Optimal control of 2-levels dissipative quantum control systems: - - PowerPoint PPT Presentation

Optimal control of 2-levels dissipative quantum control systems: analytical aspects

Nataliya Shcherbakova

ENSEEIHT, Toulouse, France

April 27, 2010

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 1 / 27

Main collaborators

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

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 2 / 27

Main publications

• 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

• B. Bonnard, N. Shcherbakova, D. Sugny, The smooth continuation

method in optimal control with an application to quantum systems, 2009, to appear in ESAIM-COCV

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

minimization problem for two-level dissipative quantum systems, submitted

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 3 / 27

Problem : control of the dynamics of finite-dimensional quantum system interacting with a dissipative environment: i∂ρ ∂t = [H0 + H1, ρ] + iL(ρ), H0 - the field-free Hamiltonian of the system H1 - the Hamiltonian of the interaction with the control field L - the dissipation operator ρ - the density matrix (i.e. positive semi-definite Hermitian

• perator) s.t.

tr(ρ) = 1, tr(ρ2) ≤ 1

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 4 / 27

Problem : control of the dynamics of finite-dimensional quantum system interacting with a dissipative environment: i∂ρ ∂t = [H0 + H1, ρ] + iL(ρ), H0 - the field-free Hamiltonian of the system H1 - the Hamiltonian of the interaction with the control field L - the dissipation operator ρ - the density matrix (i.e. positive semi-definite Hermitian

• perator) s.t.

tr(ρ) = 1, tr(ρ2) ≤ 1

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 4 / 27

˙ ρkk = −i[H0 + H1, ρ]kk −

N

• l=k

(γlkρkk + γklρll) ˙ ρlk = −i[H0 + H1, ρ]lk − Γlkρlk, l = k, = 1; γkl, Γkl are real non-negative constants; γkl - the population relaxation from state k to state l; Γkl = Γlk - de-phasing rate of the transition from state k to state l.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 5 / 27

˙ ρkk = −i[H0 + H1, ρ]kk −

N

• l=k

(γlkρkk + γklρll) ˙ ρlk = −i[H0 + H1, ρ]lk − Γlkρlk, l = k, = 1; γkl, Γkl are real non-negative constants; γkl - the population relaxation from state k to state l; Γkl = Γlk - de-phasing rate of the transition from state k to state l.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 5 / 27

Lindblad equations for 2-levels systems

Relevant physical models for 2-levels systems: control of molecular alignment by laser fields in dissipative media; spin- 1

2 particle controlled by magnetic field.

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) 2 levels dissipative quantum systems April 2010 6 / 27

Lindblad equations for 2-levels systems

Relevant physical models for 2-levels systems: control of molecular alignment by laser fields in dissipative media; spin- 1

2 particle controlled by magnetic field.

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. The state q = (x, y, z) belongs to the Bloch ball q ≤ 1 and satisfies the Lindblad equations ˙ x = −Γx + u2z, ˙ y = −Γy − u1z, ˙ z = γ− − γ+z + u1y − u2x with γ− = γ12 − γ21, γ+ = γ12 + γ21, and 2Γ ≥ γ+ ≥ |γ−|.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 6 / 27

Lindblad equations for 2-levels systems

Relevant physical models for 2-levels systems: control of molecular alignment by laser fields in dissipative media; spin- 1

2 particle controlled by magnetic field.

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. The state q = (x, y, z) belongs to the Bloch ball q ≤ 1 and satisfies the Lindblad equations ˙ x = −Γx + u2z, ˙ y = −Γy − u1z, ˙ z = γ− − γ+z + u1y − u2x with γ− = γ12 − γ21, γ+ = γ12 + γ21, and 2Γ ≥ γ+ ≥ |γ−|. (0, 0, γ−

γ+ ) - the equilibrium state of the free motion ;

q = 1 - pure state

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 6 / 27

Lindblad equations for 2-levels systems

Relevant physical models for 2-levels systems: control of molecular alignment by laser fields in dissipative media; spin- 1

2 particle controlled by magnetic field.

