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The Journal of Chemical Physics

Generalized quantum kinetic expansion: Time scale separation between intra-cluster and inter-cluster kinetics

Zhoufei Tang, Zhihao Gong, Jianlan Wu

Published Sep 11, 2015• 9 citations
Publisher

Abstract

For a general two-cluster network, a new methodology of the cluster-based generalized quantum kinetic expansion (GQKE) is developed in the matrix formalism under two initial conditions: the local cluster equilibrium and system-bath factorized states. For each initial condition, the site population evolution follows exactly a distinct closed equation, where all the four terms involved are systematically expanded over inter-cluster couplings. For the system-bath factorized initial state, the numerical investigation of the two models, a biased (2, 1)-site system and an unbiased (2, 2)-site system, verifies the reliability of the GQKE and the relevance of higher-order corrections. The time-integrated site-to-site rates and the time evolution of site population reveal the time scale separation between intra-cluster and inter-cluster kinetics. The population evolution of aggregated clusters can be quantitatively described by the approximate cluster Markovian kinetics.

DOI
10.1063/1.4930012

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For a general two-cluster network, a new methodology of the cluster-based generalized quantum kinetic expansion (GQKE) is developed in the matrix formalism under two initial conditions: the local cluster equilibrium and system-bath factorized states. For each initial condition, the site population evolution follows exactly a distinct closed equation, where all the four terms involved are systematically expanded over inter-cluster couplings. For the system-bath factorized initial state, the numerical investigation of the two models, a biased (2, 1)-site system and an unbiased (2, 2)-site system, verifies the reliability of the GQKE and the relevance of higher-order corrections. The time-integrated site-to-site rates and the time evolution of site population reveal the time scale separation between intra-cluster and inter-cluster kinetics. The population evolution of aggregated clusters can be quantitatively described by the approximate cluster Markovian kinetics.
Within the framework of the hierarchy equation of motion (HEOM), the quantum kinetic expansion (QKE) method of the spin-boson model is reformulated in the matrix representation. The equivalence between the two formulations (HEOM matrices and quantum operators) is numerically verified from the calculation of the time-integrated QKE rates. The matrix formulation of the QKE is extended to the system-bath factorized initial state. Following a one-to-one mapping between HEOM matrices and quantum operators, a quantum kinetic equation is rederived. The rate kernel is modified by an extra term following a systematic expansion over the site-site coupling. This modified QKE is numerically tested for its reliability by calculating the time-integrated rate and non-Markovian population kinetics. For an intermediate-to-strong dissipation strength and a large site-site coupling, the population transfer is found to be significantly different when the initial condition is changed from the local equilibrium to system-bath factorized state.
We apply a new formalism to derive the higher-order quantum kinetic expansion (QKE) for studying dissipative dynamics in a general quantum network coupled with an arbitrary thermal bath. The dynamics of system population is described by a time-convoluted kinetic equation, where the time-nonlocal rate kernel is systematically expanded of the order of off-diagonal elements of the system Hamiltonian. In the second order, the rate kernel recovers the expression of the noninteracting-blip approximation method. The higher-order corrections in the rate kernel account for the effects of the multi-site quantum coherence and the bath relaxation. In a quantum harmonic bath, the rate kernels of different orders are analytically derived. As demonstrated by four examples, the higher-order QKE can reliably predict quantum dissipative dynamics, comparing well with the hierarchic equation approach. More importantly, the higher-order rate kernels can distinguish and quantify distinct nontrivial quantum coherent effects, such as long-range energy transfer from quantum tunneling and quantum interference arising from the phase accumulation of interactions.
We apply a new formalism to derive the higher-order quantum kinetic expansion (QKE) for studying dissipative dynamics in a general quantum network coupled with an arbitrary thermal bath. The dynamics of system population is described by a time-convoluted kinetic equation, where the time-nonlocal rate kernel is systematically expanded of the order of off-diagonal elements of the system Hamiltonian. In the second order, the rate kernel recovers the expression of the noninteracting-blip approximation method. The higher-order corrections in the rate kernel account for the effects of the multi-site quantum coherence and the bath relaxation. In a quantum harmonic bath, the rate kernels of different orders are analytically derived. As demonstrated by four examples, the higher-order QKE can reliably predict quantum dissipative dynamics, comparing well with the hierarchic equation approach. More importantly, the higher-order rate kernels can distinguish and quantify distinct nontrivial quantum coherent effects, such as long-range energy transfer from quantum tunneling and quantum interference arising from the phase accumulation of interactions.
In the framework of real-time Green's functions, the general kinetic equations are investigated in a first-order gradient expansion. Within this approximation, the problem of the reconstruction of the two-time correlation functions from the one-time Wigner function was solved. For the Wigner function, a cluster expansion is found in terms of a quasiparticle distribution function. In equilibrium, this expansion leads to the well-known generalized Beth-Uhlenbeck expression of the second virial coefficient. As a special case, the T-matrix approximation for the self-energy is investigated. The quantum kinetic equation derived thus has, besides the (Markovian) Boltzmann collision integral, additional terms due to the retardation expansion which reflect memory effects. Special interest is paid to the case that bound states exist in the system. It is shown that the bound state contribution, which can be introduced via a bilinear expansion of the two-particle T matrix, follows from the first-order retardation term in the general kinetic equation. The full Wigner function is now a sum of one function describing the unbound particles and another one for the bound state contribution. The latter two functions have to be determined from a coupled set of kinetic equations. In contrast to the quantum Boltzmann equation, energy and density of a nonideal system are conserved. \textcopyright{} 1996 The American Physical Society.
In the framework of the quantum-classical Liouville equation (QCLE), the quantum kinetic expansion (QKE) of the spin-boson model is extended to an arbitrary combination of the bath potential and the system-bath interaction. The mixed quantum-classical estimation of the QKE rate kernels and modification functions are transformed into averages of deterministic classical trajectories over the Wigner initial distribution. For the standard spin-boson model, the QCLE-QKE method produces exactly the same result as that from full quantum dynamics and the numerical applicability of the approximate action-angle initial distribution is verified. For an anharmonic bath with the quartic potential, the QCLE-QKE calculation under the action-angle initial distribution illustrates the influence of this specific anharmonicity. With the increase of the quartic parameter, the fourth order QKE corrections are suppressed and the short-time population transfer is accelerated together with an enhanced quantum oscillation.
Abstract It is shown that the relevant part of an equilibrium two-time correlation function of operators related to a subsystem of S < N particles, selected from the whole system of N ≫ 1 quantum particles, obeys the exact completely closed (homogeneous) evolution equation. This equation differs from the conventional Nakajima–Zwanzig equation by the absence of the undesirable inhomogeneous term comprising all interparticle correlations at the initial time moment. The equation defines the subdynamics in a subspace of S selected particles and is obtained without any use of approximation like the Boltzmann ‘molecular chaos’, factorized initial state or Bogoliubov’s principle of weakening of initial correlations. In the second order in a small inter-particle interaction, this equation reduces to the Markovian evolution equation, which for a one-particle correlation function ( S = 1) takes the form of the linear generalized quantum Boltzmann-like equation for a relevant part of a correlation function (one particle momentum distribution function) with the additional terms conditioned by initial correlations taken into account. This equation makes it possible to obtain the solution for the corresponding one-particle correlation function, which is valid for all timescales and reveals the influence of initial correlations on the evolution in time process. On a large timescale t ≫ t cor ∼ ℏ / k B T (decoherence time), the initial correlations cease to influence the damping process and the evolution enters a kinetic stage. However, the factor related to initial correlations survives on the large timescale, and can contribute to the kinetic coefficients.