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arXiv:1511.03875 · Physical review. B./Physical review. B

Electron-phonon thermalization in a scalable method for real-time quantum dynamics

Valerio Rizzi, Tchavdar N. Todorov, Jorge José Kohanoff, Alfredo A. Correa

Published Jan 27, 2016• 31 citations
arXivHTMLPDF

Abstract

We present a quantum simulation method that follows the dynamics of out-of-equilibrium many-body systems of electrons and oscillators in real time. Its cost is linear in the number of oscillators and it can probe time scales from attoseconds to hundreds of picoseconds. Contrary to Ehrenfest dynamics, it can thermalize starting from a variety of initial conditions, including electronic population inversion. While an electronic temperature can be defined in terms of a nonequilibrium entropy, a Fermi-Dirac distribution in general emerges only after thermalization. These results can be used to construct a kinetic model of electron-phonon equilibration based on the explicit quantum dynamics.

DOI
10.1103/physrevb.93.024306

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We present a quantum simulation method that follows the dynamics of out-of-equilibrium many-body systems of electrons and oscillators in real time. Its cost is linear in the number of oscillators and it can probe time scales from attoseconds to hundreds of picoseconds. Contrary to Ehrenfest dynamics, it can thermalize starting from a variety of initial conditions, including electronic population inversion. While an electronic temperature can be defined in terms of a nonequilibrium entropy, a Fermi-Dirac distribution in general emerges only after thermalization. These results can be used to construct a kinetic model of electron-phonon equilibration based on the explicit quantum dynamics.
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We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I\ensuremath{-}V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.
In this work, we lay down the ab initio many-body quantum theory of electrons and phonons in equilibrium as well as in steady-state or time-varying settings.Our focus is on the harmonic approximation, but the developed tools can readily incorporate anharmonic effects.We begin by showing the necessity of determining the ab initio Hamiltonian in a self-consistent manner to ensure the existence of an equilibrium state.We then identify the correct partitioning into a "noninteracting" and an "interacting" part to carry out diagrammatic expansions in terms of dressed propagators and screened interactions.The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, showcasing the emergence of the coupling between electrons and coherent phonons through the time-local Ehrenfest diagram.The Hedin equations have limited practical utility for real-time simulations of systems driven out of equilibrium by external fields.To overcome this limitation, we leverage the versatility of the diagrammatic approach to generate a closed system of differential equations for the dressed propagators and nuclear displacements.These are the Kadanoff-Baym equations for electrons and phonons.The formalism naturally merges with the theory of conserving approximations, which guarantee the satisfaction of the continuity equation and the conservation of total energy during time evolution.As an example, we show that the popular Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving, provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electronphonon coupling.We also derive the formal solution of the Kadanoff-Baym equations for nonequilibrium steady states, which is useful for studies in photovoltaics and optoelectronics.Interestingly, the expansion of the phononic Green's function around the quasiphonon energies points to a correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
We lay down the {\em ab initio} many-body quantum theory of electrons and phonons in- and out-of-equilibrium at any temperature. We begin by addressing a fundamental issue concerning the {\em ab initio} Hamiltonian in the harmonic approximation, which we show must be determined {\em self-consistently} to avoid inconsistencies. After identifying the most suitable partitioning into a ``noninteracting'' and an ``interacting'' part we embark on the Green's function diagrammatic analysis. We single out key diagrammatic structures to carry on the expansion in terms of dressed propagators and screened interaction. The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, featuring the appearance of the time-local Ehrenfest diagram in the electronic self-energy. The Hedin equations have limited practical utility for real-time simulations of driven systems. We leverage the versatility of diagrammatic expansion to generate a closed system of integro-differential equations for the Green's functions and nuclear displacements. These are the Kadanoff-Baym equations for electrons and phonons. Another advantage of the diagrammatic derivation is the ability to use conserving approximations, which ensure the satisfaction of all fundamental conservation laws during the time evolution. As an example we show that the adiabatic Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electron-phonon coupling. We also derive the formal solution of the Kadanoff-Baym equations in the long time limit and at the steady state. The expansion of the phononic Green's function around the quasi-phonon energies points to a possible correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
We present an exact Monte Carlo method to simulate the nonequilibrium dynamics of electron-phonon models in the adiabatic limit of zero phonon frequency. The classical nature of the phonons allows us to sample the equilibrium phonon distribution and efficiently evolve the electronic subsystem in a time-dependent electromagnetic field for each phonon configuration. We demonstrate that our approach is particularly useful for charge-density-wave systems experiencing pulsed electric fields, as they appear in pump-probe experiments. For the half-filled Holstein model in one and two dimensions, we calculate the out-of-equilibrium response of the current and the energy after a pulse is applied as well as the photoemission spectrum before and after the pump. Finite-size effects are under control for chains of 162 sites (in one dimension) or 16×16 square lattices (in two dimensions).