Electron recollision and High Harmonic Generation - … · 3 May 2011 Olga Smirnova MBI-Theory 1...

3 May 2011Olga Smirnova

1MBI-Theory

Electron recollision and High Harmonic Generation

TutorialAttofel summer school 3 May 2011, Crete

Olga SmirnovaMax-Born Institute


2MBI-Theory

HHG: How to generate them fast?

• S-matrix expression for HHG dipole (one electron)• Stationary phase method and factorization of the HHG

dipole (ionization, propagation, recombination)

• Stationary phase equations for HHG: The Lewenstein model

• The classical 3-step photoelectron model: where it goes wrong and how it can be improved

• HHG dipole for many electrons, including laser-induced dynamics in the ionic core between ionization and recombination


3MBI-Theory

HHG: the non-linear optics perspective

HHG is frequency up-conversion. It results from macroscopic response of the medium:

2

2

22

2

22

2 41tP

ctE

czE

∂∂=

∂∂−

∂∂ π

The response is described by polarization P(t,z) induced in the medium:•response of atoms/molecules•response of free electrons•guiding medium (e.g. hollow core fiber)•etc

This lecture is about P(t) from individual atoms/molecules


4MBI-Theory

HHG: the non-linear optics perspective

This lecture is about P(t) from individual atoms/molecules

n- number density)()( tnDtP =

),(ˆ),()( trdtrtD rr ΨΨ=

),()(),( trtHt

tri rr

Ψ=∂

Ψ∂

The key is to find the wavefunction


5MBI-Theory

The S-matrix expressions (one electron)

),()(ˆ),( trtHt

tri rr

Ψ=∂

Ψ∂ )(ˆˆ)(ˆ0 tVHtH L+=

)()'(')(),( )'(ˆ)(ˆ)(ˆ00'

0

00 retVedtiretr gttHi

LdHi

t

tg

ttHit

trrr Ψ−Ψ=Ψ −−∫−−− ∫

ττ

Exact:The hard part Easy part


6MBI-Theory

The S-matrix expressions (one electron)

),()(ˆ),( trtHt

tri rr

Ψ=∂

Ψ∂ )(ˆˆ)(ˆ0 tVHtH L+=

Neglect the Coulomb potentialThe SFA:

)()'(')(),( )'(ˆ)(ˆ)(ˆ 00'

0

00 retVedtiretr gttHi

LdHi

t

tg

ttHit

tLAS rrr Ψ−Ψ=Ψ −−∫−− ∫

ττ

Bound part Continuum part


7MBI-Theory

S-matrix expression for HHG dipole (one electron)


..)()'('ˆ)()( )'(ˆ)(ˆ)(ˆ 00'

0

00 ccretVedtderitD gttHi

LdHi

t

t

ttHig

t

tLAS +ΨΨ−≈ −−∫−− ∫

rr ττ

Bound part Continuum part


8MBI-Theory



..)()'('ˆ)()( )'(ˆ)(ˆ)(ˆ 00'

0

00 ccretVedtderitD gttHi

LdHi

t

t

ttHig

t

tLAS +ΨΨ−≈ −−∫−− ∫

rr ττ

)'()'(1 tAktAkkdrrrrr

++= ∫+∞

∞−

Volkov functions, the length gauge:

)()'( '

2

'

)]([21

)(ˆtAketAke

t

t

t

tLAS

dAkidHirrrr

+=+∫ +−∫−

ττττ


9MBI-Theory


( ) ( ))'()'()(' ),',(* tAkdtFetAkddt LkttiS ++ −

rrrr..)(

0

cckditDt

t

+−=+∞

∞−∫∫r

Phase (action)

( ) ( )∫ −++=t

tttIdAkkttS p

''2)(

21),',( ττ

rr

Ionization? Ionization? ––Not yet!Not yet!recombinationrecombination

Lewenstein et al, 1994


10MBI-Theory


( ) ( ) ..)'()'()(')( ),',(*

0

cctAkdtFetAkddtkditD LkttiS

t

t

+++−= −+∞

∞−∫∫

rrrrr

Evaluate D(t)?

