Weakly-coupled bilinear quantum systems Thomas Chambrion Nabile - PowerPoint PPT Presentation

Bilinear quantum systems Averaging techniques Examples Conclusion Weakly-coupled bilinear quantum systems Thomas Chambrion Nabile Boussaïd (Besançon) and Marco Caponigro (Rutgers) PICOF 2012. April 2–4, 2012

Bilinear quantum systems Averaging techniques Examples Conclusion Objective of the talk Most of the quantum systems encountered in practice are governed by PDEs i ∂ψ ∂ t ( x , t ) = ( − ∆ + V ( x )) ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t ) We will try to understand how the properties (controllability) of these infinite dimensional systems can be deduced from the properties of their finite dimensional approximations. In what follows, we neglect decoherence.

Bilinear quantum systems Averaging techniques Examples Conclusion Outline of the talk Bilinear quantum systems 1 Bilinear Schrödinger equation Good Galerkin approximation Averaging techniques 2 Rotating wave approximation Examples 3 Quantum harmonic oscillator Rotation of a planar molecule Conclusion 4 Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion Outline of the talk Bilinear quantum systems 1 Bilinear Schrödinger equation Good Galerkin approximation Averaging techniques 2 Rotating wave approximation Examples 3 Quantum harmonic oscillator Rotation of a planar molecule Conclusion 4 Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion Some examples A quantum system evolving in Ω , a finite dimensional Riemannian manifold, is described by its wave function ψ in the unit sphere of L 2 (Ω , C ) . The system is in the subset ω with probability � ω | ψ | 2 d µ . The time evolution is given by the Schrödinger equation i ∂ψ ∂ t ( x , t ) = ( − ∆ + V ( x )) ψ ( x , t ) When submitted to an external field ( e.g. , a laser) with time variable intensity, ψ satisfies i ∂ψ ∂ t = ( − ∆ + V ( x )) ψ ( x , t ) + u ( t ) W ( x ) ψ ( x , t )

Bilinear quantum systems Averaging techniques Examples Conclusion Some examples Rotation of a planar molecule Ω = SO ( 2 ) ≃ R / 2 π Z i ∂ψ ∂ t ψ ( θ, t ) = − ∂ θθ ψ ( θ, t ) + u ( t ) cos θψ ( θ, t ) (Dion, Salomon, Turinici, Sugny,...) Rotation of a molecule in space Ω = S 2 i ∂ψ ∂ t ψ ( θ, ν, t ) = − ∆ ψ ( θ, ν, t ) + u ( t ) cos θψ ( θ, ν, t )

Bilinear quantum systems Averaging techniques Examples Conclusion Some examples Harmonic oscillator Ω = R i ∂ψ ∂ t ψ ( x , t ) = ( − ∂ xx + x 2 ) ψ ( x , t ) + u ( t ) x ψ ( x , t ) (Mirrahimi, Rouchon, Illner,...) Infinite square potential well Ω = ( 0 , π ) i ∂ψ ∂ t ψ ( x , t ) = ∂ xx ψ ( x , t ) + u ( t ) x ψ ( x , t ) (Beauchard, Coron, Laurent, Nersessyan,...)

Bilinear quantum systems Averaging techniques Examples Conclusion Abstract form In the Hilbert space H (= L 2 (Ω , C )) , we consider an unbounded skew-adjoint linear operator A (= − i (∆ + V )) , a skew symmetric operator B (= − i W ( x )) and the evolution equation d ψ dt = ( A + u ( t ) B ) ψ For more complicated models, see Morancey (2011).

Bilinear quantum systems Averaging techniques Examples Conclusion Well-posedness � d ψ dt = ( A + u ( t ) B ) ψ ψ ( 0 ) = ψ 0 Well-posedness is very far from obvious when A or B is unbounded (i.e., not continuous), what is the case here. In the presented examples, for every locally integrable u : R → R , we can define the solution t �→ Υ u t ( ψ 0 ) . If ψ 0 belongs to D ( A ) , then Υ u ( ψ 0 ) is absolutely continuous and d dt Υ u t ( ψ 0 ) = ( A + u ( t ) B )Υ u t ( ψ 0 ) for a.e. t

Bilinear quantum systems Averaging techniques Examples Conclusion Discrete spectrum In the presented examples, A has discrete spectrum. There exists a non-decreasing sequence ( λ n ) n ∈ N in [ 0 , + ∞ ) and an Hilbert basis ( ψ n ) n ∈ N of H such that A ψ n = − i λ n φ n for every n . Infinite dimensional matrices representation  − i λ 1 0 · · · · · ·  ...   0 − i λ 2   A =  .  ... ... .   . − i λ 3    .  ... ... . . b j , k = � φ j , B φ k � , b jk = − b kj

Bilinear quantum systems Averaging techniques Examples Conclusion Weakly coupled quantum systems Definition (Weakly coupled system) ( A , B ) is k-weakly-coupled if A is skew-adjoint with discrete spectrum ( − i λ n ) n ∈ N , ( λ n ) n ∈ N is positive, non decreasing and tends to infinity; B is skew-adjoint; for every u in R , D ( | A + uB | k / 2 ) = D ( | A | k / 2 ) ; there exists C A , B such that, for every ψ in D ( | A | k ) , |ℜ�| A | k ψ, B ψ �| ≤ C A , B |�| A | k ψ, ψ �|

