Advanced computational methodologies for solving quantum control problems Alfio Borz` ı Dipartimento e Facolt` a di Ingegneria, Universit` a degli Studi del Sannio, Italy Institute for Mathematics and Scientific Computing Karl-Franzens-University, Graz, Austria Alfio Borz` ı Advanced computational methodologies for solving quantum control
Application fields of quantum control methodologies 1. Quantum control: state transitions, laser induced chemistry, magnetic and optical trapping. 2. Quantum computing: qubits, data operations. 3. Quantum transport, superfluids of atoms, vortices. 4. Construction of barriers, channels, etc. for few atoms. 5. Amplification of material waves: laser of atoms. 6. Semiconductor nanostructures. Alfio Borz` ı Advanced computational methodologies for solving quantum control
New challenges from quantum control problems The possibility to manipulate states of atoms and molecules by means of laser pulses or magnetic fields opens new technological perspectives. The solution of quantum control problems poses new challenges involving optimal control theory, numerical analysis, and scientific computing. Quantum control models define an important class of nonlinear control mechanisms. Alfio Borz` ı Advanced computational methodologies for solving quantum control
Quantum mechanical models ◮ One-particle Schr¨ odinger equation, ψ = ψ ( x , t ) or ψ = ψ ( t ) i ∂ ∂ t ψ = ( H 0 + V 0 + V control ) ψ ◮ BEC Condensate Gross-Pitaevskii equation, ψ = ψ ( x , t ) i ∂ � − 1 � 2 ∇ 2 + V 0 + V control + g | ψ | 2 ∂ t ψ = ψ ◮ Time-dependent Kohn-Sham equation, ψ i = ψ i ( x , t ) i ∂ � − 1 � 2 ∇ 2 + V ext + V Hartree ( ρ ) + V exc ( ρ ) + V control ∂ t ψ i = ψ i i =1 | ψ i | 2 is the where ψ i , i = 1 , . . . , N are the K-S orbitals; ρ = � N one-electron density. ◮ Multi-particle ( n ) Schr¨ odinger equation, ψ = ψ ( x 1 , x 2 , . . . , x n , t ) n n n i ∂ − 1 � � � ∇ 2 ψ ∂ t ψ = i + V i + U ij + V control 2 i =1 i =1 i , j =1 Alfio Borz` ı Advanced computational methodologies for solving quantum control
Quantum mechanics structure and objectives Dynamically stable systems exist with confining potentials V 0 −∇ 2 + V 0 ( x ) − E j � � φ j ( x ) = 0 , j = 1 , 2 , . . . , where φ j ∈ H represent the eigenstates and E j represent the energy. Here, H is a complex Hilbert space. Control may be required to drive state transitions φ i − → φ j . The expectation value of a physical observable A when the system is in a state ψ is given by ( ψ, A ψ ). Control may be required to maximize observable expectation. An Hermitian operator O may represent a transformation regardless of initial and final states (e.g., quantum gates). Control may be required to obtain best performance of O . Alfio Borz` ı Advanced computational methodologies for solving quantum control
Quantum control mechanisms Laser pulses, electric fields, and magnetic fields represent physically meaningful control mechanisms. They are represented by potentials that sum up to the stationary one V ( x , t ) = V 0 ( x ) + V control ( x , t ) The dipole approximation of the electric control field modeling a laser pulse results in the form V control ( x , t ) = u ( t ) x where u : (0 , T ) → R is the modulating control amplitude. A magnetic potential for manipulating a BEC is given by V control ( x , u ( t )) = − u ( t ) 2 d 2 x 2 + 1 c x 4 8 c where u is a parameter control function. Alfio Borz` ı Advanced computational methodologies for solving quantum control
Mathematical issues of quantum control problems ◮ Finite- and infinite-dimensional quantum systems Finite-level systems are characterized by H 0 , V ∈ C n × n , while H 0 is unbounded in ∞ -dim systems and V : Ω × (0 , T ) → R . ◮ Existence and uniqueness of quantum optimal control Existence of optimal solutions can usually be proven. Uniqueness usually does not occur: for dipole control, if u ( t ) is a minimizer, then so is − u ( t ). ◮ Exact and approximate controllability A finite-level system is controllable iff Lie { i H 0 , i V } = su ( n ), the Lie algebra of zero-trace skew-Hermitian n × n matrices; see, e.g., Dirr & Helmke. For infinite-dimensional systems, see Beauchard & Coron, Chambrion, Mason, Sigalotti & Boscain, and Turinici. ◮ Accurate and fast solution schemes for optimal control Gradient schemes, monotonic schemes, Newton schemes, and multigrid schemes. Alfio Borz` ı Advanced computational methodologies for solving quantum control
