Adiabatic methods in Quantum Control Theory Gianluca Panati - PowerPoint PPT Presentation

Adiabatic methods in Quantum Control Theory Gianluca Panati Universit� di Roma � La Sapienza � Workshop on Quantum Control Institute Henri Poincar� , Paris, December 8-11, 2010

Separation of time-scales slow degrees ↔ fast degrees of freedom of freedom Fast degrees of freedom readjust ε - istantaneuosly to the evolution of the slow ones, where ε is the ratio between the two time scales.

Examples from the microphysical world: (i) molecular physics (Born-Oppenheimer approx) nuclei ↔ electrons (ii) Bloch electron: an electron in a crystal with a slowly varying external electromagnetic potential macroscale ↔ lattice scale dynamics dynamics

Adiabatic methods in Quantum Control Theory ?

Part I Adiabatic decoupling in a prototypical example: Born-Oppenheimer approximation in molecular physics

The framework x = ( x 1 , . . . , x K ) ∈ R 3 K =: X K nuclei: coordinates H n = L 2 ( X, dx ) H el = � N y = ( y 1 , . . . , y N ) ∈ R 3 N =: Y N electrons: coordinates i =1 L 2 ( R 3 , dy i ) Hilbert space : H := L 2 ( X ) ⊗ H el ∼ = L 2 ( X, H el ) Molecular dynamics is described by the Schr�dinger equation i ∂ s : microscopic time ∂s Ψ s = H mol Ψ s , with Hamiltonian K N � � h 2 h 2 ¯ ¯ H mol = − ∆ x k − ∆ y i + V e ( y ) + V n ( x ) + V en ( x, y ) 2 M k 2 m e k =1 i =1

The framework x = ( x 1 , . . . , x K ) ∈ R 3 K =: X K nuclei: coordinates H n = L 2 ( R 3 K , dx ) H el = � N y = ( y 1 , . . . , y N ) ∈ R 3 N =: Y N electrons: coordinates i =1 L 2 ( R 3 , dy i ) Hilbert space : H := L 2 ( X ) ⊗ H el ∼ = L 2 ( X, H el ) The Hamiltonian operator contains the following terms K K N N � � � � e 2 Z k Z l e 2 V n ( x ) = V e ( y ) = | x k − x l | | y i − y j | i =1 k =1 l � = k j � = i K N � � e 2 Z k V e , n ( x, y ) = − | x k − y i | k =1 i =1 where eZ k , for Z k ∈ Z , is the electric charge of the k -th nucleus. A cut-o� on the coulomb singularity is sometimes assumed to get rigorous results.

The framework The large number of degrees of freedom makes convenient to elaborate an approximation scheme , exploiting the smallness of the parameter � m e M ≃ 10 − 2 ε = h = 1 , m e = 1 ) and the adiabatic parameter By introducing atomic units ( ¯ ε the Hamiltonian H mol reads (up to a change of energy scale) K N � � ε 2 − 1 H ε = − 2 ∆ x k + V n ( x ) + 2∆ y i + V e ( y ) + V en ( x, y ) k =1 i =1 � �� � H el ( x ) For each �xed nuclei con�guration x = ( x 1 , . . . , x K ) ∈ X the operator H el ( x ) is an operator acting on the space H el .

The framework If the kinetic energies of the nuclei and the electrons are comparable, then the velocities scale as � m e | v n | ≈ M | v e | = ε | v e | . We have to wait a microscopically long time , namely O ( ε − 1 ) , in order to see a non-trivial dynamics for the nuclei. This scaling �xes the macroscopic time scale t = εs . In the macroscopic time scale, the Schr�dinger equation reads � � − ε 2 iε ∂ ∂t Ψ t = 2 ∆ x + H el ( x ) Ψ t , Ψ t =0 = Ψ 0 We are interested in the behavior of the solutions as ε ↓ 0 .

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X

A real-life example: the hydrogen quasi-molecule Credits: Eckart Wrede, University of Durham (UK)

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el = L 2 ( Y ) Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X The family { Ran P n ( x ) } x ∈ X , de�nes a complex vector bundle over X \ C , where C is the crossing manifold.

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el = L 2 ( Y ) Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X Geometric information is encoded in the Berry connection , A n ( x ) := i � χ n ( x ) , ∇ x χ n ( x ) � H el . de�ned over X \ C .