•Numerically – be careful to take care of highly oscillating integrands•Analytically – use highly oscillating integrands to your advantage

-Analytic approach supplies:• “time-energy mapping”, important for attosecond imaging • approximate picture of HHG as a 3-step process involving

ionization, propagation, recombination•• extension beyond SFA and single electron!

S (t,t’,k) is large, theintegrand is a highly oscillating function

extension beyond SFA and single electron!


11MBI-Theory

Stationary phase method

e-iS(x)

Xx*)()( xSie

b

axfdx λ−

∫ 1>>λ

The integral is accumulated in points x*, where phase is stationary:

0*)(' =xSStationary phase region 1/√S’’ Idea:

2*)(*)(''*)*)(('*)()(

2xxxSxxxSxSxS −+−+=

2*)(*)(''*)(*)()()(

2xxxSiedxxSiexfxSie

b

axfdx

−−∫∞+

∞−

−≈−∫

λλλ

Can be evaluated analytically


12MBI-Theory

(see Mathews, Walker,Mathematical Physics)Stationary phase method

e-iS(x)

Xx*)()( xSie

b

axfdx λ−

∫ 1>>λ

The integral is accumulated in points x*, where phase is stationary:

0*)(' =xSStationary phase region 1/√S’’ Idea:

2*)(*)(''*)*)(('*)()(

2xxxSxxxSxSxS −+−+=

( )[ ]*))(''(

4*)(

*)(*)(''

2)()( 1xSsignixSi

eOxfxS

xSieb

axfdx

πλλ

λπλ +−

+=−∫ −

)()( zSiezfdx λ

γ

−∫ For contour integrals in complex plane a

similar idea leads to the saddle point method


13MBI-Theory

Saddle point method for HHG dipole

tiNkttiSt

t

eekddtdtND ωω ),',(

0

')( −+∞

∞−

+∞

∞−∫∫∫∝

rHarmonic spectrum:

Phase=S(t,t’,k)-Nω must be stationary wrt t,t’,k

( ) ( )∫ −++=t

tttIdAkkttS p

''2)(

21),',( ττ

rr

0'=

∂∂tS

0||

=∂∂kS 0=

∂∂

⊥kS

ωNtS =∂∂

1) Ionization time ti

Canonical momentum ks2)

Recombination time tr3)

If we know ti, tr, ks, we know D(Nω) Lewenstein et al, 1994


14MBI-Theory

Results of saddle point integration

Now we need to find ks, tr, ti

( ) 02)(21

'=++=

∂∂

pis ItAktS rr

ionization

( ) 0)(||||

∫ =+=∂∂ r

i

t

tdAk

kS ττ

0)( =−=∂∂

⊥⊥

ir ttkkS 0=⊥k

return

( ) ωNItAktS

prs =++=∂∂ 2)(

21 rr

recombination


15MBI-Theory

Complex ionization time

( ) 02)(21 =++ pis ItAk Ionization time ti is complex ti= ti’+i ti’’1)

Real time

Imaginary time

ti

Tunnel entrance

Tunnel entrance ti=ti’+ti’’

t

Tunnel exit

Tunnel exit

( ) exexist

ex izzitAkdizi

''')'(0

''

+=++= ∫ ττ

ti’’

ti’

t’i

The coordinate at the exit is complex


16MBI-Theory

Recombination time and canonical momentum

Displacement between entering the barrier (start of ionizationstart of ionization) and recombination should be zero

Imaginary displacement “under the barrier” must be compensated: ttrr is is complex

2) Return

( ) 0)(∫ =+r

i

s

t

tdAk ττ

complex

3) Energy conservation Energy conservation dictates that electron velocity at the time of recombination is real( ) ωNItAk prs =++ 2)(

21

Since recombination time tr is complex, canonical momentum canonical momentum kkss is also complexis also complex

In general, In general, kkss, , ttrr, , ttii are all complex. Only the observable are all complex. Only the observable –– the photon the photon

energy energy NNωω –– is realis real


17MBI-Theory

Quantum orbits (Salieres et al, 2000)

