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arXiv:0812.3650 · AIP conference proceedings

Exact Stochastic Mean-Field dynamics

Denis Lacroix A, Guillaume Hupin, Karl Ernst Rehm, Birger B. Back, Henning Esbensen, C. J. Lister

Published Jan 1, 2009• 0 citations
arXivHTMLPDF

Abstract

The exact evolution of a system coupled to a complex environment can be described by a stochastic mean-field evolution of the reduced system density. The formalism developed in Ref. [D.Lacroix, Phys. Rev. E77, 041126 (2008)] is illustrated in the Caldeira-Leggett model where a harmonic oscillator is coupled to a bath of harmonic oscillators. Similar exact reformulation could be used to extend mean-field transport theories in Many-body systems and incorporate two-body correlations beyond the mean-field one. The connection between open quantum system and closed many-body problem is discussed.

DOI
10.1063/1.3108765

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The exact evolution of a system coupled to a complex environment can be described by a stochastic mean-field evolution of the reduced system density. The formalism developed in Ref. [D.Lacroix, Phys. Rev. E77, 041126 (2008)] is illustrated in the Caldeira-Leggett model where a harmonic oscillator is coupled to a bath of harmonic oscillators. Similar exact reformulation could be used to extend mean-field transport theories in Many-body systems and incorporate two-body correlations beyond the mean-field one. The connection between open quantum system and closed many-body problem is discussed.
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 show that the dynamics of interacting fermions can be exactly replaced by a quantum jump theory in the many-body density matrix space. In this theory, jumps occur between densities formed of pairs of Slater determinants, ${D}_{\mathit{ab}}=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|$, where each state evolves according to the stochastic Schr\"odinger equation given by O. Juillet and Ph. Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. A stochastic Liouville-von Neumann equation is derived as well as the associated. Bogolyubov-Born-Green-Kirwood-Yvon hierarchy. Due to the specific form of the many-body density along the path, the presented theory is equivalent to a stochastic theory in one-body density matrix space, in which each density matrix evolves according to its own mean-field augmented by a one-body noise. Guided by the exact reformulation, a stochastic mean-field dynamics valid in the weak coupling approximation is proposed. This theory leads to an approximate treatment of two-body effects similar to the extended time-dependent Hartree-Fock scheme. In this stochastic mean-field dynamics, statistical mixing can be directly considered and jumps occur on a coarse-grained time scale. Accordingly, numerical effort is expected to be significantly reduced for applications.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.
We present an application of the extended stochastic Liouville equation [G. M. G. McCaul et al., Phys. Rev. B 95, 125124 (2017)], which gives an exact solution for the reduced density matrix of an open system surrounded by a harmonic heat bath. This method considers the extended system (the open system and the bath) being thermally equilibrated prior to the action of a time-dependent perturbation, as opposed to the usual assumption that the system and the bath are initially partitioned. Our method presents an exact technique capable of accounting for arbitrary parameter regimes of the model. Here, we present our first numerical implementation of the method in the simplest case of a Caldeira-Leggett representation of the bath Hamiltonian, and apply it to a spin-boson system driven from coupled equilibrium. We observe significant behaviors in both the transient dynamics and asymptotic states of the reduced density matrix not present in the usual approximation.
The dynamics of a many-body system coupled to an external environment represents a fundamentally important problem. To this class of open quantum systems pertains the study of energy transport and dissipation, dephasing, quantum measurement and quantum information theory, phase transitions driven by dissipative effects, etc. Here, we discuss in detail an extension of time-dependent current-density-functional theory (TDCDFT), we named stochastic TDCDFT [Phys. Rev. Lett. 98, 226403 (2007)], which allows the description of such problems from a microscopic point of view. We discuss the assumptions of the theory, its relation to a density-matrix formalism, and the limitations of the latter in the present context. In addition, we describe a numerically convenient way to solve the corresponding equations of motion and apply this theory to the dynamics of a one-dimensional gas of excited bosons confined in a harmonic potential and in contact with an external bath.