Parametrizations of a two-levelquantum control system

Markku Nihtilaand Petri Kokkonen

Department of Mathematics and StatisticsUniversity of Kuopio

POB 1627, FI-70211 Kuopio, FinlandEmail: markku.nihtila; petri.kokkonen@uku.fi

Jean LevineCentre Automatique et Systemes

Ecole Nationale Superieure des Mines de Paris35 rue Saint-Honore, F-77305 Fontaineblau, France

Email: jean.levine@ensmp.fr

Abstract— A known two-level population transfer model ofa quantum system is studied. Construction of the basic four-dimensional real-variable model is repeated first. By using atechnique of underdetermined systems of ordinary differentialequations (ODE) the two scalar control variables are representedthen as functions of the state variables. This representation isused to obtain an underdetermined system of nonlinear ODEs,which does not include the control variables. Via a flatness-basedidea two state variables are used to parametrize the remainingones and the two controls. The coordinate system of the states isconverted into polar form. Then another form of parametrizationis obtained for the states and controls. A quadratic energyminimizing optimization is also studied. It is converted via thetwo parametrizations to two equivalent variational problems. Ageneral solution to the polar-form variational problem is given.Simulation results to be presented in a companion paper willcomplete results of this study.

I. INTRODUCTION

Molecular excitation, i.e. driving of an ensemble ofmolecules from one locally stable steady state to another iscarried out by using coherent light. Based on laser technologyshorter and shorter coherent pulses can be generated forcontrolling molecular excitation, see [16]–[19]. The goal isto direct molecular reactions towards unprobable but desirabledirection [1]-[6]. Then nonlinear and more and more sophis-ticated control methods are needed for properly designingdurations and forms of the control pulses. In classical N-levelproblems [4] the system to be controlled can be modelled byusing ordinary 2N -dimensional differential equation systems.Due to femto- and picosecond scale pulses feedback is notin general applicable in the control design for these systems.Flatness-based control [7]–[11] is then an ideal methodologyfor open-loop design needed in quantum control problems.

The system

dx

dt= f(x, u); x(t) ∈ Rn, u(t) ∈ Rm (1)

is called differentially flat if there exists algebraic functionsA, B, C, and finite integers α, β, and γ such that for any pair(x, u) of inputs and controls, satisfying the dynamics (1),there exists a function z, called a flat (or linearizing) output,such that

x(t) = A(z, z, . . . , z(α))u(t) = B(z, z, . . . , z(β)) (2)z(t) = C(x, u, u, . . . , u(γ)).

Here we study the problem of a two-level quantum systemfrom the flatness viewpoint. We parametrize the controls intwo different coordinates. A quadratic energy minimizingoptimization problem is posed and converted to two differentbut equivalent problems of variational calculus. A generalsolution to the polar-form variational problem is given. Simu-lation results to be presented in a companion paper [12] willcomplete results of the model studies presented here.

II. SYSTEM MODELS

We start with the standard finite-state Schrodinger equationof two energy levels and transform it into control theoreticform. Via elimination of the control variables un underdeter-mined system of differential equations is obtained, which thenserves as a basis for parametrization.

A. Model conversion from physics to control

At first, we repeat here shortly as a background the deriva-tion of our dynamics. This was given in [2]–[4]. Populationtransfer in a two-level quantum system can be described bythe time-dependent Schrodinger equation, i.e. by the dynamics

idψ

dt= H(t) ψ, H(t) =

[E1 Ω(t)

Ω∗(t) E2

], (3)

where the modified Planck’s constant ~ = h2π has been scaled

to ~ = 1, and i =√−1. The wavefunction ψ : R → C2 has

the probabilistic interpretation, in the sense that

‖ψ(t)‖2 = |ψ1(t)|2 + |ψ2(t)|2 = 1, ∀ t ∈ R, (4)

where ψ = (ψ1, ψ2). The control is given by Ω : R → C, andΩ∗ is the complex conjugate of Ω. E1 and E2 are the energylevels. The unitary transformation ψ 7→ ψ and Ω 7→ u by

