Benefits and inconveniences of
channel-specific absorption boundaries
for the solution of the TDSE
Luca Argenti & Eva Lindroth
Albanova
Universitetscentrum
Stockhom
Oslo
28 Apr 2010
Temporal scales in the microcosm
τatomic ≈ 24 as
Sub-femtosecond extreme-UV pulses
by High Harmonic Generation
Milan 2006
Sansone et al., Science 314, 443 (2006)
Garching 2008
Goulielmakis et al., Science 320, 1314 (2008)
An example: pump probe investigation
of doubly excited states in helium
PUMP
He (1s2) + ϒxuv → He** (1Po)
→ He+ (1s,2s,2p) + ePROBE
He** (1Po) + n ϒir → He** (1Lπ)
→ He+ (1s,2s,2p) + e-
Detection of photofragments
System confinement
Reflection on the box boundary
Absorbing boundaries
Possible inconveniences of the
absorbing boundaries
reflection on the absorption
potential
annihilation of long-range
bound satellites
Time propagation
Time propagation
Radiationless
hamiltonian
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
SPLIT
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
SPLIT
Magnus I + Krylov
Time propagation
Radiationless
hamiltonian
Radiation-matter
interaction (V-gauge)
SPLIT
Spectral
resolution
Magnus I + Krylov
1s ! l
2s ! l
2p ! l!1 2p ! l+1
N= oo
..
.
N=3
N=2
1
N=1
L
"L
The Basis
- Close Coupling
- B-splines
- L≤6
Absorption potentials
vα(r) = Pα zα (r - rα)nα θ(r - rα) Pα
V = Σα Vα
∀ α,β
z = -( 1+ 5i ) 10-4
n=2
ro = Rbox - 100 au
zα ∈ C
[ Vα ,Vβ ] = 0
Rbox = 400 - 800 au
Absorption potentials
vα(r) = Pα zα (r - rα)nα θ(r - rα) Pα
V = Σα Vα
● Complex
∀ α,β
zα ∈ C
[ Vα ,Vβ ] = 0
Extraction & Absorption
● Channel specific & Fully independent
●
zα freely adjustable on the run
z = -( 1+ 5i ) 10-4
n=2
ro = Rbox - 100 au
Rbox = 400 - 800 au
Effect of V on Close Coupling wave functions
v(r) = z θ( r - ro ) ( r - ro )n
Effect of V on Close Coupling wave functions
v(r) = z θ( r - ro ) ( r - ro )n
PARTIAL WAVE CHANNEL
Effect of V on Close Coupling wave functions
v(r) = z θ( r - ro ) ( r - ro )n
PARTIAL WAVE CHANNEL
LOCALIZED
CHANNEL
Effect of V on Close Coupling wave functions
v(r) = z θ( r - ro ) ( r - ro )n
PARTIAL WAVE CHANNEL
LOCALIZED
CHANNEL
Effect of V on Close Coupling wave functions
v(r) = z θ( r - ro ) ( r - ro )n
PARTIAL WAVE CHANNEL
LOCALIZED
CHANNEL
Effect of V on one-particle functions
a
b
Effect of V on one-particle functions
f
a
b
Effect of V on one-particle functions
f
a
c
b
Effect of V on one-particle functions
f[a,c]
a
f[c,b]
c
b
Projection on a B-spline basis
12 3 4
a
i i+1 ...
n n+1 ...
c
b
Projection on a B-spline basis
12 3 4
a
V Bi = 0
i i+1 ...
i = 1, 2, ..., n
n n+1 ...
c
b
Projection on a B-spline basis
12 3 4
a
V Bi = 0
i i+1 ...
i = 1, 2, ..., n
Span {Bi}i≤n ⊂ ker V
n n+1 ...
c
b
Projection on a B-spline basis
12 3 4
a
V Bi = 0
i i+1 ...
i = 1, 2, ..., n
Span {Bi}i≤n ⊂ ker V
n n+1 ...
c
b
V Bn+j ≠ 0 j = 1, 2, ..., m
Projection on a B-spline basis
12 3 4
a
i i+1 ...
V Bi = 0
i = 1, 2, ..., n
n n+1 ...
c
V Bn+j ≠ 0 j = 1, 2, ..., m
Span {Bi}i≤n ⊂ ker V
V’ = PVP
b
P : projector onto Span{Bi}i≤n+m
Projection on a B-spline basis
12 3 4
a
i i+1 ...
