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29 April 2011 COST - DCU 1

R-matrix Floquet theory for multiphoton processes

Hugo van der Hart


Overview

• Background of the approach

• Mathematical implementation– Inner region– Outer region

• Selected applications


Background

• From the 80s, increasing interest into the physics of atoms and molecules in intense laser fields.

• It is important to be able to describe a general multi-electron atom in an intense laser field.

• For many general multi-electron systems, the accurate description of single ionization already requires a multi-electron treatment.


The R-matrix Floquet approach

• Developed over the last 20 years by international collaboration.(Burke et al JPB 24 (1991) 761, Dörr et al JPB 25 (1992) 2809)

• Based on atomic R-matrix theory– Accurate description of atomic

structure

• Laser field included through Floquet Ansatz– Count net number of photons absorbed


People involved

Belfast:Phil BurkeKen TaylorJohn PurvisDavid GlassRobert GebarowskiGeorg WösteJames ColganNatalia VinciLiang FengClaire McKennaPeter BinghamMartin MadineLinda Hamonou

Brussels:Charles JoachainMartin DörrOlivier LatinneNiels KylstraMounir Bensaid

Rennes:Mariko Terao-DunseathKevin DunseathAlain Cyr

Daresbury:Cliff NobleMartin Plummer

Durham:Andrew Fearnside



• Applied to a wide range of problems– Multiphoton ionization

H-, He, Ne, Ar, F-, Cl-, Li-, Na-, K-, Ca, Sr, Xe – Harmonic generation

He, Mg, Ne– Atoms in two-colour laser fields

He, Mg


Mathematical basis ofR-matrix-Floquet theory

Inner region, N+1 electrons

All interactions between all electrons

Laser field: length gauge

R-matrix at boundary via a diagonalisation

Outer region, 1 electron

Electron feels long-range field

Laser field: velocity gauge

E, width of state:match R-matrix to asymptotic solutions


R-matrix wavefunction

• Wave functions in R-matrix theory are often written in the following form:

• Φ: Channel functions: ionic wavefunctions + ℓ, spin of the outer electron

• u: Radial wavefunction for continuum electron• χ: Additional correlation states

∑

∑

+

++++

+

=

j1N1jjk

ij1Nij1N1NN1iijk1N1k

)x,,(xχd

)(r)uσr̂,x,,(xΦc)x,,(xΨ

A


R-matrix wavefunctions

• Channel functions and correlation functions are built from input orbitals– Optimised to describe residual target states– Must be known beforehand– Modified to have 0 value and derivative at boundary

• Continuum functions are generated inside the codes– Nr lowest potential eigenfunctions (incomplete set)– B-spline based potential eigenfunctions (complete set)


The TDSE

• The time-dependent Schrodinger equation for an atom in an intense laser field:

where HN+1: field-free HamiltonianA(t): laser vector fieldPN+1: total momentum operator

• This equation is transformed into a separate TDSE for each region

ψ(t)A2c

1NPA(t)c1H

tψi 221N1N

++⋅+=

∂∂

++


Inner region TDSE

• For the inner region, transform to the length gauge

where RN+1 is the total position operator, so that

with E(t) the electric field

t),(XψRA(t)ciexpt),ψ(X 1NL1N1N +++

⋅−=

( ) t),(XψRE(t)Ht

t),(Xψi 1NL1N1N1NL ++++ ⋅+=∂∂


The Floquet Ansatz

• Time dependence can be removed for an infinitely long single-frequency “pulse”: the Floquet Ansatz

• This gives the infinite set of coupled equations

for an electric field

)(XΨeet),(XΨ 1NLn

n

itnωiEt1NL +

∞

−∞=

−−+ ∑=

( ) ( ) 0ψψRε̂2

EψnωEH L 1nL

1n1N0L

n1N =+⋅+−− +−++

tωcos E ε̂E(t) 0=


Boundary effects

• The boundary is important:it is where the inner-region and outer-region are linked together through the R-matrix

• The R-matrix can be evaluated efficiently for any energy when the inner-region eigenfunctions and associated eigenvalues are known

