Laura Moore

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Plan of Talk• Introduction to HELIUM• The PDE and the Algorithm• The computational demand of HELIUM• Post-processing capabilities• Future plans• SummaryThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUMDemand:Memory. Grown from ∼ 16 Gbytes on Cray T3D to ∼ 15Tbytes on Cray XE6The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUMDemand:Memory. Grown from ∼ 16 Gbytes on Cray T3D to ∼ 15Tbytes on Cray XE6Communications. Per 12 wall-clock hour run there are100,000 transfer events each involving 50 GbytesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Helium and a short intense laser pulsee 1zr 1LaserOe 2r 2The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Helium atom plus intense laseri ∂Ψ∂t = HΨ(r 1,r 2 ,t) TDSEFundamental equation of Quantum MechanicsAtomic units e = m = ¯h = 1The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Helium atom plus intense laseri ∂Ψ∂t = HΨ(r 1,r 2 ,t) TDSEFundamental equation of Quantum MechanicsH = − 1 2 ∇2 1− Z r 1− A(t)i c+ 1r 12− 1 2 ∇2 2− Z r 2− A(t)i c∂∂z 1∂∂z 2Atomic units e = m = ¯h = 1The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Requirements• Efficiency on massively parallel machines (minimumcommunications overhead)• High accuracy (solution maintained correct to at least 13decimal places)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Calculated (full-lines) and Ohio State measured (dashed-lines)momentum-resolved single- and double-ionization electronspectra for He exposed to 390 nm laser lightProbability Density (arbitrary units)1 (a)Double(x 7273)0.1Single0.0110.10.01I = 8 x 10 14 W/cm 2I = 11 x 10 14 W/cm 2Single(b)Double(x 3333)0 0.5 1 1.5 2 2.5 3Momentum (au)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Solving the TDSE• To solve ˙Ψ = −iHΨ chooseΨ = ∑l 1 l 2 L1r 1 r 2f l1 l 2 L(r 1 ,r 2 ,t) |l 1 l 2 L〉• Propagate over grid from Ψ(r 1 ,r 2 ,t) toΨ(r 1 ,r 2 ,t + δt) using an Arnoldi propagatorThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time propagation using an ArnoldipropagatorThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time propagation (1)• Ψ(r 1 ,r 2 ,t) - propagate over a discrete time interval δtΨ(t + δt) = Ψ(t) + δt ˙Ψ(t) + (δt)22!¨Ψ(t) + ...The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time propagation (1)• Ψ(r 1 ,r 2 ,t) - propagate over a discrete time interval δtΨ(t + δt) = Ψ(t) + δt ˙Ψ(t) + (δt)22!¨Ψ(t) + ...• But ˙Ψ = −iHΨ, where H is a very large sparse matrix,and soΨ(t + δt) = e −iHδt Ψ(t)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time propagation (2)• To n th order the Taylor series propagation isΨ TS (t + δt) =n∑k=0(−iδt) kH k Ψk!The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time propagation (2)• To n th order the Taylor series propagation isΨ TS (t + δt) =n∑k=0(−iδt) kk!H k Ψ• The Krylov sub-space K n+1 is the sub-space spannedby vectors Ψ,HΨ,...,H n Ψ.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Krylov subspace (1)• Space spanned by the vectors Ψ,HΨ,...,H n ΨThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Krylov subspace (1)• Space spanned by the vectors Ψ,HΨ,...,H n Ψ• Gram-Schmidt orthogonalization of these yields n + 1ortho-normal vectors Q 0 ,Q 1 ,...,Q n collected ascolumns of a non-square matrix QThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Krylov subspace (1)• Space spanned by the vectors Ψ,HΨ,...,H n Ψ• Gram-Schmidt orthogonalization of these yields n + 1ortho-normal vectors Q 0 ,Q 1 ,...,Q n collected ascolumns of a non-square matrix Q• Then the Arnoldi propagation isΨ AP (t + δt) = Qe −ihδt Q † Ψ(t)where h = Q † HQ is a tri-diagonal square matrix oforder n + 1 of G-S coefficients.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Arnoldi Propagator: H and Q matricesOrder of H is typically 58 × 10 9Value of n is typically 15The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Krylov subspace (2)• From h = Q † HQ where h is of order n + 1 we have˜H = QhQ †The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Krylov subspace (2)• From h = Q † HQ where h is of order n + 1 we have˜H = QhQ †• Moreover since (QhQ † ) m = Qh m Q † thene −i ˜Hδt = Qe −ihδt Q †where Qe −ihδt Q † is the Arnoldi propagator, viz:Ψ AP (t + δt) = e −i ˜Hδt = Qe −ihδt Q † Ψ(t)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Five benefits of the Arnoldi propagator• It is explicit - vital for massively parallel machines• Computational overhead rises linearly with n• A unitary operator correct to order n in δt• Very efficient