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Non-linear response from real-time simulations
Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR)
Repetita iuvant(repeating does good)
What is it non-linear optics?
P(r , t )=P0+χ(1)E+χ
(2)E2+O(E3
)
What is it non-linear optics?
P(r , t )=P0+χ(1)E+χ
(2)E2+O(E3
)
First experiments on linear-optics by P. Franken 1961
Ref: Nonlinear Optics and Spectroscopy The Nobel Prize in Physics 1981Nicolaas Bloembergen
Ref: supermaket
Why non-linear optics?
..applications..
Ref: supermaket
Why non-linear optics?..applications..
Ref: supermaket
Why non-linear optics?..applications..
Outdated slide: In 2012, Nichia and OSRAM developed and manufactured commercial high-power green laser
diodes (515/520 nm)
Why non-linear optics?
..research..
Linear
Science, 2014, vol. 344, no 6183, p. 488-490
Non-linear(COLOR)Non-linear(BW)
To see “invisible” excitations
The Optical Resonances in Carbon
Nanotubes Arisefrom 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).
Probing symmetries
Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic
Generation Nano Lett. 13, 3329 (2013)
SHG can probemagnetic transition
...and more...
photon entanglement in vivo imaging
Ref: PNAS 107, 14535 (2007)
“Self-focusing limit in terms of peak power is of the order of 4 MW in the 1-μm wavelength region. No method is known m wavelength region. No method is known for increasing the self-focusing limit of optical fibers
beyond that value.” RP Photonics
Why non-linear optics?
..applications.. (the darkside )
Real-time spectroscopy in practice 1) Choose an external perturbation E(t)
2) Evolve the Schroedinger equation
3) We calculate the P(t) from Y(t)
4) Fourier transform P(t) and E(t) and get
idY(t )dt
=[H+E(t )]Y (t )
ϵ(ω)=1+Δ P(ω)
E (ω)
The problem of bulk polarization
● How to define polarization as a bulk quantity?
● Polarization for isolated systems is well defined
P=⟨R ⟩
V=
1V∫ d r n(r )=
1V
⟨Y∣R̂∣Y⟩
Bulk polarization, the wrong way 1
1) P=⟨R ⟩sample
V sample
Bulk polarization, the wrong way 2
2) P=⟨R ⟩cell
V cell
Unfortunately Clausius-Mossottidoes not work for solids because
WF are delocalized
Bulk polarization, the wrong way 3
3) P∝∑n ,mk⟨ψn k∣r∣ψm k⟩
⟨ψnk∣r∣ψm k⟩
● intra-bands terms undefined
● diverges close to the bands crossing
● ill-defined for degenerates states
How to define Hamiltonian and polarization?
Berry's phase and non-linear response
The Berry phase
IgNobel Prize (2000) together
with A.K. Geim for flying frogs
A generic quantum Hamiltonian with a parametric dependence
… phase difference between two ground eigenstates at two different x
cannot have any physical meaning
Berry, M. V. . Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802), 45-57 (1984).
...connecting the dots...
the phase difference of a closed-path is gauge-invariant therefore is a potential physical observable
g is an “exotic” observable which cannot be expressed in termsof any Hermitian operator
Berry's geometric phase
Berry's Phase and Geometric Quantum Distance: Macroscopic Polarization and Electron LocalizationR. Resta, http://www.freescience.info/go.php?pagename=books&id=1437
−iΔ ϕ≃⟨ψ(x)∣∇x ψ(x)⟩⋅Δ x
g=∑s=1
MΔ ϕ s , s+1→∫C
i⟨ ψ(x)∣∇ x ψ(x)⟩ d x
Berry's connection
● Berry's phase exists because the system is not isolated x is a kind of coupling with the “rest of the Universe”
● In a truly isolated system, there can be no manifestation of a Berry's phase
Examples of Berry's phases
Molecular AB effect Aharonov-Bohm effect
Correction to the Wannier-Stark ladder spectra of semiclassical electrons
Ph. Dugourd et al. Chem. Phys. Lett. 225, 28 (1994)
R.G. Sadygov and D.R. YarkonyJ. Chem. Phys. 110, 3639 (1999)
J. Zak, Phys. Rev. Lett. 20, 1477 (1968)
J. Zak, Phys. Rev. Lett. 62, 2747 (1989)
Electrons in a periodic system
ϕnk (r+R)=e ik Runk (r )Born-von-Karman
boundary conditions
[ 12m
p2+V (r )]ϕ n k(r)=ϵn(k)ϕ n k(r )
Bloch orbitals solution of a mean-field Schrödinger eq.
