Optimal control of a dissipative 2-level quantum system Nataliya Shcherbakova ENSEEIHT, Toulouse, France October 25, 2010 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 1 / 41
Main collaborators Bernard Bonnard, Institut de Mathématiques de Bourgogne, UMR CNRS 5584; Dominique Sugny, Institut Carnot de Bourgogne, UMR CNRS 5209; Olivier Cots, ENSEEIHT and Institut de Mathématiques de Bourgogne. Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 2 / 41
Main publications B. Bonnard, O.Cots, N. Shcherbakova, D. Sugny, The energy minimization problem for two-level dissipative quantum systems , J. Math.Phys. 51, 2010, DOI: 10.1063/1.3479390 B. Bonnard, N. Shcherbakova, D. Sugny, The smooth continuation method in optimal control with an application to quantum systems , 2009, ESAIM-COCV, DOI: 10.1051/cocv/2010004 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 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 3 / 41
Problem 1: Control of molecular alignment by laser fields in dissipative media i � ∂ρ ∂ t = [ H 0 + H 1 , ρ ] + i L ( ρ ) , (1) ρ - the density matrix (i.e. positive semi-definite Hermitian operator) s.t. tr ( ρ 2 ) ≤ 1 tr ( ρ ) = 1 , H 0 - the field-free Hamiltonian of the system H 1 - the Hamiltonian of the interaction with the control field L - the dissipation operator Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 4 / 41
Problem 1: Control of molecular alignment by laser fields in dissipative media i � ∂ρ ∂ t = [ H 0 + H 1 , ρ ] + i L ( ρ ) , (1) ρ - the density matrix (i.e. positive semi-definite Hermitian operator) s.t. tr ( ρ 2 ) ≤ 1 tr ( ρ ) = 1 , H 0 - the field-free Hamiltonian of the system H 1 - the Hamiltonian of the interaction with the control field L - the dissipation operator Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 4 / 41
Dissipation term: N � � L ( ρ ) kk = − ( γ lk ρ kk + γ kl ρ ll ) , L ( ρ ) lk l � = k = − Γ lk ρ lk , � � l � = k where γ kl , Γ kl are real non-negative constants describing : 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 . � ρ 11 � 1 + z � � ρ 12 = 1 x + iy ρ = 2-levels systems: 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) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41
Dissipation term: N � � L ( ρ ) kk = − ( γ lk ρ kk + γ kl ρ ll ) , L ( ρ ) lk l � = k = − Γ lk ρ lk , � � l � = k where γ kl , Γ kl are real non-negative constants describing : 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 . � ρ 11 � 1 + z � � ρ 12 = 1 x + iy ρ = 2-levels systems: 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 . Control Hamiltonian: H 1 = − σ x E x − σ y E y , where E x , E y are linearly polarized laser fields Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41
Dissipation term: N � � L ( ρ ) kk = − ( γ lk ρ kk + γ kl ρ ll ) , L ( ρ ) lk l � = k = − Γ lk ρ lk , � � l � = k where γ kl , Γ kl are real non-negative constants describing : 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 . � ρ 11 � 1 + z � � ρ 12 = 1 x + iy ρ = 2-levels systems: 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 . Control Hamiltonian: H 1 = − σ x E x − σ y E y , where E x , E y are linearly polarized laser fields Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 5 / 41
Lindblad equations for 2-levels systems ( � = 1 ) ˙ x = − Γ x + u 2 z ˙ y = − Γ y − u 1 z (2) ˙ 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 − | . q = ( x , y , z ) belongs to the invariant Block ball � q � ≤ 1 � q � = 1 - pure state ( 0 , 0 , γ − γ + ) - the equilibrium state of the free motion Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 6 / 41
