Multi-polarization quantum control of rotational motion Multi-polarization quantum control of rotational motion Gabriel Turinici Universitris Dauphine 1 IHP, Paris Dec. 8-11, 2010 1 financial support from INRIA Rocquencourt, GIP-ANR C-QUID program and NSF-PICS program is acknowledged
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: R. J. Levis, G.M. Menkir, and H. Rabitz. Science , 292:709–713, 2001
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: SELECTIVE dissociation of chemical bonds (laser induced). Other examples: CF 3 or CH 3 from CH 3 COCF 3 ... (R. J. Levis, G.M. Menkir, and H. Rabitz. Science , 292:709–713, 2001).
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: Selective dissociation AND CREATION of chemical bonds (laser induced). Other examples: CF 3 or CH 3 from CH 3 COCF 3 ... (R. J. Levis, G.M. Menkir, and H. Rabitz. Science , 292:709–713, 2001).
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: Experimental High Harmonic Generation (argon gas) obtain high frequency lasers from lower frequencies input pulses ω → n ω (electron ionization that come back to the nuclear core) (R. Bartels et al. Nature, 406, 164, 2000).
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: Studying the excited states of proteins. F. Courvoisier et al., App.Phys.Lett.
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: thunder control : experimental setting ; J. Kasparian Science, 301, 61 – 64 team of J.P.Wolf @ Lyon / Geneve , ...
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: thunder control : (B) random discharges ; (C) guided by a laser filament ; J. Kasparian Science, 301, 61 – 64 team of J.P.Wolf @ Lyon / Geneve , ...
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: LIDAR = atmosphere detection; the pulse is tailored for an optimal reconstruction at the target : 20km = OK ! ; J. Kasparian Science, 301, 61 – 64
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Figure: Creation of a white light of high intensity and spectral width ; J. Kasparian Science, 301, 61 – 64
Multi-polarization quantum control of rotational motion Optical manipulation of quantum dynamics Other applications • EMERGENT technology • creation of particular molecular states • long term: logical gates for quantum computers • fast “switch” in semiconductors • ...
Multi-polarization quantum control of rotational motion Controllability Outline 1 Controllability Background on controllability criteria 2 Control of rotational motion Physical picture 3 Controllability assessment with three independently polarized field components 4 Controllability for a locked combination of lasers 5 Controllability with two lasers − → x + i − → Field shaped in the − → y z and directions √ 2 − → x + i − → − → x − i − → y y Field shaped in the and directions √ √ 2 2
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Single quantum system, bilinear control Time dependent Schr¨ odinger equation i ∂ � ∂ t Ψ( x , t ) = H 0 Ψ( x , t ) (1) Ψ( x , t = 0) = Ψ 0 ( x ) . Add external BILINEAR interaction (e.g. laser) i ∂ � ∂ t Ψ( x , t ) = ( H 0 − ǫ ( t ) µ ( x ))Ψ( x , t ) (2) Ψ( x , t = 0) = Ψ 0 ( x ) Ex.: H 0 = − ∆ + V ( x ), unbounded domain Evolution on the unit sphere: � Ψ( t ) � L 2 = 1 , ∀ t ≥ 0.
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Controllability A system is controllable if for two arbitrary points Ψ 1 and Ψ 2 on the unit sphere (or other ensemble of admissible states) it can be steered from Ψ 1 to Ψ 2 with an admissible control. Norm conservation : controllability is equivalent, up to a phase, to say that the projection to a target is = 1.
