全计数统计的电流守恒交流量子理论

Transport processes in mesoscopic systems are dominated by quantum effect and are stochastic in nature. Therefore in addition to average current, full probability distribution of charge transport called full counting statistics is needed to fully characterize the quantum transport.Indeed, the noise spectrum (the second moment of current operator) and higher order fluctuations can provide additional information about quantum effect and nature of the interaction in electronic systems.Theoretically, finite frequency full counting statistics has been studied using quantum master equation and scattering matrix approach. It is known that in the presence of ac field, the conduction current is not conserved and the displacement current due to Coulomb interaction must be considered.Therefore, the current conserving problem for full counting statistics is still an outstanding problem. Given the importance of the full counting statistics, it is timely to address this issue. In this project, we will develop a current conserving formalism for ac quantum transport that can be used for both average current and its correlations and therefore for full counting statistics. In this project, we will study the finite frequency full counting statistics in the presence of dephasing mechanism as well as disorders. We will investigate the transient full counting statistics in the presence of pulse like voltage.The waiting time distribution is an important tool to study the temporal correlation of stochastic events. The waiting time distribution can’t be obtained from finite frequency full counting statistics where the system is biased with ac voltage.In this project,we will extend the existing quantum theory for waiting time distribution into ac regime and present a finite temperature quantum theory for waiting time distribution.

介观系统的输运由量子行为的随机过程主导，除平均电流外，量子传输的所有特征需要电荷传输的全部统计分布，即全计数统计（full counting statistics，FCS）才能描述完整。除伏安特性外，噪声谱（电流算符的二阶矩）和更高阶涨落能够提供电子相互作用和量子效应的更多信息。理论上，利用主方程和散射矩阵理论，已经开展了有限频率下的全计数统计研究，然而，全计数统计的电流守恒问题仍然是个突出的需要解决的问题。本项目将要发展交流量子输运下的电流守恒公式，解决全计数统计下的交流电流各阶矩的电流守恒问题；计算有失相下和无序杂质下的介观系统的有限频率下的全计数统计，以及研究瞬态电压脉冲下系统响应的全计数统计；等待时间分布是研究随机事件的时间相关性的重要工具，本项目将要将等待时间分布的量子直流理论扩展到量子交流理论，得到有限温度下的等待时间分布。

介观系统中，为了完整表征电子的量子传输过程，需要考虑电流相干性的所有阶矩，全计数统计（full counting statistics, FCS）就是研究介观系统电流相干性的最简洁和最完美的方法，它计算了给定时间内通过一个终端的电子数的概率分布函数，包含系统中电流涨落的全部信息，不仅能给出噪声谱，也能给出所有的高阶矩。本项目研究量子电子输运行为的随机过程和电荷传输的全部统计分布，利用非平衡格林函数理论，发展量子输运下的全计数统计，通过定义传导电流算符和位移电流算符，解决了全计数统计下的电流各阶矩的电流守恒问题，将直流全计数统计理论扩展到交流，解决了交流FCS理论（包括直流FCS）中各阶矩的电流守恒问题；同时，重点计算了无序系统的有限频率下的全计数统计。本项目中，基于非平衡格林函数发展了一套直接计算FCS电导的产生函数的无序平均的方法，摒弃图式微扰，将CGF(cumulant generating function)对格林函数的非线性泛函的依赖通过映射到更高维度下变成线性问题，从而只用CPA就可以求解CGF的随机平均，避免图式方法的复杂性。在紧束缚模型下，利用我们的FCS-CPA对Anderson无序和二元无序系统进行了数值求解，得到了直到五阶矩的结果，并与传统的粗暴计算（brutal calculation）进行比较，结果完全相同。基于非平衡格林函数理论，在CPA框架下，发展了计算无序平均动力学电导、直流偏压下的频率相关散粒噪声、交流偏压下的频率相关的噪声谱，将我们的直流FCS-CPA拓展到交流偏压情形，并通过方格子的Anderson无序的数值计算，对公式进行了验证，结果与粗暴计算（brutal force calculation）相一致。