V Bi = 0
n n+1 ...
i = 1, 2, ..., n
c
V Bn+j ≠ 0 j = 1, 2, ..., m
Span {Bi}i≤n ⊂ ker V
V’ = PVP
V’ ϕ = λϕ,
P : projector onto Span{Bi}i≤n+m
λ≠0
b
ϕ∈ Span {Bi}i≤n⊥
Projection on a B-spline basis
12 3 4
a
i i+1 ...
V Bi = 0
n n+1 ...
i = 1, 2, ..., n
c
V Bn+j ≠ 0 j = 1, 2, ..., m
Span {Bi}i≤n ⊂ ker V
V’ = PVP
V’ ϕ = λϕ,
P : projector onto Span{Bi}i≤n+m
λ≠0
b
ϕ∈ Span {Bi}i≤n⊥
Projection on a B-spline basis
12 3 4
a
i i+1 ...
V Bi = 0
n n+1 ...
i = 1, 2, ..., n
c
V Bn+j ≠ 0 j = 1, 2, ..., m
Span {Bi}i≤n ⊂ ker V
V’ = PVP
V’ ϕ = λϕ,
P : projector onto Span{Bi}i≤n+m
λ≠0
b
ϕ∈ Span {Bi}i≤n⊥
Construction of the orthogonal
complement to ker v
B-spline superposition matrix
2(k-2)-th order homogeneous linear difference equation
Boundary conditions
Solution of the inner part
of the linear system
ci
M
ci+1
Associated secular problem
Symmetry of the characteristic
polynomials
λ is a root
λ-1 is also a root
therefore the eigenvalues are separated in three classes:
λ
<1
λ
=1
λ
>1
As a matter of fact, for B-splines,
all the λs are real, negative, and λ ≠ 1
Absolute values of the coefficients ci of an
eigenfunction of V on the boundary
k
λ
2
3.732
3
2.322
4
1.869
5
1.645
6
1.512
7
1.425
8
1.363
9
1.316
10
1.280
First summary
The absorbing boundaries we have described are
- easy to implement and fast to execute
- localized but with a characteristic minimal range
- channel specific
- adjustable on the run
The process under study
PUMP
He (1s2) + ϒxuv → He** (1Po)
+
→ He (1s,2s,2p) + e-
PROBE He** (1Po) + n ϒir → He** (1Lπ)
→ He+ (1s,2s,2p) + eIR PROBE
ωir = 0.057 au (1.55 eV)
I
= 1⋅1012 W/cm2 s
fwhm = 5.3 fs
0.02
0.01
E(au) 0
−0.01
−0.02
0
XUV PUMP
ωxuv = 2.23 au (60.7 eV)
I
= 2⋅1014 W/cm2 s
fwhm = 550 as
200
400
600
time (au)
800
1000
Quantum interference
Cooper, Fano, Prats, Phys. Rev. Lett. 10 5168 (1963)
Quantum interference
What if the impinging radiation
pulse were much shorter than
the lifetime of doubly excited
states ?
Cooper, Fano, Prats, Phys. Rev. Lett. 10 5168 (1963)
A single channel,
single resonance
example.
A single channel,
single resonance
example.
XUV pump
S
P
D
F
G
H
I
time (au)
XUV pump
S
P
D
F
G
H
I
time (au)
XUV pump
S
P
D
F
G
H
I
time (au)
XUV pump
He + ϒxuv → He+(1s) + e- (strongest)
S
P
D
F
G
H
I
time (au)
XUV pump
He + ϒxuv → He+(1s) + e- (strongest)
He + ϒxuv → He** → He+(1s)+eHe + ϒxuv → He+(N=2) + eS
P
D
F
G
H
I
time (au)
XUV pump
He + ϒxuv → He+(1s) + e- (strongest)
He + ϒxuv → He** → He+(1s)+eHe + ϒxuv → He+(N=2) + eHe + 2ϒxuv → He+(1s) + eHe + 2ϒxuv → He+(N=2) + e-
S
P
D
F
G
H
I
time (au)
XUV pump
He + ϒxuv → He+(1s) + e- (strongest)
He + ϒxuv → He** → He+(1s)+eHe + ϒxuv → He+(N=2) + eHe + 2ϒxuv → He+(1s) + eHe + 2ϒxuv → He+(N=2) + eHe + 3ϒxuv → He+(N=1,2) + eHe + 2ϒxuv → He+(1s) + e- + ϒxuv
He + 2ϒxuv → He+(N=2) + e- + ϒxuv
S
P
D
F
G
H
I
time (au)
XUV pump
He + ϒxuv → He+(1s) + e- (strongest)
He + ϒxuv → He** → He+(1s)+eHe + ϒxuv → He+(N=2) + eHe + 2ϒxuv → He+(1s) + eHe + 2ϒxuv → He+(N=2) + eHe + 3ϒxuv → He+(N=1,2) + eHe + 2ϒxuv → He+(1s) + e- + ϒxuv
He + 2ϒxuv → He+(N=2) + e- + ϒxuv
Four photon processes ...
