Computational Physics: Quantum Dynamics & Control

Transcription

1 Computational Physics: Quantum Dynamics & Control

2 ultrafast processes frequency domain pulse shaping using AOM s or LCD s for both amplitude and phase... the versatility of laser light available in VIS, near-ir Brixner & G. Gerber, ChemPhysChem, 4, ) T.

3 ultrafast processes: quantum control coherent control experiments Teaching lasers to control molecules H. Rabitz, R. de Vivie-Riedle, M. Motzkus, K. Kompa, Science, 288, )

4 time-dependent methods!

5 Outline 1 Introduction 2 Methods : Representation of wavefunctions and operators 3 Methods : Time propagation 4 Concepts of coherent control 5 Optimal control theory

6 Born-Oppenheimer approximation Timescales of nuclear and electronic motion Separation of total wavefunction Ψ tot r, R) = χ e r R) ψ N R) Electronic Schrödinger eq. Nuclear Schrödinger eq. Ĥ e χ e r R) = V e R)χ e r R) electronic structure theory Ĥ N R)ψ N R) = Eψ N R) internal states nuclear dynamics

7 The nuclear Schrödinger equation for each electronic) state in Born-Oppenheimer approximation Ĥ N R)ψ e,n R) = ) 2 2m R + V er) ψ 2 e,n R) = E e,n ψ e,n R) example: photoassociation of Cs 2 V g R) : a 3 Σ u 6s + 6s) ψ v, E v V e R) : 0 g 6s + 6p 3/2 ) ψ v, E v Energy cm 1 ) a 3 Σ u + iii) 1 u 0 g 1 g 0 u + X 1 Σ g + ii) Interatomic distance R a 0 ) Cs6s)+Cs6p 3/2 ) i) λ PA Cs6s)+Cs6s)

8 Nonadiabatic effects coupling between electronic) states 1. natural) breakdown of BO approximation neglected in ˆT N : χ e r R) χ e r R) χ e r R) 2 χe r R) R R 2 examples: spin-orbit coupling, rotational coupling, induced by external field magnetic field coupling different rotational states Feshbach resonances electric field coupling different electronic states photoassociation

9 The nuclear Schrödinger equation 2) if Hamiltonian time-dependent Ĥ N R, t)ψ e R; t) = ) 2 2m R + V er) ψ 2 e R; t) + k W ek R, t)ψ k R; t) example photoassociation of Cs 2 : W ge R, t) µ ge R) Et) Note: simple time-dependence expiωt)) frame rotating with ω & neglecting terms exp±2iω) RWA) time-independent Hamiltonian example: photoassociation with CW lasers ) ˆT + V Ĥ = g R) W R) W R) ˆT + V e R) ω

10 Simulation of dynamical events Methods Solution Classical molecular dynamics trajectory method) Quantum coupled channel method Quantum molecular dynamics Newton s equations sympathetic cooling in a trap: Schiller & Lämmerzahl, PRA 68, ) Stationary scattering equations Feshbach resonances: Tiesinga et al., PRA 71, ) Time-dependent Schrödinger eq. this lecture series

11 The Schrödinger equation on a grid i ψt) = Ĥt)ψt) t A discrete grid Phase space

12 Two phase spaces phase space volume: V = N h D Number of grid points position - momentum phase space : V = p L time - energy phase space : V = E T Kosloff, Quantum Molecular Dynamics on Grids review) ronnie/papers.html

13 Two phase spaces Connection t T p, q) t, E) p max 2m E max t { T V q max ) E max treat p, q) & t, E) on same level of rigour { q L q Kosloff, Quantum Molecular Dynamics on Grids review) ronnie/papers.html

14 Grid based dynamical simulation time Setting boundaries t,e) q,p) coordinates Defining the Grid Direct product grid. energy momentum qn g tn t Correlated grid time scale of event energy scale of event Initial wavefunction generation Ψ0) Operator mapping ϕ = H Ψ Propagation iht Ψt) = e Ψ0) Run time Analysis representation of wavefunctions and operators time propagation Asymptotic Analysis δ examples

15 Outline 1 Introduction 2 Methods : Representation of wavefunctions and operators 3 Methods : Time propagation 4 Concepts of coherent control 5 Optimal control theory

16 Collocation wavefunction defined at grid points : { ψq1 ), ψq 2 ),..., ψq N ) } grid points : q j = j 1) q approximate arbitrary function : N 1 Φq) ψq) = a n g n q) n=0 at grid points : N 1 Φq j )=ψq j ) = a n g n q j ) n=0

17 Collocation N linear coupled equations : ψ j = G jn a n n with ψ j = ψq j ), G jn = g n q j ) if g n linearly independent: Orthogonal collocation N 1 gnq i )g n q j ) = δ ij n=0 direct inversion for expansion coefficients : N 1 a n = ψq j )gnq j ) j=0 a n = j G 1 ) nj ψ j G jn unitary example : Fourier grid Kosloff, Quantum Molecular Dynamics on Grids review), ronnie/papers.html

