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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
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
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 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
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