Quantum Control via Adiabatic Theory and intersection of eigenvalues - PowerPoint PPT Presentation

Quantum Control via Adiabatic Theory and intersection of eigenvalues U. Boscain, F. C. Chittaro, P. Mason, M. Sigalotti L2S-Sup´ elec (Paris) Workshop on Quantum Control IHP, Paris, December 8th-11th, 2010 F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 1 / 29

Introduction The problem Want to control the Schr¨ odinger equation i ∂ ∂ t ψ ( x , t ) = ( H 0 + u 1 ( t ) H 1 + u 2 ( t ) H 2 ) ψ ( x , t ) H 0 , H 1 , H 2 self-adjoint linerar operators on a Hilbert space H u = ( u 1 , u 2 ) : R → R 2 control x ∈ Ω ⊂ R n (possibly the whole R n ) F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 2 / 29

Introduction Assumptions on the Hamiltonians (H) H 0 is a self-adjoint operator on a Hilbert space H the discrete spectrum of H 0 is nonempty (and nontrivial). H 1 and H 2 are bounded and self-adjoint linear operators on H real with respect to H 0 Typical case: H 0 = − ∆ + V ( x ) where ∆ is the Laplacian on a domain of R n , V is a L 1 loc real-valued multiplication operator H 1 and H 2 are measurable bounded real valued multiplication operators. ( Σ ) there is an open domain in ω ⊂ R 2 where H ( u ) = H 0 + u 1 H 1 + u 2 H 2 , u ∈ ω , has a separated discrete spectrum . F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 3 / 29

Introduction Assumptions on the Hamiltonians (H) H 0 is a self-adjoint operator on a Hilbert space H the discrete spectrum of H 0 is nonempty (and nontrivial). H 1 and H 2 are bounded and self-adjoint linear operators on H real with respect to H 0 Typical case: H 0 = − ∆ + V ( x ) where ∆ is the Laplacian on a domain of R n , V is a L 1 loc real-valued multiplication operator H 1 and H 2 are measurable bounded real valued multiplication operators. ( Σ ) there is an open domain in ω ⊂ R 2 where H ( u ) = H 0 + u 1 H 1 + u 2 H 2 , u ∈ ω , has a separated discrete spectrum . F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 3 / 29

Introduction Assumptions on the Hamiltonians (H) H 0 is a self-adjoint operator on a Hilbert space H the discrete spectrum of H 0 is nonempty (and nontrivial). H 1 and H 2 are bounded and self-adjoint linear operators on H real with respect to H 0 Typical case: H 0 = − ∆ + V ( x ) where ∆ is the Laplacian on a domain of R n , V is a L 1 loc real-valued multiplication operator H 1 and H 2 are measurable bounded real valued multiplication operators. ( Σ ) there is an open domain in ω ⊂ R 2 where H ( u ) = H 0 + u 1 H 1 + u 2 H 2 , u ∈ ω , has a separated discrete spectrum . F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 3 / 29

Introduction Definitions Example of separated discrete spectrum F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 4 / 29

Introduction Definitions Definition of separated discrete spectrum Definition Let ω be a domain in R 2 . 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 f 1 , f 2 : ω → R such that: f 1 ( u ) < f 2 ( u ) and Σ( u ) ⊂ [ f 1 ( u ) , f 2 ( u )] ∀ u ∈ ω . there exists Γ > 0 such that u ∈ ω dist ([ f 1 ( u ) , f 2 ( u )] , σ ( u ) \ Σ( u )) > Γ inf Notation: Σ = { λ 0 ≤ . . . ≤ λ k } , where λ 0 is not necessarily the ground state. ϕ i ( u ) , i = 0 , . . . , k real eigenfunction of H ( u ) relative to λ i ( u ). F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 5 / 29

Introduction Definitions Definition of separated discrete spectrum Definition Let ω be a domain in R 2 . 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 f 1 , f 2 : ω → R such that: f 1 ( u ) < f 2 ( u ) and Σ( u ) ⊂ [ f 1 ( u ) , f 2 ( u )] ∀ u ∈ ω . there exists Γ > 0 such that u ∈ ω dist ([ f 1 ( u ) , f 2 ( u )] , σ ( u ) \ Σ( u )) > Γ inf Notation: Σ = { λ 0 ≤ . . . ≤ λ k } , where λ 0 is not necessarily the ground state. ϕ i ( u ) , i = 0 , . . . , k real eigenfunction of H ( u ) relative to λ i ( u ). F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 5 / 29

Introduction Definitions Definition of Spread Controllability Definition Σ be a separated discrete spectrum on ω u 0 ∈ ω such that λ i ( u 0 ) � = λ j ( u 0 ) i � = j . We say that the system is approximately spread controllable in ( ω, Σ( ω )) if for every Φ in ∈ { ϕ 0 ( u 0 ) , . . . , ϕ k ( u 0 ) } , ψ (0) = Φ in p ∈ [0 , 1] k +1 such that � k i =0 p 2 i = 1 ε > 0 there exists T > 0 and a continuous control u ( · ) : [0 , T ] → ω , u (0) = u ( T ) = u 0 such that � k � 1 / 2 � ( |� ϕ i ( u 0 ) , ψ ( T ) �| − p i ) 2 ≤ ε i =0 where ψ ( T ) is the solution of the equation i ˙ ψ ( t ) = H ( u ( t )) ψ ( t ). F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 6 / 29