18 Fourier grid method Basis g k q j ) = e i 2π L kq j with k = N 2 1),..., 0,..., N 2 sampling points : q j = j 1) q periodic boundary conditions

19 Fourier grid method Basis g k q j ) = e i 2π L kq j with k = N 2 1),..., 0,..., N 2 sampling points : q j = j 1) q periodic boundary conditions ψq) FFT ψk) = ak ifft ψq) Method arbitrary wavefunction Φq) N/2 k= N/2 1) amplitude in momentum space a k = 1 N a k e i 2π L kq N ψq j )e i 2π L kq j j=1

20 Wavefunctions 1.5 example: harmonic oscillator, ground state 1 N = 32 N = coordinate q

21 Interpolation by FFT remember... Φq) N/2 k= N/2 1) a k e i 2π L kq a k = 1 N N ψq j )e i 2π L kq j j=1 Φq) 1 N N/2 k= N/2 1) j=1 N ψq j )e i 2π L kq j... interpolation scheme, ON 2 )

22 Interpolation by FFT remember... Φq) N/2 k= N/2 1) a k e i 2π L kq a k = 1 N N ψq j )e i 2π L kq j j=1 q Φq) 1 N FFT N/2 k= N/2 1) j=1 N ψq j )e i 2π L kq j... interpolation scheme, ON 2 ) k max k max ON logn)) q ifft k k max max but: no gain in accuracy!

23 Operators on a grid operators map wavefunction ψ into wavefunction φ : on a discrete grid : φ = Âψ φq i ) = N A ij ψq j ) scaling ON 2 ) j=1 what kind of operators? local operators : potential energy nonlocal operators : kinetic energy, translation

24 Local operators a local operator  depends only on the coordinate  = f q) example: potential energy ˆVq) mapping is done at each grid point: φ = ˆVq)ψ φq j ) = N V q j )ψq j ) j=1 scaling ON)

25 Non-local operators : derivatives Analyticity of expansion functions φ N q = j=1 in particular at each grid point : ψ q i = N j=1 q g jq i ) N n=1 a j q g jq) G 1 jn ψ n = N D jn ψ n j=1 with derivative matrix D = F G 1 F ij = g j q i ) q, remember G jn = g n q j )) numerically exact! unlike SOD)

26 Non-local operators : derivatives 2) On the Fourier grid G jk = 1 N e i 2π L q j k basis functions g k q) = e i 2π L qk are eigenfunctions of ˆD : D k,k = i2πk L δ k,k derivative operator local in momentum space ˆT local in momentum space scaling ON logn))

27 Kinetic energy operator ψq j ) a k k 2 2m a k ˆTψq j ) D B V A T C q P D A

28 Hamiltonian 1. potential energy evaluate action of Ĥ on wavefunction ψ φ = Ĥψ = ˆTψ + ˆVψ ˆV local in coordinate space ˆVψ evaluated in coordinate space 2. kinetic energy ˆT local in momentum space ˆTψ evaluated in momentum space overall scaling: ON logn)) efficient numerically exact

29 Sampling : Phase space of Fourier grid example: harmonic oscillator p p p p +p +p max +p max +p max p p max q q q q q too small L too large optimal +L/2 L/2 p p max +L/2 L/2 p max area of circle area of square = π ) 2 optimum: T max = V max T max = p2 max = 1 π 2m 2m q q opt = π p max What is q opt for harmonic oscillator? +L/2 L/2

30 Energy scale of problem sets grid but... what if different energy scales need to be addressed?

31 Mapped Fourier grid use adaptive grid step! π R i+1 = R i + β 2µ V max V R i ) coordinate transformation R = f x), dr = Jx)dx, J = dr dx transformed Ĥ with 4 FFTs, but N mapped N N mapped O1000) Kokoouline et al., J. Chem. Phys. 110, ) β 0 β 1) controls accuracy!

32 Mapped Fourier grid : example internuclear distance R [ units of a 0 ] last bound level of 87 Rb 2 X 1 Σ g 5s + 5s)

33 Outline 1 Introduction 2 Methods : Representation of wavefunctions and operators 3 Methods : Time propagation 4 Concepts of coherent control 5 Optimal control theory

34 remember: Two phase spaces Connection t T p, q) t, E) p max 2m E max t { T V q max ) E max treat p, q) & t, E) on same level of rigour { q L q Kosloff, Quantum Molecular Dynamics on Grids review) ronnie/papers.html

35 Time evolution Time-dependent Schrödinger equation ψr; t) i t = Ĥt)ψR; t) formal solution: ψr; t 2 ) = Ût 2, t 1 )ψr; t 1 ) 1. Ĥ time-independent Ût 2, t 1 ) = exp i ) Ĥt 2 t 1 ) 2. Ĥ time-dependent Ût 2, t 1 ) = ˆT 0 exp i t2 t 1 ) dτĥτ)

36 Stability conditions A propagation scheme is stable if 1 it obeys time-reversal symmetry: ) ψt + t) ψt t) = Û t) Û t) ψt) = ψt) 2 it conserves the norm: ψt) ψt) = 1 t 3 it conserves the total energy: ψt) Ĥ ψt) = ψt 0 ) Ĥ ψt 0 ) t

37 Split propagator & Co. Split propagator to second order Û = e i ˆV t/2 e i ˆT t e i ˆV t/2 Û = e i ˆT t/2 e i ˆV t e i ˆT t/2 higher order schemes possible very cheap Norm & energy conserved to O t 3 ) Second order differences ψt + t) = ψt t) 2iĤ tψt) stable if t 1/E max Attention: Errors accumulate!

