Correlated two-electron quantumdynamics in intense laser fields

Dieter BauerUniversity of Rostock, Germany

7th Workshop on Time-Dependent Density Functional TheoryBenasque, Spain, September 20–23, 2016

People

Julius Rapp

Adrian HanuschMartinš Brics

Outline

1 Motivation

2 Natural orbitals and their propagation

3 Results from TDRNOT

4 Conclusion

Motivation to simulate few-body systems such asHe or H+

2 in intense laser fields

Ab initio simulation far behind experiment.

N = 2 at the boundary of what is treatable on a TDSElevel.

Highly correlated systems.

Ideal test bed for “many”-body approaches.

First step towards N > 2 in a systematic bottom-upapproach.

Functional Specification Documentfor a useful strong-field ab initio code

Method must work for high (yet nonrelativistic,A = E/ω � 137) laser intensities in a wide frequencyregime and for arbitrary pulse forms.

Configurations far away from the ground state.Several electrons might be in the continuum.Interaction might be resonant.Autoionization, Auger, interatomic Coulomb decay, ...

Method must allow for the calculation of:ion yields,emitted radiation,(correlated) photoelectron spectra.

Code must have typical run times t � TPhD.

Can we be more efficient than TDSE,MCTDHF, and full TDCI but (much)more accurate than (practicable)

TDDFT?

Obvious idea:

use a basic variable that is “more differential” than

nσ(~r , t) but less than Ψ(~r1σ1, . . . ,~rNσN)

Can we be more efficient than TDSE,MCTDHF, and full TDCI but (much)more accurate than (practicable)

TDDFT?

Obvious idea:

use a basic variable that is “more differential” than

nσ(~r , t) but less than Ψ(~r1σ1, . . . ,~rNσN)

Reduced density matrices and natural orbitalsP.-O. Löwdin, Phys. Rev. 97, 1474 (1955)P.-O. Löwdin, H. Shull, Phys. Rev. 101, 1730 (1956)C.A. Coulson, Rev. Mod. Phys. 32, 170 (1960)A.J. Coleman, Rev. Mod. Phys. 35, 668 (1963)A.J. Coleman, V.I. Yukalov, Reduced Density Matrices, Coulson’s Challenge, (Springer, 2000)J. Cioslowski (Ed.), Many-electron densities and reduced density matrices, (Kluwer/Plenum, New York, 2000)N.I. Gidopoulos, S. Wilson (Eds.), Electron Density, Density Matrix and Density Functional Theory in Atoms,Molecules and the Solid State, (Kluwer, Dordrecht, 2003)T.L. Gilbert, Phys. Rev. B12, 2111 (1975)D.A. Mazziotti (Ed.), Reduced-Density-Matrix Mechanics, (Wiley, Hoboken, 2007)H. Appel, Time-Dependent Quantum Many-Body Systems: Linear Response, Electronic Transport, andReduced Density Matrices, (Doctoral Thesis, Free University Berlin, 2007)K.J.H. Giesbertz, Time-Dependent One-Body Reduced Density Matrix Functional Theory, AdiabaticApproximations and Beyond, (PhD Thesis, Free University Amsterdam, 2010)A.M.K. Müller, Phys. Lett. 105A, 446 (1984)S. Goedecker, C.J. Umrigar, Phys. Rev. Lett. 81, 866 (1998)O. Gritsenko, K. Pernal, E.J. Baerends, J. Chem. Phys. 122, 204102 (2005)M. Piris, Int. J. Quant. Chem. 113, 620 (2013)E.N. Zarkadoula, S. Sharma, J.K. Dewhurst, E.K.U. Gross, N.N. Lathiotakis, Phys. Rev. A85, 032504 (2012)D.A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)K. Pernal, Phys. Rev. Lett. 94, 233002 (2005)N. Helbig, N.N. Lathiotakis, M. Albrecht, and E.K.U. Gross, Europhys. Lett. 77, 67003 (2007)Ch. Schilling, D. Gross, M. Christandl, Phys. Rev. Lett. 110, 040404 (2013)D.A. Portes, Jr., Takeshi Kodama, A.F.R. de Toledo Piza, Phys. Rev. A54, 1889 (1996)K. Pernal, O. Gritsenko, E.J. Baerends, Phys. Rev. A75, 012506 (2007)K.J.H. Giesbertz, E.J. Baerends, O.V. Gritsenko, Phys. Rev. Lett. 101, 033004 (2008)R. Requist, O. Pankratov, Phys. Rev. A81, 042519 (2010)H. Appel, E.K.U. Gross, Europhys. Lett. 92, 23001 (2010)A.K. Rajam, I. Raczkowska, N.T. Maitra, Phys. Rev. Lett. 105, 113002 (2010)M. Brics, D. Bauer, Phys. Rev. A88, 052514 (2013)J. Rapp, M. Brics, D. Bauer, Phys. Rev. A90, 012518 (2014)M. Brics, J. Rapp, D. Bauer, Phys. Rev. A90, 053418 (2014)M. Brics, J. Rapp, D. Bauer, Phys. Rev. A93, 013404 (2016)A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

Time-dependent natural orbital (NO) theory

1-RDM everything in color is time-dependent!