• Thus we need to diagonalise the inner-region Hamiltonian


Boundary effects

• There is a problem: the inner-region Hamiltonian is not Hermitian

At the boundary: ψ, ψ' ≠ 0 in general, in which case

• We can make the Hamiltonian Hermitian by adding a Bloch operator Lb:

k2

mm2

k ψψψψ ∇≠∇

( )

−−−=++

+1N1N

1Nb r1b

drdarδ

21L


R-matrix

• We then obtain:

or

• With ψk an eigenfunction of HF+Lb, and Ek the associated eigenvalue, this can be written as

ψLELH

1ψ bbF −+

=

( ) ψLψELH bbF =−+

∑ −= km bkkkkm

mm ψLψψEEδψψψ


Hamiltonian

• Schematic structure of the Hamiltonian

• 6 angular momenta

• 4 Floquet blocks


R-matrix

• The last matrix element can be evaluated directly due to the δ-function in the Bloch operator

• We now evaluate both LHS and RHS at r=a,project onto component n of photon space,and onto channel function i.

• With Fin: the wavefunction for outer electronwink: boundary amplitude of ψk, in channel i /n

∑ −= km bkkkkm

mm ψLψψEEδψψψ

∑∑=+ +

−

−=

k arjm

1N

jm

k

jmkink

jmin

1N

bFdrdF

aEE

ww2a1(a)F


R-matrix

• This equation can be simplified as

where the R-matrix is given by

• The R-matrix can thus be seen to be a matrix linking the wavefunction to its derivative at the boundary.

∑∑=+ +

−

−=

k arjm

1N

jm

k

jmkink

jmin

1N

bFdrdF

aEE

ww2a1(a)F

arjm

1N

jm

jmjmin,in

1N

bFdrdF

aR(a)F=+ +

−= ∑

∑ −= k kjmkink

jmin, EEww

2a1R


Outer region

• The outer region is divided into two regions– A propagation region, in which the R-matrix is

propagated (velocity gauge)– An asymptotic region, in which the wavefunction can

be described well using asymptotic solutions (acceleration frame)

• These separate regions are needed, as for accurate solution, we need different propagation distances for different laser frequencies/intensities


Outer region TDSE

• For the propagation region, transform following

where RN is the total position operator for all electrons except the outermost one, so that

with E(t) the electric field

t),(Xψ)dt'(t'A2c

iRA(t)ciexpt),ψ(X 1NV

t2

2N1N ++

−⋅−= ∫

t),(XψpA(t)c1RE(t)H

tt),(Xψi 1NV1NN1N1NV ++++

⋅+⋅+=

∂∂


Propagation region wavefunction

• Similar to the inner region, we first apply the Floquet Ansatz.

• The outer-region wavefunction in the propagation region is then given by a close-coupling expansion:

∑ ++++ =i

1Nin1N1NN1i1N1n )(r)Gσr̂,x,,(xΦ)x,,(xΨ


Propagation region

• This gives the following set of coupled differential equations:

WE: electron-repulsion outer-core electronsexcluding the fully symmetric termin the 1/r12 expansion

WD: laser interactions with core electronsWP: laser interactions with outer electron

∑ ++=

+−++−

jmjmPDEin

2in2

ii2

2

(r))GWW(W2(r)Gkr

N)2(Zr

1)(drd


Propagation region

• This set of equations can be rewritten in matrix form as

– P is a coefficient matrix for the interaction between the laser field and the outer electron

– EV is the energy in the velocity gauge– V is a potential energy matrix including all long range

couplings as well as the coupling between target states

0G(r)2EV(r)drdP

drd

V2

2

=

+−+


Propagation

• Standard R-matrix propagation techniques (eg. The Light-Walker method) exist for equations of type

(ie. equations excluding the P d/dr term).• Use an operator identity

0G(r)2EV(r)drd

V2

2

=

+−

Pr/22

2

2

2

2Pr/2 e

4P

drd

drdP

drde

−=

+


Propagation

• Using this transformation, we obtain

• This equation is in the desired form with an effective potential

and an effective wavefunction

0G(r)e2EV(r)ee-4

Pdrd Pr/2

VPr/2-Pr/2

2

2

2

=

+−

Pr/2-Pr/22

eff V(r)ee4PV +=

G(r)eg(r) Pr/2=


Light-Walker propagation

• Once we have the equations in the desired form, we can propagate the R-matrix.