38 Spectral methods remember: ψt + t) = exp iĥ t)ψt) think of it as: φ = f Ĥ)ψt) with f Ĥ) = exp iĥ t) spectral decomposition N f Â) a n ˆP n with ˆP n = u n u n n=0 best choice: polynomial basis Newton polynomials, Chebychev polynomials) recursion relations

39 Chebychev propagator N Ût) = e iĥt a n P n iĥt) with P n ˆX) = Φ n ˆX) n=0 renormalization: Ĥ norm = 2 Ĥ 1 2 E+V min) E Ψt) = Ût)Ψ0) e i 1 2 E+V min)t N a n E t)φ n iĥ norm )Ψ0) n=0 a n α) = i i dx eiαx Φ n x) 1 x 2 = 2J n α) J n : Bessel functions

40 Chebychev propagation scheme 1 calculate expansion coefficients a n 2 renormalize Ĥ Ĥ norm 3 recursion relation for Chebychev polynomials φ 0 = ψ φ 1 = Ĥ norm ψ... φ n+1 = 2Ĥ norm φ n φ n 1 4 ψt + t) N n=0 a nφ n R. Kosloff. Time-Dependent Quantum-Mechanical Methods for Molecular Dynamics. J. Phys. Chem. 92, )

41 Chebychev propagator little storage : φ n 1, φ n expensive : N FFT + N ifft numerically exact! N E t/2 arbitrarily large time steps t time-dependence only in expansion coefficients error uniformly distributed in energy

42 Outline 1 Introduction 2 Methods : Representation of wavefunctions and operators 3 Methods : Time propagation 4 Concepts of coherent control 5 Optimal control theory

43 Coherent / Quantum Control wave properties of atoms/molecules superposition principle) variation of phase between different, but indistinguishable quantum pathways constructive interference in desired channel destructive interference in all other channels

44 Coherent vs Optimal Control Coherent Control Goal: improve outcome of process vary some parameters Optimal Control Goal: obtain maximal control over process tune all available parameters simple, intuitive schemes in time or in frequency domain complex outcome discovery of new schemes usually not accessible by intuition) in time / frequency phase space global or local in time

45 Control in frequency domain Brumer-Shapiro scheme analogous to double slit experiment but IF between quantum pathways n 2 lasers n 2 coherent pathways from initial to degenerate final states Example: n = 2 Ψt = 0) = c c 2 2 superposition of 2 eigenstates E f coefficients of final state superposition determined by relative phase and relative amplitude of CW lasers and initial state E 2 E 1 ω 1 ω 2

46 Brumer-Shapiro scheme εω) With pulsed lasers 1 photon vs. 3 photon absorption E f E f ω 1 ω 1 ω 2 ω 3 ω 1 ω 3 E 3 E 2 E 1 E i ω 1

47 Control in time domain based on localized wave packets coordinate vs. eigenstate representation Tannor-Rice scheme variation of phase between quantum paths variation of time delay Pump-dump control T. Brixner & G. Gerber, Physikal. Blätter April 2001

48 Control via STIRAP Counterintuitive pulse sequence: S precedes, but overlaps P coherent population trapping K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, )

49 Outline 1 Introduction 2 Methods : Representation of wavefunctions and operators 3 Methods : Time propagation 4 Concepts of coherent control 5 Optimal control theory

50 optimal control theory: basic idea time/frequency phase space picture t = 0 ϕ i t = T ϕ f define the objective : GOAL ϕ i Û + T, 0; ε) ϕ f 2 = J T as a functional of the field ε

51 optimal control theory: basic idea time/frequency phase space picture t = 0 ϕ i t = T ϕ f define the objective : GOAL ϕ i Û + T, 0; ε) ϕ f 2 = J T as a functional of the field ε include additional constraints: J = J T + T 0 J t ε, ϕ) dt

52 optimal control theory: basic idea time/frequency phase space picture t = 0 ϕ i t = T ϕ f define the objective : GOAL ϕ i Û + T, 0; ε) ϕ f 2 = J T as a functional of the field ε include additional constraints: J = J T + T 0 J t ε, ϕ) dt optimize J: ɛ J = 0 ϕt) J = 0 2 ɛ J > 0 ϕt) = Ût, 0; ɛ) ϕ i can only be fulfilled locally!

