1. Generation and control of high- order harmonics by the Interaction of infrared lasers with a thin Graphite layer Ashish K Gupta & Nimrod Moiseyev Technion-Israel Institute of Technology, Haifa, Israel

2. Photo-assisted chemical reactions Reactant A, product B are chemicals and light is a catalyst. Light – Matter Interaction Light – Matter Interaction Harmonic Generation Phenomena Reactants and product are photons and chemicals are a catalyst.

3. Mechanism for generation of high energy photons (high order harmonics) Multi-photon absorption Acceleration of electron z Radiation ħΩ Probability to get high energy photon ħΩ  ħω: E k ħωħω

4. Quantum-mechanical solution Time-dependent wave-function of electron  (t) Acceleration of electron Hamiltonian with electron-laser interaction Linearly Polarized light: Circularly Polarized light:

5. The intensity of emitted radiation is 6-8 orders of magnitude less than the incident laser intensity. Harmonic generation from atoms Highly nonlinear phenomenon: powerful laser 10 15 W/cm 2 & more Incoming laser frequency multiplied up to 300 times: Experiments

6. 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 symmetry C 6 Carbon nanotube symmetry C 178 Benzene symmetry C 6 Graphite

7. 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. CW laser or pulse laser with broad envelope (supports at least 10 oscillations) has 2 nd order time-space symmetry:

8. For atoms: An exact proof: An Exact Proof for odd Harmonic Generation Space symmetry Time symmetry Time-space symmetry:

9. Floquet Theory - Floquet State An exact proof: Floquet Hamiltonian has time-space symmetry:

10. An exact proof: Dipole moment: Probability of emitting n-th harmonic: For non-zero probability, the integral should not be zero.

11. For odd n=2m+1: An exact proof: For a non-zero integrand, following equality must hold true: For even n=2m: Therefore, no even harmonics

12. Atoms in circularly polarized light Symmetry of the Floquet Hamiltonian: Floquet Hamiltonian has infinite order time-space symmetry, N=  Hence no harmonics Selection rule for emitted harmonics: Ω=(N  1)ω, (2N  1)ω,…

13. Symmetric molecules 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 C N symmetry ħΩ, Ω=(N  1)ω, (2N  1)ω,…

14. Graphite C 6 symmetry (6 th order time-space symmetry in circularly polarized light) 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.

15. Graphite Lattice Direct Lattice with the unit vectors F A B C D E

16. Tight Binding Model Only nearest neighbor interactions are included in the calculation. σ-basis set:  j ={2s,2p x,2p y }, j=1,2,3 π-basis set:  j ={2p z }, j=1 σ- and π-basis sets do not couple. A Bloch basis set is used to describe the quasi energy states, α denotes an atom (A-F) in a unit cell. The summation goes over all the unit cells [n 1,n 2 ], generated by translation vectors. A BC D EF A BC D EF  2py,A  2px,B

17. Formula for calculating HG The probability to obtain n-th harmonic within Hartree approximation is given by 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:

18. 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

19. Intensity Comparison Minimal intensity to get plateau: 3.56  10 12 W/cm 2 Plateau: Intensity remains same for a long range of harmonics (3 rd -31 st )

20. Effect of laser frequency

21. Effect of ellipticity

22. 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.

23. Conclusions 1.High harmonics predicted from graphite. 2.Interaction of C N 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 5 th (symmetry allowed) harmonic is 100 times larger than the intensity of the 3 rd (forbidden) harmonic.

24. Thanks Prof. Nimrod Moiseyev Prof. Lorenz Cederbaum Dr. Ofir Alon Dr.Vitali Averbukh Dr. Petra Žďánská Dr. Amitay Zohar Aly Kaufman Fellowship

25. First Band of Graphite

26. HG due to acceleration in x

27. HG due to acceleration in y

28. Mean energy of 1 st Floquet State

29. First quasi energy band First quasi energy band

30. Avoided crossing for 1 st Floquet State

31. Entropy of 1 st Floquet State

32. Reciprocal Lattice Reciprocal lattice: Brillouin zone b1b1 b2b2 Potential: V(r)=V(r+d); d=d 1 a 1 +d 2 a 2 For the translation symmetry to hold good: n=n 1 b 1 +n 2 b 2

33. Bloch Function Brillouin Zone : k and k+2pi*n correspond to same physical solution hence k could be restricted. For a cubic lattice: d=d 1 a 1 +d 2 a 2