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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum particle transfer in a system with a discontinuous modulation of the intersite coupling

Elmar G. Petrov, Igor Goychuk, Volkhard May

Published Nov 1, 1996• 9 citations
Publisher

Abstract

The nonequilibrium density matrix technique is utilized to study the incoherent transfer of a quantum particle coupled to a fluctuating environment. The dynamics of the environment is assumed to split into two different types of molecular motions. The first type is associated with low-amplitude vibrations forming a thermal bath (TB). High-amplitude displacements of separated molecular groups define the second type and create stochastic fields for acting on the transferred particle. Specifying the TB by a set of harmonic oscillators and the stochastic fields by a dichotomic process averaged kinetic equations were derived. These equations enable us to construct a rate constants expressions and to find the steady-state populations for a quantum particle. Regimes of enhanced or reduced particle transfer are discussed in detail in relation to the characteristics of the stochastic field and the TB. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreve.54.4726

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arXiv:math-ph/0610033•Oct 16, 2006•0 citations
The nonequilibrium density matrix technique is utilized to study the incoherent transfer of a quantum particle coupled to a fluctuating environment. The dynamics of the environment is assumed to split into two different types of molecular motions. The first type is associated with low-amplitude vibrations forming a thermal bath (TB). High-amplitude displacements of separated molecular groups define the second type and create stochastic fields for acting on the transferred particle. Specifying the TB by a set of harmonic oscillators and the stochastic fields by a dichotomic process averaged kinetic equations were derived. These equations enable us to construct a rate constants expressions and to find the steady-state populations for a quantum particle. Regimes of enhanced or reduced particle transfer are discussed in detail in relation to the characteristics of the stochastic field and the TB. \textcopyright{} 1996 The American Physical Society.
Kinetic equations of the quantum particle motion in a symmetric dimer are studied for the case of fluctuating site energies and a fluctuating intersite coupling matrix element. Modulations of the site energies are incorporated via quantum fluctuations of a thermal bath whereas the fluctuations of the intersite coupling are taken into account by a stochastic description. In contrast to the quantum fluctuations which are considered in the weak coupling limit, the stochastic fluctuations are included in an exact manner. The exact solution of the averaged kinetic equations are obtained in the case of an Ohmic thermal bath with a white noise spectrum and dichotomous fluctuations of the intersite coupling.
A large number of multifaceted quantum transport processes in molecular systems and physical nanosystems can be treated in terms of quantum relaxation processes which couple to one or several fluctuating environments. A thermal equilibrium environment can conveniently be modelled by a thermal bath of harmonic oscillators. An archetype situation provides a two-state dissipative quantum dynamics, commonly known under the label of a spin-boson dynamics. An interesting and nontrivial physical situation emerges, however, when the quantum dynamics evolves far away from thermal equilibrium. This occurs, for example, when a charge transferring medium possesses nonequilibrium degrees of freedom, or when a strong time-dependent control field is applied externally. Accordingly, certain parameters of underlying quantum subsystem acquire stochastic character. Herein, we review the general theoretical framework which is based on the method of projector operators, yielding the quantum master equations for systems that are exposed to strong external fields. This allows one to investigate on a common basis the influence of nonequilibrium fluctuations and periodic electrical fields on quantum transport processes. Most importantly, such strong fluctuating fields induce a whole variety of nonlinear and nonequilibrium phenomena. A characteristic feature of such dynamics is the absence of thermal (quantum) detailed balance.
The methods of nonequilibrium density-matrix and coarse-temporal conception are used to obtain the kinetic equation for the parameters ${\ensuremath{\gamma}}_{\mathit{n}\mathit{m}}$(t)=Sp[\ensuremath{\rho}^(t)\ensuremath{\Vert}n〉〈m\ensuremath{\Vert}] of a quantum-dynamic system (QDS) interacting with a thermal bath and external stochastic field. It is important that the stochastic field is taken exactly into consideration. For diagonal QDS parameters ${\ensuremath{\gamma}}_{\mathit{n}\mathit{n}}$(t) this equation is reduced to the generalized Pauli equation (GPE) with stochastic time-dependent coefficients wnm{\mathit{w}}_{\mathit{n}\mathit{m}}wnm​(t). Special attention is given to the procedure of averaging over stochastic processes. It is shown that after averaging over energy fluctuations affected by the stochastic field, in the first cumulant approximation in terms of stochastic processes wnm{\mathit{w}}_{\mathit{n}\mathit{m}}wnm​(t), the GPE is transformed to the Pauli equation for the QDS state population Pn{\mathit{P}}_{\mathit{n}}Pn​(t)=〈${\ensuremath{\gamma}}_{\mathit{n}\mathit{n}}$(t)〉f{\mathrm{〉}}_{\mathit{f}}〉f​. As an example, the relaxation behavior of a two-level system interacting with a dichotomous field (dichotomous Markovian process of kangaroo type) and a harmonic oscillator coupled with a thermal bath is considered. It is shown that the probability of relaxation transitions between energy levels may be changed by several orders of magnitude under the influence of the dichotomous field.
We obtain kinetic equations for the stochastically averaged reduced density matrix of a quantum system driven by a dichotomous Markovian process and weakly coupled to a quantum thermal bath. These kinetic equations are exact in arbitrary dichotomous perturbation and provide a unified description of both small and large correlation time limits, along with an intermediate case. As an example, the problem of a dissipative two-level system with a dichotomically modulated difference in eigenergies is considered.
A master equation for a quantum system coupled to a heat bath with stochastically fluctuating energy levels is derived by making use of the ensemble averaging and the averaging with respect to a stochastic process in the bath. Relaxation terms are determined in the Born approximation with respect to the system-bath interaction and the damping parameters related to a relaxation kernel are specified. In parallel with the spectral strength of the bath, the damping parameters determine the transient times for the Markovian description creating the physical origin of the slippage [A. Suarez, R. Sibey, and I. Oppenheim, J. Chem. Phys. 97, 5101 (1992)]. The influence of energy fluctuations of the bath is analyzed for a two-level system including the solution of the corresponding non-Markovian equation for the level population difference. The conditions for the formation of Boltzmann's thermal ratio between steady-state populations are evaluated as well.
The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.