bi-linear control : multiple systems and / or perturbationsturinici/images... · motivation optical...

Bi-linear control : multiple systems and / orperturbations

Gabriel Turinici

CEREMADE,Universite Paris Dauphine

Grenoble, GDRE ConEDP Meeting 2013, April 10th, 2013

Gabriel Turinici (U Paris Dauphine) Multiple bi-linear control Grenoble, ConEDP 2013 1 / 68

Motivation

Outline1 Motivation

Mathematical questionsOptical and magnetic manipulation of quantum dynamics

2 ModelizationSingle quantum system: wavefunction formulationSeveral quantum systems: density matrix formulationSimultaneous control of quantum systemsEvolution semigroupObservables

3 ControllabilityBackground on controllability criteriaSimultaneous controllability of quantum systemsControllability of a set of identical moleculesControllability in presence of known perturbations

4 Numerical algorithms: general monotonic schemesEvolution equations and optimal control functionalMonotonic algorithmsMore non-linear situationsGeneral monotonic algorithmsInterpretation of monotonic and tracking algorithmsLyapounov (tracking) algorithmsExperimental control and related numerical algorithms

5 Inverse problemsIdentifiabilityOptimal identificationOther identification approaches


Motivation Mathematical questions

Simultaneous controllability

Isolated system: ddt x(t) = f (t, x(t), u(t)).

Collection of similar systems

d

dtxk(t) = fk(t, xk(t), u(t)), k = 1, ...,K . (1)

Similar may mean fk(t, x , u) = f (t, x , αku), αk ∈ R. The systemsdiffer in their interaction with the control e.g. for instance are placedspatially differently with respect to the controlling action.

Can one simultaneously control all systems with same u(t) ?


Motivation Mathematical questions

Controllability of “large perturbations”

Un-perturbed system: ddt x(t) = f (t, x(t), u(t)).

Perturbed system:

d

dtxk(t) = f (t, xk(t), u(t) + δuk(t)), k = 1, ...,K . (2)

Large perturbations: δuk(t), k = 1, ...,K in a fixed list (alphabet) butthere is no knowledge about which k will actually occur.

Can one still control the system ?


Motivation Optical and magnetic manipulation of quantum dynamics

Laser SELECTIVE control over quantum dynamics

Figure: Optimized laser pulses can be used to control molecular dynamics. a, An optimized laser pulse excites benzene intothe superposition state S0 + S1 with bidirectional electron motion that results in switching between two discrete Kekulestructures on a subfemtosecond timescale; b: A different optimized laser pulse excites benzene into the superposition state S0 +S2, which is triply ionic; c: The next frontier in simulating control is to excite both electronic and nuclear dynamicssimultaneously. Credits: NATURE CHEMISTRY, VOL 4, FEBRUARY 2012, p 72.



Figure: Studying the excited states of proteins. F. Courvoisier et al., App.Phys.Lett.



Quantum non-demolition measurements

Figure: Physics Nobel prize 2012 to Haroche and Wineland ”for ground-breaking experimental methods that enablemeasuring and manipulation of individual quantum systems. Serge Haroche and David Wineland have independently inventedand developed methods for measuring and manipulating individual particles while preserving their quantum-mechanical nature,in ways that were previously thought unattainable”; [these are ] ”the first tiny steps towards building a quantum computer”.Picture credits: Nature 492, 55 (06 December 2012)



Other applications

• EMERGENT technology

• NMR: spin interacting with magnetic fields; control by magnetic fields

• creation of particular molecular states

• fast “switch” in semiconductors

• ...


Modelization

Outline1 Motivation







Modelization Single quantum system: wavefunction formulation

Single quantum system

Time dependent Schrodinger equation{i ∂∂t Ψ(x , t) = H(t)Ψ(x , t)Ψ(x , t = 0) = Ψ0(x).

