Quantum Control via Adiabatic Theory U. Boscain, F. C. Chittaro, P. - - PowerPoint PPT Presentation

Quantum Control via Adiabatic Theory

• U. Boscain, F. C. Chittaro, P. Mason, M. Sigalotti

L2S-Sup´ elec (Paris)

Workshop on Nonlinear Control and Singularities Porquerolles, October 24th-28th, 2010

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 1 / 29

Introduction

The problem

i ∂ ∂tψ(x, t) = (H0 + u1(t)V1(x) + u2(t)V2(x)) ψ(x, t) (1) x ∈ Ω ⊂ Rn H0 = −∆ + V (x) has a non-empty discrete spectrum V1 and V2 are measurable bounded real valued multiplication operators. (u1, u2) =: u ∈ ω ⊂ R2

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 2 / 29

Introduction Definitions

Example of separated discrete spectrum

• Σ

f f ω

+ −

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 3 / 29

Introduction Definitions

Definition of separated discrete spectrum

Definition Let ω be a domain in R2. A map Σ defined on ω that associates to each u ∈ ω a subset Σ(u) of the discrete spectrum of H(u) is said to be a separated discrete spectrum on ω if there exist two continuous bounded functions f1, f2 : ω → R such that: f1(u) < f2(u) and Σ(u) ⊂ [f1(u), f2(u)] ∀u ∈ ω. there exists Γ > 0 such that inf

u∈ω dist([f1(u), f2(u)], σ(u) \ Σ(u)) > Γ

Notation: Σ = {Λ0 ≤ . . . ≤ Λk} ϕi(u), i = 0, . . . , k real eigenfunction of H(u) relative to Λi(u).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 4 / 29

Introduction Definitions

Definition of separated discrete spectrum

Definition Let ω be a domain in R2. A map Σ defined on ω that associates to each u ∈ ω a subset Σ(u) of the discrete spectrum of H(u) is said to be a separated discrete spectrum on ω if there exist two continuous bounded functions f1, f2 : ω → R such that: f1(u) < f2(u) and Σ(u) ⊂ [f1(u), f2(u)] ∀u ∈ ω. there exists Γ > 0 such that inf

u∈ω dist([f1(u), f2(u)], σ(u) \ Σ(u)) > Γ

Notation: Σ = {Λ0 ≤ . . . ≤ Λk} ϕi(u), i = 0, . . . , k real eigenfunction of H(u) relative to Λi(u).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 4 / 29

Introduction Definitions

Definition of spread controllable

Definition Let Σ be a separated discrete spectrum on ω and assume that it is non-degenerate at u0 ∈ ω. We say that the system is spread controllable in (ω, Σ(ω)) if for every Φin ∈ {ϕ0(u0), . . . , ϕk(u0)}, p ∈ [0, 1]k+1 such that k

i=0 p2 i = 1 and ε > 0 there exists T > 0 and a continuous control

u(·) : [0, T] → R2, u(0) = u(T) = u0 such that

k

• i=0

( |ϕi(u0), ψ(T)| − pi)2 ≤ ε where ψ(T) is the solution at time T of (1) with ψ(0) = Φin. ∃ ϑi, . . . , ϑk such that Φf = k

i=1 eiϑipiϕi(u0) and we have

Φf − ψ(T)2 ≤ ε

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 5 / 29

Introduction Definitions

Definition of spread controllable

Definition Let Σ be a separated discrete spectrum on ω and assume that it is non-degenerate at u0 ∈ ω. We say that the system is spread controllable in (ω, Σ(ω)) if for every Φin ∈ {ϕ0(u0), . . . , ϕk(u0)}, p ∈ [0, 1]k+1 such that k

i=0 p2 i = 1 and ε > 0 there exists T > 0 and a continuous control

u(·) : [0, T] → R2, u(0) = u(T) = u0 such that

k

• i=0

( |ϕi(u0), ψ(T)| − pi)2 ≤ ε where ψ(T) is the solution at time T of (1) with ψ(0) = Φin. ∃ ϑi, . . . , ϑk such that Φf = k

i=1 eiϑipiϕi(u0) and we have

Φf − ψ(T)2 ≤ ε

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 5 / 29

Introduction Results

Main result

Theorem Let Σ = {Λ0(u) ≤ . . . ≤ Λk(u)} be a separated discrete spectrum on ω. Let u0, ¯ ui ∈ ω, i = 1, . . . , k. Assume that Σ(u0) is non degenerate and that ¯ ui is a conical intersection between the eigenvalues Λi−1 and Λi. Then the system is spread controllable in (ω, Σ(ω)). Remark The proof is constructive Main tools Adiabatic Theorem Conical intersection

