Generation and control of high-Generation and control of high-order harmonics by order harmonics by the Interaction the Interaction

of infrared lasers with a thin of infrared lasers with a thin Graphite layerGraphite layer

Ashish K GuptaAshish K Gupta

&&

Nimrod MoiseyevNimrod MoiseyevTechnion-Israel Institute of Technology,Technion-Israel Institute of Technology,

Haifa, IsraelHaifa, Israel

Photo-assisted chemical reactions

nA B

Reactant A, product B are chemicals and light is a catalyst.

Light – Matter InteractionLight – Matter Interaction

Harmonic Generation Phenomena

/atoms moleculesn

Reactants and product are photons and chemicals are a catalyst.

Mechanism for generation of high Mechanism for generation of high energy photons (high order energy photons (high order

harmonics)harmonics)Multi-photon

absorption

Acceleration of electron

z

Radiation ħΩ

2

( ) i te z dt

Probability to get high energy photon ħΩ ħω:

E

k

ħω

Quantum-mechanical solutionQuantum-mechanical solution

2

2( ) ( )z t z t

t

Time-dependent wave-function of electron (t)

( )ˆ ( ) ( )t

H t t it

Acceleration of electron

0ˆ ˆ( ) ( )H t H er E t

Hamiltonian with electron-laser interaction

0

0

( ) 0,0,cos( )

( ) cos( ),sin( ),0

E t t

E t t t

Linearly Polarized light:

Circularly Polarized light:

.( , , )Rare gas atomseg He Ar Kr

n

, 3,5,7...n n

The intensity of emitted radiation is 6-8 orders of magnitude less than the incident laser intensity.

Harmonic generation from atomsHarmonic generation from atoms

Highly nonlinear phenomenon: powerful laser 1015 W/cm2 & more

2 600eV eV Incoming laser frequency multiplied up to 300 times:

Experiments

Molecular systemsMolecular systemsOur theoretical prediction of Harmonic generation from symmetric

molecules:1) Strong effect because higher induced dipole2) Selective generation caused by structure with high order symmetry

symmetry C6

Carbon nanotube

symmetry C178

Benzene symmetry C6

Graphite

Why do atoms emit only odd Why do atoms emit only odd harmonics in linearly polarized harmonics in linearly polarized

electric field ?electric field ?Non perturbative explanation (exact solution)Selection rules due to the time-space symmetry properties of Floquet operator.

0 0 coˆ ˆ s( )Floquet e z tH i Ht

CW laser or pulse laser with broad envelope (supports at least 10 oscillations)

has 2nd order time-space symmetry:

2

2ˆ , ;2

z TP z z t t T

ˆFloquetH

0 0ˆ ˆ cos( )H H r e z t

H i

t

For atoms: 0 0ˆ ˆH r H r

2ˆ ˆ( , ) ( , );2

TH z t H z t T

2ˆ ˆH t H t

2 ,2

z TP z z t t

An

exac

t pro

of: An Exact Proof for odd Harmonic GenerationAn Exact Proof for odd Harmonic Generation

Space symmetry

Time symmetry

Time-space symmetry:

Floquet TheoryFloquet Theory

- Floquet State

0 0ˆ ˆ cos( )FloquetH i H e z t

t

ˆ ( ) ( )FloquetH t t

ˆ ˆ( , ) ( , )2Floquet Floquet

TH z t H z t

2 2 2ˆ ; 1zP P P

An

exac

t pro

of:

Floquet Hamiltonian has time-space symmetry:

( ) ; ( )( () )i t tt e t t T

1 1

22 2

1

2 2

2

( ) ( )

( ) ( )

( ) ( )

in t

z in t z

z in t

z

z

z

ze

P ze P

P ze

t t

t P

P

P t

t t

2

4

0

( ) () )( inT

tt ze tn n dt

( ) ( )t e z t A

n ex

act p

roof

: Dipole moment:

Probability of emitting n-th harmonic:

For non-zero probability, the integral should not be zero.

