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 AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 1 / 44
The Time-Evolution Operator Time evolution in QM done through an operator: | Ψ( t ) � = ˆ U ( t , t 0 ) | Ψ( t 0 ) � This operator should fulfill: U ( t , t 0 ) = ˆ ˆ U ( t , t 1 )ˆ U ( t 0 , t 0 ) = ˆ ˆ U ( t 1 , t 0 ) , I, U † ( t , t 0 ) = ˆ ˆ U − 1 ( t , t 0 ) = ˆ U ( t 0 , t ) The Time-Dependent Schr¨ odinger Equation (TDSE) gives us the requirement that the time-evolution operator must satisfy: i ∂ U ( t , t 0 ) = ˆ ˆ H ( t )ˆ U ( t , t 0 ) ∂ t Which can be rewritten as: � t U ( t , t 0 ) = ˆ ˆ H ( t 1 )ˆ ˆ I − i U ( t 1 , t 0 )d t 1 AU AARHUS UNIVERSITY t 0 Kenneth Hansen December 17, 2015 2 / 44
Dividing the time-step in smaller steps so t k = t 0 + 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: ∞ ( n ) ˆ ˆ � U ( t k +1 , t k ) = U ( t k +1 , t k ) , n =0 With (0) ˆ = ˆ U I � t k +1 � t 1 � t n − 1 ( n ) ˆ d t n ˆ H ( t 1 )ˆ H ( t 2 ) . . . ˆ ( t k +1 , t k ) = ( − i ) n U d t 1 d t 2 . . . H ( t n ) t k t k t k n = 1 , 2 , ... AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 3 / 44
If the number of intervals is sufficiently large it will be a good approximation that, H ( t ) = ˆ ˆ H ( t k ) for t k ≤ t ≤ t k +1 . The expansion of the time-evolution operator then collapses to, ∞ ∞ ( − i ) ( n ) [(∆ t )ˆ H ( t k )] n ( n ) ˆ ˆ � � U ( t k +1 , t k ) = U ( t k +1 , t k ) = . n ! n =0 n =0 This is per definition an exponential function expansion and can be contracted as, U ( t k +1 , t k ) = exp[ − i ∆ t ˆ ˆ H ( t k )] . 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. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 4 / 44
The Explicit Euler Method Instead of working with ˆ H ( t k ) one can improve most methods by working with ˆ H ( t k +1 / 2 ). (Working at mid-points/in the mean) The first order expansion of the exponential results in the explicit Euler scheme: � I − i ∆ t � ˆ ˆ | Ψ( t k +1 / 2 ) � = H ( t k +1 / 2 ) | Ψ( t k ) � 2 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. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 5 / 44
The Implicit Euler Method Instead of increasing the order of the expansion we use that: U † ( t k +1 , t k ) = ˆ ˆ U − 1 ( t k +1 , t k ) = ˆ U ( t k , t k +1 ) = exp[ i ∆ t ˆ H ( t k +1 )] , and get the implicit equation: ˆ U ( t k , t k +1 ) | Ψ( t k +1 ) � = | Ψ( t k ) � ⇒ � − 1 � I + i ∆ t ˆ ˆ | Ψ( t k +1 ) � = H ( t k +1 / 2 ) | Ψ( t k ) � 2 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. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 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: � − 1 � � I + i ∆ t I − i ∆ t � ˆ ˆ ˆ ˆ | Ψ( t k +1 ) � = H ( t k +1 / 2 ) H ( t k +1 / 2 ) | Ψ( t k ) � 2 2 Which is unitary! (realize that ˆ U † ( t k +1 , t k ) = ˆ U ( t k +1 , t k )) 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. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 7 / 44