Optimal control of finite-level quantum systems Alfio Borz` ı Advanced computational methodologies for solving quantum control
��������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Optimal control of a finite-level quantum system Quantum systems with a finite number of states model artificial atoms (semiconductor quantum dots) and quantum devices (quantum gates). Consider a Λ-type three-level system with two stable states ψ 1 and ψ 2 (conservative), and one unstable state ψ 3 (dissipative). ψ 3 ������������� � ������������� � � � � � � � � � � � � u � � � � � � � � � � � � � � � � � � δ ψ 2 ψ 1 Alfio Borz` ı Advanced computational methodologies for solving quantum control
Finite-level quantum models Governed by Schr¨ odinger-type equations for a n -component wave function ψ : [0 , T ] → C n as follows i ˙ ψ ( t ) = H ( u ( t )) ψ ( t ) , ψ (0) = ψ 0 , for t ∈ [0 , T ] and T > 0 is a given terminal time. The function u : [0 , T ] → C m represents the external control field. Alternatively u = ( u 1 , . . . , u 2 m ) and u i : [0 , T ] → R The linear Hamiltonian H ( u ) = H 0 + V ( u ), consists of A free Hamiltonian H 0 ∈ C n × n describing the unperturbed (uncontrolled) system; A control Hamiltonian V ( u ) ∈ C n × n modeling the coupling of the quantum state to the control field u . Alfio Borz` ı Advanced computational methodologies for solving quantum control
The objective of the quantum control Control is applied to reach a target state at t = T . One needs to avoid population of dissipative states during the control process, while having limited laser resources. These modeling requirements may result in the following = 1 C n + γ L 2 (0 , T ; C ) + µ 2 | ψ ( T ) − ψ d | 2 2 � u � 2 u � 2 J ( ψ, u ) 2 � ˙ L 2 (0 , T ; C ) +1 � α j � ψ j � 2 L 2 (0 , T ; C ) 2 j ∈ J where ψ d is the desired terminal state; γ > 0 and µ, α i ≥ 0 are weighting factors; ψ j denotes the j -th (dissipative) component of ψ . Alfio Borz` ı Advanced computational methodologies for solving quantum control
First-order necessary optimality conditions For the quantum optimal control problem min J ( ψ, u ) , subject to i ˙ ψ ( t ) = H ( u ( t )) ψ ( t ) , ψ (0) = ψ 0 Theorem Suppose that x = ( ψ, u ) ∈ X is a local solution to the optimal control problem. Then there exist (unique) Lagrange multipliers p ∈ H 1 (0 , T ; C n ) ( µ > 0 ) satisfying i ˙ ψ = H ( u ( · )) ψ p = H ( u ( · )) ∗ p − α j ( ψ ) j i ˙ u + γ u = ℜ e ( p · ( V ′ r ( u ) ψ ) ∗ ) + i ℜ e ( p · ( V ′ i ( u ) ψ ) ∗ ) − µ ¨ where ψ (0) = ψ 0 , ip ( T ) = ψ ( T ) − ψ d , u ( T ) = u (0) = 0 . Alfio Borz` ı Advanced computational methodologies for solving quantum control
Second-order optimality conditions Consider the following optimal control problem � min u J ( ψ, u ) 2 | ψ ( T ) − ψ d | 2 + γ 1 2 � u � 2 := i ˙ c ( ψ, u ) := ψ − a ψ − u ψ = 0 The solution of the SE for a given u provides ψ = ψ ( u ). We obtain the reduced objective ˆ J ( u ) = J ( ψ ( u ) , u ). The reduced Hessian J δ u , δ u ) = ( W δ u )( W δ u ) ∗ + 2 ℜ e ( p δ u , W δ u ) + γ ( δ u , δ u ) . ( ∇ 2 ˆ where W = W ( ψ ( u ) , u ) = c ψ ( ψ ( u ) , u ) − 1 c u ( ψ ( u ) , u ). Because of unitary of evolution, we have | p ( t ) | = | p ( T ) | = | ψ ( T ) − ψ d | . Therefore, we have that |ℜ e ( p δ u , W δ u ) | ≤ C ( | u | ) | ψ ( T ) − ψ d | � δ u � 2 . For sufficiently small values of the tracking error | ψ ( T ) − ψ d | positiveness of the reduced Hessian is obtained. Alfio Borz` ı Advanced computational methodologies for solving quantum control
Control of a Λ-type three-level model Free Hamiltonian − δ 0 0 H 0 = 1 0 δ 0 2 0 0 − i Γ o where the term − i Γ o accounts for environment losses (spontaneous photon emissions, scattering of gamma rays from crystals). The coupling to the external field is given by 0 0 µ 1 u V ( u ) = − 1 0 0 µ 2 u 2 µ 1 u ∗ µ 2 u ∗ 0 where µ 1 and µ 2 describe the coupling strengths of states ψ 1 and ψ 2 to the inter-connecting state ψ 3 (e.g., optical dipole matrix elements). Initial and final states are given by 1 0 and ψ d = e − i δ T ψ 0 = 0 0 0 Alfio Borz` ı Advanced computational methodologies for solving quantum control
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