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. The state q = (x, y, z) belongs to the Bloch ball q ≤ 1 and satisfies the Lindblad equations ˙ x = −Γx + u2z, ˙ y = −Γy − u1z, ˙ z = γ− − γ+z + u1y − u2x with γ− = γ12 − γ21, γ+ = γ12 + γ21, and 2Γ ≥ γ+ ≥ |γ−|. (0, 0, γ−

γ+ ) - the equilibrium state of the free motion ;

q = 1 - pure state

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 6 / 27

Control setting

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

• I. Minimal time problem:

(P), u ≤ 1, T − → min

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 7 / 27

Control setting

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

• I. Minimal time problem:

(P), u ≤ 1, T − → min

• II. Energy minimizing problem:

(P), T - fixed, 1 2

T

• u2

1(t) + u2 2(t)

• dt → min

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 7 / 27

Control setting

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

• I. Minimal time problem:

(P), u ≤ 1, T − → min

• II. Energy minimizing problem:

(P), T - fixed, 1 2

T

• u2

1(t) + u2 2(t)

• dt → min

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 7 / 27

Minimal time problem : preliminary analysis

The Hamiltonian of minimal time problem hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ), hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

qR3,

PMP : the optimal controls are 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} defines the switching surface. Regular extremals are solutions of the Hamiltonian system associated to h(ξ) = h0(ξ) +

• h2

1(ξ) + h2 2(ξ).

(1)

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 8 / 27

Minimal time problem : preliminary analysis

The Hamiltonian of minimal time problem hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ), hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

qR3,

PMP : the optimal controls are 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} defines the switching surface. Regular extremals are solutions of the Hamiltonian system associated to h(ξ) = h0(ξ) +

• h2

1(ξ) + h2 2(ξ).

(1)

• Proposition. The Hamiltonian system associated to h admits a first

integral of the form h3(ξ) = p, [F1, F2](q), ξ = (p, q), p ∈ T∗

qR3.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 8 / 27

Minimal time problem : preliminary analysis

The Hamiltonian of minimal time problem hu(ξ) = h0(ξ) + u1h1(ξ) + u2h1(ξ), hi(ξ) = p, Fi(q), ξ = (p, q), p ∈ T∗

qR3,

PMP : the optimal controls are 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} defines the switching surface. Regular extremals are solutions of the Hamiltonian system associated to h(ξ) = h0(ξ) +

• h2

1(ξ) + h2 2(ξ).

(1)

• Proposition. The Hamiltonian system associated to h admits a first

integral of the form h3(ξ) = p, [F1, F2](q), ξ = (p, q), p ∈ T∗

qR3.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 8 / 27

In spherical coordinates x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, Properties: ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0};

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 9 / 27

In spherical coordinates x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, Properties: ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0}; h3 = pθ = const. Every time-minimal optimal trajectory is either

an extremal trajectory with pθ = 0 and contained in the meridian plane θ = const

• r an extremal consisting of sub-arcs of solutions with pθ = 0 with

possible connections in the equatorial plane ϕ = π

2 .

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 9 / 27

In spherical coordinates x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, Properties: ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0}; h3 = pθ = const. Every time-minimal optimal trajectory is either

an extremal trajectory with pθ = 0 and contained in the meridian plane θ = const

• r an extremal consisting of sub-arcs of solutions with pθ = 0 with

possible connections in the equatorial plane ϕ = π

2 .

if γ− = 0, then pr = const (where r = ln R) and the problem is Liouville integrable.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 9 / 27

In spherical coordinates x = R sin ϕ cos θ, y = R sin ϕ sin θ, z = R cos ϕ, Properties: ˙ R ≤ 0, moreover, ˙ R = 0 on the collinear set C = {q : det(F0, F1, F2)(q) = 0}; h3 = pθ = const. Every time-minimal optimal trajectory is either

an extremal trajectory with pθ = 0 and contained in the meridian plane θ = const

• r an extremal consisting of sub-arcs of solutions with pθ = 0 with

possible connections in the equatorial plane ϕ = π

2 .

