在相干态表象中求解非平衡多体系统量子主方程

Building up a framework to calculate non-equilibrium stationary states (NESS) for many-body interacting systems, and implementing it on some realistic systems, has been a challenging task to theoretical physicists in statistical physics and condensed matter for many decades. For equilibrium systems we can apply the ensemble theory, there is, however, no such a general framework for NESS. Even after we know the Hamiltonian and spectrum of the Hamiltonian, temperature and chemical potential, which are enough to define uniquely the equilibrium distribution, we still can not write down a corresponding NESS. Therefore, from the viewpoint of pure theoretical interest, a framework for calculating NESS is already an important question. Recent progresses in transport experiments on nanoscale/mesoscopic quantum systems drew also attentions of experimentalists in condensed matter and placed an urgent challenge to theorists. .During my 6-year PH.D. study in the University of British Columbia, together with Mona，I worked on tackling this challenge. Based on the projector technique and Kinetic equation, which has been successfully used in studies of relaxation towards thermal equilibrium by others, we made some essential contribution in extending the Kinetic equation to studies of non-equilibrium transport phenomena. Although the original idea of using the Kinetic equation for transport belongs to K. Saito and others, research works along this line made few progresses due to the difficulty in solving the large-dimension Kinetic equation before we discovered our methods. For example, for an N-site quantum system, the dimension of the equation is 4 to the power of N, which increases exponentially with N. Notice that Schr?dinger's equation of the same system has dimension 2 to the power of N, so the Kinetic equation is roughly square harder than the Schroedinger's equation. All available numerical techniques before we came up with new and efficient techniques can only deal with systems with very small N (roughly 20) if the systems are of interaction. Therefore, it is necessary to have much more efficient methods in solving the Kinetic equations of interacting systems. Along this line of thinking, we did came up with two new approaches to solve the equations: the BBGKY-like method and the idea of simulating the equations in coherent-state representation. .Before my graduation, we have more or less developed the first method and some works on this have been published. Referees spoke very high of the method. Now in this application, I propose to develop the second method and apply it on physical systems to discuss some potentially testable physical properties..One major technical dificulty of this approach is at dealing with the third-order derivatives in the induced generalized Fokker-Planck equation. In this proposal we suggest two methods to accomplish this goal.

非平衡定态，尤其是包含相互作用的多体系统的非平衡定态的计算，是目前物理学中一个非常重要的问题。在实验方面，量子系统输运性质的实验已经积累了大量的研究素材，迫切需要一个研究输运性质的一般理论。在理论方面，对于平衡态，我们有系综理论，而对于非平衡态，却没有一个一般的框架。在六年博士学习期间，本项目负责人与其导师一起，完善了一个研究非平衡定态的一般方法：动理学（Kinetic）方程方法（从输运系统和外界的整体的薛定谔方程出发，通过投影算符技术得到系统的量子主方程），并在此基础上提出了两种新的求解量子主方程的计算方法：类BBGKY方法以及基于相干态表象的量子随机微分方程方法。目前前者的理论框架已经取得比较好的进展，一部分论文已经发表。本项目将在集中在后者的理论框架以及计算实现。其中一个重要的技术困难是如何处理量子随机微分方程中的高阶导数项。本申请中，我们提出了两个可能可以达成此目标的方法。

本项目旨在研究用于描述量子输运过程的量子master方程在相干态表象下的求解方法。对于N-qubit的系统，相应的量子master方程的维数是4^N。因此，传统方法（直接对角化、离散波函数等）非常难以求解这样的方程。在输运问题等量子非平衡过程或者趋向平衡的耗散过程的研究中，我们有经常遇到这样的方程。于是，如何求解这样的方程就是一个非常重要的问题。能够求解这样的方程之后，我们才能从第一原理出发（而不是从某种维象方程出发）开展量子输运、量子系统的弛豫过程的研究。其中，量子输运的问题，由于理论上目前还不能写出来一个非平衡输运状态的分布函数——对于平衡态，我们有Boltzmann分布，但是对于非平衡定态我们没有——因此，本项目还具有基础理论的意义，有可能可以给出来非平衡定态的一般的理论框架，用于非平衡过程以及非平衡相变的研究。..本项目主要研究成果是4篇研究论文。其中一篇在Physical Review E发表，一篇正在Physical Review B二审（在打架中），另外一篇准备投稿Physical Review B（两篇文章有一定关系，所以在等待前一篇文章的结果）。一篇在中文核心发表。..已经发表的一篇论文研究了不依赖于序参量来识别相变点的问题。这个工作在量子系统和社会经济系统上的应用还在研究中。另外一篇论文介绍了我们的方法。正在审稿和准备提交的论文研究了相干态表象中求解两种不同的量子Master方程——Redfield和Lindblad方程，并做了对比。

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