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Focus Topic: for the TDSE Part I: The -Evolution and Crank-Nicolson

Kenneth Hansen

QUSCOPE Meeting, Aarhus University, Denmark

December 17, 2015

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Kenneth Hansen December17,2015 1/44 The Time-Evolution Operator

Time evolution in QM done through an operator:

|Ψ(t)i = Uˆ (t, t0) |Ψ(t0)i This operator should fulﬁll:

Uˆ (t, t0) = Uˆ (t, t1)Uˆ (t1, t0), Uˆ (t0, t0) = I,ˆ

† −1 Uˆ (t, t0) = Uˆ (t, t0) = Uˆ (t0, t) The Time-Dependent Schr¨odinger Equation (TDSE) gives us the requirement that the time-evolution operator must satisfy: ∂ i Uˆ (t, t ) = Hˆ (t)Uˆ (t, t ) ∂t 0 0 Which can be rewritten as:

t ˆ ˆ ˆ ˆ U(t, t0) = I − i H(t1)U(t1, t0)dt1 AARHUS AU UNIVERSITY Zt0

Kenneth Hansen December17,2015 2/44 Dividing the time-step in smaller steps so tk = t0 + k∆t for k = 0, 1, ..., N and by making a sequence by self-insertion in the integral form of the time-evolution operator we can expand the time-evolution operator:

∞ ˆ (n) Uˆ (tk+1, tk ) = U (tk+1, tk ), n X=0 With

(0) Uˆ = Iˆ

(n) tk+1 t1 tn−1 ˆ n U (tk+1, tk ) = (−i) dt1 dt2 ... dtnHˆ (t1)Hˆ (t2) ... Hˆ (tn) Ztk Ztk Ztk

n = 1, 2, ...

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Kenneth Hansen December17,2015 3/44 If the number of intervals is suﬃciently large it will be a good approximation that,

Hˆ (t) = Hˆ (tk ) for tk ≤ t ≤ tk+1.

The expansion of the time-evolution operator then collapses to,

∞ ∞ (n) [(∆t)Hˆ (t )]n Uˆ (t , t ) = Uˆ (t , t ) = (−i)(n) k . k+1 k k+1 k n! n n X=0 X=0 This is per deﬁnition an exponential function expansion and can be contracted as,

Uˆ (tk+1, tk ) = exp[−i∆tHˆ (tk )].

Numerically evolving states is now performed using this exponential operator on a state. There are diﬀerent ways of doing this and we will now present some of them. AARHUS AU UNIVERSITY

Kenneth Hansen December17,2015 4/44 The Explicit Euler Method

Instead of working with Hˆ (tk ) one can improve most methods by working ˆ with H(tk+1/2). (Working at mid-points/in the mean) The ﬁrst order expansion of the exponential results in the explicit Euler scheme: i∆t |Ψ(t )i = Iˆ− Hˆ (t ) |Ψ(t )i k+1/2 2 k+1/2 k   This scheme has an error of O((∆t)2) but is extremely unstable. One could go to a higher order in the expansion an gain an error in the norm of O((∆t)3), but since it still isn’t unitary and it still doesn’t suppress non physical states (these will grow exponentially) this is not a good path to go.

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Kenneth Hansen December17,2015 5/44 The Implicit Euler Method

Instead of increasing the order of the expansion we use that:

† −1 Uˆ (tk+1, tk ) = Uˆ (tk+1, tk ) = Uˆ (tk , tk+1) = exp[i∆tHˆ (tk+1)],

and get the implicit equation:

Uˆ (tk , tk+1) |Ψ(tk+1)i = |Ψ(tk )i ⇒ i∆t −1 |Ψ(t )i = Iˆ+ Hˆ (t ) |Ψ(t )i k+1 2 k+1/2 k   The implicit method is a lot more stable as unphysical components don’t grow exponentially like they do in the explicit method. It is still an error of O((∆t)2) method and still lacks .

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Kenneth Hansen December17,2015 6/44 The Crank-Nicolson Method

The implicit method is better than the explicit method but it still isn’t unitary. Combining the implicit and explicit method into one we get the Crank-Nicolson form:

i∆t −1 i∆t |Ψ(t )i = Iˆ+ Hˆ (t ) Iˆ− Hˆ (t ) |Ψ(t )i k+1 2 k+1/2 2 k+1/2 k     † Which is unitary! (realize that Uˆ (tk+1, tk ) = Uˆ (tk+1, tk )) This is also known as Cayley’s form of the complex-exponential for time-evolution. The Crank-Nicolson scheme has an error of O((∆t)3) and is unconditionally stable.