How to solve 3 saddle point equations? Total 6 unknowns: ti’,ti’’, tr’,tr’’, ks’,ks’’

( ) 0)(∫ =+r

i

s

t

tdAk ττ( ) 02)(

21 =++ pis ItAk

rr( ) ωNItAk prs =++ 2)(

21

Total 6 equations: for real and imaginary parts. Goal: Set N Find ti’,ti’’, tr’,tr’’, ks’,ks’’

All 6 All 6 eqseqs. do not have analytical solutions. . do not have analytical solutions.

Step 1: express everything via return time (imaginary and real),Step 1: express everything via return time (imaginary and real), using 4 using 4 eqseqs..Step 2: solve the remaining 2 equations togetherStep 2: solve the remaining 2 equations together


18MBI-Theory

Solving the saddle point equations for Nω>Ip

tvtFtA ωωω

sinsin)( 00 −=−=)cos()( 0 tFtF ω=Specify the field:

p

pp

UI

FI

22

2

22 ==

ωγ

02

''vkk s=

ii tωϕ =0

1'

vkk s=Dimensionless variables:

p

pN U

IN2

2 −=

ωγrr tωϕ =

Step 1 a.Step 1 a.

( ) prs INtAk −=+ ω2)(21 rr Real part

rrk 'cos''sinh2 ϕϕ=Nrrk γϕϕ += 'sin''cosh1

Imaginary part

For each N we have expressed k2 and k1 via ϕr’ , ϕr’’


19MBI-Theory

Solutions of the saddle point equationsStep 1 b.Step 1 b.

1''cosh'sin kii =ϕϕ

2'cos''sinh kii += γϕϕ

Real part

Imaginary part( ) 02)(

21 =++ pis ItAk

2~ k+= γγ1~22

1 ++= γkP2

12 4kPD −=

( ) )2/arcsin(' DPi +=ϕ

( ) )2/arccosh('' DPi −=ϕ

For each N we have expressed k2 and k1 via ϕr’ , ϕr’’ and we have expressed ϕi’ , ϕi’’via k1, k2

Hence ϕi’ , ϕi’’ and k1, k2 are all expressed via ϕr’ , ϕr’’

Now we can use the remaining equations to find ϕr’ , ϕr’’


20MBI-Theory

Solutions of the saddle point equations

( ) 0)(∫ =+r

i

s

t

tdAk ττStep 2: Now use the last equation:

( ) ( ) 0'cos''cosh''cosh'cos'''''' 211 =+−−−−= rriiirir kkF ϕϕϕϕϕϕϕϕReal part

( ) ( ) 0'sin''sinh''sinh'sin'''''' 212 =−+−+−= rriiirir kkF ϕϕϕϕϕϕϕϕIm part

( )[ ] ( )[ ] 0'',','',', 22

21 =+ rrrr NFNF ϕϕϕϕ

Set grid of ϕr’ and ϕr’’ and numerically find minimum of this surface for each N


21MBI-Theory

SolutionsHarmonic 11

Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV

Imaginary time of return, units of laser cycle


22MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



23MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



24MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



25MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



26MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



27MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



28MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



29MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



30MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



31MBI-Theory


Ip=15.6 eV

Rea

l tim

e of

retu

rn, u

nits

of l

aser

cyc

le

I=1.3 1014W/cm2

ω=1.5 eV



32MBI-Theory

Energy of return

Short and long trajectories: two different saddle point solutions for the same Energy of return (Harmonic number)

Cut-off

short longtrajectories

Below threshold harmonics

Saddle point method breaks down near the cut-off: 2 saddle points merge (Stt’’=0)


33MBI-Theory

Imaginary time of return

Imaginary& real time of return define integration contour in complex plane: only along this contour the energy of return is real.

Cut-off

Belowthreshold Short Long

Difficult part is here: divergence


34MBI-Theory

Imaginary time of return

Imaginary& real time of return define integration contour in complex plane: only along this contour the energy of return is real.

