This document discusses nonlinear optics and the dynamical Berry phase. It introduces nonlinear optics and summarizes early experiments. It then discusses how the Berry phase is related to nonlinear optical effects like second harmonic generation (SHG). Computational methods are presented for calculating SHG and other nonlinear optical properties from first principles using time-dependent density functional theory and the dynamical Berry phase. Examples of applying these methods to study SHG in semiconductors are provided.

1. Non-linear optics
by means of
dynamical Berry phase
C. Attaccalite, Institut Néel Grenoble
M. Grüning, Queen's University, Belfast

2. What is it non­linear optics?
References:
Nonlinear Optics and Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen
First experiments on linear­optics
by P. Franken 1961
D(r ,t)=ϵ0 E(r ,t)+P(r ,t) ∇⋅E(r ,t)=4 πρtot (r ,t)
∇⋅D(r ,t)=4 πρext (r ,t)
From Gauss's law:Materials equations:
Electric
Displacement
Electric Field Polarization
P(r ,t)=χ(1)
E+χ(2)
E2
+O(E3
)
In general:

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)

7. … and even more …..
Second harmonic generating
(SHG) nanoprobes for
in vivo imaging
PNAS 107, 14535 (2007)
Second harmonic microscopy of MoS2
PRB 87, 161403 (2013)

8. A bit of theory
Which is the link between
Berry's phase and SHG?

9. 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 ξ
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).

10. ...connecting the dots...
the phase difference of a closed-path is gauge-invariant
therefore is a potential physical observable
γ is an “exotic” observable which cannot be expressed in terms
of any Hermitian operator

11. Berry's geometric phase
Berry's Phase and Geometric Quantum Distance:
Macroscopic Polarization and Electron Localization
R. Resta, http://www.freescience.info/go.php?pagename=books&id=1437
−i Δ ϕ≃〈ψ(ξ)∣∇ξ ψ(ξ)〉⋅Δ ξ
γ=∑s=1
M
Δ ϕs, s+1→∫C
i〈 ψ(ξ)∣∇ξ ψ(ξ)〉 d ξ
Berry's connection
●
Berry's phase exists because the system is not isolated
ξ 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

12. Examples of Berry's phases
Molecular AB effectAharonov-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. Yarkony
J. Chem. Phys. 110, 3639 (1999)
J. Zak, Phys. Rev. Lett. 20, 1477 (1968)
J. Zak, Phys. Rev. Lett. 62, 2747 (1989)

13. The problem
of bulk polarization
●
How to define polarization as a bulk quantity?
●
Polarization for isolated systems is well defined
P=
〈 R〉
V
=
1
V
∫d r n(r)=
1
V
〈 Ψ∣̂R∣Ψ 〉
1) P=
〈R〉sample
V sample
2) P=
〈 R〉cell
Vcell
3) P∝∑nm k
〈 ψnk∣r∣ψmk 〉

14. Bulk polarization, the wrong way 1
1) P=
〈R〉sample
V sample

15. Bulk polarization, the wrong way 2
2) P=
〈 R〉cell
Vcell
Unfortunately Clausius-Mossotti
does not work for solids because
WF are delocalized

16. 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

17. Electrons in a periodic system
ϕnk(r+ R)=eik R
ϕn k(r) Born-von-Karman
boundary conditions
[ 1
2m
p
2
+V (r)
]ϕn k(r)=ϵn(k)ϕn k(r) Bloch orbitals solution of
a mean-field Schrödinger eq.
ϕn k(r+R)=e
ik r
unk(r)
Bloch functions
u obeys to periodic boundary conditions
[ 1
2m
(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

18. What is the Berry's phase related to k?
King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
Pα=
2ie
(2π)
3 ∫BZ
d k∑n=1
nb
〈un k∣
∂
∂ kα
∣unk 〉
Berry's connection
again!!