S
P
D
F
G
H
I
time (au)
Partial Diff. Photoelectron Spectrum
1s
2s
2p
Energy (au)
N=2
XUV pump
-1
0
cos θ
1
1au ~ 27.2 eV
Partial Diff. Photoelectron Spectrum
1s
2s
2p
Energy (au)
N=2
Direct ionization
XUV pump
-1
0
cos θ
1
1au ~ 27.2 eV
Partial Diff. Photoelectron Spectrum
1s
2s
2p
Energy (au)
Continuous
across threshold
N=2
Direct ionization
XUV pump
-1
0
cos θ
1
1au ~ 27.2 eV
Partial Diff. Photoelectron Spectrum
1s
2s
2p
Energy (au)
Continuous
across threshold
N=2
Direct ionization
XUV pump
-1
0
cos θ
1
1au ~ 27.2 eV
Partial Diff. Photoelectron Spectrum
1s
2s
2p
Energy (au)
N=2
?
-1
0
cos θ
1
What happens when
the first bunch of fast
electrons is absorbed
at the boundaries
(but the other lot still isn’t)?
Partial Diff. Photoelectron Spectrum
1s
Zero
2s
2p
The same as before
Energy (au)
N=2
-1
0
cos θ
1
Partial Diff. Photoelectron Spectrum
1s
Zero
2s
2p
The same as before
Energy (au)
N=2
He+(2s,2p)+e- continua are
comparable to the decaying
part of the DES, and yet
they are neatly resolved
-1
0
cos θ
1
Partial Diff. Photoelectron Spectrum
1s
Zero
2s
2p
The same as before
Energy (au)
N=2
sp+3
sp+2
-1
0
cos θ
1
Breathing sp+ mode
15
R (au) 0
15
+
1
o
([001] P )
Breathing sp+ mode
15
R (au) 0
15
+
1
o
([001] P )
Breathing sp+ mode
15
R (au) 0
15
+
1
o
([001] P )
Electron density at large radii
700
R (au)
0
Electron density at large radii
700
R (au)
0
MEASURING THE
WAVE NUMBER OF
THE ELECTRON
BURSTS
kbursts ≈ 0.078 au
magenta = scale saturation
Long range interference of resonances
kbursts ≈ 0.078 au
sp+3
E
sp+2
k2 = [2(E2-Eth)]1/2
k1 = [2(E1-Eth)]1/2
Long range interference of resonances
kbursts ≈ 0.078 au
sp+3
E
sp+2
k2 = [2(E2-Eth)]1/2
k1 = [2(E1-Eth)]1/2
Long range interference of resonances
kbursts ≈ 0.078 au
sp+3
E
sp+2
k2 = [2(E2-Eth)]1/2
Δk ≈ 0.074 au
k1 = [2(E1-Eth)]1/2
τbeat= 48.7 au (~1.2 fs)
Burst-breath correlation
800
Radius (au)
600
400
200
0
700
800
900
Time (au)
1000
1100
to fit (au)
421 471 518 564 621 671 717 764 817 869 920 965 1011 1073 1118
tmax-den
422 470 518 568 622 670 717 768 818 866 920 968 1012 1068 1120
Ionization-excitation yield
Pα = what is annihilated in channel α
xuv on
Pα(t) & P’α(t)
1.5E−3
field free propagation
1sEp
1E−3
2pEs x 1000
2sEp x 1000
5E−4
2pEd x 1000
0
0
2000
4000
time (au)
6000
Probe of the metastable
wave packet dynamics
with an IR pulse
On the tail of the pulse
1s
2s
2p
N=2
E
[001]+2 + ϒIR
Emerging new ang. dist.:
He** + n ϒIR → He+(2s,2p) + e-
[001]+2 - ϒIR
[001]+2 - 2ϒIR
-1
0
cos θ
1
IR PROBE
2S+1Lπ[N N m]A
1 2
1Se [100]+
1Se [010]+
1Po[100]1Po [010]-
1De [100]+
1De [001]1Fo [010]1Fo [001]+
1Ge [010]+
1Po [001]+
1De [010]+
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
XUV
2s
2s
IR
2p
2p
0
1000
2000
3000
time (au)
4000
5000
6000
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
cannot be extrapolated
XUV
2s
2s
IR
2p
2p
0
1000
2000
3000
time (au)
4000
5000
6000
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
cannot be extrapolated
XUV
slowly decaying DES
2s
2s
IR
2p
2p
0
1000
2000
3000
time (au)
4000
5000
6000
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
