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 k E z Acceleration of electron Radiation Probability to get high energy photon : () e it

z dt 2 Quantum-mechanical solution Time-dependent wave-function of electron (t)t) (t ) H (t ) (t ) i t 2 Acceleration of electron z 2 (t ) z (t ) t Hamiltonian with electron-laser interaction

H (t ) H 0 er E (t ) Linearly Polarized light: Circularly Polarized light: E (t ) 0 0,0,cos( t ) 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 Benzene symmetry C6 Carbon nanotube symmetry C178 Graphite 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. H Floquet i H 0 e 0 z cos( t ) t CW laser or pulse laser with broad envelope (supports at least 10 oscillations) H Floquet has 2nd order time-space symmetry: T 2 z P2 z z , t t ; T 2

An exact proof: An Exact Proof for odd Harmonic Generation H H 0 r e 0 z cos( t ) Space symmetry H 0 r H 0 r H i t For atoms:

2 H t H t Time-space symmetry: Time symmetry T 2 H ( z, t ) H ( z , t ); T 2 T

z P2 z z , t t 2 An exact proof: Floquet Theory (t ) e it (t ); (t ) (t T ) - Floquet State H Floquet (t ) (t ) H Floquet i H 0 e 0 z cos( t ) t Floquet Hamiltonian has time-space symmetry:

T H Floquet ( z , t ) H Floquet ( z , t ) 2 z 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 int (t ) dt 0 For non-zero probability, the integral should not be zero. (t ) ze int (t ) z 2 z 1 2 (t ) P P

z 2 (t ) P ze int ze int z 1 2 P z 1 2 P (t )

z 2 P (t ) An exact proof: For a non-zero integrand, following equality must hold true: ze int z 2 P ze int z 1 2

P ze T in( t ) 2 For even n=2m: ze T in( t ) 2 ze int ze ze

T i 2 m( t ) 2 int ze im2 t imT 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 imT e e im T 2 ze int

Atoms in circularly polarized light Symmetry of the Floquet Hamiltonian: H Floquet i H 0 e 0 x cos( t ) e 0 y sin( t ) t 2 PN 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: =(N 1), (2N 1), Hence no harmonics Symmetric molecules Can we get exclusively the very energetic photon??? YES

Low frequency photons are filtered: Circularly polarized light , =(N 1), (2N 1), CN symmetry Systems with N-th order time-space symmetry: T 2 2 2 2 PN x x cos y sin , y x sin y cos , t t N N N N N

H Floquet i H 0 e 0 x cos( t ) e 0 y sin( t ) t Graphite C6 symmetry (6th order time-space symmetry in circularly polarized light) H Floquet i H 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 Schrdinger 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 ik R , n1 , n2 n1 ,n2 j r R ,n1 ,n2 .

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 F E A a1 F a1

D B C E 2py,A A D RA,0,0 2px,B B C -basis set: j={2s,2px,2py}, j=1,2,3 -basis set: j={2pz}, j=1 - and -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) n 2 filled band in t i k , r , t ( p x ip y )e i k , r , t

2 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: 1T (k ) dt i k , r , t i i k , r , t T 0 t

Localized () vs. delocalized () ) vs. delocalized () ) basis electrons are delocalized freely moving electrons, with low potential barriers, hence low harmonics 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 where =(N 1), (2N 1), 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 nsk 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 1 Floquet State st 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 ik . r ( r ) e uk ( r ) uk ( r ) uk ( r d ) n n1b1 n2b2 k k 2 n

( r ) e ik . r uk ( r ) e 2 imr 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 ;