Numerical methods

in
Strong field dynamics II

Correlated Multielectron Dynamics in Intense Light Fields

PhD School of the INT-network CORINF – 11 - 15 June 2012

Armin Scrinzi
LMU – Computational and Plasma Physics
 Strong pulse hits an atom

Electron density of the Hydrogen atom

In a 2-cycle pulse @ 2 x 1014 W/cm2, 800nm

Complex appearance – simple basic physics

ρ
– tunnel (or so) ionization,
z – free electron in the laser field,
– dipole matrix elements

Free motion is exactly described by the

Volkov solutions
of the time-dependent Schrödinger equation

Ionization can be (reasonably well) described

Dreseden, June 11-15, 2012 / 2

by rate formulae

Dipole matrix elements are known

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from electronic structure calculations

-50 50
z
We believe we understand the physics.

Why struggle to solve the TDSE?

 Why solve the TDSE?
(Except for producing colorful pictures)

Test models:
We think we have understood – have we?

Numerical experiments:
Try, instead of thinking too hard
can be simple and inspiring

Predict:
Use models for trustworthy predictions of phenomena (not numbers)
Here we are doing physics at its best
High harmonic cutoff-energy ~ Ip + 3.2 Up
was first seen in simulation!
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Get the numbers right, get the details:

ionization rates, photo-electron spectra, high harmonics
In the end, numbers is all that matters
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Reproduce experimental findings:

The ultimate (but only ultimate) prove of correct experiment&computation
(very hard, if taken seriously!)
 The solution – t-SURFF

The time-dependent SURFace Flux method

Idea:
We know the time-evolution beyond radius Rc (“channel radius”), e.g.Volkov solutions
Integrate flux through a surface for obtaining the asymptotic solutions

Related to:
R-matrix theory (in spirit, not technically)
Spectra from auto-correlation (for time-independent Hamiltonians)

Requires:
- wave function and derivative on |r|=Rc: Ψ|Rc, ∂rΨ|Rc
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- asymptotic solution (exact or numerically “cheap”)

- perfect (or very good) absorption beyond Rc
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Can do:
- fully differential strong-field IR single photo-electron spectra any polarization
- shake-up processes in two-electron systems
- fully differential strong-field IR double photo-electron spectra
 We have understood: HHG

Time-frequency analysis of the high harmonic response

More structures
beyond naïve recollision
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Colors: complete numerical solution

Solid lines: classical recollision energies
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 Tunnel ionization - principle

Gamov factor

r0
E0
0
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Strongly non-linear (exponential !) dependence on parameters

Ammosov-Delone-Krainov (ADK) formulae

 Get the numbers right: field-ionization rates of He

Comparison of
accurate numerical data to
ADK ionization rates
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ADK works
nicely here
Factor 6 discrepancy!
 We have not (fully) understood: ionization

Laser ionization of H2

Solve the TDSE for H2

Electrons coherently leave

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from the two atoms of H2

When atomic separation matches

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½ of the phase difference Truth

(well, of TDSE)
Destructive interference, no ionization? Molecular ADK prediction

Predicted by molecular ADK formula

Thanks to A. Saenz
[Y.V.Vanne, A. Saenz PRA 82, 11404(R), (2010)]
 Numerical experiments

Two-photon double ionization of He

Basic idea:
Very large scale simulation of TDSE
Two photons
well separated in time (“long” pulse)
Ionic ground state after 1st ionization

2 well separated photo-electron energies

Two photons
at the same time (short pulse)
Both from the correlated neutral state
Both in the same energy range
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Just where exactly would we expect the [J. Feist et al., PRL 63002 (2009)]
Effect and how strongly?
 Understand experiments in detail...

Photo-electron emission and spectra

- basis for all re-scattering type experiments
High harmonic generation IR photo-electron spectra (COLTRIMS)
[Meckel et al., Science 2008]
IR photo-electron spectra

“Re-scattering imaging”

Surprisingly little is known

- Hydrogen
- Helium (to some degree)
- Single electron models
Phantastic detail – difficult interpretation!
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Experimentalists rely on ADK and its derivatives

Known to be quite wrong where we can check it:
e.g. Hydrogen atom and molecule at 800 nm, [shown by Alejandro Saenz]
 Things to come...