way to obtain Hamiltonian eigenstates• At least twice as efficient as Taylor series andperformance ratio scales linearly with nThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Computational demand of HELIUMThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Time-propagating the wavefunction•Ψ = ∑l 1 l 2 L1r 1 r 2f l1 l 2 L(r 1 ,r 2 ,t) |l 1 l 2 L〉• Propagate over grid from Ψ(r 1 ,r 2 ,t) toΨ(r 1 ,r 2 ,t + δt) using an Arnoldi propagatorThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Length of vector f l′1 l ′ 2 L′ (r 1 ,r 2 ,t) at any t• For necessary accuracy, our biggest calculations so far(for Ti:sapphire laser λ = 800 nm) using 16,110HECToR cores have demanded:– About 2,000 distinct l 1 ,l 2 ,L triplets– Typically 5,370 mesh points in each of r 1 and r 2The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Length of vector f l′1 l ′ 2 L′ (r 1 ,r 2 ,t) at any t• For necessary accuracy, our biggest calculations so far(for Ti:sapphire laser λ = 800 nm) using 16,110HECToR cores have demanded:– About 2,000 distinct l 1 ,l 2 ,L triplets– Typically 5,370 mesh points in each of r 1 and r 2• Hence length of f is∼ 2 × 10 3 × 5 × 10 3 × 5 × 10 3 = 57.7 × 10 9= 922 Gbytes memory.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Length of vector f l′1 l ′ 2 L′ (r 1 ,r 2 ,t) at any t• For necessary accuracy, our biggest calculations so far(for Ti:sapphire laser λ = 800 nm) using 16,110HECToR cores have demanded:– About 2,000 distinct l 1 ,l 2 ,L triplets– Typically 5,370 mesh points in each of r 1 and r 2• Hence length of f is∼ 2 × 10 3 × 5 × 10 3 × 5 × 10 3 = 57.7 × 10 9= 922 Gbytes memory.• to time propagate, we need approx. 16 vectors of suchlength - ∼ 14,752 GybtesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Computer code designThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011fortran90 + MPI: Distribution over coresr 130 1 2r 24 5 6 78 9 10 11Illustrative 16core example12 13 14 15• Each core is assigned a region of r 1 , r 2 space• Each region has to communicate with neighbouringregions (boundary swapping)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011fortran90 + MPI: Distribution over coresr 130 1 2r 24 5 6 78 9 10 11Illustrative 16core example12 13 14 15• Each core is assigned a region of r 1 , r 2 space• Each region has to communicate with neighbouringregions (boundary swapping)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011fortran90 + MPI: Distribution over coresr 10 1 2 35 6r 24 Illustrative 107 8core example9• The problem is symmetrical ⇒ Saving on core count• Care must be taken along the diagonalThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011fortran90 + MPI: Distribution over coresr 10 1 2 35 6r 24 Illustrative 107 8core example9• The problem is symmetrical ⇒ Saving on core count• Care must be taken while boundary swappingThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Communications• Communications occur ∼ 100,000 times per 12-hour run• Each communication transfers ∼ 50 Gbytes of data• On the Cray XE6 such communications occupy only 5% ofrunning time!The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Ti:sapphire λ = 800 nm calculations using near fullcapability of EPCC Cray XE6The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Ti:sapphire λ = 800 nm calculations using near fullcapability of EPCC Cray XE6Motivation: Much experimental data at 800 nmThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 20111D 3-step recollision mechanism for doubleionization• Step 1 - field ionization of first electron• Step 2 - this first electron moves away from the nucleusbut is subsequently driven back, by the laser field, towardsthe parent core. The first electron then recollides with theparent core and at the instant of recollision has amaximum energy of 3.2 U p .• Step 3 - the recollision brings about double ionizationThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution at 390 nmp 1(au)0.0 0.5 1.0 1.5 2.0 2.5 3.01.000.75p20.500.250.00The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Prediction verified by laboratory experiment(a) This work 390 nm, 10 ×10 14 W/cm 2Parker et al PRL 96 133001 (2006)(b) Experimental 800 nm, 4.5 ×10 14 W/cm 2Staudte et al PRL 99 263002 (2007)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011800 nm Run Size• 16,110 cores• 108 wallclock hours per pulse (nine 12-hour runs)• 9-cycle laser pulseThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Plot of two-electron radial momentum space probabilitydensity at end of 7-cycle laser pulse, peak intensity3.2×10 14 W/cm 2 at 800 nm. The colour scale is linear.