1

2 2in t z in t zze P ze P

( ) (2 1) ( )2

2 1

2

2( )

T

Tim in t

Ti

im

n t i m t

i m t Te e

z

e

e

ze z

ze

For odd n=2m+1:

An

exac

t pro

of: For a non-zero integrand, following equality must hold true:

( )2

Tin t

ze

For even n=2m:

2( ) 2 ( ) 22

Ti

Tin t i m t im Tt

n t

m

i

ze ze ze e

ze

in tze Therefore, no even harmonics

Atoms in Atoms in circularlycircularly polarized light polarized lightSymmetry of the Floquet Hamiltonian:

Floquet Hamiltonian has infinite order time-space symmetry, N=

Hence no harmonics

0 0 0ˆ ˆ cos( ) sin( )FloquetH i H e x t e y t

t

2 2 2 2ˆ cos sin , sin cos ,N

TP x x y y x y t t

N N N N N

Selection rule for emitted harmonics: Ω=(N 1)ω, (2N 1)ω,…

Symmetric moleculesSymmetric moleculesCan we get exclusively the very energetic photon???Can we get exclusively the very energetic photon???

YES

Systems with N-th order time-space symmetry:

Low frequency photons are filtered:

Circularly polarized light ħω

CN symmetry

ħΩ, Ω=(N 1)ω, (2N 1)ω,…

2 2 2 2ˆ cos sin , sin cos ,N

TP x x y y x y t t

N N N N N

0 0 0cos( ) sˆ ˆ in( )Floquet e x t e yi Ht

tH

GraphiteGraphiteC6 symmetry (6th order time-space symmetry in circularly polarized light)

0 0ˆ ˆ cos( ) sin( )Floquet graphiteH i H e x t e y 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 Graphite Lattice

1a

2a

Direct Lattice with the unit vectors

F

A

B

CD

E

Tight Binding ModelTight Binding Model

Only nearest neighbor interactions are included in the calculation.

σ-basis set: j={2s,2px,2py}, j=1,2,3

π-basis set: j={2pz}, j=1

σ- and π-basis sets do not couple.

, ,1 2

1 2

1 2

, , ,,

1, .n nik R

j j n nn n

k r e r RN

A Bloch basis set is used to describe the quasi energy states ,

, ,j k r

, ,k r t

α denotes an atom (A-F) in a unit cell. The summation goes over all the unit cells [n1,n2], generated by translation vectors .

1 2, , ,0,0 1 1 2 2n nR R n a n a

1 2[ , ]a a

A

B C

D

EF

A

B C

D

EF

,0,0AR 2py,A

1a

2px,B

1a

Formula for calculating HGFormula for calculating HGThe probability to obtain n-th harmonic within Hartreeapproximation is given by

2

( ) 2 ˆ ˆ, , ( ) , ,n in ti x y i

filled band

I n k r t p ip e k r t

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:

0

1( ) , , , ,

T

i ik dt k r t i k r tT t

Localized (σ) vs. delocalized (π) Localized (σ) vs. delocalized (π) basisbasis

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 ComparisonIntensity ComparisonMinimal intensity to get plateau: 3.56 1012 W/cm2

Plateau: Intensity remains same for a long range of harmonics (3rd-31st)

Effect of laser frequencyEffect of laser frequency

Effect of ellipticityEffect of ellipticity 0( ) 2 cos cos( ),sin sin( ),0E t t t

Graphite vs. BenzeneGraphite 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.

ConclusionsConclusions1. 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.

ThanksThanks

Prof. Nimrod MoiseyevProf. Lorenz Cederbaum Dr. Ofir AlonDr.Vitali Averbukh Dr. Petra ŽďánskáDr. Amitay Zohar

Aly Kaufman Fellowship

First Band of GraphiteFirst Band of Graphite

HG due to acceleration in xHG due to acceleration in x

HG due to acceleration in yHG due to acceleration in y

Mean energy of 1Mean energy of 1stst Floquet State Floquet State

First quasi energy bandFirst quasi energy band

Avoided crossing for 1Avoided crossing for 1stst Floquet Floquet StateState

Entropy of 1Entropy of 1stst Floquet State Floquet State

Reciprocal LatticeReciprocal Lattice

Reciprocal lattice:

Brillouin zone

b1

b2

Potential: V(r)=V(r+d); d=d1a1+d2a2

( ) exp(2 )nn

V r V in r For the translation symmetry to

hold good: n=n1b1+n2b2

i j ija b

( ) exp(2 ( ))

exp(2 )exp(2 ) ( )

nn

nn

V r d V in r d

V in r in d V r

integern d

Bloch FunctionBloch Function

.

1 1 2 2

. 2 .

( ) ( )

( ) ( )

2

( ) ( ) ( )

ik rk

k k

ik r im r ik rk k

r e u r

u r u r d

n n b n b

k k n

r e u r e e u r

Brillouin Zone : k and k+2pi*n correspond to same physical

solution hence k could be restricted. For a cubic lattice:

d=d1a1+d2a2

1 1 1 2 2 2 3 3 3; ; ;b k b b k b b k b