ϕn k(r+R)=e ik runk(r )Bloch functions
u obeys to periodic boundary conditions
[ 12m
( p+ℏk )2+V (r )]un k(r )=ϵn(k)unk(r )
We map the problem in k-dependent Hamiltonian and k-independent boundary conditions
k plays the role of an external parameter
What is the Berry's phase related to k?
King-Smith and Vanderbilt formulaPhys. Rev. B 47, 1651 (1993)
Pα=2i e
(2π)3∫BZ
d k∑n=1
nb⟨un k|
∂∂ kα
|unk ⟩
Berry's connection again!!
King-Smith and Vanderbilt formulaPhys. Rev. B 47, 1651 (1993)
Pα=2ie
(2π)3∫BZ
d k∑n=1
nb⟨un k∣
∂∂ kα
∣unk ⟩
Berry's phase !!
1) it is a bulk quantity2) time derivative gives the current3) reproduces the polarizabilities at all orders
What is the Berry's phase related to k?
King-Smith and Vanderbilt formula
Pα=−ef2π v
aα
N kα
⊥
∑k α⊥ ℑ∑i
Nkα−1tr ln S (k i ,k i+qα )
.. discretized King-Smith and Vanderbilt formula....Phys. Rev. B 47, 1651 (1993)
An exact formulation exists also for correlated wave-functions R. Resta., Phys. Rev. Lett. 80, 1800 (1998)
From Polarization to the Equations of Motion
L=i ℏN ∑n=1
M
∑k⟨ vkn|v̇ kn⟩−E
0−v Ε⋅P
i ℏ ∂∂ t
|vk n ⟩=H k0|vk n ⟩+i eΕ⋅|∂k vk n ⟩
It is an object difficult to calculate numerically due to the gauge freedom of the Bloch functions
|vk m ⟩→∑n
occU k ,nm|vkn ⟩
I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004)
r=i∂ k
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)
Non-linear optics
Advantages of the real-time approach
The Time-dependent Schrodinger equation
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)E1E2E3+O(E4)
SFG
DFG
SHG
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)E1E2E3+O(E4)
One code to rule “all” correlation effects
Equation of motionsare 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=H 1+H 2+H 3+ ...
And even more..
Non-perturbative phenomena (HHG)
Coupling with ionic motion,
other response functions..
Hamiltonians
The Hamiltonian Iindependent 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)=ϵiKS
δi , j
GW corrections
HGW=H KS+∑iΔϵi∣ψi ⟩ ⟨ψi∣
GW correction are included in the Hamiltonian as a non-local operator
Contrary to the linear-response case
even a simple scissor correction does not correspond to a rigid shift in non-linear response!!!
AlAs
Δ ϵi=ϵiGW
−ϵiKS
This operator modifies the eigenvalues only
The Hamiltonian IItime-dependent Hartree (RPA)
HTDH=T+V ion+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)
The density is written as:ρ(r , t )=∑i=1
N v
|Y(r , t )|2
ρ(t )
V h(t)Y(t )
CdTe
The Hamiltonian IIITD-DFT
HTDH=T+V ion+V h(ρ)+V xc(ρ)
We let density fluctuate in both the Hartree and the Vxc
tems
The Runge-Gross theorem guarantees that this is an exact theory for isolated systems
ρ(t )
V h(t)+V xc(t )Y(t )
We get the TD-DFT for solids
Beyond TD-DFT
HTDH=T+V ion+V h(ρ)+V xc(ρ)−ΩExc⋅∇k
We let Long range correlation that are missing in TD-DFT can be introduced through an
exchange correlation electric field
Exc(t ) is a function of the polarization(minimum ingredient to describe excitons)
Dielectrics in a time-dependent electric field: a real-time approach based on density-polarization functional theory
M. Gruning, D. Sangalli, and C. AttaccalitePhys. Rev. B 94, 035149 (2016)
Hamiltonian derived from many-body
If you want to include excitonic effect in an Hamiltonian,you just have to include the exchange (like TD-HF).
More precisely to be equivalent to the BSEwe include the out-equilibrium SEX self-energy.