ψ(t) = U(t)ψ(t), U(t) =[e−iE1t 0

0 e−iE2t

](5)

u(t) = e−i(E2−E1)t Ω(t) (6)

1-4244-0987-X/07/$25.00 ©2007 IEEE.

transforms (3) to

idψ

dt= H(t)ψ, H(t) =

[0 u(t)

u∗(t) 0

]. (7)

The componentwise representation

ψ(t) = ψ1(t)[

10

]+ ψ2(t)

[01

](8)

converts (7) to the dynamics

ψ1 = −iuψ2,

ψ2 = −iu∗ ψ1.(9)

By using the real-valued decompositionsψ1 = x1 + ix2

ψ2 = x3 + ix4

u = u1 + iu2

one obtains a state-variable representation in standard controltheoretic form

x1

x2

x3

x4

=

x4 x3

−x3 x4

x2 −x1

−x1 −x2

[u1

u2

](10)

or in another formdxdt

=(u1F1 + u1F2

)x,

x = [x1 x2 x3 x4]T,

F1 =

0 0 0 10 0 −1 00 1 0 0

−1 0 0 0

, F2 =

0 0 1 00 0 0 1

−1 0 0 00 −1 0 0

The constraint equation is

4∑k=1

x2k = 1.

Remark 1: It is clear that the system (10) is not controllablein R4. This is evident from the original Schrodinger equation(3) and its consequence (4). On the other hand, the system iscontrollable in

S3 =x ∈ R4

∣∣∣ 4∑k=1

x2k = 1

, (11)

i.e. the state x can be driven from any point x0 ∈ S3 to anyother point xT ∈ S3.

Remark 2: It can be shown that F1 and F2 together withtheir Lie product F3 = 2[F1, F2] = 2(F1F2 − F2F1) forma Lie algebra with some isomorfic ”brothers”. This can beused as a basis for differential geometric considerations of thecontrol system (10), see [3]. However, the elementary approachapplied in this paper is sufficient for our parametrizationpurposes.

B. Conversion from control model to underdetermined system

By applying the flatness ideas of [7] in the modified system[x1

x2

]=

[x4 x3

−x3 x4

] [u1

u2

](12)[

x3

x4

]=

[x2 −x1

−x1 −x2

] [u1

u2

](13)

via elimination of the controls with the aid of the equationsof the controls solved from (13)[

u1

u2

]=

[x2 −x1

−x1 −x2

]−1 [x3

x4

](14)

we arrive at the underdetermined system with two equationsand four variables and their derivatives

(x21 + x2

2)[

x4 −x3

x3 x4

] [x1

x2

](15)

= (x23 + x2

4)[

x2 −x1

−x1 −x2

] [x3

x4

].

This can be interpreted as an underdetermined differentialequation system of the form (see [7])

F (x, x) = 0. (16)

It means that there is a certain freedom to choose the variablesin such a way that the equation is satisfied. Actually, thisfreedom corresponds to choosing the control variables moreor less arbitrarily.

III. FIRST PARAMETRIZATION

When we apply the abbreviationsn2 = x2

1 + x22

f =√

1− n2 − x23 (= x4),

multiply the system (15) from the left by the matrix

C−1 =1n2

[x2 −x1

−x1 −x2

],

and eliminate x3 from the system we obtain a stateparametrization

x3 = ϕ3(x1, x2) = (x1x2 − x1x2)

√1− n2

n2(x21 + x2

2), (17)

x4 = ϕ4(x1, x2) =√

1− n2 − ϕ3(x1, x2)2. (18)

By using of these the controls are then parametrized by x1, x2

and their time derivatives[u1

u2

]=

11− x2

1 − x22

[x1ϕ4 − x2ϕ3

x1ϕ3 + x2ϕ4

]. (19)