53 objective functionals / costs encoding the desired physics J [{ϕ k t), ϕ kt)}, εt)] = J T [{ϕ k T ), ϕ kt )}] + J t [{ϕ k t), ϕ kt)}, εt)] final-time target intermediate-time target time-dependent cost state-dependent cost functionals of the field εt) explicitly implicitly through ϕ k t), ϕ k T )

54 final-time objectives J T i) state-to-state transfer CPK, Palao, Kosloff, Masnou-Seeuws, PRA 70, initial state: highly excited vibrational state of Na 2 electronic ground state objective: vibronic ground state of Na )

55 final-time objectives J T i) state-to-state transfer CPK, Palao, Kosloff, Masnou-Seeuws, PRA 70, initial state: highly excited vibrational state of Na 2 electronic ground state objective: vibronic ground state of Na )

56 final-time objectives J T i) state-to-state transfer equations of motion: Schrödinger equation of the Na 2 molecule coupled to a laser field i ) ˆT t ψt) = + V g ˆR) µ ˆR)ɛt) µ ˆR) + ɛ ψt) t) ˆT + V e ˆR) CPK, Palao, Kosloff, Masnou-Seeuws, PRA 70, initial state: highly excited vibrational state of Na 2 electronic ground state objective: vibronic ground state of Na ) numerical effort: grid representation with dim H , action of Ĥ = vector-vector multiplication plus FFT exp iĥ t) = n a n E t)p n Ĥ)

57 final-time objectives J T i) state-to-state transfer equations of motion: Schrödinger equation of the Na 2 molecule coupled to a laser field i ) ˆT t ψt) = + V g ˆR) µ ˆR)ɛt) µ ˆR) + ɛ ψt) t) ˆT + V e ˆR) CPK, Palao, Kosloff, Masnou-Seeuws, PRA 70, initial state: highly excited vibrational state of Na 2 electronic ground state objective: vibronic ground state of Na ) numerical effort: grid representation with dim H , action of Ĥ = vector-vector multiplication plus FFT exp iĥ t) = n a n E t)p n Ĥ)

58 final-time objectives J T ii) unitary transformation d ) 11 01, goal: perform a two-qubit gate on the logical basis each of the N = 4 basis states needs to transform under the action of Ĥt) for t [0, T ] such that the desired unitary operation is realized at time t = T effort: N dim H V R) [cm -1 ] B 1 Σ u + 3 Σg 1 Π g X 1 Σ g + 3 Πg µ R) [at.u.] µ internuclear distance R [at.u.] µ 0 S+ 1 P S+ 1 D S+ 3 P S+S

59 final-time objectives J T J T = λ [ }] 0 N Re Tr {Ô+ ˆP N ÛT, 0; ɛ) ˆP N real-valued, phase-sensitive functional Ô target operator λ 0 weight N = dim{ô} ˆP N projector on subspace of Ô ÛT, 0; ɛ) actual time evolution

60 final-time objectives J T J T = λ [ }] 0 N Re Tr {Ô+ ˆP N ÛT, 0; ɛ) ˆP N real-valued, phase-sensitive functional Ô target operator λ 0 weight N = dim{ô} ˆP N projector on subspace of Ô ÛT, 0; ɛ) actual time evolution state-to-state transfer: Ô = ϕ target ϕ target, N = 1 single-qubit gate: N = 2, two-qubit gate: N = 4 cf. Palao & Kosloff, PRA 68, )

61 intermediate-time objectives J t J t = assumption: additive costs T 0 {g a [ɛt)] + g b [ϕt), ϕ t)]} dt

62 intermediate-time objectives J t J t = assumption: additive costs T 0 {g a [ɛt)] + g b [ϕt), ϕ t)]} dt examples g a [ɛt)] = λ a St) [ɛt) ɛ ref t)] 2 minimization of field intensity ɛ ref t) = 0) or change in field intensity ɛ ref t) = ɛ old ) choice of ɛ ref t) determines update vs replacement rule!

63 intermediate-time objectives J t J t = assumption: additive costs T 0 {g a [ɛt)] + g b [ϕt), ϕ t)]} dt examples g a [ɛt)] = λ a St) [ɛt) ɛ ref t)] 2 minimization of field intensity ɛ ref t) = 0) or change in field intensity ɛ ref t) = ɛ old ) choice of ɛ ref t) determines update vs replacement rule! g b [ϕt), ϕ t)] = λ b ϕt) ˆDt) ϕt) ˆDt) target operator λ a, λ b weights, St) switch/shape function

64 time-dependent targets g b [ϕt), ϕ t)] = λ b ϕt) ˆDt) ϕt) prescribing a desired evolution ˆDt) = 6 6 ΘT 1 t) Θt T 1 )ΘT 2 t) Θt T 2 )ΘT 3 t) Θt T 3 )ΘT t) Ndong, Tal-Ezer, Kosloff, CPK, JCP 130, )

65 time-dependent targets g b [ϕt), ϕ t)] = λ b ϕt) ˆDt) ϕt) prescribing a desired evolution keeping the dynamics in a subspace ˆDt) = 6 6 ΘT 1 t) Θt T 1 )ΘT 2 t) Θt T 2 )ΘT 3 t) Θt T 3 )ΘT t) Ndong, Tal-Ezer, Kosloff, CPK, JCP 130, ) ˆDt) = ˆP allow Palao, Kosloff, CPK, PRA 77, )