19. King-Smith and Vanderbilt formula
Pα=
−ef
2πv
aα
Nkα
⊥
∑kα
⊥ ℑ∑i
Nk α
−1
tr ln S(ki , ki+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)

20. From Polarization to the
Equations of Motion
L=
i ℏ
N
∑n=1
M
∑k
〈 vkn∣˙vkn〉−E
0
−v Ε⋅P
i ℏ ∂
∂t
∣vk n〉=Hk
0
∣vk n〉+i e Ε⋅∣∂k vk n〉
∂k
It is an object difficult to calculate numerically
due to the gauge freedom of the Bloch functions
∣vk m〉→∑n
occ
Uk ,nm∣vkn 〉
I. Souza, J. Iniguez and D. Vanderbilt, Phys. Rev. B 69, 085106 (2004)

21. Our computational setup

22. As expected we reproduce results
obtained from linear response theory:
C. Attaccalite, M. Gruning, A. Marini, Phys. Rev. B 84, 245110 (2011)

23. Let's add some correlation in 4 steps
hk
1) We start from the Kohn-Sham Hamiltonian:
hk+Δhk
universal, parameter free approach
2) Single-particle levels are renormalized within the G0
W0
approx.
hk+Δhk+V H [Δρ]
3) Local-field effects are included in the response function
hk+Δhk+V H [Δρ]+Σsex [Δ γ]
Time-Dependent Hartree
4) Excitonic effects included by means of the Screened-Exchange

24. SHG in bulk semiconductors: SiC, AlAs, CdTe
AlAs
SiC
CdTe
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)
E. Luppi, H. Hübener, and V. Véniard
Phys. Rev. B 82, 235201 (2010)

25. THG in silicon
D. J. Moss, J. E. Sipe, and H. M. van Driel,
Phys. Rev. B 41, 1542 (1990)
D. Moss, H. M. van Driel, and J. E. Sipe,
Optics letters 14, 57 (1989)

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

27. 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?
Z. Wang et al. PRL 105, 256803 (2010)
Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011)
R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf

28. Acknowledgement
Myrta Grüning,
Queen's University Belfast
Reference:
1) Real-time approach to the optical properties of solids and nanostructures:Real-time approach to the optical properties of solids and nanostructures:
Time-dependent Bethe-Salpeter equationTime-dependent Bethe-Salpeter equation, PRB 84, 245110 (2011)
2) Nonlinear optics from ab-initio by means of the dynamical Berry-phaseNonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Grüning. http://arxiv.org/abs/1309.4012
3) Second Harmonic Generation in h-BN and MoSSecond Harmonic Generation in h-BN and MoS22
monolayers: the role of electron-hole interactionmonolayers: the role of electron-hole interaction
M. Grüning and C. Attaccalite submitted to NanoLetters

29. 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]

30. How to perform k-derivatives?
Solutions:
1) In mathematics the problem has been solved by using
second, third,... etc derivatives
SIAM, J. on Matrix. Anal. and Appl. 20, 78 (1998)
2) Global-gauge transformation
Phys. Rev. B 76, 035213 (2007)
3) Phase optimization
Phys. Rev. B 77, 045102 (2008)
4) Covariant derivative
Phys. Rev. B 69, 085106 (2004)
M (k )vk =λ(k)vk

31. 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 =
p2
2 m
+r E+V (r) Ψ(r ,t )
Velocity gauge:
H =
1
2 m
( p−e A)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, 045102 (2008)
V. N. Genkin and P. M. Mednis
Sov. Phys. JETP 27, 609 (1968)

32. Post-processing real-time data
P(t) is a periodic function of period TL
=2π/ωL
pn
is proportional to χn
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

33. SHG in frequency domain

34. King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
Pα=
2ie
(2π)
3 ∫BZ
d k∑n=1
nb
〈un k∣
∂
∂ kα
∣unk 〉
The idea of Chen, Lee, Resta.....
Berry's phase and Green's functions
Z. Wang et al. PRL 105, 256803 (2010)
Chen, K. T., & Lee, P. A. Phys. Rev. B, 84, 205137 (2011)
R. Resta, www-dft.ts.infn.it/~resta/sissa/draft.pdf