Introduction Definitions Definition of Spread Controllability � k � 1 / 2 � ( |� ϕ i ( u 0 ) , ψ ( T ) �| − p i ) 2 ≤ ε i =0 � ∃ θ 0 , . . . , θ k such that Φ f = � k i =0 e i θ i p i ϕ i ( u 0 ) and we have � Φ f − ψ ( T ) � H ≤ ε F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 7 / 29

Introduction Results Main result Theorem Σ : ω → R k +1 separated discrete spectrum on ω ⊂ R 2 ∃ u j ∈ ω, j = 0 , . . . , k − 1 , such that λ j ( u j ) = λ j +1 ( u j ) conical intersection λ i ( u j ) simple if i � = j , j + 1 . Then the system is approximately spread controllable on Σ , where the final time T in can be chosen of the order O (1 /ε ) . Remark The proof is constructive Main tools Adiabatic Theorem Conical intersection F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 8 / 29

Introduction Results Main result Theorem Σ : ω → R k +1 separated discrete spectrum on ω ⊂ R 2 ∃ u j ∈ ω, j = 0 , . . . , k − 1 , such that λ j ( u j ) = λ j +1 ( u j ) conical intersection λ i ( u j ) simple if i � = j , j + 1 . Then the system is approximately spread controllable on Σ , where the final time T in can be chosen of the order O (1 /ε ) . Remark The proof is constructive Main tools Adiabatic Theorem Conical intersection F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 8 / 29

Introduction Results The Adiabatic Theorem Consider slowly varying controls i ∂ ∂ t ψ ( x , t ) = ( H 0 + u 1 ( ε t ) H 1 + u 2 ( ε t ) H 2 ) ψ ( x , t ) , ε > 0 H a ( τ ) = H ( τ ) − i ε P Σ ( τ ) ˙ P Σ ( τ ) − i ε P ⊥ Σ ( τ ) ˙ P ⊥ Σ ( τ ) τ = ε t Theorem (Born-Fock, Kato, Nenciu, Avron, Teufel...) Assume that H ( t ) ∈ C 2 . Then there is a constant C > 0 (depending on the gap) such that for all τ, τ 0 ∈ R � U ε ( τ, τ 0 ) − U ε a ( τ, τ 0 ) � ≤ C ε (1 + | τ − τ 0 | ) ( | τ − τ 0 | = O (1)) ≤ C ε (1 + ε | t − t 0 | ) ( | t − t 0 | = O (1 /ε )) F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 9 / 29

Introduction Results The Adiabatic Theorem Consider slowly varying controls i ∂ ∂ t ψ ( x , t ) = ( H 0 + u 1 ( ε t ) H 1 + u 2 ( ε t ) H 2 ) ψ ( x , t ) , ε > 0 H a ( τ ) = H ( τ ) − i ε P Σ ( τ ) ˙ P Σ ( τ ) − i ε P ⊥ Σ ( τ ) ˙ P ⊥ Σ ( τ ) τ = ε t Theorem (Born-Fock, Kato, Nenciu, Avron, Teufel...) Assume that H ( t ) ∈ C 2 . Then there is a constant C > 0 (depending on the gap) such that for all τ, τ 0 ∈ R � U ε ( τ, τ 0 ) − U ε a ( τ, τ 0 ) � ≤ C ε (1 + | τ − τ 0 | ) ( | τ − τ 0 | = O (1)) ≤ C ε (1 + ε | t − t 0 | ) ( | t − t 0 | = O (1 /ε )) F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 9 / 29

Introduction Results The Adiabatic Theorem Consider slowly varying controls i ∂ ∂ t ψ ( x , t ) = ( H 0 + u 1 ( ε t ) H 1 + u 2 ( ε t ) H 2 ) ψ ( x , t ) , ε > 0 H a ( τ ) = H ( τ ) − i ε P Σ ( τ ) ˙ P Σ ( τ ) − i ε P ⊥ Σ ( τ ) ˙ P ⊥ Σ ( τ ) τ = ε t Theorem (Born-Fock, Kato, Nenciu, Avron, Teufel...) Assume that H ( t ) ∈ C 2 . Then there is a constant C > 0 (depending on the gap) such that for all τ, τ 0 ∈ R � U ε ( τ, τ 0 ) − U ε a ( τ, τ 0 ) � ≤ C ε (1 + | τ − τ 0 | ) ( | τ − τ 0 | = O (1)) ≤ C ε (1 + ε | t − t 0 | ) ( | t − t 0 | = O (1 /ε )) F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 9 / 29

Conical intersections Definition Conical Intersections Definition u ∈ R 2 is a conical Let H ( u ) satisfy hypothesis (H) . We say that ¯ intersection between the eigenvalues λ 1 and λ 2 if λ 1 (¯ u ) = λ 2 (¯ u ) ∃ c > 0 such that for any unit vector v ∈ R 2 and t > 0 small enough we have that λ 2 (¯ u + t v ) − λ 1 (¯ u + t v ) > ct . Remark This definition is appropriate if the Hamiltonian is smooth with respect to the controls. F. C. Chittaro (L2S) QC via Adiabatic Theory December 11th, 2010 10 / 29