Chebychev Propagator for Inhomogeneous Schrödinger Equations

Transcription

1 Chebychev Propagator for Inhomogeneous Schrödinger Equations Michael Goerz May 16, 2011

2 Solving the Schrödinger Equation Schrödinger Equation Ψ(t) = Ĥ Ψ(t) ; e.g. Ĥ = t ( V1 (R) ) µɛ(t) µɛ(t) V 2 (R)

3 Solving the Schrödinger Equation Schrödinger Equation Ψ(t) = Ĥ Ψ(t) ; e.g. Ĥ = t ( V1 (R) ) µɛ(t) µɛ(t) V 2 (R) Solution Ψ(t) = e iĥt Ψ 0 if Ĥ not time dependent

4 Solving the Schrödinger Equation Schrödinger Equation Ψ(t) = Ĥ Ψ(t) ; e.g. Ĥ = t ( V1 (R) ) µɛ(t) µɛ(t) V 2 (R) Solution Ψ(t + t) = e iĥ t Ψ(t) piecewise constant pulses

5 Solving the Schrödinger Equation Schrödinger Equation Ψ(t) = Ĥ Ψ(t) ; e.g. Ĥ = t ( V1 (R) ) µɛ(t) µɛ(t) V 2 (R) Solution Ψ(t + t) = e iĥ t Ψ(t) piecewise constant pulses Evaluation of the Time Evolution Operator Expand into series: N e iĥt a np n(ĥ) k=1

6 Solving the Schrödinger Equation Schrödinger Equation Ψ(t) = Ĥ Ψ(t) ; e.g. Ĥ = t ( V1 (R) ) µɛ(t) µɛ(t) V 2 (R) Solution Ψ(t + t) = e iĥ t Ψ(t) piecewise constant pulses Evaluation of the Time Evolution Operator Expand into series: N e iĥt a np n(ĥ) k=1 cf. Runge-Kutta: solving the differential equation, instead of evaluating the analytical solution

7 Chebychev Polynomials Properties of Chebychev polynomials P 0 = 1, P 1 (x) = x, P n(x) = 2xP n 1 (x) P n 2 (x) Defined over range [ 1, 1] normalize Hamiltonian Ĥ norm = 2 Ĥ E min1 E Fastest converging polynomial expansion 1 P n(x) = cos(nθ) with θ = arccos(x) Cosine transform for coefficients Chebychev coefficients Expansion coefficients a n for function f (x): a n = 2 δ n0 π +1 1 f (x)p n(x) 1 x 2 For f (Ĥ norm) = e iĥ norm t : a n Bessel functions. dx

8 Inhomogeneous Schrödinger Equation t Ψ(t) = Ĥ Ψ(t) + Φ(t)

9 Inhomogeneous Schrödinger Equation t Ψ(t) = Ĥ Ψ(t) + Φ(t) Note: not the same as nonlinear SE: e.g. t Ψ(t) = (Ĥ + Ψ(t) 2 ) Ψ(t)

10 OCT with State-Dependent Costs Optimization Functional J = F [Ψ t] + g a[ɛ(t)] dt + g b [Ψ(t)] dt; g b [Ψ] = λ b Ψ(t) ˆP allow Ψ(t)

11 OCT with State-Dependent Costs Optimization Functional J = F [Ψ t] + g a[ɛ(t)] dt + g b [Ψ(t)] dt; g b [Ψ] = λ b Ψ(t) ˆP allow Ψ(t) also: time-dependent targets

12 Reminder: Krotov The Chebychev Propagator 1 backward propagation with old pulse ɛ 1 ɛ 2 ɛ nt 2 ɛ nt 1 Ψ bw (t 0) Ψ bw (t) Ψ. bw..(t) Ψ bw (t) Ψ t Ψ t ˆµ ˆµ.. ˆµ. ˆµ = τ Ψ i Ψ fw (t) Ψ. fw.(t). Ψ fw (t) Ψ fw (T ) Ψ fw (T ) ɛ 1 ɛ 2 ɛ nt 2 ɛ nt 1 2 forward propagation with new pulse Pulse update by matching forward- and backward-propagated states

13 Inhomogeneous Backward-Propagation Daniel: Second Order Krotov Preprint (arxiv: v1) Optimization Functional J = F [Ψ t] + g a[ɛ(t)] dt + g b [Ψ(t)] dt; g b [Ψ] = λ b Ψ(t) ˆP allow Ψ(t) Pulse Update ɛ Im χ (0) (t) ˆµ φ (1) (t) Backward Propagation d dt χ(0) (t) = i Ĥ [ϕ (0), ɛ (0) ] χ (0) (t) + ϕ g b ϕ (0) (t) χ (0) (T ) = ϕ J T ϕ (0) (T )

