Research
Constrained Eigenproblems
Unlike in classical mechanics, quantum systems often have a discrete set of possible states, such as the "s", "d", "p", or "f" orbitals in an atom. To understand a quantum system, then, the goal is to find and characterize all of these states. But in many problems, we aren't actually interested in all of the possible states; we want to find states that satisfy certain criteria. For instance, if we want to understand how an electron scatters off a nucleus, we need to identify the quantum states that travel outward from the nucleus as waves. If we want to understand how a given electronic state responds to a local defect in a crystal, we want to find the quantum states that resemble our known solution away from the defect.
We have employed our algorithm here to identify eigenstates of the Laplacian that approximately carry a five-point symmetry. Our algorithm returns the top three states shown, but not the bottom three.
These problems – any many others – can be formulated as a certain "eigenproblem" with linear constraints. That is, if we have a Hilbert space, a Hermitian operator on that space, and a closed subspace, can we find all of the eigenvectors of our operator that lie nearby (i.e., within a given tolerance to) the closed subspace? Along with Jeffrey Ovall (Portland State University), I am developing an approach to solve this class of problems efficiently, and showing how it applies to a wide variety of practical scientific problems.
Stratified Turbulence
Many key fluid flows in nature—including the atmosphere and the deep ocean—are stratified, meaning that their density varies significantly in the vertical direction. However, stratified turbulence gives rise to a separation of scales that precludes traditional methods of simulation, and so large-scale, accurate models of these flows remain outside our reach, even with current technology.
A model flow of Froude number 0.03, Reynolds number 1100; despite the large Reynolds number, the strong stratification here brings the flow to a steady state. Our NCQL algorithm provides a significant improvement over the existing MTQL algorithm for strongly stratified flows.
However, Chini et al. recently developed a "multi-timescale quasi-linear" (MTQL) algorithm to model such flows in the stratified limit; MTQL reduces and splits the Boussinesq system into a mean and fluctuating flow, drastically reducing the computation time needed to model a stratified flow. In my Master's dissertation at Cambridge—and in ongoing work—I have developed an improved, "non-linear cascading quasi-linear" (NCQL) approach to model these flows. Using weakly nonlinear theory, NCQL directly models the forward energy cascade of stratified flows, achieving significant improvements over MTQL in the moderately and highly stratified regimes (pictured above).
Hydrodynamic Quantum Analogues
A central focus of my work at MIT is the investigation of hydrodynamic quantum analogues, and particularly the "walking" oil droplet system of Couder and Fort. These walking droplets exhibit a wide variety of quantum-like features: stable spin states and quantised orbits, Bohm-like "surreal trajectories", and Fraunhofer-like diffraction patterns, among others.
My work lies on the theoretical end of this project; though these droplets exhibit qualitative similarities to quantum mechanics, they often fall short in achieving quantitative agreement. Through analytical and numerical approaches, I am investigating which quantum phenomena can be closely approximated with techniques of classical field theory.