γ1 = N TrN−1{|Ψ〉〈Ψ|

}

1-RDM expanded in NOs

γ1 =∞∑

k=1

nk |k〉〈k |,∑

k

nk = N

NOs are eigenvectors of 1-RDM

γ1|k〉 = nk |k〉

Occupation numbers (OCs) nk are eigenvalues

Non-integer nk ⇔ correlation ∼ S ∼∑

k

nk ln nk

Time-dependent natural orbital (NO) theory

1-RDM everything in color is time-dependent!

γ1 = N TrN−1{|Ψ〉〈Ψ|

}1-RDM expanded in NOs

γ1 =∞∑

k=1

nk |k〉〈k |,∑

k

nk = N

NOs are eigenvectors of 1-RDM

γ1|k〉 = nk |k〉

Occupation numbers (OCs) nk are eigenvalues

Non-integer nk ⇔ correlation ∼ S ∼∑

k

nk ln nk

Time-dependent natural orbital (NO) theory

1-RDM everything in color is time-dependent!

γ1 = N TrN−1{|Ψ〉〈Ψ|

}1-RDM expanded in NOs

γ1 =∞∑

k=1

nk |k〉〈k |,∑

k

nk = N

NOs are eigenvectors of 1-RDM

γ1|k〉 = nk |k〉

Occupation numbers (OCs) nk are eigenvalues

Non-integer nk ⇔ correlation ∼ S ∼∑

k

nk ln nk

Equation of motion (EOM)

Our goal: Derive useful EOM for NOs and OCs

EOM for 1-RDM involves 2-RDM (BBGKY)

i ˙γ1 = 1-body part with γ1, h + Tr2

{2-body part with γ2, vee

}

2-RDM expanded in NOs

γ2 =∑ijkl

γ2,ijkl |i〉|j〉〈k |〈l |

Approximations of γ2,ijklHartree-Fock limit: γ2,ijkl = δikδjl − δilδjk , OCs nk = 1 (or 0)(others: Müller, Goedecker & Umrigar, Baerends, Piris)

→ problem of observables relaxed

Equation of motion (EOM)

Our goal: Derive useful EOM for NOs and OCs

EOM for 1-RDM involves 2-RDM (BBGKY)

i ˙γ1 = 1-body part with γ1, h + Tr2

{2-body part with γ2, vee

}2-RDM expanded in NOs

γ2 =∑ijkl

γ2,ijkl |i〉|j〉〈k |〈l |

Approximations of γ2,ijklHartree-Fock limit: γ2,ijkl = δikδjl − δilδjk , OCs nk = 1 (or 0)(others: Müller, Goedecker & Umrigar, Baerends, Piris)

→ problem of observables relaxed

Equation of motion (EOM)

Our goal: Derive useful EOM for NOs and OCs

EOM for 1-RDM involves 2-RDM (BBGKY)

i ˙γ1 = 1-body part with γ1, h + Tr2

{2-body part with γ2, vee

}2-RDM expanded in NOs

γ2 =∑ijkl

γ2,ijkl |i〉|j〉〈k |〈l |

Approximations of γ2,ijklHartree-Fock limit: γ2,ijkl = δikδjl − δilδjk , OCs nk = 1 (or 0)(others: Müller, Goedecker & Umrigar, Baerends, Piris)

→ problem of observables relaxed

Useful EOM for NOs

i ˙γ1 = 1-body part with γ1, h + Tr2

{2-body part with γ2, vee

}Left hand side:

iddt

∑k

nk |k〉〈k | = i∑

k

{nk |k〉〈k |+ nk

˙|k〉〈k |+ nk |k〉 ˙〈k |}

Introduce

i|k〉 =∑

m

αkm|m〉, αmk = α∗km = i〈k |m〉

⇒

∑k

{ink |k〉〈k |+ ink

˙|k〉〈k | − nk∑

m

αmk |k〉〈m|

}= right hand side

Determination of the αkm⟨m∣∣∣∑

l

{inl |l〉〈l|+ nl

∑p

αlp|p〉〈l| − nl

∑p

αpl |l〉〈p|

}= right hand side

∣∣∣k⟩⇒ ink δkm + (nk − nm)αkm =

⟨m∣∣∣right hand side

∣∣∣k⟩

For k = m follows with H =∑N

i=1 h(i) +∑

i<j v (i,j)ee :

nk = 4 Im∑

ijl

γ2,ijkl〈lk |vee|ji〉

EOM for nk ... but αkk is undetermined

For k 6= m and nk 6= nm:

αkm = 〈m|h|k〉+ 2nm − nk

∑pjl

{γ2,mpjl〈lj |vee|pk〉 −

[γ2,kpjl〈lj |vee|pm〉

]∗}If nk = nm despite k 6= m: αkm undetermined

Determination of the αkm⟨m∣∣∣∑

l

{inl |l〉〈l|+ nl

∑p

αlp|p〉〈l| − nl

∑p

αpl |l〉〈p|

}= right hand side

∣∣∣k⟩⇒ ink δkm + (nk − nm)αkm =

⟨m∣∣∣right hand side

∣∣∣k⟩

For k = m follows with H =∑N

i=1 h(i) +∑

i<j v (i,j)ee :

nk = 4 Im∑

ijl

γ2,ijkl〈lk |vee|ji〉

EOM for nk ... but αkk is undetermined

For k 6= m and nk 6= nm:

αkm = 〈m|h|k〉+ 2nm − nk

∑pjl

{γ2,mpjl〈lj |vee|pk〉 −

[γ2,kpjl〈lj |vee|pm〉

]∗}

If nk = nm despite k 6= m: αkm undetermined

Determination of the αkm⟨m∣∣∣∑

l

{inl |l〉〈l|+ nl

∑p

αlp|p〉〈l| − nl

∑p

αpl |l〉〈p|

}= right hand side

∣∣∣k⟩⇒ ink δkm + (nk − nm)αkm =

⟨m∣∣∣right hand side

∣∣∣k⟩

For k = m follows with H =∑N

i=1 h(i) +∑

i<j v (i,j)ee :

nk = 4 Im∑

ijl

γ2,ijkl〈lk |vee|ji〉

EOM for nk ... but αkk is undetermined

For k 6= m and nk 6= nm:

αkm = 〈m|h|k〉+ 2nm − nk

∑pjl

{γ2,mpjl〈lj |vee|pk〉 −

[γ2,kpjl〈lj |vee|pm〉

]∗}If nk = nm despite k 6= m: αkm undetermined

Useful EOM for NOs

i|k〉 =∑

m αkm|m〉 is not a useful EOM. Instead,

⟨1∣∣∣∑

k

{ink |k〉〈k |+ ink

˙|k〉〈k | − nk

∑m

αmk |k〉〈m|

}= right hand side

∣∣∣p⟩

⇒ inp∂tφp(1) = −inpφp(1) +∑

k

nkαpkφk (1) +⟨

1∣∣∣right hand side

∣∣∣p⟩

has already the useful form

i∂t ~φ(1) = H ~φ(1)

but what about αkk and degeneracy?

The general two-fermion caseP.-O. Löwdin, H. Shull, Phys. Rev. 101, 1730 (1956)

Two-fermion state |Ψ〉, expanded in orthonormalsingle-particle basis {|ψi〉} (comprising spin and otherdegrees of freedom)

|Ψ〉 =∑

ij

Ψij |ψi , ψj〉,

Ψij = 〈ψi , ψj |Ψ〉.

Define matrix Ψ =[Ψij]. Antisymmetry implies

ΨT = −Ψ.

The general two-fermion case

With

ψ =

|ψ1〉|ψ2〉

...

, ψ∗ =

〈ψ1|〈ψ2|

...

,

ψT =(|ψ1〉, |ψ2〉, . . .

), ψ† =

(〈ψ1|, 〈ψ2|, . . .

)one can write

ψ∗ψ† =

〈ψ1, ψ1| 〈ψ1, ψ2| . . .〈ψ2, ψ1| 〈ψ2, ψ2| . . .

......

. . .

Hence

Ψ = ψ∗ψ†|Ψ〉, |Ψ〉 = ψTΨψ.

The general two-fermion case

Any skew-symmetric matrix Ψ = −ΨT can be factorizedinto unitary matrices U,UT and a block-diagonal matrix,

Ψ = UΣUT, Σ = diag(Σ1,Σ3,Σ5, . . . ),

Σi =

(0 ξi−ξi 0

), i odd.

Hence,

|Ψ〉 = ψTUΣUTψ = φTΣφ, φ = UTψ

and thus

|Ψ〉 =∑i odd

ξi

[|φi , φi+1〉 − |φi+1, φi〉

].

The general two-fermion case

For the 1-RDM follows

γ1 =∑

k odd

2|ξk |2[|φk 〉〈φk |+ |φk+1〉〈φk+1|

],

which proves that |k〉 = |φk 〉, i.e., the set {|φk 〉} is a set ofNOs, and 2|ξk |2 are the pairwise degenerate ONs

nk = nk+1 = 2|ξk |2, k odd.

