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arXiv:1004.0740 · New Journal of Physics

Non-equilibrium stationary states from the equation of motion of open systems

Jinshan Wu

Published Aug 24, 2010• 12 citations
arXivHTMLPDF

Abstract

A BBGKY-like hierarchy is derived from the non-equilibrium Redfield equation. Two further approximations are introduced and each can be used to truncate and solve the hierarchy. In the first approximation such a truncation is performed by replacing two-particle Green's functions (GFs) in the hierarchy by their values at equilibrium. The second method is developed based on the cluster expansion, which constructs two-particle GFs from one-particle GFs and neglects the correlation part. A non-equilibrium Wick's Theorem is proved to provide a basis for this non-equilibrium cluster expansion. Using those two approximations, our method of solving the Redfield equation, for instance, of an N-site chain of interacting spinless fermions, involves an eigenvalue problem with dimension $2^{N}$ and a linear system with dimension $N^2$ in the first case, and a nonlinear equation with dimension $N^2$ in the second case, which can be solved iteratively via a sequence of $N^2$ linear systems. Other currently available direct methods correspond to a linear system or an eigenvalue system with dimension $4^N$ plus an eigenvalue system with dimension $2^N$. As a test of the methods, for small systems with size N=4, results are found to be consistent with results made available by other direct methods. Although not discussed here, extending both methods to their next levels is straightforward. This indicates a promising potential for this BBGKY-like approach of non-equilibrium kinetic equations.

DOI
10.1088/1367-2630/12/8/083042

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A BBGKY-like hierarchy is derived from the non-equilibrium Redfield equation. Two further approximations are introduced and each can be used to truncate and solve the hierarchy. In the first approximation such a truncation is performed by replacing two-particle Green's functions (GFs) in the hierarchy by their values at equilibrium. The second method is developed based on the cluster expansion, which constructs two-particle GFs from one-particle GFs and neglects the correlation part. A non-equilibrium Wick's Theorem is proved to provide a basis for this non-equilibrium cluster expansion. Using those two approximations, our method of solving the Redfield equation, for instance, of an N-site chain of interacting spinless fermions, involves an eigenvalue problem with dimension 2N2^{N}2N and a linear system with dimension N2N^2N2 in the first case, and a nonlinear equation with dimension N2N^2N2 in the second case, which can be solved iteratively via a sequence of N2N^2N2 linear systems. Other currently available direct methods correspond to a linear system or an eigenvalue system with dimension 4N4^N4N plus an eigenvalue system with dimension 2N2^N2N. As a test of the methods, for small systems with size N=4, results are found to be consistent with results made available by other direct methods. Although not discussed here, extending both methods to their next levels is straightforward. This indicates a promising potential for this BBGKY-like approach of non-equilibrium kinetic equations.
We establish a theoretical framework for the exploration of the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and, if applicable, symmetry conserving closure approximation is developed on the basis of a cluster expansion for these finite systems, using suitably normalized RDMs and symmetrization operators. The underlying cluster expansion is formulated recursively by means of super-operators, which allows for going to large truncation orders. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. The theoretical framework provided here is extensively applied and evaluated in our immediate follow-up work [1], involving truncation orders as large as twelve.
Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of with cubic repulsion. Equilibrium statistical mechanics of this gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the Pechukas gas particles (i.e. energy levels in the system).
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer time-scale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimal-invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.
The quantum dynamics of correlated fermionic or bosonic many-body systems following external excitation can be successfully studied using nonequilibrium Green functions (NEGFs) or reduced density matrix methods. Approximations are introduced via a proper choice of the many-particle self-energy or decoupling of the BBGKY-hierarchy, respectively. These approximations are based on Feynman's diagram approaches or on cluster expansions into single-particle and correlation operators. In a recent paper [E. Schroedter, J.-P. Joost, and M. Bonitz, Condens. Matter Phys. 25, 23401 (2022)] we presented a different approach in which, instead of equations of motion for the many-particle NEGF (or density operators), equations for the correlation functions of fluctuations are analyzed. In particular, we derived the stochastic GWGWGW and polarization approximations that are closely related to the nonequilibrium GWGWGW approximation. Here, we extend this approach to the computation of two-time observables depending on the specific ordering of the underlying operators. In particular, we apply this extension to the calculation of the density correlation function and dynamic structure factor of correlated Hubbard clusters in and out of equilibrium.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.