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 6 / 29

Introduction Results

Main result

Theorem Let Σ = {Λ0(u) ≤ . . . ≤ Λk(u)} be a separated discrete spectrum on ω. Let u0, ¯ ui ∈ ω, i = 1, . . . , k. Assume that Σ(u0) is non degenerate and that ¯ ui is a conical intersection between the eigenvalues Λi−1 and Λi. Then the system is spread controllable in (ω, Σ(ω)). Remark The proof is constructive Main tools Adiabatic Theorem Conical intersection

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 6 / 29

The adiabatic Theorem

Slow Dynamics

H(t) time dependent Hamiltonian, U(t) evolution operator, Σ(t) its separated discrete spectrum , PΣ(t) spectral projection on Σ(t). PΣ(t)U(t, t0) = U(t, t0)PΣ(t0) Consider the slow dynamics i ∂ ∂tψ(t) = H(εt)ψ(t) ε > 0 Notation: τ = εt and from now on ˙ f :=

d dτ f.

Define the adiabatic Hamiltonian Ha(τ) = H(τ) − iεPΣ(τ) ˙ PΣ(τ) − iεP ⊥

Σ (τ) ˙

P ⊥

Σ (τ)

PΣ(τ)Uε

a(τ, τ0) = Uε a(τ, τ0)PΣ(τ0)

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 7 / 29

The adiabatic Theorem

Slow Dynamics

H(t) time dependent Hamiltonian, U(t) evolution operator, Σ(t) its separated discrete spectrum , PΣ(t) spectral projection on Σ(t). PΣ(t)U(t, t0) = U(t, t0)PΣ(t0) Consider the slow dynamics i ∂ ∂tψ(t) = H(εt)ψ(t) ε > 0 Notation: τ = εt and from now on ˙ f :=

d dτ f.

Define the adiabatic Hamiltonian Ha(τ) = H(τ) − iεPΣ(τ) ˙ PΣ(τ) − iεP ⊥

Σ (τ) ˙

P ⊥

Σ (τ)

PΣ(τ)Uε

a(τ, τ0) = Uε a(τ, τ0)PΣ(τ0)

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 7 / 29

The adiabatic Theorem

The Adiabatic Theorem

Theorem (Born-Fock,Kato,Teufel...) Assume that H(t) ∈ C2. Then PΣ(τ) ∈ C2 and there is a constant C > 0 (depending on the gap) such that for all τ, τ0 ∈ R Uε(τ, τ0) − Uε

a(τ, τ0) ≤ Cε(1 + |τ − τ0|)

(|τ − τ0| = O(1)) ≤ Cε(1 + ε|t − t0|) (|t − t0| = O(1/ε))

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 8 / 29

Conical intersections Definition

Conical Intersections

Definition Let ω ⊂ R2 a domain and ¯ u ∈ ω such that Λi(¯ u) = Λi+1(¯ u). We say that ¯ u is a conical intersection if and only if for any straight line r(t) = ¯ u + tu in R2 passing through ¯ u we have that Dr (λα − λβ) |¯

u = 0.

(2) Recall (Kato): {λα(r(t)), λβ(r(t))} = {Λi(r(t)), Λi+1(r(t))} and λα, λβ are analytic along r. Remark This definition is appropriate if the Hamiltonian is smooth with respect to the controls.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 9 / 29

Conical intersections Definition

Conical Intersections

Definition Let ω ⊂ R2 a domain and ¯ u ∈ ω such that Λi(¯ u) = Λi+1(¯ u). We say that ¯ u is a conical intersection if and only if for any straight line r(t) = ¯ u + tu in R2 passing through ¯ u we have that Dr (λα − λβ) |¯

u = 0.