Henceγ1 =

∑k odd

nk[|k〉〈k |+ |k + 1〉〈k + 1|

]

The general two-fermion case

Exact 2-fermion state

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k + 1〉 − |k + 1〉|k〉

]

Exact 2-DM

|Ψ〉〈Ψ| = γ2 =∑ijkl

(−1)i−k ei(ϕi−ϕk )

√nink

2δij′δkl′︸ ︷︷ ︸

γ2,ijkl

|i〉|j〉〈k |〈l |

j ′ = j ± 1 for j odd or even, respectively

The general two-fermion case

Exact 2-fermion state

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k + 1〉 − |k + 1〉|k〉

]Exact 2-DM

|Ψ〉〈Ψ| = γ2 =∑ijkl

(−1)i−k ei(ϕi−ϕk )

√nink

2δij′δkl′︸ ︷︷ ︸

γ2,ijkl

|i〉|j〉〈k |〈l |

j ′ = j ± 1 for j odd or even, respectively

The general two-fermion case

Plugging the 2-fermion state

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k ′〉 − |k ′〉|k〉

]into the TDSE and multiplication from the left by 〈k |〈k ′|yields

αkk + αk ′k ′

= 〈k |h|k〉+ 〈k ′|h|k ′〉+2nk

Re∑

ijl

γ2,ijkl〈kl | ˆvee|ij〉

Phase freedom

1-RDM independent of NO phases

γ1 =∑

k odd

nk[|k〉〈k |+ |k ′〉〈k ′|

]

Exact 2-fermion state

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k ′〉 − |k ′〉|k〉

]not because

|j〉 = eiϑj |j〉leads to

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k ′〉 − |k ′〉|k〉

]with a new phase

ϕk = ϕk − ϑk − ϑ′k

Phase freedom

1-RDM independent of NO phases

γ1 =∑

k odd

nk[|k〉〈k |+ |k ′〉〈k ′|

]Exact 2-fermion state

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k ′〉 − |k ′〉|k〉

]not because

|j〉 = eiϑj |j〉leads to

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k ′〉 − |k ′〉|k〉

]with a new phase

ϕk = ϕk − ϑk − ϑ′k

Phase choices“Standard choice”

i〈k |k〉 = αkk = 0 ⇔ ϑk (t) = i∫ t

〈k(t ′)|∂t′ |k(t ′)〉 dt ′

leads to time-dependent phases ϕk and nk = 0 duringimaginary-time propagation

Giesbertz’s phase-including NOs (PINOs)

ϑk (t) = ϑk ′(t) =12

[ϕk (t)− ϕk (0)]

shift time-dependence to the NOs,

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k + 1〉 − |k + 1〉|k〉

],

and imaginary-time propagation yield real groundstate NOs witheiϕ1 = 1, eiϕ3 = eiϕ5 = . . . = −1 (singlet)eiϕ1 = eiϕ3 = eiϕ5 = . . . = 1 (triplet)

Phase choices“Standard choice”

i〈k |k〉 = αkk = 0 ⇔ ϑk (t) = i∫ t

〈k(t ′)|∂t′ |k(t ′)〉 dt ′

leads to time-dependent phases ϕk and nk = 0 duringimaginary-time propagation

Giesbertz’s phase-including NOs (PINOs)

ϑk (t) = ϑk ′(t) =12

[ϕk (t)− ϕk (0)]

shift time-dependence to the NOs,

|Ψ〉 =∑

k odd

√nk

2eiϕk

[|k〉|k + 1〉 − |k + 1〉|k〉

],

and imaginary-time propagation yield real groundstate NOs witheiϕ1 = 1, eiϕ3 = eiϕ5 = . . . = −1 (singlet)eiϕ1 = eiϕ3 = eiϕ5 = . . . = 1 (triplet)

TDRNOT in position space

Get rid of spin degrees, i.e., write

|Ψ〉 = |Φ0,1〉 ⊗1√2

[|+−〉 ∓ | −+〉

], |Ψ〉 = |Φ1〉 ⊗ | ± ±〉

Go to position space (φk (x) = 〈x |k〉) and derive EOM for spatialNOs

i~φ(x) = H(x) ~φ(x)

Unify φk (x) and nk into single EOM

i ~φ(x) = ˆH(x) ~φ(x)

by renormalizing |k〉 =√

nk |k〉,∫dx |φk (x)|2 = nk

TDRNOT in position space

Get rid of spin degrees, i.e., write

|Ψ〉 = |Φ0,1〉 ⊗1√2

[|+−〉 ∓ | −+〉

], |Ψ〉 = |Φ1〉 ⊗ | ± ±〉

Go to position space (φk (x) = 〈x |k〉) and derive EOM for spatialNOs

i~φ(x) = H(x) ~φ(x)

Unify φk (x) and nk into single EOM

i ~φ(x) = ˆH(x) ~φ(x)

by renormalizing |k〉 =√

nk |k〉,∫dx |φk (x)|2 = nk

TDRNOT in position space

Get rid of spin degrees, i.e., write

|Ψ〉 = |Φ0,1〉 ⊗1√2

[|+−〉 ∓ | −+〉

], |Ψ〉 = |Φ1〉 ⊗ | ± ±〉

Go to position space (φk (x) = 〈x |k〉) and derive EOM for spatialNOs

i~φ(x) = H(x) ~φ(x)

Unify φk (x) and nk into single EOM

i ~φ(x) = ˆH(x) ~φ(x)

by renormalizing |k〉 =√

nk |k〉,∫dx |φk (x)|2 = nk

NO-Hamiltonian ˆH everything in color is time-dependent!