 Approaches to solving the TDSE

 Simplifying models: 1d and restricted dynamics
 The 2-electron problem
 The few-electron problem and beyond
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 Absorption of outgoing flux & spectra

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 The TDSE
potential
kinetic
field-interaction
Single electron, length gauge

Multi-electron

Gauge-transformation

Velocity gauge ~ laser vector potential

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field-interaction
Linear partial differential equation (parabolic)
Dimensions 1 (time) + 3n (space), n... number of particles
Simple time-dependence
 Approaches to solutions

Simplified analytic models – the Volkov propagator

Classical models – classical trajectory Monte-Carlo
Single-electron models
Models in reduced dimensions
Full fledged quantum dynamics calculations
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Classical trajectory methods
 Classical trajectory methods

Motion of free electron in the field is nearly classial

→ simulate by classical trajectories

Advantages: N2 Single-to-double
Can handle 2-e situation relatively easily
Follow the trajectories – “see what happens” ionization ratio
Disadvantages:
“Auto-ionization” of the ground state
Crucial choice of initial distributions in phase space
Difficult to assess correctness

Reproduce
Double-differential photo-electron emission patterns
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Double-to-single ratios at 800nm

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Geyer, Rost, J. Phys. B 36, L107 (2003)

Ye, Chen, Liu, PRA 77, 013403 (2008) Ye et al. 2008
Shaaran, Nygren, Faria, PRA 81, 063413 (2010) Exp. : Cornaggia 2000
Numerical solutions
 Numerical solutions

What we need to do
In mathematical terms:
Solve the initial value problem
for a (parabolic) partial differential equation

Discretization
approximately represent Ψ by a finite set of numbers ci
Turns the partial differential equation
into a set of ordinary differential equations

Time-propagation
Determine an initial state
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Solve the set of ordinary differential equations in time

Obtain Ψ(t) in terms of ci(tk) for (a set of) times tk
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Analyze
Extract the information from the ci(t)
– electron spectra, ionization rates, high harmonic responses, …
 Numerics
Grid representation of and

periodic boundary conditions

ci-1
Ψ ci-1
ci+1

x1 xi-1 xi xi+1 XN XN + 1→x1

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Finite differences rep. of -Δ

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Storage ~ 3 N Storage ~ N
Accuracy T ~ Accuracy V ~

Simple & straight forward, very limited accuracy

 Numerics
Basis representation of and

General basis φi
Full matrices

Matrix elements

Storage ~ N2, accuracy can be exponential in N !

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Rapid convergence, more elaborate setup (integrations), large storage

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 Numerics
Local basis representation of and

ξi - 1 ξi ξi + 1 ξi+2
Local basis ξi .... ....

(Example B-splines)

Band matrices

Half-bandwidth n ~ order of splines

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Storage ~ nN
Accuracy ~
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Easy setup (simple integrations), tunable accuracy

 Numerics
Local basis sets: finite elements

Element boundaries

“Infinite” Finite “Infinite”

FEM – finite elements method element elements element
pn(x) exp(+αx) pn(x) pn(x) pn(x) exp(-αx)
Piece-wise approximation
by a smooth basis:
- polynomials pn(x)
- others, e.g. pn(x) e-αx
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Can be combined with Gauss quadratures to give the

“FEM-DVR” method
finite elements in “discrete variable representation”
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Mathematically and computationally

very similar to B-splines
Possibly more flexible...
 Numerics
Time-propagation

Solve a set of ordinary (usually linear) differential equations

Formal solution

Highly symbolic notation:

T … time-ordered product!
How to obtain the exponential of the matrix?
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Make numerical sense of it:

Runge-Kutta
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Crank-Nicolson
Split-step
… and many more, more elaborate methods
 Numerics
Absorption of outgoing flux

What is the problem?

During a few-cycle pulse,
electrons can move VERY far
much further than just the quiver amplitude

either
reflection from simulation box boundaries

ρ <= bad (aka wrong) results

Dreseden, June 11-15, 2012 / 24

z
or
VERY large simulation box <= VERY costly
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or
traceless absorption of flux <= non-trivial
 Numerics
Extracting information from Ψ(x,t): excitation and HHG

Excitation
Probability of finding an excited state after the pulse is over

or

Straight forward and easy to compute

High harmonic spectra

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Time-dependent dipole

Harmonic power spectrum

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Reasonably simple, some care for density of time-grid, smooth beginning and end of field
 Numerics
Extracting information from Ψ(x,t): ionization