66 optimal control theory: variants variational approach guess the right functional, including eqs. of motion & phasefactors do the variations to obtain eqs. of motion and eq. for the field guess the correct time discretization s.t. method converges W. Zhu, J. Botina, H. Rabitz, JCP 108, )

67 optimal control theory: variants variational approach guess the right functional, including eqs. of motion & phasefactors do the variations to obtain eqs. of motion and eq. for the field guess the correct time discretization s.t. method converges W. Zhu, J. Botina, H. Rabitz, JCP 108, ) Krotov method ingredients: objective + constraints) + eqs. of motion construct auxiliary functional L with auxil. potential to guarantee monoton. convergence derive the eq. for ɛt) from the minimization of L Sklarz & Tannor, PRA 66, ), Palao & Kosloff, PRA 68, )

68 optimal control theory: schemes improve the field by St) λ 0 [ ɛ j+1 t) = ] Im ϕ i Û + T, 0; ɛ j ) }{{} ϕ f ϕ f Û + t, T ; ɛ j ) }{{} ˆµ Ût, 0; ɛj+1 ) ϕ i }{{} forward backward forward propagation 1) propagation 2) propagation 3)

69 optimal control theory: schemes improve the field by St) λ 0 [ ɛ j+1 t) = ] Im ϕ i Û + T, 0; ɛ j ) }{{} ϕ f ϕ f Û + t, T ; ɛ j ) }{{} ˆµ Ût, 0; ɛj+1 ) ϕ i }{{} forward backward forward propagation 1) propagation 2) propagation 3) interference between past and future events ϕ i ε 1 εt) ε 0 0 t T ϕ f

70 optimal control theory: schemes improve the field by St) λ 0 [ ɛ j+1 t) = ] Im ϕ i Û + T, 0; ɛ j ) }{{} ϕ f ϕ f Û + t, T ; ɛ j ) }{{} ˆµ Ût, 0; ɛj+1 ) ϕ i }{{} forward backward forward propagation 1) propagation 2) propagation 3) interference between past and future events ϕ i ε 1 εt) ε 0 0 t T variational approach & Krotov method lead to similar schemes Maday & Turinici, JCP 118, ) & work by Nielsen group but: reduced Krotov method only 1st order variant) ϕ f

71 general philosophy of Krotov s method

72 basic concept ingredients: final-time target J T [ϕ T, ϕ T ] time-dep. targets / costs g a [ɛ] + g b [ϕt), ϕ t)] equations of motion Ĥt) ϕt) ϕt 0 ) = ϕ 0 i t ϕt) =

73 basic concept ingredients: final-time target J T [ϕ T, ϕ T ] time-dep. targets / costs g a [ɛ] + g b [ϕt), ϕ t)] equations of motion Ĥt) ϕt) ϕt 0 ) = ϕ 0 i t ϕt) = construction of auxiliary functional L L[ϕ, ϕ, ɛ, Φ] = J[ϕ, ϕ, ɛ] choose arbitrary scalar potential Φ[ϕ, ϕ, t] such that L[ϕ i, ϕ,i, ɛ i, Φ] L[ϕ i+1, ϕ,i+1, ɛ i+1, Φ] building in monotonic convergence

74 auxiliary functional L L[ϕ, ϕ, ɛ, Φ] = G[ϕT ), ϕ T )] Φ[ϕ0), ϕ 0), 0] T 0 R[ϕt), ϕ t), ɛt), t]dt final-time contribution: G [ϕt ), ϕ T )] = J T [ϕt ), ϕ T )] + Φ[ϕT ), ϕ T ), T ]

75 auxiliary functional L L[ϕ, ϕ, ɛ, Φ] = G[ϕT ), ϕ T )] Φ[ϕ0), ϕ 0), 0] T 0 R[ϕt), ϕ t), ɛt), t]dt final-time contribution: G [ϕt ), ϕ T )] = J T [ϕt ), ϕ T )] + Φ[ϕT ), ϕ T ), T ] intermediate-time contribution: ) R [ϕt), ϕ t), ɛt), t] = g a [ɛt)] + g b [ϕt), ϕ t)] + Φ t + N k=1 [ ϕk Φ f k [ϕ, ϕ, ɛ, t] ] + ϕ k Φ f k [ϕ, ϕ, ɛ, t]

76 auxiliary functional L L[ϕ, ϕ, ɛ, Φ] = G[ϕT ), ϕ T )] Φ[ϕ0), ϕ 0), 0] T 0 R[ϕt), ϕ t), ɛt), t]dt final-time contribution: G [ϕt ), ϕ T )] = J T [ϕt ), ϕ T )] + Φ[ϕT ), ϕ T ), T ] intermediate-time contribution: ) R [ϕt), ϕ t), ɛt), t] = g a [ɛt)] + g b [ϕt), ϕ t)] + Φ t + N k=1 [ ϕk Φ f k [ϕ, ϕ, ɛ, t] ] + ϕ k Φ f k [ϕ, ϕ, ɛ, t] the choice of Φ[ϕt), ϕ t), t] completely determines G, R, L