ˆHkm = (h +Ak)δkm + Bkm + Ckm

with

h = p2/2 + v + Ap,

Ak = − 1nk

Re∑jpl

γ2,kjpl〈pl |vee|k j〉, nk = 〈k |k〉,

Bkmm 6=k=

2nm − nk

∑jpl

{γ2,mjpl〈pl |vee|k j〉 −

[γ2,kjpl〈pl |vee|mj〉

]∗},

Ckm = 2∑

jl

γ2,mjkl 〈l |vee |j〉.

J. Rapp, M. Brics, D. Bauer; Phys. Rev. A 90, 012518 (2014)

TDRNOT in position space

Unitary propagation on a grid

Space

NO

in

de

x

TDDFT

beyond

Use favorite unconditionally stable unitary propagator

Results

Test system1D model He in intense laser fields

in TDSE:

V (x , x ′, t) = − 2√1 + x2

− 2√1 + x ′2

+ E(t)(x + x ′),

Vee(|x − x ′|) =1√

1 + (x − x ′)2

Test system1D model He in intense laser fields

in TDSE:

V (x , x ′, t) = − 2√1 + x2

− 2√1 + x ′2

+ E(t)(x + x ′),

Vee(|x − x ′|) =1√

1 + (x − x ′)2

Test system1D model He in intense laser fields

in TDSE:

V (x , x ′, t) = − 2√1 + x2

− 2√1 + x ′2

+ E(t)(x + x ′),

Vee(|x − x ′|) =1√

1 + (x − x ′)2

Ground statesvia imaginary propagation

Number of Total energy Dominant occupation numbersRNOs No E0 (a.u.) n1 n3/10−3 n5/10−5

Spin-singlet

2 (TDHF) −2.224318 1.00000004 (TDRNOT) −2.236595 0.9912665 8.73356 (TDRNOT) −2.238203 0.9909590 8.3142 72.6838 (TDRNOT) −2.238324 0.9909438 8.3221 70.229∞ (TDSE) −2.238368 0.9909473 8.3053 70.744

Spin-triplet

2 (TDHF) −1.8120524 1.000000004 (TDRNOT) −1.8160798 0.99764048 2.359526 (TDRNOT) −1.8161870 0.99760705 2.36464 2.82988 (TDRNOT) −1.8161945 0.99760656 2.36267 2.9581∞ (TDSE) −1.8161954 0.99760677 2.36220 2.9610

Ground state ONsvia imaginary propagation (spin-singlet)

ON

orbital number

10−20

10−15

10−10

10−5

100

10 20 30 40 50 60 70 80

Linear response

Energy E − E0 (a.u.)

Dip

ole

stre

ngth

(arb

.u.)

0.0 0.5 1.0 1.5 2.0 2.5 3.010−

1510−

1010−

510

0

No =∞ (TDSE)

No = 2 (TDHF)

Linear response

Energy E − E0 (a.u.)

Dip

ole

stre

ngth

(arb

.u.)

0.0 0.5 1.0 1.5 2.0 2.5 3.010−

1510−

1010−

510

0

No =∞ (TDSE)

No = 2 (TDHF)

No = 4

Linear response

Energy E − E0 (a.u.)

Dip

ole

stre

ngth

(arb

.u.)

0.0 0.5 1.0 1.5 2.0 2.5 3.010−

1510−

1010−

510

0

No =∞ (TDSE)

No = 2 (TDHF)

No = 8

Linear response

Energy E − E0 (a.u.)

Dip

ole

stre

ngth

(arb

.u.)

0.0 0.5 1.0 1.5 2.0 2.5 3.010−

1510−

1010−

510

0

No =∞ (TDSE)

No = 2 (TDHF)

No = 16

Absorption spectroscopyLorentz vs Fano C. Ott et al., Science 340, 716 (2013)

Photon energy ω (a.u.)

Spe

ctra

lem

issi

on(a

rb.u

.)

0.0 0.5 1.0 1.5 2.0 2.5 3.0−1.

0−

0.5

0.0

0.5

Em

issi

onA

bsor

ptio

n

No = 2 (TDHF)

ω = 1.84, 3-cycle sin2 + postprop., 1012 Wcm−2

Absorption spectroscopyLorentz vs Fano C. Ott et al., Science 340, 716 (2013)

Photon energy ω (a.u.)

Spe

ctra

lem

issi

on(a

rb.u

.)

0.0 0.5 1.0 1.5 2.0 2.5 3.0−1.

0−

0.5

0.0

0.5

Em

issi

onA

bsor

ptio

n

No =∞ (TDSE)No = 2 (TDHF)

ω = 1.84, 3-cycle sin2 + postprop., 1012 Wcm−2

Absorption spectroscopyLorentz vs Fano C. Ott et al., Science 340, 716 (2013)

Photon energy ω (a.u.)

Spe

ctra

lem

issi

on(a

rb.u

.)

