Non-linear response from real-time simulations · Non-linear response from real-time simulations...

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Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating does good)

Transcript of Non-linear response from real-time simulations · Non-linear response from real-time simulations...

Page 1: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Non-linear response from real-time simulations

Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR)

Repetita iuvant(repeating does good)

Page 2: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

What is it non-linear optics?

P(r , t )=P0+χ(1)E+χ

(2)E2+O(E3

)

Page 3: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

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Ref: supermaket

Why non-linear optics?

..applications..

Page 5: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Ref: supermaket

Why non-linear optics?..applications..

Page 6: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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Outdated slide: In 2012, Nichia and OSRAM developed and manufactured commercial high-power green laser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)

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Why non-linear optics?

..research..

Linear

Science, 2014, vol. 344, no 6183, p. 488-490

Non-linear(COLOR)Non-linear(BW)

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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).

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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

Page 10: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

...and more...

photon entanglement in vivo imaging

Ref: PNAS 107, 14535 (2007)

Page 11: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

“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 )

Page 12: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

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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⟩

Page 14: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Bulk polarization, the wrong way 1

1) P=⟨R ⟩sample

V sample

Page 15: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Bulk polarization, the wrong way 2

2) P=⟨R ⟩cell

V cell

Unfortunately Clausius-Mossottidoes not work for solids because

WF are delocalized

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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

Page 17: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

How to define Hamiltonian and polarization?

Berry's phase and non-linear response

Page 18: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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).

Page 19: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

...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

Page 20: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 21: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 22: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 23: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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!!

Page 24: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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?

Page 25: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

King-Smith and Vanderbilt formula

Pα=−ef2π v

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)

Page 26: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 27: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 28: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Advantages of the real-time approach

The Time-dependent Schrodinger equation

Page 29: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 30: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 31: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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+ ...

Page 32: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

And even more..

Non-perturbative phenomena (HHG)

Coupling with ionic motion,

other response functions..

Page 33: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating
Page 34: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Hamiltonians

Page 35: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 36: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 37: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 38: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 39: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 40: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

+ -=

Page 41: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Local-fields and excitonic effectsin h-BN monolayer

IPA

independent particles+time-dependent Hartree (RPA)

H=

Page 42: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Local-fields and excitonic effectsin h-BN monolayer

IPA IPA + GW

independent particles+quasi-particle corrections+time-dependent Hartree (RPA)

H=

Page 43: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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=

Page 44: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 45: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Dephasing and damping

h = 1/T= = 1/T1/TT

Page 46: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 47: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Applications

Page 48: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

Second Harmonic Generation in MoS2

Page 49: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 50: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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)

Page 51: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 52: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

Page 53: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

External and total field

Page 54: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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 ?

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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

Page 56: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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!

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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 ]

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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

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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

Page 60: Non-linear response from real-time simulations · Non-linear response from real-time simulations Claudio Attaccalite CNRS/CINaM, Aix-Marseille Universite (FR) Repetita iuvant (repeating

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

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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!

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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)

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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

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Theoretical point of view

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How to calculate non-linear response?

1) Direct evaluation of X(1),X(2)....χ=χ

0+χ

0(v+ f xc)χ

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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

....=....

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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

....=....

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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

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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