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Kenneth Hansen December17,2015 7/44 t or n

FTCS x or j (a)

(b) Fully Implicit (c) Crank-Nicolson

AARHUS AU UNIVERSITY 0Numerical Recipes, 2. ed., page 850 Kenneth Hansen December17,2015 8/44 The Crank-Nicolson Method - Numerically

The Crank-Nicolson method is used with a grid-based representation of the wave function. Remembering the Schr¨odinger Equation in a length gauge:

∂ 1 ∂2 i Ψ(x, t) = − + Vˆ (x) + Eˆ(t)ˆx Ψ(x, t) ∂t 2 ∂x2   We then use the second-order central diﬀerence formula: ∂2 Ψ(x , t) − 2Ψ(x , t) + Ψ(x , t) Ψ(x , t) = j+1 j j−1 , ∂x2 j (∆x)2

± and get evolution matrices U (tk+1/2) whose elements are:

±i∆t/[2(∆x)2], j = j ′ + 1, j ′ − 1, ± 2 ′ Uj,j′ (tk+1/2) = 1 ∓ i∆t[1/(∆x) + V (xj ) + E(tk+1/2)xj ], j = j ,  AARHUS 0, else. AU UNIVERSITY  Kenneth Hansen December17,2015 9/44 Using the Crank-Nicolson scheme consists now in solving the linear equations with appropriate boundary conditions:

− + U (tk+1/2)ΨΨΨ(tk+1) = U (tk+1/2)ΨΨΨ(tk ).

− It is seen that U (tk+1/2) is tridiagonal and an inverse can therefore be found without much computational eﬀort.

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Kenneth Hansen December17,2015 10/44 Using the Crank-Nicolson method

Good things about this method: Accurate to the second order Unconditionally stable Unitary Can be computationally eﬃcient (O(n2)) For this to be an eﬀective method it has to be brought into a tridiagonal (or band diagonal/Toeplitz) form to simplify calculating the inverse (solving the implicit equation)

This is almost done automatically in 1D, but will require eﬀort in more dimensions!

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Kenneth Hansen December17,2015 11/44 Alternating Direction Implicit Method (ADI)

For multi-dimensional systems a split-step can be performed to simplify the problem.

i,j-1 i,j i,j+1 n+1

i-1,j

n+1/2

i,j z i+1,j

n i,j-1 y x i,j AU AARHUS i,j+1 UNIVERSITY

Kenneth Hansen December17,2015 12/44 References

A good introduction to numerical integration of wave functions: Atoms in Intense Laser Fields, C.J. Joachian, N.J. Kylstra, R.M. Potvliege, 2012 General introduction to Crank-Nicolson from computational standpoint: Numerical Recipes, W.H. Press et.al.

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Kenneth Hansen December17,2015 13/44 End of Part I Thank You!

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Kenneth Hansen December17,2015 14/44 Focus Topic: Propagators for the TDSE Part II: Split-Operator Method

Haruhide Miyagi

QUSCOPE Meeting, Aarhus University, Denmark

December 17, 2015

Please have a look at the handouts!

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Haruhide Miyagi December17,2015 15/44 Focus Topic: Propagators for the TDSE Part III: Arnoldi-Lanczos

Chuan Yu

QUSCOPE Meeting, Aarhus University, Denmark

December 17, 2015

AARHUS AU UNIVERSITY

Chuan Yu December17,2015 16/44 Arnoldi-Lanczos Propagator

Short-time propagator |Ψ(t + ∆t)i = exp[−iHˆ (t)∆t]|Ψ(t)i Exact calculation of matrix exponential scales cubically with dimension, feasible only for small systems.

Idea of Arnoldi-Lanczos method 1 Truncate Taylor expansion to some order L, and span Krylov subspace by a small set of vectors Hˆ k |Ψi , (k = 0,..., L)

L n |Ψ(t + ∆t)i ≈ (n!)−1 −iHˆ (t)∆t |Ψ(t)i . n X=0 h i 2 Find an orthonormal set of vectors |Qk i , (k = 0,..., L) of Krylov

subspace with modiﬁed Gram-Schmidt algorithm, starting from a

normalized state |Q0i = |Ψ(t)i /kΨ(t)k = |Ψ(t)i / hΨ(t)|Ψ(t)iAARHUS. AU UNIVERSITY p Chuan Yu December17,2015 17/44 Arnoldi-Lanczos Propagator (cont’d)

