Numerical solution of the Schrödinger equation on unbounded domains

Transcription

1 Numerical solution of the Schrödinger equation on unbounded domains Maike Schulte Institute for Numerical and Applied Mathematics Westfälische Wilhelms-Universität Münster Cetraro, September 6

2 OUTLINE Transparent boundary conditions (TBC) OUTLINE

3 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation OUTLINE -A

4 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme OUTLINE -B

5 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme Approximation of DTBC OUTLINE -C

6 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme Approximation of DTBC Simulation of quantum wave guides OUTLINE -D

7 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme Approximation of DTBC Simulation of quantum wave guides Extension of the DTBC to (nearly) arbitrary potentials OUTLINE -E

8 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme Approximation of DTBC Simulation of quantum wave guides Extension of the DTBC to (nearly) arbitrary potentials more realistic simulations OUTLINE -G

9 OUTLINE Transparent boundary conditions (TBC) Analytical transparent boundary conditions for the Schrödinger equation Derivation of discrete transparent boundary conditions (DTBC) for the Schrödinger equation for a higher order compact 9-point scheme Approximation of DTBC Simulation of quantum wave guides Extension of the DTBC to (nearly) arbitrary potentials more realistic simulations solving the Schrödinger equation on circular domains OUTLINE -H

10 TRANSPARENT BOUNDARY CONDITIONS (IN GENERAL) Γ R n ψ Ω Definition (TBC): Consider a given whole-space initial value problem (IVP) on R n and Ω R n, Γ = Ω. We are interested in the solution of the IVP on Ω. Therefore we need new artificial boundary conditions on Γ. We call these artificial BC transparent, if the solution of the IVBP on Ω corresponds to the whole-space solution of the IVP restricted on Ω. TRANSPARENT BOUNDARY CONDITIONS (IN GENERAL) 3

11 ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION IVP: time-dependent Schrödinger equation (here: D) i «t ψ(x,t) = + V (x, t) ψ(x,t), x R, t > m x ψ(x,) = ψ I (x) L (R) on a domain of interest Ω = {x R < x < X}. ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION 4

12 ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION IVP: time-dependent Schrödinger equation (here: D) i «t ψ(x,t) = + V (x, t) ψ(x,t), x R, t > m x ψ(x,) = ψ I (x) L (R) on a domain of interest Ω = {x R < x < X}. Assumptions: supp ψ I Ω potential V (., t) L (R), V (x,.) is piecewise continuous V constant on R\Ω (here: V (x, t) = for x and V (x, t) = V X for x X) Goal: Calculate the solution ψ(x,t) C on Ω with TBC at x = and x = X. ANALYTICAL TBC FOR THE SCHRÖDINGER EQUATION 4-A

13 DERIVATION OF TBC ψ V = const V = const G Ω G ψ I X x x G : x G : i ψ t = m ψ xx + V (x, t)ψ i ψ t = m ψ xx + V (x, t)ψ ψ(x, ) = ψ I (x) v(x, ) = ψ x (, t) = (T ψ)(,t) v(x, t) = Φ(t) t >, Φ() = ψ x (X, t) = (T X ψ)(x,t) lim v(x, t) = x v x (, t) = (T X Φ)(t) DERIVATION OF TBC 5

14 Laplace-transformation on the exterior domains: ˆv xx (x, s) + im s + iv X ˆv(x, s) = x > X ˆv(X, s) = ˆΦ(s) lim ˆv(x, s) = x ˆv x (X, s) = (T X Φ)(s) r solution: ˆv(x, s) = e i + im `s+ iv X (x X) ˆΦ(s) r r m (TX Φ)(s) = e iπ 4 + s + iv X ˆΦ(s) With the inverse Laplace-Transformation follows the analytical TBC ψ x (X, t) = r π V πm e i 4 e i X t d dt Z t ψ(x,τ)e i V X τ τ t dτ. [J. S. Papadakis (98)] DERIVATION OF TBC 6

15 Application of DTBC for the Schrödinger equation: Simulation of quantum transistors in quantum waveguides (with inhomogeneous DTBC for the D Schrödinger equation) Analyse steady states and transient behaviour y V control Y ψ inc Γ Ω Γ X x Y y ψ inc V control Γ Ω Γ ψ pw X x DERIVATION OF TBC 7

