This document discusses ab-initio real-time spectroscopy and its application to non-linear optics. It begins with an overview of non-linear optics and the polarization response. It then discusses using time-dependent density functional theory to calculate nonlinear optical properties in real-time by solving the time-dependent Schrodinger equation under an external electric field. Examples are given of calculating second and third harmonic generation in materials. The document also discusses approaches to address challenges like treating bulk polarization and including many-body effects.

1. Ab-initio real-time spectroscopy:
application to non-linear optics
C. Attaccalite, Institut Néel Grenoble
M. Grüning, Queen's University, Belfast

2. What is it non­linear optics?
Materials equations:
ϵ 0 E (r , t)=D (r , t)−P (r , t)
Electric
Displacement
Electric Field
Polarization
In general:
P(r ,t )=P 0 + χ(1) E+ χ(2) E 2 +O( E3 )
First experiments on linear­optics
by P. Franken 1961
References:
Nonlinear Optics and Spectroscopy
The Nobel Prize in Physics 1981
Nicolaas Bloembergen

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 SecondHarmonic Generation Nano Lett. 13, 3329 (2013)
Second harmonic microscopy of MoS2
PRB 87, 161403 (2013)

7. … and even more …..
Photon entanglement
Second harmonic generating
(SHG) nanoprobes for
in vivo imaging
PNAS 107, 14535 (2007)

8. Spectroscopy:
experimental point of view

9. Spectroscopy:
theoretical point of view
Linear response theory:
+ fast approach
+ analysis of the results
- difficult for non-linear response
- limited to equilibrium phenomena

10. Real­time approach
External perturbation
a uniform electric field
−1 dA (t )
E( t)=
c
dt
Time-dependent Schrodinger equations
i ∂ Ψ ( x 1,. . x n)=[∑i
∂ t1
p2
i
+r⋅E+V (r 1,. .r N )] Ψ ( x 1,. . , x n)
2m
P (r ,t )=χ (1) E +χ(2) E 2 +O(E 3 )

11. Non­linear optics
Non-linear optics can calculated in the same way of TDDFT as it is done in OCTOPUS or RT-TDDFT/SIESTA codes.
Quasi-monocromatich-field
p-nitroaniline
Y.Takimoto, Phd thesis (2008)

12. The problem of bulk polarization
●
Polarization for isolated systems is well defined
〈R〉 1
1
̂
P=
= ∫ d r n( r )= 〈 Ψ∣R∣ Ψ 〉
V
V
V
●
How to define polarization as a bulk quantity?

13. Bulk polarization, the wrong way

14. The bulk polarization!!
King-Smith-Vanderbilt formula
n
2ie
Pα =
d k ∑ n=1 〈 un k∣ ∂ ∣un k 〉
3 ∫BZ
∂ kα
(2 π)
b
King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
Berry's phase !!
1) it is a bulk quantity
2) time derivative gives the current
3) reproduces the polarizabilities at all orders
4) is not an Hermitian operator

15. Our computational setup

16. Let's add some correlation in 4 steps
1)
We start from the DFT
(Kohn-Sham) Hamiltonian:
2) Renormalization of the band
structure due to correlation (GW)
universal, parameter
h k free approach
3)
Charge fluctuations
(time-dependent Hartree)
Δ ρ→Δ V H
4) Electron-hole interaction

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

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

19. 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)

20. Local-fields and excitonic effects
in h-BN monolayer
IPA
IPA + GW + TDSHF
independent particles
+quasi-particle corrections
+time-dependent Hartree (RPA)
+screend Hartree-Fock (excitonic effects)

21. MoS2 single-layer
Second harmonic microscopy of monolayer MoS2
N. Kumar et al.
Phys. Rev. B 87, 161403(R) (2013)
Observation of intense second harmonic generation from MoS2
atomic crystals
L. Malard et al.
Phys. Rev. B 87, 201401(R) (2013)
Probing Symmetry Properties of Few-Layer MoS2 and h-BN by
Optical Second-Harmonic Generation
Y. Li et al.
NanoLetters, 13, 3329 (2013)

22. What next? …
SFG, DFG, optical rectification, four-wave mixing,
electron-optical effect, Fourier spectroscopy, etc....
SHG in liquid-liquid interfaces, nanostructures
Pump and probe experiments
Dissipation, coupling with phonons.....
time resolved luminescence....

