Focus Topic: Propagators for the TDSE Part I: The Time-Evolution Operator 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 unitarity.
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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 Propagator
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
AARHUS AU UNIVERSITY
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 Hilbert space =⇒ 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 equations of motion, 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