if γ− = 0, then pr = const (where r = ln R) and the problem is Liouville integrable.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 9 / 27

Complexity of the problem

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

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 10 / 27

Optimal trajectories in the Grushin case φ

θ

φ(0) π−φ(0) Cut locus Conjugate locus C −π π

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 11 / 27

2D case

Extremal curves with pθ = 0 lie in a meridian plane θ = const. They are extremals of the single-input 2D control system (for θ = 0): ˙ ¯ q = F(¯ q) + uG(¯ q), ¯ q = (y, z), (2) F =

• Γy

γ− − γ+z

• ,

G = −z y

• .

Observations: The extremals of the general 3D problem starting from the z-axis are solutions of (2);

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 12 / 27

2D case

Extremal curves with pθ = 0 lie in a meridian plane θ = const. They are extremals of the single-input 2D control system (for θ = 0): ˙ ¯ q = F(¯ q) + uG(¯ q), ¯ q = (y, z), (2) F =

• Γy

γ− − γ+z

• ,

G = −z y

• .

Observations: The extremals of the general 3D problem starting from the z-axis are solutions of (2); The collinear set: C = {¯ q : det(F, G)(¯ q) = 0} consists of equilibrium points of R;

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 12 / 27

2D case

Extremal curves with pθ = 0 lie in a meridian plane θ = const. They are extremals of the single-input 2D control system (for θ = 0): ˙ ¯ q = F(¯ q) + uG(¯ q), ¯ q = (y, z), (2) F =

• Γy

γ− − γ+z

• ,

G = −z y

• .

Observations: The extremals of the general 3D problem starting from the z-axis are solutions of (2); The collinear set: C = {¯ q : det(F, G)(¯ q) = 0} consists of equilibrium points of R; The singular trajectories of (2) are contained in the singular set: S = {¯ q : det(G, [G, F])(¯ q) = 0} = {y = 0} ∪ {z =

−γ− 2(Γ−γ+)}

• r

S = {y = 0}.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 12 / 27

2D case

Extremal curves with pθ = 0 lie in a meridian plane θ = const. They are extremals of the single-input 2D control system (for θ = 0): ˙ ¯ q = F(¯ q) + uG(¯ q), ¯ q = (y, z), (2) F =

• Γy

γ− − γ+z

• ,

G = −z y

• .

Observations: The extremals of the general 3D problem starting from the z-axis are solutions of (2); The collinear set: C = {¯ q : det(F, G)(¯ q) = 0} consists of equilibrium points of R; The singular trajectories of (2) are contained in the singular set: S = {¯ q : det(G, [G, F])(¯ q) = 0} = {y = 0} ∪ {z =

−γ− 2(Γ−γ+)}

• r

S = {y = 0}.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 12 / 27

(a). Γ = 1.1, γ+ = 1.6, γ− = 0; (b). Γ = 4, γ+ = 1.5, γ− = 0.5; (c). Γ = 4 , γ+ = 6.5, γ− = −1.5; (d). Γ = 1, γ+ = 0.5, γ− = −0.1.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 13 / 27

Integrable problem γ− = 0

Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact, contains no singular points and admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}, then it contains a periodic trajectory (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) 2 levels dissipative quantum systems April 2010 14 / 27

Integrable problem γ− = 0

Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact, contains no singular points and admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}, then it contains a periodic trajectory (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 |γ+ − Γ| ≥ 2 the level set

• f 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. No conjugate points were observed.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 14 / 27

Integrable problem γ− = 0

Periodic extremals. If for fixed (pr, pθ) the level set of h = ε, ε ∈ {0, 1}, is compact, contains no singular points and admits a central symmetry with respect to the set {ϕ = π/2, pϕ = 0}, then it contains a periodic trajectory (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 |γ+ − Γ| ≥ 2 the level set

• f 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. No conjugate points were observed.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 14 / 27

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) 2 levels dissipative quantum systems April 2010 15 / 27