• Divide the outer region into small subsectors

• On each subsector, assume that the potential Veff(r) is constant.


Light-Walker propagation

• We need the Green’s function from

• This set of equations uncouples in the eigenbasis of the potential Veff(r). The uncoupled equations can be solved in terms of hyperbolic functions.

• In this propagation scheme, we thus– Transform to the eigenbasis– Propagate the uncoupled solutions from r to r`– Transform back to the original basis

• This is repeated for each subsector.

)r'-δ(r)r'T(r,2EV(r)drd

V2

2

=

+−


R-matrix in propagation region

• We can now need to obtain an initial R-matrix for the propagation region:

where the last form is needed to transform from length to velocity gauge

Pa/21

1

ar

Pa/2VP

1

ar

VP

eP2a(a)G

drdGae(a)R

drdgag(a)(a)R

−−

−

=

−

=

+=

=


L-to-V transform

• From the wavefunction transformations, we can deduce that

• Substitute A(t) (with sine time dependence)

t),(XψrA(t)ciexp)dt'(t'A

2ciexpt),(Xψ 1NL1N

t2

21NV +++

⋅−

= ∫

t),(Xψ) sin(tωrε̂c

iAexp

) sin(2tωc 8ω

iAexpt4ciAexpt),(Xψ

1NL1N0

2

20

2

20

1NV

++

+

⋅−×

−

=


L-to-V transform

• Insertion of the Floquet Ansatz gives

where Jl indicates a Bessel function.

⋅

=

=

−=−=

+−

∞

−∞=+

∞

−∞=+−

∑

∑

1N0

2l'll'

2

20

l'1N0l

Ln

n'1N0n'n

Vn

2

20

pV

rε̂c

AJc 8ω

AJ)r,(Af

ψ)r,(Afψ

4cAEEEE


R-matrix

• Interested in the effect on the R-matrix.With

we obtaindrdFF

a1R

P21G

drdG

a1R

drdFAF

drdA

drdG

AF,G

L

11V

=

+=

+=

=

−−

11L1V PA

21ARa

drdAA

a1R

−−−

++=


Asymptotic solutions

• Now determine the asymptotic solutions

• Outgoing waves uncouple asymptotically in the acceleration frame, so we have that

with Aasy the A-to-V transformation at r=∞ and

Θ(r)AG(r) asyr ∞→→

+−+⋅= iσ

2πiln2qr

qiςriqexp

q1Θ(r)


Asymptotic solutions

• However, we determine the asymptotic solutions at a finite distance.

• We therefore define the asymptotic solutions through an asymptotic expansion:

which is started by setting A(0)=Aasy.

∑∞

=

−=0n

(n)n Θ(r)ArG(r)


Matching procedure

• The final step in the procedure is to match, at r=a´, the asymptotic solutions to the R-matrix.

• Matching succeeds if there is a vector x such that

0xdrdGRa'

2RPGa'G =

−−


Matching procedure

• We thus look for an energy E such that

• We start with three initial guess Ein-ΔE, Ein+ΔE, EinHope for convergence via Muller’s method

• Convergence: energy E0+Δ-iΓ/2,

0drdGR

2RPGa'Gdet =

−−


Practical issues

• Floquet expansion must be truncated

• Number of angular momenta must be truncated

• Set a minimum and maximum length of propagation sectors


Numerical issues

• Number of terms in the asymptotic expansion• Propagation distance

• Accuracy of length-to-velocity-gauge transformation must be monitored– unitarity check in calculations– extend Floquet expansion if necessary– Convergence worsens with increasing intensity and

increasing box size– Will be inaccurate for outermost blocks


Atomic structure data

• Three different methods are available for generating the atomic-structure data needed to perform the calculations– R-matrix I code (old, adapted for RMF)– Two-electron B-spline code– B-spline based R-matrix II code (RMF not yet

adapted for this code)



• Overview of results over the last 10 years– Accuracy of the approach

He irradiated with 390 nm laser lightOther noble gases

– Two-photon double ionization of He– Two-photon inner-shell detachment of Li-

– Two-photon ionization of Ne+


Towards 390 nm

• RMF codes obtain rates through an iterative root-finding approach.