16 FORMER STRATEGIES: Discretization of the analytic TBCs with an numerical approximation of the convolution integral [e.g. B. Mayfield (989)] only conditionally stable, not transparent! FORMER STRATEGIES: 8

17 FORMER STRATEGIES: Discretization of the analytic TBCs with an numerical approximation of the convolution integral [e.g. B. Mayfield (989)] only conditionally stable, not transparent! Create a buffer zone Θ of the length d with a complex potential V (X) = W ia around the computational domain Ω with Dirichlet -BC at Θ and absorbing boundary conditions on Ω [e.g. L. Burgnies (997)] ψ= Θ Ω X Θ ψ= unconditionally stable, unphysical reflections at the boundary, huge numerical costs FORMER STRATEGIES: 8-A

18 SUCCESSFUL STRATEGIES Family of absorbing BCs (also for the non-linear Schrödinger equation, wave equation) [J. Szeftel (5)] discretize the whole space problem with an unconditionally stable scheme (e.g. Crank-Nicolson finite difference scheme) and calculate new discrete transparent boundary conditions for the full discretized Schrödinger equation DTBC: dx=.65, dt=e 5, t= ψ x [A. Arnold, M. Ehrhardt (since 995)] SUCCESSFUL STRATEGIES 9

19 DERIVATION OF DISCRETE TRB Discretize D Schrödinger equation: Crank-Nicolson scheme in time, t n = n t, n N, H := + V Hamilton-Operator, Ω = [,X] [, Y ] + ih t «ψ(x,y, t + t) = ih t «ψ(x,y, t) x + y «`ψn+ (x, y) + ψ n (x, y) = g n+ (x, y) `ψn+ (x, y) + ψ n (x, y) + Wψ n (x, y) compact 9-point scheme in space, x j = j x, y k = k y with j Z, k K DTBC at x = and x J = X = J x with J Z -BC at y = and y = Y = K y with K N DERIVATION OF DISCRETE TRB

20 9-point discretization scheme: (j+,k+) (j,k) (j+,k) D x + Dy + x + y Dx Dy = ψ n+ j,k I + x D x + y D y «h V n+ j,k ψ n+ j,k id + t ψ n j,k i with ψ n+ j,k = `ψn+ j,k + ψj,k n D + t ψj,k n = ψn+ j,k t ψn j,k, n D x ψ n j,k = ψn j,k ψ n j,k + ψ n j+,k x, j Z D y ψ n j,k = ψn j,k ψ n j,k + ψ n j,k+ y, k N DERIVATION OF DISCRETE TRB

21 Discrete Sine-Transformation in y-direction: ˆψ n j,m := K K X k= «πkm ψj,k n sin K m =,...,K Motivation: Solve discrete stationary Schrödinger equation in D: y χ m j,k = E m χ m j,k, k =,...,K χ m j, = χ m j,k =. The eigenfunctions χ m j,k = sin ` πkm K E m = πm cos y K provide the energies. Hence follows for m =,...,K y ψ n j,k ψ n j,k + ψ n j,k+ b m = πm cos y K ˆψn j,m. DERIVATION OF DISCRETE TRB

22 Sine-Transformation of the discrete Schrödinger equation on the exterior domains j, j J yields for the modes m =,...,K: C m j+ ˆψ n+ j+,m + Cm j ˆψ n+ j,m + Rm j ˆψ n+ j,m = (D C m j+) ˆψ n j+,m + (D C m j ) ˆψ n j,m + (B m j R m j ) ˆψ n j,m. DERIVATION OF DISCRETE TRB 3

23 Sine-Transformation of the discrete Schrödinger equation on the exterior domains j, j J yields for the modes m =,...,K: C m j+ ˆψ n+ j+,m + Cm j ˆψ n+ j,m + Rm j ˆψ n+ j,m = (D C m j+) ˆψ n j+,m + (D C m j ) ˆψ n j,m + (B m j R m j ) ˆψ n j,m. Definition [Z - Transformation]: The Z-Transformation of a sequence (ψ n ) n N is given by Z {ψ n } = Ψ(z) := ψ n z n z C, z >. n= DERIVATION OF DISCRETE TRB 3-A