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Galerkin discretization of the Time Dependent Schr¨ odinger equation i ∂ ∂ t Ψ( x , t ) = ( H 0 − ǫ ( t ) µ )Ψ( x , t ) • basis functions { ψ i ; i = 1 , ..., N } , e.g. the eigenfunctions of the H 0 : ψ k = e k ψ k • wavefunction written as Ψ = � N k =1 c k ψ k • We will still denote by H 0 and µ the matrices ( N × N ) associated to the operators H 0 and µ : H 0 kl = � ψ k | H 0 | ψ l � , µ kl = � ψ k | µ | ψ l � ,
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Lie algebra approaches To assess controllability of i ∂ ∂ t Ψ( x , t ) = ( H 0 − ǫ ( t ) µ )Ψ( x , t ) construct the “dynamic” Lie algebra L = Lie ( − iH 0 , − i µ ): � ∀ M 1 , M 2 ∈ L , ∀ α, β ∈ I R : α M 1 + β M 2 ∈ L ∀ M 1 , M 2 ∈ L , [ M 1 , M 2 ] = M 1 M 2 − M 2 M 1 ∈ L Theorem If the group e L is compact any e M ψ 0 , M ∈ L can be attained. “Proof” M = − iAt : trivial by free evolution Trotter formula: e − iB / √ n e − iA / √ n e iB / √ n e iA / √ n � n e i ( AB − BA ) = lim � n →∞
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Operator synthesis ( “lateral parking”) e − iB / √ n e − iA / √ n e iB / √ n e iA / √ n � n Trotter formula: e i [ A , B ] = lim � n →∞ e ± iA = advance/reverse ; e ± iB = turn left/right
Multi-polarization quantum control of rotational motion Controllability Background on controllability criteria Corollary. If L = u ( N ) or L = su ( N ) (the (null-traced) skew-hermitian matrices) then the system is controllable. “Proof” For any Ψ 0 , Ψ T there exists a “rotation” U in U ( N ) = e u ( N ) (or in SU ( N ) = e su ( N ) ) such that Ψ T = U Ψ 0 . • (Albertini & D’Alessandro 2001) Controllability also true for L isomorphic to sp ( N / 2) (unicity). sp ( N / 2) = { M : M ∗ + M = 0 , M t J + JM = 0 } where J is a matrix � � 0 I N / 2 unitary equivalent to and I N / 2 is the identity matrix of − I N / 2 0 dimension N / 2
Multi-polarization quantum control of rotational motion Control of rotational motion Outline 1 Controllability Background on controllability criteria 2 Control of rotational motion Physical picture 3 Controllability assessment with three independently polarized field components 4 Controllability for a locked combination of lasers 5 Controllability with two lasers − → x + i − → Field shaped in the − → y z and directions √ 2 − → x + i − → − → x − i − → y y Field shaped in the and directions √ √ 2 2
Multi-polarization quantum control of rotational motion Control of rotational motion Physical picture Physical picture • linear rigid molecule, Hamiltonian H = B ˆ J 2 , B = rotational constant, ˆ J = angular momentum operator. • control= electric field − ǫ ( t ) by the dipole operator − − → → d . Field − − → ǫ ( t ) is multi-polarized i.e. x , y , z components tuned independently Time dependent Schr¨ odinger equation ( θ, φ = polar coordinates): i � ∂ J 2 − − ǫ ( t ) · − − → → ∂ t | ψ ( θ, φ, t ) � = ( B ˆ d ) | ψ ( θ, φ, t ) � (3) | ψ (0) � = | ψ 0 � , (4)
Multi-polarization quantum control of rotational motion Control of rotational motion Physical picture Discretization J 2 with spherical harmonics ( J ≥ 0 Eigenbasis decomposition of B ˆ and − J ≤ m ≤ J ): B ˆ J 2 | Y m J � = E J | Y m J � , E J = BJ ( J + 1). highly degenerate ! Note E J +1 − E J = 2 B ( J + 1), we truncate : J ≤ J max . Refs: G.T. H. Rabitz : J Phys A (to appear), preprint http://hal.archives-ouvertes.fr/hal-00450794/en/
Multi-polarization quantum control of rotational motion Control of rotational motion Physical picture Dipole interaction Dipole in space fixed cartesian coordinates ǫ ( t ) · − − − → → d = ǫ x ( t ) x + ǫ y ( t ) y + ǫ z ( t ) z − → x , − → y and − → z , components ǫ x ( t ) , ǫ y ( t ) , ǫ z ( t ) = independent. Using as basis the J = 1 spherical harmonics � � = ∓ 1 3 x ± iy 1 = 1 3 z Y ± 1 , Y 0 r , (5) 1 2 2 π r 2 π We obtain − ǫ ( t ) · − − → → 1 Y − 1 d = ǫ 0 ( t ) d 10 Y 0 1 + ǫ +1 ( t ) d 11 Y 1 1 + ǫ − 1 ( t ) d 1 − . 1 After rescaling − − → ǫ ( t ) · − → d = ǫ 0 ( t ) Y 0 1 + ǫ +1 ( t ) Y 1 1 + ǫ − 1 ( t ) Y − 1 . (6) 1
Multi-polarization quantum control of rotational motion Control of rotational motion Physical picture Discretization D k = matrix of Y k 1 ( k = − 1 , 0 , 1). Entries: � 1 | Y m ′ 1 ( θ, φ ) Y m ′ J ) ∗ ( θ, φ ) Y k ( D k ) ( Jm ) , ( J ′ m ′ ) = � Y m J | Y k ( Y m J ′ � = J ′ ( θ, φ ) sin( θ ) d θ d � J � � J � 3(2 J + 1)(2 J ′ + 1) J ′ J ′ � 1 1 = . (7) m ′ 0 0 0 m k 4 π � J � J J ′ J ′ � � 1 1 and = Wigner 3J-symbols m ′ 0 0 0 m k Entries are zero except when m + k + m ′ = 0 and | J − J ′ | = 1.
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