• A good guess for the root is essential• Parker et al studied He ionization at 248.6 nm using RMF

and the HELIUM code and obtained excellent agreement (JPB 33 (2000) L239)

• Increase the wavelength slowly from 248.6 to 390 nm

• Combine RMF theory with a B-spline-based inner-region code.


• Ionization rates obtained over a wide range of wavelengths

• Seven-photon ionization rates at I = 3×1014 W/cm2

TDHF:Perry et alPRL 63 (1989) 1058

RMF:PRA 73 (2006) 023417


• Minimum of 9 photons required• Excellent agreement between RMF and TDNI for

I in 1 - 2.5 × 1014 W/cm2: benchmark results

TDNI:B. DohertyJ. ParkerK. TaylorJPB 38 (2005) L207


Difficulties

• Below 1014 W/cm2, difficult to find state due to small width. Initial energy guess needs to be accurate to within 5 times the width.

• Calculations extended to 4×1014 W/cm2, but increasingly affected by linear dependence in asymptotic solutions

• Comparison with TDNI very good up to the highest intensities considered


• Advantage of RMF : easy resonance analysis• Resonances strongly affected by the field• Crossing between 1sns and 1snd states


• Extend study: Ne, Ar irradiated by 390 nm light• Minimum of 6-photon absorption for Ar• Minimum of 8-photon absorption for Ne

PRA 73 (2006) 023417


Difficulties

• Argon calculations relatively “easy”

• Neon calculations difficult– Tracking the Ne ground state across N-photon

thresholds complicated– 2Po residual-ion state gives 2 channels per

angular momentum


Two-photon double ionization of He

He+ 1s

He+ n=3He2+

He+ n=2

He+He

ħω

ħω

ħωħω

e-

ħω = 45 eVHe 1s2


Two-photon double ionization cross sections

• Non-sequential DI: two photons required• Sequential DI: three photons required• Initial calculations for two-photon DI

– At 45 eV, σ ~ 7×10-52 cm4s (Mercouris et al JPB 34 (2001) 3789, Nikolopoulos et al JPB 34 (2001) 545)

• Time-dependent calculations– At 43.5 eV, σ < 2.1×10-52 cm4s (Parker et al JPB 34 (2001) L69)– At 45 eV, σ ~ 1.2×10-52 cm4s (Colgan et al PRL 88(2002) 173002)

• Combine RMF approach with a dedicated two-electron B-spline basis


• RMF confirms lower value (Feng et al JPB 36 (2003) L1) • Theory is converging


Intense-field emission of inner-shell electrons

• With the development of high-frequency laser facilities, intense-field studies need to be extended to processes involving inner-shell electrons

• Inner-shell electrons more responsive to high-frequency light

• Need for theory to develop a capability in this area.


Single-photon emission of inner-shell electrons

• Synchroton experiments performed on 1s detachment from Li-. (Berrah et al PRL 87 (2001) 253002)– Li left in autoionizing state– Measure Li+ yield obtained after autoionization

• Generally good agreement with theory. (Zhou et al PRL 87 (2001) 023001)


Li and Li- energy levels

Doubly excited states

“Dble detach.” states

Singly excited states


Li target states

• Bound states and autoionizing states of Li described by bound orbitals

• Differences between 2s orbitals in 1s22s and 1s22s requires correlation orbitals, eg 4s

• Energy of 1s24s below 1s2s2. Then 1s24s must be included in calculations.


Excitation of Li 1s2s2

These rates are small at 1012 W/cm2:0.004% of totaldetachment

Emission of d electrons suppressed

Resonances near the 1s2s2p thresholds


Excitation of Li 1s2l2l′

2 main processes:

1s2s2p2 1S resonance:Correlation in initialstate plus excitationof 1s → 2s

Excitation of 1s2s2:Direct two-photonemission of 1s


Two-photon ionization of Ne+

• Two-photon ionization of Ne+ studied experimentally at 38.4 eV

(Sorokin et al PRA 75 (2007) 051402)

• Establish importance of including all accessible channels

• What are the most important final states of Ne2+?