Idea of Arnoldi-Lanczos method (cont’d) 3 Approximate wavefunction and Hamiltonian as

L |Ψ(t + ∆t)i ≈ |Qj i hQj |Ψ(t + ∆t)i , j X=0 L L Hˆ ≈ |Qj i hQj | Hˆ |Qk i hQk | , j k X=0 X=0

where hQj | Hˆ |Qk i is matrix element of reduced Hamiltonian HL −1 4 Diagonalize HL with a similarity transformation S HLS = λ 5 c Q t t Q iHˆ t t Evaluate coeﬃcents j = h j |Ψ( + ∆ )i = h j | exp(− ∆ ) |Ψ( )i

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Chuan Yu December17,2015 18/44 Arnoldi-Lanczos Propagator (cont’d)

Final expression

L |Ψ(t + ∆t)i ≈ |Qj i hQj | exp(−iHˆ ∆t) |Ψ(t)i j X=0 L L L

≈ |Qj i hQj | |Qk i [exp (−iHL∆t)]kl hQl |Q0i kΨ(t)k j k l X=0 X=0 X=0 L L Q S i t S−1 t = | j i [ ]jk exp [− λk ∆ ] k0 kΨ( )k. j=0 k=0 X X  

Reduced Hamiltonian HL is of dimension (L + 1) × (L + 1), which is much smaller than Hamiltonian H of dimension N × N.

Krylov dimension should be chosen to ensure convergence, i.e., AARHUS AU UNIVERSITY |Ψ(t + ∆t)i should be well-described by {|Q0i , ··· , |QLi}.

Chuan Yu December17,2015 19/44 Arnoldi-Lanczos Algorithm

Starting from a normalized vector |Q0i = |Ψ(t)i / hΨ(t)|Ψ(t)i, for k = 0,..., L do p (0) ˆ |Φk+1i = H |Qk i for j = 0,..., k do (j) βj,k = hQj |Φk+1i (j+1) (j) |Φk+1 i = |Φk+1i − |Qj i βj,k end for (k+1) (k+1) (k+1) βk+1,k = kΦk+1 k = hΦk+1 |Φk+1 i (k+1) q |Qk+1i = |Φk+1 i /βk+1,k end for

gives an orthonormal basis set {|Q0i , ··· , |QLi} together with matrix elements βj,k for j ≤ k + 1, hQj | Hˆ |Qk i = 0 else. AARHUS ( AU UNIVERSITY

Chuan Yu December17,2015 20/44 Properties

General Unitary Stable Krylov dimension is related to system Hamiltonian and time step size. For small time step size, usually Krylov dimension less than 20 can give converged results. For Hamiltonian matrix of dimension N × N, computational complexity is O(N2). If Hamiltonian matrix is sparse, e.g., in ﬁnite-element discrete variable representation (FEDVR), computational complexity is O(N).

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Chuan Yu December17,2015 21/44 References

For Arnoldi-Lanczos Unitary quantum time evolution by iterative Lanczos reduction, T. J. Park and J. C. Light, J. Chem. Phys. 85, 5870 (1986) Explicit schemes for time propagating many-body wave functions, A. L. Frapiccini, A. Hamido, S. Schr¨oter, D. Pyke, F. Mota-Furtado, P. F. O’Mahony, J. Madro˜nero, J. Eiglsperger, and B. Piraux, Phys. Rev. A 89, 023418 (2014)

For FEDVR Numerical grid methods for quantum-mechanical scattering problems, T. N. Rescigno and C. W. McCurdy, Phys. Rev. A 62, 032706 (2000) Parallel solver for the time-dependent linear and nonlinear

Schr¨odinger equation, B. I. Schneider, L. A. Collins, and S. X. Hu,

Phys. Rev. E 73, 036708 (2006) AARHUS AU UNIVERSITY

Chuan Yu December17,2015 22/44 End of Part III Thank You!

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Chuan Yu December17,2015 23/44 Focus Topic: Propagators for the TDSE Part IV: Polynomial Propagators - Chebyshev (et al.)