24 One can show: Z = z ˆψ j,m + zψ m j (z) ˆψn+ j,m ψ J+,k = ψ J,k = ψ J,k = for k =,...,K V j constant for j, j J C m j = C m, R m j = R m, B m j = B m Ψ J+ (z) +» R z + R B Ψ J (z) + Ψ J (z) =. C z + C D DERIVATION OF DISCRETE TRB 4

25 One can show: Z = z ˆψ j,m + zψ m j (z) ˆψn+ j,m ψ J+,k = ψ J,k = ψ J,k = for k =,...,K V j constant for j, j J C m j = C m, R m j = R m, B m j = B m Ψ J+ (z) +» R z + R B Ψ J (z) + Ψ J (z) =. C z + C D This difference equation with constant coefficients is solved by Ψ j (z) = ν j (z):» R z + R B ν (z) + ν(z) + =, C z + C D Physical background forces decay of the solution for j ν(z) > und ν(z)ψ J (z) = Ψ J (z) Z-transformed DTBC DERIVATION OF DISCRETE TRB 4-A

26 Theorem [DTBC for the D Schrödinger equation]: Discretize the D Schrödinger-Equation with the compact 9-point difference scheme in space and with the Crank-Nicolson scheme in time. Then the DTBC at x J = J x and x = for n read ˆψ n,m s (),m ˆψ n,m = ˆψ n J,m s () J,m ˆψ n J,m = n ν= n ν= s (n ν),m ˆψ ν,m a m s (n ν) J,m ˆψ ν J,m a m J ˆψ n,m, ˆψ n J,m. The convolution coefficients s (n) j,m can be calculated by s (n) j,m = αm j (λ m j ) n n [ ] P n (µ m j ) P n (µ m j ) with the Legendre-Polynomials P n ( P P ). DERIVATION OF DISCRETE TRB 5

27 Advantages of the new DTBC: no numerical reflections 3-point recursion for s (n) these DTBC have exactly the same structure like the DTBC calculated with the 5-point scheme [A. Arnold, M. Ehrhardt] convergence: O( x 4 + y 4 + t ) same numerical effort like discretized analytical TRB s (n) j,m = O(n 3/ ) CN-FD scheme with DTBC is unconditionally stable, with A := I + x D x + y D y follows: D + t ψ A = with ψ A := ψ, Aψ. () DERIVATION OF DISCRETE TRB 6

28 some drawbacks: DTBC are non-local in time high memory costs: the solution ψ n j,m x and x J for all time steps n =,,... has to be saved in in each time step n =,,... you have to calculate K convolutions of the length n (FFT not useable, O(Kn )) DERIVATION OF DISCRETE TRB 7

29 some drawbacks: DTBC are non-local in time high memory costs: the solution ψ n j,m x and x J for all time steps n =,,... has to be saved in in each time step n =,,... you have to calculate K convolutions of the length n (FFT not useable, O(Kn )) These DTBC are given in the Sine-transformed form: BC of one mode is a linear combination of all other boundary points diagonal structure of the system matrix is destroyed DERIVATION OF DISCRETE TRB 7-A

30 Sparsity pattern of the system matrix for J = K = DERIVATION OF DISCRETE TRB 8

31 Example : free D Schrödinger equation on Ω = [, ] [, ] with the initial data: + y ik x+ik y 6 x ) ( ( x y ), (x, y) Ω ψ I (x, y) = e it = it = 6 it = x y T = D ERIVATION OF DISCRETE TRB y T = 6 t x x y T = 8 t 9

32 * APPROXIMATION OF DTBC Idea: Approximate the convolution coefficients s (n) j,m exponentials: by a sum of s (n) s (n) = L l= b l q n l, n N, q l >, L 4 APPROXIMATION OF DTBC

33 * APPROXIMATION OF DTBC Idea: Approximate the convolution coefficients s (n) j,m exponentials: by a sum of s (n) s (n) = L l= b l q n l, n N, q l >, L 4 b l, q l are calculated by the Padé - Approximation of f(x) = L n= s (n) x n, x [A.Arnold, M. Ehrhardt, I. Sofronov (3)] APPROXIMATION OF DTBC -A