(3)

• H(t) = H0+ interaction terms E.g. H0 = −∆ + V (x)• H(t)∗ = H(t) thus ‖Ψ(t)‖L2 = 1, ∀t ≥ 0.• dipole approximation: H(t) = H0 − ε(t)µ(x)• E.g. O − H bond, H0 = − ∆

2m + V , m = reduced mass

V (x) = D0[e−β(x−x0) − 1]2 − D0, µ(x) = µ0xe−x/x∗

• higher order approximation: H(t) = H0 +∑

k εk(t)µk(x)

• misc. approximations (rigid rotor interacting with two-color linearlypolarized pulse): H(t) = H0 + (E1(t)2 + E2(t)2)µ1 + E1(t)2 · E2(t)µ2


Modelization Several quantum systems: density matrix formulation

Several quantum systems: density matrix formulation

• Evolution equation for a projector

Let Pψ(t) = Ψ(t)Ψ†(t) i.e. the projector on Ψ(t), also noted in bra-ketnotation |Ψ(t)〉〈Ψ(t)| ; then

i∂

∂tPψ(t) =

(i∂

∂tΨ(t)

)Ψ†(t) + Ψ(t)

(− i

∂

∂tΨ(t)

)†(4)(

H(t)Ψ(t))

Ψ†(t) + Ψ(t)(− H(t)Ψ(t)

)†= (5)

[H(t),Ψ(t)Ψ†(t)] = [H(t),Pψ(t)]. (6)


Modelization Several quantum systems: density matrix formulation

Several quantum systems: density matrix formulation

• For a sum of projectors (density matrix): ρ(t) =∑

k ηkPψk (t). Bylinearity the equation for the density matrix evolution{

i ∂∂t ρ(x , t) = [H(t), ρ(x , t)]ρ(x , t = 0) = ρ0(x).

(7)

Usually ηk ∈ R+ define a discrete probability law with ηk the probability tobe at t = 0 in state Ψk(0). This gives interpretation in terms ofobservables.


Modelization Simultaneous control of quantum systems

Simultaneous control of quantum systems

Under simultaneous control of a unique laser field L ≥ 2 molecular species.

Initial state |Ψ(0)〉 =∏L`=1 |Ψ`(0)〉.

Any molecule evolves by its own Schrodinger equationi~ ∂∂t |Ψ`(t)〉 = [H`

0 − µ` · ε(t)]|Ψ`(t)〉


Modelization Simultaneous control of quantum systems

A set of identical molecules with different spatial positions

N ≥ 2 identical molecules, DIFFERENT orientations, simultaneous controlby one laser field.Interaction with a field µε(t)αk where αk depends on R = (r , θ, ζ) whichcharacterizes the localization of the molecule in the ensemble.


Modelization Evolution semigroup

Evolution semigroup

{i ∂∂t U(t) = H(t)U(t)U(0) = Id .

(8)

Relationship with- wavefunction forumulation Ψ(t) = U(t)Ψ(0)- density matrix version: ρ(t) = U(t)ρ(0)U†(t)


Modelization Observables

Observables: localization

Measurable quantities: 〈Ψ(t),OΨ(t)〉 for self-adjoint operators O (rq:phase invariance).

For density matrix: ρ(t) =∑

k ηkPψk (t)∑k

ηk〈Ψk(t),OΨk(t)〉 = Tr(ρO). (9)

coherent with the probabilistic interpretation.Important example: O = (sum of) projection(s) to some (eigen)state.


Modelization Observables

Observables: localizatione.g. O-H bond : O(x) = γ0√

πe−γ

20 (x−x ′)2

Figure: Successful quantum control for the localization observable.Gabriel Turinici (U Paris Dauphine) Multiple bi-linear control Grenoble, ConEDP 2013 17 / 68

Controllability

Outline1 Motivation







Controllability Background on controllability criteria

Single quantum system, bilinear control

Time dependent Schrodinger equation{i ∂∂t Ψ(x , t) = H0Ψ(x , t)Ψ(x , t = 0) = Ψ0(x).

(10)

Add external BILINEAR interaction (e.g. laser){i ∂∂t Ψ(x , t) = (H0 − ε(t)µ(x))Ψ(x , t)Ψ(x , t = 0) = Ψ0(x)

(11)

Ex.: H0 = −∆ + V (x), unbounded domainEvolution on the unit sphere: ‖Ψ(t)‖L2 = 1, ∀t ≥ 0.



Controllability

A system is controllable if for two arbitrary points Ψ1 and Ψ2 on the unitsphere (or other ensemble of admissible states) it can be steered from Ψ1

to Ψ2 with an admissible control.

Norm conservation : controllability is equivalent, up to a phase, to saythat the projection to a target is = 1.