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Results at 800 nmThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Results at 800 nm• Cut-off in the total KE:at low I : 5 U p ( I independent)I > 2.2 × 10 14 W/cm 2 : 5 to 8 U p ( I dependent)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Results at 800 nm• Cut-off in the total KE:at low I : 5 U p ( I independent)I > 2.2 × 10 14 W/cm 2 : 5 to 8 U p ( I dependent)• What’s 2.2 × 10 14 W/cm 2 ?The classical recollisional excitation threshold at 800 nm.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Results at 800 nm• Cut-off in the total KE:at low I : 5 U p ( I independent)I > 2.2 × 10 14 W/cm 2 : 5 to 8 U p ( I dependent)• What’s 2.2 × 10 14 W/cm 2 ?The classical recollisional excitation threshold at 800 nm.• Analysis ongoing!The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Post-processing of wavefunctionThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Radial probability density•Ψ = ∑l 1 l 2 L1r 1 r 2f l1 l 2 L(r 1 ,r 2 ,t) |l 1 l 2 L〉• Two-electron radial probability densityP(r 1 ,r 2 ,t) = ∑|f l1 l 2 L(r 1 ,r 2 ,t)| 2l 1 l 2 LThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Transformation of the final-statetwo-electron wavefunction intomomentum-spaceThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Momentum RepresentationHenceg(k 1 ,k 2 ,T end )∫ ∫= Ψ(r 1 ,r 2 ,T end )X(r 1 ,k 1 ,r 2 ,k 2 )dr 1 dr 2= ∑l 1 l 2 L1k 1 k 2g l1 l 2 L(k 1 ,k 2 ,T end )|l 1 l 2 L >P mom (k 1 ,k 2 ,T end ) = ∑l 1 l 2 L|g l1 l 2 L(k 1 ,k 2 ,T end )| 2The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Application to 2-photon double-ionizationof HeliumTheoretical calculations of non-sequential double-ionization(NSDI) cross-sections differ by an order of magnitude.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 20112-photon NSDI cross-sections(taken from B Piraux et. al. J Phys: Conf Series 141 012013 (2008))The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM Calculation of 2-photon NSDIcross-sections• Choose low intensity (10 13 W/cm 2 ) so as to minimizesequential ionization.• Choose laser frequency of 1.6 a.u. (43.5 eV).• 2-photon double-ionization is a direct processThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Energy level diagram of He2.0 au ω = 1.6 auHe ++He + (2p)0.9 auHe + (1s)Heground stateThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Run parameters• l 1,2 max = 4 (∼ 100 basis states)• δr = 0.29 a.u.• 4200 grid points for r 1 and r 2 (integration extends to1218 Bohr radii)• Pulse ramp-on 18 T, constant for 30 T, ramp-off 18 T.• Calculation runs for a further 30 T (field-free) to letdoubly-ionizing electrons depart the strong Coulomb fieldof the residual ion.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Radial probability distribution right after laser pulse hasramped offThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Radial probability distribution a further 30 T after the laserpulse has ramped offThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution of the twodoubly-ionizing electrons obtained by projecting the final-statewavefunction onto plane wavesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution at 390 nmp 1(au)0.0 0.5 1.0 1.5 2.0 2.5 3.01.000.75p20.500.250.00The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution at 390 nmp 1(au)0.0 0.5 1.0 1.5 2.0 2.5 3.01.000.75p20.500.250.00No rectilinear features!The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution of the twodoubly-ionizing electrons obtained by projecting the final-statewavefunction onto plane wavesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Calculation of 2-photon double-ionizationcross-sectionThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Calculation of 2-photon double-ionizationcross-section• No natural division between the arc and the linear featuresthat represent bound states of He + .The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Calculation of 2-photon double-ionizationcross-section• No natural division between the arc and the linear featuresthat represent bound states of He + .• Probability of 2-photon ionization as extracted from such aplot can vary by up to a factor of 1.5.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Calculation of 2-photon double-ionizationcross-section• No natural division between the arc and the linear featuresthat represent bound states of He + .• Probability of 2-photon ionization as extracted from such aplot can vary by up to a factor of 1.5.