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Theoretical Spectroscopy Lectures: real-time approach 1


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In this lecture, I will describe how to calculate optical response functions using real-time simulations. In particular, I will discuss td-hartree, td-dft and similar approximations.

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Theoretical Spectroscopy Lectures: real-time approach 1

  1. 1. Real­time Spectroscopy for solids Claudio Attaccalite  CNRS/CINaM, Aix­Marseille Universite (FR)  Theoretical Spectroscopy Lectures October 8­12, 2018, Lausanne Repetita iuvant (repeating does good)
  2. 2. What is it non­linear optics? P(r ,t)=P0+χ (1) E+χ (2) E 2 +O(E 3 ) First experiments on linear­optics  by P. Franken 1961 Ref: Nonlinear Optics and  Spectroscopy  The Nobel Prize in Physics 1981 Nicolaas Bloembergen
  3. 3. Ref: supermaket Why non­linear optics? ..applications..
  4. 4. Ref: supermaket Why non­linear optics? ..applications..
  5. 5. Why non­linear optics? ..applications.. New green laser appeared in 2012  without non­linear crystals
  6. 6. Why non­linear optics? ..research.. Linear Science, 2014, vol. 344, no 6183, p. 488­490 Non­linear(COLOR)Non­linear(BW)
  7. 7. To see “invisible” excitations The Optical Resonances in  Carbon  Nanotubes Arise from Excitons Feng Wang, et al. Science 308, 838 (2005); ..research.. Right interpretation of the experiment “Selection rules for one-and two-photon absorption by excitons in carbon nanotubes,” E. B. Barros et al. PRB 73, 241406 (2006).
  8. 8. Probing symmetries Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation Nano Lett. 13, 3329 (2013) SHG can probe magnetic transition
  9. 9. ...and more... photon entanglement  in vivo imaging Ref: PNAS 107, 14535 (2007) 
  10. 10. “Self­focusing limit in terms of peak power is of the order  of 4 MW in the 1­μm wavelength region. No method is known  for increasing the self­focusing limit of optical fibers  beyond that value.”  RP Photonics Why non­linear optics? ..applications.. (the darkside      )
  11. 11. Real­time spectroscopy is  much similar to experiments
  12. 12. What is real­time  spectroscopy? Choose a  perturbation E(t)=δ(t−t0) E0 E(t)=sin(ωt)E0
  13. 13. What is real­time  spectroscopy? Choose a  perturbation E(t)=δ(t−t0) E0 E(t)=sin(ωt)E0 Time­evolution  of an effective  Schroedinger  equation Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t ) Ψ(t=0)=ΨGS
  14. 14. What is real­time  spectroscopy? Choose a  perturbation E(t)=δ(t−t0) E0 E(t)=sin(ωt)E0 Time­evolution  of an effective  Schroedinger  equation Ψ(t+Δt)=Ψ(t)−i Δ t H Ψ (t ) Ψ(t=0)=ΨGS Analise the  results
  15. 15. Real time spectroscopy in practice 1 3- / D(r ,t)=E(r ,t)+P(r ,t) Materials equations: Electric Displacement Electric Field Polarization ∇⋅E(r ,t)=4 πρtot (r ,t) ∇⋅D(r ,t)=4 πρext (r ,t) From Gauss's law: Δ P(r ,t)=∫χ(t−t ' ,r ,r ')E(t ' r ')dt ' dr '+∫dt 1 dt 2 χ 2 (...)E(t 1 ) E(t 2 )+O(E 3 ) In general:
  16. 16. Real time spectroscopy in practice 2 3- / For a small perturbation we consider only the first term, the linear response regime Δ P(r ,t)=∫χ(t−t ' ,r ,r ')E(t ' r ')dt ' dr '+O(E2 ) Δ P(ω)=χ(ω)E(ω)=(ϵ(ω)−1)E(ω) And finally: ϵ(ω)=1+ Δ P(ω) E(ω) ϵ(ω)= D(ω) E(ω)
  17. 17. Real time spectroscopy in practice 3 3- / 1) Choose an external perturbation E(t) 2) Evolve the Schroedinger equation 3) We calculate the P(t) from (t) 4) Fourier transform P(t) and E(t) and get i d Ψ(t ) dt =[H + E(t )]Ψ(t ) ϵ(ω)=1+ Δ P(ω) E(ω)
  18. 18. Motivations Better scaling for large system Polarization and  Hamiltonian depend only  from valence bands. No need  of conduction bands!
  19. 19. Motivations Better scaling for large system Theory and implementation are much easier Polarization and  Hamiltonian depend only on  valence bands. No need of  conduction bands!
  20. 20. One code to rule all spectroscopy responses χ(2) (ω;ω1, ω2) P(ω)=P0+χ (1) (ω)E1(ω)+χ (2) E1(ω1) E2(ω2)+χ (3) E1 E2 E3+O(E 4 ) SFG DFG SHG
  21. 21. One code to rule all spectroscopy responses χ(3) (ω; ω1, ω2, ω3) THG P(ω)=P0+χ(1) (ω)E1(ω)+χ(2) E1(ω1) E2(ω2)+χ(3) E1 E2 E3+O(E4 )
