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Outdated slide: In 2012, Nichia and OSRAM developed and manufactured commercial high-power green laser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