Fundamentally, ionization is an asymptotic concept

– well defined only at large times

Easy to compute only when we know

the stationary states of the field free system

Single ionization yield

Easy, when all bound states Cn are available:

Double ionization yield

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Harder: need to know bound and single-electron scattering states

Alternative definition
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Yield ~ probability of finding the electron(s)

outside a small box surrounding the nucleus
(simple and efficient, reasonably reliable)
 Extracting information from Ψ(x,t): photo-electron spectra
Numerics

Fundamentally, photo-electron energy is an asymptotic concept

– well defined only at very large times and distances

Generally hard
(1) Need to keep the complete wave function until the pulse is over
=> Very large simulation boxes needed
(2) Need to asymptotically analyze Ψ(x,T)

Note: we do not have the exact double-photoelectron states!

Ways out:
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(1) Analyze only asymptotic part of wave function using free-electron states
=> need to wait even longer until also slow electrons come there
Long after the end of the pulse
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=> need an even larger box

(2) Keep track of current through a surface, perfectly absorb beyond that surface
=> need to wait long, but at least can keep the box small
 Numerics
It's no mystery: try it in Python...

Your choice of the discretization

– for simplicity start from finite differences
Choose a grid: (x0,x1,...,xN)

Set up the matrices T, V, X, H=T+V

Solve the eigenproblem for an initial state:

“C = eig(H)” (eig is a standard SciPy eigensolver)

Choose a laser pulse:

e.g. “function f = field(t)
f = cos(ωt) cos(π t/T)”

Define the derivative

“function d = hamiltonian(c,t)
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D = ( T + V + field(t)*X ) * c”

Time-propagate from t = -T to t = T
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function “ode()” [solver for moderately stiff ODEs]

Analyze: Python:
Determine ionization It's free!
High harmonic response It's a programming language
Excited state populations It's great!
Numerical approaches:
More than 1 electron
 Scaling of the computational effort

N ...number of discretization points ~ phase space covered by the solution

~ Pm a x x Xm a x x Pulse duration

Electron in the laser field:

Pm a x~ Em a x x λl a s e r and Xm a x ~ Em a x x (λl a s e r)2

Pulse durations grow ~ λlaser

Total effort
~ (λlaser)4 (linear polarization)
~ (λlaser)7 (general polarization)
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Number of particles: N grows exponentially !

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Limitations Long wave length

Several “active” electrons
Very strong pulses
 The curse of dimensions

The quantum wave function Ψ

(tells us everything we can know about the system)
Ψ(x1,y1,z1,x2,y2,z2,...,xn,yn,zn) n...number of electrons

n = 1 – one electron (3 dimensions) 1003 points store on a

100 points in each dimension 8 MB memory stick

1006 points hard disk

n = 2 – two electrons (6 dimensions)
8 TB (rather large)
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1009 points e.g. 109 DVDs

n = 3 – three electrons (9 dimensions)
8 ExaB (4000 metric tons)
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The “curse of dimensionality” strikes early on!

 Resources needed

1d, single electron problem – try run it on your smartphone

3d problem, cylindrical symmetry – laptop, desktop (minutes – hours)

3d problem, no symmetry: good workstation, small cluster (hours – days)

Larger (2 or more electrons): really large computers

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Simplifying models
 Lower-dimensional models

“1d-Helium”

Adjust a and b to obtain spectra similar to 3d Helium

NOTE: a=1/2 give the exact He+ ground state energy E0= -2 (atomic units)

Easy to program, fast solving:

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E.g. x1,x2 – grid of 1000 x 1000 ~ 16 MB for the wave function

FFT and split step very efficient for time propagation
Very nice toy!
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 Lower-dimensional models: restricted dynamics

Restrict dynamics A. Becker & group

e.g. only center-of-mass motion of two electrons

r R→Z

Jacobi coordinates...
Probability distribution
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Investigated XUV + IR ionization

Chen et al., PRA 82, 033626 (2010)
…
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XUV + IR IR only
But...
 A note of caution for lower-dimensional models

Ionization of molecules
3-d calculations vs. 1-d calculations
(solid lines vs. dashed lines)
With increasing size
2 – 4 – 6 atoms
2 – 4 – 6 electrons

A simple process!
Shouldn't 1-d models give us
a feeling for what happens?
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1-d models can be qualitatively wrong

Simple models can be inspiring, but do not base conclusions on them!