Galerkin discretization of the Time Dependent Schrodingerequation

i∂

∂tΨ(x , t) = (H0 − ε(t)µ)Ψ(x , t)

• basis functions {ψi ; i = 1, ...,N}, e.g. the eigenfunctions of the H0:H0ψk = ekψk

• wavefunction written as Ψ =∑N

k=1 ckψk

• We will still denote by H0 and µ the matrices (N × N) associated to theoperators H0 and µ : H0kl = 〈ψk |H0|ψl〉, µkl = 〈ψk |µ|ψl〉,



Lie algebra approaches

To assess controllability of

i∂

∂tΨ(x , t) = (H0 − ε(t)µ)Ψ(x , t)

construct the “dynamic” Lie algebra L = Lie(−iH0,−iµ):{∀M1,M2 ∈ L, ∀α, β ∈ IR : αM1 + βM2 ∈ L∀M1,M2 ∈ L, [M1,M2] = M1M2 −M2M1 ∈ L

Theorem If the group eL is compact any eMψ0, M ∈ L can be attained.“Proof” M = −iAt : trivial by free evolutionTrotter formula:

e i(AB−BA) = limn→∞

[e−iB/

√ne−iA/

√ne iB/

√ne iA/

√n]n



Operator synthesis ( “lateral parking”)

Trotter formula: e i [A,B] = limn→∞

[e−iB/

√ne−iA/

√ne iB/

√ne iA/

√n]n

e±iA = advance/reverse ; e±iB = turn left/right



Corollary. If L = u(N) or L = su(N) (the (null-traced) skew-hermitianmatrices) then the system is controllable.“Proof” For any Ψ0, ΨT there exists a “rotation” U in U(N) = eu(N) (orin SU(N) = esu(N)) such that ΨT = UΨ0.• (Albertini & D’Alessandro 2001) Controllability also true for Lisomorphic to sp(N/2) (unicity).sp(N/2) = {M : M∗ + M = 0,M tJ + JM = 0} where J is a matrix unitary

equivalent to

(0 IN/2

−IN/2 0

)and IN/2 is the identity matrix of dimension N/2



Results by the connectivity graph (G.T. & H. Rabitz)

Let us define the connectivity graph

G = (V ,E ),V = {ψ1, ..., ψN}; E = {(ψi , ψj), i 6= j ,Bij 6= 0}

A =

1.0 0 0 0 00 1.2 0 0 00 0 1.3 0 00 0 0 2.0 00 0 0 0 2.15

,B =

0 0 0 ∗ ∗0 0 0 ∗ ∗0 0 0 ∗ ∗∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0



Look for degenerate transitions : list all ekl = ek − el and look forrepetitions.

A =

1.0 0 0 0 00 1.2 0 0 00 0 1.3 0 00 0 0 2.0 00 0 0 0 2.15

,B =

0 0 0 ∗ ∗0 0 0 ∗ ∗0 0 0 ∗ ∗∗ ∗ ∗ 0 0∗ ∗ ∗ 0 0

e41 = 2.0− 1.0 = 1.0, e51 = 2.15− 1.0 = 1.15e42 = 2.0− 1.2 = 0.8, e52 = 2.15− 1.2 = 0.95e43 = 2.0− 1.3 = 0.7, e53 = 2.15− 1.3 = 0.85



Theorem (G.T. & H.Rabitz 2000, C.Altafini 2001) If the connectivitygraph is connected and if there are no degenerate transitions then thesystem is controllable.

Note: non connected = independent quantum systems.

A =

1.0 0 0 0 00 1.2 0 0 00 0 1.3 0 00 0 0 2.0 00 0 0 0 2.15

,B =

0 0 0 1 10 0 0 1 10 0 0 1 11 1 1 0 01 1 1 0 0


Controllability Simultaneous controllability of quantum systems

Simultaneous controllability of quantum systems

Under simultaneous control of a unique laser field L ≥ 2 molecular species.

Initial state |Ψ(0)〉 =∏L`=1 |Ψ`(0)〉.

Total controllability = can simultaneously and arbitrarily steer any state|Ψ`(0)〉 → |Ψ`(t)〉, t ≥ 0 under the influence of only one laser field ε(t).

Any molecule evolves by its own Schrodinger equationi~ ∂∂t |Ψ`(t)〉 = [H`

0 − µ` · ε(t)]|Ψ`(t)〉



Discretization spaces D` = {ψ`i (x); i = 1, ..,N`}, N`,N` ≥ 3 eigenstates ofH`

0

A` and B` = matrices of the operators H`0 and µ` respectively, with

respect to D`; N =∑L

`=1 N`,

A =

A1 0 . . . 00 A2 . . . 0...

.... . .

...0 0 . . . AL

, B =

B1 0 . . . 00 B2 . . . 0...

.... . .

...0 0 . . . BL

.

