Search for non-Abelian quantal phases and statistics

量子统计和波函数的自旋和对称性是量子力学理解世界的核心。自从量子力学出现以来，人们就知道，统计学根据两个相同粒子交换的对称性来区分费米子和玻色子。在三维空间中，只有两种可能的对称性：玻色子的波函数在粒子置换下是对称的，而费米子的波函数是反对称的。二维系统在性质上是不同的，由于两个准粒子的交换，波函数可以获得任意相位因子。这些任意子准粒子服从分数统计，并在分数量子霍尔效应的背景下产生。然而，更多奇特的相和统计已经被设想，其中准粒子状态是简并的，它们的状态向量乘以一个单位矩阵，而不是一个相因子作为准粒子交换的结果。在这种非阿贝尔多体系统中，两个粒子的交换改变了凝聚体中所有粒子的状态。这种波函数的刚性导致了基于非阿贝尔状态的量子比特的容错量子计算机的想法。虽然理论上非阿贝尔统计和非阿贝尔量子相已经在许多系统中被预测，但迄今为止还没有实验证实非阿贝尔统计状态的存在。最有希望的候选者-填充因子v = 5/2量子霍尔态尚未被证明是非阿贝尔态。与非阿贝尔统计不同，非阿贝尔量子相位已经可以是单粒子波函数的属性。我们提出探索总角动量J = 3/2重空穴态的二维空穴气体的非阿贝尔量子相。设想的耦合量子环实验有望显示非阿贝尔多体效应，并导致发现这种难以捉摸的新物质状态。提出的研究涉及先进的材料生长、纳米制造、小能量尺度的射频研究和对空穴相互作用的深入理解。剑桥大学卡文迪什实验室的半导体物理小组，由大卫·里奇教授领导，是世界上少数几个能够实现探测非阿贝尔相位这一雄心勃勃的目标的地方之一。该小组率先开发了关键的使能技术，如高质量二维气体的MBE增长和低温射频技术。来自美国普渡大学的Leonid Rokhinson教授将带来多年来对砷化镓中空穴的研究，特别是针对纳米制造技术的异质结构的发展以及最近开发的应变控制技术的专业知识。

Quantum statistics and the spin and symmetry of wavefunctions are central to a quantum mechanical understanding of the world. Since the dawn of quantum mechanics it has been known that statistics distinguishes fermions from bosons due to symmetry with respect to interchange of two identical particles. In three spatial dimensions there are only two possible symmetries: the wave function of bosons is symmetric under permutation of particles while for fermions the wavefunction is antisymmetric. Two-dimensional systems are qualitatively different and the wavefunction can acquire any phase factor owing to interchange of two quasiparticles. These anyonic quasiparticles obey fractional statistics and arise in the context of the fractional quantum Hall effect. Yet more exotic phases and statistics has been envisioned, in which quasiparticle states are degenerate and their state vectors are multiplied by a unitary matrix rather than a phase factor as a result of quasiparticle interchange. In such a non-Abelian many-body systems exchange of two particles changes the state of all the particles in the condensate. Such rigidity of the wavefunction has lead to the idea of a fault-tolerant quantum computer with qubits based on non-Abelian states.While theoretically non-Abelian statistics and non-Abelian quantal phases have been predicted in many systems, no experiment to date has confirmed the existence of states with non-Abelian statistics. The most promising candidate - the filling factor v = 5/2 quantum Hall state is yet to be shown to be a non-Abelian state. Unlike the non-Abelian statistics, the non-Abelian quantal phase can already be a property of a single-particle wavefunction. We propose to explore the non-Abelian quantal phase in two-dimensional hole gases with total angular momentum J = 3/2 heavy hole states. The envisaged experiments on coupled quantum rings promise to show non- Abelian many-body effects and lead to the discovery of this elusive new state of matter. The proposed research involves advanced material growth, nanofabrication, rf investigation of small energy scales and a deep understanding of holes interactions. The Semiconductor Physics group at the Cavendish Laboratory, Cambridge University, headed by Professor David Ritchie, is one of a few places in the word where such an ambitious goal of detecting non-Abelian phases can be achieved. The group pioneered the key enabling technologies, such as MBE growth of high quality two dimensional gases and low temperature rf techniques. Prof. Leonid Rokhinson from Purdue University, USA, will bring expertise of many years of investigation of holes in GaAs, development of heterostructures especially tailored for nanofabrication techniques, and recently developed strain control techniques.