77 central idea of Krotov s method goal: minimization of L, resp. J T 1 we need an extremum in ϕ i ϕ G ϕ i) = 0 and ϕ R ϕ i) = 0 equation for backward propagation d dt χt) = JT t) χt) + ϕ g t, ϕ i), ɛ i)) ) χ T ) = ϕ J T ϕ i) T ) with ϕ containing real & imag. parts of projections of all states ϕ k ) m t) = Re [ m ϕ k t) ], m = 1,..., dimh, k = 1,..., N, ϕ k ) dimh+m t) = Im [ m ϕ k t) ], m = 1,..., dimh, k = 1,..., N. and Jacobian J in components J ij t) = f i ϕ j t, ϕ i), ɛ i))

78 central idea of Krotov s method 2 we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in ϕ and changes in ɛ Krotov s solution

79 central idea of Krotov s method 2 we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in ϕ and changes in ɛ Krotov s solution i) choose Φ at the extremum, ϕ i, such that it is the worst possible choice with respect to any change in the states maximize L when going from ϕ i to ϕ i+1 for fixed ɛ i ii) then any change in the field from ɛ i to ɛ i+1 will lead to a minimization of L ) ϕt) i+1), ɛt), t or R ɛ ɛ i+1) t) = arg max R ɛt) ) ϕ i+1), ɛ i+1), t = 0, 2 R ) ɛ 2 ϕ i+1), ɛ i+1), t < 0

80 central idea of Krotov s method 2 we want a minimum of L, i.e. minimum of G & maximum of R but L is changed by both changes in ϕ and changes in ɛ Krotov s solution i) choose Φ at the extremum, ϕ i, such that it is the worst possible choice with respect to any change in the states maximize L when going from ϕ i to ϕ i+1 for fixed ɛ i ii) then any change in the field from ɛ i to ɛ i+1 will lead to a minimization of L ) ϕt) i+1), ɛt), t or R ɛ ɛ i+1) t) = arg max R ɛt) ) ϕ i+1), ɛ i+1), t = 0, 2 R ) ɛ 2 ϕ i+1), ɛ i+1), t < 0

81 central idea of Krotov s method Krotov s solution i) optimization with respect to change in states is translated into construction of Φ at the extremum ϕ i ii) convergence for step in field, ɛ i) ɛ i+1), assured globally for R by ɛ R = R ϕ i+1), ɛ i+1), t R ) ɛ R ) ϕ i+1), ɛ i+1), t ) ϕ i+1), ɛ i), t = 0 0 a global optimum would be found, if we could actually implement ) ɛ i+1) t) = arg max R ϕt) i+1), ɛt), t ɛt)

82 Krotov s step i) second order construction of Φ

83 maximization of L wrt ϕ locally globally ) G ψ 2 ϕg! 0 and 2 ϕr! 0 = G ϕ i) T ) + ψ ) G ϕ i) T ) ) ) R ψt), t! 0 ψ = R ϕ i) t) + ψt), ) ɛ i) t), t R ϕ i) t), ɛ i) t), t )! 0 ψ

84 maximization of L wrt ϕ locally globally ) G ψ 2 ϕg! 0 and 2 ϕr! 0 = G ϕ i) T ) + ψ ) G ϕ i) T ) ) ) R ψt), t! 0 ψ = R ϕ i) t) + ψt), ) ɛ i) t), t R ϕ i) t), ɛ i) t), t )! 0 ψ

85 Krotov s ansatz construct Φ to second order in the states ϕ i+1) k Φt, ϕ i+1), ϕ,i+1) ) = N ) χ k ϕ i+1) k + ϕ i+1) k χ k k=1 N ϕ i+1) k k,l=1 ϕ i) k ˆσ klt) ϕ i+1) l ϕ i) l choose ˆσ kl t) such that maximum condition for G and minimum condition for R are fulfilled

86 Krotov s ansatz construct Φ to second order in the states ϕ i+1) k Φt, ϕ i+1), ϕ,i+1) ) = N ) χ k ϕ i+1) k + ϕ i+1) k χ k k=1 N ϕ i+1) k k,l=1 ϕ i) k ˆσ klt) ϕ i+1) l ϕ i) l choose ˆσ kl t) such that maximum condition for G and minimum condition for R are fulfilled

87 construction of ˆσ kl t) can be done locally or globally Sklarz/Tannor s discussion local but results coincide with global derivation due to choice of J T ) Krotov: constructive proof for global conditions derivation for global conditions leads to much simpler solution for fourth-degree tensor ˆσ ˆσt) = α e γt t) 1 ) + β ) 11 σ t) 11

88 Krotov s proof main idea: assure that nothing goes wrong for very large & very small ϕ and very large ɛ If: 1 The right-hand side of the equation of motion, f t, ϕ, ɛ), is bounded. Specifically, for large values of the state vector, ϕ, the right-hand side of the equations of motion does not grow faster than quadratically with respect to ϕ for all t and possible fields ɛ. 2 The Jacobian of the right-hand side of the equations of motion, J, is bounded for any time t, field ɛ and state vector ϕ. 3 The functionals J T ϕ) and g ɛ, ϕ, t) are twice differentiable and bounded. In particular, for large values of the state vector ϕ, the functionals J T and g do not grow faster than quadratically with respect to ϕ. then ˆσt) = α e γt t) 1 ) + β ) 11 σ t) 11

