Applied and Computational Mathematics Seminar

I will describe new ideas relating quantum chaos to the complexity of time evolution.  One approach treats physical time evolution as a quantum computation, and bounds the smallest quantum circuit that can simulate this evolution.  The second approach quantifies how ergodically and rapidly a quantum state explores the accessible part of the system's Hilbert space.  I will illustrate how these measures separate integrable and chaotic quantum systems by considering examples including particles on group manifolds, spin chains, quantum billiards, and Random Matrix Theory.  I will end by describing an application of these methods to a conjecture that geometrizes complexity in quantum gravity.

The aggregated and crystalline phases of π-conjugated molecules and polymers continue to receive widespread attention as semiconducting materials for field effect transistors, light emitting diodes and solar cells. However, despite the more than five decades of intensive experimental and theoretical research following Kasha's pioneering work on H- and J-aggregates[1] there remain important questions regarding the nature of the photo-excitations in molecular assemblies and how their spectral signatures are related to crystal packing and morphology. In this talk we explore the absorption and emission of light in molecular aggregates using simplified Hamiltonians which account for electronic coupling between molecules as well as the coupling between electronic and vibrational degrees of freedom. Once a suitable basis set is chosen, such Hamiltonians can be represented as matrices which are readily analyzed numerically using linear algebraic techniques. A range of novel spectral signatures are presented which are directly related to the way molecules pack to form aggregates. Applications are made to a range of p-conjugated chromophores which have been intensively investigated for opto-electronic applications.[2]

[1] M. Kasha, Energy Transfer Mechanisms and the Molecular Exciton Model for Molecular Aggregates, Radiation Research 20 (1963) 55-70.

[2] N.J. Hestand, F.C. Spano, Expanded Theory of H- and J- Molecular Aggregates: The Effects of Vibronic Coupling and Intermolecular Charge Transfer, Chem. Rev. 118 (2018) 7069–7163.