 Single active electron approximation

Not very precisely defined: somehow “freeze” all electrons except for one
[since 1987: Kulander, Schafer,...]
Build a potential Veff that simulates the effect of the frozen electrons

“invent” a good Veff: match excitation energies, scattering properties...

Slater determinant, n-1 fixed functions, let one be time-dependent

Use exact Hamiltonian

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Ansatz: Ψ(r1,r2,...,rn;t) = det[Φ1(r1;t) Φ2(r2) ... Φn(rn)]

Only least bound orbital evolves in time Field free HF orbitals, frozen
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Note: gauge question arises!

Neither is exact, errors of both are difficult to estimate

Make the simplest choice for your purpose
 Non-linear approximations to the TDSE

As the wave function evolves, different discretizations would be optimal

Some time t0... ...some later time t1

Adjust the discretization to the evolution of the solution

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The feed-back introduces non-linearity into the equations

The current solution acts back on its representation
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→ NONLINEAR EQUATIONS
Examples:
Adaptive meshes
Time-dependent Hartree-Fock
The two-electron problem
 He in linearly polarized fields – fixed discretization

5-dimensional problem, just at the limit where full discretizations are still possible

Combine spherical harmonics with grids on r1 and r2

Finite-differences grid
[K. Taylor & group]

Double-ionization photo-electron spectra

at wavelength ~ 400 nm
one result @ 800 nm

FEM-DVR grid + split-step propagator

[J. Feist et al. PRA 77, 43420 (2008)]
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mostly at shorter wave length

numerically very solid results
Pazourek et al.
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Two-photon double-ionization

Very large computer resources

Tackling the multi-electron
problem
 Multi-electron systems

Isn't the single-electron picture sufficient?

NOT for
Inherently two- or more-electron processes:
Auger processes in the field
Double ionization

Severe “perturbations” by multi-electron effects

As in molecular orbital tomography

Getting the numbers right

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 Approaches that work:
CI and R-matrix

CI – configuration interaction
Few electron dynamics

[A. Saenz & group]:

Get a basis from a dedicated CI calculation
Select by physical insight, energy range
Expand time-dependent problem in this basis
Very efficient, very precise in some parameter ranges
Possible limitations:
Continuum difficult to include
Plays an important role in strong-field polarization

R-matrix Floquet and time-dependent R-matrix

[Burke, Joachain, H.van der Hart]

Exploit existing structure codes to compute inner region

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R-matrix:
match value and derivative at the boundary
Between outer region scattering solutions to inner region
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Question of different gauges:

inner region must be length gauge
outer region is more efficient in velocity gauge
Possibile limitations:
Similar as CI
 Continuum states and polarization

Bound/unbound is an asymptotic concept

Few electron dynamics

Spectral distinction between bound and unbound non-bound

only for time-independent Hamiltonians |field|

In CI, continuum states are excluded

from the core → play a role in polarization!

Time (a.u.)
Conceptual question: Contents of field-free non-bound states
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Can we look into the laser cycle? during ionization of H (length gauge)

How to define an ionization rate during the cycle?? “Deep tunneling regime”:
Keldysh γ ~ 0.3 !
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There is no “adiabatic” Hamiltonian

to distinguish bound-continuum during the pulse
 Approaches that work:
Very large scale full discretizations

Carry the multi-surface Born-Oppenheimer picture to the extreme (F. Martin & group)

Use a very large number of ER(i)

including: continuum states
doubly excited states
relevant couplings

[F. Martin et al., Science 315, 629 (2007)]

Obtain ΦR(i) by
(B-spline) discretization on a very large box
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Advantages:
best results results on laser-dissociation of H2
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Disadvantages:
Limited to H2
Enormous computer needs

Very large computer resources

 Approaches that do not work: TDHF and TDDFT
Few electron dynamics

(Time-dependent) Hartree-Fock method – TDHF

Approximate the electron wave function by
a single determinant of inter-dependent three-dimensional orbitals

Ψ(r1,r2,...,rn;t) = det[Φ1(r1;t) Φ2(r2;t) ... Φn(rn;t)]

Orbitals Φ are optimal for each time t

We hope/assume that the determinant ansatz is suitable
+ compact representation of the wave function (some 10 MB)
+ straight forward extraction of experimental observables
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- large computational effort (much larger than TDDFT)