Note: the system evolves on the product of spheres S =∏L`=1 S

N`−1C ,

Sk−1C = complex unit sphere of Ck .



Theorem (B. Li, G. T., V. Ramakhrishna, H. Rabitz. 2002, 2003) : If

dimRLie(−iA,−iB) = 1 +L∑`=1

(N2` − 1),

then the system is controllable (the dimension of Lie(−iA,−iB) iscomputed over R). Moreover, if the system is controllable, there exists atime T > 0 such that any target can be reached at or before time T (andthereafter for all t > T ), i.e. for any c0 ∈ S et t ≥ T the set of reachablestates from c0 is S.


Controllability Controllability of a set of identical molecules

Controllability of a set of identical molecules

N ≥ 2 identical molecules, placed under simultaneous control of only onelaser field. The molecules have DIFFERENT orientations.




Linear case:

d

dtx1 = Ax1 + Bu(t), x1(0) = 0 (12)

d

dtx2 = Ax2 + 2Bu(t), x2(0) = 0. (13)

(14)

Then for any control u(t) we have x2(t) = 2x1(t), thus there arerestrictions on the attainable set; no simultaneous control in the linearcase.




Interaction with the control field is µε(t)αk where αk depends on thelocalization R = (r , θ, ζ) of the molecule in the ensemble.

Theorem (GT & H. Rabitz, PRA 2004): Suppose |αk | 6= |αj |, H0 is withnon-degenerate transitions and graph of µ is connected. Then if one(arbitrary) molecule is controllable then the whole (discrete) ensemble is.




Interaction with the control field is µε(t)αk where αk depends on thelocalization R = (r , θ, ζ) of the molecule in the ensemble.

Theorem (GT & H. Rabitz, PRA 2004): Suppose H0 is withnon-degenerate transitions and that the connectivity graph of the systemis connected but not bi-partite. Then if a molecule is controllable then thewhole (discrete) ensemble is.



−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

P(x

)

x=1 single moleculefull ensemble

Figure: The target yield P(x) for different orientations of x = cos(θ). The yield P(x) from a field optimized at x = 1produced the quality control index Q(ε) = 49%. The yield P(x) for full ensemble control of was required to optimize a sampleof M = 31 orientations uniformly distributed over the interval [−1, 1]. The resultant quality index is Q(ε) = 85% (G.T. andH. Rabitz. PRA 2004



Other results from the literature:

• control of PDE: T. Chambrion, K. Beauchard, P. Rouchon: mostly forA + αku(t)B

• finite-dimensional, mostly for spin systems: C. Altafini, N. Khaneja:αkA + u(t)B


Controllability Controllability in presence of known perturbations

Controllability in presence of known perturbations: jointwork with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei

d

dtx = (A + u(t)B)x . (15)

What if u(t) is submitted to a random perturbation in a predefined(discrete) list {δuk , k = 1, ...,K} ?Linear systems:

d

dtx1 = Ax1 + Bu(t), x1(0) = 0. (16)

d

dtx2 = Ax2 + B[u(t) + α], x2(0) = 0. (17)

The dynamics of x2(t)− x1(t) is not influenced by the control:

d

dt(x2 − x1) = A(x2 − x1) + Bα, x2(0)− x1(0) = 0. (18)

Thus: no control of perturbations in the linear case.Gabriel Turinici (U Paris Dauphine) Multiple bi-linear control Grenoble, ConEDP 2013 37 / 68


Controllability in presence of known perturbations: jointwork with M. Belhadj, J. Salomon, C. Lefter, B. Gavrilei

Theorem (GT et al. 2013)

Suppose the bi-linear system on SU(N)

d

dtx = (A + u(t)B)x (19)

is such that the Lie algebra generated by [A,B] and B is the whole Liealgebra su(N). Then for any distinct αk ∈ R, k = 1, ..,K , the collection ofsystems

d

dtxk = (A + [u(t) + αk ]B)x , k = 1, ...,K , (20)

is controllable.



Controllability in presence of known perturbations:extensions and perspectives

• slowly varying perturbations, e.g. piecewise constant with long enoughintervals: OK

• control of more non-linear situations:Rigid rotor interacting with linearly polarized pulse:

i∂

∂tΨ(x , t) =

[H0 + u(t)µ1 + u(t)2µ2 + u(t)3µ3

]Ψ(x , t). (21)

Rigid rotor interacting with two-color linearly polarized pulse:

i∂

∂tΨ(x , t) =

[H0 + (E1(t)2 + E2(t)2)µ1 + E1(t)2 · E2(t)µ2

]Ψ(x , t).

