[PPT] - Optimal control of a dissipative 2-level quantum system Nataliya PowerPoint Presentation

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Optimal control of a dissipative 2-level quantum system

Nataliya Shcherbakova

ENSEEIHT, Toulouse, France

October 25, 2010

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 1 / 41

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Main collaborators

Bernard Bonnard, Institut de Mathématiques de Bourgogne, UMR CNRS 5584; Dominique Sugny, Institut Carnot de Bourgogne, UMR CNRS 5209; Olivier Cots, ENSEEIHT and Institut de Mathématiques de Bourgogne.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 2 / 41

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Main publications

B. Bonnard, O.Cots, N. Shcherbakova, D. Sugny, The energy

minimization problem for two-level dissipative quantum systems,

J. Math.Phys. 51, 2010, DOI: 10.1063/1.3479390
B. Bonnard, N. Shcherbakova, D. Sugny, The smooth continuation

method in optimal control with an application to quantum systems, 2009, ESAIM-COCV, DOI: 10.1051/cocv/2010004

B. Bonnard, D. Sugny, Time-minimal control of dissipative

two-level quantum systems: the integrable case, SIAM J. on Control and Optimization, vol.48 (2009), pp. 1289-1308

B. Bonnard, M. Chyba, D. Sugny, Time-minimal control of

dissipative two-level quantum systems: the generic case, IEEE Transactions on Automatic control, vol. 54, N.11 (2009), pp.2598 - 2610

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 3 / 41

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Problem 1: Control of molecular alignment by laser fields in dissipative media

i∂ρ ∂t = [H0 + H1, ρ] + iL(ρ), (1) ρ - the density matrix (i.e. positive semi-definite Hermitian

perator) s.t.

tr(ρ) = 1, tr(ρ2) ≤ 1 H0 - the field-free Hamiltonian of the system H1 - the Hamiltonian of the interaction with the control field L - the dissipation operator

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 4 / 41

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Problem 1: Control of molecular alignment by laser fields in dissipative media

i∂ρ ∂t = [H0 + H1, ρ] + iL(ρ), (1) ρ - the density matrix (i.e. positive semi-definite Hermitian

perator) s.t.

tr(ρ) = 1, tr(ρ2) ≤ 1 H0 - the field-free Hamiltonian of the system H1 - the Hamiltonian of the interaction with the control field L - the dissipation operator

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 4 / 41

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Dissipation term: L(ρ)kk = −

N

l=k

(γlkρkk + γklρll), L(ρ)lk

l=k = −Γlkρlk,

where γkl, Γkl are real non-negative constants describing : γkl - the population relaxation from state k to state l; Γkl = Γlk - de-phasing rate of the transition from state k to state l. 2-levels systems: ρ = ρ11 ρ12 ρ21 ρ22

= 1

2 1 + z x + iy x − iy 1 − z

where

x = 2ℜ[ρ12], y = 2ℑ[ρ12], z = ρ22 − ρ11.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41

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Dissipation term: L(ρ)kk = −

N

l=k

(γlkρkk + γklρll), L(ρ)lk

l=k = −Γlkρlk,

where γkl, Γkl are real non-negative constants describing : γkl - the population relaxation from state k to state l; Γkl = Γlk - de-phasing rate of the transition from state k to state l. 2-levels systems: ρ = ρ11 ρ12 ρ21 ρ22

= 1

2 1 + z x + iy x − iy 1 − z

where

x = 2ℜ[ρ12], y = 2ℑ[ρ12], z = ρ22 − ρ11. Control Hamiltonian: H1 = −σxEx − σyEy, where Ex, Ey are linearly polarized laser fields

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41

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Dissipation term: L(ρ)kk = −

N

l=k

(γlkρkk + γklρll), L(ρ)lk

l=k = −Γlkρlk,

where γkl, Γkl are real non-negative constants describing : γkl - the population relaxation from state k to state l; Γkl = Γlk - de-phasing rate of the transition from state k to state l. 2-levels systems: ρ = ρ11 ρ12 ρ21 ρ22

= 1

2 1 + z x + iy x − iy 1 − z

where

x = 2ℜ[ρ12], y = 2ℑ[ρ12], z = ρ22 − ρ11. Control Hamiltonian: H1 = −σxEx − σyEy, where Ex, Ey are linearly polarized laser fields

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41

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Lindblad equations for 2-levels systems ( = 1)