In population transfer problems from the level 1 correspondingto the situation |ψ1(0)|2 = x1(0)2 + x2(0)2 = 0 to the level2, where |ψ2(T )| = x3(T )2 + x4(T )2 = 0, where T is thetransfer time, we can parametrize the partial trajectory byusing a sufficiently smooth, but otherwise arbitrarily chosen,

parametrization x1, x2 : [0, T ] → R with the boundaryconditions

x1(0)2 + x2(0)2 = 0, (20)x1(T )2 + x2(T )2 = 1. (21)

Remark 3: It has to be noted, that in practical simulationsthe zero-divisor problem appearing in (19) is solved by usingL’Hopital’s rule (several times). Then we can conclude thatu1(0) = u2(0) = 0. These issues together with numericalsimulations are studied in a companion paper [12].

Remark 4: When considering controllability as a necessarycondition one must be careful, i.e. we have take care, what isthe manifold, where the system is controllable. It is seen thatwe have found for the system (10) the functions A, B, andC in the sense of the definition in Introduction. However, thesystem (10) is not controllable in the manifold R4, but onlyin the manifold S3, which is a proper submanifold of R4, andin any scaled manifold

αS3 =αx ∈ R4

∣∣∣ 4∑k=1

x2k = 1, α > 0

.

IV. PARAMETRIZED ENERGY APPROACH

If the laser energy applied will be optimized, the costfunctional approach can be used in some appropriate controlset U . A feasible set and the constrained optimization problemare

U = u : [0, T ] → R2 | ‖u(t)‖ ≤ 1

minu∈U

J(u) =12

∫ T

0

[u1(t)2 + u2(t)2

]dt (22)

subject to the constraint equation (10).

The optimization task can be converted to a standard vari-ational problem. From (19), and by appying (17)–(18), weobtain

u21 + u2

2 =1

(1− n2)2[(x1ϕ4 − x2ϕ3)2

+ (x1ϕ3 + x2ϕ4)2]

=1

(1− n2)2(x2

1 + x22)(ϕ

23 + ϕ2

4)

=x2

1 + x22

1− x21 − x2

2

. (23)

This in turn with (22) results in the variational problem

minz∈Z

V (z) =12

∫ T

0

K(z, z) dt (24)

z = (x1, x2)Z = z : [0, T ] → R2 | ‖z(t)‖ ≤ 1 (25)

K(z, z) =‖z‖2

1− ‖z‖2, (26)

where the function set Z must be sufficiently smooth forguaranteeing the existence of the solution.

V. POLAR REPRESENTATION

It is natural to consider quantum control problems in spheri-cal coordinates due to the wavefunction constraint ‖ψ(t)‖ = 1.

A basic polar representation in R4 is given byx1 = r sinψ cos θ cosϕx2 = r sinψ cos θ sinϕx3 = r sinψ sin θx4 = r cosψ

(27)

When the system is evolving in S3 we set r = 1. The derivativerepresentations needed in the polar conversion are

x1 = cosψ cos θ cosϕ ψ − sinψ sin θ cosϕ θ− sinψ cos θ sinϕ ϕ (28)

x2 = cosψ cos θ sinϕ ψ − sinψ sin θ sinϕ θ+ sinψ cos θ cosϕ ϕ. (29)

VI. SPHERICAL OPTIMIZATION PROBLEM

The integrand K(z, z) in (26) has to be converted into polarform by using the basic relation (27) and the derivatives (28)& (29). The two terms in K are

x21 + x2

2 =(cosψ cos θ cosϕ ψ − sinψ sin θ cosϕ θ

− sinψ cos θ sinϕ ϕ)2

+(cosψ cos θ sinϕ ψ − sinψ sin θ sinϕ θ

+sinψ cosϕ ϕ)2

= cos2 ψ cos2 θ ψ2 + sin2 ψ sin2 θ θ2

+sin2 ψ cos2 θ ϕ2

−2 cosψ sinψ cos θ sin θ ψθ= (cosψ cos θ ψ + sinψ sin θ θ)2 + sin2 ψ cos2 θ ϕ2

=[d

dt(sinψ cos θ)

