In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.

1. Real­time Spectroscopy
for solids
Claudio Attaccalite
CNRS/CINaM, Aix­Marseille Universite (FR)
Theoretical Spectroscopy Lectures
October 8­12, 2018, Lausanne
Repetita iuvant
(repeating does good)

2. What is it non­linear optics?
P(r ,t)=P0+χ
(1)
E+χ
(2)
E
2
+O(E
3
)
First experiments on linear­optics
by P. Franken 1961
Ref: Nonlinear Optics and
Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

3. Ref: supermaket
Why non­linear optics?
..applications..

4. Ref: supermaket
Why non­linear optics?
..applications..

5. Why non­linear optics?
..applications..
New green laser appeared in 2012
without non­linear crystals

6. Why non­linear optics?
..research..
Linear
Science, 2014, vol. 344, no 6183, p. 488­490
Non­linear(COLOR)Non­linear(BW)

7. To see “invisible” excitations
The Optical Resonances in
Carbon
Nanotubes Arise
from Excitons
Feng Wang, et al.
Science 308, 838 (2005);
..research..
Right interpretation of the experiment
“Selection rules for one-and two-photon absorption by excitons in
carbon nanotubes,” E. B. Barros et al. PRB 73, 241406 (2006).

8. Probing symmetries
Probing Symmetry Properties of Few-Layer MoS2
and h-BN by Optical Second-Harmonic
Generation
Nano Lett. 13, 3329 (2013)
SHG can probe
magnetic transition

9. ...and more...
photon entanglement in vivo imaging
Ref: PNAS 107, 14535 (2007)

10. “Self­focusing limit in terms of peak power is of the order
of 4 MW in the 1­μm wavelength region. No method is known
for increasing the self­focusing limit of optical fibers
beyond that value.” RP Photonics
Why non­linear optics?
..applications.. (the darkside )

11. Real­time spectroscopy is
much similar to experiments

12. What is real­time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0

13. What is real­time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0
Time­evolution
of an effective
Schroedinger
equation
Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t )
Ψ(t=0)=ΨGS

14. What is real­time
spectroscopy?
Choose a
perturbation
E(t)=δ(t−t0) E0
E(t)=sin(ωt)E0
Time­evolution
of an effective
Schroedinger
equation
Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t )
Ψ(t=0)=ΨGS
Analise the
results

15. Real time spectroscopy in practice 1 3- /
D(r ,t)=E(r ,t)+P(r ,t)
Materials equations:
Electric
Displacement
Electric Field
Polarization
∇⋅E(r ,t)=4 πρtot (r ,t)
∇⋅D(r ,t)=4 πρext (r ,t)
From Gauss's law:
Δ P(r ,t)=∫χ(t−t ' ,r ,r ')E(t ' r ')dt ' dr '+∫dt
1
dt
2
χ
2
(...)E(t
1
) E(t
2
)+O(E
3
)
In general:

16. Real time spectroscopy in practice 2 3- /
For a small perturbation we consider only the first term,
the linear response regime
Δ P(r ,t)=∫χ(t−t ' ,r ,r ')E(t ' r ')dt ' dr '+O(E2
)
Δ P(ω)=χ(ω)E(ω)=(ϵ(ω)−1)E(ω)
And finally: ϵ(ω)=1+
Δ P(ω)
E(ω)
ϵ(ω)=
D(ω)
E(ω)

17. Real time spectroscopy in practice 3 3- /
1) Choose an external perturbation E(t)
2) Evolve the Schroedinger equation
3) We calculate the P(t) from (t)
4) Fourier transform P(t) and E(t) and get
i
d Ψ(t )
dt
=[H + E(t )]Ψ(t )
ϵ(ω)=1+
Δ P(ω)
E(ω)

18. Motivations
Better scaling for large system
Polarization and
Hamiltonian depend only
from valence bands. No need
of conduction bands!

19. Motivations
Better scaling for large system
Theory and implementation
are much easier
Polarization and
Hamiltonian depend only on
valence bands. No need of
conduction bands!

20. One code to rule all
spectroscopy responses
χ(2)
(ω;ω1, ω2)
P(ω)=P0+χ
(1)
(ω)E1(ω)+χ
(2)
E1(ω1) E2(ω2)+χ
(3)
E1 E2 E3+O(E
4
)
SFG
DFG
SHG

21. One code to rule all
spectroscopy responses
χ(3)
(ω; ω1, ω2, ω3)
THG
P(ω)=P0+χ(1)
(ω)E1(ω)+χ(2)
E1(ω1) E2(ω2)+χ(3)
E1 E2 E3+O(E4
)

22. One code to rule all“ ”
correlation effects
Equation of motions
are always the same
In order to include
correlation effects just
change the Hamiltonian
Notice that the present
approach is limited to
single-particle Hamiltonians.
H=H1+H2+H3+...