(22)



Controllability in presence of known perturbations:extensions and perspectives

• control of rotational motion: time dependent Schrodinger equation(θ, φ = polar coordinates):

i~∂

∂t|ψ(θ, φ, t)〉 = (BJ2 −

−−→u(t) ·

−→d )|ψ(θ, φ, t)〉 (23)

|ψ(0)〉 = |ψ0〉, (24)


Numerical algorithms: general monotonic schemes

Outline1 Motivation







Numerical algorithms: general monotonic schemes Evolution equations and optimal control functional

Evolution equations and optimal control functional

• Time dependent Schrodinger equation w. BILINEAR interaction (e.g.laser) {

i ∂∂t Ψ(x , t) = (H0 − ε(t)µ(x))Ψ(x , t)Ψ(x , t = 0) = Ψ0(x)

(25)

H0 = −∆ + V (x), unbounded domainEvolution on the unit sphere: ‖Ψ(t)‖L2 = 1, ∀t ≥ 0.• evaluation of the quality of a control through a objective functional tomaximizeJ(ε) = 2<〈ψtarget |ψ(·,T )〉 −

∫ T0 α(t)ε2(t)dt

J(ε) = 2− ‖ψtarget − ψ(·,T )‖2L2 −

∫ T0 α(t)ε2(t)dt

J(ε) = 〈Ψ(T ),OΨ(T )〉 −∫ T

0 α(t)ε2(t)dt


Numerical algorithms: general monotonic schemes Evolution equations and optimal control functional

Euler-Lagrange critical point equation

{i ∂∂t Ψ(x , t) = (H0 − ε(t)µ)Ψ(x , t)Ψ(x , t = 0) = Ψ0(x){i ∂∂tχ(x , t) = (H0 − ε(t)µ)χ(x , t)χ(x , t = T ) = OΨ(x ,T )

αε(t) = −Im 〈χ|µ|Ψ〉 (t)

• Chose a numerical algorithm to update the field ε(t), e.g.,

εn+1 = εn +δJ(εn)

δε(26)

slow convergence =⇒ complicated objective functional surfaceRecent works by Alfio Borzi: functional surface seems to be very flat withmany almost optimal regions.


Numerical algorithms: general monotonic schemes Monotonic algorithms

Compute the optimal field ε(t) (Krotov cf. Tannor et. al 1992):(χk−1, εk−1,Ψk−1)→ (χk , εk ,Ψk){

i ∂∂t Ψk(x , t) = (H0 − εk(t)µ)Ψk(x , t)Ψk(x , t = 0) = Ψ0(x)

(27)

εk(t) = − 1

αIm〈χk−1|µ|Ψk〉(t) (28){

i ∂∂tχk(x , t) = (H0 − εk(t)µ)χk(x , t)

χk(x , t = T ) = OΨk(x ,T )(29)

In practice solve the equations (27)-(28) by propagating the non-linearequation{

i ∂∂t Ψk(x , t) = (H0 + 1α Im〈χ

k−1|µ|Ψk〉(t)µ)Ψk(x , t)Ψk(x , t = 0) = Ψ0(x)

(30)



A general class of algorithms (Y.Maday & G.T. 2002)

{i ∂∂t Ψk(x , t) = (H0 − εk(t)µ)Ψk(x , t)Ψk(x , t = 0) = Ψ0(x)

(31)

εk(t) = (1− δ)εk−1(t)− δ

αIm〈χk−1|µ|Ψk〉(t) (32){

i ∂∂tχk(x , t) = (H0 − εk(t)µ)χk(x , t)

χk(x , t = T ) = OΨk(x ,T )(33)

εk(t) = (1− η)εk(t)− η

αIm〈χk |µ|Ψk〉(t) (34)

Particular cases: Zhu & Rabitz for δ = 1 and η = 1; Krotov (Tannor et al.1992) for δ = 1 and η = 0.



THEOREM If O is an hermitian observable semi-positive definite then,for any η, δ ∈ [0, 2] J(εk+1) ≥ J(εk).

J(εk+1)− J(εk) =⟨Ψk+1(T )−Ψk(T )|O|Ψk+1(T )−Ψk(T )

⟩+

α

∫ T

0(

2

δ− 1)(εk+1 − εk)2 + (

2

η− 1)(εk − εk)2



Figure: Convergence of the monotonic algorithm.