HBSE=HGW+ΣSEX (Δ ρ̂)
ΣSEX (Δρ̂)i , j=i∑G,G' , n ,n 'WG ,G' (ω=0)Δ ρ̂n ,n '
At equilibrium: Δ ρ̂n , n '=δn ,n ' f (ϵn)
+ -=
Local-fields and excitonic effectsin h-BN monolayer
IPA
independent particles+time-dependent Hartree (RPA)
H=
Local-fields and excitonic effectsin h-BN monolayer
IPA IPA + GW
independent particles+quasi-particle corrections+time-dependent Hartree (RPA)
H=
Local-fields and excitonic effectsin h-BN monolayer
IPA IPA + GW
IPA + GW + TDSHF
independent particles+quasi-particle corrections+time-dependent Hartree (RPA)+screend Hartree-Fock (excitonic effects)
H=
Dephasing
Gauge-independent decoherence models for solids in external fields
M. S. Wismer and V. S. YakovlevPhys. 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-localoperator in the Hamiltoanian
Dephasing in post- processing~P (t)=P( t)e−λ t
See Octopus code or
Y.Takimoto, Phd thesis (2008)
Dephasing and damping
h = 1/T= = 1/T1/TT
Post-processing the non-linear reponse
P(t) is a periodic function of period TL=2p/w
L
pn is proportional to χn by the n-th order of the external field
Performing a discrete-time signalsampling 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)
Applications
Second Harmonic Generation in MoS2
Two Photon Absorption coefficients
Richardson extrapolation
P(ω)=χ(1 )
(ω)E(ω)+χ(3 )
(ω ;ω ,−ω ,ω)E(ω)E (−ω)E(ω)
Two-photons absorption in hexagonal boron nitrideC. Attaccalite et al., arXiv preprint arXiv:1803.10959
Bulk h-BN Two-photon absorption in hBN
Hexagonal boron nitride is an indirect band-gap semiconductor
G. Cassabois et al., Nature Photonics, 10, 262 (2016)
Acknowledgments
François Ducastelle Hakim AmaraMyrta Grüning
References Two-photon absorption in two-dimensional materials: The case of h-BN
C. Attaccalite, M. Grüning, H. Amara, S. Latil, and F. Ducastelle Phys. Rev. B 98, 165126 (2018)
Second harmonic generation in h-BN and MoS2 monolayers: Role of electron-hole interaction M. Grüning, C. Attaccalite, PRB 89, 081102 (2013)
Implementation of dynamical Berry phase: C. Attaccalite, M. Grüning, PRB 88, 235113 (2013)
Sylvain Latil Davide Sangalli
Non-linear response in extended systems: a real-time approachClaudio Attaccalite
https://arxiv.org/abs/1609.09639
Geometry and Topology in Electronic Structure TheoryReffaele Resta,
http://www-dft.ts.infn.it/~resta/gtse/draft.pdf
External and total field
How to calculated the dielectric constant
i∂ρ̂k (t )
∂ t=[H k+V
eff , ρ̂k ] ρ̂k (t )=∑if (ϵk , i)∣ψi , k ⟩ ⟨ ψi , k∣
The Von Neumann equation (see Wiki http://en.wikipedia.org/wiki/Density_matrix)
r t ,r ' t '=
indr , t
ext r ' , t ' =−i ⟨[ r , t r ' t ' ]⟩We want to calculate:
We expand X in an independent particle basis setχ( r⃗ t , r⃗ ' t ')= ∑
i , j , l ,m k
χi , j , l ,m, kϕ i , k (r )ϕ j ,k∗
(r )ϕ l ,k (r ')ϕm ,k∗
(r ' )
χi , j , l ,m , k=∂ρ̂i , j , k
∂V l ,m ,k
Quantum Theory of the Dielectric Constant in Real Solids
Adler Phys. Rev. 126, 413–420 (1962)
What is Veff ?
Independent Particle
Independent Particle Veff = V
ext
∂
∂V l ,m ,keff i
∂ρi , j ,k
∂ t= ∂
∂V l ,m , keff [H k+V
eff , ρ̂k ]i , j , k
Using: {H i , j ,k = δi , j ϵi(k)
ρ̂i , j , k = δi , j f (ϵi ,k)+∂ ρ̂k
∂V eff⋅Veff
+....
And Fourier transform respect to t-t', we get:
χi , j , l ,m, k (ω)=f (ϵi ,k)−f (ϵ j ,k)
ℏω−ϵ j ,k+ϵi ,k+ihδ j ,lδi ,m
i∂ρ̂k (t )
∂ t=[H k+V
eff , ρ̂k ]
χi , j , l ,m , k=∂ρ̂i , j , k
∂V l ,m ,k
Optical Absorption: IP
Non Interacting System
δρNI=χ0δV tot χ
0=∑
ij
ϕi(r)ϕ j*(r)ϕi
*(r ' )ϕ j(r ')
ω−(ϵi−ϵ j)+ ih
Hartree, Hartree-Fock, dft...
=ℑχ0=∑ij
∣⟨ j∣D∣i⟩∣2δ(ω−(ϵ j−ϵi))
ϵ''(ω)=
8 π2
ω2 ∑
i , j
∣⟨ϕ i∣e⋅v̂∣ϕ j ⟩∣2δ(ϵi−ϵ j−ℏω)
Absorption by independent Kohn-Sham particles
Particles are interacting!