           ˙ x = −Γx + u2z ˙ y = −Γy − u1z ˙ z = γ− − γ+z + u1y − u2x (2) with γ− = γ12 − γ21, γ+ = γ12 + γ21, and 2Γ ≥ γ+ ≥ |γ−|. q = (x, y, z) belongs to the invariant Block ball q ≤ 1 q = 1 - pure state (0, 0, γ−

γ+ ) - the equilibrium state of the free motion

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 6 / 41

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Lindblad equations for 2-levels systems ( = 1)

           ˙ x = −Γx + u2z ˙ y = −Γy − u1z ˙ z = γ− − γ+z + u1y − u2x (2) with γ− = γ12 − γ21, γ+ = γ12 + γ21, and 2Γ ≥ γ+ ≥ |γ−|. q = (x, y, z) belongs to the invariant Block ball q ≤ 1 q = 1 - pure state (0, 0, γ−

γ+ ) - the equilibrium state of the free motion

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 6 / 41

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Problem 2: control of a spin-1

2 particle in dissipative

media

Bloch equations: ˙ Mx = − 1 T2 Mx + u2Mz; ˙ My = − 1 T2 My − u1Mz; (3) ˙ Mz = 1 T1 (M0 − Mz) + u1My − u2Mx, M = (Mx, My, Mz) - magnetization vector; T1, T2 - longitudinal and transverse relaxation times, 2T1 ≥ T2; M0 = (0, 0, M0) - thermal equilibrium point; u = (u1, u2, 0) - controlled magnetic field. Normalization : q = (x, y, z) =

1 M0 M,

Γ = T−1

2 ,

γ+ = γ− = T−1

1

= ⇒ Lindblad equations for γ+ = γ−

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 7 / 41

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Problem 2: control of a spin-1

2 particle in dissipative

media

Bloch equations: ˙ Mx = − 1 T2 Mx + u2Mz; ˙ My = − 1 T2 My − u1Mz; (3) ˙ Mz = 1 T1 (M0 − Mz) + u1My − u2Mx, M = (Mx, My, Mz) - magnetization vector; T1, T2 - longitudinal and transverse relaxation times, 2T1 ≥ T2; M0 = (0, 0, M0) - thermal equilibrium point; u = (u1, u2, 0) - controlled magnetic field. Normalization : q = (x, y, z) =

1 M0 M,

Γ = T−1

2 ,

γ+ = γ− = T−1

1

= ⇒ Lindblad equations for γ+ = γ−

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 7 / 41

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Control setting

(P) ˙ q = F0(q) + u1F1(q) + u2F2(q), q0, qT − fixed and F0, F1, F2 ∈ Vec(R3) : F0 =   −Γx −Γy γ− − γ+z   , F1 =   −z y   , F2 =   z −x   .

I. Minimal time problem (PT) :

(P), u ≤ 1, T − → min

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41

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Control setting

(P) ˙ q = F0(q) + u1F1(q) + u2F2(q), q0, qT − fixed and F0, F1, F2 ∈ Vec(R3) : F0 =   −Γx −Γy γ− − γ+z   , F1 =   −z y   , F2 =   z −x   .

I. Minimal time problem (PT) :

(P), u ≤ 1, T − → min

II. Energy minimizing problem (PE) :

(P), T - fixed, 1 2

T

u2

1(t) + u2 2(t) dt → min

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41

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Control setting

(P) ˙ q = F0(q) + u1F1(q) + u2F2(q), q0, qT − fixed and F0, F1, F2 ∈ Vec(R3) : F0 =   −Γx −Γy γ− − γ+z   , F1 =   −z y   , F2 =   z −x   .

I. Minimal time problem (PT) :

(P), u ≤ 1, T − → min

II. Energy minimizing problem (PE) :

(P), T - fixed, 1 2

T

u2

1(t) + u2 2(t) dt → min

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41

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The Hamiltonian of (P) hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ) − p0 2 (u2

1 + u2 2),

hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

q R3,

p0 ∈ {0, 1} Case (PE): in the normal case (p0 = 1) optimal controls are ui(ξ) = hi(ξ), i = 1, 2, ξ ∈ T∗R3. Normal extremals are solutions of the Hamiltonian system associated to hE(ξ) = h0(ξ) + 1 2(h2

1(ξ) + h2 2(ξ)).

