1. Real-time approach to nonlinear reponse in solids: application to second- and third-harmonic generation
C. Attaccalite, Institut Néel Grenoble M. Grüning, Queen's University, Belfast

2. What is it non-linear optics?
Materials equations:
𝐷 𝑟,𝑡 = ϵ 0 𝐸 𝑟,𝑡 +𝑃 𝑟,𝑡
Electric Displacement
Polarization
Electric Field
From Gauss's law:
In general:
∇⋅𝐸 𝑟,𝑡 =4π ρ 𝑡𝑜𝑡 𝑟,𝑡 ∇⋅𝐷 𝑟,𝑡 =4π ρ 𝑒𝑥𝑡 𝑟,𝑡
𝑃 ω = χ 1 𝐸+ χ 2 𝐸 2 +𝑂 𝐸 3
References:
Nonlinear Optics and Spectroscopy The Nobel Prize in Physics Nicolaas Bloembergen
First experiments on linear-optics by P. Franken 1961

3. The first motivation to study non-linear optics is in your (my) hands
This is a red laser
This is not a green laser!!

4. How it works a green laser pointer

5. To see “invisible” excitations
The Optical Resonances in Carbon Nanotubes Arise
from Excitons
Feng Wang, et al.
Science 308, 838 (2005);

6. Probing symmetries and crystal structures
Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Nano Lett. 13, 3329 (2013)
Edge non-linear optics on a MoS2 atomic layer Science, 344, 488 (2014)

7. … and even more ….. Photon entanglement
Second harmonic generating (SHG) nanoprobes for in vivo imaging PNAS 107, (2007)

8. Non-linear response by means of Berry phase
Which is the link between Berry's phase and SHG?

9. IgNobel Prize (2000) together with A.K. Geim for flying frogs
The Berry phase
A generic quantum Hamiltonian with a parametric dependence
IgNobel Prize (2000) together with A.K. Geim for flying frogs
… 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), (1984).

10. ...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 terms of any Hermitian operator

11. Berry's geometric phase
−𝑖Δϕ≃ ψ ξ ∣ ∇ ξ ψ ξ ⋅Δξ
Berry's connection
γ= 𝑠=1 𝑀 Δ ϕ 𝑠,𝑠+1 → 𝐶 𝑖 ψ ξ ∣ ∇ ξ ψ ξ 𝑑ξ
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
Berry's Phase and Geometric Quantum Distance: Macroscopic Polarization and Electron Localization R. Resta,

12. Examples of Berry's phases
Aharonov-Bohm effect
Molecular AB effect
Ph. Dugourd et al. Chem. Phys. Lett. 225, 28 (1994) R.G. Sadygov and D.R. Yarkony J. Chem. Phys. 110, 3639 (1999)
Correction to the Wannier-Stark ladder spectra of semiclassical electrons
J. Zak, Phys. Rev. Lett. 20, 1477 (1968) J. Zak, Phys. Rev. Lett. 62, 2747 (1989)

13. The problem of bulk polarization
Polarization for isolated systems is well defined
𝑃= 𝑅 𝑉 = 1 𝑉 𝑑 𝐫𝑛 𝐫 = 1 𝑉 Ψ ∣ 𝑅 ∣ Ψ
How to define polarization as a bulk quantity?

14. Bulk polarization, the wrong way

15. Electrons in a periodic system
Born-von-Karman boundary conditions
ϕ 𝑛𝐤 𝐫+𝐑 = 𝑒 𝑖𝐤𝐑 ϕ 𝑛𝐤 𝐫
1 2m 𝑝 2 +𝑉 𝐫 ϕ 𝑛𝐤 𝐫 = ϵ 𝑛 𝐤 ϕ 𝑛𝐤 𝐫
Bloch orbitals solution of a mean-field Schrödinger eq.
Bloch functions u obeys to periodic boundary conditions
ϕ 𝑛𝐤 𝐫+𝐑 = 𝑒 𝑖𝐤𝐫 𝑢 𝑛𝐤 𝐫
k plays the role of an external parameter
1 2m 𝑝+ℏ𝐤 2 +𝑉 𝐫 𝑢 𝑛𝐤 𝐫 = ϵ 𝑛 𝐤 𝑢 𝑛𝐤 𝐫
We map the problem in k-dependent Hamiltonian and k-independent boundary conditions

16. The right polarization!! King-Smith-Vanderbilt formula
King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993)
Berry's connection !!
it is a bulk quantity
time derivative gives the current
reproduces the polarizabilities at all orders
is not an Hermitian operator