  22. 22. One code to rule all“ ” correlation effects Equation of motions are always the same In order to include correlation effects just change the Hamiltonian Notice that the present approach is limited to single-particle Hamiltonians. H=H1+H2+H3+...
  23. 23. And even more.. Non­perturbative  phenomena (HHG) Coupling with ionic motion, other response functions..
  24. 24. The Hamiltonian I independent particles H KS(ρ0)=T+V ion+V h(ρ0)+V xc (ρ0) We start from the Kohn-Sham Hamiltonian If we keep fixed the density in the Hamiltoanian to the ground-state one we get the independent particle approximation In the Kohn-Sham basis this reads: H KS(ρ0)=ϵi KS δi, j
  25. 25. The Hamiltonian II time­dependent Hartree (RPA) HTDH =T+Vion+V h(ρ)+V xc (ρ0) If we keep fixed the density in Vxc but not in Vh. We get the time-dependent Hartree or RPA (with local fields) Or equivalent: HTDH =H KS(ρ0)+Vh (ρ−ρ0) The density is written as: ρ(r ,t)=∑i=1 N v |Ψ(r ,t)| 2
  26. 26. The Hamiltonian III TD­DFT HTDH =T+Vion+V h(ρ)+V xc (ρ) We let density fluctuate in both the Hartree and the Vxc tems We get the TD-DFT for solids The Runge­Gross theorem guarantees that this is an exact  theory for isolated systems
  27. 27. Dephasing Gauge-independent decoherence models for solids in external fields M. S. Wismer and V. S. Yakovlev Phys. Rev. B 97, 144302 (2018) The previous Hamiltonian are Hermitian without any time-dependence (expect the external field) This means they do not introduce any dephasing! Dephasing as non-local operator in the Hamiltoanian Dephasing in post-(pre) processing ~P(t)=P(t)e−λ t See Octopus code or Y.Takimoto, Phd thesis (2008)
  28. 28. Dephasing and damping  
  29. 29. Post­processing 2:  periodic perturbation  P(t) is a periodic function of period TL =2p/wL   pn   is proportional to χn by the n­th order of the external field  Performing a discrete­time signal sampling we reduce the problem to  the solution of a systems of linear equations Ref: C. Attaccalite et al. PRB 88, 235113(2013)      F. Ding et al. JCP 138, 064104(2013) 
  30. 30. Second Harmonic Generation in MoS2
  31. 31. Non-linear optics in molecules Non-linear optics can calculated in the same way of TD-DFT as it is done in OCTOPUS or RT-TDDFT/SIESTA codes. Quasi-monocromatich-field p-nitroaniline Y.Takimoto, Phd thesis (2008)
  32. 32. Non-linear response in extended systems: a real-time approach Claudio Attaccalite
  33. 33. External and total field
  34. 34. How to calculated the dielectric constant i ∂ ̂ρk (t) ∂t =[Hk +V eff , ̂ρk ] ̂ρk (t)=∑i f (ϵk ,i)∣ψi,k 〉〈 ψi,k∣ The Von Neumann equation (see Wiki r t ,r' t' =  ind r ,t ext r' ,t ' =−i〈[ r ,t r' t ']〉We want to calculate: We expand X in an independent particle basis set χ(⃗r t ,⃗r ' t ' )= ∑ i, j,l,m k χi, j,l,m, k ϕi, k (r)ϕj ,k ∗ (r)ϕl,k (r')ϕm ,k ∗ (r') χi, j,l,m, k= ∂ ̂ρi, j, k ∂Vl,m ,k Quantum Theory of the Dielectric Constant in Real Solids Adler Phys. Rev. 126, 413–420 (1962) What is Veff ?
  35. 35. Independent Particle Independent Particle Veff = Vext ∂ ∂Vl ,m,k eff i ∂ρi, j ,k ∂t = ∂ ∂Vl ,m, k eff [Hk+V eff , ̂ρk ]i, j, k Using: { Hi, j ,k = δi, j ϵi(k) ̂ρi, j, k = δi, j f (ϵi,k)+ ∂ ̂ρk ∂V eff ⋅V eff +.... And Fourier transform respect to t-t', we get: χi, j,l,m, k (ω)= f (ϵi,k)−f (ϵj ,k) ℏ ω−ϵj ,k+ϵi ,k+i η δj ,l δi,m i ∂ ̂ρk (t) ∂t =[Hk +V eff , ̂ρk ] χi, j,l,m, k= ∂ ̂ρi, j, k ∂Vl,m ,k
  36. 36. Optical Absorption: IP Non Interacting System δρNI=χ 0 δVtot χ 0 =∑ ij ϕi(r)ϕj * (r)ϕi * (r')ϕj(r ') ω−(ϵi−ϵj)+ i η Hartree, Hartree-Fock, dft. =ℑχ0=∑ ij ∣〈 j∣D∣i〉∣2 δ(ω−(ϵj −ϵi)) ϵ'' (ω)= 8 π 2 ω2 ∑ i, j ∣〈ϕi∣e⋅̂v∣ϕj 〉∣2 δ(ϵi−ϵj−ℏ ω) Absorption by independent Kohn-Sham particles Particles are interacting!
  37. 37.  V ext= 0  V extV HV xc q ,= 0 q , 0 q,vf xc q ,q , TDDFT is an exact theory for neutral excitations! Time Dependent DFT V eff (r ,t)=V H (r ,t)+ V xc (r ,t)+ V ext (r ,t) Interacting System Non Interacting System Petersilka et al. Int. J. Quantum Chem. 80, 584 (1996)  I= NI=  I  Vext 0=  NI  V eff ... by using ... = 0 1 V H  V ext   V xc V ext  v f xc  i ∂ ̂ρk (t) ∂t =[ HKS , ̂ρk ]=[ Hk 0 +V eff , ̂ρk ]