34 * Recursion formula for the convolution coefficients: n X t= s (n t) ψ t = LX l= c (n) l with c (n) l = q l c () l = c (n ) l + b l q l ψ n, n =,...,N APPROXIMATION OF DTBC

35 * Recursion formula for the convolution coefficients: n X t= s (n t) ψ t = LX l= c (n) l with c (n) l = q l c () l = c (n ) l + b l q l ψ n, n =,...,N Advantages: If you have calculated b l, q l once for a set x, y, t, V, you ll easily derive b l, q l for any x, y, t, V by q l = q lā b a q l b b l = b l q l aā b b (a q l b)(q l ā b) Numerical effort: O(Kn ) O(KLn) Memory: O(Kn) O(KL) + q l + q l APPROXIMATION OF DTBC -A

36 * Example : free D Schrödinger-Equation in Ω = [, ] [, ] Initial function: ψ I (x, y) = sin(πy)e ik xx 6(x ), (x, y) Ω IT=5 IT=3 IT= psi(x,y), t=t.6.4 psi(x,y), t=t.6.4 psi(x,y), t=t.6.4 L = 5 : y T = 5 t x y T = 3 t x y T = 5 t x IT=5 IT=3 IT= psi(x,y), t=t.6.4 psi(x,y), t=t.6.4 psi(x,y), t=t.6.4 L = : y T = 5 t x y T = 3 t x y T = 5 t x APPROXIMATION OF DTBC

37 SIMULATION OF QUANTUM WAVEGUIDES y 3nm V control 4.5nm Y ψ inc Γ nm Ω Γ nm X x Incoming wave at x = : ψ inc (,y, t) = sin(πy)e iet, E = 9.9meV inhomogeneous DTBC at x =, DTBC at x = X SIMULATION OF QUANTUM WAVEGUIDES 3

38 Little trick to suppress oscillations in time: ψ inc oscillates like e iet in time. E big: fast oscillation of the solution in time small time step size is necessary high numerical effort for the analysis of steady state and long-time behaviour Define ϕ(x, y, t) := e iωt ψ(x,y, t) mit ω = E. ϕ solve the modified Schrödinger equation i ϕ t = m (ϕ xx + ϕ yy ) + (V ω ) {z } =:Ṽ ϕ SIMULATION OF QUANTUM WAVEGUIDES 4

39 .4 V= V= E t [ s] f (t) = ψ (x, y, t) ψ ref (x, y, t) f (t) = ψ (x, y, t) ψ ref (x, y, t) ψ is calculated with Ṽ =, ψ with Ṽ = E and ψ ref is a numerical reference solution, which has been calculated with high accuracy. SIMULATION OF QUANTUM WAVEGUIDES 5

40 Initial function: X=, Y=.86667, dx=.6667, dy=.6667, dt=., V= Quantenwellen.m, solution after T =.7998 ps, it = Quantenwellen.m, solution after T =.998 ps, it = psi(x,y), t=.5 ψ(x,y,t).5 ψ(x,y,t) x T = t y y [nm] T = t y [nm] T = 9 t Quantenwellen.m, solution after T =. ps, it = Quantenwellen.m, solution after T =.8 ps, it = 4 Quantenwellen.m, solution after T =.9 ps, it = ψ(x,y,t).5 ψ(x,y,t).5 ψ(x,y,t) y [nm] T = t y [nm] T = 4 t y [nm] T = 95 t SIMULATION OF QUANTUM WAVEGUIDES 6

41 EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS Drawbacks of the simulations: Hard walls (zero Dirichlet BCs) and edges are not practicable in industry! Geometry of computational domain shall be realized by potentials also in the simulations. How to choose the incoming wave and the initial function then? Potentials are NOT constant in the exterior domains! Decoupling of the modes for V (x, y) const. after Sine- Transformation is not possible EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 7

