Control of bilinear systems: multiple systems and perturbations - …turinici/images/... · 2014-04-11 · Control of bilinear systems: multiple systems and perturbations Gabriel

Control of bilinear systems: multiple systems andperturbations

Gabriel Turinici

CEREMADE,Universite Paris Dauphine

ESF OPTPDE Workshop InterDyn2013, Modeling and Control ofLarge Interacting Dynamical Systems September 10-12, Paris, 2013

Gabriel Turinici (U Paris Dauphine) Multiple and perturbed bi-linear control InterDyn2013, Paris 1 / 42

Motivation

Outline

1 MotivationMathematical 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


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

Outline





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 ∂∂tU(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 and perturbed bi-linear control InterDyn2013, Paris 17 / 42

Controllability

Outline





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; recent works by M. Morancey, V. Nersesyan

• 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 and perturbed bi-linear control InterDyn2013, Paris 37 / 42



Theorem (M. Belhadj, J. Salomon, GT 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.




Theorem (M. Belhadj, J. Salomon, GT 2013)

Consider the collection of control systems on SU(N):dYk (t)

dt ={A + (u(t) + δku(t))B

}Yk(t),

Yk(0) = Yk,0 ∈ SU(N).

(21)

Suppose that there exists 0 < t1 < t2 <∞ such that δku(t) = αk

(constant) ∀t ∈ [t1, t2]. Then there exists TA,B,α1,··· ,αKsuch that if

t2 − t1 ≥ TA,B,α1,··· ,αKthe collection of systems (21) is simultaneously

controllable at any time T ≥ t2.




Different model:dZk (t)dt = AZk(t) + [u(t) + αk ]ξ(t)BZk(t),

Zk(0) = Zk,0 ∈ SU(N),

(22)

Controls : u(t), ξ(t) with ξ(t) ∈ {0, 1}, ∀t ≥ 0 (t 7→ ξ(t) measurable).

Theorem

The system (22) is simultaneously controllable if and only if LA,B = su(N).



Controllability in presence of known perturbations:extensions and perspectives

• slowly varying perturbations, (beyond piecewise constant )

• 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). (23)

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).

(24)



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)〉 (25)

|ψ(0)〉 = |ψ0〉, (26)

• numerics (joint works with M. Belhadj, J. Salomon, C. Lefter, B.Gavrilei) : several approaches : monotonic algorithms, Lyapunovapproaches, multi-criterion optimization. In general it is difficult to findthe field robust to control.


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Control of bilinear systems: multiple systems and perturbations - …turinici/images/... · 2014-04-11 · Control of bilinear systems: multiple systems and perturbations Gabriel