Lindblad equations for 2-levels systems ( � = 1 ) ˙ x = − Γ x + u 2 z ˙ y = − Γ y − u 1 z (2) ˙ 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 − | . q = ( x , y , z ) belongs to the invariant Block ball � q � ≤ 1 � q � = 1 - pure state ( 0 , 0 , γ − γ + ) - the equilibrium state of the free motion Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 6 / 41
Problem 2: control of a spin- 1 2 particle in dissipative media Bloch equations: M x = − 1 M y = − 1 ˙ ˙ M x + u 2 M z ; M y − u 1 M z ; (3) T 2 T 2 M z = 1 ˙ ( M 0 − M z ) + u 1 M y − u 2 M x , T 1 M = ( M x , M y , M z ) - magnetization vector; T 1 , T 2 - longitudinal and transverse relaxation times, 2 T 1 ≥ T 2 ; M 0 = ( 0 , 0 , M 0 ) - thermal equilibrium point; u = ( u 1 , u 2 , 0 ) - controlled magnetic field. 1 Γ = T − 1 γ + = γ − = T − 1 Normalization : q = ( x , y , z ) = M 0 M , 2 , 1 = ⇒ Lindblad equations for γ + = γ − Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 7 / 41
Problem 2: control of a spin- 1 2 particle in dissipative media Bloch equations: M x = − 1 M y = − 1 ˙ ˙ M x + u 2 M z ; M y − u 1 M z ; (3) T 2 T 2 M z = 1 ˙ ( M 0 − M z ) + u 1 M y − u 2 M x , T 1 M = ( M x , M y , M z ) - magnetization vector; T 1 , T 2 - longitudinal and transverse relaxation times, 2 T 1 ≥ T 2 ; M 0 = ( 0 , 0 , M 0 ) - thermal equilibrium point; u = ( u 1 , u 2 , 0 ) - controlled magnetic field. 1 Γ = T − 1 γ + = γ − = T − 1 Normalization : q = ( x , y , z ) = M 0 M , 2 , 1 = ⇒ Lindblad equations for γ + = γ − Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 7 / 41
Control setting ( P ) q = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ q 0 , q T − fixed and F 0 , F 1 , F 2 ∈ Vec ( R 3 ) : − Γ x 0 z , , . F 0 = − Γ y F 1 = − z F 2 = 0 γ − − γ + z y − x I. Minimal time problem ( P T ) : ( P ) , � u � ≤ 1 , T − → min Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41
Control setting ( P ) q = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ q 0 , q T − fixed and F 0 , F 1 , F 2 ∈ Vec ( R 3 ) : − Γ x 0 z , , . F 0 = − Γ y F 1 = − z F 2 = 0 γ − − γ + z y − x I. Minimal time problem ( P T ) : ( P ) , � u � ≤ 1 , T − → min II. Energy minimizing problem ( P E ) : ( P ) , T - fixed, T � 1 u 2 1 ( t ) + u 2 2 ( t ) dt → min 2 0 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41
Control setting ( P ) q = F 0 ( q ) + u 1 F 1 ( q ) + u 2 F 2 ( q ) , ˙ q 0 , q T − fixed and F 0 , F 1 , F 2 ∈ Vec ( R 3 ) : − Γ x 0 z , , . F 0 = − Γ y F 1 = − z F 2 = 0 γ − − γ + z y − x I. Minimal time problem ( P T ) : ( P ) , � u � ≤ 1 , T − → min II. Energy minimizing problem ( P E ) : ( P ) , T - fixed, T � 1 u 2 1 ( t ) + u 2 2 ( t ) dt → min 2 0 Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 8 / 41
The Hamiltonian of ( P ) h u ( ξ ) = h 0 ( ξ ) + u 1 h 1 ( ξ ) + u 2 h 1 ( ξ ) − p 0 2 ( u 2 1 + u 2 2 ) , p ∈ T ∗ q R 3 , h i ( ξ ) = � p , F i ( q ) � , ξ = ( p , q ) , p 0 ∈ { 0 , 1 } Case ( P E ) : in the normal case ( p 0 = 1 ) optimal controls are u i ( ξ ) = h i ( ξ ) , i = 1 , 2 , ξ ∈ T ∗ R 3 . Normal extremals are solutions of the Hamiltonian system associated to h E ( ξ ) = h 0 ( ξ ) + 1 2 ( h 2 1 ( ξ ) + h 2 2 ( ξ )) . Nataliya Shcherbakova (N7, Toulouse) Optimal control of a dissipative 2-level quantum system October 2010 9 / 41
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