]2

+ (sinψ cos θ)2 ϕ2 (30)

1− x21 − x2

2 = 1− sin2 ψ cos2 θ (31)

By definingw = sinψ cos θ (32)

and using the polar parametrization for (w,ϕ) we obtain anew optimization problem, which is, however, equivalent withthe minimization problem of (24):

minq∈Q

J(q) =12

∫ T

0

L(q, q) dt (33)

q = (q1, q2) = (w,ϕ)Q = (w,ϕ) : [0, T ] → [−1, 1]× [0, 2π) (34)

L(q, q) =w2 + w2ϕ2

1− w2, (35)

where the mappings in Q must be sufficiently differentiable.Then the standard Euler-Lagrange technique can be used tosolve the optimization problem.

VII. POLAR CONTROL PARAMETRIZATION

By using the polar parameters (w,ϕ) the parameter-statesand their derivatives are

x1 = θ1(w,ϕ) = w cosϕx2 = θ2(w,ϕ) = w sinϕ

(36)x1 = w cosϕ− w sinϕ ϕx2 = w sinϕ+ w cosϕ ϕ

(37)

Substituting the expressions (38)–(40)

x21 + x2

2 = w2 (38)x2

1 + x22 = w2 + w2ϕ2 (39)

x1x2 − x2x1 = w cosϕ(w sinϕ+ w cosϕ ϕ)+ w sinϕ(−w cosϕ+ w sinϕ ϕ)= w2ϕ (40)

to the formulae (17) & (18) of x3 and x4 one obtains theparametrizations

x3 = θ3(w,ϕ) = wϕγ(w,ϕ) (41)x4 = θ4(w,ϕ) = wγ(w,ϕ) (42)

γ(w,ϕ) =

√1− w2

w2 + w2ϕ2. (43)

Polar parametrization of the controls are obtained via substi-tution of the expressions of θi(w,ϕ), i = 1, ..., 4 to (19)

u1 = c(w,ϕ)[(w2 − w2ϕ2) cosϕ− 2wwϕ sinϕ

](44)

u2 = c(w,ϕ)[(w2 − w2ϕ2) sinϕ+ 2wwϕ cosϕ

](45)

c(w,ϕ) =1√

(1− w2)(w2 + w2ϕ2)=

γ

1− w2.

VIII. EULER-LAGRANGE EQUATIONS OF THEPOLAR-FORM MINIMIZATION

The variational problem (33)–(35) of the polar form givesthe Euler-Lagrange equations

d

dt

∂L

∂qk− ∂L

∂qk= 0 ; k = 1, 2 (46)

which can be obtained in explicit form as

∂L

∂ϕ= 0

∂L

∂ϕ=

w2

1− w2ϕ

d

dt

∂L

∂ϕ=

w2

1− w2ϕ+

2ww(1− w2)2

ϕ

d

dt

∂L

∂ϕ− ∂L

∂ϕ= 0

∴ ϕ+2w

w(1− w2)ϕ = 0 (47)

∂L

∂w=

w

(1− w2)2(ϕ2 + w2

)∂L

∂w=

w

1− w2

d

dt

∂L

∂w=

w

1− w2+

2ww2

(1− w2)2

d

dt

∂L

∂w− ∂L

∂w= 0

∴ w +w

1− w2(w2 − ϕ2) = 0. (48)

A complete solution of the spherical problem is derived inAppendix. The solution of (47) & (48), which depends on theinitial values (w0, w0, ϕ0, ϕ0) is given in the form

w(t) =√α+ (α− 1) sin(2at+ β) (49)

ϕ(t) = ϕ0 + c

∫ t

0

1− w(s)2

w(s)2ds (50)

α =b+ 1

2(51)

β = arcsin2w2

0 − b− 1b− 1

(52)

a2 =(1− w2

0)w20 + w2

0ϕ20

(1− w20)2

(53)

b = (c/a)2 (54)

c =w2

0

1− w20

ϕ0. (55)

The values of the parameters a, b, and c come as follows. Thevalue of c is the value of ψ(0) multiplied by w2

0/(1 − w20)

obtained from (57). In Appendix the formulae of a and b arederived.