42 EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS Drawbacks of the simulations: Hard walls (zero Dirichlet BCs) and edges are not practicable in industry! Geometry of computational domain shall be realized by potentials also in the simulations. How to choose the incoming wave and the initial function then? Potentials are NOT constant in the exterior domains! Decoupling of the modes for V (x, y) const. after Sine- Transformation is not possible Discrete Schrödinger equation: 5-point scheme in space, Crank-Nicolson in time: i D + t ψj,k n = `D m x + Dy ψn+ j,k + V n+ j,k ψ n+ j,k Calculate new Eigenfunctions, which take the potential into account! EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 7-A

43 Solve the eigenvalue equation on the exterior domains (j, j J): with `χn y j,k,m χ n j,k,m + χ n n+ j,k+,m + V k χ n j,k,m = Ej,mχ n n j,k,m y KX χ n j,k,m = and χ n j,,m = χ n j,k,m = k= for k, m K, n >. Transformation w.r.t. the eigenfunctions yields ˆψ n j,m = y K X k= χ n j,k,mψ n j,k, m K. i D + t ˆψ j,m n = m D ˆψ n+ x j,m + En ˆψ n+ j,m j,m same structure as the sine-transformed Schrödinger equation! [N. Ben Abdallah, M.S. (5)] EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 8

44 Example 3: D channel Potential V(,y) Eigenfunction χ for j=, χ j,k,m of the m = st mode with the Eigenvalue λ = V(,y k ) χ j,k,m y k y k potential V (y) = 4y( y) eigenfunction χ of m = initial function: incoming wave: ψ I j,k = e ik xj x χ j,k,m ψ inc j,k,n = e ik xj x χ j,k,me ie xn t with E x = cos(k x x) x EXTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS 9

45 t = 5 t =. t= ψ(x,y,t) ψ(x,y,t) ψ(x,y,t) x y.8 x T = t t = 5 t = T = 5 t y y ψ(x,y,t) ψ(x,y,t) t = 7 t =.34.5 x.6 T = 5 t.4.4 x..5 y. T = 5 t t = 5 t =.5 ψ(x,y,t) t = 5 t =.3 x.8.5 y T = 5 t E XTENSION OF THE DTBC TO MORE ARBITRARY POTENTIALS x.8 y T = 7 t 3

46 SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS Solve the Schrödinger equation (in polar coordinates) iψ t = r (rψ r) r + «r ψ θθ +V (r,θ, t)ψ, r >, θ π, t > on a circular domain Ω = [,R] [,π] with TBC at x = R. Problems: solve a second order difference equation with varying coefficients: a j Ψ J+ (z) + b j (z)ψ J (z) + c j Ψ J (z) = calculation of the convolution coefficients for the DTBC "recursion from infinity" singularity at r = not equidistant offset-grid r j = r j+ approximation of the convolution coefficients and the -sum [A.Arnold, M. Ehrhardt, M. S., I. Sofronov (6)] SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 3

47 Example 4: free Schrödinger equation on unit disc Ω = [, ] [, π] ψ I (r,θ) = αx α y e ik xr cos θ+ik y r sin θ (r cos θ) (r sin θ) αx αy initial function on Ω = [,]x[,π], r = /8, θ = π /8 solution on Ω = [,]x[,π] at time t=.5, t = r = /8, θ = π /8 solution on Ω = [,]x[,π] at time t=.5, t = r = /8, θ = π / ψ(r,θ,) ψ(r,θ,t) 4 3 ψ(r,θ,t) initial function T =.5 T =.5 initial function on Ω = [,]x[,π], r = /8, θ = π /8 solution on Ω = [,]x[,π] at time t=.5, t = r = /8, θ = π /8 solution on Ω = [,]x[,π] at time t=.5, t = r = /8, θ = π / SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 3

48 Error due to the scheme/tbc: L(ψ, ϕ, t n, Ω) := ψ(r j, θ k, t n ) ϕ(r j, θ k, t n ) Ω, ϕ(r j, θ k,t n ) Ω, ψ: numerical solution ϕ: exact solution or numerical reference solution relative L error due to the scheme J=K=64, t=/64 J=K=8, t=/8 J=K=56, t=/56 relative L error due to the TBC t 5 J=K=64, t=/64 J=K=8, t=/8 J=K=56, t=/ t SCHRÖDINGER EQUATION ON CIRCULAR DOMAINS 33