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 9 / 41

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The Hamiltonian of (P) hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ) − p0 2 (u2

1 + u2 2),

hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

q R3,

p0 ∈ {0, 1} Case (PE): in the normal case (p0 = 1) optimal controls are ui(ξ) = hi(ξ), i = 1, 2, ξ ∈ T∗R3. Normal extremals are solutions of the Hamiltonian system associated to hE(ξ) = h0(ξ) + 1 2(h2

1(ξ) + h2 2(ξ)).

Case (PT) : p0 = 0, ui(ξ) =

hi(ξ)

√

h2

1(ξ)+h2 2(ξ), i = 1, 2, ξ ∈ T∗R3 \ Σ, where

Σ = {ξ : h1(ξ) = h2(ξ) = 0} is the switching surface. Regular extremals are solutions of the Hamiltonian system associated to h(ξ) = h0(ξ) +

h2

1(ξ) + h2 2(ξ).

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 9 / 41

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The Hamiltonian of (P) hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ) − p0 2 (u2

1 + u2 2),

hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

q R3,

p0 ∈ {0, 1} Case (PE): in the normal case (p0 = 1) optimal controls are ui(ξ) = hi(ξ), i = 1, 2, ξ ∈ T∗R3. Normal extremals are solutions of the Hamiltonian system associated to hE(ξ) = h0(ξ) + 1 2(h2

1(ξ) + h2 2(ξ)).

Case (PT) : p0 = 0, ui(ξ) =

hi(ξ)

√

h2

1(ξ)+h2 2(ξ), i = 1, 2, ξ ∈ T∗R3 \ Σ, where

Σ = {ξ : h1(ξ) = h2(ξ) = 0} is the switching surface. Regular extremals are solutions of the Hamiltonian system associated to h(ξ) = h0(ξ) +

h2

1(ξ) + h2 2(ξ).

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 9 / 41

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In spherical coordinates: x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, hT(R, θ, ϕ, pR, pθ, pϕ) = h0(R, ϕ, pR, pϕ) +

p2

ϕ + p2 θ cot2 ϕ

hE(R, θ, ϕ, pR, pθ, pϕ) = h0(R, ϕ, pR, pϕ) + 1 2(p2

ϕ + p2 θ cot2 ϕ)

Proposition θ is a cyclic variable and pθ = p, [F1, F2](q) is the first integral for hT and hE. Moreover, pθ = 0 = ⇒ θ = const and the problem is 2D ; in particular, the orbits starting from the z-axis are 2D (pθ = 0) ; ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0}; if γ− = 0, then pr = const (for r = ln R) and Hamiltonian systems for hT and hE are Liouville integrable.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 10 / 41

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In spherical coordinates: x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, hT(R, θ, ϕ, pR, pθ, pϕ) = h0(R, ϕ, pR, pϕ) +

p2

ϕ + p2 θ cot2 ϕ

hE(R, θ, ϕ, pR, pθ, pϕ) = h0(R, ϕ, pR, pϕ) + 1 2(p2

ϕ + p2 θ cot2 ϕ)

Proposition θ is a cyclic variable and pθ = p, [F1, F2](q) is the first integral for hT and hE. Moreover, pθ = 0 = ⇒ θ = const and the problem is 2D ; in particular, the orbits starting from the z-axis are 2D (pθ = 0) ; ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0}; if γ− = 0, then pr = const (for r = ln R) and Hamiltonian systems for hT and hE are Liouville integrable.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 10 / 41

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Abnormal (singular) case: p0 = 0, h1 = 0, h2 = 0. Abnormal extremals are characterized by the property pφ = 0, and can be of two kinds: ◮ the z -axis in a fixed meridian plane θ = θ0 (corresponding to 0

control) ;

◮ pθ = 0, and θ is non-controllable: ˙ θ = v cot φ for any admissible control v. The problem reduces to a 2D problem: ˙ ¯ q = F(¯ q) + uG(¯ q), ¯ q = (y, z), (4) F =

Γy

γ− − γ+z

,

G = −z y

.

The abnormal trajectories satisfy y = 0 (ua = 0) or z = γ−

2δ , where

δ = γ+ − Γ (ua = γ−(δ−Γ)

2δy

).