cannot be extrapolated
XUV
slowly decaying DES
2s
2s
IR
slow electrons
(right above threshold)
2p
2p
0
1000
2000
3000
time (au)
4000
5000
6000
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
cannot be extrapolated
XUV
slowly decaying DES
2s
2s
IR
slow electrons
(right above threshold)
2p
2p
0
1000
2000
3000
time (au)
4000
5000
absorption of
metastable satellites
in the wrong channel
6000
Yield of the
parent ions
IONIZATION-EXCITATION RATES
1s
cannot be extrapolated
XUV
slowly decaying DES
2s
2s
IR
slow electrons
(right above threshold)
2p
2p
0
1000
2000
3000
time (au)
4000
5000
absorption of
metastable satellites
in the wrong channel
6000
polarization effects
resulting in wrong
channel assignement
Corrections to the yields
for “large” times
dP/dt, for all channels, converges to zero very slowly. In the
1s channel, because of the long lived doubly excited states.
In the N=2 channels, because of the accumulation of density
at threshold (extremely slow electrons). Moreover, it is
dangerous! As soon as the slowest electrons above
threshold get to the boundary, also the transiently bound
electrons with long trajectories get there and get absorbed.
Effects of the parent ion polarization on yields
The IR field polarizes the 1s
core, and thus it causes
instantaneously part of
the 1s ionization rate to be
recored as pertaining to the
2p channel
2s
2p
1s + 2p c(t)
For an IR with I=1012 W/cm2
the interference turns out to
be negligible. If necessary,
though, it can be singled out
time (au)
Quantum beats in the
ionization-excitation yields
Quantum beats in the
ionization-excitation yields
2s/2p yields are out of phase by
o
~60
Buildup of the photoelectron
angular distribution
-0.425
2s
2p
Energy
(au)
-0.5
N=2
Buildup of the photoelectron
angular distribution
-0.425
2s
2p
Energy
(au)
-0.5
N=2
Interference fringes at time delays
differing by half IR period
! t=15.53 fs
after XUV
!t=16.87 fs
2s
Energy (au)
−0.425
−0.5
−1
2p
0
cos
Argenti, Lindroth
1
"
arXiv:1004.1266v1 [physics.atom-ph] 8 Apr 2010
Conclusions
- incomplete absorption
- when metastable states and thresholds are involved, the
convergence may be very slow
- reflection of low energy components
- absorption of long-range satellites
- channel partition possibly compromised by external fields
- the angular and energy distribution of the absorbed electrons
is lost
- low energy components can be accelerated
- propagation can be stopped before low energy components
and long-range satellites reach the absorbing region and the
rest be extracted by projection onto scattering states
- the effect of external fields can be singled out
- the loss of information in one channel does not influence the
other channels
The Computational Atomic
Physics Group of Stockholm
http://www.atom.physto.se/~genkin/Grouphomepage/group.html
Group leader: Eva Lindroth (lindroth@physto.se)
Luca Argenti (argenti@physto.se)
Jakob Bengtsson (jakbe@physto.se)
Michael Genkin (genkin@physto.se)
Erik Waltersson (walter@physto.se)
Luca Argenti
Jakob Bengtsson
Eva Lindroth
Michael Genkin
Erik Waltersson
!ank y"