IbortFest 2018 March 2018 ICMAT Quantum Control on the Boundary Juan Manuel Pérez-Pardo
J.M. Pérez-Pardo IbortFest 2018 Outline Controllability of Quantum Systems Unbounded operators and self-adjointness Quantum Control on the boundary
J.M. Pérez-Pardo IbortFest 2018 Controllability for Quantum Systems Time dependent Schrödinger Equation The solution of the equation is given in terms of a unitary propagator i∂ Ψ ∂t = H ( t )Ψ H ( t ) is a family of self-adjoint operators U : R × R → U ( H ) U ( t, t ) = I H U ( t, s ) U ( s, r ) = U ( t, r ) Ψ( t ) = U ( t, t 0 )Ψ 0 is a solution of Schrödinger’s Equation with initial value Ψ 0
J.M. Pérez-Pardo IbortFest 2018 Controllability of fjnite Dimensional Quantum Systems Simple situation. Linear controls: Space of controls Use the controls to steer the state of the system Finite dimensional quantum System H = C n i∂ Ψ ∂t = ( H 0 + c ( t ) H 1 ) Ψ H 0 , H 1 self-adjoint operators (Hermitean matrices). c : R → C from Ψ 0 → Ψ f .
J.M. Pérez-Pardo IbortFest 2018 Controllability of fjnite dimensional Quantum systems must minimize some functional. Minimal time Minimal energy First of all: Decide wether or not the system is controllable. Ultimate Objective (not today): Find a curve c ( t ) ⊂ C that drives the system from Ψ 0 → Ψ f . Optimal control: The solution Ψ( t ) = U ( t, t 0 )Ψ 0 If there exists c ( t ) ⊂ C such that for some T Ψ f = Ψ( T ) = U ( t, t 0 )Ψ 0
J.M. Pérez-Pardo IbortFest 2018 Controllability of fjnite dimensional Quantum systems Study the dynamical Lie algebra: the exponential map of the dynamical Lie algebra. The fjnite dimensional quantum system is control- lable if the dynamical Lie algebra is the Lie algebra Lie { iH 0 , iH 1 } The reachable set of Ψ 0 is the orbit through Ψ 0 of of U ( N ) .
J.M. Pérez-Pardo Harmonic Oscillator algebra: Example: Truncation of the Harmonic Oscillator Harmonic Oscillator IbortFest 2018 d 2 Ψ i∂ Ψ ∂t = − 1 d x 2 + 1 � 1 � 2( p 2 + q 2 ) + c ( t ) q 2 x 2 Ψ + c ( t ) x Ψ = Ψ 2 p Ψ = − i d Ψ q Ψ = x Ψ( x ) d x a † = 1 a = 1 √ √ 2( q − ip ) 2( q + ip ) N = a † a √ a | n � = √ n | n − 1 � a † | n � = N | n � = n | n � n + 1 | n + 1 �
J.M. Pérez-Pardo Harmonic Oscillator algebra: Example: Truncation of the Harmonic Oscillator Harmonic Oscillator Generators of the dynamic: IbortFest 2018 H 0 H 1 d 2 Ψ i∂ Ψ ∂t = − 1 d x 2 + 1 � 1 � 2( p 2 + q 2 ) + c ( t ) q 2 x 2 Ψ + c ( t ) x Ψ = Ψ 2 p Ψ = − i d Ψ q Ψ = x Ψ( x ) d x a † = 1 a = 1 √ √ 2( q − ip ) 2( q + ip ) N = a † a √ a | n � = √ n | n − 1 � a † | n � = N | n � = n | n � n + 1 | n + 1 � H 1 = 1 H 0 = N + 1 2( a † + a ) √ 2
J.M. Pérez-Pardo IbortFest 2018 Example: Truncation of the Harmonic Oscillator states Finite-dimensional approximation by the fjrst n eigen- ( H n ( H n 0 ) ij = � i | H 0 | j � 1 ) ij = � i | H 0 | j �
J.M. Pérez-Pardo IbortFest 2018 Example: Truncation of the Harmonic Oscillator states The fjnite dimensional approximation is control- Finite-dimensional approximation by the fjrst n eigen- ( H n ( H n 0 ) ij = � i | H 0 | j � 1 ) ij = � i | H 0 | j � lable for all n dim Lie { iH 0 , iH 1 } = n 2
J.M. Pérez-Pardo IbortFest 2018 Dynamical Lie Algebra of the Harmonic Oscillator Controllability of the Harmonic Oscillator Generators of the dynamic: H 1 = 1 H 0 = N + 1 2( a † + a ) √ 2 [ a, a † ] = I [ N, a † ] = a † [ N, a ] = − a [ iH 0 , iH 1 ] = − 1 2[ N, a † + a ] = − 1 2( a † − a ) = ip = iH 2 √ √
J.M. Pérez-Pardo IbortFest 2018 Controllability of the Harmonic Oscillator Generators of the dynamic: Dynamical Lie Algebra of the Harmonic Oscillator H 1 = 1 H 0 = N + 1 2( a † + a ) √ 2 [ a, a † ] = I [ N, a † ] = a † [ N, a ] = − a [ iH 0 , iH 1 ] = iH 2 [ iH 0 , iH 2 ] = iH 1 [ iH 1 , iH 2 ] = i I = iH 3
J.M. Pérez-Pardo IbortFest 2018 The infjnite dimensional Harmonic Oscillator is not Four dimensional Lie algebra! Dynamical Lie Algebra of the Harmonic Oscillator Controllability of the Harmonic Oscillator Generators of the dynamic: controllable. H 1 = 1 H 0 = N + 1 2( a † + a ) √ 2 [ a, a † ] = I [ N, a † ] = a † [ N, a ] = − a [ iH 0 , iH 1 ] = iH 2 [ iH 0 , iH 2 ] = iH 1 [ iH 1 , iH 2 ] = i I = iH 3