Cut-off

Belowthreshold Short Long

Would be niceto have it like so


35MBI-Theory

Treating the cut-off regionStep1: Step1: Find tr =t0 (and ti=ti0, ks=ks0), such that Stt’’ (tr0,ti0,ks0) =0, i.e. dEret/dt=0(Pick the cut-off (real) return time for t0)

Step2:Step2:Expand the action S(tr,ti,ks) around t0 ( the uniform approximation( the uniform approximation )

6)(),,())(,,(),,(),,(

30

000'''

0000'

00000ttkttSttkttSkttSkttS iitttiitiiii

−+−+=

Step3:Step3:

The resulting integral for dipole: ..)( ),,( 00 cceedtND tiNkttiS si +∝ −+∞

∞−∫ ωω

can be calculated analytically using Airy function:

( ) ( ) ⎥⎦

⎤⎢⎣

⎡±≡±∫

+∞

∞−3/13/1

3

33)cos(

axAi

axtat π


36MBI-Theory

Treating the cut-off region

Introduce the cut-off harmonic number NN0 0 and “the distance from cut-off” ∆∆N=NN=N-- NN00 :

(N0 does not have to be integer )ω000' )()( NItEtS prett ≡+=

The dipole near cut-off is expressed via Airy function :

ξωξξχω ω dNcceedtND tiNkttiS si )6

cos(..)( 3),,( 00 ∆+=+∝ ∫∫+∞

∞−

−+∞

∞−

[ ] ( ) ⎥⎦

⎤⎢⎣

⎡ ∆∝ 3131 222)(

χω

χπω NAiND

∆N<0 before cut-off: Ai~cos[-(∆Nω)3/2 (8/9χ)1/2 ]

∆N>0 after cut-off: Ai~exp[-(∆Nω)3/2 (8/9χ)1/2 ]

(oscillations are due to interference of short and long)

(exponential decrease of HH spectrum after cut-off)ωχ 00 )( Ftv≅ ω00' )( FtFt ≅

0)( 0 ≅tF

)( 0''' tSttt−=χ


37MBI-Theory

Ionization times: sub-cycle dynamics of ionizationBelowthreshold Short Long

Ionization window

time

~250asec

670asec

0 1

Imaginary ionization time defines ionization probability:short trajectories are suppressedReal ionization time defines “duration of ionization window”


38MBI-Theory

Canonical momenta

Belowthreshold

Short Long

Belowthreshold

Short Long

Long trajectories: imaginary canonical momentum is very smallShort trajectories: substantial imaginary canonical momentumPhotoelectrons: registered at the detector - canonical momentum is real


39MBI-Theory

Classical 3-step model: photoelectron perspective

Define “classical ionization time” – “time of birth”, when electron velocity is zero.

( ) 02)()(21 =+− pBi ItAtAv(tB)=k+A(tB)=0 v(ti)=A(ti)-A(tB)=-i γ

k=-A(tB) Different from “quantum ionization times”, since k is forced to be real.

Real time

Imaginary time

Tunnel entrance ti=ti’+ti’’

Tunnel exit

t

ti’’

ti’ tB

v=0

Imaginary ionization time

Real ionization time


40MBI-Theory

Classical 3-step model: photoelectron perspective

( ) 0)()(∫ =−R

B

B

t

tdtAA ττDefine classical return time:

Electron returns to the point (coordinate), where it had zero velocity

ti t’i tB

tB>ti’, v(tB)=0v(ti’)=k+A(ti’)<0

The electron is not born with v=0.

Classical return energy: E=(A(tR)-A(tB))2/2


41MBI-Theory

Classical energy of returnClassical cut-off corresponds to lower energies:electron does not return to the core

Cut-off: 3.17 Up+1.32 Ip

short longtrajectories

Below threshold harmonics

Cut-off: 3.17 Up+ Ip

Can we improve these results? Let’s let the photo-electrons return to the core.


