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Journal of Physics C Solid State Physics

Higher-order effects in non-Markovian theory of electronic relaxation in interacting electron-phonon systems

Vladislav Čápek

Published Jul 10, 1987• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/20/19/015

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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.
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.
Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.
A reduced-density-matrix description is developed for coherent linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems and semiconductor materials. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach are developed in a unified manner. The general non-Markovian formulations provide a rigorous foundation for the systematic introduction of the commonly used Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is introduced, based on a coupled hierarchy of relationships for the field components of the reduced density operator.
Abstract A reduced-density-matrix description is developed for coherent linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems and semiconductor materials. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the reduced-density-matrix approach are developed in a unified manner. The general non-Markovian formulations provide a rigorous foundation for the systematic introduction of the commonly used Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations. A preliminary semiclassical perturbation-theory treatment of the electromagnetic interaction is introduced, based on a coupled hierarchy of relationships for the field components of the reduced density operator.
A reduced-density-matrix description is developed for linear and non-linear electromagnetic interactions of quantized electronic systems in the presence of environmental decoherence and relaxation phenomena. Applications of interest include many-electron atomic systems (in electron-ion beam interactions, gases, and high-temperature plasmas) and semiconductor materials (bulk crystals and nanostructures). Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order perturbation-theory) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semiclassical treatment of the electromagnetic interaction is introduced. Compact Liouville-space operator expressions are derived for the linear and the general (n’th order) non-linear electromagnetic-response tensors, allowing for coherent initial electronic excitations and for the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions. It is emphasized that quantum-coherent many-body interactions cannot be adequately treated as environmentally induced phenomena.
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.