14 Inhomogeneous Backward-Propagation Daniel: Second Order Krotov Preprint (arxiv: v1) Optimization Functional J = F [Ψ t] + g a[ɛ(t)] dt + g b [Ψ(t)] dt; g b [Ψ] = λ b Ψ(t) ˆP allow Ψ(t) Pulse Update ɛ Im χ (0) (t) ˆµ φ (1) (t) Backward Propagation d dt χ(0) (t) = i Ĥ [ϕ (0), ɛ (0) ] χ (0) ϕ (t) + ˆP (0) allow (t) χ (0) (T ) = ϕ J T ϕ (0) (T )

15 Solving the Inhomogeneous Schrödinger Equation Numerically

16 References The Chebychev Propagator [1] JCP 130, (2009) A Chebychev propagator for inhomogeneous Schrödinger equations Mamadou Ndong, 1 Hillel Tal-Ezer, 2 Ronnie Kosloff, 3 and Christiane P. Koch 1,a 1 Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, Berlin, Germany 2 School of Computer Sciences, The Academic College of Tel-Aviv Yaffo, 2 Rabenu Yeruham St., Tel-Aviv 61803, Israel 3 Department of Physical Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel THE JOURNAL OF CHEMICAL PHYSICS 130, [2] JCP 132, (2010) A Chebychev propagator with iterative time ordering for explicitly time-dependent Hamiltonians Mamadou Ndong, 1 Hillel Tal-Ezer, 2 Ronnie Kosloff, 3 and Christiane P. Koch 1,a 1 Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, Berlin, Germany 2 School of Computer Sciences, The Academic College of Tel Aviv-Yaffo, Rabenu Yeruham St., Tel-Aviv 61803, Israel 3 Institute of Chemistry and The Fritz Haber Research Center, The Hebrew University, Jerusalem 91904, Israel THE JOURNAL OF CHEMICAL PHYSICS 132,

17 Treating the Inhomogeneity in Order m Inhomogeneous SE Expansion of Φ(t) Ψ(t) = Ĥ Ψ(t) + Φ(t) t m 1 Φ(t) m = Φj Pj ( t) j=0 Expand inhomogeneous term in Chebychev series

18 Treating the Inhomogeneity in Order m Inhomogeneous SE Expansion of Φ(t) Ψ(t) = Ĥ Ψ(t) + Φ(t) t m 1 m 1 t j Φ(t) m = Φj Pj ( t) Φ (j) j! j=0 j=0 Expand inhomogeneous term in Chebychev series Reorder into power series (or use Taylor to begin with)

19 Treating the Inhomogeneity in Order m Inhomogeneous SE Expansion of Φ(t) Ψ(t) = Ĥ Ψ(t) + Φ(t) t m 1 m 1 t j Φ(t) m = Φj Pj ( t) Φ (j) j! j=0 j=0 Expand inhomogeneous term in Chebychev series Reorder into power series (or use Taylor to begin with) Decide on which order to solve: 1, 2, 3, maybe 4 The smaller the order, the smaller t has to be

20 The Analytical Solution Inhomogeneous SE (Φ to order m) m 1 Ψ(t) = Ĥ Ψ(t) + t j=0 t j Φ (j) j! Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) m 1 f m = ( iĥ) m e iĥt ( iĥt) j j! j=0 λ (0) = Ψ 0 λ (j) = iĥ λ (j 1) + Φ (j 1) e.g. Ψ(t) (3) = Ψ 0 + t λ (1) + t2 2 λ (2) + f 3 (Ĥ) λ 3, with ( ) 3 ( ) f 3 (Ĥ) = iĥ e iĥt 1 iĥt

21 The Analytical Solution Inhomogeneous SE (Φ to order m) m 1 Ψ(t) = Ĥ Ψ(t) + t j=0 t j Φ (j) j! Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) m 1 f m = ( iĥ) m e iĥt ( iĥt) j j! j=0 λ (0) = Ψ 0 λ (j) = iĥ λ (j 1) + Φ (j 1) e.g. Ψ(t) (3) = Ψ 0 + t λ (1) + t2 2 λ (2) + f 3 (Ĥ) λ 3, with ( ) 3 ( ) f 3 (Ĥ) = iĥ e iĥt 1 iĥt

22 The Analytical Solution Inhomogeneous SE (Φ to order m) m 1 Ψ(t) = Ĥ Ψ(t) + t j=0 t j Φ (j) j! Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) m 1 f m = ( iĥ) m e iĥt ( iĥt) j j! j=0 λ (0) = Ψ 0 λ (j) = iĥ λ (j 1) + Φ (j 1) e.g. Ψ(t) (0) = e iĥt Ψ 0 homogeneous propagation

23 Chebychev Propagation Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) ; { } Φ (j) λ Idea Evaluate f m(ĥ) by expanding it into Chebychev Polynomials (just like for the standard Chebychev propagator with f 0 (Ĥ) = e iĥt )

24 Chebychev Propagation Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) ; { } Φ (j) λ Idea Evaluate f m(ĥ) by expanding it into Chebychev Polynomials (just like for the standard Chebychev propagator with f 0 (Ĥ) = e iĥt ) Algorithm Outline (for fixed m) For each time step: determine { Φ (j) } and from that { λ (j)} run through the Chebychev series for f m sum everything up, yielding Ψ(t) (m)

25 Details of the Algorithm

26 Global Initialization (before any actual propagation) Calculate Chebychev Coefficients Calculate the Chebychev expansion coefficients a n for f m(ĥ), for the chosen order m.