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) 2 levels dissipative quantum systems April 2010 16 / 27

Non-integrable case

a). |Γ − γ+| > 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) 2 levels dissipative quantum systems April 2010 17 / 27

b). |Γ − γ+| < 2. |pϕ| → +∞ oscillating while ϕ → 0 ( mod π) as t → +∞. The asymptotic limit of the dynamics is characterized by R∗ = |γ−| γ+ , ϕ∗ =  0 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) 2 levels dissipative quantum systems April 2010 18 / 27

Energy - minimizing problem

(P2)    ˙ q = F0(q) + u1F1(q) + u2F2(q), q ∈ R3,

1 2 T

• u2

1(t) + u2 2(t)

• dt → min,

T − fixed. . normal case: h = h0 + 1

2(h2 1 + h2 2),

ui = hi, i = 1, 2; abnormal case: h1 = h2 = 0. Observation.If γ− = 0, then the associated Hamiltonian system is Liouville integrable. Moreover, ϕ(t) is a solution of a natural mechan- ical system ˙ ϕ2 2 + V(φ) = h, (3) for V(ϕ) = −pr(δ cos2 ϕ + Γ) − δ2 8 sin2 2ϕ + 1 2p2

θ cot2 ϕ,

where δ = γ+ − Γ

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 19 / 27

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 (3);

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 20 / 27

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 (3); long periodic orbits crossing the equatorial plane;

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 20 / 27

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 (3); long periodic orbits crossing the equatorial plane; In addition, if p2

θ < δ2 + 2δpr one can have

short periodic orbits (in each hemisphere)

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 20 / 27

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 (3); long periodic orbits crossing the equatorial plane; In addition, if p2

θ < δ2 + 2δpr one can have

short periodic orbits (in each hemisphere) separatrices.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 20 / 27

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 (3); long periodic orbits crossing the equatorial plane; In addition, if p2

θ < δ2 + 2δpr one can have

short periodic orbits (in each hemisphere) separatrices.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 20 / 27

Parametrization of periodic solutions

• Observation. In the integrable case the normal extremals can be

found explicitly in terms of Jacobi Elliptic functions. Set x = sin2 ϕ. Then V(ϕ) − h = 0 = ⇒ P3(x) = (x − x1)(x − x2)(x − x3) = 0 and x3 < 0, 0 < x1 ≤ x2.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 21 / 27

Parametrization of periodic solutions

• Observation. In the integrable case the normal extremals can be

found explicitly in terms of Jacobi Elliptic functions. Set x = sin2 ϕ. Then V(ϕ) − h = 0 = ⇒ P3(x) = (x − x1)(x − x2)(x − x3) = 0 and x3 < 0, 0 < x1 ≤ x2. Denote z(t) = (Mt, k) and assume ϕ(0) = ϕmin.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 21 / 27

Parametrization of periodic solutions

• Observation. In the integrable case the normal extremals can be

found explicitly in terms of Jacobi Elliptic functions. Set x = sin2 ϕ. Then V(ϕ) − h = 0 = ⇒ P3(x) = (x − x1)(x − x2)(x − x3) = 0 and x3 < 0, 0 < x1 ≤ x2. Denote z(t) = (Mt, k) and assume ϕ(0) = ϕmin.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 21 / 27

Short periodic orbits: k2 = (1 − x3)(x2 − x1) (x2 − x3)(1 − x1), M = p δ2(1 − x1)(x2 − x3), ϕ(t) = arcsin »−x1(x2 − x3) + x3(x2 − x1)z2(t) −(x2 − x3) + x3(x2 − x1)z2(t) – . θ(t) = θ(0) + (1 − x3)pθt x3 + pθ(x3 − x1) x1x3M Π „x3(x2 − x1) x1(x2 − x3), (Mt, k), k « r(t) = r(0) + (δx3 − γ+)t + δ(x1 − x3) M Π „x2 − x1 x2 − x3 , (Mt, k), k « . Long periodic orbits: k2 = (x2 − x3)(1 − x1) (1 − x3)(x2 − x1), M = p δ2(x2 − x1)(1 − x3). φ(t) = arcsin »−x1(1 − x3) + x3(1 − x1)z2(t) −(1 − x3) + x3(1 − x1)z2(t) – θ(t) = θ(0) + (1 − x3)pθt x3 + pθ(x3 − x1) x1x3M Π „x3(1 − x1) x1(1 − x3), (Mt, k), k « r(t) = −r(0) + (δx3 − γ+)t + δ(x1 − x3) M Π „1 − x1 1 − x3 , (Mt, k), k « .