• Need to remove single ionization contributions (byprojecting onto eigenstates of He + ).The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Projecting out singly ionizing statesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Projecting out singly ionizing states• Very high accuracy in the treatment of the He + boundstates is essential.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Projecting out singly ionizing states• Very high accuracy in the treatment of the He + boundstates is essential.• He + bound states obtained through eigen-decompositionof the field-free finite-difference Hamiltonian.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Projecting out singly ionizing states• Very high accuracy in the treatment of the He + boundstates is essential.• He + bound states obtained through eigen-decompositionof the field-free finite-difference Hamiltonian.• Eigenvectors calculated using a Lanczos/Arnoldi methodon a discrete finite-difference grid.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Projecting out singly ionizing states• Very high accuracy in the treatment of the He + boundstates is essential.• He + bound states obtained through eigen-decompositionof the field-free finite-difference Hamiltonian.• Eigenvectors calculated using a Lanczos/Arnoldi methodon a discrete finite-difference grid.• All components of the spatial final-state wavefunction inthe directions of these orthogonal bound states areremoved.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011After removing all singly ionizing components of the spatialfinal-state wavefunction, project onto plane wavesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

Methods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011OJoint momentum-space probability distribution obtained byprojecting onto plane waves:(a) the final-state wavefunction(b) the final-state wavefunction after removal of He + boundstates(a)(b)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011A further check ...• Remove components from the spatial wavefunctionΨ(r 1 ,r 2 ,T) involving bound states of the residual He +ion.• Project remainder onto a basis of Coulomb states.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Joint momentum-space probability distribution obtained byprojecting onto Coulomb waves the final-state wavefunctionafter removal of He + bound statesThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

Methods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011OJoint momentum-space probability distribution obtained byprojecting the final-state wavefunction, after removal of He +bound states, onto:(a) Plane waves(b) Coulomb waves(a)(b)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-section atω = 1.6 a.u. (43.5 eV)The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-section atω = 1.6 a.u. (43.5 eV)• Any remaining linear feature now contains negligiblepopulation.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-section atω = 1.6 a.u. (43.5 eV)• Any remaining linear feature now contains negligiblepopulation.• Both methods (projection onto plane waves and ontoCoulomb waves) give same cross-section to within 3%.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-section atω = 1.6 a.u. (43.5 eV)• Any remaining linear feature now contains negligiblepopulation.• Both methods (projection onto plane waves and ontoCoulomb waves) give same cross-section to within 3%.• HELIUM cross-section value is 7.6 × 10 −53 cm 4 sThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-section atω = 1.6 a.u. (43.5 eV)• Any remaining linear feature now contains negligiblepopulation.• Both methods (projection onto plane waves and ontoCoulomb waves) give same cross-section to within 3%.• HELIUM cross-section value is 7.6 × 10 −53 cm 4 s• Still to test effects of some parameters (pulse shape).The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 20112-photon NSDI cross-sections(taken from B Piraux et. al. J Phys: Conf Series 141 012013 (2008))The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 20112-photon NSDI cross-sectionsHELIUM(taken from B Piraux et. al. J Phys: Conf Series 141 012013 (2008))The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011HELIUM 2-photon NSDI cross-sectioncalculations• HELIUM cross-section at ω = 1.6 a.u. (43.5eV) in thethicket!• Still to test effects of some parameters (pulse shape).• Intend to calculate cross-sections at other photonenergies too.