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 11 / 41

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Structure of the problem

General case: γ− = 0, δ = 0 3D case: pθ = 0, 2D case: pθ = 0 Integrable case: γ− = 0 Grushin case: γ− = δ = 0 (PT) ∼ (PE) R is not controllable; extremals = geodesics on the 2-sphere with a Grushin-type metric g1 = dϕ2 + tan ϕ2dθ2

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 12 / 41

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Structure of the problem

General case: γ− = 0, δ = 0 3D case: pθ = 0, 2D case: pθ = 0 Integrable case: γ− = 0 Grushin case: γ− = δ = 0 (PT) ∼ (PE) R is not controllable; extremals = geodesics on the 2-sphere with a Grushin-type metric g1 = dϕ2 + tan ϕ2dθ2

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 12 / 41

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Structure of the problem

General case: γ− = 0, δ = 0 3D case: pθ = 0, 2D case: pθ = 0 Integrable case: γ− = 0 Grushin case: γ− = δ = 0 (PT) ∼ (PE) R is not controllable; extremals = geodesics on the 2-sphere with a Grushin-type metric g1 = dϕ2 + tan ϕ2dθ2

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 12 / 41

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Structure of the problem

General case: γ− = 0, δ = 0 3D case: pθ = 0, 2D case: pθ = 0 Integrable case: γ− = 0 Grushin case: γ− = δ = 0 (PT) ∼ (PE) R is not controllable; extremals = geodesics on the 2-sphere with a Grushin-type metric g1 = dϕ2 + tan ϕ2dθ2

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 12 / 41

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Grushin’s metric

˙ ϕ2 + p2

θ cot2 ϕ = 2h

meridian circles θ = const; periodic trajectories with branches of the form: ϕ(t) = π

2 − arcsin

sin mt

m

,

m2 = p2

θ + 1

θ(t) = pθ

1

√ 1−m2 arctan

√

1−m2 m

tan mt

− t
Π

4 Π 2 3 Π 4

Π Θ min max

Nataliya Shcherbakova (N7, Toulouse)

Optimal control of a dissipative 2-level quantum system October 2010 13 / 41

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Integrable (PT) problem

Observation The fixed level of the Hamiltonian admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 14 / 41

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Integrable (PT) problem

Observation The fixed level of the Hamiltonian admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}. Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact (which is always the case for |δ| < 2), then it contains a periodic trajectory in the plane (ϕ, pϕ) of period T and we have two generically distinct extremals curves q+(t), q−(t), starting from the same point and intersecting with the same length T/2 at a point such that ϕ(T/2) = π − φ(0);

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 14 / 41

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Integrable (PT) problem

Observation The fixed level of the Hamiltonian admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}. Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact (which is always the case for |δ| < 2), then it contains a periodic trajectory in the plane (ϕ, pϕ) of period T and we have two generically distinct extremals curves q+(t), q−(t), starting from the same point and intersecting with the same length T/2 at a point such that ϕ(T/2) = π − φ(0); Non - periodic extremals. In the case |δ| ≥ 2 the level set of the Hamiltonian may be non compact and a new class of extremals appears: ϕ → ϕ∗, |pϕ| → +∞ when t → +∞, while ˙ θ → 0. The stationary state ϕ∗ solves the equation (γ+ − Γ)2 sin2 2ϕ∗ = 4.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 14 / 41

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Integrable (PT) problem

Observation The fixed level of the Hamiltonian admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}. Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact (which is always the case for |δ| < 2), then it contains a periodic trajectory in the plane (ϕ, pϕ) of period T and we have two generically distinct extremals curves q+(t), q−(t), starting from the same point and intersecting with the same length T/2 at a point such that ϕ(T/2) = π − φ(0); Non - periodic extremals. In the case |δ| ≥ 2 the level set of the Hamiltonian may be non compact and a new class of extremals appears: ϕ → ϕ∗, |pϕ| → +∞ when t → +∞, while ˙ θ → 0. The stationary state ϕ∗ solves the equation (γ+ − Γ)2 sin2 2ϕ∗ = 4.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 14 / 41

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Γ = 2.5, γ+ = 2 Γ = 4.5, γ+ = 2 ϕ(0) = π

4 , pr = 1, pθ = 2

ϕ(0) = 2π

5 , pr = 0.25, pθ = 8 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 15 / 41

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Evolution of the conjugate and cut loci

Γ = γ+ = 2; Γ = 2.5, γ+ = 2 ϕ(0) = π

4 , pr = 1, pθ = 2

ϕ(0) = π

4 , pr = 1, pθ = 2 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 16 / 41

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Non-integrable case

a). Case |δ| < 2 : |pϕ| → +∞ oscillating while ϕ → 0 ( mod π) as t → +∞. The asymptotic limit of the dynamics is characterized by R∗ = |γ−| γ+ , ϕ∗ = if γ− > 0 π if γ− < 0 . The conjugate points occurs before the beginning of the oscillating regime.