Daniel Reich

QUSCOPE Meeting, Aarhus University, Denmark

December 17, 2015

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Daniel Reich December17,2015 24/44 Many Ways to Solve the Schr¨odinger Equation

solve TDSE ≡ compute unitary propagator Uˆ (t) = e−iHtˆ

but we learned: matrix exponentials are expensive (O N3 ) 

1 Taylor series to ﬁrst order, adjust for unitarity =⇒ Crank-Nicolson

2 split exponential into two parts for which computation is simple (Tˆ diagonal in ~k-space, Vˆ diagonal in ~r-space) =⇒ Split-Step

3 compute Taylor series to higher order but employ projected Hamiltonian on a subspace of full =⇒ Arnoldi-Lanczos

4 AARHUS ??? AU UNIVERSITY

Daniel Reich December17,2015 25/44 Polynomial Propagators

idea: expand exponential in polynomial

∞ −iHtˆ e |Ψi = cnpn −iHtˆ |Ψi n X=0   where pn is a polynomial of degree n matrix exponentiation problem (O n3 ) transformed to (repeated) matrix-vector multiplication (O n2 )  important from a numerical perspective  (O n3 vs. O n2 )   Hˆ 2 |Ψi = Hˆ · Hˆ |Ψi ˆ 2 ˆ ˆ  H |Ψi = H H |Ψi

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Daniel Reich December17,2015 26/44 Taylor Series

expand exponential in polynomial ...

... Taylor series!

n ∞ −iHtˆ ˆ e−iHt |Ψi = |Ψi  n!  n X=0

simple form, convergence radius = ∞, trivial to compute

corresponds to Arnoldi-Lanczos but without any Hilbert space reduction

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Daniel Reich December17,2015 27/44 We’re done!

n ∞ −iHtˆ ˆ e−iHt |Ψi = |Ψi  n!  n X=0 DONE!

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Daniel Reich December17,2015 28/44 We’re done?

n ∞ −iHtˆ ˆ e−iHt |Ψi = |Ψi  n!  n X=0

DONE?

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Daniel Reich December17,2015 29/44

Uniformity Matters!

many polynomial approximations are troublesome at the domain boundary (cf. Runge’s phenomenon) issue with Taylor: each order has very unequal distribution along the interval (monomials xn have decreasing contribution in the center for increasing n)

n ﬁnd a polynomial πn (x) = k=0 ck pk (x) that approximates a function f uniformly, i.e. minimise for all n P

kπn − f k∞ = max kπn (x) − f (x)k x∈[−1,1]

make sure that increasing the order leads to uniform improvement

make sure that kpk k∞ is small, i.e. added orders pk stay close around

zero everywhere AARHUS AU UNIVERSITY

Daniel Reich December17,2015 34/44 Chebyshev Optimality

Chebyshev Optimality

Let Πn be the set of monic polynomials of order n in the interval [−1, 1], n n−1 k i.e. polynomials of the form pn (x) = x + k=0 ck x . Then, the normalised Chebyshev polynomials, P 1 Tn (x) = cos (n arccos (x)) , 2n−1

1 minimise kπnk∞ among all members of Πn (with kTn (x) k∞ = 2n−1 ).

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Daniel Reich December17,2015 35/44 Chebyshev Propagator

∞ −iHtˆ e |Ψi = cnpn −iHtˆ |Ψi n X=0   domain of pn? what does pn(Hˆ ) mean anyway? ∞ −iHtˆ e |Ψi = cnpn (−iEl t) |El i hEl | Ψi n l X=0 X for Hermitian Hˆ , domain of pn should be the imaginary axis map real axis to imaginary axis =⇒ complex Chebyshev polynomials

φn (x) = Tn (−ix)

Tn ∈ [−i, i] 6= iR?, but in numerics the spectrum of Hˆ is compact! normalise Hamiltonian (spectral radius ∆, minimal eigenvalue Emin) 1 Hˆ − ∆ + Emin ˆ 2 AARHUS Hnorm = 2 AU UNIVERSITY ∆  Daniel Reich December17,2015 36/44 Chebyshev Propagator

−iHdt e |Ψi = cnφn −iHˆnorm |Ψi n X   ∆ ∆ −i( +Emin)dt cn = (2 − δn ) e 2 Jn dt 0 2   Jn (x) are the Bessel functions of ﬁrst kind ∆ ∆ for n > 2 dt : Jn 2 dt −→ 0 exponentially fast =⇒ together with φ ∈ [−1, 1] a priori knowledge of truncation n  threshold (roughly nmax ≃ 2∆dt for standard applications)

easy evaluation of φn with recursion relation

φn(Aˆ) = 2Aˆφn−1(Aˆ) − φn−2(Aˆ), φ1(Aˆ) = Aˆ, φ0(Aˆ) = 1 only one additional appication of Hamiltonian per order, coeﬃcients can be reused when propagation is continued (but watch the AU AARHUS spectral radius for TDSE solving!) UNIVERSITY