0.0 0.5 1.0 1.5 2.0 2.5 3.0−1.

0−

0.5

0.0

0.5

Em

issi

onA

bsor

ptio

n

No =∞ (TDSE)No = 2 (TDHF)No = 8 (TDRNOT)

ω = 1.84, 3-cycle sin2 + postprop., 1012 Wcm−2

High-order harmonic generation (HOHG) in He

10−20

10−15

10−10

10−5

100

Sre

sp(ω

) [a

rb.

un

its]

1RNOTDSE

2RNOTDSE

10−20

10−15

10−10

10−5

100

0 20 40 60 80 100

Sre

sp(ω

) [a

rb.

un

its]

Harmonic order

3RNOTDSE

0 20 40 60 80 100

Harmonic order

6RNOTDSE

ω = 0.057, 15-cycle trapez. + 2-cycle sin2 ramping, 1014 Wcm−2

M. Brics, J. Rapp, D. Bauer, Phys. Rev. A93, 013404 (2016)

High-order harmonic generation (HOHG) in HeHe+ extension is a two-electron effect

10−20

10−15

10−10

10−5

100

0 20 40 60 80 100

Sre

sp(ω

) (a

rb.

un

its)

Harmonic order

SAE

TDSE

He+

High-order harmonic generation (HOHG) in HeHe+ extension is a two-electron effect, but not NSDR

P. Koval, F. Wilken, D. Bauer, and C. H. Keitel, Phys. Rev. Lett. 98, 043904 (2007)Kenneth K. Hansen and Lars Bojer Madsen, Phys. Rev. A 93, 053427 (2016)

First results on HOHG in 3D He

0 50 100 150

−12

−10

−8

−6

−4

−2

0

HHG spectra for 3D He (λ=800 nm, A=2 a.u. ⇒ ω=0.057 a.u., Up=1 a.u.)

Harmonic order

norm

aliz

ed in

tens

ity

3.17Up + 1.3Ip

3.17Up + 1.3Ip+

4.7Up + Ip + Ip+

5.55Up + Ip + Ip+

SAETDRNOT (No = 1)TDRNOT (No = 2)TDRNOT (No = 4)

calculations by Julius Rapp

Nonsequential double-ionization (NSDI)

huge effect

https://en.wikipedia.org/wiki/Double_ionization B. Walker et al., Phys. Rev. Lett. 73, 1227 (1994)

Nonsequential double-ionization (NSDI)The ”knee“ in the double-ionization probability

10−4

10−3

10−2

10−1

1

2−

Ioniz

ation P

rob.

1 RNO

TDSE

2 RNO

TDSE

3 RNO

TDSE

10−4

10−3

10−2

10−1

1

1015

1016

2−

Ioniz

ation P

rob.

Laser Intensity (W/cm2)

4 RNO

TDSE

1015

1016

Laser Intensity (W/cm2)

6 RNO

TDSE

1015

1016

Laser Intensity (W/cm2)

10 RNO

TDSE

M. Brics, J. Rapp, D. Bauer, Phys. Rev. A 90, 053418 (2014)

ω = 0.058, 3-cycle sin2

Nonsequential double-ionization (NSDI)The ”knee“ in the double-ionization probability

10−4

10−3

10−2

10−1

1

2−

Ioniz

ation P

rob.

1 RNO

TDSE

TDSE 1 NO

2 RNO

TDSE

TDSE 2 NO

3 RNO

TDSE

TDSE 3 NO

10−4

10−3

10−2

10−1

1

1015

1016

2−

Ioniz

ation P

rob.

Laser Intensity (W/cm2)

4 RNO

TDSE

TDSE 4 NO

1015

1016

Laser Intensity (W/cm2)

6 RNO

TDSE

TDSE 6 NO

1015

1016

Laser Intensity (W/cm2)

10 RNO

TDSE

TDSE 10 NO

M. Brics, J. Rapp, D. Bauer, Phys. Rev. A 90, 053418 (2014)

ω = 0.058, 3-cycle sin2

Nonsequential double-ionization (NSDI)Correlated momentum spectra from TDSE

p2 [

a.u

.]

p1 [a.u.]

1RNO

−4

−2

0

2

4

−4 −2 0 2 410

−6

10−5

10−4

10−3

10−2

TDSEM. Brics, J. Rapp, D. Bauer, Phys. Rev. A 90, 053418 (2014)

ω = 0.058, 3-cycle sin2, 2.25× 1015 Wcm−2

Nonsequential double-ionization (NSDI)Correlated momentum spectra from TRNOT?

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 1 RNO 1RNO

10−6

10−5

10−4

10−3

10−2

2 RNOs

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 5 RNOs 1RNO

10−6

10−5

10−4

10−3

10−2

6 RNOs

p2 [

a.u

.]

p1 [a.u.]

1RNO

−4

−2

0

2

4

−4 −2 0 2 4

18 RNOs

p1 [a.u.]