23. Acknowledgement
Myrta Grüning,
Queen's University Belfast
Reference:
1) Real-time approach to the optical properties of solids and nanostructures:
Time-dependent Bethe-Salpeter equation
C. Attaccalite, M. Gruning and A. Marini PRB 84, 245110 (2011)
2) Nonlinear optics from ab-initio by means of the dynamical Berry-phase
C. Attaccalite and M. Grüning, Phys. Rev. B 88, 235113 (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, 081102(R) (2014)

24. 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]

25. How to perform k-derivatives?
M (k )v k =λ (k )v k
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)

26. 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:
p2
H=
+ r E +V (r )
2m
Velocity gauge:
1
H=
( p−e A) 2 +V (r )
2m
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)
Ψ (r ,t )
e −r⋅A (t ) Ψ (r , t)
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)

27. 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

28. SHG in frequency domain

29. Berry's phase and Green's functions
n
2ie
Pα =
d k ∑ n=1 〈 un k∣ ∂ ∣un k 〉
3 ∫BZ
∂ kα
(2 π)
b
King-Smith and Vanderbilt formula
Phys. Rev. B 47, 1651 (1993)
The idea of Chen, Lee, Resta.....
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

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

31. 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 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).

32. ...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

33. Berry's geometric phase
−i Δ ϕ≃〈 ψ( ξ)∣∇ ξ ψ(ξ)〉⋅Δ ξ
Berry's connection
M
γ=∑ s=1 Δ ϕs , s +1 →∫C i〈 ψ(ξ)∣∇ ξ ψ( ξ)〉 d ξ
●
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
Berry's Phase and Geometric Quantum Distance:
Macroscopic Polarization and Electron Localization
R. Resta, http://www.freescience.info/go.php?pagename=books&id=1437

34. Examples of Berry's phases
Aharonov-Bohm effect
Molecular AB effect
Ph. Dugourd et al.
Chem. Phys. Lett. 225, 28 (1994)
Correction to the Wannier-Stark ladder
spectra of semiclassical electrons
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)

35. Let's add some correlation in 4 steps
1) We start from the Kohn-Sham Hamiltonian:
hk
universal, parameter free approach
2) Single-particle levels are renormalized within the G0W0 approx.
hk+ Δ hk
3) Local-field effects are included in the response function
h k + Δ h k +V H [Δ ρ]
Time-Dependent Hartree
4) Excitonic effects included by means of the Screened-Exchange
h k + Δ h k +V H [Δ ρ]+ Σ sex [Δ γ ]

36. Bulk polarization, the wrong way 3
3) P∝
∑n , m k 〈 ψn k∣r∣ψm k 〉
●
〈 ψ n k∣r∣ ψm k 〉
intra-bands terms undefined
●
diverges close to the bands crossing
●
ill-defined for degenerates states

37. Electrons in a periodic system
ϕ n k (r + R)=e i k R ϕn k (r )
[
]
1 2
p +V ( r ) ϕn k ( r)=ϵ n ( k)ϕn k ( r ) Bloch orbitals solution of
2m
a mean-field Schrödinger eq.
ϕn k ( r + R)=e
[
Born-von-Karman
boundary conditions
ik r
unk( r )
Bloch functions
u obeys to periodic boundary conditions
]
1
(p +ℏ k )2 +V (r ) un k (r )=ϵ n (k)u n k (r )
2m
We map the problem in k-dependent Hamiltonian
and k-independent boundary conditions
k plays the role of
an external parameter

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