Daniel Reich December17,2015 37/44 Algorithm

[from M. H. Goerz, “Optimizing Robust Quantum Gates in Open Quantum Systems”]

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Daniel Reich December17,2015 38/44 Advantages and Disadvantages

mathematically optimal polynomial propagator exact solution of the TDSE for time-independent Hamiltonians, no further information required (except for spectral range) independent on representation of wave function, just requires knowledge of application of Hˆ simple, easy-to-implement algorithm generalisable to imaginary time propagation (Tk (x) instead of φk (x))

very sensitive to wrongly estimated spectral range (evidence: loss of norm conservation, can be caught during algorithm with some overhead) only usable for purely real (or purely imaginary) spectrum of Hˆ requires approximation of explicit time-dependence in TDSE not applicable when inhomogeneities or non-linearities in the AU AARHUS TDSE appear (e.g. Gross-Pitaevskii equation) UNIVERSITY

Daniel Reich December17,2015 39/44 You might also ﬁnd interesting ...

R. Kosloﬀ, “Time-Dependent Quantum-Mechanical Methods for Molecular Dynamics”, J. Phys. Chem. 92, 2087 (1988) (nice, but slightly dated overwiew on propagation methods in the context of atomic and molecular physics) H. Tal-Ezer, R. Kosloﬀ, “An accurate and eﬃcient scheme for propagating the time dependent Schr¨odinger equation”, JCP 81, 3967 (1984) (reference paper for the Chebyshev propagator) M. H. Goerz, “Optimizing Robust Quantum Gates in Open Quantum Systems”, Dissertation Universit¨at Kassel (2015) (concise discussion of Chebyshev, Newton and Newton-Arnoldi propagator with pseudocodes for all algorithms)

A. Gil et al., “Numerical Methods for Special Functions”, Society for Industrial and Applied Mathematics (2007) (mathematical discussion on properties of Chebyshev polynomials and why they are useful) H. Tal-Ezer, “Polynomial approximation of functions of matrices and applications”, Journal of Scientiﬁc Computing 4, 25 (1989) (mathematical discussion on

polynomial approximation method for a variety of applications)

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Daniel Reich December17,2015 40/44 Extensions

Chebychev method can also be applied to other , as long as an analytical solution can be written down and expanded in Chebychev polynomials expansion coeﬃcients are not Bessel functions anymore, they must be derived using a cosine transform

example: Chebychev propagator for the inhomogeneous TDSE [M. Ndong et al., JCP 130, 124108 (2009)]

remarkable extension: use iterative time ordering to rewrite time-dependent TDSE as time-independent Hamiltonian TDSE with inhomogeneous source term =⇒Chebyshev propagator for inhomogeneous Schr¨odinger equation

can eliminate requirement of approximating explicit time-dependence ˆ in H [M. Ndong et al., JCP 132, 064105 (2010)] AARHUS AU UNIVERSITY

Daniel Reich December17,2015 41/44 Newton Polynomials

what can we do for a truly complex spectrum, e.g. open quantum system evolutions following a Liouvillian instead of a Hamiltonian? ∂ ρ = Lρ ∂t solution still exponential (time-independent Liouvillian for simplicity) ρ (t) = eLt ρ (0)

approximate in complex plane by interpolating Newton polynomial n−1 f (z) = anRn (z) , Rn (z) = (z − zj ) n j X Y=0 with sampling points {zj } fastest convergence when complex eigenvalue of L are used for zj but: diagonalisation problem for Liouvillians is even more AU AARHUS complicated than for Hamiltonians! UNIVERSITY

Daniel Reich December17,2015 42/44 Newton-Arnoldi

solution: estimate spectral domain, encircle it with a rectangle or ellipse, calculate expansion coeﬃcients from sampling points on that boundary even better: use Arnoldi algorithm to obtain some approximate eigenvalues

further reading: G. Ashkenazi et al., “Newtonian propagation methods applied to the − photodissociation dynamics of I3 ”, JCP 103, 10005 (1995) (introduction to Newton propagator) H. Tal-Ezer, “On Restart and Error Estimation for Krylov Approximation of w = f (A) v”, SIAM J. Sci. Comput. 29, 2426 (2007) (modiﬁcation towards restarted Arnoldi-Newton propagator)

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Daniel Reich December17,2015 43/44 Thank You!

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Daniel Reich December17,2015 44/44