tSURFF - Photo-Electron Emission tSURFF - Photo-Electron Emission - PowerPoint PPT Presentation

tSURFF - Photo-Electron Emission tSURFF - Photo-Electron Emission from from One-, Two- and Few-Electron Systems One-, Two- and Few-Electron Systems Armin Scrinzi Armin Scrinzi Ludwig Maximilians University, Munich Ludwig Maximilians University, Munich Quantum Dynamics - Theory Quantum Dynamics - Theory Hamburg, March 24-26, 2014 Hamburg, March 24-26, 2014 Munich Advanced Photonics Vienna Computational Materials Science Marie Curie ITN Excellence Cluster FWF Special Research Program

A wealth of measured photo-electron spectra... A wealth of measured photo-electron spectra... in “attosecond physics”: RABITT spectrogram RABITT spectrogram Few-cycle IR pulse Few-cycle IR pulse Complete diagnosis Direct image of a laser pulse of sub-fs pulses Double emission Double emission Emission from surface Emission from surface Quantum Dynamics, Hamburg s. 2 from conduction band March 24 – 26, 2014 from 4f states Details of molecular Correlation electronic structure Miniscule (10 as) time differences

...but modelling and calculation are hard ...but modelling and calculation are hard Two main difficulties: Single electron density Single electron density (1) solution covers large (phase) space during 2.6 fs during 2.6 fs (2) complexity of few-electron calculations Large box sizes due to ionization: Large box sizes due to ionization: irECS perfect absorption method irECS perfect absorption method 2.6 fs Spectra from a truncated calcuation: Spectra from a truncated calcuation: Quantum Dynamics, Hamburg tSurff time dependent surface flux tSurff time dependent surface flux s. 3 160 Bohr ! March 24 – 26, 2014 Complexity of quantum chemistry wave function: Complexity of quantum chemistry wave function: → Integrate quantum chemisty with strong field dynamics Integrate quantum chemisty with strong field dynamics →

One-electron systems One-electron systems One-electron systems One-electron systems  Discretization beyond basis sets (high order FEM)  Discretization beyond basis sets (high order FEM)  Control box size by perfect absorption (irECS)  Control box size by perfect absorption (irECS)  Determine spectra from truncated calculation (tSURFF)  Determine spectra from truncated calculation (tSURFF)  Efficiency example: strong field photo-emission  Efficiency example: strong field photo-emission

Discretization: why finite elements? Discretization: why finite elements? Required basis sizes Required basis sizes d degrees of freedom phase space volume V There are no smart tricks to beat this number unless we have additional information Additional information Additional information E.g. perturbative ionization, i.e. initial state or free motion or SFA: initial state or Volkov wave packet or: we “know” only bound states play a role or … Basis sets Basis sets Pseudo-spectral (e.g. field-free eigenstates, momentum-space) Pseudo-spectral (e.g. field-free eigenstates, momentum-space) Build energy- or momentum -information into ansatz Quantum Dynamics, Hamburg Local basis sets (B-splines, finite-element, FEM-DVR) Local basis sets (B-splines, finite-element, FEM-DVR) s. 5 Exploit locality of operators (differentiation, multiplication) Numerically robust March 24 – 26, 2014 High order finite elements High order finite elements Locally adjustable (→ irECS) Well-defined points of non-analyticity (element boudaries) Rapid convergence due to high order (e.g. 10-20) Parallelization: communication independent of order

Exterior complex scaling (ECS) Exterior complex scaling (ECS) General approach for perfect absorbers (PML, ECS) General approach for perfect absorbers (PML, ECS) [A.S., H-P. Stimming, N. Mauser, J. Comp. Phys., to appear] Outside some inner region [0,R 0 ] analytically continue a unitary transformation U λ (e.g. coordinate scaling) to contractive (non-unitary) U Θ Unitarity + analyticity guarantee unchanged solution Ψ on [0,R 0 ] θ !!! Caution: Domain issues for U θ H(t)U θ !!! - 1 Quantum Dynamics, Hamburg Translates into: s. 6 Complex coordinates Im r March 24 – 26, 2014 beyond a finite distance R 0 r -> r for r < R 0 θ Re r r -> R 0 + e iθ (r-R 0 ) for r > R 0 0 r R 0

