Dynamics of quantum particles and waves in noisy environments

The goal of this project is to investigate wave propagation in random media and noisy quantum systems. Waves are a central feature of the natural world, from the sound and light waves that shape everyday experience to quantum wave functions manipulated in physics laboratories. Although the basic phenomenology of waves is well understood, deep mathematical questions remain about the transport of waves in complex environments. Disordered variations in a material can cause waves to scatter and diffuse in ways that are difficult to predict, for example, radio waves used by cell phones are scattered by the urban environment, the quantum wave function of an electron in a semiconductor is scattered by dopant impurities, and subcellular structures scatter photons in optical coherence tomography. In quantum systems, noisy interactions between quantum waves and their external environment suppress delicate interference effects and can cause waves to behave more like classical particles. This project will develop new mathematical tools to understand these phenomena and their consequences for science and technology. The project will also involve training graduate and undergraduate students. This project will leverage recent developments in probability and mathematical physics to better understand the dynamics of random waves. The first thrust of the project will build on the recent connection made between random matrix theory and the random Schrödinger equation to address the more general setting of waves in random media. The approach replaces perturbative expansions and diagrammatic arguments with ideas from random matrix theory and will be used to study waves in random media without the commonly used paraxial approximation. The second thrust of the proposal involves the study of open quantum systems, which include the effect of noisy interactions with an external environment. This will build on recent work of the investigator which establishes the semiclassical limit of such systems and quantifies the amount of environmental noise necessary for the semiclassical limit to be valid. The investigator will develop and implement an efficient classical algorithm for the simulation of open quantum systems and make code publicly available. Applications to non-adiabatic quantum systems arising in chemistry will also be investigated. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.