- no checks of accuracy
- fails for strong time-dependent interactions
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Time-dependent density functional theory - TDDFT

suffers from similar problems, possibly more severely
 Why time-dependet Hartree-Fock fails
Few electron dynamics

A two-electron system Symbolic Quantum wave

in various states representation Ψ(r1,r2)
≈

Superposition state
Initial state (a configuration) Aab Φa(r1)Φb(r2)

Partial ionization +

Ionized state (another configuration) AcdΦc(r1)Φd(r2)

+
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Another state (more configurations) Aef Φe(r1)Φf(r2)

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+
.
Yet another state.... .
.
 A non-linear method: MCTDHF
Few electron dynamics

Solve the time-dependent Schrödinger equation for a few particles

Multi-Configuration Time-dependent Hartree -Fock

Hartree: Few -particle wave function ≈ Product of single particle wave function

Multi-Configuration: use linear combination of products

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Time-dependent: single-particle functions and

linear coefficients depend on time

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[Review of MCTDH: Beck et al. Physics Reports 324, 1 (2000)]

Fock: restrict to anti-symmetric:

-
 System of non-linear PDEs
Few electron dynamics

arbitrary self-adjoint single-particle op.

“Gauge fixing” fixes non-uniqueness of the ansatz

Hamiltonian with two-body interactions Projector

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Single particle density matrix Two particle density matrix

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Mean field operators

 Example
HHG – a dynamic multi-electron process
Few electron dynamics

HHG for the following system:


Diatomic, homonuclear molecule
Internuclear distance R = 2.8 a.u.
3d with cylindrical symmetry 4 active electrons
ungerade outer orbital

Screened Coulomb potential
Ip = 0.58 a.u. (N2)

Laser pulse:
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800 nm, 1 cycle, 3 x 1014 W/cm2

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[G. Jordan et al., NJP 10, 025035 (2008)]

 Do single active electron models work?
Few electron dynamics

MCTDHF

SFA: HOMO + plane waves

Effective 1e Hamiltonian
Heff = |0>E0<0| + (1-|0><0|)Hm(1-|0><0|)

Multi-electron corrections to the

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transition dipole [Patchkovskii et. al]

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(Known) single active electron models do not work.

 More than one electron moves!
Few electron dynamics

Factor MCTDHF wave function

into (core) x (single electron)

ΨMCTDHF~Ψfactor = Φcoreφe
choose Φcore
determine φe
Static core:
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Φcore = ion without field

Dynamic core:
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Φcore(t) = ion in the field

[G. Jordan et al., NJP 10, 025035 (2008)]

 Why does it matter?
Few electron dynamics

If we want to learn from harmonics about electronic structure and dynamics...

Molecular orbital tomography

Idea: extract electronic structure from high harmonics
measured under different polarization directions...

2σg HOMO orbital of N2 ?

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Do we have “single electron” [Itatani, Nature 432, 867 (2004)]

Information in the harmonics?
 Electron energy spectrum: 2Up

Assume release at time t0

Momentum at t0 ~ 0
Acceleration by the laser field

=0
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Maximal acceleration = ± A0 = ± ε0/ω

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Maximal electron energy ~ 2Up = 2 ε02/4ω2

“direct electrons”
 The 10 Up cutoff

Acceleration of the electron

from t0 until t1 -

from t1 until end of pulse -

final momentum without scattering A(t0)

final momentum if it
reverses momentum at t1
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maximal momentum
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numerically determine pmax

=>
Emax = 10.00.. Up
Absorption of outgoing flux
Photo-electron spectra
 Numerics
Absorption of outgoing flux

What is the problem?

During a few-cycle pulse,
electrons can move VERY far
much further than just the quiver amplitude

either
reflection from simulation box boundaries

ρ <= bad (aka wrong) results

Dreseden, June 11-15, 2012 / 57

z
or
VERY large simulation box <= VERY costly
CORINF PhD school

or
traceless absorption of flux <= non-trivial
 Numerics
Methods for absorption

Absorption by mask
After each time step, multiply the wave function that comes close to the boundary
By some function that goes gently to 0 towards the box boundaries

M(r)
1

Straight forward, but crude Box boundary

Produces artefacts

Complex absorbing potentials

Dreseden, June 11-15, 2012 / 58

Add a potential with a (negative) imaginary part near the box boundary
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Closer analysis shows that this is the differential version of an absorption by mask

