An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.

With a decomposition scheme for the bath correlation function, the hierarchy equation of motion (HEOM) is extended to the zero-temperature sub-Ohmic spin-boson model, providing a numerically accurate prediction of quantum dynamics. As a dynamic approach, the extended HEOM determines the delocalized-localized (DL) phase transition from the extracted rate kernel and the coherent-incoherent dynamic transition from the short-time oscillation. As the bosonic bath approaches from the strong to weak sub-Ohmic regimes, a crossover behavior is identified for the critical Kondo parameter of the DL transition, accompanied by the transition from the coherent to incoherent dynamics in the localization.

Following a recently proposed decomposition technique [C. R. Duan et al., Phys. Rev. B 95, 214308 (2017)], we inspect the zero-temperature spin-boson model for five different cutoff functions of the spectral density. With oscillatory and non-oscillatory exponentially decaying functions to decompose the bath correlation function, the hierarchy equation of motion is reliably extended to each spectral density under our investigation. The predicted spin dynamics is gradually converged with the increase of the hierarchic expansion order and the number of decomposing basis functions. Our systematic study of different cutoff functions expands previous results of the delocalized-localized phase transition with the exponential and sudden cutoffs in the spectral density.

Through a decomposition of the bath correlation function, the hierarchical equations of motion are extended to the Ohmic spin-boson model at zero temperature. For two typical cutoff functions of the bath spectral density, the rate kernel of spin dynamics is numerically extracted by a time-convolution equation of the average magnetic moment. A characteristic time is defined accordingly as the inverse of the zeroth-order moment of the rate kernel. For a given Kondo parameter in the incoherent regime, the time evolution of average magnetic moments gradually collapses onto a master curve after rescaling the time variable with the characteristic time. The rescaled spin dynamics is nearly independent of the cutoff frequency and the form of cutoff functions. For a given cutoff frequency, the characteristic time with the change of the Kondo parameter is fitted excellently as a function of the renormalized tunneling amplitude. Despite a significant difference in definition, our result is in good agreement with the characteristic time of the noninteracting blip approximation.

The energy absorbed in a light-harvesting protein complex is often transferred collectively through aggregated chromophore clusters. For population evolution of chromophores, the time-integrated effective rate matrix allows us to construct quantum kinetic clusters quantitatively and determine the reduced cluster-cluster transfer rates systematically, thus defining a minimal model of energy-transfer kinetics. For Fenna-Matthews-Olson (FMO) and light-havrvesting complex II (LCHII) monomers, quantum Markovian kinetics of clusters can accurately reproduce the overall energy-transfer process in the long-time scale. The dominant energy-transfer pathways are identified in the picture of aggregated clusters. The chromophores distributed extensively in various clusters can assist a fast and long-range energy transfer.