89 quantum control state vectors ϕ i), ϕ i+1) inherently bounded boundedness conditions already guaranteed if f t, ϕ, ɛ), J, J T ϕ) and g ɛ, ϕ, t) regular change in states compact subset of R 2NM

90 quantum control state vectors ϕ i), ϕ i+1) inherently bounded boundedness conditions already guaranteed if f t, ϕ, ɛ), J, J T ϕ) and g ɛ, ϕ, t) regular change in states compact subset of R 2NM

91 fulfilling G ψ) 0 ) G ψ = for ψ = ) 0: G 0 = 0 for ψ 0: χt ) ψ ) + 1 ) 2 σt ) ψ ψ + J T ϕt ) i) + ψ ) J T ϕt ) i))

92 fulfilling G ψ) 0 ) G ψ = for ψ = ) 0: G 0 = 0 for ψ 0: χt ) ψ ) + 1 ) 2 σt ) ψ ψ + J T ϕt ) i) + ψ ) J T ϕt ) i)) ) ) [ G ψ = ψ ψ 1 2 σt ) + χt ) ψ ) + J T ϕt ) i) + ψ ) J ) T ϕt ) i) ] ) ψ ψ A = sup ψ χ T ψ ) + J T ϕt ) i) + ψ ) J ) T ϕt ) i) ) ψ ψ

93 fulfilling G ψ) 0 ) G ψ = for ψ = ) 0: G 0 = 0 for ψ 0: χt ) ψ ) + 1 ) 2 σt ) ψ ψ + J T ϕt ) i) + ψ ) J T ϕt ) i)) ) ) [ G ψ = ψ ψ 1 2 σt ) + χt ) ψ ) + J T ϕt ) i) + ψ ) J ) T ϕt ) i) ] ) ψ ψ A = sup ψ χ T ψ ) + J T ϕt ) i) + ψ ) J ) T ϕt ) i) ) ψ ψ σt ) < 2A

94 fulfilling R ψ) 0 ) R ψt), t = for ψ = ) 0: R 0, t = 0 t for ψ 0: χt) ψt) ) σt) ψt) ψt) ) + χt) + σt) ψt) ) )) ) f ψt), t g ψt), t 1 σ t) σ t) B + C > 0 2

95 fulfilling R ψ) 0 ) R ψt), t for ψ = 0: R for ψ 0: ) R ψt), t χt) ψt) ) σt) ψt) ψt) ) = + χt) + σt) ψt) ) ) 0, t = 0 t = ψt) ψt) ) f )) ) ψt), t g ψt), t [ 1 ψt) 2 σt) + σt) f ) + χt) ψt) + χt) ] f g ) ψt) ψt) ψt) ψt) B = sup ψt) R 2NM ;t [0,T ] C = inf ψt) R 2NM ;t [0,T ] ψt) f ψt) ψt) χt) ψt) + χt) f g ψt) ψt) )

96 fulfilling R ψ) 0 ) R ψt), t = for ψ = ) 0: R 0, t = 0 t for ψ 0: χt) ψt) ) σt) ψt) ψt) ) + χt) + σt) ψt) ) )) ) f ψt), t g ψt), t B = sup ψt) R 2NM ;t [0,T ] C = inf ψt) R 2NM ;t [0,T ] ψt) f ψt) ψt) χt) ψt) + χt) f g ψt) ψt) ) 1 σ t) σ t) B + C > 0 2

97 maximizing L wrt ϕ σt ) < 2A 1 σ t) σ t) B + C > 0 2 one solution ) t) C σt) = e BT B Ā C B with B = 2B + δ, C = min δ, 2C δ) and Ā = max ε, 2A + ε) or more generally σ t) = α e γt t) 1 ) + β

98 how to get A, B, C? A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder Lagrange s form) W ϕ) = α n 1 1 α! α W ) ϕ i)) ψ ) α + R ϕ i),n ψ

99 how to get A, B, C? A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder Lagrange s form) W ϕ) = α n 1 R ϕ i),n 1 α! α W ) ϕ i)) ψ ) α + R ϕ i),n ψ ) ψ 1 α! MW n ϕ i)) ψ α, α = n M W n ϕ i)) = sup ψ, α =n α W ϕ i) + ψ),

100 how to get A, B, C? A, B, C are Taylor expansions of certain quantities starting at the first or second order estimate the remainder Lagrange s form) W ϕ) = α n 1 R ϕ i),n 1 α! α W ) ϕ i)) ψ ) α + R ϕ i),n ψ ) ψ 1 α! MW n ϕ i)) ψ α, α = n M W n ϕ i)) = sup ψ, α =n α W ϕ i) + ψ), estimate that is independent of ϕ i) M W n = sup M n ϕ i)) ϕ i) X

101 estimate of A A = sup ψ J T,2 ψ ψ ). estimate J T,2 by its Lagrange remainder: A 1 2 MJ T 2 = 1 2 sup ψ, α =2 ) α J T ψ