• Initial studies have focused on M=0.


Inclusion of 2s emission increases ionisation rates substantially

2s22p4(3Pe)nℓ series decreases rates if 2s emission excluded

Hamonou et al, J. Phys. B 41 (2008) 121001

Exp. Sorokin et al PRA 75 (2007) 051402


• Can compare RMF calculations with time-dependent R-matrix calculations

• Shows how rates emerge from TD calculations

Hamonou et al, J. Phys. B 43 (2010) 045601


TDRM approach• The TDRM approach was developed to

transfer R-matrix philosophy to time-dependent studies (Lysaght et al, PRA 79 (2009) 053411)


Pump-probe Scheme for C+

• Excitation by pump laser with energy 10.88 eV

• Creates wavepacket between 2s2p2 2Se and 2De states for M=0

• Vary delay δt• Interpret dynamics via

uncoupled 2p02p0 and (2p12p-1)S states

• Ionized by a probe laser pulse with energy 16.33eV or 21.77eV

XUV pump, 2.28 fs

δt

1s22s2 1Se

2s2p2 2Se

2s22p 2Po

C+

CONTINUUM

XUV probe

2s2p2 2De

0.0 eV

9.29 eV

11.96 eV

24.38 eV

2s2p 3Po

2s2p 1Po37.07 eV

30.88 eV

C2+

Ionization Probability, M=0

m = (0,0)

m = (1,-1)

yield

Lysaght et al, Phys. Rev. Lett. 102, 193001 (2009)


Momentum Distributions(C2+ in 3Po)

M. A. Lysaght et al, New J. Phys. 11 093014 (2009)

m=0 emission when 2p2 in (0,0) m=±1 emission when 2p2 in (1,-1)

kz


Conclusions

• R-matrix Floquet theory can provide an accurate description of multiphoton ionization processes

• Wide range of multiphoton ionization processes can be investigated

• Can be applied to a wide range of atomic systems

• Provides important information on strong-field dynamics, complementary to time-dependent calculations


N-photon emission of inner electrons

• Investigate the behaviour of 1s2s 1S in HeM. Madine

• How does the ratio 1s / 2s emission change with number of photons?

• What is the influence of resonances?


• Ratio 1s/2s decreases by factor 5 per photon • 1s and 2s emission similar for 3-photon absorption • 4-photon emission affected by resonances


• Three photons required for 1s emission• Intermediate resonances enhance 1s emission• Higher 2lnl´ states → He+ left in higher n


Is experimental observation possible?

• Li+ creation used in previous experiments

• Li+ creation in present study at 1012 W/cm2– Simultaneous detachment of both 2s electrons:

~5% of total detachment– Detachment of 1s : 0.004% of total detachment

• Study electrons due to autoionization of Li

R-matrix Floquet theory for multiphoton processesOverviewBackgroundThe R-matrix Floquet approachPeople involvedSlide 6Mathematical basis of R-matrix-Floquet theoryR-matrix wavefunctionR-matrix wavefunctionsThe TDSEInner region TDSEThe Floquet AnsatzBoundary effectsSlide 14R-matrixHamiltonianSlide 17Slide 18Outer regionOuter region TDSEPropagation region wavefunctionPropagation regionSlide 23PropagationSlide 25Light-Walker propagationSlide 27R-matrix in propagation regionL-to-V transformSlide 30Slide 31Asymptotic solutionsSlide 33Matching procedureSlide 35Practical issuesNumerical issuesAtomic structure dataSlide 39Towards 390 nmSlide 41Slide 42DifficultiesSlide 44Slide 45Slide 46Two-photon double ionization of HeTwo-photon double ionization cross sectionsSlide 49Intense-field emission of inner-shell electronsSingle-photon emission of inner-shell electronsLi and Li- energy levelsLi target statesExcitation of Li 1s2s2Excitation of Li 1s2l2lTwo-photon ionization of Ne+Slide 57Slide 58TDRM approachPump-probe Scheme for C+Ionization Probability, M=0Momentum Distributions (C2+ in 3Po)ConclusionsN-photon emission of inner electronsSlide 65Slide 66Is experimental observation possible?

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