(2) Recall (Kato): {λα(r(t)), λβ(r(t))} = {Λi(r(t)), Λi+1(r(t))} and λα, λβ are analytic along r. Remark This definition is appropriate if the Hamiltonian is smooth with respect to the controls.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 9 / 29

Conical intersections Definition

Example (true conical intersection)

H(u) = u1 u2 u2 −u1

• The eigenvalues are

Λ±(u) = ±

• u2

1 + u2 2.

The eigenvectors at u are ϕ+(u) = (cos(θ/2), sin(θ/2)) ϕ−(u) = (− sin(θ/2), cos(θ/2)) where u = ρ(cos θ, sin θ). At the intersection u = (0, 0) there is no choice that makes ϕ+, ϕ− continuous in a neighbourhood of (0, 0).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 10 / 29

Conical intersections Definition

Example (false conical intersection)

H(u) =

• u2

1 + u2 2

−

• u2

1 + u2 2

• (3)

The eigenvalues are Λ±(u) = ±

• u2

1 + u2 2.

The eigenvectors are constant with respect to u. Remark Actually (3) is not smooth.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 11 / 29

Conical intersections Dynamics through conical intersections

Passage through a conical intersection

u

2 1

u

u0 conical intersection between Λ1 and Λ2 u(τ) = ¯ u + (τ − 1)(cos α, sin α) τ ∈ [0, 2] ψ(T) solution of (1) at time T = 2/ε with ψ(0) = ϕ1(u0) From adiabatic theory |1 − |ϕ2(u(2)), ψ(T)|| ≤ C√ε

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 12 / 29

Conical intersections Dynamics through conical intersections

Passage through a conical intersection

u

2 1

u

u0 conical intersection between Λ1 and Λ2 u(τ) = ¯ u + (τ − 1)(cos α, sin α) τ ∈ [0, 2] ψ(T) solution of (1) at time T = 2/ε with ψ(0) = ϕ1(u0) From adiabatic theory |1 − |ϕ2(u(3)), ψ(T)|| ≤ C√ε

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 13 / 29

Conical intersections Dynamics through conical intersections

Passage through a conical intersection

u u2

1

u(τ) =

• ¯

u + τ(w1, w2), τ ∈ [−1, 0] ¯ u + τ(v1, v2), τ ∈ [0, 1] Is it possible to spread the probability

• f occupation of ϕ1 and ϕ2?
• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 14 / 29

Conical intersections Geometry of conical intersections

Behaviour around a conical intersection

Theorem (Kato) ”Along analytic curves the eigenfunctions and the eigenvalues are analytic.” For any analytic curve γ such that γ(¯ t) = ¯ u, there is a choice of ϕ1, ϕ2

• rthonormal eigenfunctions of H(¯

u) relative to Λ1(¯ u) = Λ2(¯ u) such that lim

t→¯ t ϕi(γ(t)) = ϕi,

i = 1, 2. In particular, if γ(t) = ¯ u + t(w1, w2), then the basis satisfies ϕ1(γ(t)), (w1V1 + w2V2)ϕ2(γ(t)) = 0 (4) The relation (4) is invariant for α → α + π and for exchanges between ϕ1 and ϕ2.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 15 / 29

Conical intersections Geometry of conical intersections

Behaviour around a conical intersection

Proposition Let γ be a C1 curve such that γ(¯ t) = ¯ u. Let r(t) be the tangent line to γ at ¯ u, r(¯ t) = ¯

• u. Then

lim

t→¯ t ϕi(γ(t)) = lim t→¯ t ϕi(r(t)),

i = 1, 2. Proposition Let γ be a C2 curve such that γ(¯ t) = ¯

• u. Then there is a choice of ϕ1, ϕ2
• rthonormal eigenfunctions of H(¯

u) relative to Λ1(¯ u) = Λ2(¯ u) that makes the eigenfunctions of H (with respect to the selected eigenspace) continuous and differentiable along γ at ¯ u.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 16 / 29

Conical intersections Geometry of conical intersections

The Conicity Matrix

Definition Let ϕ1, ϕ2 two real eigenfunctions of H(u) associated to Λ1, Λ2. The conicity matrix associated to the pair (ϕ1, ϕ2) is the matrix M(ϕ1, ϕ2) = ϕ1, V1ϕ2