IX. CONCLUSION

This paper is the first one in the series where our results onparametrization of finite-state quantum control problems arerepresented. In a companion paper to be published [12] numer-ical issues of finite-state systems are considered in detail. Ear-lier we have been concentrating on studying parametrizationof systems described by partial differential control systems,see [13]–[15]. Flatness-based ideas, originally developed byMichel Fliess and his co-workers, see the seminal paper [10],have been developed for open-loop control design. These,however, form a means for control design of a large class ofnonlinear differential systems. In quantum control problems,where laser pulses are used for the control, the dynamics is sofast that, at least at the present level of the speed of possiblecomputations, feedback control seems to be impossible toimplement.

Here we study as a toy example, a two-level populationtransfer problem, which is described by a 4-dimensionalordinary differential system, bilinear in the two scalar controls.Without more advanced differential geometric considerations,which might be helpful in understanding quantum phenomenain general, we use the formulation found generally in theliterature, to obtain our basic driftless system model of the

form x = g(x)u, where g is linear in the state x.By using the elimination technique of [7] we obtain an

underdetermined system of differential equations, i.e. a systemwith a lesser number of equations than variables, of the formF (x, x) = 0, where the controls have been eliminated. Thissystem is manipulated in such a way that we can represent thecontrols and two state variables as functions of the other twostates and their time derivatives. These states then serve asso-called flat outputs, or parameter functions. Via a sphericalstate representation we obtain another parametrization for thecontrols.

An energy minimizing optimization problem is also formu-lated for the control design. This optimization problem is con-verted into two different but equivalent variational problems,where the parameter functions serve as the variables to beoptimized. Then, a closed-form solution is constructed for thespherical variational problem via the standard Euler-Lagrangetechnique.

ACKNOWLEDGMENT

This work was supported by the European Commission,in Marie Curie programme’s Transfer of Knowledgeproject Parametrization in the Control of Dynamic Systems(PARAMCOSYS, MTKD-CT-2004-509223). The work of M.Nihtila was initiated while he was working as Marie CurieFellow at Scuola Internazionale Superiore di Studi Avanzati(SISSA-ISAS), Italy in 2004-2005. The work of J. Levinewas in part carried out when he also was working as MarieCurie Fellow but now at University of Kuopio, Finland in2006. All this support is greatly acknowledged.

APPENDIX: SOLVING OF THE EULER-LAGRANGEEQUATIONS OF THE SPHERICAL SYSTEM

By redefining the derivative of ϕ one obtains from (47)

ψ +2w

w(1− w2)ψ = 0 ; ψ

.= ϕ (56)

The solution is obtained as follows

dψψ

= − 2ww(1− w2)

dt∫ ψ(t)

ψ0

dψψ

= −∫ t

t0

2ww(1− w2)

dt

= −∫ w(t)

w0

2 dww(1− w2)

lnψ − lnψ0 =∫ w

w0

(− 2w− 1

1− w+

11 + w

)dw

ψ = ψ0w2

0

1− w20

1− w2

w2(57)

ϕ = c1− w2

w2(58)

which can be directly integrated, when the solution of w isobtained in closed form. The second parameter function w is

obtained via more or less elementary calculations from (48)by substituting into it the expression of ϕ

w +w

1− w2(w2 − ϕ2) = 0

w = − ww2

1− w2+ c2

1− w2

w3.

Redefinition of w by z and by considering z as a function ofw one obtains

w = f(w, w) ; z.= w , z = z(w)

dz

dw=

f(w, z)z

; z′.=dz

dw

z′ = − w

1− w2z + c2

1− w2

w3z.