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 22 / 27

Short periodic orbits: k2 = (1 − x3)(x2 − x1) (x2 − x3)(1 − x1), M = p δ2(1 − x1)(x2 − x3), ϕ(t) = arcsin »−x1(x2 − x3) + x3(x2 − x1)z2(t) −(x2 − x3) + x3(x2 − x1)z2(t) – . θ(t) = θ(0) + (1 − x3)pθt x3 + pθ(x3 − x1) x1x3M Π „x3(x2 − x1) x1(x2 − x3), (Mt, k), k « r(t) = r(0) + (δx3 − γ+)t + δ(x1 − x3) M Π „x2 − x1 x2 − x3 , (Mt, k), k « . Long periodic orbits: k2 = (x2 − x3)(1 − x1) (1 − x3)(x2 − x1), M = p δ2(x2 − x1)(1 − x3). φ(t) = arcsin »−x1(1 − x3) + x3(1 − x1)z2(t) −(1 − x3) + x3(1 − x1)z2(t) – θ(t) = θ(0) + (1 − x3)pθt x3 + pθ(x3 − x1) x1x3M Π „x3(1 − x1) x1(1 − x3), (Mt, k), k « r(t) = −r(0) + (δx3 − γ+)t + δ(x1 − x3) M Π „1 − x1 1 − x3 , (Mt, k), k « .

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 22 / 27

Example: δ = −1, pr = −2, pθ = 1, ϕ(0) = π/4

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

φ θ

1 2 3 1 2 3

φ θ Short extremals p1

ϕ(0) = 0, p2 ϕ(0) = 1,

the same orbits up to the first conjugate point h = 5.5 Long Extremals p1

ϕ(0) = −1, p2 ϕ(0) = 2,

h = 6.5.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 23 / 27

Non-integrable case: γ− = 0

Proposition The asymptotic stationary points (R∗, ϕ∗, θ∗) are char- acterized by the following relations R∗ = |γ−| γ+ , ϕ∗ = 0 if γ− > 0, and ϕ∗ = π if γ− < 0. Numerical observation: every extremal contains conjugate points.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 24 / 27

Non-integrable case: γ− = 0

Proposition The asymptotic stationary points (R∗, ϕ∗, θ∗) are char- acterized by the following relations R∗ = |γ−| γ+ , ϕ∗ = 0 if γ− > 0, and ϕ∗ = π if γ− < 0. Numerical observation: every extremal contains conjugate points.

Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 24 / 27

2 4 6 8 1 2 3

φ θ

1 2 3 4 0.2 0.4 0.6 0.8 1

ρ t

1 2 3 4 −40 −20 20 40

v1 t

1 2 3 4 −40 −20

v2 t

Γ = 2.5, γ+ = 2, γ− = −0.1, ϕ(0) = π

4 , pR(0) = −10, pθ = 1, pϕ(0) = −1, 0, 1. Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 25 / 27

2 4 6 8 1 2 3

φ θ

1 2 3 4 0.2 0.4 0.6 0.8 1

ρ t

1 2 3 4 −8 −6 −4 −2 2 4 6 8

v1 t

1 2 3 4 2 4 6 8 10 12 14

v2 t

Γ = 2.5, γ+ = 2, γ− = 0.1, ϕ(0) = π

4 , pr(0) = −10, pθ = 1, pϕ(0) = −1, 0, 1. Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 26 / 27

Meridian 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) 2 levels dissipative quantum systems April 2010 27 / 27