J.M. Pérez-Pardo IbortFest 2018 Controllability of infjnite dimensional systems Approximate Controllability: A linear control system is ap- Reasonable for infjnite dimensions Space is defjned as equivalence classes of con- vergent sequences Is natural to expect this if one has exact controlla- bility of every fjnite dimensional subsystem proximately controllable if for every Ψ 0 , Ψ 1 ∈ S and every ǫ > 0 there exist T > 0 and c ( t ) ⊂ C such that � Ψ 1 − U ( T, t 0 )Ψ 0 � < ǫ
J.M. Pérez-Pardo IbortFest 2018 Outline Controllability of Quantum Systems Unbounded operators and self-adjointness Quantum Control on the boundary
J.M. Pérez-Pardo IbortFest 2018 Unbounded operators and self-adjointness Unbounded operators are not continuous One needs to defjne a suitable domain There are two major probems The domain of the adjoint operator The sum of operators is not well defjned in general � Φ , T Ψ � = � T † Φ , Ψ � D ( T ) = D ( A ) ∩ D ( B ) T = A + B
J.M. Pérez-Pardo Quadratic Forms Quadratic Form defjned by the operator If the operator is lower semibounded, there exists a unique self-adjoint operator that represents it. Q.F. Laplace operator Splitting of the boundary conditions IbortFest 2018 Q (Φ , Ψ) = � Φ , T Ψ � Q (Φ , Ψ) = � d Φ , d Ψ � − � ϕ, ˙ ϕ � � U − I � ϕ = − i ˙ ϕ = A U ϕ U + I = � d Φ , d Ψ � − � ϕ, A U ϕ � Pϕ = 0 U = − 1 P + P ⊥ U P ⊥ ˙ ϕ = A U P ⊥ ϕ
J.M. Pérez-Pardo IbortFest 2018 Approximate Controllability Consider the Linear Control System: The linear control system is approximately con- trollable with piecewise constant controls if [Cham- brion, Mason, Sigalotti, Boscain 2009]: i∂ Ψ ∂t = ( H 0 + c ( t ) H 1 ) Ψ H 0 , H 1 are self-adjoint. { Φ n } n ∈ N O.N.B of eigenvectors of H 0 Φ n ∈ D ( H 1 ) for every n ∈ N ( λ n +1 − λ n ) n ∈ N are Q -linearly independent. � B Φ n , Φ n +1 � � = 0 for any n ∈ N
J.M. Pérez-Pardo IbortFest 2018 Outline Controllability of Quantum Systems Unbounded operators and self-adjointness Quantum Control on the boundary
J.M. Pérez-Pardo IbortFest 2018 Quantum Control at the boundary Time dependent Schrödinger Equation of the same operator Advantage. There is no need to apply an external fjeld Problem. Even the existence of solutions of the dynamics is compromised. i∂ Ψ ∂t = H ( t )Ψ H ( t ) is a family of difgerent self-adjoint extension � � H, D ( c ( t ))
J.M. Pérez-Pardo Defjne the unitary operator Quantum Control at the Boundary Assumption: The spectrum of only contains eigenvalues with fjnite degeneracy. IbortFest 2018 Fix a reference extension � � H, D ( c ) Then { Φ c n } n ∈ N forms a complete orthonormal base. � � H, D ( c 0 ) V c : H → H n → Φ 0 Φ c n One needs to require additionally that V c : D ( c ) → D ( c 0 )
J.M. Pérez-Pardo The Schrödinger Equation becomes is uniformly semibounded in time and under If the family Theorem (J. Kisynski ’63): IbortFest 2018 gator solving the time dependent Schrödinger equation. Quantum Control at the boundary Write the Schrödinger Equation on the reference domain: Ψ ∈ D ( c ) Ψ 0 := V c Ψ ∈ D ( c 0 ) V c HV † c is a self-adjoint operator with domain D ( c 0 ) i ∂ V † = HV † � � c Ψ 0 c Ψ 0 ∂t i ∂ c Ψ 0 − iV c ˙ ∂t Ψ 0 = V c HV † V † c Ψ 0 � � H, D ( c ( t )) suitable difgerentiability conditions of V c ( t ) there exists a unitary propa-
J.M. Pérez-Pardo IbortFest 2018 Example: Varying Quasiperiodic Boundary Conditions By the theorem [Chambrion, Mason, Sigalotti, Boscain 2009] this system is approximately controllable by piecewise constant controls. Particle Moving in a circular wire Quantum Faraday Law d θ − α ] 2 Ψ + θ ˙ i d d t Ψ = [ i d α Ψ α Magnetic fmux of intensity α
J.M. Pérez-Pardo IbortFest 2018 References A. Ibort, F. Lledó, JMPP. Self-Adjoint Extensions of the Laplace- Beltrami Operator and unitaries at the boundary. J. Funct. Anal. 268 , 634-670, (2015). M. Barbero-Liñán, A. Ibort, JMPP. Boundary dynamics and topology change in quantum mechanics. Int. J. Geom. Methods in Modern Phys. 12 156011, (2015). A. Ibort, JMPP. Quantum control and representation theory. J. Phys. A: Math. Theor. 42 205301, (2009). A.P. Balachandran, G. Bimonte, G. Marmo, A. Simoni. Nucl. Phys. B 446 299-214, (1995).
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