CAREER: Wave Function Structure and Transport in Quantum Chaotic Systems

研究的重点是具有不可积经典极限的系统中波函数和量子输运的统计性质。推动这项工作的实验和应用来自各种领域，如通过二维纳米结构的电流，量子点中的库仑阻塞电导，不规则形状电磁谐振器中的微波，大型结构声学系统中的能量传输，化学反应统计，非对称光学谐振器，以及非平凡几何形状的卡西米尔力。本提案所涵盖的密切相关的具体研究主题如下。(1)量子点中的相互作用矩阵元素波动：对于现实的混沌几何结构和实验上相关的点大小，根据单电子波函数计算的相互作用矩阵方差可能超过随机波模型预测的方差的3或4倍，这对于将实验测量的电导峰间距统计与Hartree-Fock计算相一致具有重要意义。未来的工作包括更好地理解波函数关联，而不是单纯的前序半经典近似，以及研究混沌激光谐振腔中与模式竞争的有趣联系。(2)弱随机势中的分支流动：二维电子气中的电子流动图像显示出意外的分支行为，这可以用经典流动中的奇点来解释。正在进行的工作包括将该理论扩展到有限波长和均方根势高的实验上重要的参数区域，从而能够与纳米结构实验和目前正在开发的微波腔实验以及最近对远程海洋声学的研究进行比较。(3)长时间输运行为的自举：短时量子行为，包括相位信息，可以用来预测非随机的长时间输运和稳态性质；应用包括弹道和扩散量子点，以及声学系统中的能量输运。(4)长期半经典精度：正在进行的工作表明，长时间的半经典传播子在二维混沌系统中比在规则系统中更精确，这意味着量子保真度和量子化模糊。未来的工作包括将方法推广到高维和相互作用系统，以及高阶半经典近似，以及探索Casimir能量的半经典计算的有效性。该提案的更广泛影响包括：(1)建立一个由本科生和研究生组成的多元化研究小组，性别、种族和残疾代表不足的群体积极参与，包括参与LSAMP等促进多样性的计划，并鼓励与原子物理、凝聚态物理和化学的研究人员合作；(2)开发“计算物理”和“混沌和非线性动力学”的新课程，目标是高年级本科生和初级研究生，包括与数学和工程学教师合作；(3)通过发表学生可以获得的教学文章，促进本科生参与研究；(4)教授量子力学和经典力学的相关入门研究生课程和高级本科生课程，重点是经典/量子对应；以及(5)在第二学期继续有效、高评价的入门物理课程教学，重点是现代物理，并结合创新技术，以鼓励学生对物理和相关科学领域的职业感兴趣。

The research focus is on statistical properties of wave functions and quantum transport in systems with a non-integrable classical limit. Experiments and applications motivating this work come from fields as diverse as current flow through two-dimensional nanostructures, Coulomb blockade conductance in quantum dots, microwaves in irregularly-shaped electromagnetic resonators, energy transport in large structural acoustic systems, chemical reaction statistics, asymmetric optical resonators, and Casimir forces for nontrivial geometries. Specific closely interrelated research topics covered by this proposal are as follows. (1) Interaction matrix element fluctuations in quantum dots: For realistic chaotic geometries and experimentally relevant dot sizes, interaction matrix variance as computed from single-electron wave functions may exceed by a factor of 3 or 4 the variance predicted by a random wave model, with important implications for reconciling experimentally measured conductance peak spacing statistics with Hartree-Fock calculations. Future work includes obtaining a better understanding of wave function correlations beyond naive leading-order semiclassical approximation, and investigation of intriguing connections with mode competition in chaotic laser resonators. (2) Branched flow through weak random potentials: Images of electron flow in a two-dimensional electron gas show unexpected branching behavior, which may be explained by singularities in the classical flow. Ongoing work includes extension of the theory to the experimentally important parameter regimes of finite wave-length and rms potential height, allowing comparison with nanostructure experiments and with microwave cavity experiments that are now under development, as well as with recent studies of long-range ocean acoustics. (3) Bootstrapping of long-time transport behavior: Short-time quantum behavior, including phase information, may be used to predict nonrandom long-time transport and stationary properties; applications include ballistic and diffusive quantum dots, as well as energy transport in acoustic systems. (4) Long-time semiclassical accuracy: Ongoing work suggests the semiclassical propagator at long times is more accurate in chaotic than in regular systems in two dimensions, with implications for quantum fidelity and quantization ambiguity. Future work includes extending the methods to higher-dimensional and interacting systems, and to higher-order semi-classical approximations, as well as exploring the validity of semiclassical calculations for Casimir energies. Broader impacts of the proposal include: (1) building of a diverse research group of undergraduate and graduate students, with active participation of gender, racially, and disability underrepresented groups, including involvement in diversity-enhancement programs such as LSAMP, and encouraging collaboration with researchers in atomic physics, condensed matter physics, and chemistry; (2) development of new courses in "Computational physics" and "Chaos and nonlinear dynamics", targeted toward upper division undergraduate and beginning graduate students, and including collaboration with faculty in mathematics and engineering; (3) promoting undergraduate research involvement through publication of pedagogical articles accessible to students; (4) teaching of relevant introductory graduate and upper level undergraduate courses in quantum and classical mechanics, with emphasis on classical/quantum correspondence; and (5) continuation of effective, highly rated teaching of the introductory physics curriculum, with a focus on modern physics in the second semester and incorporating the use of innovative technology to encourage students interested in careers in physics and related fields of science.