1RNO

−4 −2 0 2 410

−6

10−5

10−4

10−3

10−2

30 RNOs

TDRNOT

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 1 NO 1RNO

10−6

10−5

10−4

10−3

10−2

2 NOs

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 5 NOs 1RNO

10−6

10−5

10−4

10−3

10−2

6 NOs

p2 [

a.u

.]

p1 [a.u.]

1RNO

−4

−2

0

2

4

−4 −2 0 2 4

18 NOs

p1 [a.u.]

1RNO

−4 −2 0 2 410

−6

10−5

10−4

10−3

10−2

30 NOs

from TDSEM. Brics, J. Rapp, D. Bauer, Phys. Rev. A 90, 053418 (2014)

Nonsequential double-ionization (NSDI)Correlated momentum spectra from TRNOT?

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 5 RNOs 1RNO

10−6

10−5

10−4

10−3

10−2

6 NOs

TDRNOT (30 NOs)p

2 [

a.u

.]

1RNO

−4

−2

0

2

4 1 NO 1RNO

10−6

10−5

10−4

10−3

10−2

2 NOs

p2 [

a.u

.]

1RNO

−4

−2

0

2

4 5 NOs 1RNO

10−6

10−5

10−4

10−3

10−2

6 NOs

p2 [

a.u

.]

p1 [a.u.]

1RNO

−4

−2

0

2

4

−4 −2 0 2 4

18 NOs

p1 [a.u.]

1RNO

−4 −2 0 2 410

−6

10−5

10−4

10−3

10−2

30 NOs

from TDSEM. Brics, J. Rapp, D. Bauer, Phys. Rev. A 90, 053418 (2014)

Moving photoelectron peak in TDDFT... unless derivative discontinuity is taken care of

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Moving photoelectron peak in TDDFTTDRNOT nails it

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

TDHFTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

TDHFTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

TDHFTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

TDHFTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

2−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

2−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

2−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

2−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

3−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

3−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

3−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

3−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

4−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

4−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

4−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

4−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

5−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

5−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

5−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

5−RNOTDSE

0

0.002

0.004

0.006

0.008

0.01

0.012

n(p)

[a.u

.]

I0=9.0×1010

W/cm2

6−RNOTDSE

0

0.2

0.4

0.6

0.8

1

1.2

I0=1.3×1013

W/cm2

6−RNOTDSE

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

n(p)

[a.u

.]

p [a.u.]

I0=2.3×1013

W/cm2

6−RNOTDSE

0.0

1.0

2.0

3.0

4.0

5.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

p [a.u.]

I0=8.9×1013

W/cm2

6−RNOTDSE

(2,46,2)-pulse, ω = 1

Single-photon double ionization in He20-cycle sin2, 7.6 nm (ω = 6), 2.4× 1014 Wcm−2

TDSE reference spectrum

p2 [a.u

.]

p1 [a.u.]

1RNO

−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 810

−12

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

Single-photon double ionization in He20-cycle sin2, 7.6 nm (ω = 6), 2.4× 1014 Wcm−2

TDSE, with various numbers of NOs taken into account

−8

−6

−4

−2

0

2

4

6

8

p2 [a.u

.]

1RNO1 NO 1RNO2 NOs 1RNO

10−12

10−10

10−8

10−6

10−4

10 NOs

−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8

p2 [a.u

.]

p1 [a.u.]

1RNO20 NOs

−8 −6 −4 −2 0 2 4 6 8

p1 [a.u.]

1RNO38 NOs

−8 −6 −4 −2 0 2 4 6 8

p1 [a.u.]

1RNO

10−12

10−10

10−8

10−6

10−4

60 NOs

Single-photon double ionization in He20-cycle sin2, 7.6 nm (ω = 6), 2.4× 1014 Wcm−2

TDRNOT, with various numbers of NOs propagatedp

2 [

a.u

.]

1RNO

−8

−6

−4

−2

0

2

4

6

81 RNO 1RNO

10−12

10−10

10−8

10−6

10−4

10 RNOs

p2 [

a.u

.]

p1 [a.u.]

1RNO

−8

−6

−4

−2

0

2

4

6

8

−8 −6 −4 −2 0 2 4 6 8

20 RNOs

p1 [a.u.]

1RNO

−8 −6 −4 −2 0 2 4 6 8

10−12

10−10

10−8

10−6

10−4

38 RNOs

TRNOT for simplest multi-component system: H+2

Usual 1D-model of H+2 :

H(x ,R, t) = he + hn + Ven(x ,R),

where

he(x , t) = − 12µe

∂2x + qe x E(t)

hn(R) = − 12µn

∂2R + Vnn(R)

Ven(x ,R) = − 1√(x − R

2 )2 + ε2en

− 1√(x + R

2 )2 + ε2en

Vnn(R) =1√

R2 + ε2nn

TRNOT for simplest multi-component system: H+2

Two types of 1-RDM from pure γ1,1(t) = |Ψ(t)〉〈Ψ(t)|

γ1,0(t) = Trn γ1,1(t), γ0,1(t) = Tre γ1,1(t)