Straight forward, less crude

Produces artefacts
 A perfect absorber
irECS
infinite-range Exterior Complex Scaling
[A.S., Phys. Rev. A81, 53845 (2010)]

Triggered by COST STMS visit to F. Martin & group

 Exterior complex scaling

Hamiltonian is analytic only beyond R0

complex scale only

Im r
beyond a finite distance R0
Re r
θ
r -> r for r < R0
r -> R0 + eiθ (r-R0) for r > R0 0 R0 r

Outgoing wave boundary conditions

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exp( ikx ) → exp (i cos θ kx – sin θ kx)

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Outoing waves k > 0 → exponentially decaying functions, L2 integrable

Ingoing waves k < 0 → exponentially exploding functions, not in L2
 Implementation of exterior complex scaling

Important technical complication

Bra and ket functions are not from the same space!!!

Exterior scaled Laplacian ΔR0,Θ

R0,Θ
is defined on discontinuous functions

Discontinuity arises because we need to keep the norm constant

Discontinuity is reversed for the left hand functions
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Matrix elements of ΔR0,Θ

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are computed by piece-wise integration [0,R0] + [R0,∞)

Conditions easy to implement with a local basis set

 IrECS – infinite range exterior complex scaling
[A.S., Phys. Rev. A81, 53845 (2010)]

Exterior complex scaling

Im r
r -> r for r < R0
r -> R0 + eiθ (r-R0) for r > R0 Re r θ
0 R0 r
Usually done with a finite discretization box...

Example:
order 2 finite elements

R0
Infinite range exterior complex scaling
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Use an infinite size last element [R0,∞)

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Example:
order 2 infinite last element

R0
In practice: 20 ~ 30 functions on last element
Does it work?
 Exterior complex scaling for the TDSE

Solve

Hope that

ΨθR0 (x)= Ψ(x) for |x| < R0

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 Does it work?

Atom in a strong dipole field

Pulse parameters: 2 x 1014 W/cm2, 760 nm, 5 opt. cycl. FWHM

Massive ionization:
At the end of the pulse ~ 80% of the probability is outside [-40,40]

log 10(error)
0
Accuracy of the solution (1-dimensional)
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-8
Relative error

|ΨθR0 (x) – Ψ(x)| / |ΨθR0 (x) + Ψ(x)|

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Inside box ~ 10-7

Works very nicely!

[Phys. Rev. A81, 53845 (2010)]
 irECS is a perfect absorber
Errors are determined by the accuracy of the discretization
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Mathematically speaking there are NO errors introduced by ECS itself

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ch line +++ punch line +++ punch line +++ punch line +++ punch line +++ punch line +++ pun
 How efficient is irECS?
How many discretization points do we need for perfect absorption?
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 How many discretization points are needed?

Error measure: ||ΨθR0 (x) – Ψ(x)||2x<R0 / ||Ψ(x)||2x<R0

Points in unscaled region 160

Points for absorption

MA = 10

MA=20
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20 Absorption points are perfect!

Quite insensitive to scaling angle θ and exponential exp(-αx)

[Phys. Rev. A81, 53845 (2010)]

 Compare to complex absorbing potentials (CAPs)

Errors of ECS and CAPs with identical discretizations

Dreseden, June 11-15, 2012 / 69

Number of points Accuracy

CORINF PhD school

[Phys. Rev. A81, 53845 (2010)]

 In 3 dimensions

Works just the same way and equally robust and efficient
Perfect high harmonic responses for H and model Ne
Very stable wave functions when varying scaling parameters R0 and θ

High harmonic spectrum

of the Hydrogen atom
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CORINF PhD school

Relative error
 Can we hope to backscale the irECS solution?

Observation:
Wave function is not just absorbed in the scaled region
It returns from the exterior scaled region without distortions

~ 1% probability returns
from scaled region |x|>5
into unscaled region |x|<5 Probability in [-5,5]

Wave-function in |x|<5
accuracy < 10-11 at all times
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CORINF PhD school

Dynamics is correctly encoded also in the scaled region!