42MBI-Theory

Photoelectrons vs Lewenstein modelEnergy at closest approach: not all electrons can return exactly to the core since we limited k=-A(tB)

Belowthreshold:n/a

Short Long

Lewenstein model

Improved 3 Improved 3 ––step model (k<1) +step model (k<1) +relaxed return condition:relaxed return condition:

•Neglect Imag ∆z =0 •Minimize Real ∆z

The strict requirement of perfect return is an artefact of neglecting the size of the ground state. Relaxing this requirement seems quite reasonable!


43MBI-Theory

Photoelectrons: return coordinate

Not all electrons can return exactly to the core since we limited k=-A(tB)

Return is notpossible(with k<1)

Exactreturn on real axis Re

al ∆

z

Real time of return, units of laser cycle


44MBI-Theory

Photoelectrons: saddle point region

Real k should be within the saddle point region of the exact complex saddle point. This region is ~ |S’’kk|-1/2

Inside the stationary phase region

|∆k|<|tr-ti|1/2

|∆k|=∆z/(tr-ti)

|∆k|<|S’’kk|-1/2

|∆z|<(tr-ti) 1/2

Real time of return, units of laser cycle

|∆k|- difference between the exact ks and photolelctron ks|∆z|- distance of the closest approach to the core

|∆z|<(tr-ti) 1/2

(tr-ti) 1/2

|∆z|


45MBI-Theory

Photoelectrons vs Lewenstein model

Ionization time

Belowthreshold:n/a

Short Long

ImaginaryionizationtimeReal

Ionizationtime

~100as

Lewenstein model

Photoelectrons, k is not restricted (k can be >1)

Improved 3 –step model +relaxed return condition

Long trajectories: good agreement due to small imaginary displacement


46MBI-Theory

Factorization of the dipole

Dipole = product of 3 amplitudes: ionization, propagation, recombination

);(),;();()( isioniRspropRsrec tkattkatkaND =ω

This factorization is rigorous within the photo-electron picture, i.e. if the imaginary part of the canonical momentum is negligible

The next step is to take each amplitude separately and improve it beyond the SFA and the simple model of an ion without internal

states.


47MBI-Theory




( ) ( )

'',

')(21

'2

)]([~);(ii

it

itiips

tt

ttiIdAki

issionS

etAkdtka πττ∫ −−+−

+rr

The amplitudes are (within saddle point): 1. Ionization

Re time

Im timeti=Re[ti]+ i*Im[ti ]=ti’+ i ti’’


48MBI-Theory




The amplitudes are (within saddle point):

Propagation: wavepacket spreading and phase accumulation

( ) ( )

2/3

3')(21

][~)';;( '

2

iR

ttiIdAki

iRsprop ttettka

Rt

itiRps

−

∫ −−+− πττrr


49MBI-Theory



);()',;();()( isioniRspropRsrec tkattkatkaND =ω

The amplitudes are (within saddle point):

Recombination: proportional to the recombination dipole

''))((*~);(

RRtt

RsRsrecS

tAkdtka π+


50MBI-Theory

Ionization amplitude

( ) ( )

'',

')(21

'2

)]([);(ii

it

itiips

tt

ttiIdAki

issionS

etAkdtka πττ∫ −−+−

+=rr

This expression came from applying the saddle point approximation to

( ))'()'('),( '),',(

0

tAkdtFedtitka sLtIkttiS

t

ts

prr

+−= +−∫This integral has been extensively studied by Keldysh, PPT (Popov, Perelomov, Teren’tev), and others. The SFA result can be

significantly improved!


51MBI-Theory

Ionization amplitude

),(tSFAexponen, ),(),( is tk

pmlision eFIRtka −=

The recipe for atoms: (PPT, Keldysh, Yudin-Ivanov, Becker-Faisal, Popruzhenko-Bauer) SubSub--cycle ratescycle rates

-Calculate exponential dependence with SFA- Add Coulomb correction Rlm to account for the core.