NOs and ONs

γ1,0(t)|k(t)〉 = nk (t)|k(t)〉, γ0,1(t)|K (t)〉 = NK (t)|K (t)〉

Schmidt decomposition

Ψ(x ,R, t) =∑

k

ck (t)ϕk (x , t) ηk (R, t).

where ϕk (x , t) = 〈x |k(t)〉, ηk (R, t) = 〈R|K (t)〉One also finds nk (t) = NK (t)Two—via matrix elements coupled—sets of EOMs for|k(t)〉 and |K (t)〉 can be derived

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Two types of 1-RDM from pure γ1,1(t) = |Ψ(t)〉〈Ψ(t)|

γ1,0(t) = Trn γ1,1(t), γ0,1(t) = Tre γ1,1(t)

NOs and ONs

γ1,0(t)|k(t)〉 = nk (t)|k(t)〉, γ0,1(t)|K (t)〉 = NK (t)|K (t)〉

Schmidt decomposition

Ψ(x ,R, t) =∑

k

ck (t)ϕk (x , t) ηk (R, t).

where ϕk (x , t) = 〈x |k(t)〉, ηk (R, t) = 〈R|K (t)〉One also finds nk (t) = NK (t)Two—via matrix elements coupled—sets of EOMs for|k(t)〉 and |K (t)〉 can be derived

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Two types of 1-RDM from pure γ1,1(t) = |Ψ(t)〉〈Ψ(t)|

γ1,0(t) = Trn γ1,1(t), γ0,1(t) = Tre γ1,1(t)

NOs and ONs

γ1,0(t)|k(t)〉 = nk (t)|k(t)〉, γ0,1(t)|K (t)〉 = NK (t)|K (t)〉

Schmidt decomposition

Ψ(x ,R, t) =∑

k

ck (t)ϕk (x , t) ηk (R, t).

where ϕk (x , t) = 〈x |k(t)〉, ηk (R, t) = 〈R|K (t)〉

One also finds nk (t) = NK (t)Two—via matrix elements coupled—sets of EOMs for|k(t)〉 and |K (t)〉 can be derived

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Two types of 1-RDM from pure γ1,1(t) = |Ψ(t)〉〈Ψ(t)|

γ1,0(t) = Trn γ1,1(t), γ0,1(t) = Tre γ1,1(t)

NOs and ONs

γ1,0(t)|k(t)〉 = nk (t)|k(t)〉, γ0,1(t)|K (t)〉 = NK (t)|K (t)〉

Schmidt decomposition

Ψ(x ,R, t) =∑

k

ck (t)ϕk (x , t) ηk (R, t).

where ϕk (x , t) = 〈x |k(t)〉, ηk (R, t) = 〈R|K (t)〉One also finds nk (t) = NK (t)

Two—via matrix elements coupled—sets of EOMs for|k(t)〉 and |K (t)〉 can be derived

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Two types of 1-RDM from pure γ1,1(t) = |Ψ(t)〉〈Ψ(t)|

γ1,0(t) = Trn γ1,1(t), γ0,1(t) = Tre γ1,1(t)

NOs and ONs

γ1,0(t)|k(t)〉 = nk (t)|k(t)〉, γ0,1(t)|K (t)〉 = NK (t)|K (t)〉

Schmidt decomposition

Ψ(x ,R, t) =∑

k

ck (t)ϕk (x , t) ηk (R, t).

where ϕk (x , t) = 〈x |k(t)〉, ηk (R, t) = 〈R|K (t)〉One also finds nk (t) = NK (t)Two—via matrix elements coupled—sets of EOMs for|k(t)〉 and |K (t)〉 can be derived

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Linear response spectrum shows discrete peaks due to truncation error

10-25

10-20

10-15

10-10

10-5

100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Dip

ole

stre

ngth

[arb

. uni

ts]

Energy E - E0 [a.u.]

TDSE

TDSE No = 1

No = 1

No = 2

No = 4

No = 6

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

TRNOT for simplest multi-component system: H+2

Nuclear density during 800-nm four-cycle sin2-shaped 1014 Wcm−2 pulse

0

10

20

30(a) (b) (c)

R [a.

u.]

0

10

20

30

0 1000 2000 3000

(d)

t [a.u.]

0 1000 2000 3000

(e)

0 1000 2000 3000 10-12

10-10

10-8

10-6

10-4

10-2

100

Pnuc(

R,t) [a

.u.]

(f)

A. Hanusch, J. Rapp, M. Brics, D. Bauer, Phys. Rev. A93, 043414 (2016)

Conclusion

TDRNOT so far

method to simulate correlated dynamicsexact for N = 2, requires approx. γ2,ijkl for N > 2no problem to calculate observables (if γ2 is sufficient)3D He (→ Julius Rapp)To do: TDRNOT 6= MCTDHF (but work out connection,also to TDDMRG)No-free-lunch theorem confirmed yet another time