Yes, we can hope... in practice: neither easy nor very important

Computation of photo-electron spectra
t-SURFF
Time-dependent surface flux method
[L. Tao and A.S., New. J. Phys. 14, 013021 (2012)]
 Spectra from a finite range wave function

Scattering spectra = asymptotic information by definition

a rd
is c
d
b/
sor
Ab
Finite range Asymptotic part

Problem:
Dreseden, June 11-15, 2012 / 73

If we solve only on a finite range, exactly the asymptotic information is missing

Solution
CORINF PhD school

Continue beyond the box using some known solution - Volkov

[Caillat et al,, Rev. A 71 , 012712 (2005)]

[L. Tao and A.S., New. J. Phys. 14, 013021 (2012)]
 How we usually calculate spectra from TDSE

Get Ψ(r,t) at the end of the pulse t=T: Ψ(r,T) ← problem 1

Scattering solution ψk ← problem 2

With asymptotics

Spectrally analyze Ψ(x,T)

Dreseden, June 11-15, 2012 / 74

Spectral density
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 Solve problem 2: only use asymptotic states

Choose distance Rc where we (approximately) know the scattering solution

Propagate until large T where bound Ψb and scattering Ψs parts separate

Ψb Rc Ψs
Dreseden, June 11-15, 2012 / 75
CORINF PhD school

Only need χk
asymptotic scattering solution

Solves problem 2, but worsens problem 1: even larger T needed

 Solve problems 1 & 2:
t-SURFF – time-dependent surface flux method

Same Rc as before
Asympotitic “comparison Hamiltonian” Hc(t)

with known solutions χk

E.g. with χk ...Volkov solutions

Convert volume-integral → time-integral + surface-integral

Dreseden, June 11-15, 2012 / 76

TDSE for χc and Ψs

CORINF PhD school

Hc ≡ H for r > Rc

Commutator depends only on Ψ(Rc,t)!!!

 Commutator in terms of the angular basis

Φ(k,t)..Volkov phase
Ylm ...spherical harmonics
Zln ...angular expansion of Ψ
jl ...spherical Bessel functions at |r|=Rc
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cln ...radial expansion of Ψ at |r|=Rc

dln ...radial expansion of ∂rΨ at |r|=Rc
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Note: Bln, Cln, Dln are time-independent

 Photo-electron spectra – short range potential

Truncated Coulomb potential

10 optical cyles, 800 nm, 2 x 1014 W/cm2

Dreseden, June 11-15, 2012 / 78
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Errors for 90 discretization points relative to a well-converged calculation

 Total box must accommodate quiver radius

Box 30

75 pts
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100 pts Box 40

CORINF PhD school

Quiver radius ~ 23 a.u.

Quiver radius ~ 36 a.u.
Potential range R= R0 = 15 120 radial points
Box size ~ 25 a.u.
Box size ~ 40 a.u. (black line)
Radial points 75 (black line)
~ 30 a.u. (red line)
100 (red line)
 Hydrogen atom

Infinite range of the Coulomb potential

→ Volkov solutions never become exact
Choose larger Rc

Short range R = Rc = R0 =20

Coulomb Rc > 100

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CORINF PhD school

Note: Rc can be reduced with significant extra effort

– maybe useful for multi-electron
CORINF PhD school
Dreseden, June 11-15, 2012 / 81
Spectra: 800nm, 1014W/cm2, 40 opt. cycl.
 Photoelectron spectra
IR ionization

Photo-electron momentum density

If you like glossy pictures... pz

Photo-electron momentum distribution

(10Up)1/2 “rescatterd electrons”

20 optical cycles FWHM
800 nm @ 2 x 1014 W/cm2

Fastest electrons move to ~ 4000 au!

electrons”
(2Up)1/2
“direct
Simulation box size 30 au !

- All visible structures

are accurate to ~ 1% in density
Dreseden, June 11-15, 2012 / 82

(mostly much better)

- Dynamical range is ~ 108
CORINF PhD school

[Work by Liang Tao] py

Note: Coulomb causes trouble, requires larger boxes ~ 100 au

CORINF PhD school
Dreseden, June 11-15, 2012 / 83
Angle-resolved spectra
 General polarization

Problem grows from 2 to 3 dimensions

But only for |r| < Rc!

For I = 2 x 1014 W/cm2 @ 800 nm
Needs lmax=30, i.e. ~ 900 angular functions and ~ 90 radial functions
→ Workstation size problem

Additional term in the commutator:

Dreseden, June 11-15, 2012 / 84
CORINF PhD school

In the process of implementation...