17. From Polarization to the Equations of Motion
𝐿= 𝑖ℏ 𝑁 𝑛=1 𝑀 𝐤 𝑣 𝐤𝑛 ∣ 𝑣 𝐤𝑛 − 𝐻 0 −𝑣Ε⋅𝑃
𝑖ℏ ∂ ∂𝑡 ∣ 𝑣 𝐤𝑛 = 𝐇 𝐤 0 ∣ 𝑣 𝐤𝑛 +𝑖𝑒Ε⋅ ∣ ∂ 𝐤 𝑣 𝐤𝑛
It is an object difficult to calculate numerically due to the gauge freedom of the Bloch functions
∂ 𝐤
∣ 𝑣 𝐤𝑚 → 𝑛 𝑜𝑐𝑐 𝑈 𝐤,𝑛𝑚 ∣ 𝑣 𝐤𝑛
I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, (2004)

18. Our computational setup

19. Non-linear optics
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)

20. Let's add some correlation in 4 steps1)
We start from the DFT (Kohn-Sham) Hamiltonian: 2)
Renormalization of the band structure due to correlation (GW)
universal, parameter free approach
𝐡 𝐤 4)
Electron-hole interaction 3)
Charge fluctuations (time-dependent Hartree)
Δρ→Δ 𝑉 𝐇

21. We reproduce results obtained from linear response theory:
C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, (2011)

22. SHG in bulk semiconductors: local fields
SiC
AlAs
E. Luppi, H. Hübener, and V. Véniard Phys. Rev. B 82, (2010)
CdTe
CdTe
The main objective of this section is to validate the computational approach described in Secs. II and III against results in the literature for SHG obtained by the response theory based approach in frequency domain.
he minor discrepancies between the curves are due to the different choice for the k-grid used for integration in momentum space: we used a Γ-centered uniform grid (for which we can implement the numerical derivative) whereas Ref. 6 used a shifted grid.
In order to interpret those spectra, note that SHG resonances occur when either
w or 2w equals the difference between two single-particle energies. Then one can distinguish two energy region: below the single-particle minimum direct gap where only resonances at 2ω can occur, and above where both resonance can occur.
Local-field reduce from 15% to 30%
Cadmium telluride
E. Ghahramani, D. J. Moss, and J. E. Sipe, Phys. Rev. B 43, 9700 (1991) I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, J. Opt. Soc. Am. B 14, 2268 (1997)
J. I. Jang, et al. J. Opt. Soc. Am. B 30, 2292 (2013)

23. THG in silicon
D. J. Moss, J. E. Sipe, and H. M. van Driel, Phys. Rev. B 41, 1542 (1990)
For ener-gies below 1 eV, our QPA spectra is in good agreementwith results obtained from semi-ab-initio tight-bindingand with the experimental measurement.
For higher energies our spectra are less structured with respect both the semi-ab-initio tight-binding and the experiment, in particular missing the peak at eV. The intensities of the spectra however are more consistent with the ex-
periment than the previous theoretical results
D. Moss, H. M. van Driel, and J. E. Sipe, Optics letters 14, 57 (1989)

24. Excitonic effects in SHG?

25. Local-fields and excitonic effects in h-BN monolayer
IPA
IPA + GW + TDSHF
In fact, the IPA+GW shows two peaks: the first at about 4 eV is the shifted two-photon π → π∗ resonances peak which is attenuated by 40% with respect
to IPA [Fig. 2 (a)]; the second very pronounced peak at about 8 eV comes from the interference of π → π ∗ one-photon resonances and σ → σ∗ two-photon resonances.
independent particles +quasi-particle corrections +time-dependent Hartree (RPA) +screend Hartree-Fock (excitonic effects)

26. MoS2 single-layer
10) Second harmonic microscopy of monolayer MoS2 Phys. Rev. B 87, (R) (2013) ) Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation NanoLetters, 13, 3329 (2013)
1000) Observation of intense second harmonic generation from MoS2 atomic crystals Phys. Rev. B 87, (R) (2013)
MoS2 differs from h-BN in several aspects. First, while the h-BN has an indirect minimum band gap as its bulk counterpart, in MoS2 an indirect-to-direct bandgap transition occurs passing from the bulk to the monolayer due to the vanishing interlayer interaction. Second, spin-orbit coupling plays an important role in this material, splitting the top valence bands, as visible from the absorption spectrum, presenting a double peak at the onset.7 Third, Mo and S atoms in the MoS2 monolayer are on different planes resulting in a larger inhomogeneity than for the
H-BN.
gap at the K point; a larger peak around 1.5 eV, which originates from two-photon resonances with transitions along the high symmetry axis between Γ and K where the highest valence and lowest conduction bands are flat and there is a high density of states; a broad structure between 2−3.5 eV which originates from one-photon res-