By redefining the square of z by q one proceeds

q.= z2 , q′ = 2zz′

q′ = − 2w1− w2

q + 2c21− w2

w3. (59)

This is a linear equation for q, where standard techniquesapply.

q′h = − 2w1− w2

qh (homogeneous part)

qh =q0

1− w20

(1− w2) .= d (1− w2)

q = d(w)(1− w2) (variation of constant)q′ = d′ (1− w2)− 2w d

= − 2w1− w2

d (1− w2) + 2c21− w2

w3

∴ d′ = c22w3

d(w) = −c2 1w2

+ C.

The complete solution for q is then

q(w) = −c2 1− w2

w2+ C (1− w2)

=(Cw2 − c2

) 1− w2

w2. (60)

The initial value constant C turns out to be positive.

q = z2 ⇒ C > 0 ; C.= a2 , a > 0. (61)

Returning back to the second parameter function w = w(t)one obtains a differential equation for it

q(w) = (a2w2 − c2)1− w2

w2

z(w) =

√(a2w2 − c2)(1− w2)

w

w =

√(a2w2 − c2)(1− w2)

w. (62)

The equation is solved by redefining the square of w by v.

v.= w2 ; v = 2ww

v = 2√

(a2v − c2)(1− v)

v = 2a√

(v − b)(1− v) ; b = (c/a)2. (63)

Via elementary integration one obtains

dv√−v2 + (b+ 1)v − b

= 2a dt∫ v

v0

dv√−v2 + (b+ 1)v − b

=∫ t

t0

2a dt/v

v0

arcsin2v − b− 1b− 1

= 2a(t− t0)

2v − b− 1b− 1

= sin(2at+ β)

v =12

[b+ 1 + (b− 1) sin(2at+ β)]

β = arcsin2v0 − b− 1

b− 1; t0 = 0.

Returning back to w the solution is obtained as functions ofthe initial values (w0, w0, ϕ0, ϕ0)

w(t) =√α+ (α− 1) sin(2at+ β) (64)

α =b+ 1

2(65)

β = arcsin2w2

0 − b− 1b− 1

(66)

a2 =(1− w2

0)w20 + w2

0ϕ20

(1− w20)2

(67)

b = (c/a)2 (68)

c =w2

0

1− w20

ϕ0. (69)

The solution of the first parameter function ϕ is obtained bydirectly integrating its derivative (58) by taking into accountits initial value ϕ(0) = ϕ0. Because q = z2 = w2 we obtainfrom (60) & (61) with some rearrangements

q = (a2w2 − c2)1− w2

w2= w2,

a2w2 = c2 +w2w2

1− w2, ∀ t ∈ [0, T ],

a2 =c2

w20

+w2

0

1− w20

, for t = 0,

a2 =w2

0

(1− w20)2

ϕ20 +

w20

1− w20

,

a2 =(1− w2

0)w20 + w2

0ϕ20

(1− w20)2

,

b =c2

a2=

w40ϕ

20

(1− w20)w

20 + w2

0ϕ20

.

REFERENCES

[1] A. D. Bandrauk, M. C. Delfour, and C. Le Bris, eds., Quantum Control:Mathematical and Numerical Challenges, CRM Proc. & Lecture Notes,American Mathematical Society, vol. 33, 211 pp., 2002.

[2] H. Boscain, G. Charlot, J.-P. Gauthier, S. Guerin, and H.-R. Jauslin,”Optimal control in laser-induced population transfer for two- and three-level quantum systems,” J. Math. Phys., vol. 43, no. 5, pp. 2107–2132,May 2002.

[3] H. Boscain, G. Charlot, J.-P. Gauthier, ”Optimal control of the Schrdingerequation with two or three levels,” in: Nonlinear and Adaptive ControlNCN4 2001, Lecture Notes in Control and Information Sciences, vol.281, pp. 33–43, 2002.

[4] H. Boscain, T. Chambrion, J.-P. Gauthier, ”Optimal control on a N-levelquantum system,” in: Lagrangian and Hamiltonian Methods in NonlinearControl, IFAC Proc., Seville, Spain, pp. 151–156, 2003.