1 2 (ϕ2, V1ϕ2 − ϕ1, V1ϕ1)

ϕ1, V2ϕ2

1 2 (ϕ2, V2ϕ2 − ϕ1, V2ϕ1)

• Proposition

Actually the conicity matrix is defined on every pair of orthonormal functions φ1, φ2 ∈ L2(Ω). It continuous as a function defined on the 2-Grassmannian of L2(Ω).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 17 / 29

Conical intersections Geometry of conical intersections

The Conicity Matrix

Definition Let ϕ1, ϕ2 two real eigenfunctions of H(u) associated to Λ1, Λ2. The conicity matrix associated to the pair (ϕ1, ϕ2) is the matrix M(ϕ1, ϕ2) = ϕ1, V1ϕ2

1 2 (ϕ2, V1ϕ2 − ϕ1, V1ϕ1)

ϕ1, V2ϕ2

1 2 (ϕ2, V2ϕ2 − ϕ1, V2ϕ1)

• Proposition

Actually the conicity matrix is defined on every pair of orthonormal functions φ1, φ2 ∈ L2(Ω). It continuous as a function defined on the 2-Grassmannian of L2(Ω).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 17 / 29

Conical intersections Geometry of conical intersections

The Conicity Matrix

Proposition The determinant of M(φ1, φ2) is invariant for any rotation of the pair (φ1, φ2). Corollary The function u → det M(ϕ1(u), ϕ2(u)) is well defined as a function of u ∈ ω.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 18 / 29

Conical intersections Geometry of conical intersections

Characterisation of conical intersections

Theorem The intersection is conical if and only if the conicity matrix is non-degenerate at the intersection. Theorem Let Ω be a either a bounded connected domain of Rn or the whole Rn, and let us fix a Schr¨

• dinger operator H0 on L2(Ω) with a double

eigenvalue Λ1 = Λ2. Then u = (0, 0) corresponds to a conical intersection for H0 + u1V1 + u2V2 generically with respect to the choice of the pair {V1, V2} ∈ L∞(Ω) × L∞(Ω).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 19 / 29

Conical intersections Geometry of conical intersections

Characterisation of conical intersections

Theorem The intersection is conical if and only if the conicity matrix is non-degenerate at the intersection. Theorem Let Ω be a either a bounded connected domain of Rn or the whole Rn, and let us fix a Schr¨

• dinger operator H0 on L2(Ω) with a double

eigenvalue Λ1 = Λ2. Then u = (0, 0) corresponds to a conical intersection for H0 + u1V1 + u2V2 generically with respect to the choice of the pair {V1, V2} ∈ L∞(Ω) × L∞(Ω).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 19 / 29

Conical intersections Geometry of conical intersections

Relation between the limit bases

Proposition Let ¯ u + t(w1, w2) and ¯ u + t(v1, v2), t ≥ 0, be two semiaxis. Call ϕw

1 , ϕw 2

and ϕv

1, ϕv 2 the limit eigenfunction bases along t(w1, w2) (respectively

t(v1, v2)), as t → 0+. Then ϕv

1 = cos θϕw 1 + sin θϕw 2

ϕv

2 = − sin θϕw 1 + cos θϕw 2

where θ satisfies the equation (v1, v2)M(ϕw

1 , ϕw 2 )

cos 2θ sin 2θ

• .
• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 20 / 29

The effective dynamics Reduction to 2-d

Reduction to 2-d

• f

ω

− Λ Λ Λ Λ Λ Λ Λ

1 2 3 5 4 5 1 Γ

Σ

Γ

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 21 / 29

The effective dynamics Reduction to 2-d

Representation in C2

Assumptions: Σ(ω′) = {Λ1 ≤ Λ2} γ C2 curve in R2 such that:

λα, λβ : ω′ → R continuous such that {λα(u), λβ(u)} = {Λ1(u), Λ2(u)} ∀ u ∈ ω′ ϕα, ϕβ C1-smooth

The operator U(t) : PΣ(γ(t))(D(H)) → C2 defined as U(t) = 1

• ϕα(γ(t)), · +

1

• ϕβ(γ(t)), ·

establishes an isomorphism between PΣ(γ(t))(D(H)) = C{ϕ1(u), ϕ2(u)} ≃ C2 {ϕ1(u), ϕ2(u)} ↔