27. Nonlinear optics in semiconductors from first-principles real-time simulations
TDSE

28. Open questions? What next? …
SFG, DFG, optical rectification, four-wave mixing, electron-optical effect, Fourier spectroscopy, etc....
SHG in liquid-liquid interfaces, nanostructures
Dissipation, coupling with phonons.....
luminescence, light emission,strong fields...
Open questions?
Dissipative effects? How?
Coupling dynamical Berry phase with Green's functions?
Coupling dynamical Berry phase with density matrix hierarchy equations, BBGKY?
R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf‎

29. Acknowledgement References:
Myrta Grüning, Queen's University Belfast
References:
1) Real-time approach to the optical properties of solids and nanostructures C. Attaccalite, M. Grüning and A. Marini PRB 84, (2011) ) Nonlinear optics from ab-initio by means of the dynamical Berry-phase C. Attaccalite and M. Grüning. Phys. Rev. B 88, (2013)
3) Second Harmonic Generation in h-BN and MoS2 monolayers: the role of electron-hole interaction M. Grüning and C. Attaccalite Phys. Rev. B 89, (R) (2014)

30. SHG in bulk semiconductors: quasi-particle effects
Validation of our the computational approach against results
in the literature for SHG obtained by the response theory based approach in frequency domain.
SiC
AlAs
The main objective of this section is to validate the computational approach described in Secs. II and III against results in the literature for SHG obtained by the response theory based approach in frequency domain.
he minor discrepancies between the curves are due to the different choice for the k-grid used for integration in momentum space: we used a Γ-centered uniform grid (for which we can implement the numerical derivative) whereas Ref. 6 used a shifted grid.
In order to interpret those spectra, note that SHG resonances occur when either
w or 2w equals the difference between two single-particle energies. Then one can distinguish two energy region: below the single-particle minimum direct gap where only resonances at 2ω can occur, and above where both resonance can occur.
Local-field reduce from 15% to 30%
Cadmium telluride
E. Luppi, H. Hübener, and V. Véniard Phys. Rev. B 82, (2010)

31. The King-Smith and Vanderbilt formula
We introduce the Wannier functions Blount, 1962
We express the density in terms of
Wannier functions
Polarization in terms of Wannier functions [Blount 62]

32. How to perform k-derivatives?
𝑀 𝑘 𝑣 𝑘 =λ 𝑘 𝑣 𝑘
Solutions:
In mathematics the problem has been solved by using second, third,... etc derivatives SIAM, J. on Matrix. Anal. and Appl. 20, 78 (1998)
Global-gauge transformation Phys. Rev. B 76, (2007)
Phase optimization Phys. Rev. B 77, (2008)
Covariant derivative Phys. Rev. B 69, (2004)

33. Wrong ideas on velocity gauge
In recent years different wrong papers using velocity gauge have been published (that I will not cite here) on: 1) real-time TD-DFT 2) Kadanoff-Baym equations + GW self-energy 3) Kadanoff-Baym equations + DMFT self-energy
Length gauge:
𝐇= 𝐩 𝟐 2𝑚 +𝐫𝐸+𝑉 𝐫
Ψ 𝐫,𝑡
Velocity gauge:
𝐇= 1 2𝑚 𝐩−𝑒𝐴 2 +𝑉 𝐫
𝑒 −𝐫⋅𝐴 𝑡 Ψ 𝐫,𝑡
Analitic demostration:
K. Rzazewski and R. W. Boyd, Journal of modern optics 51, 1137 (2004) W. E. Lamb, et al. Phys. Rev. A 36, 2763 (1987)
Well done velocity gauge:
M. Springborg, and B. Kirtman
Phys. Rev. B 77, (2008) V. N. Genkin and P. M. Mednis Sov. Phys. JETP 27, 609 (1968)

34. Post-processing real-time data
P(t) is a periodic function of period TL=2p/wL
pn is proportional to cn by the n-th order of the external field
Performing a discrete-time signal sampling we reduce the problem to the solution of a systems of linear equations

35. SHG in frequency domain

36. Berry's phase and Green's functions
King-Smith and Vanderbilt formula Phys. Rev. B 47, 1651 (1993)
The idea of Chen, Lee, Resta.....
Z. Wang et al. PRL 105, (2010) Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, (2011) R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf‎