Numerical algorithms: general monotonic schemes More non-linear situations

Evolution equations and optimal control functional

• quadratic nonlinearity in the field{i ∂∂t Ψ(x , t) = (H0 − ε(t)2µ(x))Ψ(x , t)Ψ(x , t = 0) = Ψ0(x)

(35)

ε(t) = − ε(t)α Im〈χ|µ|Ψ〉(t) (singular)

• quartic nonlinearity in the field{i ∂∂t Ψ(x , t) = (H0 − ε(t)3µ(x))Ψ(x , t)Ψ(x , t = 0) = Ψ0(x)

(36)

ε(t) = − αIm〈χ|µ|Ψ〉(t)

• vectorial case (rotation control, NMR):i ∂∂t Ψ(x , t) = [H0 + (v1(t)2 + v2(t)2)µ1 + v1(t)2 · v2(t)µ2]Ψ(x , t).


Numerical algorithms: general monotonic schemes General monotonic algorithms

General monotonic algorithms

X =state, v= control.

• ∂tX + A(t, v(t))X = B(t, v(t))

• minv J(v), J(v) :=∫ T

0 F(t, v(t),X (t)

)dt + G

(X (T )

).

• F , G : C 1 + concavity with respect to X∀X ,X ′ ∈ H, G (X ′)− G (X ) ≤ 〈∇XG (X ),X ′ − X 〉

∀t ∈ R, ∀v ∈ E , ∀X ,X ′ ∈ H,F (t, v ,X ′)− F (t, v ,X ) ≤〈∇XF (t, v ,X ),X ′ − X 〉.



Mean field games (w. A. Lachapelle & J. Salomon)

• Mean field games: limits of Nash equilibriums for infinite number ofplayers (P.L.Lions & J.M.Lasry)• equation for each player dXt = αdt + σdWt , α(t, x) = control• m(t, x) = the density of players at time t and position x ∈ Q• evolution equation

∂

∂tm(t, x)− ν∆m(t, x) + div(α(t, x)m(t, x)) = 0,

m(0, x) = m0(x).

• minα J(α)

J(α) := Ψ(m(·,T )) +∫ T

0

{Φ(m(t, ·)) +

∫Q L(x , α)m(t, x)dx

}dt

• Φ,Ψ are often concave, typical L : L(x , α) = α2

2 .



Direct-adjoint equations and first lemma

∂tX + A(t, v(t))X = B(t, v(t))X (0) = X0

∂tYv − A∗(t, v(t)

)Yv +∇XF

(t, v(t),Xv (t)

)= 0

Yv (T ) = ∇XG(Xv (T )

).

Lemma

Suppose that A,B,F are differentiable everywhere in v ∈ E , then thereexists ∆(·, ·; t,X ,Y ) ∈ C 0(E 2,E ) such that, for all v , v ′ ∈ E

J(v ′)− J(v) ≤∫ T

0∆(v ′, v ; t,Xv ′ ,Yv ) ·E

(v ′ − v

)dt (37)



Well-posedness

J(v ′)− J(v) ≤∫ T

0∆(v ′, v ; t,Xv ′ ,Yv ) ·E

(v ′ − v

)dt (38)

Remark: useful factorisation because can test at each step if J goes theright way; also can choose v ′(t∗) = v(t∗) if pb.

Lemma

Under hypothesis on A,B,F ,G , θ > 0

∆(v ′, v ; t,X ,Y ) = −θ(v ′ − v) (39)

has an unique solution v ′ = Vθ(t, v ,X ,Y ) ∈ E .



Application

A nonlinear vectorial case ∂tX + A(t, v(t))X = B(t, v(t))

v(t) =

(v1

v2

)∈ E = R2 and

A(t, v) = i [H0 + (v1(t)2 + v2(t)2)µ1 + v1(t)2v2(t)µ2]. Denoteξ1 = −Re〈Y , iµ1X 〉V + α(t), ξ2 = −Re〈Y , iµ2X 〉V then

∆(v ′, v ; t,X ,Y ) = ξ1

(v1 + v ′1v2 + v ′2

)+ ξ2

((v1 + v ′1)v ′2

(v1)2

)(40)

and the equation in v ′ is: ∆(v ′, v ; t,X ,Y ) = −θ(v ′ − v) and has for θlarge enough a unique solution

v ′ = Vθ(t, v ,X ,Y ) =

(θ−ξ1)v2−ξ2v2

1θ+ξ1

−θ−ξ1+ξ2

(θ−ξ1)v2−ξ2v21

θ+ξ1

θ+ξ1+ξ2(θ−ξ1)v2−ξ2v

21

θ+ξ1

v1

.