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X We focus on an isolated (non degenerate) energy band. We assume the initial state is concentrated on the n-th band , i. e. in the closed subspace Ran P n = { Ψ ∈ H : Ψ( x, y ) = ϕ ( x ) χ n ( x, y ) for ϕ ∈ L 2 ( X ) }

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el = L 2 ( Y ) Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X Transitions from an isolated band are O ( ε ) : � (1 − P n ) e − iH ε t/ε P n Ψ 0 � = O ( ε ) We say that an isolated band is adiabatically protected against tran- sitions. ⊲ Note: the upper bound holds for any Ψ 0 such that �− iε ∇ x Ψ 0 � = O (1) ≤ E , corresponding to the fact that the kinetic energy of the nuclei is supposed to be O (1) , i. e. comparable with that of the electrons.

The band structure Solution of the electronic structure problem: σ( ) H (x) E H el ( x ) χ n ( x, y ) = E n ( x ) χ n ( x, y ) (x) Σ E (x) 3 Eigenvalue: E (x) E n ( x ) 2 E (x) 1 Eigenfunction: χ n ( x, · ) ∈ H el = L 2 ( Y ) Eigenprojector: P n ( x ) = | χ n ( x ) �� χ n ( x ) | x Total projector: P n = { P n ( x ) } x ∈ X For a �xed band, the dynamics of the nuclei is governed by the Hamiltonian K � P n H ε P n = − ε 2 ∆ x k + E n ( x ) + O ( ε ) 2 k =1 in Ran P n ∼ = H n = L 2 ( X ) . Notice the impressive dimensional reduction! This is the time-dependent Born-Oppenheimer approximation .

References (i) Predecessors: time-adiabatic theorems ⊲ [Kato, Nenciu, Avron, Seiler, Simon, Sj�strand . . . and many others] (ii) Dynamical Born-Oppenheimer approximation ⊲ Propagation of generalized Gaussian wavepackets [Hagedorn and Joye] ⊲ Matrix valued pseudodi�erential operators [Brummelhaus, Nourrigat; Martinez, Nenciu, Sordoni; Panati, Spohn, Teufel] ⊲ Scattering theory including resonances [Martinez, Nakamura, Nenciu, Sordoni] ⊲ Exponentially small transitions [Hagedorn and Joye] ⊲ Optimal truncation [Betz and Teufel] (iii) Stationary Born-Oppenheimer approximation ⊲ [Combes, Duclos and Seiler; Klein, Martinez, Seiler, Wang] (iv) Dynamics near conical eigenvalue intersections ⊲ [P. Gerard, Fermannian, Lasser, Teufel, Colin de Verdi�re]

To prove the claim, one has to bound the di�erence � e − i H ε t/ε − e − i P n H ε P n t/ε � P n . The Duhamel formula yields � t/ε � e − i H ε t/ε − e − i P n H ε P n t/ε � ds e i H ε s ( P n H ε P n − H ε ) e − i P n H ε P n s P n P n = ie − i H ε t/ε 0 � t/ε ds e i H ε s ( P n H ε P n − H ε ) P n e − i P n H ε P n s = ie − i H ε t/ε 0 � t/ε ds e i H ε s [ P n , H ε ] P n e − i P n H ε P n s . = ie − i H ε t/ε � �� � 0 O ( ε ) The commutator is � � | χ n ( x ) �� χ n ( x ) | , − ε 2 [ P n , H ε ] P n = 2 ∆ x P n = O ( ε ) but the integration interval is O ( ε − 1 ) . Thus the na�f approach fails . A rigorous proof has been provided by [Spohn Teufel 2001], elaborating on [Kato 1950].

For a �xed band, the dynamics of the nuclei is governed by the Hamiltonian K � P n H ε P n = − ε 2 ∆ x k + E n ( x ) + O ( ε ) 2 k =1 acting in Ran P n ∼ = H n = L 2 ( X ) . What about higher-order corrections?

For a �xed band, the dynamics of the nuclei is governed by the Hamiltonian K � P n H ε P n = − ε 2 ∆ x k + E n ( x ) + O ( ε ) 2 k =1 acting in Ran P n ∼ = H n = L 2 ( X ) . What about higher-order corrections? The na�f expansion has no physical meaning since � (1 − P n ) e − iH ε t/ε P n Ψ 0 � = O ( ε ) ≥ Cε

For a �xed band, the dynamics of the nuclei is governed by the Hamiltonian K � P n H ε P n = − ε 2 ∆ x k + E n ( x ) + O ( ε ) 2 k =1 acting in Ran P n ∼ = H n = L 2 ( X ) . What about higher-order corrections? The na�f expansion has no physical meaning since � (1 − P n ) e − iH ε t/ε P n Ψ 0 � = O ( ε ) ≥ Cε Questions: (i) almost-invariant subspace: is there a subspace of H = H n ⊗H el which is almost-invariant under the dynamics, up to errors ε N ?