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arXiv:cond-mat/0606497 · The Journal of Chemical Physics

Time-convolutionless master equation for mesoscopic electron-phonon systems

Andrey Pereverzev, Eric R. Bittner

Published Sep 14, 2006• 66 citations
arXivHTMLPDF

Abstract

The time-convolutionless master equation for the electronic populations is derived for a generic electron-phonon Hamiltonian. The equation can be used in the regimes where the golden rule approach is not applicable. The equation is applied to study the electronic relaxation in several models with the finite number of normal modes. For such mesoscopic systems the relaxation behavior differs substantially from the simple exponential relaxation. In particular, the equation shows the appearance of the recurrence phenomena on a time scale determined by the slowest mode of the system. The formal results are quite general and can be used for a wide range of physical systems. Numerical results are presented for a two level system coupled to Ohmic and super-Ohmic baths, as well as for a model of charge-transfer dynamics between semiconducting organic polymers.

DOI
10.1063/1.2348869

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The time-convolutionless master equation for the electronic populations is derived for a generic electron-phonon Hamiltonian. The equation can be used in the regimes where the golden rule approach is not applicable. The equation is applied to study the electronic relaxation in several models with the finite number of normal modes. For such mesoscopic systems the relaxation behavior differs substantially from the simple exponential relaxation. In particular, the equation shows the appearance of the recurrence phenomena on a time scale determined by the slowest mode of the system. The formal results are quite general and can be used for a wide range of physical systems. Numerical results are presented for a two level system coupled to Ohmic and super-Ohmic baths, as well as for a model of charge-transfer dynamics between semiconducting organic polymers.
A new general non-perturbative method for the solution of the generalized master equation for a simple two-level model of the electron-phonon system coupled to a thermodynamic reservoir is suggested. The model includes a two-level system interacting linearly with the single-mode phonons coupled to the thermodynamic bath. For the interaction of the bath with the vibrational mode we use a stochastic model generalized to finite temperatures. The results are compared with a direct numerical solution.
The time-convolutionless quantum master equation is an exact description of the nonequilibrium dynamics of open quantum systems, with the advantage of being local in time. We derive a perturbative expansion to arbitrary order in the system-reservoir coupling for its generator, which contains significantly fewer terms than the ordered-cumulant expansion. We show that the derived expansion also admits a simple recursive formulation. The derived series is then used to describe the nonequilibrium dynamics of a quantum dot, including coherences. We find a relation between the generator and a generalization of the TTT-matrix. Even though the TTT-matrix rate equations are plagued by divergences, we show that these cancel order by order in the generator of the time-convolutionless master equation. This generalizes previous work on the time-convolutionless Pauli master equation, which does not include coherences, and lays the ground for possible numerical evaluations and analytical resummations.
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The time-convolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak system-bath coupling regime. In this paper, we derive an exact time-convolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical two-level system linearly coupled to a harmonic bath.
Despite recent advances in quantum sciences, a quantum master equation that accurately and simply characterizes open quantum dynamics across extremely long timescales and in dispersive environments is still needed. In this study, we optimize the computation of the fourth-order time-convolutionless master equation to meet this need. Early versions of this master equation required computing a multidimensional integral, limiting its use. Our master equation accounts for simultaneous relaxation and dephasing, resulting in coefficients proportional to the system's spectral density over frequency derivative. In sub-Ohmic environments, this derivative induces infrared divergence in the master equation, invalidating the second-order Bloch-Redfield master equation findings. We analyze the approach to a ground state in a generic open quantum system and demonstrate that it is not reliably computed by the Bloch-Redfield equation alone. The optimized fourth-order equation shows that the ground-state approach is accurate to second order in bath coupling regardless of the dispersion, even though it can diverge in the fourth order.
Steady-state observables, such as occupation numbers and currents, are crucial experimental signatures in open quantum systems.The time-convolutionless (TCL) master equation, which is both exact and time-local, is an ideal candidate for the perturbative computation of such observables.We develop a diagrammatic approach to evaluate the steady-state TCL generator based on operators rather than superoperators.We obtain the steady-state occupation numbers, extend our formulation to the calculation of currents, and provide a simple physical interpretation of the diagrams.We further benchmark our method on a single noninteracting level coupled to Fermi reservoirs, where we recover the exact expansion to next-to-leading order.The low number of diagrams appearing in our formulation makes the extension to higher orders accessible.Combined, these properties make the steady-state time-convolutionless master equation an effective tool for the calculation of steady-state properties in open quantum systems.