Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

Abstract

Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker-Planck equation and a quantum master equation from a system-bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.

Long-Lived Electronic Coherence in Dissipative Exciton Dynamics of Light-Harvesting Complexes

“…Realistic models of the exciton dynamics in the FMO complex have to include higher order phonon processes as well as the finite time scale of the reorganization process [17]. This requires to go beyond approximative rate equations [23] and non-perturbative techniques [24][25][26][27][28][29][30] are necessary to study the dissipative transfer dynamics.One key parameter determining the duration of coherent oscillations is the spectral density, which encodes the the mode-dependent exciton-bath coupling. The spectral density has important implications for the two contributions to the decoherence rate γ which is the sum of the relaxation rate γ r and the pure-dephasing rate γ d ([33], ch.…”

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

“…[11][12][13][14][15] As an example, we employ a model describing the thermal effects in resonant tunneling diodes (RTDs). 16,17) Although the validity of the Caldeira-Leggett Hamiltonian for describing electron transport phenomena has not yet been investigated thoroughly, because it has a firm quantum mechanical foundation, and because we feel that it is necessary to investigate the validity of the previous theoretical results in a fully quantum mechanical treatment, we believe that the present study will be helpful in constructing a general understanding of electron transport phenomena.…”

“…We then employ the reduced hierarchy equations of motion (HEOM) approach, which can be used to treat non-Markovian and non-perturbative system-bath interactions at finite temperature with no approximation. 15) In the HEOM formalism, the reduced density operator is expressed in terms of the auxiliary hierarchy density matrix elements, ðnÞ j 1 ;...;j K , where the indices n and j k arise from the hierarchal expansion of the decay functions e À t and e À k t , where k ¼ 2 k= h " is the kth Matsubara frequency. 15,26) Then the 0th element is identical to^ð 0Þ 0;0;...;0 ðtÞ ¼ Tr B f^t ot ðtÞg.…”

“…15) In the HEOM formalism, the reduced density operator is expressed in terms of the auxiliary hierarchy density matrix elements, ðnÞ j 1 ;...;j K , where the indices n and j k arise from the hierarchal expansion of the decay functions e À t and e À k t , where k ¼ 2 k= h " is the kth Matsubara frequency. 15,26) Then the 0th element is identical to^ð 0Þ 0;0;...;0 ðtÞ ¼ Tr B f^t ot ðtÞg. The HEOM has been used to study chemical reactions, 12,27) linear and nonlinear spectroscopy, 28,29) exciton and electron transfer, [30][31][32][33][34] and quantum information.…”

“…35,36) The HEOM is ideal for studying quantum transport systems when we employ the Wigner representation, because it allows us to treat continuous systems utilizing open boundary conditions and periodic boundary conditions. 12,22) The equations of motion are written in hierarchical form as follows: 15,28) @ @t W ðnÞ j 1 ;...;j K ðtÞ Here, the quantum Liouvillian in the Wigner representation is defined by 22) ÀL qm Wðp; qÞ ¼ À p m @ @q Wðp; qÞ À Z 1 À1 dp 0 2 h " 2 U w ðp À p 0 ; q; tÞWðp 0 ; qÞ with U w ðp; qÞ ¼ 2…”