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.
The goal of this project is to study quantum and classical wave behavior in complicated geometries, with a particular emphasis on the relationship between wave dynamics and the dynamics of the corresponding ray system. This work combines methods, insights, and examples from diverse areas, such as condensed matter and mesoscopic physics, atomic, optical, chemical, nuclear, and microwave physics, nonlinear dynamics, and mathematical physics. We aim for broadly applicable results, that will spur further theoretical and experimental progress in specific areas, while at the same time furthering our fundamental understanding of the classical-quantum interface. Examples of the research work supported by this project include: 1) We have studied the formation of rogue waves in the ocean due to scattering by random currents, and obtained expressions describing the effect of current strength and other variables on the likelihood of rogue wave formation. For conditions commonly appearing in nature, the probability of encountering am extreme wave can be enhanced by a factor of 10 to 1000. Even greater enhancement is obtained when scattering by currents is combined with nonlinearity in the wave equation. The work may contribute to the long-term goal of rogue wave forecasting. 2) The Casimir force is an effect associated with the vacuum energy of quantum fields, e.g. the attraction between two uncharged metal plates due to the quantum fluctuations of the electromagnetic field in the "empty" space surrounding the plates. We have investigated Casimir forces in non-trivial geometries and shown that the approximate value of the Casimir forces can often be computed using only information from short periodic ray trajectories in the space surrounding the objects. 3) We have shown that a simple method utilizing only short-time dynamical information is sufficient for accurate quantitative prediction of long-time behavior in quantum systems whose classical analogue is chaotic. 4) We have investigated the implementation of quantum logic gates (needed for building a quantum computer) in a scheme that combines linear optics with quantum measurement. Linear optics logic gates are non-deterministic, i.e., the measurement produces the desired outcome only with some success probability <1. The goal in this approach is to maximize the success probability of the gate or transformation. We have shown that direct optimization of the success probability can lead to much better results than the traditional approach of constructing a given gate out of standard building block "universal" gates. Examples of broader impacts include: 1) The PI has led the development of new four-year Engineering Physics undergraduate program offered by the School of Science and Engineering at Tulane, which combines a broad study of physics, chemistry, and mathematics with an emphasis on applications to 21st century technology, especially novel materials. 2) In addition, the PI has led the development of a five-year Dual-Degree Engineering program, in which a student will receive a physics degree from Tulane and a degree from a top engineering school in Mechanical, Civil, Electrical, or Environmental Engineering (i.e., in one of the engineering programs that were phased out as part of Tulane's renewal plan). Agreements have been signed with Vanderbilt and Johns Hopkins as partners 3) The PI has developed and taught a new course in Computational Physics and Engineering, aimed at upper-level undergraduates and beginning graduate students with little prior experience in the use of computers in physics. The course provides an interdisciplinary grounding in relevant aspects of physics, mathematics, statistics, and computer science. 4) The PI has regularly taught Physics 101, Great Ideas in Science, a course is aimed at exposing liberal arts majors to the basic principles of physical science, the applications of these principles in our world, and the relation of science to philosophy, politics and other aspects of human activity. Additionally, together with Research Assistant Professor Dmitry Uskov, the PI has developed and co-taught three times a new interdisciplinary class for first-year students: 'Mysteries of the Quantum World,' in which students are exposed to topics ranging from the basic of quantum information and computation to the physics of lasers and quantum chaos.
