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Philosophical Magazine B

Higher-order memory effects in phonon-assisted hopping

Vladislav Čápek

Published Sep 1, 1986• 5 citations
Publisher

Abstract

To lowest order in the electron-phonon coupling causing electron transitions among electronic eigenstates, the extent (on the time axis) of the memory functions wrs(t) governing the state population via the generalized master equations for the diagonal elements of the electron density matrix ρ(t) is independent of the coupling, being given by the decay time of the bath correlation functions. However, a typical reciprocal decay time of some higher-order contributions to wrs(t) has been found to be given by the reciprocal decay time of the off-diagonal elements of ρ(t), i.e. to be proportional to the square of the electron-phonon coupling constant. This feature makes such weakly damped contributions non-negligible even in lowest-order dynamics, as is illustrated on a dimer.

DOI
10.1080/13642818608239014

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To lowest order in the electron-phonon coupling causing electron transitions among electronic eigenstates, the extent (on the time axis) of the memory functions wrs(t) governing the state population via the generalized master equations for the diagonal elements of the electron density matrix ρ(t) is independent of the coupling, being given by the decay time of the bath correlation functions. However, a typical reciprocal decay time of some higher-order contributions to wrs(t) has been found to be given by the reciprocal decay time of the off-diagonal elements of ρ(t), i.e. to be proportional to the square of the electron-phonon coupling constant. This feature makes such weakly damped contributions non-negligible even in lowest-order dynamics, as is illustrated on a dimer.
Abstract. We study the dynamics of an electron weakly coupled to a phonon gas. The initial state of the electron is the superposition of two spatially localized distant bumps moving towards each other, and the phonons are in a thermal state. We investigate the dynamics of the system in the kinetic regime and show that the time evolution makes the non-diagonal terms of the density matrix of the electron decay, destroying the interference between the two bumps. We show that such a damping effect is exponential in time, and the related decay rate is proportional to the total scattering cross section of the electron-phonon interaction. 1. Introduction and
Electromagnetically coupled two-level systems play a central role in several condensed matter components being considered for quantum information processing applications. If the states couple to phonon excitations, their electromagnetic response is altered via phonon-assisted transitions and lifetime broadening. The former has been treated extensively for a number of specific two-level systems (e.g., excitons in artificial quantum dots, localized states associated with impurities or defects, etc.), but the latter has received less attention. Here we study a microscopic model of the dipole transition of a two-level system under the influence of both diagonal and nondiagonal interactions with a bath of phonons. Our results capture both the influence of the frequency distribution of phonons on the relative spectral weight of the zero-phonon transition and the phonon sidebands, and the broadening of the zero-phonon line due to nondiagonal electron-phonon coupling. We use a formalism that includes non-Markovian effects related to the feedback mechanism between the two-level system and the phonon bath. For simplified forms of the phonon spectral functions we provide analytical expressions up to second order in the coupling strength that demonstrate the importance of including both forms of electron-phonon coupling in studies of these systems. Our formalism can be generalized to higher orders of coupling and for realistic phonon spectral functions.
A finite coupling generalised master equation theory of the electronic relaxation among electronic eigen-states is presented. When converted into a closed time-convolution form (TCGME) for just diagonal elements of the density matrix, the theory is appreciably non-Markovian with the memory kernel summed partially up to the infinite order in the electron-phonon coupling. On soluble examples, it is shown that: (a), higher-order effects may influence even the lowest-order relaxation rates; (b), weakness of the finite coupling is not sufficient for reducing the theory to the standard lowest-order form; (c), the Markov approximation to the TCGME is in general misleading; (d), the initial condition term may appreciably influence the long-time development of the relaxation; (e), the theory is able to yield quasi-coherent effects (quantum beats) due to the electron coupling to the bath; (f), very effective tunnelling through barriers between equivalent sites appears with non-zero tunnelling currents starting from the second order in the electron-phonon coupling. Limitations from above (below) for the time (frequency) are found for the theory to reduce to the standard lowest-order form.
Dynamical mean-field theory computations of the electron self energy of the Hubbard-Holstein model as a function of electron-phonon and electron-electron interactions are analyzed to gain insight into the dependence of electron-phonon couplings on correlation strength in quantum materials. We find that the electron-phonon interaction is strongly suppressed by electronic correlations, while electron-electron correlation effects at Fermi liquid scales are only weakly modified by coupling to phonons, with phonon-induced modifications most evident at high frequencies on the order of the electronic bandwidth. Implications for beyond-density functional theories of the electron-phonon interaction are discussed.
The time evolution of the purity (the trace of the square of the reduced density matrix) and von Neumann entropy in a symmetric two-level system coupled to a dissipative harmonic bath is investigated through analytical arguments and accurate path integral calculations on simple models and the singly excited bacteriochlorophyll dimer. A simple theoretical analysis establishes bounds and limiting behaviors. The contributions to purity from a purely incoherent term obtained from the diagonal elements of the reduced density matrix, a term associated with the difference of the two eigenstate populations, and a third term related to the square of the time derivative of a site population, are discussed in various regimes. In the case of tunneling dynamics from a localized initial condition, the complex interplay among these contributions leads to the recovery of purity under low-temperature, weakly dissipative conditions. Memory effects from the bath are found to play a critical role to the dynamics of purity. It is shown that the strictly quantum mechanical decoherence process associated with spontaneous phonon emission is responsible for the long-time recovery of purity. These analytical and numerical results show clearly that the loss of quantum coherence during the evolution toward equilibrium does not necessarily imply the decay of purity, and that the time scales relevant to these two processes may be entirely different.
Electron-phonon interactions underpin a variety of phenomena, ranging from transport and superconductivity to polarons and ultrafast carrier dynamics. Despite being one of the most intensely studied subjects in condensed matter physics, research on electron-phonon physics mostly focused on linear, first-order couplings. Second- and higher-order nonlinear couplings are commonly ignored because their calculations are too demanding and we lack computational frameworks that can provide both diagonal and off-diagonal coupling matrix elements. In this work, we report a theory and computational method for computing nonlinear electron-phonon interactions of any order in real materials. Our approach combines the advantages of unit-cell calculations of electron wavefunctions and supercell calculations of phonon perturbations, is systematically improvable, can be used with either semilocal or nonlocal exchange-correlation functionals, and is amenable to Wannier-Fourier interpolation. As a first proof of concept, we illustrate this method by computing second-order electron-phonon coupling matrix elements in diamond, lithium fluoride, and graphite as representative nonpolar semiconductors, polar semiconductors, and metals, respectively. Furthermore, we generalize the ab initio polaron equations to second-order electron-phonon couplings, and we show that second-order couplings are essential to achieve quantitative accuracy in polaron formation energies and hopping barriers. The present methodology will find application in the study of all properties and phenomena that are currently being investigated within the linear electron-phonon coupling approximation, from phonon-mediated superconductivity to excited-states dynamics, both in harmonic and anharmonic systems.