Lemma 8) lets invariant the subspace generated by φ1 and φ2 and the result of Proposition 3 applies. We illustrate the notion of efficiency on some examples of control, namely u∗ : t 7→ cosl (3t) for l ∈ {1, 2, 3, 4, 5}. The efficiency is zero when l is even. In numerical simulations, the quantity (22),u∗ |hφ2 , Xt φ1 i| is less than 2.10−5 for every t < 500 (see Figure below for l = 2). When l is odd, the efficiency is not zero. To estimate numerically the efficiency, one considers, for n = 10 and n = 30, the first maximum p† of (N ),u∗ /n t 7→ |hφ2 , Xt φ1 i|, reached at time t† , and computes (1 − p† )nπ . R t† 2|hφ1 , Bφ2 i| 0 |u∗ (τ )|dτ The Scilab source codes used for the simulation are available on the web page http://www.iecn.u-nancy.fr/∼chambrio/PreprintUK.html. We sum up the results in the following tabular. Control u∗ n Time t† Precision Numerical (Theoretical Efficiency) 1 − p† Efficiency n=1 6.8 2.10−2 73% t 7→ cos(3t) n = 10 63 4.10−4 78% π/4 ≈ 79% n = 30 189 3.10−5 78% n=1 8.9 2.10−2 83% t 7→ cos(3t)3 n = 10 84 2.10−4 88% 9π/32 ≈ 88% n = 30 252 2.10−5 88% n=1 10 7.10−3 93% t 7→ cos(3t)5 n = 10 101 2.10−4 92% 75π/256 ≈ 92% n = 30 302 2.10−5 92%
3.3
Not uniquely resonant case
We concentrate on the space He . The restriction A of i∆ to He is skew adjoint, with simple eigenvalues (ik 2 )k∈N∪{0} associated to the eigenvectors (φk )k∈N∪{0} , with φk : θ √1π 7→ cos(kθ) for k in N and φ0 : θ 7→ √12π . The restriction 14
B of ψ 7→ cos(θ)ψ to He is skew-symmetric. The couple (A + i, B) satisfies Assumption 1. The translation from A to A + i just induces a phase shift and will be neglected in the following. The Galerkyn approximation of A and B at order N are √i 0 0 · · · 0 2 0 0 ··· 0 . .. √i i . .. 0 . . 2 .. .. 2 0 (N ) i . ,B .. .. A(N ) = = i . . . . 0 0 0 .. ... .. 0 .. .. i i 2 . 0 . 0 ··· 0 (N − 1) i 0
···
2
0
i 2
2
0
Our aim is to transfer the population from the first eigenspace, associated with eigenvalue 0, to the second one, associated with eigenvalue i. The transition (1, 2) is not uniquely resonant, and in contrary to what happens on the space of odd eigenfunctions, the limit matrix M † does not stabilize the space spanned by φ1 and φ2 . Note however that B only connects level 2 to levels 1 and 3. In other words, it is enough to find a 2π-periodic function u∗ such that E (2,3) (u∗ ) is zero and E (1,2) (u∗ ) is not zero (and as large as possible) P to induce the desired transfer. This is achieved, for instance, with u∗ = n 2δ2πn + δ2πn−π/3 + δ2πn+π/3 (this control is the limit of the sequence of piecewise constant controls build in [BCCS11]), for which E (1,2) (u∗ ) = 1+cos(π/3) = 34 and E (2,3) (u∗ ) = 0. 2 Another example is presented on Figure 2.
4
Conclusion
This main result of this paper will not surprise anyone already familiar with quantum control. Its interest lies rather, first in the theoretical rigorous proof of the convergence and second, in the interpretation of the notion of efficiency, seen as a measure of the wasted L1 norm of the control to achieve a given transition. Despite some recent advances in the field of bilinear control of skew-adjoint operators, many questions remain open. Among other topics, future works may concentrate on the design of time-efficient controls or on the control of systems where operators have continuous spectrum.
Acknowledgments This research has been supported by the European Research Council, ERC StG 2009 “GeCoMethods”, contract number 239748, by the ANR project GCM, program “Blanche”, project number NT09-504490, and by the Inria NancyGrand Est “COLOR” project. It is a pleasure for the author to thank Pierre Rouchon who suggested him this question, Ugo Boscain, Marco Caponigro, Julien Salomon, Mario Sigalotti 15
and Dominique Sugny for many discussions and advices, Nabile Boussa¨ıd for his most valuable help on the functional analysis topics and Denzil Millichap for many corrections. The author wishes also to thank the Institut Henri Poincar´e (Paris, France) for providing research facilities and a stimulating environment during the “Control of Partial and Differential Equations and Applications” program in the Fall 2010.
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17
1.8e−05
1.6e−05
1.4e−05
1.2e−05
1.0e−05
8.0e−06
6.0e−06
4.0e−06
2.0e−06
0.0e+00 0
50
100
150
200
250
300
0
50
100
150
200
250
300
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Figure 1: Evolution of the modulus of the second coordinate when applying the controls : t 7→ cos2 (3t)/30 (top) and t 7→ cos3 (3t)/30 (bottom) on the planar molecule (odd subspace) with initial condition φ1 . The simulations have been done on a Galerkyn approximation of size N = 22.
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1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
Figure 2: Evolution of the modulus of coordinate when applying the the second 1 2 t+π/3 control 20 2 cos2 2t + cos2 t−π/3 + cos on the planar molecule 2 2 (even subspace) with initial condition φ1 . The simulation has been done on a Galerkyn approximation of size N = 22. Precision 1 − p† is equal to 2.10−3 . Numerical efficiencies are 38% (theoretical: 3/8) for the transition (1, 2) and less than 5.10−4 for the transition (2, 3) (theoretical: 0).
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