- The Coulomb correction depends on l, m - With reasonable accuracy and for linearly polarized or low-

frequency fields the Coulomb correction can be taken from statictunneling The recipe for molecules: Take the Coulomb correction from static

tunneling rates (MO-ADK (Tong&Lin), recently Murray & Ivanov)


52MBI-Theory

The Multi-Channel Case

Exact ‘continuum’ part: electron+ion

gttHi

LdHi

t

tc etVedtit

t

t Ψ−=Ψ −−∫−∫)'(ˆ)(ˆ

00'

0

)'(')( ττ

)'()'(1 tAknntAkkdn

rrrrr++= ∫∑

+∞

∞− Polarized Core states

Continuum states

gttHi

LdHi

t

tnc etVtknntkedtkdit

t

t Ψ−=Ψ −−∫−+∞

∞−∫∫∑ )'(ˆ)(ˆ

00'

0

)'()'()'(')(rrr ττ

nD

ttHiL etVtk Ψ−− )'(ˆ

00)'()'(r

gnD n Ψ∝ΨDyson orbital:


53MBI-Theory



gttHi

LdHi

t

tc etVedtit

t

t Ψ−=Ψ −−∫−∫)'(ˆ)(ˆ

00'

0

)'(')( ττ

)'()'(1 tAknntAkkdn

rrrrr++= ∫∑

+∞

∞− Polarized Core states

Continuum states

nD

ttHiL

dHit

tnc etVtkntkedtkdit

t

t Ψ−=Ψ −−∫−+∞

∞−∫∫∑ )'(ˆ)(ˆ

00'

0

)'()'()'(')(rrr ττ

Neglect e-e correlation after ionization: the continuum electron moves in a self-consistent field of the core


54MBI-Theory



nD

ttHiL

dHit

tnc etVtkntkedtkdit

t

t Ψ−=Ψ −−∫−+∞

∞−∫∫∑ )'(ˆ)(ˆ

00'

0

)'()'()'(')(rrr ττ

Neglect e-e correlation after ionization: the continuum electron moves in a self-consistent field of the core

nD

ttHiL

dHidHit

tnc etVtknetkedtkdit

t

ti

t

tc Ψ−=Ψ −−∫−∫−

+∞

∞−∫∫∑ )'(ˆ)(ˆ)(ˆ

00''

0

)'()'()'(')(rrr ττττ

continuum ion

Evolution in the continuum – like beforeEvolution in the ion – start in |n> at t’, end in |m> at t, amplitude amn(t,t’)


55MBI-Theory


nD

ttHiL

dHidHit

tnc etVtknetkedtkdit

t

ti

t

tc Ψ−=Ψ −−∫−∫−

+∞

∞−∫∫∑ )'(ˆ)(ˆ)(ˆ

00''

0

)'()'()'(')(rrr ττττ

continuum ion

Ion |n>

Ion |m>-

ionization

recombination


56MBI-Theory

The harmonic dipole in the multi-channel case

The harmonic dipole is a sum over all ionic states at tion and trec

);(),;();()( ,,,,

isnioniRsmnpropRsmrecmn

tkattkatkaND ∑=ω

The key change is in the propagation amplitude— it includes transitions between the initial (n) and the final (m) states of the ionic core:

Recombination –the electron recombines with the ion in the state m:

( ) ( )

2/3

3')(21

,, ][)',(~)';;( '

,2

iR

ttiIdAki

iRmncoreiRsmnprop ttettattka

Rt

itiRnps

−

∫ −−+− πττrr

'', ))((*~);(RRtt

RsmRsmrec StAkdtka π+


57MBI-Theory

Example of core dynamics in the multi-channel case

Populations of ionic states

X 2Σ+g

A 2Πu

B 2Σ+u

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

|A|2

|X|2

|B|2

3.5eV1.3 eV

Ip~15.6 eV

Time, laser cycleInitial condition: population of the polarized ground state of N2

+ upon ionization, I=1014W/cm2

Find dipole couplings between the states A,B,X. Solve 3-level system numerically

Date post:	05-Jun-2018
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Electron recollision and High Harmonic Generation - … · 3 May 2011 Olga Smirnova MBI-Theory 1...

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