• (1, 0)T , (0, 1)T
• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 22 / 29

The effective dynamics Reduction to 2-d

Representation in C2

Assumptions: Σ(ω′) = {Λ1 ≤ Λ2} γ C2 curve in R2 such that:

λα, λβ : ω′ → R continuous such that {λα(u), λβ(u)} = {Λ1(u), Λ2(u)} ∀ u ∈ ω′ ϕα, ϕβ C1-smooth

The operator U(t) : PΣ(γ(t))(D(H)) → C2 defined as U(t) = 1

• ϕα(γ(t)), · +

1

• ϕβ(γ(t)), ·

establishes an isomorphism between PΣ(γ(t))(D(H)) = C{ϕ1(u), ϕ2(u)} ≃ C2 {ϕ1(u), ϕ2(u)} ↔

• (1, 0)T , (0, 1)T
• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 22 / 29

The effective dynamics Reduction to 2-d

The effective Hamiltonian

Heff(τ) = U(τ)Ha(τ)U+(τ) + iε ˙ U(τ)U+(τ) = λα(τ) λβ(τ)

• + iε
• ˙

ϕα(τ), ϕβ(τ) ˙ ϕα(τ), ϕβ(τ)

• Uε

eff(τ, τ0) evolution operator (on C2) associated to Heff, represented back

to PΣ(γ(t))(D(H0)). (Uε(τ, τ0) − Uε

eff(τ, τ0)) PΣ(γ(t)) ≤ Cε(1 + |τ − τ0|)

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 23 / 29

The effective dynamics Reduction to 2-d

The effective Hamiltonian

Heff(τ) = U(τ)Ha(τ)U+(τ) + iε ˙ U(τ)U+(τ) = λα(τ) λβ(τ)

• + iε
• ˙

ϕα(τ), ϕβ(τ) ˙ ϕα(τ), ϕβ(τ)

• Uε

eff(τ, τ0) evolution operator (on C2) associated to Heff, represented back

to PΣ(γ(t))(D(H0)). (Uε(τ, τ0) − Uε

eff(τ, τ0)) PΣ(γ(t)) ≤ Cε(1 + |τ − τ0|)

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 23 / 29

The effective dynamics The non-mixing dynamics

The non-mixing Field

˙ ϕ1(τ), ϕ2(τ) = ϕ1, ( ˙ u1V1 + ˙ u2V2)ϕ2 ˙ ϕ1(τ), ϕ2(τ) ≡ 0 along the solutions of the equation

• ˙

u1 = −ϕ1, V2ϕ2 ˙ u2 = ϕ1, V1ϕ2

• ˙

u1 = ϕ1, V2ϕ2 ˙ u2 = −ϕ1, V1ϕ2

• Definition

The field X(u) = (±)(−ϕ1(u), V2ϕ2(u), ϕ1(u), V1ϕ2(u)) is called the non-mixing field. X is well defined and continuous in ω′ \ {¯ u}; it is multivalued at ¯ u.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 24 / 29

The effective dynamics The non-mixing dynamics

The non-mixing Field

˙ ϕ1(τ), ϕ2(τ) = ϕ1, ( ˙ u1V1 + ˙ u2V2)ϕ2 ˙ ϕ1(τ), ϕ2(τ) ≡ 0 along the solutions of the equation

• ˙

u1 = −ϕ1, V2ϕ2 ˙ u2 = ϕ1, V1ϕ2

• ˙

u1 = ϕ1, V2ϕ2 ˙ u2 = −ϕ1, V1ϕ2

• Definition

The field X(u) = (±)(−ϕ1(u), V2ϕ2(u), ϕ1(u), V1ϕ2(u)) is called the non-mixing field. X is well defined and continuous in ω′ \ {¯ u}; it is multivalued at ¯ u.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 24 / 29

The effective dynamics The non-mixing dynamics

The non-mixing Field

˙ ϕ1(τ), ϕ2(τ) = ϕ1, ( ˙ u1V1 + ˙ u2V2)ϕ2 ˙ ϕ1(τ), ϕ2(τ) ≡ 0 along the solutions of the equation