27 Global Initialization (before any actual propagation) Calculate Chebychev Coefficients Calculate the Chebychev expansion coefficients a n for f m(ĥ), for the chosen order m. a n cannot be calculated analytically (like for the standard Chebychev) Calculation of a n is done through a fast cosine-transform: a n = 2 δ n0 N N 1 f m(θ k ) cos(nθ k ) k=0 Ĥ needs to be normalized a n might have to be re-calculated if spectral radius changes (after each OCT iteration) On a non-equidistant time grid, the a n would have to be re-calculated at every time step For small Ĥ, the term ( iĥ) m might lead to numerical instability. We could use Taylor instead....?

28 Local Initialization (at every time step) Calculate Expansion of Inhomogeneous Term Calculate all necessary Φ (j) (i.e. up to order m) to approximate the local Φ(t i ), either by an intermediate Chebychev expansion, fallowed by calculation of coefficients in the power series, or by a direct Taylor series.

29 Local Initialization (at every time step) Calculate Expansion of Inhomogeneous Term Calculate all necessary Φ (j) (i.e. up to order m) to approximate the local Φ(t i ), either by an intermediate Chebychev expansion, fallowed by calculation of coefficients in the power series, or by a direct Taylor series. Calculation via Taylor: Calculate derivatives through FFT Calculation via Chebychev: Sample Φ(t) at intermediate points around t by interpolation (splining should be fine) Calculate Chebychev coefficients Φ j by cosine transform Calculate Φ (j) from Φ j by formulas in References (just collect the powers)

30 Propagation Step The Chebychev Propagator Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) Calculate λ (j), j = 0... m 1 λ (j) = iĥ λ (j 1) + Φ (j 1) Calculate f m(ĥ) λ (m) by Chebychev recursion (Choose N to reach machine precision) f m(ĥ) λ (m) N = a np n(ĥ) λ (m) n=1 P n(ĥ) = 2ĤP n 1 (Ĥ) P n 2 (Ĥ) Calculate Ψ(t) (m)

31 in QDYN

32 Propagation Routines (prop.f90) prop subroutine prop(psi, grid, ham, work, para, from ti, to ti, bw, & & info hook, pulses, alt pulse, upd hook, storage) prop step subroutine prop step(psi, grid, ham, work, para, ti, bw, & & pulses, alt pulses, alt pulse, upd hook)

33 Propagation Routines (prop.f90) prop subroutine prop(psi, grid, ham, work, para, from ti, to ti, bw, & & info hook, pulses, alt pulse, upd hook, storage) prop with optional inhomogeneous term subroutine prop(psi, grid, ham, work, para, from ti, to ti, bw, inh psi, & & get inh phi, info hook, pulses, alt pulse, upd hook, storage) inh psi: stored array of forward-propagated states get inh phi: function that calculates Φ from Ψ (e.g. ˆP allow Ψ ). prop step subroutine prop step(psi, grid, ham, work, para, ti, bw, & & pulses, alt pulses, alt pulse, upd hook) prop step with optional inhomogeneous term subroutine prop step(psi, grid, ham, work, para, ti, bw, inh psi, & & get inh phi, pulses, alt pulses, alt pulse, upd hook)

34 Calculation of Chebychev Coefficients Calculate Chebychev Coefficients Calculate the Chebychev expansion coefficients a n for f m(ĥ), for the chosen order m. in inhom cheby.f90: subroutine init inh cheby(ham, wcheby, order, para) Use the same work array (wcheby) as for the homogeneous Chebychev propagation. Sufficient a n are calculated and stored in wcheby to reach machine precision Watch out for numerical instability ( Taylor)

35 Expansion of Inhomogeneous Term Calculate Expansion of Inhomogeneous Term Calculate all necessary Φ (j) (i.e. up to order m) to approximate the local Φ(t i ), either by an intermediate Chebychev expansion, fallowed by calculation of coefficients in the power series, or by a direct Taylor series. in inhom cheby.f90: subroutine expand inh phi(inh psi, get inh phi, order, phi coeffs) phi coeffs stores the Φ (j) (power series) invoke monic transf to calculate Φ (j) from Φ j Continue with calculation of λ (j) : subroutine get inh lambda(lambda, phi, ham,...)

36 Performing the Propagation Step Solution m 1 Ψ(t) (m) = j=0 t j j! λ (j) + f m(ĥ) λ (m) in inhom cheby.f90: subroutine inhom cheby(psi, work, ham, grid, lambda, para, dt, ti, & & alt pulses, alt pulse) Identical interface to cheby, except for λ coefficients are implicit in work