1. Adiabatic Green’s function technique and
transient behavior in time-dependent
fermion-boson coupled models
Sungkyunskwan university
Yun-Tak Oh, Yoichi Higashi, Ching-Kit Chen, and Jung Hoon Han
arXiv:

2. Contents 1
Backgrounds 2
Research Results 3
Summary

3. 1. Backgrounds
Wang, Steinberg, Jarillo-Herrero and Gedik, Science 2013
Mahmood, Chan, Alpichshev, Gardner, Lee, Lee and Gedik, Nat. Phys. 2016
Replicas of surface Dirac state of 3D TI were first realized by Gedik’s group in pump-probe ARPES experiment in 2013 and refined in 2016
These replicas, named as sidebands, were predicted and can be explained by Floquet theory

4. 1. Backgrounds
Floquet theory
Treating the light as classical field makes minimally coupled Hamiltonian periodic in time
Solution of Floquet Hamiltonian gives the quasi energy band and its replicas (sidebands)
Lindner, Refael, & Galitski, Nature Physics (2011)
Oka & Aoki, phys. Rev. B (2009)
Energy gap opening
Sidebands

5. Limits of the Floquet theory
1. Backgrounds
Limits of the Floquet theory
Floquet theory fails to provide a clear picture of the sidebands formed as a result of the coupling of photons of pump laser with the electrons in the solid
It fails to describe the transient dynamics of photo-induced Bloch band structure

6. Keldysh Green’s function
1. Backgrounds
Keldysh Green’s function
Keldysh Green’s function is a common tool to solve the nonequilibrium problem
We proposed a new method to calculate non-equilibrium Green’s function for time-dependent Hamiltonian

7. Model: Lang-Frisov Hamiltonian
2. Results
Model: Lang-Frisov Hamiltonian
Well known case with exactly solved sidebands is Lang-Firsov model (1962)
Lang-Firsov model resembles the laser driven Dirac Hamiltonian
Dirac
Spin-boson
Lang-Firsov

8. Time-dependent Lang-Firsov Hamiltonian
2. Results
Time-dependent Lang-Firsov Hamiltonian
We give explicit time dependence to the Lang-Firsov Hamiltonian by
Adiabatic assumption:
Electron spectral function of time-dependent Lang-Firsov Hamiltonian can be obtained by solving two-time Green’s function
Source of difficulty: time evolution operator U(t,t’) is hard to compute

9. ! Adiabatic Green’s function technique
2. Results
Adiabatic Green’s function technique
We split the propagator into products of small time slices as in Feynman’s path integral technique
Exact propagator U(t,t’) is obtained from path-ordered exponential of the new effective Hamiltonian !

10. Photo-current intensity P(t,ω)
2. Results
Photo-current intensity P(t,ω)
Freericks et al made the formula representing photo-current in pump-probe experiment
Freericks, Krishnamurthy, Pruschke, PRL 102, (2009)
We obtained the lesser Green’s function for Lang-Firsov model by the scheme just introduced

11. Photo-current intensity P(t,ω)
2. Results
Photo-current intensity P(t,ω)
Sidebands gradually diminish in intensity over time while a new peak centered at the bare energy emerges like a bamboo sprout
α =0
α =1

12. Photo-current intensity P(t,ω) - spin-boson model
2. Results
Photo-current intensity P(t,ω)
- spin-boson model
Zeroth order contribution to photo-intensity P(t,ω) is obtained
Spin-boson
Lang-Firsov
The sidebands Intensities are monotonically diminishing as intensity for the bare energy band increases

13. –Driven Dirac Hamiltonian (with dissipation)
3. Summary
With the idea of `instantaneous basis unitary operator’, we developed “Adiabatic Green’s function” technique
Photo-current intensity spectra for two fermion-boson coupled models are obtained quasi-exactly
Our technique may also can be applied to the realistic transient dynamics problem
–Driven Dirac Hamiltonian (with dissipation)

14. Thank you!

15. Lang-Firsov Hamiltonian; exact diagonalization
Supplementary
Lang-Firsov Hamiltonian; exact diagonalization
electron-photon interaction can be easily decoupled by simple unitary transformation
The unitary operator transforms each boson and fermion operator as

16. Lang-Firsov Hamiltonian; exact diagonalization
2. Results
Lang-Firsov Hamiltonian; exact diagonalization
Spectral weight of electron (=imaginary part of Green’s function) shows clear sideband features