Γ = 2.5, γ+ = 2, γ− = −0.5, ϕ(0) = π

4 , pR = 0.1, R(0) = 1, pθ = 2,

pϕ(0) = −10, −2.5, 0, 2.5, 10.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 17 / 41

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b). Case |δ| > 2 : |pϕ(t)| → +∞ as t → +∞, and the asymptotic stationary points (R∗, ϕ∗, θ∗) of the dynamics are given by R∗ = |γ−| √ 1 + Γ2 1 + γ+Γ , ϕ∗ =

arctan 1

Γ

if γ− > 0 π − arctan 1

Γ

if γ− < 0 . No conjugate points were observed numerically.

Γ = 4.5, γ+ = 2, γ− = −0.5, ϕ(0) = π

4 , pR = 0.1, R(0) = 1, pθ = 2,

pϕ(0) = −10, −2.5, 0, 2.5, 10.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 18 / 41

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(PE): integrable case for δ = 0

Observation. If γ− = 0, the problem can be integrated in quadra-
tures. More precisely, ϕ(t) is a solution of a natural mechanical sys-

tem ˙ ϕ2 2 + V(φ) = h, (5) for V(ϕ) = −pr(δ cos2 ϕ + Γ) − δ2 8 sin2 2ϕ + 1 2p2

θ cot2 ϕ,

and θ and r satisfy ˙ θ = pθ

1

sin2 ϕ − 1

,

˙ r = δ sin2 ϕ − γ+.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 19 / 41

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(PE): integrable case for δ = 0

Observation. If γ− = 0, the problem can be integrated in quadra-
tures. More precisely, ϕ(t) is a solution of a natural mechanical sys-

tem ˙ ϕ2 2 + V(φ) = h, (5) for V(ϕ) = −pr(δ cos2 ϕ + Γ) − δ2 8 sin2 2ϕ + 1 2p2

θ cot2 ϕ,

and θ and r satisfy ˙ θ = pθ

1

sin2 ϕ − 1

,

˙ r = δ sin2 ϕ − γ+.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 19 / 41

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3D integrable case : γ− = 0, pθ = 0

a)

Π
2

Π

b)

Π

2
p2

θ < δ2 + 2δpr

p2

θ ≥ δ2 + 2δpr

parallel orbits: ϕ = ϕ∗ = const - equilibrium point of (5);

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 20 / 41

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3D integrable case : γ− = 0, pθ = 0

a)

Π
2

Π

b)

Π

2
p2

θ < δ2 + 2δpr

p2

θ ≥ δ2 + 2δpr

parallel orbits: ϕ = ϕ∗ = const - equilibrium point of (5); long periodic orbits crossing the equatorial plane;

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 20 / 41

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3D integrable case : γ− = 0, pθ = 0

a)

Π
2

Π

b)

Π

2
p2

θ < δ2 + 2δpr

p2

θ ≥ δ2 + 2δpr

parallel orbits: ϕ = ϕ∗ = const - equilibrium point of (5); long periodic orbits crossing the equatorial plane; short periodic orbits (in each hemisphere)

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 20 / 41

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3D integrable case : γ− = 0, pθ = 0

a)

Π
2

Π

b)

Π

2
p2

θ < δ2 + 2δpr

p2

θ ≥ δ2 + 2δpr

parallel orbits: ϕ = ϕ∗ = const - equilibrium point of (5); long periodic orbits crossing the equatorial plane; short periodic orbits (in each hemisphere) separatrices.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 20 / 41

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3D integrable case : γ− = 0, pθ = 0

a)

Π
2

Π

b)

Π

2
p2

θ < δ2 + 2δpr

p2

θ ≥ δ2 + 2δpr

parallel orbits: ϕ = ϕ∗ = const - equilibrium point of (5); long periodic orbits crossing the equatorial plane; short periodic orbits (in each hemisphere) separatrices.

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 20 / 41

SLIDE 42

2D integrable case: γ− = 0, pθ = 0

a)

Π
2

Π Π

b)
Π
2
Π
2
− 1

2 < pr δ < 0

0 < pr

δ < 1 2

c)

Π

2

Π

d)

Π

2

Π

2
pr

δ < − 1 2 pr δ > 1 2

Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 21 / 41

SLIDE 43