Generation and control of highorder harmonics by the Interaction
of infrared lasers with a thin
Graphite layer
Ashish K Gupta
&
Nimrod Moiseyev
Technion-Israel Institute of Technology,
Haifa, Israel
Light – Matter Interaction
Photo-assisted chemical reactions

nA  B
Reactant A, product B are chemicals and light is a catalyst.
Harmonic Generation Phenomena
n  



atoms / molecules
Reactants and product are photons and chemicals are a
catalyst.
Mechanism for generation of high
energy photons (high order
harmonics)
Multi-photon
absorption ħω
k
E
z
Radiation ħΩ
Acceleration of electron
Probability to get high energy photon ħΩ ħω:
()   e
 it
2
z dt
Quantum-mechanical solution
Time-dependent wave-function of electron (t)
 (t )
ˆ
H (t )  (t )  i
t

z  2  (t ) z  (t )
t
2
Acceleration of electron
Hamiltonian with electron-laser interaction
Hˆ (t )  Hˆ 0  er  E (t )
Linearly Polarized light:
E (t )   0 0,0,cos( t ) 
Circularly Polarized light:
E (t )   0 cos( t ),sin( t ),0
Harmonic generation from atoms
Rare gas atoms
n  


eg .( He , Ar , Kr )
  n , n  3,5, 7...
Experiments
Highly nonlinear phenomenon: powerful laser 1015 W/cm2 & more
Incoming laser frequency multiplied up to 300 times:
  2 eV    600eV
The intensity of emitted radiation is 6-8 orders of magnitude less
than the incident laser intensity.
Molecular systems
Our theoretical prediction of Harmonic generation from symmetric
molecules:
1) Strong effect because higher induced dipole
2) Selective generation caused by structure with high order symmetry
Graphite
Benzene
symmetry C6
Carbon nanotube
symmetry C178
symmetry C6
Why do atoms emit only odd
harmonics in linearly polarized
electric field ?
Non perturbative explanation (exact solution)
Selection rules due to the time-space symmetry properties of Floquet
operator.

ˆ
H Floquet  i
 Hˆ 0  e  0 z cos( t )
t
CW laser or pulse laser with broad envelope
(supports at least 10 oscillations)
Hˆ Floquet has 2nd order time-space symmetry:
T
2

z
ˆ
P2   z   z, t  t   ; T 
2


An exact proof:
An Exact Proof for odd Harmonic Generation

Ĥ   i
t
For atoms:

 r 
Hˆ  Hˆ 0 r  e 0 z cos(t )

Hˆ 0 r  Hˆ 0
 2 
ˆ
ˆ
H t   H  t 



Time-space symmetry:
Space symmetry
Time symmetry
T
2

Hˆ ( z, t )  Hˆ (  z, t  ); T 
2

T

z
ˆ
P2   z   z, t  t  
2

An exact proof:
Floquet Theory
(t )  e
it
(t ); (t )  (t  T ) - Floquet State
Hˆ Floquet (t )   (t )

ˆ
H Floquet  i
 Hˆ 0  e  0 z cos( t )
t
Floquet Hamiltonian has time-space symmetry:
T
Hˆ Floquet ( z, t )  Hˆ Floquet (  z, t  )
2
z
ˆ
P2   P2 ; P2  1
An exact proof:
  (t ) e z (t )
Dipole moment:
Probability of emitting n-th harmonic:
2
T
(  n)  n
4

(t ) ze
 int
(t ) dt
0
For non-zero probability, the integral should not be zero.
(t ) ze
 int
(t )
 (t ) P  P
z
2
 (t ) P ze
z
2

z 1
2
 int
ze
 int
P 
z 1
2
P 
z 1
2
(t )
P (t )
z
2
An exact proof:
For a non-zero integrand, following equality must hold true:
ze
 int
 P ze
z
2
 int
P 
z 1
2
  ze
T
 in( t  )
2
For even n=2m:
 ze
T
 in( t  )
2
  ze
 int
  ze
 ze
T
 i 2 m( t  )
2
 int
  ze
 im2 t  imT
e
Therefore, no even harmonics
For odd n=2m+1:
 ze
T
 in( t  )
2
  ze
  ze
T
 i (2 m 1) ( t  )
2
 i ( 2 m 1) t  imT
e
e
 im
T
2
 ze
 int
Atoms in circularly polarized light
Symmetry of the Floquet Hamiltonian:

ˆ
H Floquet  i
 Hˆ 0  e  0 x cos( t )  e  0 y sin( t )
t

 2
PˆN   x  x cos 
 N


 2

y
sin



 N

 2
,
y

x
sin



 N

 2

y
cos



 N
T

,
t

t


N 

Floquet Hamiltonian has infinite order time-space symmetry, N=
Selection rule for emitted harmonics: Ω=(N  1)ω, (2N  1)ω,…

Hence no harmonics
Symmetric molecules
Can we get exclusively the very energetic photon???
YES
Low frequency photons are filtered:
Circularly polarized light ħω
ħΩ, Ω=(N  1)ω, (2N  1)ω,…
CN symmetry
Systems with N-th order time-space symmetry:
T