• ˙

u1 = −ϕ1, V2ϕ2 ˙ u2 = ϕ1, V1ϕ2

• ˙

u1 = ϕ1, V2ϕ2 ˙ u2 = −ϕ1, V1ϕ2

• Definition

The field X(u) = (±)(−ϕ1(u), V2ϕ2(u), ϕ1(u), V1ϕ2(u)) is called the non-mixing field. X is well defined and continuous in ω′ \ {¯ u}; it is multivalued at ¯ u.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 24 / 29

The effective dynamics The non-mixing dynamics

The integral curves of X

DX (λα − λβ)(u) = det M(u) There is a neighbourhood of the conical intersection ¯ u such that for any u ∈ U the integral curve of (±)X starting from u reaches ¯ u in finite time at the conical intersection X(¯ u) covers all possible directions the integral curves of X are C1 (¯ u included) and ϕ is C1 along them

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 25 / 29

The effective dynamics The non-mixing dynamics

The integral curves of X

DX (λα − λβ)(u) = det M(u) There is a neighbourhood of the conical intersection ¯ u such that for any u ∈ U the integral curve of (±)X starting from u reaches ¯ u in finite time at the conical intersection X(¯ u) covers all possible directions the integral curves of X are C1 (¯ u included) and ϕ is C1 along them

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 25 / 29

The effective dynamics The non-mixing dynamics

The integral curves of X

DX (λα − λβ)(u) = det M(u) There is a neighbourhood of the conical intersection ¯ u such that for any u ∈ U the integral curve of (±)X starting from u reaches ¯ u in finite time at the conical intersection X(¯ u) covers all possible directions the integral curves of X are C1 (¯ u included) and ϕ is C1 along them

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 25 / 29

Results 2 levels controllability

Main results

Theorem Let Σ = {Λ1(u) ≤ Λ2(u)} be a separated discrete spectrum on ω, and u0 ∈ ω. Assume that Λ1(u0) = Λ2(u0) and that ¯ u is a conical intersection between Λ1 and Λ2. Consider a closed curve γ(·) : [0, 1] → ω starting from u0 and such that there exists a ¯ t ∈ (0, 1) with γ(¯ t) = ¯ u and λ0(γ(t)) = λ1(γ(t)) ∀t = ¯ t. Assume that γ is C2 in (0, ¯ t) ∪ (¯ t, 1); ˙ γ(t) = X(γ(t)) t ∈ [t1, ¯ t) ∪ (¯ t, t2] in a neighbourhood of ¯ u there are two unit vectors w, v such that lim

t→¯ t+

˙ γ(t) ˙ γ(t) = w lim

t→¯ t−

˙ γ(t) ˙ γ(t) = v.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 26 / 29

Results 2 levels controllability

Main results

Then for any ε > 0, there is a T > 0, T = O(1/ε) such that (|ϕ1(u0), ψ(T)| − p1)2 ≤ ε, (|ϕ2(u0), ψ(T)| − p2)2 ≤ ε. where ψ is the solution of equation (1) with ψ(0) = ϕ1(u0) corresponding to the control u : [0, T] → ω defined by u(t) = γ(t/T), and p2

1 = cos2 θ

and p2

2 = sin2 θ satisfy the equation

(v1, v2) M(ϕw

1 , ϕw 2 )

cos 2θ sin 2θ

• = 0,
• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 27 / 29

Results 2 levels controllability

Main results

Theorem Let Σ = {Λ1(u) ≤ . . . ≤ Λk(u)} be a separated discrete spectrum on ω. Assume that there is u0 ∈ ω such that Λi(u0) = Λj(u0), i = j, and that for every i = 0, . . . , k − 1 there is ¯ ui ∈ ω conical intersection between Λi and Λi+1 such that ¯ ui = ¯ uj if i = j. Then the system (1) is spread controllable in (ω, Σ(ω)).

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 28 / 29

Results Conclusions

Further Perspectives

study the case H(u) nonlinear w.r.t. u. try to obtain a stronger controllability result, that is allowing |ϕi, ψ(0)| = πi with k

i=1 π2 i = 1

looking for a good approximation of the integral curves of X which are more easily computable.

• F. C. Chittaro (L2S)

Quantum Control via Adiabatic Theory October 23rd, 2010 29 / 29