102 estimate of A A = sup ψ J T,2 ψ ψ ). estimate J T,2 by its Lagrange remainder: A 1 2 MJ T 2 = 1 2 sup ψ, α =2 ) α J T ψ for functionals J T that are quadratic in ϕ A = 0

103 estimate of B B X sup ψ + sup ) ω ψ ψt);t [0,T ] ω ψt), ɛ i), t ) mat

104 estimate of B B X sup ψ + sup ) ω ψ ψt);t [0,T ] ω ψt), ɛ i), t ) mat for Hamiltonians that do not depend on the state ψt) ω ɛ i), t ) ψt) B = sup ψ;t [0,T ] ψt) ψt) = sup ω ɛ i), t t [0,T ] ) mat max. eigenvalue of Ĥ

105 estimate of C C χt) ω 1 sup ψt) g ) + sup 2 ) ψt);t [0,T ] ψt) ψt) ψt);t [0,T ] ψt) ψt) C sup M1 ω χt) ) + 1 t [0,T ] 2 sup ψ, α =2 ) α g ψ.

106 estimate of C C χt) ω 1 sup ψt) g ) + sup 2 ) ψt);t [0,T ] ψt) ψt) ψt);t [0,T ] ψt) ψt) C sup M1 ω χt) ) + 1 t [0,T ] 2 sup ψ, α =2 ) α g ψ. for state-independent Ĥ and g depending on ϕ only up to linear order : ω 1 = 0 and g 2 = 0 C = 0

107 estimate of C C χt) ω 1 sup ψt) g ) + sup 2 ) ψt);t [0,T ] ψt) ψt) ψt);t [0,T ] ψt) ψt) C sup M1 ω χt) ) + 1 t [0,T ] 2 sup ψ, α =2 ) α g ψ. for state-independent Ĥ and g depending on ϕ only up to linear order : ω 1 = 0 and g 2 = 0 C = 0 for certain!) functionals and EoMs: A = 0 & C = 0 σt) = 0 and the second order contribution to Φ vanishes: Palao-Kosloff version of Krotov method Krotov-PK) still ensuring monotonic convergence globally)

108 Krotov s step ii) second order construction of ɛ

109 equation for the field remember R ) ϕ i+1), ɛ i+1), t = 0 ) ɛ ) ɛ R = R ϕ i+1), ɛ i+1), t R ϕ i+1), ɛ i), t 0 equations of motion in basis set expansion: 2M 2M matrix Ω k = H k,r H k,i ) H k,i H k,r first order condition yields: ɛ i+1) t) = ɛ ref t) + 1 2λ a St) { N 2M k=1 m,n=1 +σt) N Ω k mn χ km ɛ ϕi+1) kn 2M k=1 m,n=1 Ω k } mn ϕ km ɛ ϕ kn

110 assuring convergence locally 2 g ɛ 2 ɛ i+1), ϕ i+1)) > N 2M k=1 m,n=1 +σt) N 2 Ω k mn χ km ɛ 2 ϕ kn 2M k=1 m,n=1 2 Ω k mn ϕ km ɛ 2 ϕ kn

111 assuring convergence locally 2 g ɛ 2 ɛ i+1), ϕ i+1)) > N 2M k=1 m,n=1 +σt) N 2 Ω k mn χ km ɛ 2 ϕ kn 2M k=1 m,n=1 for linear dependency of Ĥ on ɛ 2 Ω k mn ϕ km ɛ 2 ϕ kn then all we need: 2 Ω k nm ɛ 2 = 0 m, n 2 g ɛ 2 ɛ i+1)) > 0! fulfilled for typical choice of g a quadratic in ɛ i+1)!

112 assuring convergence globally ɛr = g t, ɛ i), ϕ i+1)) g t, ɛ i+1), ϕ i+1)) + 0 N 2M k=1 m,n=1 +σt) N k=1 m,n=1 Ω k mn χ km [Ω k mn 2M ) ) ] ϕ i+1), ɛ i+1), t Ω k mn ϕ i+1), ɛ i), t ϕ i+1) t) kn [ ) ϕ i+1) Ω k km mn ϕ i+1), ɛ i+1), t ϕ i+1), ɛ i), t ) ] ϕ i+1) t) kn

113 assuring convergence globally ɛr = g t, ɛ i), ϕ i+1)) g t, ɛ i+1), ϕ i+1)) + 0 N 2M k=1 m,n=1 +σt) N k=1 m,n=1 Ω k mn χ km [Ω k mn 2M ) ) ] ϕ i+1), ɛ i+1), t Ω k mn ϕ i+1), ɛ i), t ϕ i+1) t) kn [ ) ϕ i+1) Ω k km mn ϕ i+1), ɛ i+1), t ϕ i+1), ɛ i), t ) ] ϕ i+1) t) kn can be fulfilled by estimating λ a St) in terms of Lagrange remainder of the Taylor expansion of Ω k mn ϕ i+1), ɛ, t ) around ɛ = ɛ i) for linear dependence of Ĥ on ɛ, global and local 2nd order conditions coincide