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUMThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUM• Non-dipole interactionsThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUM• Non-dipole interactions– Enables calculations to be carried out at x-raywavelengths.– M is not conserved, so basis set has been extendedfrom |l 1 l 2 L〉 to |l 1 l 2 L,M〉.– New Hamiltonian terms.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUMThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUM• Crossed laser fieldsThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011New extensions to HELIUM• Crossed laser fields– Enables calculations to be carried out with twoperpendicular laser fields (with two arbitraryfrequencies), or with one laser field of circular/ellipticpolarization.– M is not conserved, so basis set has been extendedfrom |l 1 l 2 L〉 to |l 1 l 2 L,M〉.– New Hamiltonian terms.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011A new post-processing codeThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011A new post-processing code• Transformation of wavefunction from spherical tocylindrical geometry.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011A new post-processing code• Transformation of wavefunction from spherical tocylindrical geometry.– Enables direct comparison with data from manyexperiments.z 2z 1The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Future plansThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Carry over of HELIUM MethodsThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Carry over of HELIUM Methods• There is a crucial need to handle the TDSE accurately formulti-electron atoms and molecules coupled toIR/visible/UV and VUV laser fields and undergoingdouble- and/or single-ionization.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Carry over of HELIUM Methods• There is a crucial need to handle the TDSE accurately formulti-electron atoms and molecules coupled toIR/visible/UV and VUV laser fields and undergoingdouble- and/or single-ionization.• The R-matrix concept allows the carry over of HELIUMmethods – development of the RMT (R-Matrixincorporating Time) code – next talk by Michael Lysaght.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011People currently working directly withHELIUM• Ken Taylor• Jonathan Parker• Laura Moore• Gregory Armstrong• David RobinsonThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Acknowledgments• UK EPSRC• The distributed Computational Science and Engineeringservice operated by the Numerical Algorithms Group LtdThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011SummaryThe HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Summary• The Arnoldi propagator has allowed us to handle thehigh-dimensionality TDSE for two-electron atoms inintense laser fields efficiently and to the accuracynecessary to complement laboratory experiment.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Summary• The Arnoldi propagator has allowed us to handle thehigh-dimensionality TDSE for two-electron atoms inintense laser fields efficiently and to the accuracynecessary to complement laboratory experiment.• The EPCC Cray XE6 (HECToR) with this propagator inuse has allowed first double-ionization spectra to becalculated for fundamental Ti:sapphire laser light.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011Summary• The Arnoldi propagator has allowed us to handle thehigh-dimensionality TDSE for two-electron atoms inintense laser fields efficiently and to the accuracynecessary to complement laboratory experiment.• The EPCC Cray XE6 (HECToR) with this propagator inuse has allowed first double-ionization spectra to becalculated for fundamental Ti:sapphire laser light.• Separating degenerate doubly and singly ionizing statesin the final-state wavefunction has enabled calculation of2-photon NSDI cross-sections with quantitative accuracy.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011And finally!The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011And finally!The HELIUM methods are being carried over to multi-electronatoms and molecules: the RMT (R-Matrix incorporating Time)code – next talk by Michael Lysaght.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O

OMethods and Codes for Atoms and Molecules in Strong Laser Fields, Dublin, April 2011And finally!The HELIUM methods are being carried over to multi-electronatoms and molecules: the RMT (R-Matrix incorporating Time)code – next talk by Michael Lysaght.Thanks for listening.The HELIUM code: E S Smyth, J S Parker and KTT Comp Phys Commun 114 (1998) 1-14O