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

Partial-trace-free time-convolutionless equation of motion for the reduced density matrix

I. Knežević, David K. Ferry

Published Jul 30, 2002• 26 citations
Publisher

Abstract

Evolution of a system, coupled to its environment and influenced by external driving fields, is an old problem that remains of interest. In this paper, we derive an equation of motion for the reduced system density matrix, which is time convolutionless and free of the partial trace with respect to the environment states. This new approach uses an extension of the projection-operator technique, which incorporates an isomorphism between the system's Liouville space and the unit eigenspace of the projection operator induced by the uniform environment density matrix. Numerical application of the present approach is particularly useful in large externally driven systems, as the partial-trace-free equation is given in terms of submatrices significantly smaller than the matrices in the conventional time-convolutionless approaches, which alleviates the computational burden. We also show that all time-convolutionless approaches, conventional or partial-trace-free, are based upon a hidden underlying assumption of time reversibility of the system's evolution. This feature puts significant constraints on applicability of time-convolutionless approaches when employing approximations that yield time irreversibility. Also, we investigate the application of the approach in the description of far-from-equilibrium systems.

DOI
10.1103/physreve.66.016131

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Due to recent advances in quantum information, as well as in mesoscopic and nanoscale physics, the interest in the theory of open systems and decoherence has significantly increased. In this paper, we present an interesting approach to solving a time-convolutionless equation of motion for the open system reduced density matrix beyond the limit of weak coupling with the environment. Our approach is based on identifying an effective, memory-containing interaction in the equations of motion for the representation submatrices of the evolution operator (these submatices are written in a special basis, adapted for the ``partial-trace-free'' approach, in the $\mathrm{system}+\mathrm{environment}$ Liouville space). We then identify the ``memory dressing,'' a quantity crucial for solving the equation of motion for the reduced density matrix, which separates the effective from the real physical interaction. The memory dressing obeys a self-contained nonlinear equation of motion, which we solve exactly. The solution can be represented in a diagrammatic fashion after introducing an ``information exchange propagator,'' a quantity that describes the transfer of information to and from the system, so the cumulative effect of the information exchange results in the memory dressing. In the case of weak system-environment coupling, we present the expansion of the reduced density matrix in terms of the physical interaction up to the third order. However, our approach is capable of going beyond the weak-coupling limit, and we show how short-time behavior of an open system can be analyzed for arbitrary coupling strength. We illustrate the approach with a simple numerical example of single-particle level broadening for a two-particle interacting system on short time scales. Furthermore, we point out a way to identify the structure of decoherence-free subspaces using the present approach.

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Evolution of a system, coupled to its environment and influenced by external driving fields, is an old problem that remains of interest. In this paper, we derive an equation of motion for the reduced system density matrix, which is time convolutionless and free of the partial trace with respect to the environment states. This new approach uses an extension of the projection-operator technique, which incorporates an isomorphism between the system's Liouville space and the unit eigenspace of the projection operator induced by the uniform environment density matrix. Numerical application of the present approach is particularly useful in large externally driven systems, as the partial-trace-free equation is given in terms of submatrices significantly smaller than the matrices in the conventional time-convolutionless approaches, which alleviates the computational burden. We also show that all time-convolutionless approaches, conventional or partial-trace-free, are based upon a hidden underlying assumption of time reversibility of the system's evolution. This feature puts significant constraints on applicability of time-convolutionless approaches when employing approximations that yield time irreversibility. Also, we investigate the application of the approach in the description of far-from-equilibrium systems.
Due to recent advances in quantum information, as well as in mesoscopic and nanoscale physics, the interest in the theory of open systems and decoherence has significantly increased. In this paper, we present an interesting approach to solving a time-convolutionless equation of motion for the open system reduced density matrix beyond the limit of weak coupling with the environment. Our approach is based on identifying an effective, memory-containing interaction in the equations of motion for the representation submatrices of the evolution operator (these submatices are written in a special basis, adapted for the ``partial-trace-free'' approach, in the system+environment\mathrm{system}+\mathrm{environment}system+environment Liouville space). We then identify the ``memory dressing,'' a quantity crucial for solving the equation of motion for the reduced density matrix, which separates the effective from the real physical interaction. The memory dressing obeys a self-contained nonlinear equation of motion, which we solve exactly. The solution can be represented in a diagrammatic fashion after introducing an ``information exchange propagator,'' a quantity that describes the transfer of information to and from the system, so the cumulative effect of the information exchange results in the memory dressing. In the case of weak system-environment coupling, we present the expansion of the reduced density matrix in terms of the physical interaction up to the third order. However, our approach is capable of going beyond the weak-coupling limit, and we show how short-time behavior of an open system can be analyzed for arbitrary coupling strength. We illustrate the approach with a simple numerical example of single-particle level broadening for a two-particle interacting system on short time scales. Furthermore, we point out a way to identify the structure of decoherence-free subspaces using the present approach.
Reduced equations of motion (REM) are derived for the reduced density matrix of some of the degrees of freedom taking part in a scattering process. The derivation is made making use of the Zwanzig–Mori projection operator technique and utilizing a partial time ordering prescription (POP) which results in simple REM which contain no convolution in time. The present formulation is convenient for the direct calculation of reduced information on scattering events since it avoids the necessity of calculating the complete S matrix. The entire dynamical information relevant for the present description is expressed in terms of a hierarchy of correlation functions which in turn depend on the potential surface only in the region of interest. We further develop an exponential approximation for the reduced tetradic scattering S matrix in Liouville space, which forms a convenient framework for systematic approximations.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (rN (t), pN (t)) for a given initial condition (rN (0), pN (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1.2.2)) that describes the time evolution of the phase space probability density f (rN , pN ; t). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1–5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenbergtype uncertainty principles, the Schrödinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (rN , pN ; t) is now the quantum mechanical density operator (often referred to as the “density matrix”), whose time evolution is determined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or a reduced description of part of the overall system is desired. Such situations are considered later in this chapter.
We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the extended stochastic Liouville--von Neumann equation. Our approach generalizes previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions. The applicability of this model and the potential for numerical implementations are also discussed.
We derive an exact equation of motion for the reduced density matrices of individual subsystems of quantum many-body systems of any lattice dimension and arbitrary system size. Our projection operator based theory yields a highly efficient analytical and numerical approach. Besides its practical use it provides an interpretation and systematic extension of mean-field approaches and an adaption of open quantum systems theory to settings where a dynamically evolving environment has to be taken into account. We show its high accuracy for two significant classes of complex quantum many-body dynamics, unitary evolutions of nonequilibrium states in closed and stationary states in driven-dissipative systems.