Well-posedness

Theorem (GT, Julien Salomon ’10)

Under hypothesis ... • the following eq. has a solution:

∂tXv ′(t) + A(t, v ′)Xv ′(t) = B(t, v ′) (41)

v ′(t) = Vθ(t, v(t),Xv ′(t),Yv (t)) (42)

Xv ′(0) = X0 (43)

• ∃ (θk)k∈N such that vk+1(t) = Vθk (t, vk(t),Xvk+1(t),Yvk (t))• J(vk+1)− J(vk) ≤ −θk‖vk+1 − vk‖2

L2([0,T ]);

• if vk+1(t) = vk(t) : ∇vJ(vk) = 0.


Numerical algorithms: general monotonic schemes Interpretation of monotonic and tracking algorithms

Interpretation of monotonic and tracking algorithms

J(ε, ψ, χ) = 2<〈ψtarget |ψ(.,T )〉 −∫ T

0 α(t)ε2(t)dt

−2<∫ T

0 〈χ(., t)|∂t + iH − µε(t)|ψ(., t)〉dtEuler-Lagrange equations:

∇χJ →

i ∂∂tψ(x , t) = (H − ε(t)µ(x))ψ(x , t)

ψ(x , t = 0) = ψ0(x)

∇ψJ →

i ∂∂tχ(x , t) = (H − ε(t)µ(x))χ(x , t)

χ(x , t = T ) = ψtarget(x)∇εJ → α(t)ε(t) = −= < χ(., t)|µ|ψ(., t) >




At time t, “best guess for a solution” is ε = ε · χ[0,t] + εref · χ[t,T ].

Forward objective functional (easily to compute at time “t”):i ∂∂tψref (x , t) = (H − εref (t)µ)ψref (x , t), ψref (T ) = ψtarget .

Jfwd(ε, t; εref ) =∫ t

0 αε2(t)dt +

∫ Tt αεref

2(t)dt + ‖ψref (t)− ψ(t)‖2.

Theorem (G.T., Proc. 44th IEEE CDC-ECC Sevilla, Spain, Dec. 2005. ; G. T., J. Salomon, J. Chem. Phys. 124:074102

(2006).): J(ε) = Jfwd(ε, t; εref ).Decisions on the optimality of some control can be made locally e.g. bymonotonic algorithms that ensure Jfwd(ε, t; εref ) is decreasing locally intime.




Evolving state ψk approaches monotonically reference ψkref . No optimization during the backward propagation (i.e.

εk+1 = εk+1), imply ‖ψk+1ref− ψk+1‖ = cst. The shrinking distance between the two trajectories ensures the progression of

the quality functional toward optimal values. The optimal couple of trajectories will be a tube whose nonzero width is related to

the driving laser field fluence penalty α.


Numerical algorithms: general monotonic schemes Lyapounov (tracking) algorithms

Lyapunov algorithms (joint works with M. Mirrahimi, P.Rouchon)

Let us consider V (t) = ‖ψ(t)− ψtarget(t)‖2 with ψtarget(t) a stationarystate i.e. i ∂∂tψtarget(x , t) = H0ψtarget(x , t)

dV

dt= 2ε(t)Im〈µψ, ψtarget〉 (44)

e.g. ε(t) = −aIm〈µψ, ψtarget〉 (a ≥ 0) dVdt will be negative thus the

Lyapunov function V decreases.Remark: we can characterize the limit points by computing all derivativesof V which have the form Im〈[H0...[H0, µ]...]ψ, φ〉.


Numerical algorithms: general monotonic schemes Experimental control and related numerical algorithms

Experimental control

Figure: Experimental quantum control: In practice H and/or the dipole may not be known: one uses then a zero orderalgorithm (no gradients): genetic algorithms, Nelder-Mead simplex (C. Le Bris, H.Rabitz & G.T. PRE 2004) ... This is possible

due to a high laboratory experimental repetition rate (1Hz − 105Hz or more).


Inverse problems

Outline1 Motivation







Inverse problems

Inverse problem coupled with control

d

dtx = f (t, x , u). (45)

Unknown : f and / or x(0).

Measurements: quantities depending on x(t), t ∈ [0,T ].

Idea: use several u(t) to identify f and / or x(0).

Is it possible ?Some controls u(t) may be more discriminant than others ... which u touse ?