 Two-electron systems
Multi-channel single ionization
Shake-up
[A. S., submitted to New. J. Phys., arXiv:1201.3590v1]
 Defining surfaces...

Split two-electron coordinate space

B... |r1|,|r2| < Rc “bound” region

Numerical solutions on r1 and r2

S... |r2| < Rc, |r2| > Rc “singly asympotic” region

Numerical ionic solution on r2: Φc(r2,t)
Volkov solution on r1

D... |r1|,|r2| > Rc “doubly asymptotic” region

Volkov solutions on r1 and r2
Dreseden, June 11-15, 2012 / 86

Simplest case: neglect double ionization

Channel solution
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Ionic TDSE
 Multi-channel case

E.g. different ionic states channels

Play the same trick as before

“Channel asymptotic functions” χc,k

Channel spectral density

Channel asymptotic TDSE

Dreseden, June 11-15, 2012 / 87

Channel Hamiltonian
CORINF PhD school

… further the same reasoning as before

 Numerical demonstration: 2 x 1d model

M(x) … mask function turning off all potentials for |rα|> Cp

→ Volkov solution exact beyond Cp
Neutral energy: -2.88 a.u.
Ion energy: -2.0 a.u.
“helium”

Shake-up photo-electron spectra

Single cycle, 2 x 1014W/cm2 @800 nm
Dreseden, June 11-15, 2012 / 88

Black lines:
Channel method,
double ionization neglected
CORINF PhD school

Color:
including double ionization
(discussed below)
Two-electron systems
Double ionization

[A. S., submitted to New. J. Phys., arXiv:1201.3590v1]

 Extending t-SURFF to double-ionization

Do not neglect flux S → D

For Ψ and ∂rΨ on interface S|D
on area S
expand into Volkov χk(r1,t) x basis ξn(r2,t)

Equation for the expansion coefficients b(k1,n,t)

…ionic time-evolution
Dreseden, June 11-15, 2012 / 90

…flux B → S
CORINF PhD school

For each k1, solve one ionic problem in |r2|<Rc !

Integrate flux S → D by t-SURFF

Spectrum of k2 for each k1
 Numerical demonstration for 2 x 1d model

cos2 – pulse @ 800nm, I=2 x 1014 W/cm2, FWHM = 1,2,3 cycles

Discretization size
49 points on [0,∞)
total of 97 x 97 points
Dreseden, June 11-15, 2012 / 91
CORINF PhD school
 Coulomb potential: importance of long-range

M(x) restricts the potential to |rα|<Cp

Double ionization Single ionization
E1=E2 1st excited channel

Double ionization Double ionization

E1=3 eV E1=13 eV
Dreseden, June 11-15, 2012 / 92
CORINF PhD school

Satisfactory approximation at Cp ≈ 80 a.u.

 Scaling to 2 x 3 dimensions

In 2 x 1d two-electron calculations
→ 2 directions -x and +x
→ 50 linear coefficients for x in [0,∞)

Discretization size: 2 x 2 x 50 x 50 = 104

Single Intel i5 core @ 2.6GHz ~ 2 CPU hours per pulse

From 3d single electron calculations

Discretization at laser parameters 800nm, I=2 x 1014 W/cm2
→ Lmax ≈ 30 angular momenta
→ 100 ~ 200 linear coefficients for r in [0,∞) (Coulomb potential!)

Discretization size: 30 x 30 x 30 x 100 x 100 ≈ 3 x 108

Dreseden, June 11-15, 2012 / 93

Resource estimate for accurate fully differential 2-electron spectra

CORINF PhD school

6 x 104 CPU hours

~ 2 days on 1000 CPUs

Not small... but rather high accuracy standards
 Summary & outlook

Prospects are that

t-SURFF completely solves IR single and double ionization

- single-electron spectra in linear polarization

- single-electron spectra in general polarization (in preparation)

- shake-up and double ionization in 2x1d (demonstrated)

- 3d double-ionization @ 800 nm: large but feasible problem

Dreseden, June 11-15, 2012 / 94

~ Thanks ~
CORINF PhD school

[A.S., Phys. Rev. A81, 53845 (2010)]

[L. Tao and A.S., New. J. Phys. 14, 013021 (2012)]
[A. S., submitted to New. J. Phys., arXiv:1201.3590v1]