t or n FTCS x or j (a) (b) Fully Implicit (c) Crank-Nicolson AU AARHUS UNIVERSITY 0 Numerical Recipes, 2. ed., page 850 Kenneth Hansen December 17, 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: ∂ 2 � � i ∂ − 1 ∂ x 2 + ˆ V ( x ) + ˆ ∂ t Ψ( x , t ) = E ( t )ˆ x Ψ( x , t ) 2 We then use the second-order central difference formula: ∂ 2 ∂ x 2 Ψ( x j , t ) = Ψ( x j +1 , t ) − 2Ψ( x j , t ) + Ψ( x j − 1 , t ) , (∆ x ) 2 and get evolution matrices U ± ( t k +1 / 2 ) whose elements are: j = j ′ + 1 , j ′ − 1 , ± i ∆ t / [2(∆ x ) 2 ] , 1 ∓ i ∆ t [1 / (∆ x ) 2 + V ( x j ) + E ( t k +1 / 2 ) x j ] , U ± j , j ′ ( t k +1 / 2 ) = j = j ′ , 0 , else . AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 9 / 44
Using the Crank-Nicolson scheme consists now in solving the linear equations with appropriate boundary conditions: U − ( t k +1 / 2 )Ψ Ψ( t k +1 ) = U + ( t k +1 / 2 )Ψ Ψ Ψ Ψ( t k ) . It is seen that U − ( t k +1 / 2 ) is tridiagonal and an inverse can therefore be found without much computational effort. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 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 ( n 2 )) 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! AU AARHUS UNIVERSITY Kenneth Hansen December 17, 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 i,j x AU AARHUS UNIVERSITY i,j+1 Kenneth Hansen December 17, 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. AU AARHUS UNIVERSITY Kenneth Hansen December 17, 2015 13 / 44
End of Part I Thank You! AU AARHUS UNIVERSITY Kenneth Hansen December 17, 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! AU AARHUS UNIVERSITY Haruhide Miyagi December 17, 2015 15 / 44
Focus Topic: Propagators for the TDSE Part III: Arnoldi-Lanczos Propagator Chuan Yu QUSCOPE Meeting, Aarhus University, Denmark December 17, 2015 AU AARHUS UNIVERSITY Chuan Yu December 17, 2015 16 / 44
Arnoldi-Lanczos Propagator Short-time propagator | Ψ( t + ∆ t ) � = exp[ − i ˆ H ( t ) ∆ t ] | Ψ( t ) � 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 H k | Ψ � , ( k = 0 , . . . , L ) by a small set of vectors ˆ L � n ( n !) − 1 � − i ˆ � | Ψ( t + ∆ t ) � ≈ H ( t ) ∆ t | Ψ( t ) � . n =0 2 Find an orthonormal set of vectors | Q k � , ( k = 0 , . . . , L ) of Krylov subspace with modified Gram-Schmidt algorithm, starting from a � normalized state | Q 0 � = | Ψ( t ) � / � Ψ( t ) � = | Ψ( t ) � / � Ψ( t ) | Ψ( t ) � . AU AARHUS UNIVERSITY Chuan Yu December 17, 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 ) � ≈ | Q j � � Q j | Ψ( t + ∆ t ) � , j =0 L L ˆ � � | Q j � � Q j | ˆ H ≈ H | Q k � � Q k | , j =0 k =0 where � Q j | ˆ H | Q k � is matrix element of reduced Hamiltonian H L 4 Diagonalize H L with a similarity transformation S − 1 H L S = λ 5 Evaluate coefficents c j = � Q j | Ψ( t + ∆ t ) � = � Q j | exp( − i ˆ H ∆ t ) | Ψ( t ) � AU AARHUS UNIVERSITY Chuan Yu December 17, 2015 18 / 44
Arnoldi-Lanczos Propagator (cont’d) Final expression L � | Q j � � Q j | exp( − i ˆ | Ψ( t + ∆ t ) � ≈ H ∆ t ) | Ψ( t ) � j =0 L L L � � � ≈ | Q j � � Q j | | Q k � [exp ( − i H L ∆ t )] kl � Q l | Q 0 � � Ψ( t ) � j =0 k =0 l =0 L L � � S − 1 � � = | Q j � [ S ] jk exp [ − i λ k ∆ t ] k 0 � Ψ( t ) � . j =0 k =0 Reduced Hamiltonian H L 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., AU AARHUS | Ψ( t + ∆ t ) � should be well-described by {| Q 0 � , · · · , | Q L �} . UNIVERSITY Chuan Yu December 17, 2015 19 / 44
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