Question is related to simultaneous control of several systems but here thegoal is to be as discriminant as possible in a list of candidates for f and /or x(0).


Inverse problems Identifiability

Identifiability

Here µ and/or H0 and/or ψ(0) are unknown, need to recoved frommeasurements

Theorem [C. Le Bris, M. Mirrahimi H. Rabitz, G.T.; 2006 ] Considertwo finite dimensional Hamiltonians H1, H2 and two dipole momentsµ1, µ2, with Ψ1,Ψ2 ,solutions of :

iΨk = (Hk + ε(t) µk) Ψk

Suppose ∀t ≥ 0, ∀ε(t) ∈ L2

|〈Ψ1(t) | ei 〉|2 = |〈Ψ2(t) | ei 〉|2 i = 1, ..,N, (46)

( {ei}Ni=1 = canonical basis of CN).


Inverse problems Identifiability

Assume for H1, H2 :1 Equations are controllable

2 H1 and H2 have the same eigenvalues λi .

3 λi1 − λj1 6= λi2 − λj2 for (i1, j1) 6= (i2, j2).

4 〈φi | µ | φi 〉 = 0, i = 1, ..,N.

5 there does not exist a subspace of dimension one or two spannedby the vectors {ei}, which remains invariant during the freeevolution (ε ≡ 0) of the first system (H1 and µ1).

Then, there exist {αi}Ni=1 such that, for all 1 ≤ i , j ≤ N,

(µ1)ij = e i(αi−αj )(µ2)ij , (H1)ij = e i(αi−αj )(H2)ij . (47)


Inverse problems Optimal identification

Optimal identification

1 Objective functional J(ε, µ,V ) the distance between the measures with the field ε(t) and the

numerical simulation with the potential V , dipole µ and field ε.“Canonical” formulation of the inversion problem : solve

minV ,µ

∫L2

J(ε, µ,V )dε(t) (48)

BUT: space L2 too large ... introduce a discriminating field approachthat allows to distinguish between the admissible candidates.

2 Inversion problem: S(ε) = {µ,V ; J(ε, µ,V ) ≤ tolerance}. m(ε) = ameasure of S(ε).

3 Optimization problem: minimize m(ε). The solution ε is thediscriminating field; S(ε) = set of possible solutionsIn practice: m(ε)= diameter, S(ε) not explicitly computed (!). Algorithms

can use fields coming from the identifiability theory.


Inverse problems Optimal identification

Figure: Optimal identification machine : implementable experimental +numerical algorithms (J.M. Geremia & H. Rabitz JCP, 2003)


Inverse problems Local convexity for inversion

Local convexity for inversion

ı~∂

∂t|ψ(t)〉 = (H0 − ε(t)µ)|ψ(t)〉, |ψ(0)〉 = |i〉, Pif (ε) = |〈f , ψ(T )〉|2 .

(49)For all T > 0, we suppose that we can create any field ε ∈ ET and that wecan measure Pif (ε). For M fields {ε1, .., εM} we can collect themeasurements {Pif (ε1), ..,Pif (εM)}. Through (49) Pif is a function of µand a functional of ε, and when necessary this explicit dependence will bewritten as Pif (ε, µ).GOAL: explore the feasibility of estimating the dipole moment matrix µfrom the measured data {Pif (ε1), ..,Pif (εM)} using well chosen controls{ε1, .., εM}.


Inverse problems Local convexity for inversion

Local convexity for inversion

Theorem (Z. Leghtas, P. Rouchon, G.T. 2012)

Consider a real symmetric matrix µ with zero diagonal entries and a realdiagonal matrix H0 with non-degenerate transitions. Suppose that thesystem state in (49) is controllable. Then for any positive constant α thereexists a time T > 0 and M fields (ε1, .., εM) ∈ ETM such that the costfunction

J : µ ∈M→M∑k=1

(Pif (εk , µ)− Pif (εk , µ))2 (50)

is in C2(M,R) and locally α-convex around µ.


Inverse problems Numerical construction of discriminating sequence

Numerical construction of discriminating sequence

A suitable set of discriminating controls seems to exist. How to constructit in practice ?

Y. Maday and J. Salomon (2009) construct, by a greedy algorithm, asequence of controls uk , k ≥ 1 that generate the most discriminantmesurements for a given observable operator:• u1: maximizes the norm of measurement• u2 maximizes the interpolation error of the reconstruction done with u1

• and so on ...


bi-linear control : multiple systems and / or perturbationsturinici/images... · motivation optical...

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