V ext=0V extV HV xc
q ,=0q ,
0q ,vf 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
V ext
0=NI
V eff
... by using ...
=01
V H
V ext
V xc
V ext
vf xc
i∂ρ̂k (t )
∂ t=[H KS , ρ̂k ]=[H k
0+V eff , ρ̂k ]
Runge-Gross theorem does not hold in periodic system (PRB 68, 045109)
Current-density functional theory of the response of solidsNeepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68, 045109(2003)
charge densityexternal potential current
This part of the RG does not hold
jq (ω)=ωnq(ω)
q
A simple example
L
Independent Electrons in 1D
Current-density functional theory of the response of solids
Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68,
045109(2003)
ϕm(x ,0)=ei2πm x/L
√(L)At equilibrium the solution is H (t )=
p̂2
2
L
Independent Electrons in 1D
ϕm(x ,0)=ei2πm x/L
√LAt equilibrium the solution is
Turn on a constant electric field: A=−cε t E=−1c
∂ A∂ t
H (t )=( p̂−ε t)2
2ϕm(x , t)=
e−i km t /2−km ε t 2/ 2+ε
2 t3 /6 ei2πm x /L
√L
Current-density functional theory of the response of solids
Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68,
045109(2003)
A simple example
km=2 πm /L
LIndependent Electrons in 1D
ϕm(x ,0)=e2πm x/L
√LAt equilibrium the solution is
Turn on a constant electric field: A=−cε t E=−1c
∂ A∂ t
H (t )=( p̂−ε t)2
2ϕm(x , t)=
e−i km t /2−km ε t 2/2+ε
2 t3 /6 e2πm x/ L
√L
n(r , t)=|ϕm( x , t)2|=n(r ,0) The density does not change!!!!!
Current-density functional theory of the response of solids
Neepa T. Maitra, Ivo Souza, and Kieron Burke, Phys. Rev. B 68,
045109(2003)
A simple example
… but the current does it!
When the density is not enough
Current-density-Funtional Theory works :-)
If there are not magnetic fields and transverse reponse is not important it reduces to
Density-Polarization-FT
Exc(P , n)
Does the velocity gauge make the life easier?
It is true if you have a local Hamiltonian
Length gauge:
H=p2
2m+r E+V (r ) Y(r ,t )
Velocity gauge:
H=1
2m( p−e A)
2+V (r )
e−i r⋅A(t )Y(r , t)
Analitic demostration:K. Rzazewski and R. W. Boyd, J. of Mod. optics 51, 1137 (2004)W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
Well done velocity gauge:M. Springborg, and B. KirtmanPhys. Rev. B 77, 045102 (2008) V. N. Genkin and P. M. MednisSov. Phys. JETP 27, 609 (1968)
... but when you have non-local pseudo, exchange,damping term etc..
they do not commute with the phase factor
Theoretical point of view
How to calculate non-linear response?
1) Direct evaluation of X(1),X(2)....χ=χ
0+χ
0(v+ f xc)χ
How to calculate non-linear response?
1) Direct evaluation of X(1),X(2)....
2) Sternheimer equation R. M. Sternheimer, Phys. Rev. 96, 951(1954)
H0ψn
0=ϵn
0ψn
0
χ=χ0+χ
0(v+ f xc)χ
(H0−ϵn
0 ) ψn1=(H1
−ϵn1 )ψn
0
....=....
How to calculate non-linear response?
1) Direct evaluation of X(1),X(2)....
2) Sternheimer equation R. M. Sternheimer, Phys. Rev. 96, 951(1954)
3) Real-time propagation
P(t)=P0+χ(1)E+χ
(2)E2+O (E3
)
H0ψn
0=ϵn
0ψn
0
χ=χ0+χ
0(v+ f xc)χ
−i∂tψ=H ksψ
(H0−ϵn
0 ) ψn1=(H1
−ϵn1 )ψn
0
....=....
1) Equations become more and more complex with the response order
2) Difficult to include more than one external field (FWM, etc..)
Ref: PRB 80, 165318(2009), PRA 83, 062122(2001), JCP 126, 184106(2007)
Direct evaluation of X(1),X(2)....
Disadvantages:
Sternheimer equation
Advantages
Ref: JCP, 127, 154114(2007) PRB, 89, 081102(2014)
Real-time propagation
1) The same equation for all response functions
2) Can deal with complex spectroscopic tecniques (SFG, FWM, etc...)
Disadvantages
1) Results are more difficult to analize