[5] E. Brown and H. Rabitz, ”Some mathematical and algorithmic challengesin the control of quantum dynamics phenomena,” J. Mathematical Chem-istry, vol. 31, pp. 17–63, 2002.

[6] C. Le Bris, ”Control theory applied to quantum chemistry: Some tracks,”ESAIM: Proceedings of Controle des Systemes Gouvernes par desEquations aux Derivees Partielles, vol. 8, pp. 77–94, 2000.

[7] J. Levine, ”On necessary and sufficient conditions for differential flat-ness,” [on-line], Available: http://arxiv.org/abs/math.OC/0605405, Dec.2006

[8] M. Fliess, ”Generalized controller canonical forms for linear and non-linear dynamics,” IEEE Trans. Automatic Control, vol. 35, no. 9, pp.994–1001, Sept. 1990.

[9] M. Fliess, M., J. Levine, P. Martin, and P. Rouchon, ”On differentiallyflat nonlinear systems,” in: Proc. IFAC Symp. Nonlinear Control SystemsDesign. Bordeaux, France, M.Fliess, ed., pp. 408-412, June 23-26, 1992.

[10] M. Fliess, M., J. Levine, P. Martin, and P. Rouchon, ”Flatness anddefect of non-linear systems: Introductory theory and applications,” Int.J. Control, vol. 61, no. 6, pp. 1327–1361, 1995.

[11] M. Fliess, M., J. Levine, P. Martin, and P. Rouchon, ”A Lie-Backlundapproach to equivalence and flatness of nonlinear systems,” IEEE Trans.Automatic Control, vol. 44, no. 5, pp. 922–937, May 1999.

[12] M. Nihtila, J. Levine, and P. Kokkonen, ”Two parametrizations in finite-state quantum system simulations,” (in preparation, to appear in 2007)

[13] M. Nihtila, J. Tervo, and P. Kokkonen, ”Control of Burgers’ systemvia parametrization,” in: Preprints of 6th IFAC Symp. Nonlinear ControlSystems, NOLCOS 2004, , Stuttgart, Germany, Frank Allgower, ed., vol.I, pp. 423-428, 1-3 Sept. 2004.

[14] M. Nihtila, J. Tervo, and P. Kokkonen, ”Pseudo-differential operatorsin parametrization of boundary-value control systems,” in: CD-ROMProceedings of the 34rd IEEE Conf. Decision and Control, CDC’04,(IEEE Catalog number 04CH37601C, ISBN 0-7803-8683-3), ParadiseIslands, The Bahamas, pp. 1958-1963, 14-17 Dec. 2004.

[15] M. Nihtila, J. Tervo, P. Kokkonen, and O. Sarafanov, ”Parametrizationand control of partial differential systems,” in: Proc. IV Int. Conf. SystemIdentification and Control Problems, SICPRO’05, K.R. Chernyskov, ed.,V.A. Trapeznikov Institute of Control Sciences, Moscow, Russia, ISBN5-201-14975-8, pp. 767-785, 25-28 Jan. 2005.

[16] J. A. Pople, ”Nobel lecture: Quantum chemical models,” Reviews ofModern Physics, vol. 71, no. 5, pp. 1267–1274, 1999.

[17] H. Rabitz, R. de Vivie-Riedle, M. Motzkus, M., and K. Kompa, ”Whitherthe future of controlling quantum phenomena?” Science, vol. 288, pp.824-828, 2000.

[18] M. Shapiro and P. Brumer, Principles of Quantum Control of Molecularprocesses, John Wiley & Sons Inc., Hoboken, 2003.

[19] M. Tian, Z. W. Barber, J. A. Fischer, and Wm. R. Babbitt, ”Ge-ometric quantum operations in two-level atomic systems,” QuantumInformation and Quantum Control Conference, University of Toronto andFields Institute, poster presentation, 19-23 July 2004, [on-line], avail-able: http://www.fields.utoronto.ca/programs/scientific/04-05/quantumIC/posters/tian.pdf

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