23. And even more..
Non­perturbative
phenomena (HHG)
Coupling with ionic motion,
other response functions..

24. The Hamiltonian I
independent particles
H KS(ρ0)=T+V ion+V h(ρ0)+V xc (ρ0)
We start from
the Kohn-Sham Hamiltonian
If we keep fixed the
density in the Hamiltoanian
to the ground-state one
we get the independent
particle approximation
In the Kohn-Sham basis
this reads:
H KS(ρ0)=ϵi
KS
δi, j

25. The Hamiltonian II
time­dependent Hartree (RPA)
HTDH =T+Vion+V h(ρ)+V xc (ρ0)
If we keep fixed the
density in Vxc but not in
Vh. We get the
time-dependent Hartree or
RPA (with local fields)
Or equivalent:
HTDH =H KS(ρ0)+Vh (ρ−ρ0)
The density is written as:
ρ(r ,t)=∑i=1
N v
|Ψ(r ,t)|
2

26. The Hamiltonian III
TD­DFT
HTDH =T+Vion+V h(ρ)+V xc (ρ)
We let density fluctuate in
both the Hartree and the Vxc
tems
We get the TD-DFT for
solids
The Runge­Gross theorem guarantees that this is an exact
theory for isolated systems

27. Dephasing
Gauge-independent decoherence models for
solids in external fields
M. S. Wismer and V. S. Yakovlev
Phys. Rev. B 97, 144302 (2018)
The previous Hamiltonian are Hermitian
without any time-dependence
(expect the external field)
This means they do not introduce any dephasing!
Dephasing as non-local
operator in the
Hamiltoanian
Dephasing in post-(pre)
processing
~P(t)=P(t)e−λ t
See Octopus code or
Y.Takimoto, Phd thesis (2008)

28. Dephasing and damping


29. Post­processing 2:
periodic perturbation
P(t) is a periodic function of period TL
=2p/wL
pn
is proportional to χn
by the n­th order of the external field
Performing a discrete­time signal
sampling we reduce the problem to
the solution of a systems of linear equations
Ref: C. Attaccalite et al. PRB 88, 235113(2013)
F. Ding et al. JCP 138, 064104(2013)

30. Second Harmonic Generation in MoS2

31. Non-linear optics in molecules
Non-linear optics can calculated in the same way of TD-DFT as it
is done in OCTOPUS or RT-TDDFT/SIESTA codes.
Quasi-monocromatich-field
p-nitroaniline
Y.Takimoto, Phd thesis (2008)

32. Non-linear response in extended systems: a real-time approach
Claudio Attaccalite
https://arxiv.org/abs/1609.09639

33. External and total field

34. How to calculated the dielectric constant
i
∂ ̂ρk (t)
∂t
=[Hk +V
eff
, ̂ρk ] ̂ρk (t)=∑i
f (ϵk ,i)∣ψi,k 〉〈 ψi,k∣
The Von Neumann equation
(see Wiki http://en.wikipedia.org/wiki/Density_matrix)
r t ,r'
t'
=

ind
r ,t
ext r' ,t '
=−i〈[ r ,t r' t ']〉We want to calculate:
We expand X in an independent particle basis set
χ(⃗r t ,⃗r
'
t
'
)= ∑
i, j,l,m k
χi, j,l,m, k ϕi, k (r)ϕj ,k
∗
(r)ϕl,k (r')ϕm ,k
∗
(r')
χi, j,l,m, k=
∂ ̂ρi, j, k
∂Vl,m ,k
Quantum Theory of the
Dielectric Constant in Real Solids
Adler Phys. Rev. 126, 413–420 (1962)
What is Veff
?

35. Independent Particle
Independent Particle Veff
= Vext
∂
∂Vl ,m,k
eff
i
∂ρi, j ,k
∂t
= ∂
∂Vl ,m, k
eff
[Hk+V eff
, ̂ρk ]i, j, k
Using:
{
Hi, j ,k = δi, j ϵi(k)
̂ρi, j, k = δi, j f (ϵi,k)+
∂ ̂ρk
∂V
eff
⋅V eff
+....
And Fourier transform respect to t-t', we get:
χi, j,l,m, k (ω)=
f (ϵi,k)−f (ϵj ,k)
ℏ ω−ϵj ,k+ϵi ,k+i η
δj ,l δi,m
i
∂ ̂ρk (t)
∂t
=[Hk +V eff
, ̂ρk ]
χi, j,l,m, k=
∂ ̂ρi, j, k
∂Vl,m ,k

36. Optical Absorption: IP
Non Interacting System
δρNI=χ
0
δVtot χ
0
=∑
ij
ϕi(r)ϕj
*
(r)ϕi
*
(r')ϕj(r ')
ω−(ϵi−ϵj)+ i η
Hartree, Hartree-Fock, dft.
=ℑχ0=∑
ij
∣〈 j∣D∣i〉∣2
δ(ω−(ϵj −ϵi))
ϵ''
(ω)=
8 π
2
ω2 ∑
i, j
∣〈ϕi∣e⋅̂v∣ϕj 〉∣2
δ(ϵi−ϵj−ℏ ω)
Absorption by independent
Kohn-Sham particles
Particles are interacting!

37.  V ext=
0
 V extV HV xc
q ,=
0
q ,
0
q,vf xc q ,q ,
TDDFT is an exact
theory for neutral
excitations!
Time Dependent DFT
V eff (r ,t)=V H (r ,t)+ V xc (r ,t)+ V ext (r ,t)
Interacting System
Non Interacting System
Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)
 I= NI=
 I
 Vext
0=
 NI
 V eff
... by using ...
=
0
1
V H
 V ext

 V xc
V ext

v
f xc 
i
∂ ̂ρk (t)
∂t
=[ HKS , ̂ρk ]=[ Hk
0
+V eff
, ̂ρk ]