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:1601.04872

Contents 1 Backgrounds 2 Research Results 3 Summary

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

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

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

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

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

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

! 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 !

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, 136401 (2009) We obtained the lesser Green’s function for Lang-Firsov model by the scheme just introduced

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

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

–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)

Thank you!

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

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