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 = [ H 0 + H 1 , ρ ] + i L ( ρ ) , H 0 - the field-free Hamiltonian of the system H 1 - the Hamiltonian of the interaction with the control field L - the dissipation operator ρ - the density matrix (i.e. positive semi-definite Hermitian operator) s.t. tr ( ρ 2 ) ≤ 1 tr ( ρ ) = 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 = [ H 0 + H 1 , ρ ] + i L ( ρ ) , H 0 - the field-free Hamiltonian of the system H 1 - the Hamiltonian of the interaction with the control field L - the dissipation operator ρ - the density matrix (i.e. positive semi-definite Hermitian operator) s.t. tr ( ρ 2 ) ≤ 1 tr ( ρ ) = 1 , Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 4 / 27

N � ρ kk = − i [ H 0 + H 1 , ρ ] kk − ˙ ( γ lk ρ kk + γ kl ρ ll ) l � = k ρ lk = − i [ H 0 + H 1 , ρ ] lk − Γ lk ρ lk , ˙ l � = k , � = 1 ; γ kl , Γ kl are real non-negative constants; γ kl - the population relaxation from state k to state l ; Γ kl = Γ lk - 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

N � ρ kk = − i [ H 0 + H 1 , ρ ] kk − ˙ ( γ lk ρ kk + γ kl ρ ll ) l � = k ρ lk = − i [ H 0 + H 1 , ρ ] lk − Γ lk ρ lk , ˙ l � = k , � = 1 ; γ kl , Γ kl are real non-negative constants; γ kl - the population relaxation from state k to state l ; Γ kl = Γ lk - 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 11 � 1 + z � � ρ 12 = 1 x + iy ρ = ρ 21 ρ 22 x − iy 1 − z 2 where x = 2 ℜ [ ρ 12 ] , y = 2 ℑ [ ρ 12 ] , z = ρ 22 − ρ 11 . 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 11 ρ 12 = 1 1 + z x + iy ρ = ρ 21 ρ 22 x − iy 1 − z 2 where x = 2 ℜ [ ρ 12 ] , y = 2 ℑ [ ρ 12 ] , z = ρ 22 − ρ 11 . The state q = ( x , y , z ) belongs to the Bloch ball � q � ≤ 1 and satisfies the Lindblad equations x = − Γ x + u 2 z , ˙ y = − Γ y − u 1 z , ˙ z = γ − − γ + z + u 1 y − u 2 x ˙ with γ − = γ 12 − γ 21 , γ + = γ 12 + γ 21 , and 2 Γ ≥ 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 11 ρ 12 = 1 1 + z x + iy ρ = ρ 21 ρ 22 x − iy 1 − z 2 where x = 2 ℜ [ ρ 12 ] , y = 2 ℑ [ ρ 12 ] , z = ρ 22 − ρ 11 . The state q = ( x , y , z ) belongs to the Bloch ball � q � ≤ 1 and satisfies the Lindblad equations x = − Γ x + u 2 z , ˙ y = − Γ y − u 1 z , ˙ z = γ − − γ + z + u 1 y − u 2 x ˙ with γ − = γ 12 − γ 21 , γ + = γ 12 + γ 21 , and 2 Γ ≥ 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 11 ρ 12 = 1 1 + z x + iy ρ = ρ 21 ρ 22 x − iy 1 − z 2 where x = 2 ℜ [ ρ 12 ] , y = 2 ℑ [ ρ 12 ] , z = ρ 22 − ρ 11 . The state q = ( x , y , z ) belongs to the Bloch ball � q � ≤ 1 and satisfies the Lindblad equations x = − Γ x + u 2 z , ˙ y = − Γ y − u 1 z , ˙ z = γ − − γ + z + u 1 y − u 2 x ˙ with γ − = γ 12 − γ 21 , γ + = γ 12 + γ 21 , and 2 Γ ≥ 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 = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ where F 0 , F 1 , F 2 ∈ Vec ( R 3 ) :       − Γ x 0 z  ,  ,  . − Γ y − z F 0 = F 1 = F 2 = 0    γ − − γ + z − x y 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 = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ where F 0 , F 1 , F 2 ∈ Vec ( R 3 ) :       − Γ x 0 z  ,  ,  . − Γ y − z F 0 = F 1 = F 2 = 0    γ − − γ + z − x y I. Minimal time problem: ( P ) , � u � ≤ 1 , T − → min II. Energy minimizing problem: ( P ) , T - fixed, T � 1 � u 2 1 ( t ) + u 2 � 2 ( t ) dt → min 2 0 Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 7 / 27

Control setting Lindblad equations: ( P ) q = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ where F 0 , F 1 , F 2 ∈ Vec ( R 3 ) :       − Γ x 0 z  ,  ,  . − Γ y − z F 0 = F 1 = F 2 = 0    γ − − γ + z − x y I. Minimal time problem: ( P ) , � u � ≤ 1 , T − → min II. Energy minimizing problem: ( P ) , T - fixed, T � 1 � u 2 1 ( t ) + u 2 � 2 ( t ) dt → min 2 0 Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 7 / 27

Minimal time problem : preliminary analysis The Hamiltonian of minimal time problem h u ( ξ ) = h 0 ( ξ ) + u 1 h 1 ( ξ ) + u 2 h 1 ( ξ ) , p ∈ T ∗ q R 3 , h i ( ξ ) = � p , F i ( q ) � , ξ = ( p , q ) , h i ( ξ ) √ PMP : the optimal controls are u i ( ξ ) = 2 ( ξ ) , i = 1 , 2 , h 2 1 ( ξ )+ h 2 ξ ∈ T ∗ R 3 \ Σ , where Σ = { ξ : h 1 ( ξ ) = h 2 ( ξ ) = 0 } defines the switching surface . Regular extremals are solutions of the Hamiltonian system associated to � h 2 1 ( ξ ) + h 2 h ( ξ ) = h 0 ( ξ ) + 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 h u ( ξ ) = h 0 ( ξ ) + u 1 h 1 ( ξ ) + u 2 h 1 ( ξ ) , p ∈ T ∗ q R 3 , h i ( ξ ) = � p , F i ( q ) � , ξ = ( p , q ) , h i ( ξ ) √ PMP : the optimal controls are u i ( ξ ) = 2 ( ξ ) , i = 1 , 2 , h 2 1 ( ξ )+ h 2 ξ ∈ T ∗ R 3 \ Σ , where Σ = { ξ : h 1 ( ξ ) = h 2 ( ξ ) = 0 } defines the switching surface . Regular extremals are solutions of the Hamiltonian system associated to � h 2 1 ( ξ ) + h 2 h ( ξ ) = h 0 ( ξ ) + 2 ( ξ ) . (1) Proposition. The Hamiltonian system associated to h admits a first integral of the form h 3 ( ξ ) = � p , [ F 1 , F 2 ]( q ) � , ξ = ( p , q ) , p ∈ T ∗ q R 3 . Nataliya Shcherbakova (N7, Toulouse) 2 levels dissipative quantum systems April 2010 8 / 27