Bilinear quantum systems Averaging techniques Examples Conclusion Growth of | A | k / 2 norms k / 2 = �| A | k / 2 ψ � 2 = |�| A | k ψ, ψ �| = � � ψ � 2 λ k n |� φ n , ψ �| n ∈ N The 1 / 2 norm is the expected value of the energy at ψ . | d dt �| A | k ψ, ψ �| | 2 u ( t ) ℜ�| A | k ψ, B ψ �| = 2 | u ( t ) | C A , B |�| A | k ψ, ψ �| ≤ By Gronwall’s lemma: � t |�| A | k ψ ( t ) , ψ ( t ) �| ≤ e 2 C A , B 0 | u ( τ ) | d τ |�| A | k ψ ( 0 ) , ψ ( 0 ) �|

Bilinear quantum systems Averaging techniques Examples Conclusion Size of velocity tail Define π N : H → H , the orthogonal projection on the first N eigenstates of A . � � ( Id − π N ) ψ ( t ) � 2 λ r N |� φ n , ( Id − π N ) ψ ( t ) , �| 2 ≤ r n ≥ N 1 � λ k n |� φ n , ( Id − π N ) ψ ( t ) , �| 2 ≤ λ k − r N n ≥ N 1 |�| A | k ( Id − π N ) ψ ( t ) , ( Id − π N ) ψ ( t ) �| ≤ λ k − r N e 2 C A , B � u � L 1 |�| A | k ψ ( 0 ) , ψ ( 0 ) �| ≤ → 0 λ k − r N

Bilinear quantum systems Averaging techniques Examples Conclusion Good Galerkin approximation π N ψ ′ ( t ) = A ( N ) π N ψ ( t ) + u ( t ) π N B π N ψ ( t ) + u ( t ) π N B ( 1 − π N ) ψ ( t ) Denoting with X ( N ) ( t ) the propagator of the N -dimensional system u x ′ = ( A ( N ) + u ( t ) B ( N ) ) x , � t π N ψ ( t ) = X ( N ) X ( N ) ( t ) π N ψ ( 0 ) + ( t , s ) u ( τ ) π N B ( 1 − π N ) ψ ( τ ) d τ u u 0 Proposition (Boussaid-Caponigro-Chambrion, 2011) Let ( A , B ) be k- weakly-coupled and B be bounded relatively to | A | r for r < k. For every ǫ > 0 , for every K > 0 , for every ψ 0 , for every s < k, there exists N = N ( ǫ, K , ψ 0 ) such that ⇒ � π N Υ u t ( ψ 0 ) − X ( N ) � u � L 1 ≤ K = ( t ) π N ψ 0 � s < ǫ. u

Bilinear quantum systems Averaging techniques Examples Conclusion Outline of the talk Bilinear quantum systems 1 Bilinear Schrödinger equation Good Galerkin approximation Averaging techniques 2 Rotating wave approximation Examples 3 Quantum harmonic oscillator Rotation of a planar molecule Conclusion 4 Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion Non degenerate transitions Definition A transition ( j , k ) , j � = k, is non degenerate if b j , k � = 0 and, for every l 1 , l 2 , | λ l 1 − λ l 2 | = | λ j − λ k | = ⇒ { l 1 , l 2 } = { j , k } or b l 1 , l 2 = 0 .

Bilinear quantum systems Averaging techniques Examples Conclusion Periodic control laws Proposition Let ( j , k ) be a non-degenerate transition of ( A , B ) and u ∗ be a 2 π | λ j − λ k | 2 π � u ∗ ( τ ) e i | λ j − λ k | τ d τ � = 0 and | λ j − λ k | -periodic function. If 0 2 π | λ j − λ k | � u ∗ ( τ ) e i | λ l − λ m | τ d τ = 0 for every l , m such that 0 | λ l − λ m | ∈ ( N \ { 1 } ) | λ j − λ k | , then there exists T ∗ such that � � � φ k , X u ∗ / n � n →∞ nT ∗ φ j � − → 1 . � � � π T T ∗ = . � T � � 0 u ∗ ( τ ) e i ( λ j − λ k ) τ d τ 2 | b j , k | � � � �

Bilinear quantum systems Averaging techniques Examples Conclusion Some estimates L 1 norm needed to achieve the transition from level j to k π 1 Eff jk ( u ∗ ) 2 | b jk | with 2 π � � | λ j − λ k | � � u ∗ ( τ ) e i ( λ j − λ k ) τ d τ � � � 0 � � 0 ≤ Eff jk ( u ∗ ) = ≤ 1 . 2 π | λ j − λ k | � | u ∗ ( τ ) | d τ 0 Error estimates (for bounded B ) C ( u ∗ , B ) 1 − |� φ k , X u ∗ / n nT ∗ φ j �| ≤ � � | λ l 1 − λ l 2 | n inf l 1 , l 2 | λ j − λ k | − 1 � � � � Error × Time ≤ Const

Bilinear quantum systems Averaging techniques Examples Conclusion Outline of the talk Bilinear quantum systems 1 Bilinear Schrödinger equation Good Galerkin approximation Averaging techniques 2 Rotating wave approximation Examples 3 Quantum harmonic oscillator Rotation of a planar molecule Conclusion 4 Perspectives

Bilinear quantum systems Averaging techniques Examples Conclusion Quantum harmonic oscillator: Good Galerkin approximation i ∂ψ ∂ t ψ ( x , t ) = ( − ∂ xx + x 2 ) ψ ( x , t ) + u ( t ) x ψ ( x , t ) x ∈ R A = − i diag ( 1 / 2 , 3 / 2 , 5 / 2 , . . . ) √   0 1 0 . . . √ √ ...   1 0 2  √ √  B = − i   0 2 0 3    . √  ... ... . . 3 B is not bounded. However, B is bounded relatively to A . The system is 2-weakly coupled and admits a sequence of Good Galerkin approximations. For ψ 0 = φ 1 , ε = 10 − 4 and K = 3, one finds N = 420.