 2 
 2 
 2 
 2 
PˆN   x  x cos    y sin   , y  x sin    y cos   , t  t  
N
N
N
N
N


ˆ
H Floquet  i
 Hˆ 0  e  0 x cos( t )  e  0 y sin( t )
t
Graphite
C6 symmetry (6th order time-space
symmetry in circularly polarized light)

ˆ
H Floquet  i
 Hˆ graphite  e  0 x cos( t )  e  0 y sin( t )
t
Numerical Method:
1) Choose the convenient unit cell
2) Tight binding basis set
3) Bloch theory for periodic solid structure
4) Floquet operator for description of time periodic system
5) Propagate Floquet states with time-dependent Schrödinger equation.
Graphite Lattice
a1
A
F
B
E
C
D
a2
Direct Lattice with the unit vectors
Tight Binding Model
A Bloch basis set   k , r  is used to describe the quasi energy


states  k , r , t ,
 j ,
 
j ,
1
k,r 
N
e
 j  r  R ,n ,n  .
ik R , n1 , n2
1
F
2
n1 ,n2
A
α denotes an atom (A-F) in a unit cell.
The summation goes over all the unit cells [n1,n2],
generated by translation vectors [a1 , a2 ] .
R ,n1 ,n2  R ,0,0  n1a1  n2a2
E
a1
F
a1
A
D
B
C
E
RA,0,0
D
2py,A
2px,B
σ-basis set: j={2s,2px,2py}, j=1,2,3
B
C
π-basis set: j={2pz}, j=1
σ- and π-basis sets do not couple.
Only nearest neighbor interactions are included in the calculation.
Formula for calculating HG
The probability to obtain n-th harmonic within Hartree
approximation is given by
I (n)  n2




i k , r , t ( pˆ x  ipˆ y )eint i k , r , t

filled band
The triple bra-ket stands for integration over time (t), space (r),
and crystal quasi-momentum (k) within first Brillouin zone. The
summation is over filled quasi-energy bands.
The structure of bands in the field:
1

 (k )   dt i k , r , t  i
i k , r , t
T0
t
T




2
Localized (σ) vs. delocalized (π)
basis
π – electrons are delocalized freely moving electrons, with low potential
barriers, hence low harmonics
σ – electrons tightly bound in the lattice potential, hence high harmonics
Intensity Comparison
Minimal intensity to get plateau: 3.56 1012 W/cm2
Plateau: Intensity remains same for a
long range of harmonics (3rd-31st)
Effect of laser frequency
Effect of ellipticity
E (t )  2 0 cos   cos( t ),sin   sin( t ), 0 
Graphite vs. Benzene
HG from Benzene-like structure dies faster than HG from Graphite.
No enhancement of the intensity using circularly vs. linearly polarized
light is obtained, Hence it is a filter, not an amplifier.
Conclusions
1.
High harmonics predicted from graphite.
2.
Interaction of CN symmetry molecules/materials with circularly polarized
light rather than with linearly polarized light, generates photons with
energy ħΩ where Ω=(N  1)ω, (2N  1)ω,…
3. Circularly polarized light filters the low energy photons, however no
amplification effect is predicted.
4. Extended structure produces longer plateau as seen in the case of Graphite
vs. benzene-like systems .
5. HG in graphite is stable to distortion of symmetry. For 1% distortion of the
polarization the intensity of the emitted 5th (symmetry allowed) harmonic
is 100 times larger than the intensity of the 3rd (forbidden) harmonic.
Thanks
Prof. Nimrod Moiseyev
Prof. Lorenz Cederbaum
Dr. Ofir Alon
Dr.Vitali Averbukh
Dr. Petra Žďánská
Dr. Amitay Zohar
Aly Kaufman Fellowship
First Band of Graphite
HG due to acceleration in x
HG due to acceleration in y
Mean energy of 1st Floquet State
First quasi energy band
Avoided crossing for 1st Floquet
State
Entropy of 1st Floquet State
Reciprocal Lattice
Potential: V(r)=V(r+d); d=d1a1+d2a2
b1
V ( r )  Vn exp(2 in  r )
n
For the translation symmetry to
hold good: n=n1b1+n2b2
ai  b j   ij
V ( r  d )  Vn exp(2 in  ( r  d ))
n
 Vn exp(2 in  r )exp(2 in  d )  V ( r )
n
n  d  integer
b2
Reciprocal lattice:
Brillouin zone
Bloch Function
d=d1a1+d2a2
 ( r )  e uk ( r )
ik . r
uk ( r )  uk ( r  d )
n  n1b1  n2b2
k   k  2 n
 (r)  e
ik . r
uk ( r )e
2 imr
e
ik . r
uk  ( r )
Brillouin Zone : k and k+2pi*n correspond to same physical
solution hence k could be restricted. For a cubic lattice:
 b1  k1   b1;  b2  k2   b2 ;  b3  k3   b3 ;