Implementation of exterior complex scaling Implementation of exterior complex scaling Important technical complication Bra and ket functions are not from the same set!!! Exterior scaled Laplacian Δ R0,Θ is defined on discontinuous functions Exterior scaled Laplacian Δ R0,Θ is defined on discontinuous functions Discontinuity because start from unitary transformation Discontinuity is reversed for the left hand functions Discontinuity is reversed for the left hand functions Quantum Dynamics, Hamburg s. 7 Matrix elements of Δ R0,Θ March 24 – 26, 2014 are computed by piece-wise integration [0,R 0 ] + [R 0 ,∞) Conditions easy to implement with a local basis set

a perfect absorber ECS – a perfect absorber irECS – ir [A.S., Phys. Rev. A81, 53845 (2010)] [A.S., Phys. Rev. A81, 53845 (2010)] Infinite nfinite r range ange E Exterior xterior C Complex omplex S Scaling caling I r → r for r < R 0 Im r ECS: ECS: r → R 0 + e iθ (r-R 0 ) for r > R 0 Re r θ 0 r R 0 Discretization Discretization High (~8 th ) order finite elements “infinite range” infinite size last element [R 0 ,∞) R 0 Accuracy Efficiency Accuracy Efficiency |Ψ θR0 (x,t) – Ψ(x,t)| / |Ψ(x,t)| Quantum Dynamics, Hamburg Number of points Accuracy 0 s. 8 Log 10 -8 March 24 – 26, 2014 CAP = complex absorbing potential Accuracy inside R 0 ~ 10 -7

tSURFF – how to obtain spectra from a finite range wave function tSURFF – how to obtain spectra from a finite range wave function Scattering spectra = asymptotic information by definition Scattering spectra = asymptotic information by definition Absorb / discard Finite range Asymptotic part Quantum Dynamics, Hamburg If we solve only on a finite range, exactly the asymptotic information is missing s. 9 Solution: Solution: Continue beyond the box using some known solution – Volkov March 24 – 26, 2014 [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 How we usually calculate spectra from TDSE Needs ← problem 1 Get Ψ(r,t) at the end of the pulse t=T: Ψ(r,T) very large box Time-independent ← problem 2 Scattering solution ψ k scattering With asymptotics Spectrally analyze Ψ(x,t) Quantum Dynamics, Hamburg s. 10 Spectral density Spectral density March 24 – 26, 2014

Solve by using additional information Solve by using additional information (1) TDSE is a 2 nd order PDE (1) TDSE is a 2 nd order PDE Value and derivative at a surface r = R c suffice to continue the solution beyond the surface (2) Beyond distances R c ~ 50 a.u. motion is free (2) Beyond distances R c ~ 50 a.u. motion is free Use Volkov solution for free motion in the field instead of numerically solving Compare R-matrix theory! How things are done... How things are done... ➔ for a given pulse, solve with irECS absorption (box size ~ 50 a.u., laser-dependent) Quantum Dynamics, Hamburg ➔ save surface values and derivatives at surface(s) as function of time s. 11 ➔ properly time-integrate surface values for asymptotic momenta p of your choice (one integration for each p, ordinary integrals, very cheap!) March 24 – 26, 2014 ➔ can zoom in onto areas of interest (important for 2-electron problems) ➔ Effort grows only linearly with pulse duration T (cf. T 2 ~ T 4 if time and box-size grow)

t-SURFF – time-dependent surface flux method t-SURFF – time-dependent surface flux method [L. T ao and A.S., New. J. Phys. 14, 013021 (2012)] [L. T ao and A.S., New. J. Phys. 14, 013021 (2012)] Propagate until large T large T where bound where bound Ψ Ψ b and scattering and scattering Ψ Ψ s parts separate parts separate Propagate until b s Beyond Beyond distance R distance R c c scattering solutions scattering solutions χ χ k k are known are known θ(r,R c ) 1 Ψ b + Ψ s Ψ(T) = R c 0 Spectral amplitude σ(k): σ(k): Spectral amplitude with Volkov solutions χ k Quantum Dynamics, Hamburg s. 12 Volume integral → Time-integral Time-integral & & surface integral surface integral Volume integral → March 24 – 26, 2014 Commutator depends only on Ψ(R c ,t) and ∂Ψ(R c ,t) Note: need time-dependent bra-solutions ≈ Volkov (or better, if available)

Single photo-electron spectra Single photo-electron spectra

Photo-electron spectra – single electron, 3d Photo-electron spectra – single electron, 3d [L. T ao and A.S., New. J. Phys. 14, 013021 (2012)] [L. T ao and A.S., New. J. Phys. 14, 013021 (2012)] Hydrogen atom, Hydrogen atom, Laser: 2 x 10 14 W/cm 2 @ 800 nm, 20 opt.cyc. FWHM Hydrogen atom, Hydrogen atom, Linear polarization Angle resolved Quantum Dynamics, Hamburg s. 14 March 24 – 26, 2014 90 radial discretization points, 30 angular momenta

Attoclock – ionization by elliptically polarized IR Attoclock – ionization by elliptically polarized IR [Pfeiffer et al, Nat. Phys. 8, 76 (2012)] Angle-resolved photo-electron spectra Peak emission direction deviates from Peak field direction => deduce delay in release of electron Solution of the TDSE Quantum Dynamics, Hamburg Use oppositely handed polarizations to calibrate peak field direction s. 15 θ March 24 – 26, 2014